# Patent

## Title
Backlash compensation with filtered prediction in discrete time nonlinear systems by dynamic inversion using neural networks

## Abstract

Methods and apparatuses for backlash compensation. A dynamics inversion compensation scheme is designed for control of nonlinear discrete-time systems with input backlash. The techniques of this disclosure extend the dynamic inversion technique to discrete-time systems by using a filtered prediction, and shows how to use a neural network (NN) for inverting the backlash nonlinearity in the feedforward path. The techniques provide a general procedure for using NN to determine the dynamics preinverse of an invertible discrete time dynamical system. A discrete-time tuning algorithm is given for the NN weights so that the backlash compensation scheme guarantees bounded tracking and backlash errors, and also bounded parameter estimates. A rigorous proof of stability and performance is given and a simulation example verifies performance. Unlike standard discrete-time adaptive control techniques, no certainty equivalence (CE) or linear-in-the-parameters (LIP) assumptions are needed.

## Background
This application claims priority to provisional patent application Ser. No. 60/237,580 filed Oct. 3, 2000, entitled, “Backlash Compensation with Filtered Prediction in Discrete Time Nonlinear Systems by Dynamic Inversion Using Neural Networks” by Javier Campos and Frank L. Lewis. That entire disclosure is specifically incorporated by reference herein without disclaimer.


The government may own rights to portions of the present invention pursuant to Army Research Office Grant 39657-MA, ARO Grant DAAD19-99-1-0137, and Texas ATP Grant 003656-027.


BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates generally to the fields of neural networks. More particularly, it concerns using neural networks for backlash compensation in mechanical systems.
2. Description of Related Art
Many physical components of control systems have nonsmooth nonlinear characteristics such as deadzone and backlash. These are particularly common in actuators, such as mechanical connections, hydraulic servovalves and electric servomotors. The difference between toothspace and tooth width in mechanical system is known as backlash, and it is necessary to allow two gears mesh without jamming. Any amount of backlash greater than the minimum amount necessary to ensure satisfactory meshing of gears can result in instability in dynamics situations and position errors in gear trains. Backlash often severely limits the performance of feedback systems by causing delays, oscillations, and inaccuracy. In fact, there are many applications such as instrument differential gear trains and servomechanisms that require the complete elimination of backlash in order to function properly.
Many mechanical solutions have been developed to overcome backlash, for example spring-loaded split gear assemblies and dual motor systems. These mechanical solutions can satisfactorily handle some backlash problems, but they give rise to other problems like decreased accuracy and reduced bandwidth. They are also expensive, energy consuming and increase the overall weight of the system. A backlash compensation scheme not based on mechanical devices would be more convenient.
In most applications, backlash parameters are either poorly known or completely unknown, which represents a challenge for the control design engineer. Proportional-derivative (PD) controllers have been observed to result in limit cycles if the actuators have nonlinearities such as backlash or deadzones. To overcome the PD controller limitations, several techniques have been applied to compensate for the actuator nonlinearities. These techniques include adaptive control, fuzzy logic and neural networks.
Many systems with actuator nonlinearities such as deadzone and backlash are modeled in discrete time. Moreover, for implementation as a digital controller, a discrete-time actuator nonlinearity compensator is needed. To address discrete-time deadzone compensation, at least one group has proposed a specific adaptive control approach. Also a specific fuzzy logic (FL) deadzone compensation discrete time scheme has been proposed. Adaptive control approaches for backlash compensation in discrete time have also been proposed. These conventional methods, however, all require a linear in the parameter assumption.
Neural Networks (NN) have been used extensively in feedback control systems. Specifically, it has been observed that the use of PD controllers may result in limit cycles if actuators have deadzones or backlash. Rigorous results for motion tracking of such systems are notably sparse, though ad hoc techniques relying on simulations for verification of effectiveness are prolific.
Although some NN applications show at least a degree of utility, most applications, unfortunately, are ad hoc with no demonstration of stability. The stability proofs that do exist rely almost invariably on the universal approximation property for NN. However, in most real industrial control systems there are nonsmooth functions (piecewise continuous) for which approximation results in the literature are sparse or nonexistent. Examples of phenomena involving nonsmooth functions include, but are not limited to, deadzone, friction, and backlash. Though there do exist some results for piecewise continuous functions, traditional attempts to approximate jump functions using smooth activation functions require many NN nodes and many training iterations, and still do not yield very good results. It would therefore be advantageous to have the ability to estimate and compensate nonlinearities, including backlash nonlinearities, involving piecewise continuous functions with guaranteed close-loop stability.
The use of NN has accelerated in recent years in many areas, including feedback control applications. Particularly important in NN control are the universal function approximation capabilities of neural network systems. NN systems offer significant advantages over adaptive control, including no requirement for linearity in the parameters assumptions and no need to compute a regression matrix for each specific system. Dynamics inversion in continuous-time using NN has been proposed, where a NN is used for cancellation of the system inversion error. A continuous time dynamic inversion approach using NN for backlash compensation has also been proposed. A continuous time inverse dynamics approach using adaptive and robust control technique has been proposed as well. A preliminary approach using dynamic inversion for backlash compensation in discrete-time system has been proposed, but was incomplete in that its proof did not rigorously include the effects of a certain predictive filter needed to actually implement the technique.
Dynamic inversion is a form of backstepping, which has been extended to discrete-time systems with limited results. The difficulty with applying those results to discrete-time dynamic inversion is that a future value of a certain ideal control input is needed.
Certain problems facing the field enumerated above are not intended to be exhaustive but rather are among many which tend to impair the effectiveness of previously known estimation and compensation schemes. Other noteworthy problems may and do exist; however, those presented above are sufficient to demonstrate that methods of backlash compensation appearing in the art have not been altogether satisfactory.
SUMMARY OF THE INVENTION
This disclosure provides a discrete-time actuator nonlinearity compensator. In most applications, backlash models are either poorly known or completely unknown. Inverting the backlash nonlinearity even for known backlash models is not an easy task, since the nonlinearity appears in the feedforward path. This disclosure explains the design of an intelligent control system that cancels the unknown backlash nonlinearity, while still keeping the closed loop system stable. This allows faster and more precise control of industrial positioning systems in applications including, but not limited to, robotics, CNC machine tools, and vehicle suspension systems.
Backlash nonlinearities occur in almost all motion control systems. Unfortunately, it introduces errors. The problems are particularly exacerbated when the required accuracy and the speed of motion are high. There are many applications such as instrument differential gear trains and servomechanisms that require the complete elimination of backlash in order to function properly. The backlash compensator by dynamic inversion using a neural network (NN) of this disclosure may be applied in order to improve system performance and cancel the backlash effects. Potential markets are vast and include, but are not limited to, the auto industry, military applications, robotics, mechanical processes, and biomedical engineering.
This disclosure involves a discrete-time dynamic inversion compensation with a filtered prediction for backlash compensation in nonlinear systems. In one embodiment, the compensator uses the dynamic inversion technique with neural networks (NN) for inverting the backlash nonlinearity in the feedforward path. This disclosure shows how to tune the NN weights in discrete-time so that the unknown backlash parameters are learned on-line, resulting in a discrete-time adaptive backlash compensator. Using discrete-time nonlinear stability techniques, the tuning algorithm is shown to guarantee small tracking errors as well as bounded parameter estimates. Since the tracking error, backlash error and the parameter estimation error are weighted in the same Lyapunov function, no certainty equivalence assumption is needed.
The present disclosure extends dynamic inversion techniques to discrete-time systems by using a filtered prediction, and it shows how to use and tune a neural network for inverting backlash nonlinearity in the feedforward path. The techniques of this disclosure offer important improvements to controllers, neural net or otherwise, in that:

    
    
        (1) The techniques extend the dynamic inversion technique to discrete-time systems;
        (2) This is a ‘model-free’ approach. A neural net is used for inverting the backlash nonlinearity in the feedforward path so that a mathematical model of the backlash is not needed;
        (3) The techniques apply for a large class of actuator nonlinearities, not only backlash;
        (4) The neural network design algorithm is based on rigorous mathematical stability proofs, and so shows how to guarantee closed-loop stability and performance;
        (5) Unlike standard discrete-time adaptive control, no linear-in-the-parameters (LIP) or certainty of equivalence (CE) assumption is required. This requires an exceedingly complex proof, but obviates the need for any sort of LIP or CE assumption. It also allows the parameter-tuning algorithm to be derived during the proof process, not selected a priori in an ad hoc manner;
        (6) The tuning algorithms for the neural net weights are innovative, and no preliminary off-line tuning is needed. The weights may be tuned on-line, and the algorithms guarantee stability of the controlled system; and
        (7) The neural network weights are easily initialized so that the neural net output is zero, and a proportional-derivative (PD) control loop keeps the system stable initially until the weights begin to learn.
    
    


The advantages and features of the present invention differentiate it from previously-known methodology. Some of these advantages and features include, but are not limited to, the following:

    
    
        (1) Other techniques based on discrete-time adaptive control require a certainty equivalence assumption, which often does not hold in practice;
        (2) Most backlash compensators using older control technology do not have any performance or stability guarantees. This makes their acceptance by industry questionable.
        (3) Other modem backlash compensators that do not use neural nets, e.g. those based on adaptive control, require a strong ‘linear in the parameters assumption’ that does not often hold for actual industrial systems. This restricts the types of backlash nonlinearities that can be canceled.
        (4) Other backlash compensation techniques based on neural networks do not provide design algorithms based on mathematical proofs that guarantee stability of the controlled system. They use standard ‘backpropagation’ weight tuning, which cannot provide stability in all situations.
        (5) Other backlash compensation techniques based on neural networks do not show how to initialize the weights to guarantee performance.
        (6) Other techniques based on dynamic inversion have been developed for continuous-time systems, not for discrete-time systems. A discrete-time formulation is needed for digital control.
    
    


As shown in the Examples section of this disclosure, simulation results show that a NN backlash compensator according to the present invention may significantly reduce the degrading effect of backlash nonlinearity.



BRIEF DESCRIPTION OF THE DRAWINGS
The following drawings form part of the present specification and are included to further demonstrate certain aspects of the present invention. The invention may be better understood by reference to one or more of these drawings in combination with the detailed description of specific embodiments presented herein.
FIG. 1A shows a backlash model according to one embodiment of the present disclosure.
FIG. 1B shows backlash in mechanical connections according to one embodiment of the present disclosure.
FIG. 2 shows backlash inverse according to one embodiment of the present disclosure.
FIGS. 3A and 3B show backlash inverse decomposition according to one embodiment of the present disclosure.
FIG. 4 shows a discrete time neural network compensator for compensating backlash according to one embodiment of the present disclosure.
FIG. 5 illustrates data concerning a PD controller without backlash.
FIG. 6 illustrates data concerning a PD controller with backlash.
FIG. 7 illustrates data concerning a PD controller with backlash compensation according to one embodiment of the present disclosure.



DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS
This disclosure provides the complete solution for extending dynamics inversion to discrete-time systems by using a filtered prediction approach for backlash compensation. The general case of nonsymmetric backlash is treated. A rigorous design procedure is given that results in a PD tracking loop with an adaptive NN in the feedforward loop for dynamic inversion of the backlash nonlinearity. The NN feedforward compensator is adapted in such a way as to estimate on-line the backlash inverse. Unlike standard discrete-time adaptive control techniques, no certainty equivalence (CE) assumption is needed since the tracking error and the estimation error are weighted in the same Lyapunov function. Unlike previous conventional techniques, no linearity in the parameters is needed.
Background
General
Let R denote the real numbers, Rn denote the real n-vectors, Rm×n the real m×n matrices. Let S be a compact simply connected set of Rn. With maps ƒ:S→Rk, define Ck(S) as the space such that ƒ is continuous. We denote by ∥·∥ a suitable vector norm. Given a matrix A=[aij], A∈Rn×m the Frobenius norm is defined by




  
    
      
        
          
            
              
              A
              
            
            F
            2
          
          =
          
            
              tr
              ⁡
              
                (
                
                  
                    A
                    T
                  
                  ⁢
                  A
                
                )
              
            
            =
            
              
                ∑
                
                  i
                  ,
                  j
                
                
                  
                
              
              ⁢
              
                
              
              ⁢
              
                a
                ij
                2
              
            
          
        
        ,
      
    
    
      
        (
        1
        )
      
    
  




with tr( ) the trace operation. The associated inner product is (A,B)F=tr(ATB). The Frobenius norm ∥A∥F2, which is denoted by ∥·∥ throughout this disclosure unless specified explicitly, is nothing but the vector 2-norm over the space defined by stacking the matrix columns into a vector, so that it is compatible with the vector 2-norm, that is, ∥Ax∥≦∥A∥·∥x∥.

DEFINITION 1: Given a dynamical system x(k+1)=ƒ(x(k),u(k)),y(k)=h(x(k)), where x(k) is a state vector, u(k) is the input vector and y(k) is the output vector. The solution is Globally Uniformly Ultimately Bounded (GUUB) if for all x(k0)=x0, there exists an ε>0 and a number N(ε,x0) such that ∥x(k)∥<ε for all k≧k0+N.
Dynamics of an mn-th Order MIMO System
Consider an mnth-order multi-input and multi-output discrete-time system given by




  
    
      
        
          
            
              
                
                  x
                  1
                
                ⁡
                
                  (
                  
                    k
                    +
                    1
                  
                  )
                
              
              =
              
                
                  x
                  2
                
                ⁡
                
                  (
                  k
                  )
                
              
            
          
        
        
          
            ⋮
          
        
        
          
            
              
                
                  x
                  
                    n
                    -
                    1
                  
                
                ⁡
                
                  (
                  
                    k
                    +
                    1
                  
                  )
                
              
              =
              
                
                  x
                  n
                
                ⁡
                
                  (
                  k
                  )
                
              
            
          
        
        
          
            
              
                
                  
                    x
                    n
                  
                  ⁡
                  
                    (
                    
                      k
                      +
                      1
                    
                    )
                  
                
                =
                
                  
                    f
                    ⁡
                    
                      (
                      
                        x
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      )
                    
                  
                  +
                  
                    τ
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  +
                  
                    d
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




where x(k)=[x1(k),x2(k) . . . , xn(k)]T with xi(k)∈n; i=1,2, . . . ,n, u(k)∈Rm, and d(k)∈Rm denotes a disturbance vector acting on the system at the instant k with ∥d(k)∥≦dM a known constant. The actuator output τ(k) is related to the control input u(k) through the backlash nonlinearity. τ(k)=Backlash(u(k)) as discussed in the next section. Given a desired trajectory xnd(k) and its delayed values, define the tracking error as

en(k)=xn(k)−xnd(k).  (3)

It is typical in robotics to define a so-called the filtered tracking error, as r(k)∈Rm, and given by

r(k)=en(k)+λ1en−1(k)+ . . . +λn−1e1(k),  (4)

where en−1(k), . . . , e1(k) are the delayed values of the error en(k), and λ1, . . . , λn−1 are constant matrices selected so that |zn−1+λ1zn−2+ . . . +λn−1| is stable or Hurwitz (i.e. en(k)→0 exponentially as r(k)→0). Equation (4) can be further expressed as

r(k+1)=en(k+1)+λ1en−1(k+1)+ . . . +λn−1e1(k+1).  (5)

Using Eq. (2) in Eq. (5), the dynamics of the mn-th order MIMO system can be written in terms of the tracking error as

r(k+1)=f(x(k))−xnd(k+1)+λ1en−1(k)+ . . . +λn−1e1(k)+τ(k)+d(k).  (6)

Neural Networks

Given x∈N, a one-layer feedforward NN has a net output given by




  
    
      
        
          
            
              y
              i
            
            =
            
              
                ∑
                
                  j
                  =
                  1
                
                
                  N
                  h
                
              
              ⁢
              
                
              
              ⁢
              
                [
                
                  
                    
                      w
                      ij
                    
                    ⁢
                    
                      φ
                      ⁡
                      
                        (
                        x
                        )
                      
                    
                  
                  +
                  
                    θ
                    wi
                  
                
                ]
              
            
          
          ;
          
            
          
          ⁢
          
            i
            =
            1
          
        
        ,
        ...
        ⁢
        
          
        
        ,
        m
      
    
    
      
        (
        7
        )
      
    
  




with φ(·) the activation functions and wij the output-layer weights. The θwi,i=1,2, . . . , are threshold offsets, and Nh is the number of hidden-layer neurons. In the NN we should like to adapt the weights and thresholds on-line in real time to provide suitable performance of the net. That is, the NN should exhibit “on-line learning while controlling” behavior. The output of a one-layer can be also expressed in matrix form as

y(x)=WTφ(x),  (8)

where 1 is included as the first element of φ(x) in order to incorporate the thresholds θwi as the first column of WT. Thus, any tuning of W includes tuning of the thresholds as well.

One property of NN we are concerned with for control and estimation purposes is the function approximation property. Let ƒ(x) be a smooth function from n→m. Then it can be shown that, as long as x is restricted to a compact set S∈n, for some sufficiently large number of hidden-layer neurons Nh, there exist weights and thresholds such that one has

ƒ(x)=WTφ(x)+ε(x).  (9)

This equation means that a neural network can approximate any continuous function in a compact set. The value of ε(x) is called the neural network functional approximation error. In fact, for any choice of a positive number εN, one can find a neural network such that ε(x)≦εN for all x∈S. For suitable NN approximation properties, φ(x) must be a basis:

DEFINITION 2: Let S be a compact simply connected set of n and let φ(x):S→Nh be integrable and bounded. Then φ(x) is said to provide a basis for Cm(S) if:

    
    
        1. A constant function on S can be expressed as (7) for finite Nh.
        2. The functional range of neural network (7) is dense in Cm(S) for countable Nh.
    
    


It has been shown that the neural network approximation error ε(x) for one-layer NN is fundamentally bounded below by a term of the order (1/n)2/d, where n is the number of fixed basis functions and d is the dimension of the input to the NN. This does not limit the tracking performance in controllers of this disclosure because of the control system structure selected.
It is not straightforward to pick a basis φ(x). CMAC, RBF, and other structured NN approaches allow one to choose a basis by partitioning the compact set S. However, this can be tedious. If one selects

y(x)=WTσ(VTx),  (10)

with, for instance,





  
    σ
    ⁡
    
      (
      x
      )
    
  
  =
  
    1
    
      1
      +
      
        e
        
          α
          ⁢
          
            
          
          ⁢
          x
        
      
    
  




the sigmoid, then it has been shown that σ((VTx) is a basis if V is selected randomly. Once selected, V is fixed and only W is tuned. Then, the only design parameter in constructing the 1-layer NN is the number of hidden layer neurons Nh. A larger Nh results in a smaller ε(x).

Backlash Nonlinearity and Backlash Inverse

Backlash nonlinearity is shown in FIG. 1, and the mathematical model for continuous time is known in the art. For the discrete-time case, one has




  
    
      
        
          τ
          ⁡
          
            (
            
              k
              +
              1
            
            )
          
        
        =
        
          
            B
            ⁡
            
              (
              
                
                  τ
                  ⁡
                  
                    (
                    k
                    )
                  
                
                ,
                
                  u
                  ⁡
                  
                    (
                    k
                    )
                  
                
                ,
                
                  u
                  ⁡
                  
                    (
                    
                      k
                      +
                      1
                    
                    )
                  
                
              
              )
            
          
          =
          
            {
            
              
                
                  
                    
                      
                        m
                        ·
                        
                          u
                          ⁡
                          
                            (
                            
                              k
                              +
                              1
                            
                            )
                          
                        
                      
                      ,
                    
                  
                  
                    
                      
                        
                          
                            
                              
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  u
                                  ⁡
                                  
                                    (
                                    
                                      k
                                      +
                                      1
                                    
                                    )
                                  
                                
                              
                              >
                              
                                0
                                ⁢
                                
                                  
                                
                                ⁢
                                and
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  u
                                  ⁡
                                  
                                    (
                                    k
                                    )
                                  
                                
                              
                            
                            =
                            
                              
                                m
                                ·
                                
                                  τ
                                  ⁡
                                  
                                    (
                                    k
                                    )
                                  
                                
                              
                              -
                              
                                m
                                ·
                                
                                  d
                                  +
                                
                              
                            
                          
                        
                      
                      
                        
                          
                            
                              
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  u
                                  ⁡
                                  
                                    (
                                    
                                      k
                                      +
                                      1
                                    
                                    )
                                  
                                
                              
                              <
                              
                                0
                                ⁢
                                
                                  
                                
                                ⁢
                                and
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  u
                                  ⁡
                                  
                                    (
                                    k
                                    )
                                  
                                
                              
                            
                            =
                            
                              
                                m
                                ·
                                
                                  τ
                                  ⁡
                                  
                                    (
                                    k
                                    )
                                  
                                
                              
                              -
                              
                                m
                                ·
                                
                                  d
                                  -
                                
                              
                            
                          
                        
                      
                    
                  
                
                
                  
                    
                      0
                      ,
                    
                  
                  
                    otherwise
                  
                
              
              .
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




It can be seen that backlash is a first-order velocity driven dynamic system, with inputs u(k) and u(k+1), and state τ(k). Backlash contains its own dynamics, therefore its compensation requires the design of a dynamic compensator.

Whenever the motion u(k) changes its direction, the motion τ(k) is delayed from motion of u(k). One objective of a backlash compensator is to make this delay as small as possible, i.e. to make the throughput from u(k) to τ(k) be unity. The backlash precompensator needs to generate the inverse of the backlash nonlinearity. The backlash inverse function is shown in FIG. 2.
The dynamics of the NN backlash compensator is given by

u(k+1)=Binv(u(k), w(k), w(k+1)),  (12)

The backlash inverse characteristic shown in the FIG. 2 can be decomposed into two functions: a direct feedforward term plus an additional modified backlash inverse term as shown in FIG. 3. This decomposition allows design of a compensator that has a better structure than when a NN is used directly in the feedforward path.
Discrete Time NN Backlash Compensator
Portions of a discrete time NN backlash compensator according to one embodiment of this disclosure may be designed using backstepping techniques known in the art. In this section it is shown how to tune the NN weights on-line so that the tracking error is guaranteed small and all internal states are bounded. It is assumed that the actuator output τ(k) is measurable. Unlike conventional techniques, no linearity in the parameters assumption is needed.
Dynamics of Nonlinear System with Backlash
Equation (2) is in the companion form and represents a large class of multi-input multi-output (MINO) nonlinear systems. The overall system dynamics consist of (2) and the backlash dynamics (11).
The following assumptions are needed and they are true in every practical situation and are standard in the existing literature.
Assumption 1 (Bounded disturbance): The unknown disturbance satisfies ∥d(k)∥≦dM, with dM a known positive constant.
Assumption 2 (Bounded estimation error): The nonlinear function is assumed to be unknown, but a fixed estimate {circumflex over (ƒ)}(x(k)) is assumed known such that the functional estimation error, {tilde over (ƒ)}(x(k))=ƒ(x(k))−{circumflex over (f)}(x(k)), satisfies ∥{tilde over (ƒ)}(x(k))∥≦ƒM(x(k)), for some known bounding function ƒM(x(k)).
This assumption is not unreasonable, as in practical systems the bound ƒM(x(k)) can be computed knowing the upper bound on payload masses, frictional effects, and so on.
Assumption 3 (Bounded desired trajectories): The desired trajectory is bounded in the sense, for instance that




  
    
    
      
        
          
            
              
                
                  
                    
                      
                        
                          x
                          
                            1
                            ⁢
                            d
                          
                        
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          x
                          
                            2
                            ⁢
                            d
                          
                        
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                    
                  
                
              
            
            
              
                ⋮
              
            
          
        
      
      
        
          
            
              x
              nd
            
            ⁡
            
              (
              k
              )
            
          
        
      
    
    
  
  ≤
  
    
      X
      d
    
    ·
  




Backstepping Controller

A robust compensation scheme for unknown terms in ƒ(x(k)) is provided by selecting the tracking controller

τdes(k)=Kv·r(k)−{circumflex over (ƒ)}(x(k))+xnd(k+1)−λ1·en−1(k)−λ2·en−2(k)− . . . λn−1·e1(k),  (13)

with {circumflex over (ƒ)}(x(k)) an estimate for the nonlinear terms ƒ(x(k)). The feedback gain matrix Kv>0 is often selected diagonal. The problem of finding {circumflex over (ƒ)}(x(k)) is not the main concern of this disclosure, and it may be found by methods known in the art. This function ƒ(x(k)) can be estimated, for instance, using adaptive control techniques or neural network controllers.

Using (13) as a control input, the system dynamics in (6) can be rewritten as

r(k+1)=Kv·r(k)+{tilde over (ƒ)}(x(k))+d(k).  (14)

The next theorem is the first step in the backstepping design; and it shows that the desired control law (13) will keep the filtered tracking error small if there is no backlash.
Theorem 1 (Control Law for Outer Tracking Loop)
Consider the system given by equation (2). Assume that Assumptions 1 and 2 hold, and let the control action by provided by (13) with 0<Kv<I being a design parameter.
Then the filtered tracking error r(k) is UUB.
Proof
Let us consider the following Lyapunov function candidate

L1(k)=r(k)Tr(k).  (15)

The first difference is





  
    
      
        
          
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  
                    L
                    1
                  
                  ⁡
                  
                    (
                    k
                    )
                  
                
              
              =
              
                
                  
                    
                      
                        r
                        ⁡
                        
                          (
                          
                            k
                            +
                            1
                          
                          )
                        
                      
                      T
                    
                    ⁢
                    
                      r
                      ⁡
                      
                        (
                        
                          k
                          +
                          1
                        
                        )
                      
                    
                  
                  -
                  
                    
                      
                        r
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      T
                    
                    ⁢
                    
                      r
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                  
                
                =
              
            
          
        
        
          
            
              =
              
                
                  
                    
                      (
                      
                        
                          
                            K
                            v
                          
                          ·
                          
                            r
                            ⁡
                            
                              (
                              k
                              )
                            
                          
                        
                        +
                        
                          
                            f
                            ~
                          
                          ⁡
                          
                            (
                            
                              x
                              ⁡
                              
                                (
                                k
                                )
                              
                            
                            )
                          
                        
                        +
                        
                          d
                          ⁡
                          
                            (
                            k
                            )
                          
                        
                      
                      )
                    
                    T
                  
                  ⁢
                  
                    (
                    
                      
                        
                          K
                          v
                        
                        ·
                        
                          r
                          ⁡
                          
                            (
                            k
                            )
                          
                        
                      
                      +
                      
                        
                          f
                          ~
                        
                        ⁡
                        
                          (
                          
                            x
                            ⁡
                            
                              (
                              k
                              )
                            
                          
                          )
                        
                      
                      +
                      
                        d
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                    
                    )
                  
                
                -
                
                  
                    
                      r
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                    T
                  
                  ⁢
                  
                    
                      r
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                    .
                  
                
              
            
          
        
        
          
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  
                    L
                    1
                  
                  ⁡
                  
                    (
                    k
                    )
                  
                
                ⁢
                
                  
                
                ⁢
                is
                ⁢
                
                  
                
                ⁢
                negative
                ⁢
                
                  
                
                ⁢
                if
                ⁢
                
                  
                
                ⁢
                
                  
                  
                    
                      
                        K
                        v
                      
                      ⁢
                      
                        r
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                    
                    +
                    
                      
                        f
                        ~
                      
                      ⁡
                      
                        (
                        
                          x
                          ⁡
                          
                            (
                            k
                            )
                          
                        
                        )
                      
                    
                    +
                    
                      d
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                  
                  
                
              
              ≤
              
                
                  
                    K
                    
                      v
                      max
                    
                  
                  ⁢
                  
                    
                    
                      r
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                    
                  
                
                +
                
                  f
                  M
                
                +
                
                  d
                  M
                
              
              <
              
                
                
                  r
                  ⁡
                  
                    (
                    k
                    )
                  
                
                
              
            
          
        
        
          
            
              
                ⇒
                
                  
                    
                      (
                      
                        1
                        -
                        
                          K
                          
                            v
                            max
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                      
                        r
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      
                    
                  
                  >
                  
                    
                      f
                      M
                    
                    +
                    
                      d
                      M
                    
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        16
        )
      
    
  




which is true as long as





  
    
      
        
          
          
            r
            ⁡
            
              (
              k
              )
            
          
          
        
        >
        
          
            
              
                f
                M
              
              +
              
                d
                M
              
            
            
              1
              -
              
                K
                
                  v
                  ⁢
                  
                    
                  
                  ⁢
                  max
                
              
            
          
          .
        
      
    
    
      
        (
        17
        )
      
    
  




Therefore, ΔL1(k) is negative outside a compact set. According to standard Lyapunov theory extension, this demonstrates the UUB of r(k).
NN Backlash Compensation using Dynamic Inversion
Theorem 1 gives the control law that guarantees stability in term of the filtered tracking error, assuming that no nonlinearity besides the system nonlinear function plus some bounded external disturbances are present. In the presence of unknown backlash nonlinearity, the desired and actual value of the control signal τ(k) will be different. A dynamics inversion technique by neural networks may be used for compensation of the inversion error. This is a form of backstepping.
The actuator output given by (13) is the desired signal. The complete error system dynamics can be found defining the error

{tilde over (τ)}(k)=τdes(k)−τ(k).  (18)

Using the desired control input (13), under the presence of unknown backlash the system dynamics (6) can be rewritten as

r(k+1)=Kv·r(k)+{tilde over (ƒ)}(x(k))+d(k)−{tilde over (τ)}(k).  (19)

Evaluating (18) at the following time interval

{tilde over (τ)}(k+1)=τdes(k+1)−τ(k+1)=τdes(k+1)−B(τ(k),u(k),u(k+1)).  (20)

which together with (19) represents the complete system error dynamics. The dynamics of the backlash nonlinearity can be written as:

τ(k+1)=φ(k),  (21)

φ(k)=B(τ(k),u(k),u(k+1)),  (22)

where φ(k) is a pseudo-control input. In the case of known backlash, the ideal backlash inverse is given by

u(k+1)=B−1(u(k),τ(k),φ(k)).  (23)

Since the backlash and therefore its inverse are not known, one can only approximate the backlash inverse as

û(k+1)={circumflex over (B)}−1(u(k),τ(k),φ(k)).  (24)

The backlash dynamics can now be written as





  
    
      
        
          
            
              
                τ
                ⁡
                
                  (
                  
                    k
                    +
                    1
                  
                  )
                
              
              =
              
                B
                ⁡
                
                  (
                  
                    
                      τ
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                    ,
                    
                      
                        u
                        ^
                      
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                    ,
                    
                      
                        u
                        ^
                      
                      ⁡
                      
                        (
                        
                          k
                          +
                          1
                        
                        )
                      
                    
                  
                  )
                
              
            
          
        
        
          
            
              
                =
                
                  
                    
                      B
                      ^
                    
                    ⁡
                    
                      (
                      
                        
                          τ
                          ⁡
                          
                            (
                            k
                            )
                          
                        
                        ,
                        
                          
                            u
                            ^
                          
                          ⁡
                          
                            (
                            k
                            )
                          
                        
                        ,
                        
                          
                            u
                            ^
                          
                          ⁡
                          
                            (
                            
                              k
                              +
                              1
                            
                            )
                          
                        
                      
                      )
                    
                  
                  +
                  
                    
                      B
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          τ
                          ⁡
                          
                            (
                            k
                            )
                          
                        
                        ,
                        
                          
                            u
                            ^
                          
                          ⁡
                          
                            (
                            k
                            )
                          
                        
                        ,
                        
                          
                            u
                            ^
                          
                          ⁡
                          
                            (
                            
                              k
                              +
                              1
                            
                            )
                          
                        
                      
                      )
                    
                  
                
              
              ,
            
          
        
        
          
            
              =
              
                
                  
                    φ
                    ^
                  
                  ⁡
                  
                    (
                    k
                    )
                  
                
                +
                
                  
                    B
                    ~
                  
                  ⁡
                  
                    (
                    
                      
                        τ
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      ,
                      
                        
                          u
                          ^
                        
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      ,
                      
                        
                          u
                          ^
                        
                        ⁡
                        
                          (
                          
                            k
                            +
                            1
                          
                          )
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        25
        )
      
    
  




where φ(k)={circumflex over (B)}(τ(k),û(k),û(k+1)) and therefore its inverse is given by û(k+1)={circumflex over (B)}−1(τ(k),û(k),φ(k)). The unknown function {tilde over (B)}(τ(k),û(k),û(k+1)), which represents the backlash inversion error, will be approximated using a neural network.

In order to design a stable closed-loop system with backlash compensation, one selects a nominal backlash inverse û(k+1)={circumflex over (φ)}(k) and pseudo-control input as

{circumflex over (φ)}(k)=−Kb{tilde over (τ)}(k)+τfilt(k)+Ŵ(k)Tσ(VTxnn(k)),  (26)

where Kb>0 is a design parameter, and τfilt is a discrete-time filtered version of τdes. τfilt is a filtered prediction that approximates τdes(k+1), and is obtained using the discrete-time filter az/(z+a) as shown in FIG. 4. This is the equivalent of using a filtered derivative instead of a pure derivative in continuous-time dynamics inversion, which is standard in industrial control systems. The filter dynamics shown in FIG. 4 can be written as





  
    
      
        
          
            
              τ
              filt
            
            ⁡
            
              (
              k
              )
            
          
          =
          
            
              -
              
                
                  
                    τ
                    filt
                  
                  ⁡
                  
                    (
                    
                      k
                      +
                      1
                    
                    )
                  
                
                a
              
            
            +
            
              
                τ
                des
              
              ⁡
              
                (
                
                  k
                  +
                  1
                
                )
              
            
          
        
        ,
      
    
    
      
        (
        27
        )
      
    
  




where a is a design parameter. It can be seen that when the filter parameter a is large enough we have τflit(k)≈τdes(k+1). The mismatch term





  -
  
    
      
        τ
        filt
      
      ⁡
      
        (
        
          k
          +
          1
        
        )
      
    
    a
  




can be approximated along with the backlash inversion error using the NN.

Based on the NN approximation property, the backlash inversion plus the filter error dynamics can be represented as




  
    
      
        
          
            
              
                B
                ~
              
              ⁡
              
                (
                
                  
                    τ
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  ,
                  
                    
                      u
                      ^
                    
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  ,
                  
                    
                      u
                      ^
                    
                    ⁡
                    
                      (
                      
                        k
                        +
                        1
                      
                      )
                    
                  
                
                )
              
            
            +
            
              
                
                  τ
                  filt
                
                ⁡
                
                  (
                  
                    k
                    +
                    1
                  
                  )
                
              
              a
            
          
          =
          
            
              
                
                  W
                  ⁡
                  
                    (
                    k
                    )
                  
                
                T
              
              ⁢
              
                σ
                ⁡
                
                  (
                  
                    
                      V
                      T
                    
                    ⁢
                    
                      
                        x
                        nn
                      
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                  
                  )
                
              
            
            +
            
              ɛ
              ⁡
              
                (
                k
                )
              
            
          
        
        ,
      
    
    
      
        (
        28
        )
      
    
  




where the NN input vector is chosen to be xnn(k)=[1 r(k)Txd(k)T{tilde over (τ)}(k)Tτ(k)T]T, and ε(k) represents the NN approximation error. It can be seen that the first layer of weights is not time dependant since it is selected randomly at initial time to provide a basis and then it is kept constant through the tuning process.

Define the NN weight estimation error as

{tilde over (W)}(k)=W(k)−Ŵ(k),  (29)

where Ŵ(k) is the estimate of the ideal NN weights W(k).

Using the proposed controller shown in FIG. 4, the error dynamics can be written as




  
    
      
        
          
            
              
                
                  
                    τ
                    ~
                  
                  ⁡
                  
                    (
                    
                      k
                      +
                      1
                    
                    )
                  
                
                =
                
                ⁢
                
                  
                    
                      τ
                      des
                    
                    ⁡
                    
                      (
                      
                        k
                        +
                        1
                      
                      )
                    
                  
                  -
                  
                    
                      φ
                      ^
                    
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  +
                  
                    
                      B
                      ~
                    
                    ⁡
                    
                      (
                      
                        
                          τ
                          ⁡
                          
                            (
                            k
                            )
                          
                        
                        ,
                        
                          
                            u
                            ^
                          
                          ⁡
                          
                            (
                            k
                            )
                          
                        
                        ,
                        
                          
                            u
                            ^
                          
                          ⁡
                          
                            (
                            
                              k
                              +
                              1
                            
                            )
                          
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      K
                      b
                    
                    ⁢
                    
                      
                        τ
                        ~
                      
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                  
                  +
                  
                    
                      
                        τ
                        filt
                      
                      ⁡
                      
                        (
                        
                          k
                          +
                          1
                        
                        )
                      
                    
                    a
                  
                  -
                  
                    
                      
                        
                          W
                          ^
                        
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      T
                    
                    ⁢
                    
                      σ
                      ⁡
                      
                        (
                        
                          
                            V
                            T
                          
                          ⁢
                          
                            
                              x
                              nn
                            
                            ⁡
                            
                              (
                              k
                              )
                            
                          
                        
                        )
                      
                    
                  
                  +
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    B
                    ~
                  
                  ⁡
                  
                    (
                    
                      
                        τ
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      ,
                      
                        
                          u
                          ^
                        
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      ,
                      
                        
                          u
                          ^
                        
                        ⁡
                        
                          (
                          
                            k
                            +
                            1
                          
                          )
                        
                      
                    
                    )
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      K
                      b
                    
                    ⁢
                    
                      
                        τ
                        ~
                      
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                  
                  -
                  
                    
                      
                        
                          W
                          ^
                        
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      T
                    
                    ⁢
                    
                      σ
                      ⁡
                      
                        (
                        
                          
                            V
                            T
                          
                          ⁢
                          
                            
                              x
                              nn
                            
                            ⁡
                            
                              (
                              k
                              )
                            
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      
                        W
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      T
                    
                    ⁢
                    
                      σ
                      ⁡
                      
                        (
                        
                          
                            V
                            T
                          
                          ⁢
                          
                            
                              x
                              nn
                            
                            ⁡
                            
                              (
                              k
                              )
                            
                          
                        
                        )
                      
                    
                  
                  +
                  
                    ɛ
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        30
        )
      
    
  




Using (29),

{tilde over (τ)}(k+1)=Kb{tilde over (τ)}(k)+{tilde over (W)}(k)Tσ(VTxnn(k))+ε(k).  (31)

The next theorem is an important result, which shows how to tune the neural network weights so the tracking error r(k) and backlash estimation error {tilde over (τ)}(k) achieve small values while the NN weights estimation errors {tilde over (W)}(k) are bounded.

Theorem 2 (Control Law for Backstepping Loop)

Consider the system given by (2). Provided that assumptions 1, 2, and 3 hold, let the control action {circumflex over (φ)}(k) by provided by (26) with Kb>0 being a design parameter.
Let u(k+1)=φ(k), and the estimated NN weights be provided by the NN tuning law

Ŵ(k+1)=Ŵ(k)+ασ(k)r(k+1)T+ασ(k){tilde over (τ)}(k+1)T−Γ∥I−ασ(k)σ(k)T∥Ŵ(k).  (32)

where α>0 is a constant learning rate parameter or adaptation gain, Γ>0 is a design parameter, and for simplicity purposes σ(VTxnn(k)) is represented as σ(k). Then, the filtered tracking error r(k), the backlash estimation error {tilde over (τ)}(k), and the NN weight estimation error {tilde over (W)}(k) are UUB, provided the following conditions hold:

1) 0<ασ(k)Tσ(k)<1⁄2,  (33)

2) 0<Γ<1,  (34)





  
    
      
        
          
            
              3
              )
            
            ⁢
            
              
            
            ⁢
            0
          
          <
          
            K
            v
          
          <
          
            I
            ⁢
            
              
            
            ⁢
            and
            ⁢
            
              
            
            ⁢
            
              K
              vmax
            
          
          <
          
            1
            
              
                η
                +
                2
              
            
          
        
        ,
      
    
    
      
        (
        35
        )
      
    
  




where

β=Kv−1(2I−Kv)+(1−ασ(k)Tσ(k)Kv−TKv−1>0.  (36)

ρ=(1−ασ(k)Tσ(k))I−β−1(ασ(k)Tσ(k)+Γ∥I−ασ(k)σ(k)T∥)2>0.  (37)

η=(1+ασ(k)Tσ(k))I+ρ−1(ασ(k)Tσ(k)+Γ∥I−ασ(k)σ(k)T∥)2>0.  (38)

Theorem Proof: See Appendix B.

Note that condition (36) is true because of (33) and (35). Note also that (38) is satisfied because of conditions (33) and (37). A proof for condition (37) is given in Appendix A.
Remarks
It is important to note that in this theorem, there is no certainty equivalence (CE) assumption, in contrast to standard work in discrete-time adaptive control. In the latter, a parameter identifier is first selected and the parameter estimation errors are assumed small. In the tracking proof, it is assumed that the parameter estimates are exact (the CE assumption), and a Lyapunov function is selected that weights only the tracking error to demonstrate close-loop stability and tracking performance. By contrast, in this disclosure, the Lyapunov function in Appendix B is of the form




  
    
      
        
          
            J
            ⁡
            
              (
              k
              )
            
          
          =
          
            
              
                
                  
                    [
                    
                      
                        r
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      +
                      
                        
                          τ
                          ~
                        
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                    
                    ]
                  
                  T
                
                ·
                
                  [
                  
                    
                      r
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                    +
                    
                      
                        τ
                        ~
                      
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                  
                  ]
                
              
              +
              
                
                  
                    r
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  T
                
                ⁢
                
                  r
                  ⁡
                  
                    (
                    k
                    )
                  
                
              
              +
              
                
                  1
                  α
                
                ⁢
                tr
                ⁢
                
                  {
                  
                    
                      
                        
                          W
                          ~
                        
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      T
                    
                    ·
                    
                      
                        W
                        ~
                      
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                  
                  }
                
              
            
            >
            0
          
        
        ,
      
    
    
      
        (
        
          B
          ⁢
          .4
        
        )
      
    
  




which weights the tracking error r(k), backlash estimation error {tilde over (τ)}(k) and the NN weight estimation error {tilde over (W)}(k). This requires an exceedingly complex proof, but obviates the need for any sort of CE assumption. It also allows the parameter-tuning algorithm to be derived during the proof process, not selected a priori in an ad hoc manner.

The third term in (32) is a discrete-time version of Narendra's e-mod, which is required to provide robustness due to the coupling in the proof between tracking error, backlash error terms and weight estimation error terms in the Lyapunov function. This is called a ‘forgetting term’ in NN weight-tuning algorithms. These are required in that context to prevent parameter overtraining.
The following examples are included to demonstrate specific embodiments of the present disclosure. Those of skill in the art should, in light of the present disclosure, appreciate that many changes may be made in specific embodiments that are disclosed and still obtain a like or similar result without departing from the spirit and scope of the invention.
EXAMPLE 1
Simulation Results
In this section, a discrete-time NN backlash compensator according to an embodiment of this disclosure is simulated on a digital computer. It is found to be very efficient at canceling the deleterious effects of actuator backlash.
Simulation
We simulate the response for the known plant with input backlash, both with and without the NN compensator. Consider the following nonlinear plant

x1(k+1)=x2(k),





  
    
      x
      2
    
    ⁡
    
      (
      
        k
        +
        1
      
      )
    
  
  =
  
    
      -
      
        
          3
          16
        
        ⁡
        
          [
          
            
              
                x
                1
              
              ⁡
              
                (
                k
                )
              
            
            
              1
              +
              
                
                  x
                  2
                  2
                
                ⁡
                
                  (
                  k
                  )
                
              
            
          
          ]
        
      
    
    +
    
      
        x
        2
      
      ⁡
      
        (
        k
        )
      
    
    +
    
      
        u
        ⁡
        
          (
          k
          )
        
      
      .
    
  




The deadband widths for the backlash nonlinearity were selected as d+=d−=0.2 and the slope as m=0.5.
Trajectory Tracking
In this subsection we simulate the trajectory tracking performance of the system for sinusoidal reference signals. The reference signal used was selected to be

xd(k)=sin(w·tk+φ),

w=0.5, φ=π/2.

The sampling period was selected as T=0.001 s.
FIG. 5 shows the system response without backlash using a standard PD controller. The PD controller does a good job on the tracking which is achieved at about 2 seconds. FIG. 6 shows the system response with input backlash. The system backlash destroys the tracking and the PD controller by itself is not capable of compensating for that. FIG. 7 shows the same situation but using the discrete-time NN backlash compensator of the present disclosure. The backlash compensator takes care of the system backlash and the tracking is achieved in less than 0.5 seconds.
Appendix A
Note: For simplicity purposes, in this appendix, the k sub-index is omitted. So, every variable has a k sub-index unless specified otherwise. This statement is valid only for the proofs shown in Appendices A and B.
Proof of Condition (37).
Because of condition (33), we have that




  
    
      (
      
        1
        -
        
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
        
      
      )
    
    ⁢
    I
  
  >
  
    
      1
      2
    
    ⁢
    
      I
      .
    
  




Also using (33), (34) we have that





  
    
      (
      
        
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
        
        +
        
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ⁢
                
                  
                
                ⁢
                σ
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
              
            
            
          
        
      
      )
    
    2
  
  <
  
    
      1
      4
    
    ⁢
    I
  




Using (35) we have that β>I (i.e., β−1<I). Then we can conclude that





  
    
      
        β
        
          -
          1
        
      
      (
      
        
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
        
        +
        
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ⁢
                
                  
                
                ⁢
                σ
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
              
            
            
          
        
      
      )
    
    2
  
  <
  
    
      1
      4
    
    ⁢
    
      I
      .
    
  




Finally, using this last result we can show that





  ρ
  =
  
    
      
        
          (
          
            1
            -
            
              α
              ⁢
              
                
              
              ⁢
              
                σ
                T
              
              ⁢
              σ
            
          
          )
        
        ⁢
        I
      
      -
      
        
          
            β
            
              -
              1
            
          
          (
          
            
              α
              ⁢
              
                
              
              ⁢
              
                σ
                T
              
              ⁢
              σ
            
            +
            
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    σ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      σ
                      T
                    
                  
                
                
              
            
          
          )
        
        2
      
    
    >
    
      
        1
        4
      
      ⁢
      I
    
    >
    0.
  




Appendix B

Proof of Theorem 2

For simplicity purposes let us rewrite the system dynamics as

rk+1=Kv·r+D−{tilde over (τ)}.  (B.1)

where D={tilde over (ƒ)}+d. And let us rewrite the backlash dynamics as

{tilde over (τ)}k+1=Kb{tilde over (τ)}+{tilde over (W)}Tσ(VTxnn)+ε.  (B.2)

Select the Lyapunov function candidate





  
    
      
        L
        =
        
          
            
              
                
                  [
                  
                    
                      
                        
                          r
                          T
                        
                      
                      
                        
                          
                            τ
                            ~
                          
                          T
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        2
                      
                      
                        1
                      
                    
                    
                      
                        1
                      
                      
                        1
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      r
                    
                  
                  
                    
                      
                        τ
                        ~
                      
                    
                  
                
                ]
              
            
            +
            
              
                1
                α
              
              ⁢
              tr
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      W
                      ~
                    
                    T
                  
                  ⁢
                  
                    W
                    ~
                  
                
                )
              
            
          
          >
          0.
        
      
    
    
      
        (
        
          B
          ⁢
          .3
        
        )
      
    
  




This can be rewritten as





  
    
      
        L
        =
        
          
            
              2
              ⁢
              
                r
                T
              
              ⁢
              r
            
            +
            
              2
              ⁢
              
                r
                T
              
              ⁢
              
                τ
                ~
              
            
            +
            
              
                
                  τ
                  ~
                
                T
              
              ⁢
              
                τ
                ~
              
            
            +
            
              
                1
                α
              
              ⁢
              tr
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      W
                      ~
                    
                    T
                  
                  ⁢
                  
                    W
                    ~
                  
                
                )
              
            
          
          =
          
            
              L
              1
            
            +
            
              L
              2
            
            +
            
              L
              3
            
            +
            
              
                L
                4
              
              .
            
          
        
      
    
    
      
        (
        
          B
          ⁢
          .4
        
        )
      
    
  




Taking the first difference





  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            L
            1
          
        
        =
        
          
            
              2
              ⁢
              
                
              
              ⁢
              
                r
                
                  k
                  +
                  1
                
                T
              
              ⁢
              
                r
                
                  k
                  +
                  1
                
              
            
            -
            
              2
              ⁢
              
                r
                T
              
              ⁢
              r
            
          
          =
          
            
              
                
                  
                    2
                    ⁡
                    
                      [
                      
                        
                          
                            K
                            v
                          
                          ⁢
                          r
                        
                        +
                        D
                        -
                        
                          τ
                          ~
                        
                      
                      ]
                    
                  
                  T
                
                ⁡
                
                  [
                  
                    
                      
                        K
                        v
                      
                      ⁢
                      r
                    
                    +
                    D
                    -
                    
                      τ
                      ~
                    
                  
                  ]
                
              
              -
              
                2
                ⁢
                
                  r
                  T
                
                ⁢
                r
              
            
            =
          
        
      
    
  
  
    
      
        =
        
          
            
              -
              2
            
            ⁢
            
              
                r
                T
              
              ⁡
              
                [
                
                  I
                  -
                  
                    
                      K
                      v
                      T
                    
                    ⁢
                    
                      K
                      v
                    
                  
                
                ]
              
            
            ⁢
            r
          
          +
          
            4
            ⁢
            
              r
              T
            
            ⁢
            
              K
              v
              T
            
            ⁢
            D
          
          -
          
            4
            ⁢
            
              r
              T
            
            ⁢
            
              K
              v
              T
            
            ⁢
            
              τ
              ~
            
          
          +
          
            2
            ⁢
            
              D
              T
            
            ⁢
            D
          
          -
          
            4
            ⁢
            
              D
              T
            
            ⁢
            
              τ
              ~
            
          
          +
          
            2
            ⁢
            
              
                τ
                ~
              
              T
            
            ⁢
            
              
                τ
                ~
              
              .
            
          
        
      
    
  
  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            L
            2
          
        
        =
        
          
            
              2
              ⁢
              
                
              
              ⁢
              
                r
                
                  k
                  +
                  1
                
                T
              
              ⁢
              
                
                  τ
                  ~
                
                
                  k
                  +
                  1
                
              
            
            -
            
              2
              ⁢
              
                r
                T
              
              ⁢
              
                τ
                ~
              
            
          
          =
          
            
              2
              ⁢
              
                
                  (
                  
                    
                      
                        K
                        v
                      
                      ⁢
                      r
                    
                    +
                    D
                    -
                    
                      τ
                      ~
                    
                  
                  )
                
                T
              
              ⁢
              
                (
                
                  
                    
                      K
                      b
                    
                    ⁢
                    
                      τ
                      ~
                    
                  
                  +
                  
                    
                      
                        W
                        ~
                      
                      T
                    
                    ⁢
                    σ
                  
                  +
                  ɛ
                
                )
              
            
            -
            
              2
              ⁢
              
                r
                T
              
              ⁢
              
                τ
                ~
              
            
          
        
      
    
  
  
    
      
        =
        
          
            2
            ⁢
            
              r
              T
            
            ⁢
            
              K
              v
              T
            
            ⁢
            
              K
              b
            
            ⁢
            
              τ
              ~
            
          
          +
          
            2
            ⁢
            
              r
              T
            
            ⁢
            
              K
              v
              T
            
            ⁢
            
              
                W
                ~
              
              T
            
            ⁢
            σ
          
          +
          
            2
            ⁢
            
              r
              T
            
            ⁢
            
              K
              v
              T
            
            ⁢
            ɛ
          
          +
          
            2
            ⁢
            
              D
              T
            
            ⁢
            
              K
              b
            
            ⁢
            
              τ
              ~
            
          
          +
          
            2
            ⁢
            
              D
              T
            
            ⁢
            
              
                W
                ~
              
              T
            
            ⁢
            σ
          
          +
          
            2
            ⁢
            
              D
              T
            
            ⁢
            ɛ
          
          -
          
            2
            ⁢
            
              
                τ
                ~
              
              T
            
            ⁢
            
              K
              b
              T
            
            ⁢
            
              K
              b
            
            ⁢
            
              τ
              ~
            
          
          +
        
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          ɛ
        
        -
        
          2
          ⁢
          
            r
            T
          
          ⁢
          
            
              τ
              ~
            
            .
          
        
      
    
  
  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            L
            3
          
        
        =
        
          
            
              
                
                  τ
                  ~
                
                
                  k
                  +
                  1
                
                T
              
              ⁢
              
                
                  τ
                  ~
                
                
                  k
                  +
                  1
                
              
            
            -
            
              
                
                  τ
                  ~
                
                T
              
              ⁢
              
                τ
                ~
              
            
          
          =
          
            
              
                
                  (
                  
                    
                      
                        K
                        b
                      
                      ⁢
                      
                        τ
                        ~
                      
                    
                    +
                    
                      
                        
                          W
                          ~
                        
                        T
                      
                      ⁢
                      σ
                    
                    +
                    ɛ
                  
                  )
                
                T
              
              ⁢
              
                (
                
                  
                    
                      K
                      b
                    
                    ⁢
                    
                      τ
                      ~
                    
                  
                  +
                  
                    
                      
                        W
                        ~
                      
                      T
                    
                    ⁢
                    σ
                  
                  +
                  ɛ
                
                )
              
            
            -
            
              
                
                  τ
                  ~
                
                T
              
              ⁢
              
                τ
                ~
              
            
          
        
      
    
  
  
    
      
        =
        
          
            
              
                τ
                ~
              
              T
            
            ⁢
            
              K
              b
              T
            
            ⁢
            
              K
              b
            
            ⁢
            
              τ
              ~
            
          
          +
          
            2
            ⁢
            
              
                τ
                ~
              
              T
            
            ⁢
            
              K
              b
              T
            
            ⁢
            
              
                W
                ~
              
              T
            
            ⁢
            σ
          
          +
          
            2
            ⁢
            
              
                τ
                ~
              
              T
            
            ⁢
            
              K
              b
              T
            
            ⁢
            ɛ
          
          +
          
            
              σ
              T
            
            ⁢
            
              W
              ~
            
            ⁢
            
              
                W
                ~
              
              T
            
            ⁢
            σ
          
          +
          
            2
            ⁢
            
              ɛ
              T
            
            ⁢
            
              
                W
                ~
              
              T
            
            ⁢
            σ
          
          +
          
            
              ɛ
              T
            
            ⁢
            ɛ
          
          -
          
            
              
                τ
                ~
              
              T
            
            ⁢
            
              
                τ
                .
              
              ~
            
          
        
      
    
  
  
    
      
        
          Λ
          ⁢
          
            
          
          ⁢
          
            L
            4
          
        
        =
        
          
            
              1
              α
            
            ⁢
            tr
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  
                    
                      W
                      ~
                    
                    
                      k
                      +
                      1
                    
                    T
                  
                  ⁢
                  
                    
                      W
                      ~
                    
                    
                      k
                      +
                      1
                    
                  
                
                -
                
                  
                    
                      W
                      ~
                    
                    T
                  
                  ⁢
                  
                    W
                    ~
                  
                
              
              )
            
          
          =
          
            
              1
              α
            
            ⁢
            tr
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  
                    (
                    
                      
                        W
                        
                          k
                          +
                          1
                        
                      
                      -
                      
                        
                          W
                          ^
                        
                        
                          k
                          +
                          1
                        
                      
                    
                    )
                  
                  T
                
                ⁢
                
                  (
                  
                    
                      W
                      
                        k
                        +
                        1
                      
                    
                    -
                    
                      
                        W
                        ^
                      
                      
                        k
                        +
                        1
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      -
                      
                        
                          W
                          ~
                        
                        T
                      
                    
                    ⁢
                    
                      W
                      ~
                    
                  
                  )
                
              
            
          
        
      
    
  
  
    
      
        =
        
          
            1
            α
          
          ⁢
          tr
          ⁢
          
            
          
          ⁢
          
            
              (
              
                
                  
                    
                      W
                      ^
                    
                    
                      k
                      +
                      1
                    
                    T
                  
                  ⁢
                  
                    
                      W
                      ^
                    
                    
                      k
                      +
                      1
                    
                  
                
                +
                
                  
                    W
                    T
                  
                  ⁢
                  W
                
                -
                
                  2
                  ⁢
                  
                    W
                    T
                  
                  ⁢
                  
                    
                      W
                      ^
                    
                    
                      k
                      +
                      1
                    
                  
                
                -
                
                  
                    
                      W
                      ~
                    
                    T
                  
                  ⁢
                  
                    W
                    ~
                  
                
              
              )
            
            .
          
        
      
    
  




Select the tuning law

Ŵk+1=Ŵ+ασ·rk+1T+ασ·{tilde over (τ)}k+1T−Γ∥I−α·σ·σT∥Ŵ.

Then,





  
    
      
        
          
            
              
                
                  
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  
                                                                    
                                                                      
                                                                        
                                                                          
                                                                            
                                                                              
                                                                                
                                                                                  
                                                                                    
                                                                                      
                                                                                        Δ
                                                                                        ⁢
                                                                                        
                                                                                          
                                                                                        
                                                                                        ⁢
                                                                                        
                                                                                          L
                                                                                          4
                                                                                        
                                                                                      
                                                                                      =
                                                                                      
                                                                                        
                                                                                          1
                                                                                          α
                                                                                        
                                                                                        ⁢
                                                                                        tr
                                                                                        ⁢
                                                                                        
                                                                                          {
                                                                                          
                                                                                            
                                                                                              
                                                                                                
                                                                                                  [
                                                                                                  
                                                                                                    
                                                                                                      W
                                                                                                      ^
                                                                                                    
                                                                                                    +
                                                                                                    
                                                                                                      α
                                                                                                      ⁢
                                                                                                      
                                                                                                        
                                                                                                      
                                                                                                      ⁢
                                                                                                      
                                                                                                        σ
                                                                                                        ⁡
                                                                                                        
                                                                                                          (
                                                                                                          
                                                                                                            
                                                                                                              r
                                                                                                              
                                                                                                                k
                                                                                                                +
                                                                                                                1
                                                                                                              
                                                                                                              T
                                                                                                            
                                                                                                            +
                                                                                                            
                                                                                                              
                                                                                                                τ
                                                                                                                ~
                                                                                                              
                                                                                                              
                                                                                                                k
                                                                                                                +
                                                                                                                1
                                                                                                              
                                                                                                              T
                                                                                                            
                                                                                                          
                                                                                                          )
                                                                                                        
                                                                                                      
                                                                                                    
                                                                                                    -
                                                                                                    
                                                                                                      Γ
                                                                                                      ⁢
                                                                                                      
                                                                                                        
                                                                                                        
                                                                                                          I
                                                                                                          -
                                                                                                          
                                                                                                            α
                                                                                                            ⁢
                                                                                                            
                                                                                                              
                                                                                                            
                                                                                                            ⁢
                                                                                                            
                                                                                                              σσ
                                                                                                              T
                                                                                                            
                                                                                                          
                                                                                                        
                                                                                                        
                                                                                                      
                                                                                                      ⁢
                                                                                                      
                                                                                                        W
                                                                                                        ^
                                                                                                      
                                                                                                    
                                                                                                  
                                                                                                  ]
                                                                                                
                                                                                                T
                                                                                              
                                                                                              ⁡
                                                                                              
                                                                                                [
                                                                                                
                                                                                                  
                                                                                                    W
                                                                                                    ^
                                                                                                  
                                                                                                  +
                                                                                                  
                                                                                                    α
                                                                                                    ⁢
                                                                                                    
                                                                                                      
                                                                                                    
                                                                                                    ⁢
                                                                                                    
                                                                                                      σ
                                                                                                      ⁡
                                                                                                      
                                                                                                        (
                                                                                                        
                                                                                                          
                                                                                                            r
                                                                                                            
                                                                                                              k
                                                                                                              +
                                                                                                              1
                                                                                                            
                                                                                                            T
                                                                                                          
                                                                                                          +
                                                                                                          
                                                                                                            
                                                                                                              τ
                                                                                                              ~
                                                                                                            
                                                                                                            
                                                                                                              k
                                                                                                              +
                                                                                                              1
                                                                                                            
                                                                                                            T
                                                                                                          
                                                                                                        
                                                                                                        )
                                                                                                      
                                                                                                    
                                                                                                  
                                                                                                  -
                                                                                                  
                                                                                                    Γ
                                                                                                    ⁢
                                                                                                    
                                                                                                      
                                                                                                      
                                                                                                        I
                                                                                                        -
                                                                                                        
                                                                                                          α
                                                                                                          ⁢
                                                                                                          
                                                                                                            
                                                                                                          
                                                                                                          ⁢
                                                                                                          
                                                                                                            σσ
                                                                                                            T
                                                                                                          
                                                                                                        
                                                                                                      
                                                                                                      
                                                                                                    
                                                                                                    ⁢
                                                                                                    
                                                                                                      W
                                                                                                      ^
                                                                                                    
                                                                                                  
                                                                                                
                                                                                                ]
                                                                                              
                                                                                            
                                                                                            +
                                                                                          
                                                                                        
                                                                                      
                                                                                    
                                                                                  
                                                                                
                                                                                
                                                                                  
                                                                                    
                                                                                      
                                                                                        
                                                                                          
                                                                                            
                                                                                              +
                                                                                              
                                                                                                W
                                                                                                T
                                                                                              
                                                                                            
                                                                                            ⁢
                                                                                            W
                                                                                          
                                                                                          -
                                                                                          
                                                                                            
                                                                                              
                                                                                                W
                                                                                                ~
                                                                                              
                                                                                              T
                                                                                            
                                                                                            ⁢
                                                                                            
                                                                                              W
                                                                                              ~
                                                                                            
                                                                                          
                                                                                          -
                                                                                          
                                                                                            2
                                                                                            ⁢
                                                                                            
                                                                                              
                                                                                                W
                                                                                                T
                                                                                              
                                                                                              ⁡
                                                                                              
                                                                                                [
                                                                                                
                                                                                                  
                                                                                                    W
                                                                                                    ^
                                                                                                  
                                                                                                  +
                                                                                                  
                                                                                                    α
                                                                                                    ⁢
                                                                                                    
                                                                                                      
                                                                                                    
                                                                                                    ⁢
                                                                                                    
                                                                                                      σ
                                                                                                      ⁡
                                                                                                      
                                                                                                        (
                                                                                                        
                                                                                                          
                                                                                                            r
                                                                                                            
                                                                                                              k
                                                                                                              +
                                                                                                              1
                                                                                                            
                                                                                                            T
                                                                                                          
                                                                                                          +
                                                                                                          
                                                                                                            
                                                                                                              τ
                                                                                                              ~
                                                                                                            
                                                                                                            
                                                                                                              k
                                                                                                              +
                                                                                                              1
                                                                                                            
                                                                                                            T
                                                                                                          
                                                                                                        
                                                                                                        )
                                                                                                      
                                                                                                    
                                                                                                  
                                                                                                  -
                                                                                                  
                                                                                                    Γ
                                                                                                    ⁢
                                                                                                    
                                                                                                      
                                                                                                      
                                                                                                        I
                                                                                                        -
                                                                                                        
                                                                                                          α
                                                                                                          ⁢
                                                                                                          
                                                                                                            
                                                                                                          
                                                                                                          ⁢
                                                                                                          
                                                                                                            σσ
                                                                                                            T
                                                                                                          
                                                                                                        
                                                                                                      
                                                                                                      
                                                                                                    
                                                                                                    ⁢
                                                                                                    
                                                                                                      W
                                                                                                      ^
                                                                                                    
                                                                                                  
                                                                                                
                                                                                                ]
                                                                                              
                                                                                            
                                                                                          
                                                                                        
                                                                                        }
                                                                                      
                                                                                      .
                                                                                    
                                                                                  
                                                                                
                                                                              
                                                                            
                                                                          
                                                                          
                                                                            
                                                                              
                                                                                =
                                                                                
                                                                                  
                                                                                    1
                                                                                    α
                                                                                  
                                                                                  ⁢
                                                                                  tr
                                                                                  ⁢
                                                                                  
                                                                                    {
                                                                                    
                                                                                      
                                                                                        
                                                                                          
                                                                                            W
                                                                                            ^
                                                                                          
                                                                                          T
                                                                                        
                                                                                        ⁢
                                                                                        
                                                                                          W
                                                                                          ^
                                                                                        
                                                                                      
                                                                                      +
                                                                                      
                                                                                        2
                                                                                        ⁢
                                                                                        
                                                                                          
                                                                                            W
                                                                                            ^
                                                                                          
                                                                                          T
                                                                                        
                                                                                        ⁢
                                                                                        α
                                                                                        ⁢
                                                                                        
                                                                                          
                                                                                        
                                                                                        ⁢
                                                                                        
                                                                                          σ
                                                                                          ·
                                                                                          
                                                                                            r
                                                                                            
                                                                                              k
                                                                                              +
                                                                                              1
                                                                                            
                                                                                            T
                                                                                          
                                                                                        
                                                                                      
                                                                                      +
                                                                                      
                                                                                        2
                                                                                        ⁢
                                                                                        
                                                                                          
                                                                                            W
                                                                                            ^
                                                                                          
                                                                                          T
                                                                                        
                                                                                        ⁢
                                                                                        α
                                                                                        ⁢
                                                                                        
                                                                                          
                                                                                        
                                                                                        ⁢
                                                                                        
                                                                                          σ
                                                                                          ·
                                                                                          
                                                                                            
                                                                                              τ
                                                                                              ~
                                                                                            
                                                                                            
                                                                                              k
                                                                                              +
                                                                                              1
                                                                                            
                                                                                            T
                                                                                          
                                                                                        
                                                                                      
                                                                                      -
                                                                                      
                                                                                        2
                                                                                        ⁢
                                                                                        
                                                                                          
                                                                                            W
                                                                                            ^
                                                                                          
                                                                                          T
                                                                                        
                                                                                        ⁢
                                                                                        Γ
                                                                                        ⁢
                                                                                        
                                                                                          
                                                                                          
                                                                                            I
                                                                                            -
                                                                                            
                                                                                              α
                                                                                              ⁢
                                                                                              
                                                                                                
                                                                                              
                                                                                              ⁢
                                                                                              
                                                                                                σ
                                                                                                ·
                                                                                                
                                                                                                  σ
                                                                                                  T
                                                                                                
                                                                                              
                                                                                            
                                                                                          
                                                                                          
                                                                                        
                                                                                        ⁢
                                                                                        
                                                                                          W
                                                                                          ^
                                                                                        
                                                                                      
                                                                                      +
                                                                                    
                                                                                  
                                                                                
                                                                              
                                                                            
                                                                          
                                                                        
                                                                      
                                                                    
                                                                    
                                                                      
                                                                        
                                                                          
                                                                            
                                                                              +
                                                                              
                                                                                α
                                                                                2
                                                                              
                                                                            
                                                                            ⁢
                                                                            
                                                                              r
                                                                              
                                                                                k
                                                                                +
                                                                                1
                                                                              
                                                                            
                                                                            ⁢
                                                                            
                                                                              σ
                                                                              T
                                                                            
                                                                            ⁢
                                                                            
                                                                              σ
                                                                              ·
                                                                              
                                                                                r
                                                                                
                                                                                  k
                                                                                  +
                                                                                  1
                                                                                
                                                                                T
                                                                              
                                                                            
                                                                          
                                                                          +
                                                                          
                                                                            2
                                                                            ⁢
                                                                            
                                                                              α
                                                                              2
                                                                            
                                                                            ⁢
                                                                            
                                                                              r
                                                                              
                                                                                k
                                                                                +
                                                                                1
                                                                              
                                                                            
                                                                            ⁢
                                                                            
                                                                              σ
                                                                              T
                                                                            
                                                                            ⁢
                                                                            
                                                                              σ
                                                                              ·
                                                                              
                                                                                
                                                                                  τ
                                                                                  ~
                                                                                
                                                                                
                                                                                  k
                                                                                  +
                                                                                  1
                                                                                
                                                                                T
                                                                              
                                                                            
                                                                          
                                                                          -
                                                                          
                                                                            2
                                                                            ⁢
                                                                            α
                                                                            ⁢
                                                                            
                                                                              
                                                                            
                                                                            ⁢
                                                                            
                                                                              r
                                                                              
                                                                                k
                                                                                +
                                                                                1
                                                                              
                                                                            
                                                                            ⁢
                                                                            
                                                                              σ
                                                                              T
                                                                            
                                                                            ⁢
                                                                            Γ
                                                                            ⁢
                                                                            
                                                                              
                                                                              
                                                                                I
                                                                                -
                                                                                
                                                                                  α
                                                                                  ·
                                                                                  
                                                                                    
                                                                                  
                                                                                  ⁢
                                                                                  σ
                                                                                  ·
                                                                                  
                                                                                    σ
                                                                                    T
                                                                                  
                                                                                
                                                                              
                                                                              
                                                                            
                                                                            ⁢
                                                                            
                                                                              W
                                                                              ^
                                                                            
                                                                          
                                                                          +
                                                                        
                                                                      
                                                                    
                                                                  
                                                                
                                                              
                                                              
                                                                
                                                                  
                                                                    
                                                                      
                                                                        +
                                                                        
                                                                          α
                                                                          2
                                                                        
                                                                      
                                                                      ⁢
                                                                      
                                                                        
                                                                          τ
                                                                          ~
                                                                        
                                                                        
                                                                          k
                                                                          +
                                                                          1
                                                                        
                                                                      
                                                                      ⁢
                                                                      
                                                                        σ
                                                                        T
                                                                      
                                                                      ⁢
                                                                      
                                                                        σ
                                                                        ·
                                                                        
                                                                          
                                                                            τ
                                                                            ~
                                                                          
                                                                          
                                                                            k
                                                                            +
                                                                            1
                                                                          
                                                                          T
                                                                        
                                                                      
                                                                    
                                                                    -
                                                                    
                                                                      2
                                                                      ⁢
                                                                      α
                                                                      ⁢
                                                                      
                                                                        
                                                                      
                                                                      ⁢
                                                                      
                                                                        
                                                                          τ
                                                                          ~
                                                                        
                                                                        
                                                                          k
                                                                          +
                                                                          1
                                                                        
                                                                      
                                                                      ⁢
                                                                      
                                                                        σ
                                                                        T
                                                                      
                                                                      ⁢
                                                                      Γ
                                                                      ⁢
                                                                      
                                                                        
                                                                        
                                                                          I
                                                                          -
                                                                          
                                                                            α
                                                                            ·
                                                                            
                                                                              
                                                                            
                                                                            ⁢
                                                                            σ
                                                                            ·
                                                                            
                                                                              σ
                                                                              T
                                                                            
                                                                          
                                                                        
                                                                        
                                                                      
                                                                      ⁢
                                                                      
                                                                        W
                                                                        ^
                                                                      
                                                                    
                                                                    +
                                                                    
                                                                      
                                                                        Γ
                                                                        2
                                                                      
                                                                      ⁢
                                                                      
                                                                        
                                                                          
                                                                          
                                                                            I
                                                                            -
                                                                            
                                                                              α
                                                                              ·
                                                                              
                                                                                
                                                                              
                                                                              ⁢
                                                                              σ
                                                                              ·
                                                                              
                                                                                σ
                                                                                T
                                                                              
                                                                            
                                                                          
                                                                          
                                                                        
                                                                        2
                                                                      
                                                                      ⁢
                                                                      
                                                                        
                                                                          W
                                                                          ^
                                                                        
                                                                        T
                                                                      
                                                                      ⁢
                                                                      
                                                                        W
                                                                        ^
                                                                      
                                                                    
                                                                    +
                                                                    
                                                                      
                                                                        W
                                                                        T
                                                                      
                                                                      ⁢
                                                                      W
                                                                    
                                                                    -
                                                                    
                                                                      
                                                                        
                                                                          W
                                                                          ~
                                                                        
                                                                        T
                                                                      
                                                                      ⁢
                                                                      
                                                                        W
                                                                        ~
                                                                      
                                                                    
                                                                    +
                                                                  
                                                                
                                                              
                                                            
                                                          
                                                        
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  
                                                                    -
                                                                    2
                                                                  
                                                                  ⁢
                                                                  
                                                                    W
                                                                    T
                                                                  
                                                                  ⁢
                                                                  
                                                                    W
                                                                    ^
                                                                  
                                                                
                                                                -
                                                                
                                                                  2
                                                                  ⁢
                                                                  
                                                                    W
                                                                    T
                                                                  
                                                                  ⁢
                                                                  α
                                                                  ⁢
                                                                  
                                                                    
                                                                  
                                                                  ⁢
                                                                  
                                                                    σ
                                                                    ·
                                                                    
                                                                      r
                                                                      
                                                                        k
                                                                        +
                                                                        1
                                                                      
                                                                      T
                                                                    
                                                                  
                                                                
                                                                +
                                                                
                                                                  2
                                                                  ⁢
                                                                  
                                                                    W
                                                                    T
                                                                  
                                                                  ⁢
                                                                  α
                                                                  ⁢
                                                                  
                                                                    
                                                                  
                                                                  ⁢
                                                                  
                                                                    σ
                                                                    ·
                                                                    
                                                                      
                                                                        τ
                                                                        ~
                                                                      
                                                                      
                                                                        k
                                                                        +
                                                                        1
                                                                      
                                                                      T
                                                                    
                                                                  
                                                                
                                                                +
                                                                
                                                                  2
                                                                  ⁢
                                                                  
                                                                    W
                                                                    T
                                                                  
                                                                  ⁢
                                                                  Γ
                                                                  ⁢
                                                                  
                                                                    
                                                                    
                                                                      I
                                                                      -
                                                                      
                                                                        α
                                                                        ⁢
                                                                        
                                                                          
                                                                        
                                                                        ·
                                                                        σ
                                                                        ·
                                                                        
                                                                          σ
                                                                          T
                                                                        
                                                                      
                                                                    
                                                                    
                                                                  
                                                                  ⁢
                                                                  
                                                                    W
                                                                    ^
                                                                  
                                                                
                                                              
                                                              }
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                  
                                                    
                                                      
                                                        =
                                                        
                                                          
                                                            2
                                                            ⁢
                                                            
                                                              r
                                                              
                                                                k
                                                                +
                                                                1
                                                              
                                                              T
                                                            
                                                            ⁢
                                                            
                                                              
                                                                W
                                                                ^
                                                              
                                                              T
                                                            
                                                            ⁢
                                                            σ
                                                          
                                                          +
                                                          
                                                            2
                                                            ⁢
                                                            
                                                              
                                                                τ
                                                                ~
                                                              
                                                              
                                                                k
                                                                +
                                                                1
                                                              
                                                              T
                                                            
                                                            ⁢
                                                            
                                                              
                                                                W
                                                                ^
                                                              
                                                              T
                                                            
                                                            ⁢
                                                            σ
                                                          
                                                          +
                                                          
                                                            
                                                              α
                                                              ·
                                                              
                                                                r
                                                                
                                                                  k
                                                                  +
                                                                  1
                                                                
                                                                T
                                                              
                                                            
                                                            ⁢
                                                            
                                                              σ
                                                              T
                                                            
                                                            ⁢
                                                            σ
                                                            ⁢
                                                            
                                                              
                                                            
                                                            ⁢
                                                            
                                                              r
                                                              
                                                                k
                                                                +
                                                                1
                                                              
                                                            
                                                          
                                                          +
                                                          
                                                            2
                                                            ⁢
                                                            α
                                                            ⁢
                                                            
                                                              
                                                            
                                                            ⁢
                                                            
                                                              
                                                                τ
                                                                ~
                                                              
                                                              
                                                                k
                                                                +
                                                                1
                                                              
                                                              T
                                                            
                                                            ⁢
                                                            
                                                              σ
                                                              T
                                                            
                                                            ⁢
                                                            σ
                                                            ⁢
                                                            
                                                              
                                                            
                                                            ⁢
                                                            
                                                              r
                                                              
                                                                k
                                                                +
                                                                1
                                                              
                                                            
                                                          
                                                          +
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                            
                                              
                                                
                                                  
                                                    
                                                      -
                                                      2
                                                    
                                                    ⁢
                                                    Γ
                                                    ⁢
                                                    
                                                      
                                                      
                                                        I
                                                        -
                                                        
                                                          α
                                                          ·
                                                          
                                                            
                                                          
                                                          ⁢
                                                          
                                                            σσ
                                                            T
                                                          
                                                        
                                                      
                                                      
                                                    
                                                    ⁢
                                                    
                                                      r
                                                      
                                                        k
                                                        +
                                                        1
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      
                                                        W
                                                        ^
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    σ
                                                  
                                                  +
                                                  
                                                    α
                                                    ⁢
                                                    
                                                      
                                                        τ
                                                        ~
                                                      
                                                      
                                                        k
                                                        +
                                                        1
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      σ
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      σ
                                                      ·
                                                      
                                                        
                                                          τ
                                                          ~
                                                        
                                                        
                                                          k
                                                          +
                                                          1
                                                        
                                                        T
                                                      
                                                    
                                                  
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    Γ
                                                    ⁢
                                                    
                                                      
                                                      
                                                        I
                                                        -
                                                        
                                                          α
                                                          ·
                                                          
                                                            
                                                          
                                                          ⁢
                                                          
                                                            σσ
                                                            T
                                                          
                                                        
                                                      
                                                      
                                                    
                                                    ⁢
                                                    
                                                      
                                                    
                                                    ⁢
                                                    
                                                      
                                                        τ
                                                        ~
                                                      
                                                      
                                                        k
                                                        +
                                                        1
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      
                                                        W
                                                        ^
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    σ
                                                  
                                                  +
                                                
                                              
                                            
                                          
                                        
                                      
                                      
                                        
                                          
                                            
                                              
                                                -
                                                2
                                              
                                              ⁢
                                              
                                                r
                                                
                                                  k
                                                  +
                                                  1
                                                
                                                T
                                              
                                              ⁢
                                              
                                                W
                                                T
                                              
                                              ⁢
                                              σ
                                            
                                            -
                                            
                                              2
                                              ⁢
                                              
                                                
                                                  τ
                                                  ~
                                                
                                                
                                                  k
                                                  +
                                                  1
                                                
                                                T
                                              
                                              ⁢
                                              
                                                W
                                                T
                                              
                                              ⁢
                                              σ
                                            
                                            +
                                            
                                              
                                                1
                                                α
                                              
                                              ⁢
                                              tr
                                              ⁢
                                              
                                                {
                                                
                                                  
                                                    
                                                      
                                                        W
                                                        ^
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      W
                                                      ^
                                                    
                                                  
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    Γ
                                                    ⁢
                                                    
                                                      
                                                      
                                                        I
                                                        -
                                                        
                                                          α
                                                          ·
                                                          
                                                            
                                                          
                                                          ⁢
                                                          σ
                                                          ·
                                                          
                                                            σ
                                                            T
                                                          
                                                        
                                                      
                                                      
                                                    
                                                    ⁢
                                                    
                                                      
                                                        W
                                                        ^
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      W
                                                      ^
                                                    
                                                  
                                                  +
                                                  
                                                    
                                                      W
                                                      T
                                                    
                                                    ⁢
                                                    W
                                                  
                                                  -
                                                  
                                                    
                                                      
                                                        W
                                                        ~
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      W
                                                      ~
                                                    
                                                  
                                                  +
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        
                                          
                                            -
                                            2
                                          
                                          ⁢
                                          
                                            W
                                            T
                                          
                                          ⁢
                                          W
                                        
                                        +
                                        
                                          
                                            Γ
                                            2
                                          
                                          ⁢
                                          
                                            
                                              
                                              
                                                I
                                                -
                                                
                                                  α
                                                  ·
                                                  
                                                    
                                                  
                                                  ⁢
                                                  σ
                                                  ·
                                                  
                                                    σ
                                                    T
                                                  
                                                
                                              
                                              
                                            
                                            2
                                          
                                          ⁢
                                          
                                            
                                              W
                                              ^
                                            
                                            T
                                          
                                          ⁢
                                          
                                            W
                                            ^
                                          
                                        
                                        +
                                        
                                          2
                                          ⁢
                                          
                                            W
                                            T
                                          
                                          ⁢
                                          Γ
                                          ⁢
                                          
                                            
                                            
                                              I
                                              -
                                              
                                                α
                                                ·
                                                
                                                  
                                                
                                                ⁢
                                                σ
                                                ·
                                                
                                                  σ
                                                  T
                                                
                                              
                                            
                                            
                                          
                                          ⁢
                                          
                                            W
                                            ^
                                          
                                        
                                      
                                      }
                                    
                                  
                                
                              
                            
                          
                          
                            
                              
                                =
                                
                                  
                                    2
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            r
                                            T
                                          
                                          ⁢
                                          
                                            K
                                            v
                                            T
                                          
                                        
                                        +
                                        
                                          D
                                          T
                                        
                                        -
                                        
                                          
                                            τ
                                            ~
                                          
                                          T
                                        
                                      
                                      )
                                    
                                    ⁢
                                    
                                      
                                        W
                                        ^
                                      
                                      T
                                    
                                    ⁢
                                    σ
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            
                                              τ
                                              ~
                                            
                                            T
                                          
                                          ⁢
                                          
                                            K
                                            b
                                            T
                                          
                                        
                                        +
                                        
                                          
                                            σ
                                            T
                                          
                                          ⁢
                                          
                                            W
                                            ~
                                          
                                        
                                        +
                                        
                                          ɛ
                                          T
                                        
                                      
                                      )
                                    
                                    ⁢
                                    
                                      
                                        W
                                        ^
                                      
                                      T
                                    
                                    ⁢
                                    σ
                                  
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                +
                                α
                              
                              ·
                              
                                (
                                
                                  
                                    
                                      r
                                      T
                                    
                                    ⁢
                                    
                                      K
                                      v
                                      T
                                    
                                  
                                  +
                                  
                                    D
                                    T
                                  
                                  -
                                  
                                    
                                      τ
                                      ~
                                    
                                    T
                                  
                                
                                )
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            
                              σ
                              ⁡
                              
                                (
                                
                                  
                                    
                                      K
                                      v
                                    
                                    ⁢
                                    r
                                  
                                  +
                                  D
                                  -
                                  
                                    τ
                                    ~
                                  
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                  
                
              
              
                
                  
                    
                      
                        +
                        2
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        α
                        ⁡
                        
                          (
                          
                            
                              
                                
                                  τ
                                  ~
                                
                                T
                              
                              ⁢
                              
                                K
                                b
                                T
                              
                            
                            +
                            
                              
                                σ
                                T
                              
                              ⁢
                              
                                W
                                ~
                              
                            
                            +
                            
                              ɛ
                              T
                            
                          
                          )
                        
                      
                      ⁢
                      
                        σ
                        T
                      
                      ⁢
                      
                        σ
                        ⁡
                        
                          (
                          
                            
                              
                                K
                                v
                              
                              ⁢
                              r
                            
                            +
                            D
                            -
                            
                              τ
                              ~
                            
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  -
                  2
                
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  (
                  
                    
                      
                        r
                        T
                      
                      ⁢
                      
                        K
                        v
                        T
                      
                    
                    +
                    
                      D
                      T
                    
                    -
                    
                      
                        τ
                        ~
                      
                      T
                    
                  
                  )
                
                ⁢
                
                  
                    W
                    ^
                  
                  T
                
                ⁢
                σ
              
              +
            
          
        
      
    
  
  
    
      
        
          
            +
            
              α
              ⁡
              
                (
                
                  
                    
                      
                        τ
                        ~
                      
                      T
                    
                    ⁢
                    
                      K
                      b
                      T
                    
                  
                  +
                  
                    
                      σ
                      T
                    
                    ⁢
                    
                      W
                      ~
                    
                  
                  +
                  
                    ɛ
                    T
                  
                
                )
              
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            σ
            ·
            
              (
              
                
                  
                    K
                    b
                  
                  ⁢
                  
                    τ
                    ~
                  
                
                +
                
                  
                    
                      W
                      ~
                    
                    T
                  
                  ⁢
                  σ
                
                +
                ɛ
              
              )
            
          
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            (
            
              
                
                  
                    τ
                    ~
                  
                  T
                
                ⁢
                
                  K
                  b
                  T
                
              
              +
              
                
                  σ
                  T
                
                ⁢
                
                  W
                  ~
                
              
              +
              
                ɛ
                T
              
            
            )
          
          ⁢
          
            
              W
              ^
            
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          
            (
            
              
                
                  r
                  T
                
                ⁢
                
                  K
                  v
                  T
                
              
              +
              
                D
                T
              
              -
              
                
                  τ
                  ~
                
                T
              
            
            )
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          
            (
            
              
                
                  
                    τ
                    ~
                  
                  T
                
                ⁢
                
                  K
                  b
                  T
                
              
              +
              
                
                  σ
                  T
                
                ⁢
                
                  W
                  ~
                
              
              +
              
                ɛ
                T
              
            
            )
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
        
          
            1
            α
          
          ⁢
          tr
          ⁢
          
            {
            
              
                
                  -
                  2
                
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  
                    W
                    ^
                  
                  T
                
                ⁢
                
                  W
                  ^
                
              
              +
            
          
        
      
    
  
  
    
      
        
          
            
              +
              2
            
            ⁢
            
              W
              T
            
            ⁢
            Γ
            ⁢
            
              
              
                I
                -
                
                  α
                  ·
                  
                    
                  
                  ⁢
                  σ
                  ·
                  
                    σ
                    T
                  
                
              
              
            
            ⁢
            
              W
              ^
            
          
          +
          
            
              Γ
              2
            
            ⁢
            
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              2
            
            ⁢
            
              
                W
                ^
              
              T
            
            ⁢
            
              W
              ^
            
          
        
        }
      
    
  








  
    
      
        
          
            
              
                
                  
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  
                                                                    Combining
                                                                    ⁢
                                                                    
                                                                      
                                                                    
                                                                    ⁢
                                                                    all
                                                                    ⁢
                                                                    
                                                                      
                                                                    
                                                                    ⁢
                                                                    the
                                                                    ⁢
                                                                    
                                                                      
                                                                    
                                                                    ⁢
                                                                    terms
                                                                  
                                                                
                                                              
                                                              
                                                                
                                                                  
                                                                    
                                                                      Δ
                                                                      ⁢
                                                                      
                                                                        
                                                                      
                                                                      ⁢
                                                                      L
                                                                    
                                                                    =
                                                                    
                                                                      
                                                                        
                                                                          -
                                                                          2
                                                                        
                                                                        ⁢
                                                                        
                                                                          
                                                                            r
                                                                            T
                                                                          
                                                                          ⁡
                                                                          
                                                                            [
                                                                            
                                                                              I
                                                                              -
                                                                              
                                                                                
                                                                                  K
                                                                                  v
                                                                                  T
                                                                                
                                                                                ⁢
                                                                                
                                                                                  K
                                                                                  v
                                                                                
                                                                              
                                                                            
                                                                            ]
                                                                          
                                                                        
                                                                        ⁢
                                                                        r
                                                                      
                                                                      +
                                                                      
                                                                        4
                                                                        ⁢
                                                                        
                                                                          r
                                                                          T
                                                                        
                                                                        ⁢
                                                                        
                                                                          K
                                                                          v
                                                                          T
                                                                        
                                                                        ⁢
                                                                        D
                                                                      
                                                                      -
                                                                      
                                                                        4
                                                                        ⁢
                                                                        
                                                                          r
                                                                          T
                                                                        
                                                                        ⁢
                                                                        
                                                                          K
                                                                          v
                                                                          T
                                                                        
                                                                        ⁢
                                                                        
                                                                          τ
                                                                          ~
                                                                        
                                                                      
                                                                      +
                                                                      
                                                                        2
                                                                        ⁢
                                                                        
                                                                          D
                                                                          T
                                                                        
                                                                        ⁢
                                                                        D
                                                                      
                                                                      -
                                                                      
                                                                        4
                                                                        ⁢
                                                                        
                                                                          D
                                                                          T
                                                                        
                                                                        ⁢
                                                                        
                                                                          τ
                                                                          ~
                                                                        
                                                                      
                                                                      +
                                                                      
                                                                        2
                                                                        ⁢
                                                                        
                                                                          
                                                                            τ
                                                                            ~
                                                                          
                                                                          T
                                                                        
                                                                        ⁢
                                                                        
                                                                          
                                                                            τ
                                                                            +
                                                                          
                                                                          ~
                                                                        
                                                                      
                                                                    
                                                                  
                                                                
                                                              
                                                            
                                                          
                                                        
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  +
                                                                  2
                                                                
                                                                ⁢
                                                                
                                                                  r
                                                                  T
                                                                
                                                                ⁢
                                                                
                                                                  K
                                                                  v
                                                                  T
                                                                
                                                                ⁢
                                                                
                                                                  K
                                                                  b
                                                                
                                                                ⁢
                                                                
                                                                  τ
                                                                  ~
                                                                
                                                              
                                                              +
                                                              
                                                                2
                                                                ⁢
                                                                
                                                                  r
                                                                  T
                                                                
                                                                ⁢
                                                                
                                                                  K
                                                                  v
                                                                  T
                                                                
                                                                ⁢
                                                                
                                                                  
                                                                    W
                                                                    ~
                                                                  
                                                                  T
                                                                
                                                                ⁢
                                                                σ
                                                              
                                                              +
                                                              
                                                                2
                                                                ⁢
                                                                
                                                                  r
                                                                  T
                                                                
                                                                ⁢
                                                                
                                                                  K
                                                                  v
                                                                  T
                                                                
                                                                ⁢
                                                                ɛ
                                                              
                                                              +
                                                              
                                                                2
                                                                ⁢
                                                                
                                                                  D
                                                                  T
                                                                
                                                                ⁢
                                                                
                                                                  K
                                                                  b
                                                                
                                                                ⁢
                                                                
                                                                  τ
                                                                  ~
                                                                
                                                              
                                                              +
                                                              
                                                                2
                                                                ⁢
                                                                
                                                                  D
                                                                  T
                                                                
                                                                ⁢
                                                                
                                                                  
                                                                    W
                                                                    ~
                                                                  
                                                                  T
                                                                
                                                                ⁢
                                                                σ
                                                              
                                                              +
                                                              
                                                                2
                                                                ⁢
                                                                
                                                                  D
                                                                  T
                                                                
                                                                ⁢
                                                                ɛ
                                                              
                                                              -
                                                              
                                                                2
                                                                ⁢
                                                                
                                                                  
                                                                    τ
                                                                    ~
                                                                  
                                                                  T
                                                                
                                                                ⁢
                                                                
                                                                  K
                                                                  b
                                                                  T
                                                                
                                                                ⁢
                                                                
                                                                  K
                                                                  b
                                                                
                                                                ⁢
                                                                
                                                                  τ
                                                                  ~
                                                                
                                                              
                                                              +
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                  
                                                    
                                                      
                                                        
                                                          
                                                            -
                                                            2
                                                          
                                                          ⁢
                                                          
                                                            
                                                              τ
                                                              ~
                                                            
                                                            T
                                                          
                                                          ⁢
                                                          
                                                            
                                                              W
                                                              ~
                                                            
                                                            T
                                                          
                                                          ⁢
                                                          σ
                                                        
                                                        -
                                                        
                                                          2
                                                          ⁢
                                                          
                                                            
                                                              τ
                                                              ~
                                                            
                                                            T
                                                          
                                                          ⁢
                                                          ɛ
                                                        
                                                        -
                                                        
                                                          2
                                                          ⁢
                                                          
                                                            r
                                                            T
                                                          
                                                          ⁢
                                                          
                                                            τ
                                                            ~
                                                          
                                                        
                                                        +
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                            
                                              
                                                
                                                  
                                                    
                                                      
                                                        τ
                                                        ~
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      K
                                                      b
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      K
                                                      b
                                                    
                                                    ⁢
                                                    
                                                      τ
                                                      ~
                                                    
                                                  
                                                  +
                                                  
                                                    2
                                                    ⁢
                                                    
                                                      
                                                        τ
                                                        ~
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      K
                                                      b
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      
                                                        W
                                                        ~
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    σ
                                                  
                                                  +
                                                  
                                                    2
                                                    ⁢
                                                    
                                                      
                                                        τ
                                                        ~
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      K
                                                      b
                                                      T
                                                    
                                                    ⁢
                                                    ɛ
                                                  
                                                  +
                                                  
                                                    
                                                      σ
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      W
                                                      ~
                                                    
                                                    ⁢
                                                    
                                                      
                                                        W
                                                        ~
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    σ
                                                  
                                                  +
                                                  
                                                    2
                                                    ⁢
                                                    
                                                      ɛ
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      
                                                        W
                                                        ~
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    σ
                                                  
                                                  +
                                                  
                                                    
                                                      ɛ
                                                      T
                                                    
                                                    ⁢
                                                    ɛ
                                                  
                                                  -
                                                  
                                                    
                                                      
                                                        τ
                                                        ~
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      τ
                                                      ~
                                                    
                                                  
                                                  +
                                                
                                              
                                            
                                          
                                        
                                      
                                      
                                        
                                          
                                            
                                              
                                                +
                                                2
                                              
                                              ⁢
                                              
                                                (
                                                
                                                  
                                                    
                                                      r
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      K
                                                      v
                                                      T
                                                    
                                                  
                                                  +
                                                  
                                                    D
                                                    T
                                                  
                                                  -
                                                  
                                                    
                                                      τ
                                                      ~
                                                    
                                                    T
                                                  
                                                
                                                )
                                              
                                              ⁢
                                              
                                                
                                                  W
                                                  ~
                                                
                                                T
                                              
                                              ⁢
                                              σ
                                            
                                            +
                                            
                                              2
                                              ⁢
                                              
                                                (
                                                
                                                  
                                                    
                                                      
                                                        τ
                                                        ~
                                                      
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      K
                                                      b
                                                      T
                                                    
                                                  
                                                  +
                                                  
                                                    
                                                      σ
                                                      T
                                                    
                                                    ⁢
                                                    
                                                      W
                                                      ~
                                                    
                                                  
                                                  +
                                                  
                                                    ɛ
                                                    T
                                                  
                                                
                                                )
                                              
                                              ⁢
                                              
                                                
                                                  W
                                                  ^
                                                
                                                T
                                              
                                              ⁢
                                              σ
                                            
                                            +
                                          
                                        
                                      
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        
                                          
                                            +
                                            α
                                          
                                          ·
                                          
                                            (
                                            
                                              
                                                
                                                  r
                                                  T
                                                
                                                ⁢
                                                
                                                  K
                                                  v
                                                  T
                                                
                                              
                                              +
                                              
                                                D
                                                T
                                              
                                              -
                                              
                                                
                                                  τ
                                                  ~
                                                
                                                T
                                              
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          σ
                                          T
                                        
                                        ⁢
                                        
                                          σ
                                          ⁡
                                          
                                            (
                                            
                                              
                                                
                                                  K
                                                  v
                                                
                                                ⁢
                                                r
                                              
                                              +
                                              D
                                              -
                                              
                                                τ
                                                ~
                                              
                                            
                                            )
                                          
                                        
                                      
                                      +
                                    
                                  
                                
                              
                            
                          
                          
                            
                              
                                
                                  
                                    +
                                    2
                                  
                                  ⁢
                                  
                                    α
                                    ·
                                    
                                      (
                                      
                                        
                                          
                                            
                                              τ
                                              ~
                                            
                                            T
                                          
                                          ⁢
                                          
                                            K
                                            b
                                            T
                                          
                                        
                                        +
                                        
                                          
                                            σ
                                            T
                                          
                                          ⁢
                                          
                                            W
                                            ~
                                          
                                        
                                        +
                                        
                                          ɛ
                                          T
                                        
                                      
                                      )
                                    
                                  
                                  ⁢
                                  
                                    σ
                                    T
                                  
                                  ⁢
                                  
                                    σ
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            K
                                            v
                                          
                                          ⁢
                                          r
                                        
                                        +
                                        D
                                        -
                                        
                                          τ
                                          ~
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              -
                              2
                            
                            ⁢
                            Γ
                            ⁢
                            
                              
                              
                                I
                                -
                                
                                  α
                                  ·
                                  
                                    
                                  
                                  ⁢
                                  σ
                                  ·
                                  
                                    σ
                                    T
                                  
                                
                              
                              
                            
                            ⁢
                            
                              (
                              
                                
                                  
                                    r
                                    T
                                  
                                  ⁢
                                  
                                    K
                                    v
                                    T
                                  
                                
                                +
                                
                                  D
                                  T
                                
                                -
                                
                                  
                                    τ
                                    ~
                                  
                                  T
                                
                              
                              )
                            
                            ⁢
                            
                              
                                W
                                ~
                              
                              T
                            
                            ⁢
                            σ
                          
                          +
                        
                      
                    
                  
                
              
              
                
                  
                    
                      
                        +
                        
                          α
                          ⁡
                          
                            (
                            
                              
                                
                                  
                                    τ
                                    ~
                                  
                                  T
                                
                                ⁢
                                
                                  K
                                  b
                                  T
                                
                              
                              +
                              
                                
                                  σ
                                  T
                                
                                ⁢
                                
                                  W
                                  ~
                                
                              
                              +
                              
                                ɛ
                                T
                              
                            
                            )
                          
                        
                      
                      ⁢
                      
                        σ
                        T
                      
                      ⁢
                      
                        σ
                        ·
                        
                          (
                          
                            
                              
                                K
                                b
                              
                              ⁢
                              
                                τ
                                ~
                              
                            
                            +
                            
                              
                                
                                  W
                                  ~
                                
                                T
                              
                              ⁢
                              σ
                            
                            +
                            ɛ
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  -
                  2
                
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  (
                  
                    
                      
                        
                          τ
                          ~
                        
                        T
                      
                      ⁢
                      
                        K
                        b
                        T
                      
                    
                    +
                    
                      
                        σ
                        T
                      
                      ⁢
                      
                        W
                        ~
                      
                    
                    +
                    
                      ɛ
                      T
                    
                  
                  )
                
                ⁢
                
                  
                    W
                    ^
                  
                  T
                
                ⁢
                σ
              
              -
              
                2
                ⁢
                
                  (
                  
                    
                      
                        r
                        T
                      
                      ⁢
                      
                        K
                        v
                        T
                      
                    
                    +
                    
                      D
                      T
                    
                    -
                    
                      
                        τ
                        ~
                      
                      T
                    
                  
                  )
                
                ⁢
                
                  W
                  T
                
                ⁢
                σ
              
              +
            
          
        
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          
            (
            
              
                
                  
                    τ
                    ~
                  
                  T
                
                ⁢
                
                  K
                  b
                  T
                
              
              +
              
                
                  σ
                  T
                
                ⁢
                
                  W
                  ~
                
              
              +
              
                ɛ
                T
              
            
            )
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
        
          
            1
            α
          
          ⁢
          tr
          ⁢
          
            {
            
              
                
                  -
                  2
                
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  
                    W
                    ^
                  
                  T
                
                ⁢
                
                  W
                  ^
                
              
              +
            
          
        
      
    
  
  
    
      
        
          
            
              +
              2
            
            ⁢
            
              W
              T
            
            ⁢
            Γ
            ⁢
            
              
              
                I
                -
                
                  α
                  ·
                  
                    
                  
                  ⁢
                  σ
                  ·
                  
                    σ
                    T
                  
                
              
              
            
            ⁢
            
              W
              ^
            
          
          +
          
            
              Γ
              2
            
            ⁢
            
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              2
            
            ⁢
            
              
                W
                ^
              
              T
            
            ⁢
            
              W
              ^
            
          
        
        }
      
    
  
  
    
      Simplifying
    
  
  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          L
        
        =
        
          
            
              -
              2
            
            ⁢
            
              
                r
                T
              
              ⁡
              
                [
                
                  I
                  -
                  
                    
                      K
                      v
                      T
                    
                    ⁢
                    
                      K
                      v
                    
                  
                
                ]
              
            
            ⁢
            r
          
          +
          
            4
            ⁢
            
              r
              T
            
            ⁢
            
              K
              v
              T
            
            ⁢
            D
          
          -
          
            4
            ⁢
            
              r
              T
            
            ⁢
            
              K
              v
              T
            
            ⁢
            
              τ
              ~
            
          
          +
          
            2
            ⁢
            
              D
              T
            
            ⁢
            D
          
          -
          
            4
            ⁢
            
              D
              T
            
            ⁢
            
              τ
              ~
            
          
          +
          
            
              
                τ
                ~
              
              T
            
            ⁢
            
              
                τ
                +
              
              ~
            
          
        
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            K
            b
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          ɛ
        
        +
        
          2
          ⁢
          
            D
            T
          
          ⁢
          
            K
            b
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          
            D
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          
            D
            T
          
          ⁢
          ɛ
        
        -
        
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            K
            b
          
          ⁢
          
            τ
            ~
          
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          ɛ
        
        -
        
          2
          ⁢
          
            r
            T
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          ɛ
        
        +
        
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          
            ɛ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          
            ɛ
            T
          
          ⁢
          ɛ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          
            (
            
              
                
                  r
                  T
                
                ⁢
                
                  K
                  v
                  T
                
              
              +
              
                D
                T
              
              -
              
                
                  τ
                  ~
                
                T
              
            
            )
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          
            (
            
              
                
                  
                    τ
                    ~
                  
                  T
                
                ⁢
                
                  K
                  b
                  T
                
              
              +
              
                
                  σ
                  T
                
                ⁢
                
                  W
                  ~
                
              
              +
              
                ɛ
                T
              
            
            )
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            
              +
              α
            
            ·
            
              (
              
                
                  
                    r
                    T
                  
                  ⁢
                  
                    K
                    v
                    T
                  
                
                +
                
                  D
                  T
                
                -
                
                  
                    τ
                    ~
                  
                  T
                
              
              )
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            σ
            ⁡
            
              (
              
                
                  
                    K
                    v
                  
                  ⁢
                  r
                
                +
                D
                -
                
                  τ
                  ~
                
              
              )
            
          
        
        +
        
          2
          ⁢
          
            α
            ⁡
            
              (
              
                
                  
                    
                      τ
                      ~
                    
                    T
                  
                  ⁢
                  
                    K
                    b
                    T
                  
                
                +
                
                  
                    σ
                    T
                  
                  ⁢
                  
                    W
                    ~
                  
                
                +
                
                  ɛ
                  T
                
              
              )
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            σ
            ⁡
            
              (
              
                
                  
                    K
                    v
                  
                  ⁢
                  r
                
                +
                D
                -
                
                  τ
                  ~
                
              
              )
            
          
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σσ
                  T
                
              
            
            
          
          ⁢
          
            (
            
              
                
                  r
                  T
                
                ⁢
                
                  K
                  v
                  T
                
              
              +
              
                D
                T
              
              -
              
                
                  τ
                  ~
                
                T
              
            
            )
          
          ⁢
          
            
              (
              
                W
                -
                
                  W
                  ~
                
              
              )
            
            T
          
          ⁢
          σ
        
        +
        
          
            α
            ⁡
            
              (
              
                
                  
                    
                      τ
                      ~
                    
                    T
                  
                  ⁢
                  
                    K
                    b
                    T
                  
                
                +
                
                  
                    σ
                    T
                  
                  ⁢
                  
                    W
                    ~
                  
                
                +
                
                  ɛ
                  T
                
              
              )
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            (
            
              
                
                  K
                  b
                
                ⁢
                
                  τ
                  ~
                
              
              +
              
                
                  
                    W
                    ~
                  
                  T
                
                ⁢
                σ
              
              +
              ɛ
            
            )
          
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            (
            
              
                
                  
                    τ
                    ~
                  
                  T
                
                ⁢
                
                  K
                  b
                  T
                
              
              +
              
                
                  σ
                  T
                
                ⁢
                
                  W
                  ~
                
              
              +
              
                ɛ
                T
              
            
            )
          
          ⁢
          
            
              (
              
                W
                -
                
                  W
                  ~
                
              
              )
            
            T
          
          ⁢
          σ
        
        +
        
          
            1
            α
          
          ⁢
          tr
          ⁢
          
            {
            
              
                
                  -
                  2
                
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  
                    W
                    ^
                  
                  T
                
                ⁢
                
                  W
                  ^
                
              
              +
            
          
        
      
    
  
  
    
      
        
          
            
              +
              2
            
            ⁢
            
              W
              T
            
            ⁢
            Γ
            ⁢
            
              
              
                I
                -
                
                  α
                  ·
                  
                    
                  
                  ⁢
                  σ
                  ·
                  
                    σ
                    T
                  
                
              
              
            
            ⁢
            
              W
              ^
            
          
          +
          
            
              Γ
              2
            
            ⁢
            
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              2
            
            ⁢
            
              
                W
                ^
              
              T
            
            ⁢
            
              W
              ^
            
          
        
        }
      
    
  
  
    
      
        Multiplying
        ⁢
        
          
        
        ⁢
        out
        ⁢
        
          
        
        ⁢
        terms
      
    
  
  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          L
        
        =
        
          
            
              -
              2
            
            ⁢
            
              
                r
                T
              
              ⁡
              
                [
                
                  I
                  -
                  
                    
                      K
                      v
                      T
                    
                    ⁢
                    
                      K
                      v
                    
                  
                
                ]
              
            
            ⁢
            r
          
          +
          
            4
            ⁢
            
              r
              T
            
            ⁢
            
              K
              v
              T
            
            ⁢
            D
          
          -
          
            4
            ⁢
            
              r
              T
            
            ⁢
            
              K
              v
              T
            
            ⁢
            
              τ
              ~
            
          
          +
          
            2
            ⁢
            
              D
              T
            
            ⁢
            D
          
          -
          
            4
            ⁢
            
              D
              T
            
            ⁢
            
              τ
              ~
            
          
          +
          
            
              
                τ
                ~
              
              T
            
            ⁢
            
              τ
              ~
            
          
          +
        
      
    
  
  
    
      
        
          2
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            K
            b
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          ɛ
        
        +
        
          2
          ⁢
          
            D
            T
          
          ⁢
          
            K
            b
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          
            D
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          
            D
            T
          
          ⁢
          ɛ
        
        -
        
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            K
            b
          
          ⁢
          
            τ
            ~
          
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          ɛ
        
        -
        
          2
          ⁢
          
            r
            T
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          ɛ
        
        +
        
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          
            ɛ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          
            ɛ
            T
          
          ⁢
          ɛ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          
            D
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          
            ɛ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            +
            α
          
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            K
            v
          
          ⁢
          r
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          D
        
        -
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            τ
            ~
          
        
        +
        
          α
          ⁢
          
            
          
          ⁢
          
            D
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          D
        
        -
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            D
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            τ
            ~
          
        
        +
      
    
  
  
    
      
        
          
            +
            α
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            K
            v
          
          ⁢
          r
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          D
        
        -
        
          2
          ⁢
          α
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            τ
            ~
          
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          
            ασ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            K
            v
          
          ⁢
          r
        
        +
        
          2
          ⁢
          
            ασ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          D
        
        -
        
          2
          ⁢
          α
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          
            
          
          ⁢
          
            αɛ
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            K
            v
          
          ⁢
          r
        
        ⁢
        
          
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            ɛ
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          D
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          
            
          
          ⁢
          
            αɛ
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            τ
            ~
          
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            D
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            D
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          α
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            K
            b
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σɛ
        
        +
      
    
  
  
    
      
        
          
            +
            α
          
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            τ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          ɛ
        
        +
        
          
            αɛ
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          
            σ
            ·
            ɛ
          
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ^
            
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            ɛ
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            ɛ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          
            1
            α
          
          ⁢
          tr
          ⁢
          
            {
            
              
                
                  -
                  2
                
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  
                    W
                    ^
                  
                  T
                
                ⁢
                
                  W
                  ^
                
              
              +
            
          
        
      
    
  
  
    
      
        
          
            
              +
              2
            
            ⁢
            
              W
              T
            
            ⁢
            Γ
            ⁢
            
              
              
                I
                -
                
                  α
                  ·
                  
                    
                  
                  ⁢
                  σ
                  ·
                  
                    σ
                    T
                  
                
              
              
            
            ⁢
            
              W
              ^
            
          
          +
          
            
              Γ
              2
            
            ⁢
            
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              2
            
            ⁢
            
              
                W
                ^
              
              T
            
            ⁢
            
              W
              ^
            
          
        
        }
      
    
  
  
    
      Simplifying
    
  
  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          L
        
        =
        
          
            
              -
              
                
                  r
                  T
                
                ⁡
                
                  [
                  
                    
                      2
                      ⁢
                      I
                    
                    -
                    
                      
                        (
                        
                          2
                          +
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            σ
                          
                        
                        )
                      
                      ⁢
                      
                        K
                        v
                        T
                      
                      ⁢
                      
                        K
                        v
                      
                    
                  
                  ]
                
              
            
            ⁢
            r
          
          +
          
            2
            ⁢
            
              (
              
                2
                +
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
            ⁢
            
              r
              T
            
            ⁢
            
              K
              v
              T
            
            ⁢
            D
          
          +
          
            
              (
              
                2
                +
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
            ⁢
            
              D
              T
            
            ⁢
            D
          
          +
        
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          ɛ
        
        +
        
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            ɛ
            T
          
          ⁢
          ɛ
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            ɛ
            T
          
          ⁢
          D
        
        +
      
    
  
  
    
      
        
          
            +
            
              
                τ
                ~
              
              T
            
          
          ⁢
          
            τ
            ~
          
        
        -
        
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            K
            b
          
          ⁢
          
            τ
            ~
          
        
        +
        
          α
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            K
            b
          
          ⁢
          
            τ
            ~
          
        
        +
        
          α
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            τ
            ~
          
        
        -
        
          2
          ⁢
          α
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            K
            b
          
          ⁢
          
            τ
            ~
          
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          
            r
            T
          
          ⁢
          
            
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            D
            T
          
          ⁢
          
            K
            b
          
          ⁢
          
            τ
            ~
          
        
        -
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          ɛ
        
        -
        
          2
          ⁢
          
            (
            
              2
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            τ
            ~
          
        
        -
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            τ
            ~
          
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          
            (
            
              2
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            D
            T
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          ɛ
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          
            ασ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          
            (
            
              1
              -
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            ɛ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          
            ασ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          D
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            D
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            D
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            ɛ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            ɛ
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            b
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ^
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          +
          
            1
            α
          
        
        ⁢
        tr
        ⁢
        
          {
          
            
              
                -
                2
              
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                
                  W
                  ^
                
                T
              
              ⁢
              
                W
                ^
              
            
            +
            
              2
              ⁢
              
                W
                T
              
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                W
                ^
              
            
            +
            
              
                Γ
                2
              
              ⁢
              
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                2
              
              ⁢
              
                
                  W
                  ^
                
                T
              
              ⁢
              
                W
                ^
              
            
          
          }
        
      
    
  




Pick Kb=(I+Kv−1)T=I+Ka and define β=2Ka+(1−ασTσ)KaTKa−I>0 (condition (36)) which is true as long as Kv−1<I (condition (35)).
It can be seen that β>I and β is a diagonal matrix since Ka is diagonal.




  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          L
        
        =
        
          
            
              -
              
                
                  r
                  T
                
                ⁡
                
                  [
                  
                    
                      2
                      ⁢
                      I
                    
                    -
                    
                      
                        (
                        
                          2
                          +
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            σ
                          
                        
                        )
                      
                      ⁢
                      
                        K
                        v
                        T
                      
                      ⁢
                      
                        K
                        v
                      
                    
                  
                  ]
                
              
            
            ⁢
            r
          
          +
          
            2
            ⁢
            
              (
              
                2
                +
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
            ⁢
            
              r
              T
            
            ⁢
            
              K
              v
              T
            
            ⁢
            D
          
          +
          
            
              (
              
                2
                +
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
            ⁢
            
              D
              T
            
            ⁢
            D
          
          +
        
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          ɛ
        
        +
        
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            ɛ
            T
          
          ⁢
          ɛ
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            ɛ
            T
          
          ⁢
          D
        
        +
      
    
  
  
    
      
        
          
            -
            
              (
              
                β
                +
                I
              
              )
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            r
            T
          
          ⁢
          
            
              K
              v
              T
            
            ⁡
            
              (
              
                I
                +
                
                  K
                  a
                
              
              )
            
          
          ⁢
          
            τ
            ~
          
        
        -
        
          2
          ⁢
          
            r
            T
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            
              D
              T
            
            ⁡
            
              (
              
                I
                +
                
                  K
                  a
                
              
              )
            
          
          ⁢
          
            τ
            ~
          
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          ɛ
        
        -
        
          2
          ⁢
          
            (
            
              2
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            τ
            ~
          
        
        -
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            τ
            ~
          
        
        -
        
          2
          ⁢
          
            (
            
              2
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            D
            T
          
          ⁢
          
            τ
            ~
          
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          α
          ⁢
          
            
              
                τ
                ~
              
              T
            
            ⁡
            
              (
              
                I
                +
                
                  K
                  a
                  T
                
              
              )
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            
              
                τ
                ~
              
              T
            
            ⁡
            
              (
              
                I
                +
                
                  K
                  a
                  T
                
              
              )
            
          
          ⁢
          ɛ
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          
            (
            
              1
              -
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            σɛ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          D
        
        +
        
          2
          ⁢
          
            
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            D
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            D
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            ɛ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            ɛ
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ^
            
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              
                τ
                ~
              
              T
            
            ⁡
            
              (
              
                I
                +
                
                  K
                  a
                  T
                
              
              )
            
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              
                τ
                ~
              
              T
            
            ⁡
            
              (
              
                I
                +
                
                  K
                  a
                  T
                
              
              )
            
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          1
          α
        
        ⁢
        tr
        ⁢
        
          {
          
            
              
                -
                2
              
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                
                  W
                  ^
                
                T
              
              ⁢
              
                W
                ^
              
            
            +
            
              2
              ⁢
              
                W
                T
              
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                W
                ^
              
            
            +
            
              
                Γ
                2
              
              ⁢
              
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                2
              
              ⁢
              
                
                  W
                  ^
                
                T
              
              ⁢
              
                W
                ^
              
            
          
          }
        
      
    
  
  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          L
        
        =
        
          
            
              -
              
                
                  r
                  T
                
                ⁡
                
                  [
                  
                    
                      2
                      ⁢
                      I
                    
                    -
                    
                      
                        (
                        
                          2
                          +
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            σ
                          
                        
                        )
                      
                      ⁢
                      
                        K
                        v
                        T
                      
                      ⁢
                      
                        K
                        v
                      
                    
                  
                  ]
                
              
            
            ⁢
            r
          
          +
          
            2
            ⁢
            
              (
              
                2
                +
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
            ⁢
            
              r
              T
            
            ⁢
            
              K
              v
              T
            
            ⁢
            D
          
          +
          
            
              (
              
                2
                +
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
            ⁢
            
              D
              T
            
            ⁢
            D
          
          +
        
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          ɛ
        
        +
        
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            ɛ
            T
          
          ⁢
          ɛ
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            ɛ
            T
          
          ⁢
          D
        
        +
      
    
  
  
    
      
        
          
            -
            
              (
              
                I
                +
                β
              
              )
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          τ2
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            τ
            ~
          
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            D
            T
          
          ⁢
          
            K
            a
          
          ⁢
          
            τ
            ~
          
        
        -
        
          2
          ⁢
          
            D
            T
          
          ⁢
          
            τ
            ~
          
        
        +
      
    
  
  
    
      
        
          2
          ⁢
          α
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            a
            T
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            a
          
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            a
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              τ
              ~
            
            T
          
          ⁢
          
            K
            a
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          
            (
            
              1
              -
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            σɛ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          D
        
        +
        
          2
          ⁢
          
            
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            D
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            D
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            ɛ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            ɛ
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ^
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          +
          
            1
            α
          
        
        ⁢
        tr
        ⁢
        
          {
          
            
              
                -
                2
              
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                
                  W
                  ^
                
                T
              
              ⁢
              
                W
                ^
              
            
            +
            
              2
              ⁢
              
                W
                T
              
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                W
                ^
              
            
            +
            
              
                Γ
                2
              
              ⁢
              
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                2
              
              ⁢
              
                
                  W
                  ^
                
                T
              
              ⁢
              
                W
                ^
              
            
          
          }
        
      
    
  
  
    
      
        Completing
        ⁢
        
          
        
        ⁢
        squares
        ⁢
        
          
        
        ⁢
        for
        ⁢
        
          
        
        ⁢
        
          K
          a
        
        ⁢
        
          τ
          ~
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        
          τ
          ~
        
      
    
  
  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          L
        
        =
        
          
            
              -
              
                
                  r
                  T
                
                ⁡
                
                  [
                  
                    
                      2
                      ⁢
                      I
                    
                    -
                    
                      
                        (
                        
                          2
                          +
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            σ
                          
                        
                        )
                      
                      ⁢
                      
                        K
                        v
                        T
                      
                      ⁢
                      
                        K
                        v
                      
                    
                  
                  ]
                
              
            
            ⁢
            r
          
          +
          
            2
            ⁢
            
              (
              
                2
                +
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
            ⁢
            
              r
              T
            
            ⁢
            
              K
              v
              T
            
            ⁢
            D
          
          +
          
            
              (
              
                2
                +
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
            ⁢
            
              D
              T
            
            ⁢
            D
          
          +
        
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          ɛ
        
        +
        
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            ɛ
            T
          
          ⁢
          ɛ
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            ɛ
            T
          
          ⁢
          D
        
        +
      
    
  
  
    
      
        
          -
          
            
              {
              
                
                  
                    K
                    a
                  
                  ⁢
                  
                    τ
                    ~
                  
                
                -
                
                  
                    β
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          (
                          
                            1
                            +
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                          
                          )
                        
                        ⁢
                        
                          (
                          
                            D
                            +
                            ɛ
                          
                          )
                        
                      
                      -
                      
                        
                          (
                          
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                            +
                            
                              Γ
                              ⁢
                              
                                
                                
                                  I
                                  -
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      σσ
                                      T
                                    
                                  
                                
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                            W
                            ~
                          
                          T
                        
                        ⁢
                        σ
                      
                      +
                      
                        Γ
                        ⁢
                        
                          
                          
                            I
                            -
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σσ
                                T
                              
                            
                          
                          
                        
                        ⁢
                        
                          W
                          T
                        
                        ⁢
                        σ
                      
                    
                    ]
                  
                
              
              }
            
            T
          
        
        ·
        β
        ·
      
    
  
  
    
      
        
          {
          
            
              
                K
                a
              
              ⁢
              
                τ
                ~
              
            
            -
            
              
                β
                
                  -
                  1
                
              
              ⁡
              
                [
                
                  
                    
                      (
                      
                        1
                        +
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        D
                        +
                        ɛ
                      
                      )
                    
                  
                  -
                  
                    
                      (
                      
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                        
                        +
                        
                          Γ
                          ⁢
                          
                            
                            
                              I
                              -
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σσ
                                  T
                                
                              
                            
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        W
                        ~
                      
                      T
                    
                    ⁢
                    σ
                  
                  +
                  
                    Γ
                    ⁢
                    
                      
                      
                        I
                        -
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σσ
                            T
                          
                        
                      
                      
                    
                    ⁢
                    
                      W
                      T
                    
                    ⁢
                    σ
                  
                
                ]
              
            
          
          }
        
        +
      
    
  
  
    
      
        
          +
          
            
              {
              
                
                  β
                  
                    -
                    1
                  
                
                ⁡
                
                  [
                  
                    
                      
                        (
                        
                          1
                          +
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            σ
                          
                        
                        )
                      
                      ⁢
                      
                        (
                        
                          D
                          +
                          ɛ
                        
                        )
                      
                    
                    +
                    
                      
                        (
                        
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            σ
                          
                          +
                          
                            Γ
                            ⁢
                            
                              
                              
                                I
                                -
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    σσ
                                    T
                                  
                                
                              
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        
                          W
                          ~
                        
                        T
                      
                      ⁢
                      σ
                    
                    -
                    
                      Γ
                      ⁢
                      
                        
                        
                          I
                          -
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σσ
                              T
                            
                          
                        
                        
                      
                      ⁢
                      
                        W
                        T
                      
                      ⁢
                      σ
                    
                  
                  ]
                
              
              }
            
            T
          
        
        ·
        β
        ·
      
    
  
  
    
      
        
          {
          
            
              β
              
                -
                1
              
            
            ⁡
            
              [
              
                
                  
                    (
                    
                      1
                      +
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σ
                          T
                        
                        ⁢
                        σ
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      D
                      +
                      ɛ
                    
                    )
                  
                
                +
                
                  
                    (
                    
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σ
                          T
                        
                        ⁢
                        σ
                      
                      +
                      
                        Γ
                        ⁢
                        
                          
                          
                            I
                            -
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σσ
                                T
                              
                            
                          
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                      W
                      ~
                    
                    T
                  
                  ⁢
                  σ
                
                -
                
                  Γ
                  ⁢
                  
                    
                    
                      I
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σσ
                          T
                        
                      
                    
                    
                  
                  ⁢
                  
                    W
                    T
                  
                  ⁢
                  σ
                
              
              ]
            
          
          }
        
        +
      
    
  
  
    
      
        
          -
          
            
              
                [
                
                  
                    τ
                    ~
                  
                  -
                  
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    
                      σ
                      T
                    
                    ⁢
                    σ
                    ⁢
                    
                      
                    
                    ⁢
                    r
                  
                  +
                  D
                  +
                  
                    
                      K
                      v
                    
                    ⁢
                    r
                  
                
                ]
              
              T
            
            ⁡
            
              [
              
                
                  τ
                  ~
                
                -
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                  ⁢
                  
                    
                  
                  ⁢
                  r
                
                +
                D
                +
                
                  
                    K
                    v
                  
                  ⁢
                  r
                
              
              ]
            
          
        
        +
        
          
            
              (
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
              )
            
            2
          
          ⁢
          
            r
            T
          
          ⁢
          r
        
        +
        
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            K
            v
          
          ⁢
          r
        
        +
        
          
            D
            T
          
          ⁢
          D
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          D
        
        +
        
          2
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          D
        
        -
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
          
          ⁢
          r
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          
            (
            
              1
              -
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            σɛ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          D
        
        +
        
          2
          ⁢
          
            
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            D
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            D
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            ɛ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            ɛ
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ^
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          +
          
            1
            α
          
        
        ⁢
        tr
        ⁢
        
          {
          
            
              
                -
                2
              
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                
                  W
                  ^
                
                T
              
              ⁢
              
                W
                ^
              
            
            +
            
              2
              ⁢
              
                W
                T
              
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                W
                ^
              
            
            +
            
              
                Γ
                2
              
              ⁢
              
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                2
              
              ⁢
              
                
                  W
                  ^
                
                T
              
              ⁢
              
                W
                ^
              
            
          
          }
        
      
    
  
  
    
      
        
          
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                L
              
              =
              
                
                  -
                  
                    
                      r
                      T
                    
                    ⁡
                    
                      [
                      
                        
                          2
                          ⁢
                          I
                        
                        -
                        
                          
                            (
                            
                              3
                              +
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σ
                                  T
                                
                                ⁢
                                σ
                              
                            
                            )
                          
                          ⁢
                          
                            K
                            v
                            T
                          
                          ⁢
                          
                            K
                            v
                          
                        
                      
                      ]
                    
                  
                
                -
                
                  
                    (
                    
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      
                        σ
                        T
                      
                      ⁢
                      σ
                    
                    )
                  
                  2
                
                +
                
                  2
                  ⁢
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    K
                    v
                  
                
              
            
            ]
          
          ⁢
          r
        
        +
        
          2
          ⁢
          
            (
            
              3
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          D
        
        +
      
    
  
  
    
      
        
          
            +
            
              (
              
                3
                +
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
          
          ⁢
          
            D
            T
          
          ⁢
          D
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          ɛ
        
        +
        
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            ɛ
            T
          
          ⁢
          ɛ
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            ɛ
            T
          
          ⁢
          D
        
        +
      
    
  
  
    
      
        
          -
          
            
              {
              
                
                  
                    K
                    a
                  
                  ⁢
                  
                    τ
                    ~
                  
                
                -
                
                  
                    β
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          (
                          
                            1
                            +
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                          
                          )
                        
                        ⁢
                        
                          (
                          
                            D
                            +
                            ɛ
                          
                          )
                        
                      
                      -
                      
                        
                          (
                          
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                            +
                            
                              Γ
                              ⁢
                              
                                
                                
                                  I
                                  -
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      σσ
                                      T
                                    
                                  
                                
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                            W
                            ~
                          
                          T
                        
                        ⁢
                        σ
                      
                      +
                      
                        Γ
                        ⁢
                        
                          
                          
                            I
                            -
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σσ
                                T
                              
                            
                          
                          
                        
                        ⁢
                        
                          W
                          T
                        
                        ⁢
                        σ
                      
                    
                    ]
                  
                
              
              }
            
            T
          
        
        ·
        β
        ·
      
    
  
  
    
      
        
          {
          
            
              
                K
                a
              
              ⁢
              
                τ
                ~
              
            
            -
            
              
                β
                
                  -
                  1
                
              
              ⁡
              
                [
                
                  
                    
                      (
                      
                        1
                        +
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        D
                        +
                        ɛ
                      
                      )
                    
                  
                  -
                  
                    
                      (
                      
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                        
                        +
                        
                          Γ
                          ⁢
                          
                            
                            
                              I
                              -
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σσ
                                  T
                                
                              
                            
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        W
                        ~
                      
                      T
                    
                    ⁢
                    σ
                  
                  +
                  
                    Γ
                    ⁢
                    
                      
                      
                        I
                        -
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σσ
                            T
                          
                        
                      
                      
                    
                    ⁢
                    
                      W
                      T
                    
                    ⁢
                    σ
                  
                
                ]
              
            
          
          }
        
        +
      
    
  
  
    
      
        
          -
          
            
              
                [
                
                  
                    τ
                    ~
                  
                  -
                  
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    
                      σ
                      T
                    
                    ⁢
                    σ
                    ⁢
                    
                      
                    
                    ⁢
                    r
                  
                  +
                  D
                  +
                  
                    
                      K
                      v
                    
                    ⁢
                    r
                  
                
                ]
              
              T
            
            ⁡
            
              [
              
                
                  τ
                  ~
                
                -
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                  ⁢
                  
                    
                  
                  ⁢
                  r
                
                +
                D
                +
                
                  
                    K
                    v
                  
                  ⁢
                  r
                
              
              ]
            
          
        
        +
        
          
            
              
                β
                
                  -
                  1
                
              
              ⁡
              
                (
                
                  1
                  +
                  
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    
                      σ
                      T
                    
                    ⁢
                    σ
                  
                
                )
              
            
            2
          
          ⁢
          
            
              (
              
                D
                +
                ɛ
              
              )
            
            2
          
          ⁢
          
            (
            
              D
              +
              ɛ
            
            )
          
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          D
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          
            
              β
              
                -
                1
              
            
            ⁡
            
              (
              
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                
                +
                
                  Γ
                  ⁢
                  
                    
                    
                      I
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σσ
                          T
                        
                      
                    
                    
                  
                
              
              )
            
          
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            
              (
              
                D
                +
                ɛ
              
              )
            
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          
            
              
                β
                
                  -
                  1
                
              
              ⁡
              
                (
                
                  
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    
                      σ
                      T
                    
                    ⁢
                    σ
                  
                  +
                  
                    Γ
                    ⁢
                    
                      
                      
                        I
                        -
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σσ
                            T
                          
                        
                      
                      
                    
                  
                
                )
              
            
            2
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            +
            
              β
              
                -
                1
              
            
          
          ⁢
          
            Γ
            2
          
          ⁢
          
            
              
              
                I
                -
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σσ
                    T
                  
                
              
              
            
            2
          
          ⁢
          
            σ
            T
          
          ⁢
          W
          ⁢
          
            
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          
            
          
          ⁢
          
            
              β
              
                -
                1
              
            
            ⁡
            
              (
              
                1
                +
                
                  
                    ασ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σσ
                  T
                
              
            
            
          
          ⁢
          
            
              (
              
                D
                +
                ɛ
              
              )
            
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          
            
              β
              
                -
                1
              
            
            ⁡
            
              (
              
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                
                +
                
                  Γ
                  ⁢
                  
                    
                    
                      I
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σσ
                          T
                        
                      
                    
                    
                  
                
              
              )
            
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σσ
                  T
                
              
            
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        -
        
          
            (
            
              1
              -
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            σɛ
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            +
            2
          
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          D
        
        +
        
          2
          ⁢
          
            
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            D
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            
              W
              ~
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            
              
                
                  -
                  2
                
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  r
                  T
                
                ⁢
                
                  K
                  v
                  T
                
                ⁢
                
                  W
                  T
                
                ⁢
                σ
              
              -
              
                2
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  D
                  T
                
                ⁢
                
                  W
                  T
                
                ⁢
                σ
              
              +
              
                2
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  ɛ
                  T
                
                ⁢
                
                  
                    W
                    ~
                  
                  T
                
                ⁢
                σ
              
              +
            
          
        
        
          
            
              
                
                  -
                  2
                
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  ɛ
                  T
                
                ⁢
                
                  W
                  T
                
                ⁢
                σ
              
              -
              
                2
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                
                  W
                  ~
                
                ⁢
                
                  
                    W
                    ^
                  
                  T
                
                ⁢
                σ
              
              +
            
          
        
        
          
            
              
                +
                
                  1
                  α
                
              
              ⁢
              tr
              ⁢
              
                {
                
                  
                    
                      -
                      2
                    
                    ⁢
                    Γ
                    ⁢
                    
                      
                      
                        I
                        -
                        
                          α
                          ·
                          
                            
                          
                          ⁢
                          σ
                          ·
                          
                            σ
                            T
                          
                        
                      
                      
                    
                    ⁢
                    
                      
                        W
                        ^
                      
                      T
                    
                    ⁢
                    
                      W
                      ^
                    
                  
                  +
                  
                    2
                    ⁢
                    
                      W
                      T
                    
                    ⁢
                    Γ
                    ⁢
                    
                      
                      
                        I
                        -
                        
                          α
                          ·
                          
                            
                          
                          ⁢
                          σ
                          ·
                          
                            σ
                            T
                          
                        
                      
                      
                    
                    ⁢
                    
                      W
                      ^
                    
                  
                  +
                  
                    
                      Γ
                      2
                    
                    ⁢
                    
                      
                        
                        
                          I
                          -
                          
                            α
                            ·
                            
                              
                            
                            ⁢
                            σ
                            ·
                            
                              σ
                              T
                            
                          
                        
                        
                      
                      2
                    
                    ⁢
                    
                      
                        W
                        ^
                      
                      T
                    
                    ⁢
                    
                      W
                      ^
                    
                  
                
                }
              
            
          
        
        
          
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                L
              
              =
              
                
                  
                    -
                    
                      
                        r
                        T
                      
                      ⁡
                      
                        [
                        
                          
                            2
                            ⁢
                            I
                          
                          -
                          
                            
                              (
                              
                                3
                                +
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    σ
                                    T
                                  
                                  ⁢
                                  σ
                                
                              
                              )
                            
                            ⁢
                            
                              K
                              v
                              T
                            
                            ⁢
                            
                              K
                              v
                            
                          
                          -
                          
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σ
                                  T
                                
                                ⁢
                                σ
                              
                              )
                            
                            2
                          
                          +
                          
                            2
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            σ
                            ⁢
                            
                              
                            
                            ⁢
                            
                              K
                              v
                            
                          
                        
                        ]
                      
                    
                  
                  ⁢
                  r
                
                +
                
                  2
                  ⁢
                  
                    (
                    
                      1
                      +
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σ
                          T
                        
                        ⁢
                        σ
                      
                    
                    )
                  
                  ⁢
                  
                    r
                    T
                  
                  ⁢
                  
                    
                      K
                      v
                      T
                    
                    ⁡
                    
                      (
                      
                        D
                        +
                        ɛ
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
                4
                ⁢
                
                  r
                  T
                
                ⁢
                
                  K
                  v
                  T
                
                ⁢
                D
              
              +
              
                2
                ⁢
                
                  D
                  T
                
                ⁢
                D
              
              +
            
          
        
        
          
            
              
                
                  
                    
                      -
                      
                        
                          {
                          
                            
                              
                                K
                                a
                              
                              ⁢
                              
                                τ
                                ~
                              
                            
                            -
                            
                              
                                β
                                
                                  -
                                  1
                                
                              
                              ⁡
                              
                                [
                                
                                  
                                    
                                      (
                                      
                                        1
                                        +
                                        
                                          α
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            σ
                                            T
                                          
                                          ⁢
                                          σ
                                        
                                      
                                      )
                                    
                                    ⁢
                                    
                                      (
                                      
                                        D
                                        +
                                        ɛ
                                      
                                      )
                                    
                                  
                                  -
                                  
                                    
                                      (
                                      
                                        
                                          α
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            σ
                                            T
                                          
                                          ⁢
                                          σ
                                        
                                        +
                                        
                                          Γ
                                          ⁢
                                          
                                            
                                            
                                              I
                                              -
                                              
                                                α
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  σσ
                                                  T
                                                
                                              
                                            
                                            
                                          
                                        
                                      
                                      )
                                    
                                    ⁢
                                    
                                      
                                        W
                                        ~
                                      
                                      T
                                    
                                    ⁢
                                    σ
                                  
                                  +
                                  
                                    Γ
                                    ⁢
                                    
                                      
                                      
                                        I
                                        -
                                        
                                          α
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            σσ
                                            T
                                          
                                        
                                      
                                      
                                    
                                    ⁢
                                    
                                      W
                                      T
                                    
                                    ⁢
                                    σ
                                  
                                
                                ]
                              
                            
                          
                          }
                        
                        T
                      
                    
                    ·
                    β
                    ·
                  
                
              
              
                
                  
                    
                      {
                      
                        
                          
                            K
                            a
                          
                          ⁢
                          
                            τ
                            ~
                          
                        
                        -
                        
                          
                            β
                            
                              -
                              1
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  (
                                  
                                    1
                                    +
                                    
                                      α
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        σ
                                        T
                                      
                                      ⁢
                                      σ
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  (
                                  
                                    D
                                    +
                                    ɛ
                                  
                                  )
                                
                              
                              -
                              
                                
                                  (
                                  
                                    
                                      α
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        σ
                                        T
                                      
                                      ⁢
                                      σ
                                    
                                    +
                                    
                                      Γ
                                      ⁢
                                      
                                        
                                        
                                          I
                                          -
                                          
                                            α
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              σσ
                                              T
                                            
                                          
                                        
                                        
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                  
                                    W
                                    ~
                                  
                                  T
                                
                                ⁢
                                σ
                              
                              +
                              
                                Γ
                                ⁢
                                
                                  
                                  
                                    I
                                    -
                                    
                                      α
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        σσ
                                        T
                                      
                                    
                                  
                                  
                                
                                ⁢
                                
                                  W
                                  T
                                
                                ⁢
                                σ
                              
                            
                            ]
                          
                        
                      
                      }
                    
                    -
                    
                      2
                      ⁢
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      
                        σ
                        T
                      
                      ⁢
                      σ
                      ⁢
                      
                        
                      
                      ⁢
                      
                        r
                        T
                      
                      ⁢
                      D
                    
                    +
                  
                
              
              
                
                  
                    
                      -
                      
                        
                          
                            [
                            
                              
                                τ
                                ~
                              
                              -
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σ
                                  T
                                
                                ⁢
                                σ
                                ⁢
                                
                                  
                                
                                ⁢
                                r
                              
                              +
                              D
                              +
                              
                                
                                  K
                                  v
                                
                                ⁢
                                r
                              
                            
                            ]
                          
                          T
                        
                        ⁡
                        
                          [
                          
                            
                              τ
                              ~
                            
                            -
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                              ⁢
                              
                                
                              
                              ⁢
                              r
                            
                            +
                            D
                            +
                            
                              
                                K
                                v
                              
                              ⁢
                              r
                            
                          
                          ]
                        
                      
                    
                    +
                    
                      
                        
                          (
                          
                            1
                            +
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                          
                          )
                        
                        ⁡
                        
                          [
                          
                            1
                            +
                            
                              
                                β
                                
                                  -
                                  1
                                
                              
                              ⁡
                              
                                (
                                
                                  1
                                  +
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      σ
                                      T
                                    
                                    ⁢
                                    σ
                                  
                                
                                )
                              
                            
                          
                          ]
                        
                      
                      ⁢
                      
                        
                          (
                          
                            D
                            +
                            ɛ
                          
                          )
                        
                        T
                      
                      ⁢
                      
                        (
                        
                          D
                          +
                          ɛ
                        
                        )
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      2
                      ⁢
                      
                        (
                        
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            σ
                          
                          +
                          
                            Γ
                            ⁢
                            
                              
                              
                                I
                                -
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    σσ
                                    T
                                  
                                
                              
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        r
                        T
                      
                      ⁢
                      
                        K
                        v
                        T
                      
                      ⁢
                      
                        
                          W
                          ~
                        
                        T
                      
                      ⁢
                      σ
                    
                    -
                    
                      
                        [
                        
                          1
                          -
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            σ
                          
                          -
                          
                            
                              
                                β
                                
                                  -
                                  1
                                
                              
                              ⁡
                              
                                (
                                
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      σ
                                      T
                                    
                                    ⁢
                                    σ
                                  
                                  +
                                  
                                    Γ
                                    ⁢
                                    
                                      
                                      
                                        I
                                        -
                                        
                                          α
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            σσ
                                            T
                                          
                                        
                                      
                                      
                                    
                                  
                                
                                )
                              
                            
                            2
                          
                        
                        ]
                      
                      ⁢
                      
                        σ
                        T
                      
                      ⁢
                      
                        W
                        ~
                      
                      ⁢
                      
                        
                          W
                          ~
                        
                        T
                      
                      ⁢
                      σ
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        +
                        
                          β
                          
                            -
                            1
                          
                        
                      
                      ⁢
                      
                        Γ
                        2
                      
                      ⁢
                      
                        
                          
                          
                            I
                            -
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σσ
                                T
                              
                            
                          
                          
                        
                        2
                      
                      ⁢
                      
                        σ
                        T
                      
                      ⁢
                      W
                      ⁢
                      
                        
                      
                      ⁢
                      
                        W
                        T
                      
                      ⁢
                      σ
                    
                    -
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          β
                          
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            1
                            +
                            
                              
                                ασ
                                T
                              
                              ⁢
                              σ
                            
                          
                          )
                        
                      
                      ⁢
                      Γ
                      ⁢
                      
                        
                        
                          I
                          -
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σσ
                              T
                            
                          
                        
                        
                      
                      ⁢
                      
                        
                          (
                          
                            D
                            +
                            ɛ
                          
                          )
                        
                        T
                      
                      ⁢
                      
                        W
                        T
                      
                      ⁢
                      σ
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        -
                        2
                      
                      ⁢
                      
                        
                          β
                          
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                            +
                            
                              Γ
                              ⁢
                              
                                
                                
                                  I
                                  -
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      σσ
                                      T
                                    
                                  
                                
                                
                              
                            
                          
                          )
                        
                      
                      ⁢
                      Γ
                      ⁢
                      
                        
                        
                          I
                          -
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σσ
                              T
                            
                          
                        
                        
                      
                      ⁢
                      
                        σ
                        T
                      
                      ⁢
                      
                        W
                        ~
                      
                      ⁢
                      
                        W
                        T
                      
                      ⁢
                      σ
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        +
                        2
                      
                      ⁢
                      
                        
                          (
                          
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                            +
                            
                              Γ
                              ⁢
                              
                                
                                
                                  I
                                  -
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      σσ
                                      T
                                    
                                  
                                
                                
                              
                            
                          
                          )
                        
                        [
                        
                          1
                          +
                          
                            
                              β
                              
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                1
                                +
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    σ
                                    T
                                  
                                  ⁢
                                  σ
                                
                              
                              )
                            
                          
                        
                        ⁢
                        
                          
                        
                        ]
                      
                      ⁢
                      
                        
                          (
                          
                            D
                            +
                            ɛ
                          
                          )
                        
                        T
                      
                      ⁢
                      
                        
                          W
                          ~
                        
                        T
                      
                      ⁢
                      σ
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        -
                        2
                      
                      ⁢
                      Γ
                      ⁢
                      
                        
                        
                          I
                          -
                          
                            α
                            ·
                            
                              
                            
                            ⁢
                            σ
                            ·
                            
                              σ
                              T
                            
                          
                        
                        
                      
                      ⁢
                      
                        r
                        T
                      
                      ⁢
                      
                        K
                        v
                        T
                      
                      ⁢
                      
                        W
                        T
                      
                      ⁢
                      σ
                    
                    -
                    
                      2
                      ⁢
                      Γ
                      ⁢
                      
                        
                        
                          I
                          -
                          
                            α
                            ·
                            
                              
                            
                            ⁢
                            σ
                            ·
                            
                              σ
                              T
                            
                          
                        
                        
                      
                      ⁢
                      
                        
                          (
                          
                            D
                            +
                            ɛ
                          
                          )
                        
                        T
                      
                      ⁢
                      
                        W
                        T
                      
                      ⁢
                      σ
                    
                    -
                    
                      2
                      ⁢
                      Γ
                      ⁢
                      
                        
                        
                          I
                          -
                          
                            α
                            ·
                            
                              
                            
                            ⁢
                            σ
                            ·
                            
                              σ
                              T
                            
                          
                        
                        
                      
                      ⁢
                      
                        σ
                        T
                      
                      ⁢
                      
                        W
                        ~
                      
                      ⁢
                      
                        
                          W
                          ^
                        
                        T
                      
                      ⁢
                      σ
                    
                    +
                  
                
              
              
                
                  
                    
                      +
                      
                        1
                        α
                      
                    
                    ⁢
                    tr
                    ⁢
                    
                      {
                      
                        
                          
                            -
                            2
                          
                          ⁢
                          Γ
                          ⁢
                          
                            
                            
                              I
                              -
                              
                                α
                                ·
                                
                                  
                                
                                ⁢
                                σ
                                ·
                                
                                  σ
                                  T
                                
                              
                            
                            
                          
                          ⁢
                          
                            
                              W
                              ^
                            
                            T
                          
                          ⁢
                          
                            W
                            ^
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            W
                            T
                          
                          ⁢
                          Γ
                          ⁢
                          
                            
                            
                              I
                              -
                              
                                α
                                ·
                                
                                  
                                
                                ⁢
                                σ
                                ·
                                
                                  σ
                                  T
                                
                              
                            
                            
                          
                          ⁢
                          
                            W
                            ^
                          
                        
                        +
                        
                          
                            Γ
                            2
                          
                          ⁢
                          
                            
                              
                              
                                I
                                -
                                
                                  α
                                  ·
                                  
                                    
                                  
                                  ⁢
                                  σ
                                  ·
                                  
                                    σ
                                    T
                                  
                                
                              
                              
                            
                            2
                          
                          ⁢
                          
                            
                              W
                              ^
                            
                            T
                          
                          ⁢
                          
                            W
                            ^
                          
                        
                      
                      }
                    
                  
                
              
            
          
        
      
    
  




Define ρ=(1−ασTσ)I−β−1(ασTσ+Γ∥I−ασσT∥)2>0 (condition (36)).

Completing squares for {tilde over (W)}Tσ




  
    
      
        
          
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                L
              
              =
              
                
                  -
                  
                    
                      r
                      T
                    
                    ⁡
                    
                      [
                      
                        
                          2
                          ⁢
                          I
                        
                        -
                        
                          
                            (
                            
                              3
                              +
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σ
                                  T
                                
                                ⁢
                                σ
                              
                            
                            )
                          
                          ⁢
                          
                            K
                            v
                            T
                          
                          ⁢
                          
                            K
                            v
                          
                        
                      
                      ]
                    
                  
                
                -
                
                  
                    (
                    
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      
                        σ
                        T
                      
                      ⁢
                      σ
                    
                    )
                  
                  2
                
                +
                
                  2
                  ⁢
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    K
                    v
                  
                
              
            
            ]
          
          ⁢
          r
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            r
            T
          
          ⁢
          
            
              K
              v
              T
            
            ⁡
            
              (
              
                D
                +
                ɛ
              
              )
            
          
        
        +
      
    
  
  
    
      
        
          
            +
            4
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          D
        
        +
        
          2
          ⁢
          
            D
            T
          
          ⁢
          D
        
        +
      
    
  
  
    
      
        
          -
          
            
              {
              
                
                  
                    K
                    a
                  
                  ⁢
                  
                    τ
                    ~
                  
                
                -
                
                  
                    β
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          (
                          
                            1
                            +
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                          
                          )
                        
                        ⁢
                        
                          (
                          
                            D
                            +
                            ɛ
                          
                          )
                        
                      
                      -
                      
                        
                          (
                          
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                            +
                            
                              Γ
                              ⁢
                              
                                
                                
                                  I
                                  -
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      σσ
                                      T
                                    
                                  
                                
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                            W
                            ~
                          
                          T
                        
                        ⁢
                        σ
                      
                      +
                      
                        Γ
                        ⁢
                        
                          
                          
                            I
                            -
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σσ
                                T
                              
                            
                          
                          
                        
                        ⁢
                        
                          W
                          T
                        
                        ⁢
                        σ
                      
                    
                    ]
                  
                
              
              }
            
            T
          
        
        ·
        β
        ·
      
    
  
  
    
      
        
          {
          
            
              
                K
                a
              
              ⁢
              
                τ
                ~
              
            
            -
            
              
                β
                
                  -
                  1
                
              
              ⁡
              
                [
                
                  
                    
                      (
                      
                        1
                        +
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        D
                        +
                        ɛ
                      
                      )
                    
                  
                  -
                  
                    
                      (
                      
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                        
                        +
                        
                          Γ
                          ⁢
                          
                            
                            
                              I
                              -
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σσ
                                  T
                                
                              
                            
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      
                        W
                        ~
                      
                      T
                    
                    ⁢
                    σ
                  
                  +
                  
                    Γ
                    ⁢
                    
                      
                      
                        I
                        -
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σσ
                            T
                          
                        
                      
                      
                    
                    ⁢
                    
                      W
                      T
                    
                    ⁢
                    σ
                  
                
                ]
              
            
          
          }
        
        +
      
    
  
  
    
      
        
          -
          
            
              
                [
                
                  
                    τ
                    ~
                  
                  -
                  
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    
                      σ
                      T
                    
                    ⁢
                    σ
                    ⁢
                    
                      
                    
                    ⁢
                    r
                  
                  +
                  D
                  +
                  
                    
                      K
                      v
                    
                    ⁢
                    r
                  
                
                ]
              
              T
            
            ⁡
            
              [
              
                
                  τ
                  ~
                
                -
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                  ⁢
                  
                    
                  
                  ⁢
                  r
                
                +
                D
                +
                
                  
                    K
                    v
                  
                  ⁢
                  r
                
              
              ]
            
          
        
        -
        
          2
          ⁢
          α
          ⁢
          
            
          
          ⁢
          
            σ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          D
        
        +
      
    
  
  
    
      
        
          
            +
            
              (
              
                1
                +
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
          
          ⁢
          
            (
            
              1
              +
              
                
                  1
                  +
                  
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    
                      σ
                      T
                    
                    ⁢
                    σ
                  
                
                β
              
            
            )
          
          ⁢
          
            
              (
              
                D
                +
                ɛ
              
              )
            
            T
          
          ⁢
          
            (
            
              D
              +
              ɛ
            
            )
          
        
        +
      
    
  
  
    
      
        
          -
          
            
              {
              
                
                  
                    
                      W
                      ~
                    
                    T
                  
                  ⁢
                  σ
                
                -
                
                  
                    
                      ρ
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                        
                        +
                        
                          Γ
                          ⁢
                          
                            
                            
                              I
                              -
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σσ
                                  T
                                
                              
                            
                            
                          
                        
                      
                      )
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          K
                          
                            v
                            ⁢
                            
                              
                            
                          
                        
                        ⁢
                        r
                      
                      +
                      
                        
                          (
                          
                            1
                            +
                            
                              
                                β
                                
                                  -
                                  1
                                
                              
                              ⁡
                              
                                (
                                
                                  1
                                  +
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      σ
                                      T
                                    
                                    ⁢
                                    σ
                                  
                                
                                )
                              
                            
                          
                          )
                        
                        ⁢
                        
                          (
                          
                            D
                            +
                            ɛ
                          
                          )
                        
                      
                      -
                      
                        
                          β
                          
                            -
                            1
                          
                        
                        ⁢
                        Γ
                        ⁢
                        
                          
                          
                            I
                            -
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σσ
                                T
                              
                            
                          
                          
                        
                        ⁢
                        
                          W
                          T
                        
                        ⁢
                        σ
                      
                    
                    ]
                  
                
              
              }
            
            T
          
        
        ⁢
        ρ
      
    
  
  
    
      
        {
        
          
            
              
                W
                ~
              
              T
            
            ⁢
            σ
          
          -
          
            
              
                ρ
                
                  -
                  1
                
              
              ⁡
              
                (
                
                  
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    
                      σ
                      T
                    
                    ⁢
                    σ
                  
                  +
                  
                    Γ
                    ⁢
                    
                      
                      
                        I
                        -
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σσ
                            T
                          
                        
                      
                      
                    
                  
                
                )
              
            
            ⁡
            
              [
              
                
                  
                    K
                    
                      v
                      ⁢
                      
                        
                      
                    
                  
                  ⁢
                  r
                
                +
                
                  
                    (
                    
                      1
                      +
                      
                        
                          β
                          
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            1
                            +
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      D
                      +
                      ɛ
                    
                    )
                  
                
                -
                
                  
                    β
                    
                      -
                      1
                    
                  
                  ⁢
                  Γ
                  ⁢
                  
                    
                    
                      I
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σσ
                          T
                        
                      
                    
                    
                  
                  ⁢
                  
                    W
                    T
                  
                  ⁢
                  σ
                
              
              ]
            
          
        
        }
      
    
  
  
    
      
        +
        
          
            
              
                
                  ρ
                  
                    -
                    1
                  
                
                ⁡
                
                  (
                  
                    
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      
                        σ
                        T
                      
                      ⁢
                      σ
                    
                    +
                    
                      Γ
                      ⁢
                      
                        
                        
                          I
                          -
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σσ
                              T
                            
                          
                        
                        
                      
                    
                  
                  )
                
              
              2
            
            ⁡
            
              [
              
                
                  
                    K
                    
                      v
                      ⁢
                      
                        
                      
                    
                  
                  ⁢
                  r
                
                +
                
                  
                    (
                    
                      1
                      +
                      
                        
                          β
                          
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            1
                            +
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      D
                      +
                      ɛ
                    
                    )
                  
                
                -
                
                  
                    β
                    
                      -
                      1
                    
                  
                  ⁢
                  Γ
                  ⁢
                  
                    
                    
                      I
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σσ
                          T
                        
                      
                    
                    
                  
                  ⁢
                  
                    W
                    T
                  
                  ⁢
                  σ
                
              
              ]
            
          
          T
        
      
    
  
  
    
      
        
          [
          
            
              
                K
                
                  v
                  ⁢
                  
                    
                  
                
              
              ⁢
              r
            
            +
            
              
                (
                
                  1
                  +
                  
                    
                      β
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        1
                        +
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                        
                      
                      )
                    
                  
                
                )
              
              ⁢
              
                (
                
                  D
                  +
                  ɛ
                
                )
              
            
            -
            
              
                β
                
                  -
                  1
                
              
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    
                      σσ
                      T
                    
                  
                
                
              
              ⁢
              
                W
                T
              
              ⁢
              σ
            
          
          ]
        
        +
      
    
  
  
    
      
        
          
            +
            
              β
              
                -
                1
              
            
          
          ⁢
          
            Γ
            2
          
          ⁢
          
            
              
              
                I
                -
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σσ
                    T
                  
                
              
              
            
            2
          
          ⁢
          
            σ
            T
          
          ⁢
          W
          ⁢
          
            
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          
            
          
          ⁢
          
            
              β
              
                -
                1
              
            
            ⁡
            
              (
              
                1
                +
                
                  
                    ασ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σσ
                  T
                
              
            
            
          
          ⁢
          
            
              (
              
                D
                +
                ɛ
              
              )
            
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              (
              
                D
                +
                ɛ
              
              )
            
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            σ
            T
          
          ⁢
          
            W
            ~
          
          ⁢
          
            
              W
              ^
            
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          +
          
            1
            α
          
        
        ⁢
        tr
        ⁢
        
          {
          
            
              
                -
                2
              
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                
                  W
                  ^
                
                T
              
              ⁢
              
                W
                ^
              
            
            +
            
              2
              ⁢
              
                W
                T
              
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                W
                ^
              
            
            +
            
              
                Γ
                2
              
              ⁢
              
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                2
              
              ⁢
              
                
                  W
                  ^
                
                T
              
              ⁢
              
                W
                ^
              
            
          
          }
        
      
    
  




Putting the term −2Γ∥I−α·ΓTσ∥σT{tilde over (W)}ŴTσ back on the trace term and bounding the trace term.




  
    
      
        
          1
          α
        
        ⁢
        tr
        ⁢
        
          {
          
            
              
                -
                2
              
              ⁢
              αΓ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                
                  W
                  ~
                
                T
              
              ⁢
              
                σσ
                T
              
              ⁢
              
                W
                ^
              
            
            +
            
              2
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                
                  W
                  ~
                
                T
              
              ⁢
              
                W
                ^
              
            
            +
            
              
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                2
              
              ⁢
              
                
                  W
                  ^
                
                T
              
              ⁢
              
                Γ
                T
              
              ⁢
              Γ
              ⁢
              
                W
                ^
              
            
          
          }
        
      
    
  
  
    
      
        =
        
          
            
              -
              
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                α
              
            
            ⁢
            tr
            ⁢
            
              {
              
                
                  2
                  ⁢
                  αΓ
                  ⁢
                  
                    
                      W
                      ~
                    
                    T
                  
                  ⁢
                  
                    σσ
                    T
                  
                  ⁢
                  
                    W
                    ^
                  
                
                -
                
                  2
                  ⁢
                  Γ
                  ⁢
                  
                    
                      W
                      ~
                    
                    T
                  
                  ⁢
                  
                    W
                    ^
                  
                
                +
                
                  
                    
                    
                      I
                      -
                      
                        α
                        ·
                        
                          
                        
                        ⁢
                        σ
                        ·
                        
                          σ
                          T
                        
                      
                    
                    
                  
                  ⁢
                  
                    
                      W
                      ^
                    
                    T
                  
                  ⁢
                  
                    Γ
                    T
                  
                  ⁢
                  Γ
                  ⁢
                  
                    W
                    ^
                  
                
              
              }
            
          
          =
        
      
    
  
  
    
      
        =
        
          
            -
            
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              α
            
          
          ⁢
          tr
          ⁢
          
            {
            
              
                2
                ⁢
                αΓ
                ⁢
                
                  
                    W
                    ~
                  
                  T
                
                ⁢
                
                  
                    σσ
                    T
                  
                  ⁡
                  
                    (
                    
                      W
                      -
                      
                        W
                        ^
                      
                    
                    )
                  
                
              
              -
              
                2
                ⁢
                Γ
                ⁢
                
                  
                    
                      W
                      ~
                    
                    T
                  
                  ⁡
                  
                    (
                    
                      W
                      -
                      
                        W
                        ^
                      
                    
                    )
                  
                
              
              +
            
          
        
      
    
  
  
    
      
        
          
            -
            
              
              
                I
                -
                
                  α
                  ·
                  
                    
                  
                  ⁢
                  σ
                  ·
                  
                    σ
                    T
                  
                
              
              
            
          
          ⁢
          
            (
            
              W
              -
              
                W
                ^
              
            
            )
          
          ⁢
          
            Γ
            T
          
          ⁢
          
            Γ
            ⁡
            
              (
              
                W
                -
                
                  W
                  ^
                
              
              )
            
          
        
        =
      
    
  
  
    
      
        =
        
          
            -
            
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              α
            
          
          ⁢
          tr
          ⁢
          
            {
            
              
                2
                ⁢
                αΓ
                ⁢
                
                  
                    W
                    ~
                  
                  T
                
                ⁢
                
                  σσ
                  T
                
                ⁢
                W
              
              -
              
                2
                ⁢
                αΓ
                ⁢
                
                  
                    W
                    ~
                  
                  T
                
                ⁢
                
                  σσ
                  T
                
                ⁢
                
                  W
                  ~
                
              
              -
              
                2
                ⁢
                Γ
                ⁢
                
                  
                    W
                    ~
                  
                  T
                
                ⁢
                W
              
              +
              
                2
                ⁢
                
                  
                
                ⁢
                Γ
                ⁢
                
                  
                    W
                    ~
                  
                  T
                
                ⁢
                
                  W
                  ~
                
              
              +
            
          
        
      
    
  
  
    
      
        
          
            
              
                -
                
                  Γ
                  2
                
              
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                
              
              ⁢
              
                W
                T
              
              ⁢
              W
            
            +
            
              2
              ⁢
              
                
              
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                W
                T
              
              ⁢
              
                W
                ~
              
            
            -
            
              
                Γ
                2
              
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              ⁢
              
                
                  W
                  ~
                
                T
              
              ⁢
              
                W
                ~
              
            
          
          }
        
        =
      
    
  
  
    
      
        
          <
          
            
              -
              
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                α
              
            
            ⁢
            
              {
              
                
                  
                    Γ
                    ⁡
                    
                      (
                      
                        2
                        -
                        Γ
                      
                      )
                    
                  
                  ⁢
                  
                    
                    
                      I
                      -
                      
                        α
                        ·
                        
                          
                        
                        ⁢
                        σ
                        ·
                        
                          σ
                          T
                        
                      
                    
                    
                  
                  ⁢
                  
                    
                      
                      
                        W
                        ~
                      
                      
                    
                    2
                  
                
                -
                
                  
                    Γ
                    2
                  
                  ⁢
                  
                    
                    
                      I
                      -
                      
                        α
                        ·
                        
                          
                        
                        ⁢
                        σ
                        ·
                        
                          σ
                          T
                        
                      
                    
                    ⁢
                    
                      
                      ~
                    
                  
                  ⁢
                  
                    W
                    M
                    2
                  
                
              
              }
            
          
        
        =
      
    
  
  
    
      
        <
        
          
            -
            
              
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                2
              
              α
            
          
          ⁢
          
            {
            
              
                
                  Γ
                  ⁡
                  
                    (
                    
                      2
                      -
                      Γ
                    
                    )
                  
                
                ⁢
                
                  
                    
                    
                      W
                      ~
                    
                    
                  
                  2
                
              
              -
              
                
                  Γ
                  2
                
                ⁢
                
                  W
                  M
                  2
                
              
            
            }
          
        
      
    
  
  
    
      Bounding
    
  
  
    
      
        
          
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                L
              
              <
              
                
                  -
                  
                    
                      r
                      T
                    
                    ⁡
                    
                      [
                      
                        
                          2
                          ⁢
                          I
                        
                        -
                        
                          
                            (
                            
                              3
                              +
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σ
                                  T
                                
                                ⁢
                                σ
                              
                            
                            )
                          
                          ⁢
                          
                            K
                            v
                            T
                          
                          ⁢
                          
                            K
                            v
                          
                        
                      
                      ]
                    
                  
                
                -
                
                  
                    (
                    
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      
                        σ
                        T
                      
                      ⁢
                      σ
                    
                    )
                  
                  2
                
                +
                
                  2
                  ⁢
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    K
                    v
                  
                
              
            
            ]
          
          ⁢
          r
        
        +
        
          2
          ⁢
          
            (
            
              1
              +
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
              
            
            )
          
          ⁢
          
            r
            T
          
          ⁢
          
            
              K
              v
              T
            
            ⁡
            
              (
              
                D
                +
                ɛ
              
              )
            
          
        
        +
      
    
  
  
    
      
        
          
            +
            4
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          D
        
        +
        
          2
          ⁢
          
            D
            T
          
          ⁢
          D
        
        +
      
    
  
  
    
      
        
          
            +
            
              (
              
                1
                +
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
          
          ⁢
          
            ⌊
            
              1
              +
              
                
                  β
                  
                    -
                    1
                  
                
                ⁢
                
                  (
                  
                    1
                    +
                    
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      
                        σ
                        T
                      
                      ⁢
                      σ
                    
                  
                  )
                
              
            
            ⌋
          
          ⁢
          
            
              (
              
                D
                +
                ɛ
              
              )
            
            T
          
          ⁢
          
            (
            
              D
              +
              ɛ
            
            )
          
        
        -
        
          2
          ⁢
          
            ασ
            T
          
          ⁢
          σ
          ⁢
          
            
          
          ⁢
          
            r
            T
          
          ⁢
          D
        
        +
      
    
  
  
    
      
        +
        
          
            
              
                
                  ρ
                  
                    -
                    1
                  
                
                ⁡
                
                  (
                  
                    
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      
                        σ
                        T
                      
                      ⁢
                      σ
                    
                    +
                    
                      Γ
                      ⁢
                      
                        
                        
                          I
                          -
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σσ
                              T
                            
                          
                        
                        
                      
                    
                  
                  )
                
              
              2
            
            ⁡
            
              [
              
                
                  
                    K
                    
                      v
                      ⁢
                      
                        
                      
                    
                  
                  ⁢
                  r
                
                +
                
                  
                    (
                    
                      1
                      +
                      
                        
                          β
                          
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            1
                            +
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      D
                      +
                      ɛ
                    
                    )
                  
                
                -
                
                  
                    β
                    
                      -
                      1
                    
                  
                  ⁢
                  Γ
                  ⁢
                  
                    
                    
                      I
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σσ
                          T
                        
                      
                    
                    
                  
                  ⁢
                  
                    W
                    T
                  
                  ⁢
                  σ
                
              
              ]
            
          
          T
        
      
    
  
  
    
      
        
          [
          
            
              
                K
                
                  v
                  ⁢
                  
                    
                  
                
              
              ⁢
              r
            
            +
            
              
                (
                
                  1
                  +
                  
                    
                      β
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        1
                        +
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                        
                      
                      )
                    
                  
                
                )
              
              ⁢
              
                (
                
                  D
                  +
                  ɛ
                
                )
              
            
            -
            
              
                β
                
                  -
                  1
                
              
              ⁢
              Γ
              ⁢
              
                
                
                  I
                  -
                  
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    
                      σσ
                      T
                    
                  
                
                
              
              ⁢
              
                W
                T
              
              ⁢
              σ
            
          
          ]
        
        +
      
    
  
  
    
      
        
          
            β
            
              -
              1
            
          
          ⁢
          
            Γ
            2
          
          ⁢
          
            
              
              
                I
                -
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  
                    σσ
                    T
                  
                
              
              
            
            2
          
          ⁢
          
            σ
            T
          
          ⁢
          W
          ⁢
          
            
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          
            
          
          ⁢
          
            
              β
              
                -
                1
              
            
            ⁡
            
              (
              
                1
                +
                
                  
                    ασ
                    T
                  
                  ⁢
                  σ
                
              
              )
            
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ⁢
                
                  
                
                ⁢
                
                  σσ
                  T
                
              
            
            
          
          ⁢
          
            
              (
              
                D
                +
                ɛ
              
              )
            
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            2
          
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            r
            T
          
          ⁢
          
            K
            v
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        -
        
          2
          ⁢
          Γ
          ⁢
          
            
            
              I
              -
              
                α
                ·
                
                  
                
                ⁢
                σ
                ·
                
                  σ
                  T
                
              
            
            
          
          ⁢
          
            
              (
              
                D
                +
                ɛ
              
              )
            
            T
          
          ⁢
          
            W
            T
          
          ⁢
          σ
        
        +
      
    
  
  
    
      
        
          
            -
            
              
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                2
              
              α
            
          
          ⁢
          
            Γ
            ⁡
            
              (
              
                2
                -
                Γ
              
              )
            
          
          ⁢
          
            
              
              
                W
                ~
              
              
            
            2
          
        
        +
        
          
            
              
                
                
                  I
                  -
                  
                    α
                    ·
                    
                      
                    
                    ⁢
                    σ
                    ·
                    
                      σ
                      T
                    
                  
                
                
              
              2
            
            α
          
          ⁢
          
            Γ
            2
          
          ⁢
          
            W
            M
            2
          
        
      
    
  
  
    
      
        
          
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                L
              
              <
              
                
                  
                    -
                    
                      
                        r
                        T
                      
                      [
                      
                        
                          2
                          ⁢
                          I
                        
                        -
                        
                          
                            (
                            
                              3
                              +
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σ
                                  T
                                
                                ⁢
                                σ
                              
                            
                            )
                          
                          ⁢
                          
                            K
                            v
                            T
                          
                          ⁢
                          
                            K
                            v
                          
                        
                        -
                        
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                            )
                          
                          2
                        
                        +
                        
                          2
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            K
                            v
                          
                        
                      
                      ]
                    
                  
                  ⁢
                  r
                
                +
                
                  2
                  ⁢
                  
                    (
                    
                      1
                      +
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σ
                          T
                        
                        ⁢
                        σ
                      
                    
                    )
                  
                  ⁢
                  
                    r
                    T
                  
                  ⁢
                  
                    
                      K
                      v
                      T
                    
                    ⁡
                    
                      (
                      
                        D
                        +
                        ɛ
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  +
                  4
                
                ⁢
                
                  r
                  T
                
                ⁢
                
                  K
                  v
                  T
                
                ⁢
                D
              
              +
              
                2
                ⁢
                
                  D
                  T
                
                ⁢
                D
              
              +
            
          
        
        
          
            
              
                
                  +
                  
                    (
                    
                      1
                      +
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σ
                          T
                        
                        ⁢
                        σ
                      
                    
                    )
                  
                
                ⁢
                
                  ⌊
                  
                    1
                    +
                    
                      
                        β
                        
                          -
                          1
                        
                      
                      ⁢
                      
                        (
                        
                          1
                          +
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            σ
                          
                        
                        )
                      
                    
                  
                  ⌋
                
                ⁢
                
                  
                    (
                    
                      D
                      +
                      ɛ
                    
                    )
                  
                  T
                
                ⁢
                
                  (
                  
                    D
                    +
                    ɛ
                  
                  )
                
              
              -
              
                2
                ⁢
                
                  ασ
                  T
                
                ⁢
                σ
                ⁢
                
                  
                
                ⁢
                
                  r
                  T
                
                ⁢
                D
              
              +
            
          
        
        
          
            
              +
              
                
                  
                    
                      ρ
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                        
                        +
                        
                          Γ
                          ⁢
                          
                            
                            
                              I
                              -
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σσ
                                  T
                                
                              
                            
                            
                          
                        
                      
                      )
                    
                  
                  2
                
                [
                
                  
                    
                      r
                      T
                    
                    ⁢
                    
                      K
                      v
                      T
                    
                    ⁢
                    
                      K
                      
                        v
                        ⁢
                        
                          
                        
                      
                    
                    ⁢
                    r
                  
                  +
                  
                    
                      
                        (
                        
                          1
                          +
                          
                            
                              β
                              
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                1
                                +
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    σ
                                    T
                                  
                                  ⁢
                                  σ
                                
                              
                              )
                            
                          
                        
                        )
                      
                      2
                    
                    ⁢
                    
                      
                        (
                        
                          D
                          +
                          ɛ
                        
                        )
                      
                      T
                    
                    ⁢
                    
                      (
                      
                        D
                        +
                        ɛ
                      
                      )
                    
                  
                  +
                
              
            
          
        
      
    
  
  
    
      
        
          
            
              
                
                  β
                  
                    -
                    2
                  
                
                ⁢
                
                  Γ
                  2
                
                ⁢
                
                  
                    
                    
                      I
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σσ
                          T
                        
                      
                    
                    
                  
                  2
                
                ⁢
                
                  σ
                  T
                
                ⁢
                W
                ⁢
                
                  
                
                ⁢
                
                  W
                  T
                
                ⁢
                σ
              
              +
              
                2
                ⁢
                
                  (
                  
                    1
                    -
                    
                      
                        β
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          1
                          +
                          
                            
                              ασ
                              T
                            
                            ⁢
                            σ
                          
                        
                        )
                      
                    
                  
                  )
                
                ⁢
                
                  
                    (
                    
                      D
                      +
                      ɛ
                    
                    )
                  
                  T
                
                ⁢
                
                  K
                  
                    v
                    ⁢
                    
                      
                    
                  
                
                ⁢
                r
              
              ⁢
              
                
              
              -
              
                2
                ⁢
                
                  
                
                ⁢
                
                  β
                  
                    -
                    1
                  
                
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      
                        σσ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                W
                ⁢
                
                  
                
                ⁢
                
                  K
                  v
                
                ⁢
                r
              
              +
            
          
        
        
          
            
              
                
                  
                    -
                    2
                  
                  ⁢
                  
                    (
                    
                      1
                      +
                      
                        
                          β
                          
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            1
                            +
                            
                              
                                ασ
                                T
                              
                              ⁢
                              σ
                            
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  
                    β
                    
                      -
                      1
                    
                  
                  ⁢
                  Γ
                  ⁢
                  
                    
                    
                      I
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σσ
                          T
                        
                      
                    
                    
                  
                  ⁢
                  
                    
                      (
                      
                        D
                        +
                        ɛ
                      
                      )
                    
                    T
                  
                  ⁢
                  
                    W
                    T
                  
                  ⁢
                  σ
                
                ]
              
              +
            
          
        
        
          
            
              
                
                  +
                  
                    β
                    
                      -
                      1
                    
                  
                
                ⁢
                
                  Γ
                  2
                
                ⁢
                
                  
                    
                    
                      I
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σσ
                          T
                        
                      
                    
                    
                  
                  2
                
                ⁢
                
                  σ
                  T
                
                ⁢
                W
                ⁢
                
                  
                
                ⁢
                
                  W
                  T
                
                ⁢
                σ
              
              -
              
                2
                ⁢
                
                  
                    β
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      1
                      +
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σ
                          T
                        
                        ⁢
                        σ
                      
                    
                    )
                  
                
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      
                        σσ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  
                    (
                    
                      D
                      +
                      ɛ
                    
                    )
                  
                  T
                
                ⁢
                
                  W
                  T
                
                ⁢
                σ
              
              +
            
          
        
        
          
            
              
                
                  -
                  2
                
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  r
                  T
                
                ⁢
                
                  K
                  v
                  T
                
                ⁢
                
                  W
                  T
                
                ⁢
                σ
              
              -
              
                2
                ⁢
                Γ
                ⁢
                
                  
                  
                    I
                    -
                    
                      α
                      ·
                      
                        
                      
                      ⁢
                      σ
                      ·
                      
                        σ
                        T
                      
                    
                  
                  
                
                ⁢
                
                  
                    (
                    
                      D
                      +
                      ɛ
                    
                    )
                  
                  T
                
                ⁢
                
                  W
                  T
                
                ⁢
                σ
              
              +
            
          
        
        
          
            
              
                
                  -
                  
                    
                      
                        
                        
                          I
                          -
                          
                            α
                            ·
                            
                              
                            
                            ⁢
                            σ
                            ·
                            
                              σ
                              T
                            
                          
                        
                        
                      
                      2
                    
                    α
                  
                
                ⁢
                
                  Γ
                  ⁡
                  
                    (
                    
                      2
                      -
                      Γ
                    
                    )
                  
                
                ⁢
                
                  
                    
                    
                      W
                      ~
                    
                    
                  
                  2
                
              
              +
              
                
                  
                    
                      
                      
                        I
                        -
                        
                          α
                          ·
                          
                            
                          
                          ⁢
                          σ
                          ·
                          
                            σ
                            T
                          
                        
                      
                      
                    
                    2
                  
                  α
                
                ⁢
                
                  Γ
                  2
                
                ⁢
                
                  W
                  M
                  2
                
              
            
          
        
        
          
            Define
          
        
        
          
            
              η
              =
              
                
                  
                    
                      (
                      
                        1
                        +
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                        
                      
                      )
                    
                    ⁢
                    I
                  
                  +
                  
                    
                      
                        ρ
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              ασ
                              T
                            
                            ⁢
                            σ
                          
                          +
                          
                            Γ
                            ⁢
                            
                              
                              
                                I
                                -
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    σσ
                                    T
                                  
                                
                              
                              
                            
                          
                        
                        )
                      
                    
                    2
                  
                
                >
                
                  0
                  ⁢
                  
                    (
                    
                      condition
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        38
                        )
                      
                    
                    )
                  
                  ⁢
                  
                    
                  
                  ⁢
                  and
                
              
            
          
        
        
          
            
              γ
              =
              
                
                  η
                  +
                  
                    
                      β
                      
                        -
                        1
                      
                    
                    ⁢
                    
                      
                        ρ
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          1
                          +
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            σ
                          
                        
                        )
                      
                    
                    ⁢
                    
                      
                        (
                        
                          
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            σ
                          
                          +
                          
                            Γ
                            ⁢
                            
                              
                              
                                I
                                -
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    σσ
                                    T
                                  
                                
                              
                              
                            
                          
                        
                        )
                      
                      2
                    
                  
                
                >
                0
              
            
          
        
        
          
            
              S
              ⁢
              ubstituting
            
          
        
        
          
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                J
              
              <
              
                
                  
                    -
                    
                      
                        r
                        T
                      
                      ⁡
                      
                        [
                        
                          
                            2
                            ⁢
                            I
                          
                          -
                          
                            
                              (
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σ
                                  T
                                
                                ⁢
                                σ
                              
                              )
                            
                            2
                          
                          +
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            α
                            ⁢
                            
                              
                            
                            ⁢
                            
                              σ
                              T
                            
                            ⁢
                            σ
                            ⁢
                            
                              
                            
                            ⁢
                            
                              K
                              v
                            
                          
                          -
                          
                            
                              (
                              
                                η
                                +
                                2
                              
                              )
                            
                            ⁢
                            
                              K
                              v
                              T
                            
                            ⁢
                            
                              K
                              v
                            
                          
                        
                        ]
                      
                    
                  
                  ⁢
                  r
                
                +
                
                  2
                  ⁢
                  
                    γ
                    ·
                    
                      r
                      T
                    
                  
                  ⁢
                  
                    
                      K
                      v
                      T
                    
                    ⁡
                    
                      (
                      
                        D
                        +
                        ɛ
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  +
                  γ
                
                ⁢
                
                  ⌊
                  
                    1
                    +
                    
                      
                        β
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          1
                          +
                          
                            
                              ασ
                              T
                            
                            ⁢
                            σ
                          
                        
                        )
                      
                    
                  
                  ⌋
                
                ⁢
                
                  
                    (
                    
                      D
                      +
                      ɛ
                    
                    )
                  
                  T
                
                ⁢
                
                  (
                  
                    D
                    +
                    ɛ
                  
                  )
                
              
              +
              
                4
                ⁢
                
                  r
                  T
                
                ⁢
                
                  K
                  v
                  T
                
                ⁢
                D
              
              +
              
                2
                ⁢
                
                  D
                  T
                
                ⁢
                D
              
              -
              
                2
                ⁢
                
                  
                
                ⁢
                α
                ⁢
                
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                σ
                ⁢
                
                  
                
                ⁢
                
                  r
                  T
                
                ⁢
                D
              
              +
            
          
        
        
          
            
              
                
                  -
                  2
                
                ⁢
                Γ
                ⁢
                
                  
                    
                    
                      I
                      -
                      
                        α
                        ·
                        
                          
                        
                        ⁢
                        σ
                        ·
                        
                          σ
                          T
                        
                      
                    
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          β
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          
                            
                              ρ
                              
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    σ
                                    T
                                  
                                  ⁢
                                  σ
                                
                                +
                                
                                  Γ
                                  ⁢
                                  
                                    
                                    
                                      I
                                      -
                                      
                                        α
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          σσ
                                          T
                                        
                                      
                                    
                                    
                                  
                                
                              
                              )
                            
                          
                          2
                        
                      
                      +
                      1
                    
                    ]
                  
                
                ⁢
                
                  σ
                  T
                
                ⁢
                W
                ⁢
                
                  
                
                ⁢
                
                  K
                  
                    v
                    ⁢
                    
                      
                    
                  
                
                ⁢
                r
              
              -
            
          
        
        
          
            
              
                2
                ⁢
                Γ
                ⁢
                
                  
                    
                      
                      
                        I
                        -
                        
                          α
                          ·
                          
                            
                          
                          ⁢
                          σ
                          ·
                          
                            σ
                            T
                          
                        
                      
                      
                    
                    ⁡
                    
                      [
                      
                        1
                        +
                        
                          
                            β
                            
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              1
                              +
                              
                                
                                  ασ
                                  T
                                
                                ⁢
                                σ
                              
                            
                            )
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      1
                      +
                      
                        
                          β
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          
                            
                              ρ
                              
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    σ
                                    T
                                  
                                  ⁢
                                  σ
                                
                                +
                                
                                  Γ
                                  ⁢
                                  
                                    
                                    
                                      I
                                      -
                                      
                                        α
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          σσ
                                          T
                                        
                                      
                                    
                                    
                                  
                                
                              
                              )
                            
                          
                          2
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  
                    (
                    
                      D
                      +
                      ɛ
                    
                    )
                  
                  T
                
                ⁢
                
                  
                
                ⁢
                
                  W
                  T
                
                ⁢
                σ
              
              +
            
          
        
        
          
            
              
                
                  -
                  
                    
                      
                        
                        
                          I
                          -
                          
                            α
                            ·
                            
                              
                            
                            ⁢
                            σ
                            ·
                            
                              σ
                              T
                            
                          
                        
                        
                      
                      2
                    
                    α
                  
                
                ⁢
                
                  Γ
                  ⁡
                  
                    (
                    
                      2
                      -
                      Γ
                    
                    )
                  
                
                ⁢
                
                  
                    
                    
                      W
                      ~
                    
                    
                  
                  2
                
              
              +
              
                
                  
                    
                      
                      
                        I
                        -
                        
                          α
                          ·
                          
                            
                          
                          ⁢
                          σ
                          ·
                          
                            σ
                            T
                          
                        
                      
                      
                    
                    2
                  
                  α
                
                ⁢
                
                  Γ
                  2
                
                ⁢
                
                  W
                  M
                  2
                
              
              +
            
          
        
        
          
            
              
                
                  +
                  
                    β
                    
                      -
                      2
                    
                  
                
                ⁢
                
                  
                    
                      ρ
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                        
                        +
                        
                          Γ
                          ⁢
                          
                            
                            
                              I
                              -
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σσ
                                  T
                                
                              
                            
                            
                          
                        
                      
                      )
                    
                  
                  2
                
                ⁢
                
                  Γ
                  2
                
                ⁢
                
                  
                    
                    
                      I
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σσ
                          T
                        
                      
                    
                    
                  
                  2
                
                ⁢
                
                  σ
                  T
                
                ⁢
                W
                ⁢
                
                  
                
                ⁢
                
                  W
                  T
                
                ⁢
                σ
              
              +
              
                
                  β
                  
                    -
                    1
                  
                
                ⁢
                
                  Γ
                  2
                
                ⁢
                
                  
                    
                    
                      I
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σσ
                          T
                        
                      
                    
                    
                  
                  2
                
                ⁢
                
                  σ
                  T
                
                ⁢
                W
                ⁢
                
                  
                
                ⁢
                
                  W
                  T
                
                ⁢
                σ
              
            
          
        
        
          
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                J
              
              <
              
                
                  
                    -
                    
                      [
                      
                        2
                        -
                        
                          
                            (
                            
                              α
                              ⁢
                              
                                
                              
                              ⁢
                              
                                σ
                                T
                              
                              ⁢
                              σ
                            
                            )
                          
                          2
                        
                        +
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            K
                            
                              V
                              ⁢
                              
                                
                              
                              ⁢
                              min
                            
                          
                        
                        -
                        
                          
                            (
                            
                              η
                              +
                              2
                            
                            )
                          
                          ⁢
                          
                            K
                            
                              v
                              ⁢
                              
                                
                              
                              ⁢
                              max
                            
                            2
                          
                        
                      
                      ]
                    
                  
                  ⁢
                  
                    
                      
                      r
                      
                    
                    2
                  
                
                +
                
                  2
                  ⁢
                  
                    γ
                    ·
                    
                      
                      r
                      
                    
                  
                  ⁢
                  
                    
                      K
                      
                        v
                        ⁢
                        
                          
                        
                        ⁢
                        max
                      
                    
                    ⁡
                    
                      (
                      
                        
                          D
                          M
                        
                        +
                        
                          ɛ
                          M
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  
                    
                      
                        +
                        γ
                      
                      ⁢
                      
                        ⌊
                        
                          1
                          +
                          
                            
                              β
                              
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                1
                                +
                                
                                  
                                    ασ
                                    T
                                  
                                  ⁢
                                  σ
                                
                              
                              )
                            
                          
                        
                        ⌋
                      
                      ⁢
                      
                        
                          (
                          
                            
                              D
                              M
                            
                            +
                            
                              ɛ
                              M
                            
                          
                          )
                        
                        2
                      
                    
                    +
                    
                      4
                      ⁢
                      
                        
                        r
                        
                      
                      ⁢
                      
                        K
                        
                          v
                          ⁢
                          
                            
                          
                          ⁢
                          max
                        
                      
                      ⁢
                      
                        D
                        M
                      
                    
                    +
                    
                      2
                      ⁢
                      
                        D
                        M
                        2
                      
                    
                    +
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      α
                      ⁢
                      
                        
                      
                      ⁢
                      
                        σ
                        M
                        2
                      
                      ⁢
                      
                        
                        r
                        
                      
                      ⁢
                      
                        D
                        M
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        +
                        2
                      
                      ⁢
                      Γ
                      ⁢
                      
                        
                          
                          
                            I
                            -
                            
                              α
                              ·
                              
                                
                              
                              ⁢
                              σ
                              ·
                              
                                σ
                                T
                              
                            
                          
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                β
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                
                                  
                                    ρ
                                    
                                      -
                                      1
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        α
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          σ
                                          T
                                        
                                        ⁢
                                        σ
                                      
                                      +
                                      
                                        Γ
                                        ⁢
                                        
                                          
                                          
                                            I
                                            -
                                            
                                              α
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              
                                                σσ
                                                T
                                              
                                            
                                          
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                                2
                              
                            
                            +
                            1
                          
                          ]
                        
                      
                      ⁢
                      
                        σ
                        M
                      
                      ⁢
                      
                        W
                        M
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        K
                        
                          
                            v
                            ⁢
                            
                              
                            
                            ⁢
                            max
                          
                          ⁢
                          
                            
                          
                        
                      
                      ⁢
                      
                        
                        r
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  +
                  2
                
                ⁢
                Γ
                ⁢
                
                  
                    
                      
                      
                        I
                        -
                        
                          α
                          ·
                          
                            
                          
                          ⁢
                          σ
                          ·
                          
                            σ
                            T
                          
                        
                      
                      
                    
                    ⁡
                    
                      [
                      
                        1
                        +
                        
                          
                            β
                            
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              1
                              +
                              
                                
                                  ασ
                                  T
                                
                                ⁢
                                σ
                              
                            
                            )
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      1
                      +
                      
                        
                          ρ
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          
                            
                              β
                              
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  α
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    σ
                                    T
                                  
                                  ⁢
                                  σ
                                
                                +
                                
                                  Γ
                                  ⁢
                                  
                                    
                                    
                                      I
                                      -
                                      
                                        α
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          σσ
                                          T
                                        
                                      
                                    
                                    
                                  
                                
                              
                              )
                            
                          
                          2
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  (
                  
                    
                      D
                      M
                    
                    +
                    
                      ɛ
                      M
                    
                  
                  )
                
                ⁢
                
                  
                
                ⁢
                
                  W
                  M
                
                ⁢
                
                  σ
                  M
                
              
              +
            
          
        
        
          
            
              
                
                  -
                  
                    
                      
                        
                        
                          I
                          -
                          
                            α
                            ·
                            
                              
                            
                            ⁢
                            σ
                            ·
                            
                              σ
                              T
                            
                          
                        
                        
                      
                      2
                    
                    α
                  
                
                ⁢
                
                  Γ
                  ⁡
                  
                    (
                    
                      2
                      -
                      Γ
                    
                    )
                  
                
                ⁢
                
                  
                    
                    
                      W
                      ~
                    
                    
                  
                  2
                
              
              +
              
                
                  
                    
                      
                      
                        I
                        -
                        
                          α
                          ·
                          
                            
                          
                          ⁢
                          σ
                          ·
                          
                            σ
                            T
                          
                        
                      
                      
                    
                    2
                  
                  α
                
                ⁢
                
                  Γ
                  2
                
                ⁢
                
                  W
                  M
                  2
                
              
              +
            
          
        
        
          
            
              
                
                  +
                  
                    ρ
                    
                      -
                      1
                    
                  
                
                ⁢
                
                  
                    
                      β
                      
                        -
                        2
                      
                    
                    ⁡
                    
                      (
                      
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            T
                          
                          ⁢
                          σ
                        
                        +
                        
                          Γ
                          ⁢
                          
                            
                            
                              I
                              -
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σσ
                                  T
                                
                              
                            
                            
                          
                        
                      
                      )
                    
                  
                  2
                
                ⁢
                
                  Γ
                  2
                
                ⁢
                
                  
                    
                    
                      I
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σσ
                          T
                        
                      
                    
                    
                  
                  2
                
                ⁢
                
                  σ
                  M
                  2
                
                ⁢
                
                  W
                  M
                  2
                
              
              +
              
                
                  β
                  
                    -
                    1
                  
                
                ⁢
                
                  Γ
                  2
                
                ⁢
                
                  
                    
                    
                      I
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σσ
                          T
                        
                      
                    
                    
                  
                  2
                
                ⁢
                
                  σ
                  M
                  2
                
                ⁢
                
                  W
                  M
                  2
                
              
            
          
        
        
          
            Define
          
        
        
          
            
              
                ρ
                1
              
              =
              
                
                  2
                  -
                  
                    
                      (
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          σ
                          T
                        
                        ⁢
                        σ
                      
                      )
                    
                    2
                  
                  +
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    α
                    ⁢
                    
                      
                    
                    ⁢
                    
                      σ
                      T
                    
                    ⁢
                    σ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      K
                      
                        V
                        ⁢
                        
                          
                        
                        ⁢
                        min
                      
                    
                  
                  -
                  
                    
                      (
                      
                        η
                        +
                        2
                      
                      )
                    
                    ⁢
                    
                      K
                      
                        v
                        ⁢
                        
                          
                        
                        ⁢
                        max
                      
                      2
                    
                  
                
                >
                
                  0
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      see
                      ⁢
                      
                        
                      
                      ⁢
                      
                        conditions
                        ⁡
                        
                          (
                          33
                          )
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      and
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        35
                        )
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              
                ρ
                2
              
              =
              
                
                  Γ
                  ⁢
                  
                    
                      
                      
                        I
                        -
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σσ
                            T
                          
                        
                      
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            β
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              
                                ρ
                                
                                  -
                                  1
                                
                              
                              ⁡
                              
                                (
                                
                                  
                                    α
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      σ
                                      T
                                    
                                    ⁢
                                    σ
                                  
                                  +
                                  
                                    Γ
                                    ⁢
                                    
                                      
                                      
                                        I
                                        -
                                        
                                          α
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            σσ
                                            T
                                          
                                        
                                      
                                      
                                    
                                  
                                
                                )
                              
                            
                            2
                          
                        
                        +
                        1
                      
                      ]
                    
                  
                  ⁢
                  
                    σ
                    M
                  
                  ⁢
                  
                    W
                    M
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    K
                    
                      
                        v
                        ⁢
                        
                          
                        
                        ⁢
                        max
                      
                      ⁢
                      
                        
                      
                    
                  
                
                +
                
                  γ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    K
                    
                      v
                      ⁢
                      
                        
                      
                      ⁢
                      max
                    
                  
                  ⁢
                  
                    ɛ
                    M
                  
                
                +
              
            
          
        
        
          
            
              
                +
                
                  [
                  
                    
                      
                        (
                        
                          γ
                          +
                          2
                        
                        )
                      
                      ⁢
                      
                        K
                        
                          
                            v
                            ⁢
                            
                              
                            
                            ⁢
                            max
                          
                          ⁢
                          
                            
                          
                        
                      
                    
                    +
                    
                      ασ
                      M
                      2
                    
                  
                  ]
                
              
              ⁢
              
                D
                M
              
            
          
        
        
          
            
              
                ρ
                3
              
              =
              
                
                  γ
                  ⁢
                  
                    ⌊
                    
                      1
                      +
                      
                        
                          
                          
                            β
                            
                              -
                              1
                            
                          
                          
                        
                        ⁢
                        
                          (
                          
                            1
                            +
                            
                              
                                ασ
                                T
                              
                              ⁢
                              σ
                            
                          
                          )
                        
                      
                    
                    ⌋
                  
                  ⁢
                  
                    
                      (
                      
                        
                          D
                          M
                        
                        +
                        
                          ɛ
                          M
                        
                      
                      )
                    
                    2
                  
                
                +
                
                  2
                  ⁢
                  
                    D
                    M
                    2
                  
                
                +
              
            
          
        
        
          
            
              
                
                  +
                  2
                
                ⁢
                Γ
                ⁢
                
                  
                    
                      
                      
                        I
                        -
                        
                          α
                          ·
                          
                            
                          
                          ⁢
                          σ
                          ·
                          
                            σ
                            T
                          
                        
                      
                      
                    
                    ⁡
                    
                      [
                      
                        1
                        +
                        
                          
                            
                            
                              β
                              
                                -
                                1
                              
                            
                            
                          
                          ⁢
                          
                            (
                            
                              1
                              +
                              
                                
                                  ασ
                                  T
                                
                                ⁢
                                σ
                              
                            
                            )
                          
                        
                      
                      ]
                    
                  
                  ·
                  
                    [
                    
                      1
                      +
                      
                        
                          
                          
                            
                              ρ
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              β
                              
                                -
                                1
                              
                            
                          
                          
                        
                        ⁢
                        
                          
                            (
                            
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σ
                                  T
                                
                                ⁢
                                σ
                              
                              +
                              
                                Γ
                                ⁢
                                
                                  
                                  
                                    I
                                    -
                                    
                                      α
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        σσ
                                        T
                                      
                                    
                                  
                                  
                                
                              
                            
                            )
                          
                          2
                        
                      
                    
                    ]
                  
                  ·
                  
                    (
                    
                      
                        D
                        M
                      
                      +
                      
                        ɛ
                        M
                      
                    
                    )
                  
                
                ⁢
                
                  
                
                ⁢
                
                  W
                  M
                
                ⁢
                
                  σ
                  M
                
              
              +
            
          
        
        
          
            
              
                
                  +
                  
                    
                      
                        
                        
                          I
                          -
                          
                            α
                            ·
                            
                              
                            
                            ⁢
                            σ
                            ·
                            
                              σ
                              T
                            
                          
                        
                        
                      
                      2
                    
                    α
                  
                
                ⁢
                
                  Γ
                  2
                
                ⁢
                
                  W
                  M
                  2
                
              
              +
              
                
                  
                    
                    
                      β
                      
                        -
                        1
                      
                    
                    
                  
                  ·
                  
                    Γ
                    2
                  
                
                ⁢
                
                  
                    
                      
                      
                        I
                        -
                        
                          α
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σσ
                            T
                          
                        
                      
                      
                    
                    2
                  
                  ⁡
                  
                    [
                    
                      1
                      +
                      
                        
                          
                          
                            
                              ρ
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              β
                              
                                -
                                1
                              
                            
                          
                          
                        
                        ⁢
                        
                          
                            (
                            
                              
                                α
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  σ
                                  T
                                
                                ⁢
                                σ
                              
                              +
                              
                                Γ
                                ⁢
                                
                                  
                                  
                                    I
                                    -
                                    
                                      α
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        σσ
                                        T
                                      
                                    
                                  
                                  
                                
                              
                            
                            )
                          
                          2
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  σ
                  M
                  2
                
                ⁢
                
                  W
                  M
                  2
                
              
            
          
        
        
          
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                J
              
              <
              
                
                  
                    -
                    
                      ρ
                      1
                    
                  
                  ⁢
                  
                    
                      
                      r
                      
                    
                    2
                  
                
                +
                
                  2
                  ⁢
                  
                    ρ
                    2
                  
                  ⁢
                  
                    
                    r
                    
                  
                
                +
                
                  ρ
                  3
                
                -
                
                  
                    
                      
                        
                        
                          I
                          -
                          
                            α
                            ·
                            
                              
                            
                            ⁢
                            σ
                            ·
                            
                              σ
                              T
                            
                          
                        
                        
                      
                      2
                    
                    α
                  
                  ⁢
                  
                    Γ
                    ⁡
                    
                      (
                      
                        2
                        -
                        Γ
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                      
                        W
                        ~
                      
                      
                    
                    2
                  
                
              
            
          
        
        
          
            
              completing
              ⁢
              
                
              
              ⁢
              squares
              ⁢
              
                
              
              ⁢
              for
              ⁢
              
                
              
              ⁢
              
                
                r
                
              
            
          
        
        
          
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                J
              
              <
              
                
                  -
                  
                    
                      
                        ρ
                        1
                      
                      ⁡
                      
                        [
                        
                          
                            
                            r
                            
                          
                          -
                          
                            
                              ρ
                              2
                            
                            
                              ρ
                              1
                            
                          
                        
                        ]
                      
                    
                    2
                  
                
                +
                
                  
                    ρ
                    2
                    2
                  
                  
                    ρ
                    1
                  
                
                +
                
                  ρ
                  3
                
                -
                
                  
                    
                      
                        
                        
                          I
                          -
                          
                            α
                            ·
                            
                              
                            
                            ⁢
                            σ
                            ·
                            
                              σ
                              T
                            
                          
                        
                        
                      
                      2
                    
                    α
                  
                  ⁢
                  
                    Γ
                    ⁡
                    
                      (
                      
                        2
                        -
                        Γ
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                      
                        W
                        ~
                      
                      
                    
                    2
                  
                
              
            
          
        
        
          
            
              which
              ⁢
              
                
              
              ⁢
              is
              ⁢
              
                
              
              ⁢
              negative
              ⁢
              
                
              
              ⁢
              as
              ⁢
              
                
              
              ⁢
              long
              ⁢
              
                
              
              ⁢
              as
            
          
        
        
          
            
              
                
                  
                    
                      
                        
                        
                          I
                          -
                          
                            α
                            ·
                            
                              
                            
                            ⁢
                            σ
                            ·
                            
                              σ
                              T
                            
                          
                        
                        
                      
                      2
                    
                    α
                  
                  ⁢
                  
                    Γ
                    ⁡
                    
                      (
                      
                        2
                        -
                        Γ
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                      
                        W
                        ~
                      
                      
                    
                    2
                  
                
                >
                
                  
                    
                      ρ
                      2
                      2
                    
                    
                      ρ
                      1
                    
                  
                  +
                  
                    ρ
                    3
                  
                
              
              ⇒
              
                
                  
                  
                    W
                    ~
                  
                  
                
                >
                
                  
                    1
                    
                      
                      
                        I
                        -
                        
                          α
                          ·
                          
                            
                          
                          ⁢
                          σ
                          ·
                          
                            σ
                            T
                          
                        
                      
                      
                    
                  
                  ⁢
                  
                    
                      
                        α
                        ⁡
                        
                          (
                          
                            
                              ρ
                              2
                              2
                            
                            +
                            
                              
                                ρ
                                1
                              
                              ⁢
                              
                                ρ
                                3
                              
                            
                          
                          )
                        
                      
                      
                        
                          ρ
                          1
                        
                        ⁢
                        
                          Γ
                          ⁡
                          
                            (
                            
                              2
                              -
                              Γ
                            
                            )
                          
                        
                      
                    
                  
                
              
            
          
        
        
          
            or
          
        
        
          
            
              
                
                  
                    
                      ρ
                      1
                    
                    ⁡
                    
                      [
                      
                        
                          
                          r
                          
                        
                        -
                        
                          
                            ρ
                            2
                          
                          
                            ρ
                            1
                          
                        
                      
                      ]
                    
                  
                  2
                
                >
                
                  
                    
                      ρ
                      2
                      2
                    
                    
                      ρ
                      1
                    
                  
                  +
                  
                    ρ
                    3
                  
                
              
              ⇒
              
                
                  
                  r
                  
                
                >
                
                  
                    
                      ρ
                      2
                    
                    +
                    
                      
                        
                          ρ
                          2
                          2
                        
                        +
                        
                          
                            ρ
                            1
                          
                          ⁢
                          
                            ρ
                            3
                          
                        
                      
                    
                  
                  
                    ρ
                    1
                  
                
              
            
          
        
      
    
  




From the above results, ΔL is negative outside a compact set. According to a standard Lyapunov theorem extension, it can be concluded that the tracking error r(k), the actuator error {tilde over (τ)}(k) and the NN weights estimates {tilde over (W)}(k) are Globally Uniformly Ultimately Bounded (GUUB).
REFERENCES
The following materials are incorporated herein by reference.

    1. Pending U.S. patent application Ser. No. 09/553,601 entitled “Backlash compensation using neural network” by Selmic et al., filed Apr. 20, 2000.
    2. U.S. Pat. No. 6,185,469.
    3. U.S. Pat. No. 6,064,997.
    4. U.S. Pat. No. 5,943,660.
    5. A. R. Barron, “Universal approximation bounds for superposition of a sigmoidal function,” IEEE Trans. Inform. Theory, vol. 39, no. 3, pp. 930–945, 1993.
    6. L. A. Bernotas, P. E. Crago, and H. J. Chizeck, “Adaptive control of electrically stimulated muscle,” IEEE Trans. on Biomedical Engineering, no. 34, pp. 140–147, 1987.
    7. C. I. Byrnes and W. Lin, “Losslessness, feedback equivalence, and the global stabilization of discrete-time nonlinear systems,” IEEE Transactions of Automatic Control, vol. 39, no. 1, pp. 83–98, January 1994.
    8. J. Campos, F. L. Lewis, and R. Selmic, “Backlash compensation in discrete time nonlinear systems using dynamic inversion by neural networks: A preliminary approach,” submitted to the special issue on Developments in Intelligent Control for Industrial Applications of the Int. Journal of Adaptive Control and Signal Processing, November, 1999.
    9. J. Campos and F. L. Lewis, “Deadzone compensation in discrete time using adaptive fuzzy logic,” Proc. IEEE Conference on Decision and Control, pp. 2920–2926, Tampa, Fla., 1998.
    10. F. C. Chen and H. K. Khalil, “Adaptive control of nonlinear discrete-time systems using neural networks,” IEEE Trans. on Automatic Control, vol. 450, no. 5, pp. 791–801, May 1995.
    11. C. A. Desoer and S. M. Shahruz, “Stability of dithered nonlinear systems with backlash or hysteresis,” Int. J. Control, vol. 43, no. 4, pp. 1045–1060, 1986.
    12. M. Grundelius and D. Angelli, “Adaptive control of systems with backlash acting on the input,” Proceedings of the 35th Conference on Decision and Control, pp. 4689–4694, Kobe, Japan, 1996.
    13. V. Gullapalli, J. A. Franklin, and H. Benbrahuim, “Acquiring robot skills via reinforcement learning,” IEEE Cont. Syst., pp. 13–24, February 1994.
    14. W. M. Haddad, J. L. Fausz, and V. Chellaboina, “Optimal discrete-time control for nonlinear cascade systems,” Proceedings of the American Control Conference, pp. 2175–2176, Albuquerque, N.Mex., 1997.
    15. C. W. Han and Y. X. Zhong, “Robust adaptive control of time-varying systems with unknown backlash nonlinearity,” Proceedings of the American Control Conference, pp. 763–767, Albuquerque, N.Mex., 1997.
    16. K. Hornik, M. Stinchombe, and H. White. “Multilayer feedforward networks are universal approximators,” Neural Networks vol. 2, pp. 359–366, 1989.
    17. B. Igelnik and Y-H. Pao, “Stochastic choice of basis functions in adaptive function approximation and the functional-link net,” IEEE Trans. on Neural Networks, vol. 6, no. 6, pp. 1320–1329, November 1995.
    18. P. A. Ioannou and A. Datta, “Robust adaptive control: a unified approach, ” Proc. IEEE, vol. 790, no. 12, pp. 1736–1768, 1991.
    19. S. Jagannathan, and F. L. Lewis, “Discrete-Time Control of a Class of Nonlinear Dynamical Systems,” Int. Journal of Intelligent Control and Systems, vol. 1, no. 3, pp. 297–326, 1996.
    20. S. Jagannathan, “Robust backstepping control of robotic systems using neural networks,” Proc. Of the IEEE Conf. on Decision & Control, pp. 943–948, December 1998.
    21. J.-H. Kim, S.-W. Lee, and K.-C Kim, “Fuzzy precompensation of PID controllers,” Proc. IEEE Conf. Control Applications, pp. 183–188, September. 1993.
    22. J.-H. Kim, J.-H. Park, S.-W. Lee, and E. K. P Chong, “Fuzzy precompensation of PD controllers for systems with deadzones,” J. Int. And Fuzzy Systems, vol. 1, pp. 125–133, 1993.
    23. J.-H. Kim, J.-H. Park, S.-W. Lee, and E. K. P Chong, “A two-layered fuzzy logic controller for systems with deadzones,” IEEE Trans. Industrial Electron., vol. 41, no. 2, pp. 155–162, April. 1994.
    24. B. Kosko, Neural Networks and Fuzzy Systems, Prentice Hall, N.J., 1992.
    25. M. Krstic, I. Kanellahopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, New York, N.Y., 1995.
    26. J. Leitner, A. Calise and J. V. R. Prasad, “Analysis of adaptive neural networks for helicopter flight control,” Journal of Guidance, Control, and Dynamics, vol. 20, no. 5, pp. 972–979, September–October 1997.
    27. F. L. Lewis, C. T. Abdallah, and D. M. Dawson, Control of Robot Manipulators, MacMillan, N.Y., 1993.
    28. F. L. Lewis, S. Jagannathan, and A. Yesildirek, Neural Network Control of Robot Manipulators and Nonlinear Systems, Taylor & Francis, Philadelphia, Pa. 1999.
    29. F. L. Lewis, K. Liu, R. Selmic, and L.-X. Wang, “Adaptive fuzzy logic compensation of actuator deadzones,” J. Robotic Systems, pp. 501–511, 1997.
    30. F. L. Lewis, A. Yesildirek, and K. Liu, “Neural net robot controller with guaranteed tracking performance,” IEEE Trans. Neural Networks, vol. 6, no. 3, pp. 703–715, 1995.
    31. F. L. Lewis, A. Yesildirek, and K. Liu, “Multilayer Neural-Net Robot Controller with Guaranteed Tracking Performance,” IEEE Trans. Neural Networks, vol. 7, no. 2, pp. 388–399, 1996.
    32. M. B. McFarland and A. J. Calise, “Multilayer neural network and adaptive nonlinear control of agile anti-air missiles,” Preprint, 1999.
    33. K. S. Narendra and A. M Annaswamy, “A new adaptive law for robust adaptation without persistent excitation,” IEEE Trans. Automatic Control, vol. 32, no. 2, pp. 4–27, 1990.
    34. K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Trans. Neural Networks, vol. 4, no. 6, pp. 982–988, 1993.
    35. M. M. Polycarpou and P. A. Ioannou., “Identification and control using neural networks models: Design and stability analysis,” Technical Report 91-09-01, Department of Electrical Engineering Systems, University of Southern California, 1991.
    36. N. Sadegh, “A perceptron network for functional identification and control of nonlinear systems,” IEEE Trans. Neural Networks, vol. 4, pp. 1823–1836, 1992.
    37. R. M. Sanner and J-J. E. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Networks, vol. 3, pp. 837–863, 1992.
    38. R. Selmic and F. L. Lewis, “Backlash Compensation in Nonlinear Systems using Dynamic Inversion by Neural Networks,” To appear at the Conference in Control and Automation, Kona, Hi., August 1999.
    39. R. Selmic and F. L. Lewis, “Deadzone compensation in nonlinear systems using neural networks,” Proc. IEEE Conference Decision and Control, Tampa, Fla., 1998.
    40. J. J. E, Slotine, and W. Li, Applied Nonlinear Control, New Jersey: Prentice-Hall, 1991.
    41. J. J. E. Slotine and W. Li., “Adaptive manipulator control: a case study,” IEEE Trans. Automat. Control, vol. 33, no. 11, pp. 995–1003, November. 1988.
    42. Y. D. Song, T. L. Mitchell, and H. Y. Lai, “Control of a class of nonlinear uncertain systems via compensated inverse dynamics approach,” IEEE Trans. Automatic Control, vol. 39, no. 9, pp. 1866–1871, September 1994.
    43. G. Tao, “Adaptive backlash compensation for robot control,” Proceedings IFAC World Congress, pp. 307–312, San Francisco, 1996.
    44. G. Tao and P. V. Kokotovic, “Adaptive control of plants with unknown dead-zones,” Proceedings, American Control Conf., pp. 2710–2714, Chicago, 1992.
    45. G. Tao and P. V. Kokotovic, Adaptive Control of Systems with Actuator and Sensor Nonlinearities, John Wiley & Sons, Inc., New York, 1996.
    46. G. Tao and P. V. Kokotovic, “Continuous-time adaptive control of system with unknown backlash,” IEEE Trans. Automatic Control, vol. 40, no. 6, pp. 1083–1087, June 1995.
    47. G. Tao and P. V. Kokotovic, “Discrete-time adaptive control of systems with unknown nonsmooth input nonlinearities,” Proceedings of the 33rd Conference on Decision and Control, pp. 1171–1176, Lake Buena Vista, Fla., 1994.
    48. G. Tao and P. V. Kokotovic, “Discrete-time adaptive control of nonlinear systems with unknown deadzones,” Int. J. Control, vol. 61, no. 1, pp. 1–17, 1995.
    49. M. Vandergrift, F. L. Lewis, S. Jagannathan, and K. Liu, “Adaptive fuzzy logic control of discrete-time dynamical systems,” Proc. IEEE Int. Symp. Intelligent Control, pp. 395–401, August. 1995.
    50. P. J. Werbos, “Backpropagation through time: what it does and how to do it,” Proc. IEEE, vol. 78, no. 10, pp. 1550–1560, 1990.
    51. P. J. Werbos, “Neurocontrol and supervised learning: an overview and evaluation,” in Handbook of Intelligent Control, D. A. White and D. A. Sofge, Eds. NY: Van Nostrand Reinhold, 1992, pp. 65–89.
    52. P.-C. Yeh and P. Kokotovic, “Adaptive control of a class of nonlinear discrete-time systems,” International Journal of Control, vol. 62, no. 2, pp. 303–324, 1995.
    53. Y. Zhang and Y. C. Soh, “Robust adaptive control of uncertain discrete-time systems,” Automatica, pp. 321–329, September 1998.

## Claims

1. A discrete-time adaptive neural network compensator for compensating backlash of a mechanical system, comprising:
a feedforward path;
a proportional derivative tracking loop in the feedforward path;
a filter in the feedforward path;
a neural network in the feedforward path and coupled to the filter, the neural network configured to compensate the backlash by estimating an inverse of the backlash and applying the inverse to an input of the mechanical system; and
wherein a tracking error r(k), a backlash estimation error {tilde over (τ)}(k), and a weight estimation error {tilde over (W)}(k) of the neural network are each weighted in the same Lyapunov function.

2. The compensator of claim 1, wherein the tracking error, the backlash estimation error, and the weight estimation error are uniformly ultimately bounded.

3. The compensator of claim 1, wherein unknown backlash parameters are learned in real time.

4. The compensator of claim 1, wherein the mechanical system comprises an actuator or robot.

5. A discrete time adaptive neural network compensator for compensating backlash of a mechanical system, comprising:
a filter in a feedforward path;
a neural network in the feedforward path, the neural network configured to compensate the backlash by estimating an inverse of the backlash and applying the inverse to an input of the mechanical system; and
means for tuning the neural network in discrete time without a certainty equivalence assumption.

6. The compensator of claim 5, wherein unknown backlash parameters are learned in real time.

7. The compensator of claim 5, wherein the mechanical system comprises an actuator or robot.

8. A method for compensating backlash in a mechanical system, comprising:
estimating an inverse of the backlash using a discrete-time neural network in a feedforward path;
weighting a tracking error r(k), a backlash estimation error {tilde over (τ)}(k), and a weight estimation error {tilde over (W)}(k) of the neural network in the same Lyapunov function; and
applying the inverse to an input of the mechanical system to compensate the backlash.

9. The method of claim 8, wherein the tracking error, the backlash estimation error, and the weight estimation error are uniformly ultimately bounded.

10. The method of claim 8, wherein unknown backlash parameters are learned in real time.

11. The method of claim 8, wherein the mechanical system comprises an actuator or robot.

12. A discrete-time method of adaptively compensating backlash in a mechanical system, comprising:
estimating an inverse of the backlash using a neural network in a feedforward path;
adjusting weights of the neural network using an algorithm to achieve closed loop stability without a certainty equivalence assumption; and
applying the inverse to an input of the mechanical system to compensate the backlash.

13. The method of claim 12, wherein a tracking error, backlash estimation error, and weight estimation error are uniformly ultimately bounded.

14. The method of claim 12, wherein unknown backlash parameters are learned in real time.

15. The method of claim 12, wherein the mechanical system comprises an actuator or robot.