# Patent

## Title
Method and apparatus for adaptive cascade proportional-integral-derivative controller

## Abstract

An adaptive cascade proportional-integral-derivative controller produces a fixed controller output including a fixed proportional-integral-derivative and a fixed feedforward controller command, and an adaptive controller output including an adaptive cascade PID and an adaptive feedforward command all from a reference command. The fixed controller output and the adaptive controller output are added to produce a control command for a controlled system, which provides a measure of an output and a rate of change of the output as feedback for the controller.

## Background
RELATED APPLICATION
This U.S. patent application is related to U.S. patent application Ser. No. 12/057,814, co-filed herewith, and incorporated herein by reference.
FIELD OF THE INVENTION
This invention relates generally to controlled systems, in particular, to real-time adaptive proportional-integral-derivative (PID) controllers.
BACKGROUND OF THE INVENTION
Many controlled systems use proportional-integral-derivative (PID) controllers. However, a performance and stability of these controlled systems is sensitive to system parameters, such as inertia or stiffness and selected PID gains.
Adaptive control is one possible method for improving the performance of these controlled systems. However, adaptive control usually requires detailed process models or an approximation of these models to estimate the system parameters.
For example, U.S. Pat. No. 5,444,612 and U.S. Pat. No. 5,691,615 describe motion controllers with adaptive control based on a motion system model to estimate and compensate for inertia, damping, friction, and gravity parameters and perform model-based adaptive control. U.S. Pat. No. 6,055,524 and U.S. Pat. No. 5,687,077 describe function approximation methods, such as neural networks and Laguerre functions, to approximate the system model and estimate the corresponding parameters. Other approaches, such as U.S. Pat. No. 6,658,370, and references therein, describe some type of adaptive control by using a finite set of pre-designed sets of tuning constants, and a method to determine which set of tuning constants are optimum. That approach requires that at least one set of pre-designed tuning constants yields acceptable performance for an unknown system in operation. Other types of PID controllers use rule based adjustment of controller gains, such as fuzzy logic conditions.
Related approaches are described for adaptive parallel PID by Chang, W.-D., and J.-J Yan, “Adaptive robust PID controller design based on a sliding mode for uncertain chaotic systems,” Chaos, Solitons and Fractals, 26, pp. 167-175, 2005, Iwai, Z., Mizumoto, L., Liu, L.; Shah, S. L.; Jiang, H., “Adaptive Stable PID Controller with Parallel Feedforward Compensator,” Conference on Control, Automation, Robotics and Vision, December, 2006, Pirabakaran, K., and V. M. Bacerra, “Automatic Tuning of PID Controllers Using Model Reference Adaptive Control Techniques,” Conference of the IEEE Industrial Electronics Society, December, 2001, and Xiong, A. and Y. Fan, “Application of a PID Controller using MRAC Techniques for Control of the DC Electromotor Drive,”, IEEE International Conference on Mechatronics and Automation, August, 2007.
SUMMARY OF THE INVENTION
An adaptive cascade proportional-integral-derivative controller produces a fixed controller output including a fixed proportional-integral-derivative and a fixed feedforward controller command, and an adaptive controller output including an adaptive cascade PID and an adaptive feedforward command all from a reference command. The fixed controller output and the adaptive controller output are added to produce a control command for a controlled system, which provides a measure of an output and a rate of change of the output as feedback for the controller.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a block diagram of an adaptive cascade PID controller with feedforward control according to an embodiment of the invention;
FIG. 2 is a block diagram of an adaptive cascade PID controller according to an embodiment of the invention;
FIG. 3 is a block diagram of an adaptive cascade PID controller with overall proportional gain according to an embodiment of the invention; and
FIG. 4 is a block diagram of an adaptive cascade PID controller with overall integral gain according to an embodiment of the invention.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
As shown in FIG. 1, the embodiments of the invention provide an apparatus and method 100 for adaptive proportional-integral-derivative (PID) control with feedforward control. The method and apparatus comprise a fixed controller 110 and an adaptive controller 120. The fixed controller includes a fixed PID controller, and a fixed feedforward controller. The adaptive controller includes an adaptive cascade PID controller and an adaptive feedforward controller.
Input to the fixed and adaptive controllers is a reference command r 101. An output of the fixed controller is a fixed PID and a fixed feedforward 111. An output 121 of the adaptive cascade controller 120 is an adaptive cascade PID command 123 and an adaptive feedforward 122, see FIG. 2-4, which in combination form an adaptive cascade control command uadapt, as described in further detail below.
The fixed output 111 and the adaptive output are added 130 to form a control command u 131 for a controlled system 140. The controlled system can provide a measure of an output y 141 and a rate of change 142 of the output 141, using some sensing or approximation means 143, which are fed back to the fixed and adaptive cascade controllers. The controlled system can be any system as known in the art.
The controlled system 140 can be of dominant order n, with n=1 for first order dominant processes, such as most temperature or velocity controlled systems and n=2 for second order dominant processes, such as most position controlled systems. In the preferred embodiment, n≦2, because most controlled systems have dominant first and second order dynamics.
The following equalities are defined:

z=−(d/dt+Kpp)n−1e 

zI=∫zdt 

where t is time, Kpp>0 is a selected scalar gain, e=r−y is a tracking error for the reference command r 101.

The control command u 131 for the preferred embodiment is




  
    
      
        u
        =
        
          
            
              K
              pv
            
            ⁢
            z
          
          +
          
            
              K
              iv
            
            ⁢
            
              z
              1
            
          
          +
          
            
              K
              ff
            
            ⁢
            
              w
              ff
            
          
          +
          
            u
            adapt
          
        
      
    
    
      
        (
        
          1
          ⁢
          a
        
        )
      
    
  
  
    
      
        
          u
          adapt
        
        =
        
          
            
              
                K
                ^
              
              ff
            
            ⁢
            
              w
              ff
            
          
          +
          
            u
            a
          
        
      
    
    
      
        (
        
          1
          ⁢
          b
        
        )
      
    
  




where Kpv>0 is a fixed proportional gain, Kiv>0 is a fixed integral gain, Kff is a fixed feedforward gain. The adaptive cascade control command uadapt 121 includes an adaptive cascade PID command ua 123 and an adaptive feedforward command 122 {circumflex over (K)}ffwff. The fixed and adaptive feedforward gains Kff and {circumflex over (K)}ff, respectively are multiplied by a combined feedforward and feedback signal wff=y(n)−Ż.

For n=1, the fixed controller is a PI controller, and signal wff={dot over (r)}. For n=2, the fixed controller is a PID controller, and the signal wff={umlaut over (r)}+Kppė. The (.) and (..) superscripts of the variables denote first and second derivatives with respect to time.
The adaptive feedforward gain {circumflex over (K)}ff is updated according to:

{circumflex over ({dot over (K)}ff=−γffwffz−Lff{circumflex over (K)}ff,  (2)

where γff>0 is an adaptation gain for the adaptive feedforward gain, and Lff≧0 is a filter gain.

The adaptive PID control command ua and a method for updating the adaptive gains depends on the embodiment of the adaptive cascade PID controller, as described below for different embodiments.
Cascade PID Controller
FIG. 2 shows the adaptive cascade PID controller 200 in greater detail for one embodiment. The adaptive cascade PID command ua 123 is:

ua={circumflex over (K)}iv(∫ėdt+{circumflex over (K)}pp∫edt)+{circumflex over (K)}pv(ė+{circumflex over (K)}ppe),  (3)

where {circumflex over (K)}pp 210 is an adaptive outer proportional loop gain, {circumflex over (K)}pv 220 is an adaptive inner proportional loop gain, {circumflex over (K)}iv 230 is an adaptive integral loop gain. The module 250 is a differentiator, in which s is a Laplace variable.

The adaptive gains according to Equation (3) are updated according to




  
    
      
        
          
            
              K
              ^
            
            .
          
          pp
        
        =
        
          
            
              -
              
                
                  γ
                  pp
                
                (
                
                  
                    
                      
                        
                          K
                          ^
                        
                        pv
                      
                      2
                    
                    ⁢
                    e
                  
                  +
                  
                    
                      
                        
                          K
                          ^
                        
                        iv
                      
                      2
                    
                    ⁢
                    
                      ∫
                      
                        e
                        ⁢
                        
                          d
                          t
                        
                      
                    
                  
                
                )
              
            
            ⁢
            z
          
          -
          
            
              L
              pp
            
            ⁢
            
              
                K
                ^
              
              pp
            
          
        
      
    
    
      
        (
        4
        )
      
    
  
  
    
      
        
          
            
              K
              ^
            
            .
          
          pv
        
        =
        
          
            
              -
              
                
                  γ
                  pv
                
                ⁡
                
                  (
                  
                    
                      e
                      .
                    
                    +
                    
                      
                        
                          K
                          pp
                        
                        2
                      
                      ⁢
                      e
                    
                  
                  )
                
              
            
            ⁢
            z
          
          -
          
            
              L
              pv
            
            ⁢
            
              
                K
                ^
              
              pv
            
          
        
      
    
    
      
        (
        5
        )
      
    
  
  
    
      
        
          
            
              K
              ^
            
            .
          
          iv
        
        =
        
          
            
              -
              
                
                  γ
                  iv
                
                (
                
                  
                    ∫
                    
                      e
                      ⁢
                      
                        d
                        t
                      
                    
                  
                  +
                  
                    
                      
                        
                          K
                          ^
                        
                        pp
                      
                      2
                    
                    ⁢
                    
                      ∫
                      
                        e
                        ⁢
                        
                          d
                          t
                        
                      
                    
                  
                
                )
              
            
            ⁢
            z
          
          -
          
            
              L
              iv
            
            ⁢
            
              
                K
                ^
              
              iv
            
          
        
      
    
    
      
        (
        6
        )
      
    
  




where γpp, γpv and γiv>0 are adaptation gains for outer proportional inner proportional, and integral gains, respectively. Filter gains Lpp, Lpv, and Liv≧0 are used to adjust an adaptation response for adaptive PID gains. Therefore, the overall design of the adaptive cascade PID and feedforward controller according to Equations (1-6) in this embodiment with the adaptive cascade PID controller as shown in FIG. 2.

Cascade PID Controller with Overall Integral Gain
FIG. 3 shows an adaptive cascade controller 300 with overall integral gain. In this embodiment, the adaptive cascade PID control command ua, is

ua={circumflex over (K)}iv(∫ėdt+{circumflex over (K)}pp∫edt)+{circumflex over (K)}iv{circumflex over (K)}pvi(ė+{circumflex over (K)}ppe),  (7)

where {circumflex over (K)}pp 210 is an adaptive proportional outer loop gain, {circumflex over (K)}pvi 320 is an adaptive scaled proportional inner loop gain, {circumflex over (K)}iv 230 is an adaptive integral loop gain. The adaptive gains for adaptive cascade PID controller according to Equation (7) are updated according to





  
    
      
        
          
            
              K
              ^
            
            .
          
          pp
        
        =
        
          
            
              -
              
                γ
                pp
              
            
            ⁢
            
              
                
                  K
                  ^
                
                iv
              
              2
            
            ⁢
            
              (
              
                
                  
                    
                      
                        K
                        ^
                      
                      pvi
                    
                    3
                  
                  ⁢
                  e
                
                +
                
                  ∫
                  
                    e
                    ⁢
                    
                      d
                      t
                    
                  
                
              
              )
            
            ⁢
            z
          
          -
          
            
              L
              pp
            
            ⁢
            
              
                K
                ^
              
              pp
            
          
        
      
    
    
      
        (
        8
        )
      
    
  
  
    
      
        
          
            
              K
              ^
            
            .
          
          pvi
        
        =
        
          
            
              -
              
                γ
                pvi
              
            
            ⁢
            
              
                
                  K
                  ^
                
                iv
              
              2
            
            ⁢
            
              (
              
                e
                +
                
                  
                    
                      
                        K
                        ^
                      
                      pp
                    
                    3
                  
                  ⁢
                  e
                
              
              )
            
            ⁢
            z
          
          -
          
            
              L
              pvi
            
            ⁢
            
              
                K
                ^
              
              pvi
            
          
        
      
    
    
      
        (
        9
        )
      
    
  
  
    
      
        
          
            
              
                K
                ^
              
              .
            
            iv
          
          =
          
            
              
                -
                
                  
                    γ
                    iv
                  
                  (
                  
                    
                      ∫
                      
                        
                          e
                          .
                        
                        ⁢
                        
                          d
                          t
                        
                      
                    
                    +
                    
                      
                        
                          
                            K
                            ^
                          
                          pp
                        
                        2
                      
                      ⁢
                      
                        ∫
                        
                          e
                          ⁢
                          
                            d
                            t
                          
                        
                      
                    
                  
                  )
                
              
              ⁢
              z
            
            -
            
              
                γ
                iv
              
              ⁢
              
                
                  
                    K
                    ^
                  
                  pvi
                
                2
              
              ⁢
              
                (
                
                  
                    e
                    .
                  
                  +
                  
                    
                      
                        
                          K
                          ^
                        
                        pp
                      
                      3
                    
                    ⁢
                    e
                  
                
                )
              
              ⁢
              z
            
            -
            
              
                L
                iv
              
              ⁢
              
                
                  K
                  ^
                
                iv
              
            
          
        
        ,
      
    
    
      
        (
        10
        )
      
    
  




where γpp, γpvi, and γiv>0 are adaptation gains for outer proportional, scaled inner proportional, and integral gains, respectively. In addition, Lpp Lpvi, Liv≧0 are filter gains used to adjust adaptation response for adaptive PID gains. Therefore, the overall design for the adaptive cascade PID controller with feedforward controller 100 is given by Equations (1), (2), (7) and, (8)-(10) in this embodiment with the adaptive cascade PID as shown in FIG. 3

Cascade PID Controller with Overall Proportional Gain
FIG. 4 shows an adaptive cascade PID controller 400 with overall proportional gain. In this embodiment, the adaptive cascade PID control command is:

ua={circumflex over (K)}pv{circumflex over (K)}ivv(∫ėdt+{circumflex over (K)}pp∫edt)+{circumflex over (K)}pv(ė+{circumflex over (K)}ppe),  (11)

where {circumflex over (K)}pp 210 is an adaptive proportional outer loop gain, {circumflex over (K)}pv 220 is an adaptive proportional inner loop gain, {circumflex over (K)}ivv 430 is an adaptive scaled integral gain.

The adaptive gains for the adaptive cascade PID of Equation (11) are updated using the following adaptation Equations:




  
    
      
        
          
            
              K
              ^
            
            .
          
          pp
        
        =
        
          
            
              -
              
                γ
                pp
              
            
            ⁢
            
              
                
                  K
                  ^
                
                pv
              
              2
            
            ⁢
            
              (
              
                e
                +
                
                  
                    
                      
                        K
                        ^
                      
                      ivv
                    
                    2
                  
                  ⁢
                  
                    ∫
                    
                      e
                      ⁢
                      
                        d
                        t
                      
                    
                  
                
              
              )
            
            ⁢
            z
          
          -
          
            
              L
              pp
            
            ⁢
            
              
                K
                ^
              
              pp
            
          
        
      
    
    
      
        (
        12
        )
      
    
  
  
    
      
        
          
            
              K
              ^
            
            .
          
          pv
        
        =
        
          
            
              -
              
                γ
                pv
              
            
            ⁢
            
              
                
                  K
                  ^
                
                ivv
              
              2
            
            ⁢
            
              (
              
                
                  ∫
                  
                    
                      e
                      .
                    
                    ⁢
                    
                      d
                      t
                    
                  
                
                +
                
                  
                    
                      
                        K
                        ^
                      
                      pp
                    
                    3
                  
                  ⁢
                  
                    ∫
                    
                      e
                      ⁢
                      
                        d
                        t
                      
                    
                  
                
              
              )
            
            ⁢
            z
          
          -
          
            
              
                γ
                pv
              
              (
              
                
                  e
                  .
                
                +
                
                  
                    
                      
                        K
                        ^
                      
                      pp
                    
                    2
                  
                  ⁢
                  e
                
              
              )
            
            ⁢
            z
          
          -
          
            
              L
              pv
            
            ⁢
            
              
                K
                ^
              
              pv
            
          
        
      
    
    
      
        (
        13
        )
      
    
  
  
    
      
        
          
            
              K
              ^
            
            .
          
          ivv
        
        =
        
          
            
              -
              
                γ
                ivv
              
            
            ⁢
            
              
                
                  K
                  ^
                
                pv
              
              2
            
            ⁢
            
              (
              
                
                  ∫
                  
                    
                      e
                      .
                    
                    ⁢
                    
                      d
                      t
                    
                  
                
                +
                
                  
                    
                      
                        K
                        ^
                      
                      pp
                    
                    3
                  
                  ⁢
                  
                    ∫
                    
                      e
                      ⁢
                      
                        d
                        t
                      
                    
                  
                
              
              )
            
            ⁢
            z
          
          -
          
            
              L
              ivv
            
            ⁢
            
              
                K
                ^
              
              ivv
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




where γpp, γpvi, and γiv>0 are the adaptation gains for outer proportional, inner proportional, and scaled integral gains, respectively. In addition, Lpp, Lpvi, Liv≧0 are filter gains used to an adjust adaptation response for the adaptive PID gains. Therefore, the overall design for the adaptive cascade PID controller with feedforward controller 100 is given by Equations (1), (2), (11) and, (12)-(14) in this embodiment as shown in FIG. 4.

Design Considerations
For a class of controlled systems:

ay(n)=u,  (15)

where y(n) is the nth derivative of the output y, where n is the order of the system. An unknown constant parameter a>0 is a high frequency gain. The controller of Equations (1)-(2) is substituted into Equation (15), which yields:

aż=−Kpvz−KivzI+{tilde over (K)}ffwff+ua,

where {circumflex over (K)}ff={circumflex over (K)}ff−a+Kff is the feedforward gain estimation error. Consider the following Lyapunov potential function:

V=az2+KivzI2+γff−1{tilde over (K)}ff2+{circumflex over (K)}′Γ−1{circumflex over (K)}, 

where {circumflex over (K)} is a three element vector including the three adaptive PID gains for any of the three realizations of the adaptive control command ua described above. A diagonal adaptation gain matrix Γ includes adaptation gains, such as γpp, γpv and γiv for the realization in Equation (3), and so on. The design of the adaptive controller is based on obtaining negativity of the function derivative {dot over (V)}

{dot over (V)}=−Kpvz2+zua+{circumflex over (K)}′Γ−1{circumflex over ({dot over (K)}

Using the formulation for the adaptive PID control command ua in Equation (3), and the corresponding adaptation Equations (4)-(6) for {circumflex over (K)}, and substituting into the above Equation for {dot over (V)}, shows that {dot over (V)}≦0, and thus proves system stability.
The same procedure is repeated for the two other adaptive PID realizations in Equations (7) and (11) and their corresponding adaptation Equations to prove the stability of the system with {dot over (V)}≦0 according to the Lyapunov stability theory.
The adaptation Equations used to update the gains are obtained using the approach described above. In particular, a general formula for updating the adaptation gain vector V including the adaptive PID gains associated with an the adaptive PID term adaptive control command ua is




  
    
      
        u
        a
      
      ⁡
      
        (
        
          K
          ,
          t
        
        )
      
    
    =
    
      
        
          u
          a
        
        ⁡
        
          (
          
            0
            ,
            t
          
          )
        
      
      +
      
        
          
            K
            ^
          
          ′
        
        ⁢
        
          ∇
          
            
              u
              a
            
            ⁡
            
              (
              
                0
                ,
                t
              
              )
            
          
        
      
      +
      
        
          1
          2
        
        ⁢
        
          
            K
            ^
          
          ′
        
        ⁢
        
          
            ∇
            2
          
          ⁢
          
            
              u
              a
            
            ⁡
            
              (
              
                0
                ,
                t
              
              )
            
          
        
        ⁢
        
          K
          ^
        
      
      +
      
        
          1
          6
        
        ⁢
        
          K
          ′
        
        ⁢
        
          
            ∇
            3
          
          ⁢
          
            u
            a
          
        
        ⁢
        
          K
          ^
        
        ⁢
        
          
        
        ⁢
        
          K
          ^
        
      
    
  
  ,




where ua({circumflex over (K)}, e, ė, ∫e) is denoted by ua({circumflex over (K)}, t), where ∇ua(0, t) is the gradient, i.e., the first order derivative, of the adaptive PID control command ua with respect to {circumflex over (K)}, and evaluated at {circumflex over (K)}=0. Whereas, ∇2ua(0, t) is the Hessian, i.e., the second order derivative, of ua with respect to {circumflex over (K)}, and evaluated at {circumflex over (K)}=0. The third derivative tensor of order three is ∇3ua is independent of {circumflex over (K)} in this case.

EFFECT OF THE INVENTION
The invention provides adaptive PID controller and method for dynamically adjusting adaptive PID gains for cascade PID with coupled gain adaptation using output and output rate feedback signals and a reference command. The invention can operate without using detailed process models or their approximation for parameter estimation, or predetermined gain values as in most conventional controllers. The embodiments of the invention can use an overall proportional gain or overall integral gain. The adaptive PID controller can be used to compensate for possibly unknown and varying system parameters such as stiffness and inertia of a controlled system.
Although the invention has been described with reference to certain preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope of the invention. Therefore, it is the object of the append claims to cover all such variations and modifications as come within the true spirit and scope of the invention.

## Claims

1. An apparatus for controlling a system, comprising:
a fixed controller comprising:
a fixed proportional-integral-derivative (PID) controller; and
a fixed feedforward controller, in which the fixed PID controller and the fixed feedforward controller are configured to receive a reference command and to produce a fixed controller output comprising a fixed PID and a fixed feedforward controller command;

an adaptive cascade controller comprising:
an adaptive cascade PID controller; and
an adaptive feedforward controller, in which the adaptive cascade PID controller and the adaptive feedforward controller are configured to receive the reference command and to produce adaptive controller output comprising an adaptive cascade PID and an adaptive feedforward command; and

means for adding the fixed controller output and the adaptive cascade controller output to produce a control command for a controlled system, the controlled system providing a measure of an output and a rate of change of the output as feedback to the fixed controller and the adaptive controller, and wherein





  
    
      
        z
        =
        
          
            -
            
              
                (
                
                  
                    d
                    
                      /
                      
                        d
                        t
                      
                    
                  
                  +
                  
                    K
                    pp
                  
                
                )
              
              
                n
                -
                1
              
            
          
          ⁢
          e
        
      
    
  
  
    
      
        
          
            z
            I
          
          =
          
            ∫
            
              z
              ⁢
              
                d
                t
              
            
          
        
        ,
      
    
  




where n is a dominant order of a process, I is time, Kpp>0 is a selected scalar gain, e=r−y is a tracking error for the reference command r, and y is the output.

2. The apparatus of claim 1, in which gains of the adaptive cascade controller and the adaptive feedforward controller are adjusted dynamically.

3. The apparatus of claim 1, in which the controlled system has the dominant order n, with n=1 for first order dominant processes.

4. The apparatus of claim 1, in which the controlled system has the dominant order n, with n=2 for second order dominant processes.

5. The apparatus of claim 1, in which the control command is




  
    
      
        u
        =
        
          
            
              K
              pv
            
            ⁢
            z
          
          +
          
            
              K
              iv
            
            ⁢
            
              z
              1
            
          
          +
          
            
              K
              ff
            
            ⁢
            
              w
              ff
            
          
          +
          
            u
            adapt
          
        
      
    
  
  
    
      
        
          u
          adapt
        
        =
        
          
            
              
                K
                ^
              
              ff
            
            ⁢
            
              w
              ff
            
          
          +
          
            u
            a
          
        
      
    
  




where Kpv>0 is a fixed proportional gain, Kiv>0 is a fixed integral gain, Kff is a fixed feedforward gain, and an adaptive cascade control command uadapt includes an adaptive cascade PID command ua and the adaptive feedforward command {circumflex over (K)}ffwff, and in which the fixed feedforward gain Kff and the adaptive feedforward gains {circumflex over (K)}ff are multiplied by a combined feedforward and feedback signal wff=y(n)−ż.

6. The apparatus of claim 1, in which the adaptive cascade controller compensates for unknown and varying system parameters.

7. The apparatus of claim 5, in which the controlled system has a dominant order n=1 for first order dominant processes, and the fixed controller is a PI controller, and wff={dot over (r)}, where a superscript (.) on the variable r denote a first derivative with respect to time t.

8. The apparatus of claim 5, in which the controlled system has a dominant order n=2 for second order dominant processes, and the fixed controller is a PID controller, and wff={umlaut over (r)}+Kppė, where a superscript (..) on the variable r denote a second derivative with respect to time t.

9. The apparatus of claim 5, in which the adaptive feedforward gain {circumflex over (K)}ff is updated according to:


{circumflex over (K)}ff=−γffωffz−Lff{circumflex over (K)}ff,


where γff>0 is an adaptation gain for the adaptive feedforward gain, and Lff≧0 is a filter gain.

10. The apparatus of claim 5, in which the adaptive cascade PID control command ua




  
    
      u
      a
    
    =
    
      
        
          
            K
            ^
          
          iv
        
        ⁡
        
          (
          
            
              ∫
              
                
                  e
                  .
                
                ⁢
                
                  d
                  t
                
              
            
            +
            
              
                
                  K
                  ^
                
                pp
              
              ⁢
              
                ∫
                
                  e
                  ⁢
                  
                    d
                    t
                  
                
              
            
          
          )
        
      
      +
      
        
          
            K
            ^
          
          iv
        
        ⁢
        
          
            
              K
              ^
            
            pvi
          
          ⁡
          
            (
            
              
                e
                .
              
              +
              
                
                  
                    K
                    ^
                  
                  pp
                
                ⁢
                e
              
            
            )
          
        
      
    
  
  ,




where {circumflex over (K)}p is an adaptive outer proportional loop gain, {circumflex over (K)}pv 220 is an adaptive inner proportional loop gain, {circumflex over (K)}iv is an adaptive integral loop gain.

11. The apparatus of claim 10, in which the gains of the adaptive cascade PID control command ua are updated according to



  
    
      
        
          K
          ^
        
        .
      
      pp
    
    =
    
      
        
          -
          
            
              γ
              pp
            
            (
            
              
                
                  
                    
                      K
                      ^
                    
                    pv
                  
                  2
                
                ⁢
                e
              
              +
              
                
                  
                    
                      K
                      ^
                    
                    iv
                  
                  2
                
                ⁢
                
                  ∫
                  
                    e
                    ⁢
                    
                      d
                      t
                    
                  
                
              
            
            )
          
        
        ⁢
        z
      
      -
      
        
          L
          pp
        
        ⁢
        
          
            K
            ^
          
          pp
        
      
    
  




  
    
      
        
          K
          ^
        
        .
      
      pv
    
    =
    
      
        
          -
          
            
              γ
              pv
            
            ⁡
            
              (
              
                
                  e
                  .
                
                +
                
                  
                    
                      K
                      pp
                    
                    2
                  
                  ⁢
                  e
                
              
              )
            
          
        
        ⁢
        z
      
      -
      
        
          L
          pv
        
        ⁢
        
          
            K
            ^
          
          pv
        
      
    
  




  
    
      
        
          
            K
            ^
          
          .
        
        iv
      
      =
      
        
          
            -
            
              
                γ
                iv
              
              (
              
                
                  ∫
                  
                    e
                    ⁢
                    
                      d
                      t
                    
                  
                
                +
                
                  
                    
                      
                        K
                        ^
                      
                      pp
                    
                    2
                  
                  ⁢
                  
                    ∫
                    
                      e
                      ⁢
                      
                        d
                        t
                      
                    
                  
                
              
              )
            
          
          ⁢
          z
        
        -
        
          
            L
            iv
          
          ⁢
          
            
              K
              ^
            
            iv
          
        
      
    
    ,
  



where γpp, γpv and γiv>0 are adaptation gains for outer proportional, inner proportional, and integral gains, respectively.

12. The apparatus of claim 5, in which the adaptive cascade PID control command ua is




  
    
      u
      a
    
    =
    
      
        
          
            K
            ^
          
          iv
        
        ⁡
        
          (
          
            
              ∫
              
                
                  e
                  .
                
                ⁢
                
                  d
                  t
                
              
            
            +
            
              
                
                  K
                  ^
                
                pp
              
              ⁢
              
                ∫
                
                  e
                  ⁢
                  
                    d
                    t
                  
                
              
            
          
          )
        
      
      +
      
        
          
            K
            ^
          
          iv
        
        ⁢
        
          
            
              K
              ^
            
            pvi
          
          ⁡
          
            (
            
              
                e
                .
              
              +
              
                
                  
                    K
                    ^
                  
                  pp
                
                ⁢
                e
              
            
            )
          
        
      
    
  
  ,




where {circumflex over (K)}pp is an adaptive proportional outer loop gain, {circumflex over (K)}pvi is an adaptive scaled proportional inner loop gain, {circumflex over (K)}iv is an adaptive integral loop gain.

13. The apparatus of claim 12, the gains for the adaptive cascade PID controller command are updated according to



  
    
      
        
          K
          ^
        
        .
      
      pp
    
    =
    
      
        
          -
          
            γ
            pp
          
        
        ⁢
        
          
            
              K
              ^
            
            iv
          
          2
        
        ⁢
        
          (
          
            
              
                
                  
                    K
                    ^
                  
                  pvi
                
                3
              
              ⁢
              e
            
            +
            
              ∫
              
                e
                ⁢
                
                  d
                  t
                
              
            
          
          )
        
        ⁢
        z
      
      -
      
        
          L
          pp
        
        ⁢
        
          
            K
            ^
          
          pp
        
      
    
  




  
    
      
        
          K
          ^
        
        .
      
      pvi
    
    =
    
      
        
          -
          
            γ
            pvi
          
        
        ⁢
        
          
            
              K
              ^
            
            iv
          
          2
        
        ⁢
        
          (
          
            e
            +
            
              
                
                  
                    K
                    ^
                  
                  pp
                
                3
              
              ⁢
              e
            
          
          )
        
        ⁢
        z
      
      -
      
        
          L
          pvi
        
        ⁢
        
          
            K
            ^
          
          pvi
        
      
    
  




  
    
      
        
          
            K
            ^
          
          .
        
        iv
      
      =
      
        
          
            -
            
              
                γ
                iv
              
              (
              
                
                  ∫
                  
                    
                      e
                      .
                    
                    ⁢
                    
                      d
                      t
                    
                  
                
                +
                
                  
                    
                      
                        K
                        ^
                      
                      pp
                    
                    2
                  
                  ⁢
                  
                    ∫
                    
                      e
                      ⁢
                      
                        d
                        t
                      
                    
                  
                
              
              )
            
          
          ⁢
          z
        
        -
        
          
            γ
            iv
          
          ⁢
          
            
              
                K
                ^
              
              pvi
            
            2
          
          ⁢
          
            (
            
              
                e
                .
              
              +
              
                
                  
                    
                      K
                      ^
                    
                    pp
                  
                  3
                
                ⁢
                e
              
            
            )
          
          ⁢
          z
        
        -
        
          
            L
            iv
          
          ⁢
          
            
              K
              ^
            
            iv
          
        
      
    
    ,
  



where γpp, γpvi, and γiv>0 are adaptation gains for outer proportional, scaled inner proportional, and integral gains, respectively, Lpp, Lpvi, Liv≧0 are filter gains used to adjust an adaptation response for adaptive PID gains.

14. The apparatus of claim 5, in which the adaptive cascade PID control command is




  
    
      u
      a
    
    =
    
      
        
          
            K
            ^
          
          pv
        
        ⁢
        
          
            
              K
              ^
            
            ivv
          
          ⁡
          
            (
            
              
                ∫
                
                  
                    e
                    .
                  
                  ⁢
                  
                    d
                    t
                  
                
              
              +
              
                
                  
                    K
                    ^
                  
                  pp
                
                ⁢
                
                  ∫
                  
                    e
                    ⁢
                    
                      d
                      t
                    
                  
                
              
            
            )
          
        
      
      +
      
        
          
            K
            ^
          
          pv
        
        ⁡
        
          (
          
            
              e
              .
            
            +
            
              
                
                  K
                  ^
                
                pp
              
              ⁢
              e
            
          
          )
        
      
    
  
  ,




where {circumflex over (K)}pp is an adaptive proportional outer loop gain, {circumflex over (K)}pv is an adaptive proportional inner loop gain, {circumflex over (K)}ivv is an adaptive scaled integral gain.

15. The apparatus of claim 14, in which adaptive gains for the adaptive cascade PID control command are updated according to



  
    
      
        
          K
          ^
        
        .
      
      pp
    
    =
    
      
        
          -
          
            γ
            pp
          
        
        ⁢
        
          
            
              K
              ^
            
            pv
          
          2
        
        ⁢
        
          (
          
            e
            +
            
              
                
                  
                    K
                    ^
                  
                  ivv
                
                2
              
              ⁢
              
                ∫
                
                  e
                  ⁢
                  
                    d
                    t
                  
                
              
            
          
          )
        
        ⁢
        z
      
      -
      
        
          L
          pp
        
        ⁢
        
          
            K
            ^
          
          pp
        
      
    
  




  
    
      
        
          K
          ^
        
        .
      
      pv
    
    =
    
      
        
          -
          
            γ
            pv
          
        
        ⁢
        
          
            
              K
              ^
            
            ivv
          
          2
        
        ⁢
        
          (
          
            
              ∫
              
                
                  e
                  .
                
                ⁢
                
                  d
                  t
                
              
            
            +
            
              
                
                  
                    K
                    ^
                  
                  pp
                
                3
              
              ⁢
              
                ∫
                
                  e
                  ⁢
                  
                    d
                    t
                  
                
              
            
          
          )
        
        ⁢
        z
      
      -
      
        
          
            γ
            pv
          
          (
          
            
              e
              .
            
            +
            
              
                
                  
                    K
                    ^
                  
                  pp
                
                2
              
              ⁢
              e
            
          
          )
        
        ⁢
        z
      
      -
      
        
          L
          pv
        
        ⁢
        
          
            K
            ^
          
          pv
        
      
    
  




  
    
      
        
          
            K
            ^
          
          .
        
        ivv
      
      =
      
        
          
            -
            
              γ
              ivv
            
          
          ⁢
          
            
              
                K
                ^
              
              pv
            
            2
          
          ⁢
          
            (
            
              
                ∫
                
                  
                    e
                    .
                  
                  ⁢
                  
                    d
                    t
                  
                
              
              +
              
                
                  
                    
                      K
                      ^
                    
                    pp
                  
                  3
                
                ⁢
                
                  ∫
                  
                    e
                    ⁢
                    
                      d
                      t
                    
                  
                
              
            
            )
          
          ⁢
          z
        
        -
        
          
            L
            ivv
          
          ⁢
          
            
              K
              ^
            
            ivv
          
        
      
    
    ,
  



where γpp, γpvi, and γiv>0 are adaptation gains for outer proportional, inner proportional, and scaled integral gains, respectively, Lpp, Lpvi, Liv≧0 are filter gains used to adjust an adaptation response for adaptive PID gains.

16. A method for controlling a system, comprising the steps of:
producing a fixed controller output comprising a fixed proportional-integral-derivative (PID) a fixed feedforward controller command from a reference command;
producing an adaptive controller output comprising an adaptive cascade PID and an adaptive feedforward command from the reference command; and
adding the fixed controller output and the adaptive controller output to produce a control command for a controlled system providing a measure of an output and a rate of change of the output as feedback for the producing steps, and wherein





  
    
      
        z
        =
        
          
            -
            
              
                (
                
                  
                    d
                    
                      /
                      
                        d
                        t
                      
                    
                  
                  +
                  
                    K
                    pp
                  
                
                )
              
              
                n
                -
                1
              
            
          
          ⁢
          e
        
      
    
  
  
    
      
        
          
            z
            I
          
          =
          
            ∫
            
              z
              ⁢
              
                d
                t
              
            
          
        
        ,
      
    
  




where n is a dominant order of a process, t is time, Kpp>0 is a selected scalar gain, e=r−y is a tracking error for the reference command r, and y is the output.