Patent Publication Number: US-9893666-B2

Title: Method and system for controlling angular rotor speeds of sensorless induction motors

Description:
RELATED APPLICATION 
     The U.S. Patent Application is related to U.S. Ser. No. 14/612,429, co-filed herewith on Feb. 3, 2015, and incorporated herein. Both Applications disclose a method and system for controlling the angular rotor speed of sensorless induction motors. 
     FIELD OF THE INVENTION 
     This invention relates generally to controlling an angular rotor speed of induction motors, and more particularly to controlling sensorless variable speed induction motors. 
     BACKGROUND OF THE INVENTION 
     Methods for controlling the angular rotor speed of induction motors have evolved from variable frequency control to vector control and other variants, e.g., direct and indirect field oriented state feedback control, sensorless speed control, and adaptive field oriented control. Vector control with a full state or rotor speed measurements can provide adequate performance at the expenses of additional sensors, which can limit their application in practice. 
     Sensorless speed control includes sensorless adaptive control, where the rotor speed is typically treated as an unknown parameter to avoid nonlinearity in dynamics. Controls relying on parameter assumptions generally have unsatisfactory transient performance inherent to the adaptation. 
     Other methods that use a high gain state estimator, sliding mode estimator, or an extended Kalman filter (EKF) avoid the parameter assumptions, but fail to address performance. As an example, resorting to nonlinear estimator techniques entails a system in certain normal forms, which turns out to be difficult. Well-known high gain estimators designs assume an observable canonical form. 
       FIG. 1  shows a prior art sensorless speed motor drive for an induction motor. The reference numerals for  FIG. 1  are:
           101  Speed control     102  Flux control     103  Current control     104  Clarke Transformation/Power electronics     105  Induction motor     106  Flux estimator     107  Speed estimator     111  desired speed     112  estimated speed     113  estimated speed error     114  desired q-axis current     115  estimated/measured q-axis current     116  q-axis current tracking error     117  desired flux modulus     118  estimated flux modulus     119  estimated flux error     120  desired d-axis current     121  estimated/measured d-axis current     122  d-axis current tracking error     123  desired voltage input commands     124  measured voltage inputs     125  current measurements     126  estimated flux.       

     Input to the motor drive is a reference rotor flux amplitude signal  111 . An estimate  112  from a flux estimator block  106  is added to the signal  111  so that the signal  113  represents a difference between signals  111  and  112 . 
     A flux control block  101  determines a stator current  114  used to control the rotor flux linkage in the d-axis. A signal  115  is an estimate or true stator current, in the d-axis, produced by a flux estimator  106 . A difference  116  between the signals  115  and  114  is used by a current control block  103  to determine a reference stator voltage  123  in the d-axis. Similarly, a signal  117  denotes the desired rotor speed reference of the induction motor. 
     A signal  118  denotes an estimated rotor speed produced by a speed estimator  107  based on output signal  126  of the flux estimator  106 . A difference  119  between signals  117  and  118  is used to determine a reference stator current  120 , in the q-axis, by the flux control block  102 . 
     An estimated or true stator current  121 , in the q-axis, is compared to the reference stator current  120 , in an imaginary q-axis used to control the torque of the motor, to produce a difference signal  122 . The current control block  103  determines the stator voltage signal  123 , in d- and q-axes, on the basis of difference signals  116  and  122 . A Clarke or Park transformation  104  converts the desired stator voltage, in d- and q-axes, into three-phase voltages  124  to drive the induction motor  105 . 
     Note that the flux estimator  106  takes the three-phase voltages  124  and sensed phase currents  125  as input signals, and outputs estimated or true stator currents  115  and  121 , estimated rotor flux amplitude  112 , and estimated rotor speed signal  118  to produce the difference signals  113 ,  116 ,  119 , and  122 . The estimated stator current and the estimated rotor flux amplitude, and the estimated rotor speed can be determined independently. The signal  119  is used for speed control  102 . 
     SUMMARY OF THE INVENTION 
     The embodiments of the invention provide a method and system for controlling an angular rotor speed of an induction motor without measuring the angular rotor speed. With a general state estimator, the tracking control explicitly takes parametric uncertainties and state estimation errors into consideration by backstepping-based adaptive control. 
     In control theory, state estimators are also known as state observers. The state observer provides an estimate of internal states of a given system from measurements of the input and output of the system. Herein, the terms estimator and observer are used interchangeably. 
     The method results in the tracking control where the ranges of gains are practical. Thus, the method can be used by motor drives for high bandwidth speed tracking tasks. Moreover, the gain selection for the controller is relatively straightforward. An extended Kalman filter (EKF) is used to estimate the state of the induction motor. 
     Different from prior art, where the EKF design is typically based on a model of an induction motor model a constant rotor speed, the invention considers and validates an EKF state estimator for the induction motor that can operate at variable speeds. The state estimation errors are explicitly dealt with back-stepping control design, so that the final control accuracy is guaranteed. There is no assumption on time-scale separation for both the estimation and the control design, so that high overall bandwidth is achieved even with practical constrains. 
     The method can achieve a high bandwidth, e.g., 200 Hz, and high precision speed tracking of the induction motor under practical implementation constraints, e.g., a 20 kHz sampling rate and a 400 V input saturation. The gain selection for the method is straightforward and does not depend on trajectories. The method can track various trajectories as long as the state and control input constraints are considered. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of a prior art sensorless speed motor drive and induction motor; 
         FIG. 2A  is a block diagram of a sensorless speed motor drive and induction motor according to embodiments of the invention; 
         FIG. 2B  is an alternative embodiment of the sensorless speed motor drive and induction motor according to embodiments of the invention; 
         FIG. 3  is a flow diagram of model compensation and stabilizing for speed control according to embodiments of the invention; 
         FIG. 4A  is a flow diagram of torque control according to embodiments of the invention; 
         FIG. 4B  is a flow diagram of a flux control outer loop according to embodiments of the invention; 
         FIG. 4C  is a flow diagram of a flux control inner loop according to embodiments of the invention; and 
         FIG. 5  is a schematic a flux and speed estimator according to embodiments of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The embodiments of the invention provide a method and system for controlling an angular rotor speed of an induction motor based on sensing stator currents in the motor. 
     Model of Induction Motor 
     With the assumption of linear magnetic circuits and balanced operating conditions, a two-phase equivalent model of an induction motor, represented in a fixed a-b reference frame, can be written as 
                       ω   .     =       μ   ⁡     (         ϕ   ra     ⁢     i   sb       -       ϕ   rb     ⁢     i   sa         )       -         T   L     ⁡     (   ω   )       /   J         ⁢     
     ⁢       i   sa     =       α   ⁢           ⁢   β   ⁢           ⁢     ϕ   ra       +       n   p     ⁢   β   ⁢           ⁢   ω   ⁢           ⁢     ϕ   rb       -     γ   ⁢           ⁢     i   sa       +       1   σ     ⁢     u   sa           ⁢     
     ⁢       i   sb     =       α   ⁢           ⁢   β   ⁢           ⁢     ϕ   rb       -       n   p     ⁢   β   ⁢           ⁢   ω   ⁢           ⁢     ϕ   ra       -     γ   ⁢           ⁢     i   sb       +       1   σ     ⁢     u   sb           ⁢     
     ⁢         ϕ   .     ra     =         -   α     ⁢           ⁢     ϕ   ra       -       n   p     ⁢   ω   ⁢           ⁢     ϕ   rb       +     α   ⁢           ⁢     Mi   sa           ⁢     
     ⁢         ϕ   .     rb     =         -   α     ⁢           ⁢     ϕ   rb       +       n   p     ⁢   ω   ⁢           ⁢     ϕ   ra       +     α   ⁢           ⁢     Mi   sb           ⁢     
     ⁢       y   =       (       i   sa     ,     i   sb       )     T       ,             (   1   )               
where r and s represent the rotor and stator, a and b represent the a and b axis, ω is the angular speed of the rotor, i denotes the current, φ denotes the flux linkage, u is the input stator voltage, R, L, M, J, T L (ω) and n p  denote the resistance, inductance, mutual inductance, rotor inertia, load torque, and number of pole pairs, respectively, and
 
     
       
         
           
             
               α 
               = 
               
                 
                   R 
                   r 
                 
                 
                   L 
                   r 
                 
               
             
             , 
             
               β 
               = 
               
                 M 
                 
                   σ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     L 
                     r 
                   
                 
               
             
             , 
             
               σ 
               = 
               
                 
                   L 
                   s 
                 
                 - 
                 
                   
                     M 
                     2 
                   
                   
                     L 
                     r 
                   
                 
               
             
             , 
             
               γ 
               = 
               
                 
                   
                     
                       M 
                       2 
                     
                     ⁢ 
                     
                       R 
                       r 
                     
                   
                   + 
                   
                     
                       L 
                       r 
                       2 
                     
                     ⁢ 
                     
                       R 
                       s 
                     
                   
                 
                 
                   σ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     L 
                     r 
                     2 
                   
                 
               
             
             , 
             
               μ 
               = 
               
                 
                   3 
                   2 
                 
                 ⁢ 
                 
                   
                     
                       
                         n 
                         p 
                       
                       ⁢ 
                       M 
                     
                     
                       JL 
                       r 
                     
                   
                   . 
                 
               
             
           
         
       
     
     For simplicity, the load torque T L  is assumed to be parameterized as T L (ω)=T 0 +C f ω with T 0  and C f  constants. With the following change of variables 
     x 1 =ω, x 2 =i sa , x 3 =i sb , x 4 =φ ra , x 5 =φ rb , u 1 =u sa , u 2 =u sb , 
               b   =     1   σ       ,       a   1     =   μ     ,       a   2     =       T   0     J       ,       a   3     =       C   f     J       ,         
a 5 =αβ, a 6 =n p β, a 7 =γ, a 8 =α, a 9 =n p , a 10 =αM,
 
equation (1) can be rewritten as
 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       . 
                     
                     = 
                     
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             x 
                             , 
                             u 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ℊ 
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         + 
                         Bu 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       y 
                       = 
                       Cx 
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         ℊ 
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                     a 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           x 
                                           3 
                                         
                                         ⁢ 
                                         
                                           x 
                                           4 
                                         
                                       
                                       - 
                                       
                                         
                                           x 
                                           2 
                                         
                                         ⁢ 
                                         
                                           x 
                                           5 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   a 
                                   2 
                                 
                                 - 
                                 
                                   
                                     a 
                                     3 
                                   
                                   ⁢ 
                                   
                                     x 
                                     1 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     a 
                                     5 
                                   
                                   ⁢ 
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     6 
                                   
                                   ⁢ 
                                   
                                     x 
                                     1 
                                   
                                   ⁢ 
                                   
                                     x 
                                     5 
                                   
                                 
                                 - 
                                 
                                   
                                     a 
                                     7 
                                   
                                   ⁢ 
                                   
                                     x 
                                     2 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     a 
                                     5 
                                   
                                   ⁢ 
                                   
                                     x 
                                     5 
                                   
                                 
                                 - 
                                 
                                   
                                     a 
                                     6 
                                   
                                   ⁢ 
                                   
                                     x 
                                     1 
                                   
                                   ⁢ 
                                   
                                     x 
                                     4 
                                   
                                 
                                 - 
                                 
                                   
                                     a 
                                     7 
                                   
                                   ⁢ 
                                   
                                     x 
                                     3 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     - 
                                     
                                       a 
                                       8 
                                     
                                   
                                   ⁢ 
                                   
                                     x 
                                     4 
                                   
                                 
                                 - 
                                 
                                   
                                     a 
                                     9 
                                   
                                   ⁢ 
                                   
                                     x 
                                     1 
                                   
                                   ⁢ 
                                   
                                     x 
                                     5 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     10 
                                   
                                   ⁢ 
                                   
                                     x 
                                     2 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     - 
                                     
                                       a 
                                       8 
                                     
                                   
                                   ⁢ 
                                   
                                     x 
                                     5 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     9 
                                   
                                   ⁢ 
                                   
                                     x 
                                     1 
                                   
                                   ⁢ 
                                   
                                     x 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     a 
                                     10 
                                   
                                   ⁢ 
                                   
                                     x 
                                     3 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                       B 
                       = 
                       
                         [ 
                         
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               b 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               b 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       C 
                       = 
                       
                         [ 
                         
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         with 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         x 
                       
                       = 
                       
                         
                           [ 
                           
                             
                               x 
                               1 
                             
                             , 
                             
                               x 
                               2 
                             
                             , 
                             
                               x 
                               3 
                             
                             , 
                             
                               x 
                               4 
                             
                             , 
                             
                               x 
                               5 
                             
                           
                           ] 
                         
                         T 
                       
                     
                     , 
                     
                       u 
                       = 
                       
                         
                           
                             [ 
                             
                               
                                 u 
                                 1 
                               
                               , 
                               
                                 u 
                                 2 
                               
                             
                             ] 
                           
                           T 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Problem Formulation 
     Many sensorless speed motor drives use vector control to regulate two variables: the rotor speed x 1  and the amplitude x 4   2 +x 5   2  of the rotor flux. The invention synthesizes the control input u such that x 1  and x 4   2 +x 5   2  track the angular speed ω d  (t), and the flux linkage φ d   2 , respectively using the induction motor model of equation (2). 
     General State Estimator 
     A general state estimator can be expressed as
 
 {circumflex over ({dot over (x)})}={circumflex over (f)}+O ( {tilde over (y)} )
 
 ŷ=C{circumflex over (x)}   (3)
 
where {circumflex over (x)}ε   5 , {circumflex over (f)}=f({circumflex over (x)}, u), {tilde over (y)}=y−ŷ, and O is an output injection term. Equation (3) is an abstraction of well-known state estimators. For example, extended Kalman filter (EKF) and Luenberger based estimators correspond to the following expression of the vector field I as
 
 O   i ( {tilde over (x)}   2   ,{tilde over (x)}   3 )= l   i1   {tilde over (y)}   1   +l   i2   {tilde over (y)}   2 ,1≦ i≦ 5,
 
     and a sliding mode observer (SMO) corresponds to
 
 O   i ( {tilde over (x)}   2   ,{tilde over (x)}   3 )= h   i1   S ( {tilde over (y)}   1 )+ h   i2   S ( {tilde over (y)}   2 ),1≦ i ≦5.
 
     We define estimation errors as {tilde over (x)}=x−{circumflex over (x)}. Given information of previous states, we can construct a projection mapping to bound the state estimates
 
 {circumflex over (x)}   i =Proj {circumflex over (x)}     i   (τ i )
 
τ i   ={circumflex over (f)}   i   +O   i   (4)
 
where {circumflex over (f)} i  is the i th  component of {circumflex over (f)}. The projection mapping, when applied to a scalar, is
 
     
       
         
           
             
               
                 
                   
                     
                       Proj 
                       
                         
                           x 
                           ^ 
                         
                         i 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         • 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             0 
                           
                           
                             
                               
                                 if 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   i 
                                 
                               
                               = 
                               
                                 
                                   
                                     x 
                                     imax 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   and 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     • 
                                     i 
                                   
                                 
                                 &gt; 
                                 0 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               
                                 if 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   i 
                                 
                               
                               = 
                               
                                 
                                   
                                     x 
                                     imin 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   and 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     • 
                                     i 
                                   
                                 
                                 &lt; 
                                 0 
                               
                             
                           
                         
                         
                           
                             
                               • 
                               i 
                             
                           
                           
                             otherwise 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     When applied to a vector {circumflex over (x)}, the component-wise projection mapping is
 
Proj {circumflex over (x)} (•)=[Proj {circumflex over (x)}     1   (• 1 ), . . . ,Proj {circumflex over (x)}     5   (• 5 )] T .
 
     For any function τ i , the projection mapping of equation (5) guarantees 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       ^ 
                     
                     ∈ 
                     
                       Ω 
                       x 
                     
                   
                   ⁢ 
                   
                     = 
                     Δ 
                   
                   ⁢ 
                   
                     
                       { 
                       
                         
                           
                             x 
                             ^ 
                           
                           ⁢ 
                           
                             : 
                           
                           ⁢ 
                           
                             x 
                             imin 
                           
                         
                         ≤ 
                         
                           x 
                           ^ 
                         
                         ≤ 
                         
                           x 
                           imax 
                         
                       
                       } 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     A slowly varying load torque can be treated by augmenting the state x with a 2 , which is adapted in the estimator. Other slowly changing quantities, R r , a 5  a 7  a 8  a 10 , can be treated in a similar manner. 
     State Estimator 
     We select the EKF as a state estimator to avoid a complicated gain tuning process. The idea of using EKF and its variants in speed sensorless motor drives are not new. Numerous methods use the EKF with a simplified induction motor model and a constant rotor speed, i.e., as expressed in equation (1) with {dot over (ω)}=0, see Kim et al., “Speed sensorless vector control of induction motor using extended Kalman filter,” IEEE Trans. Ind. Appl., vol. 30, no. 5, pp. 1225-1233, September/October 1994, and Barut et al. “Speed-sensorless estimation for induction motors using extended Kalman filter,” IEEE Trans. Ind. Electron., vol. 54, no. 1, pp. 272-280, January 2007. 
     Using a simplified induction motor model does simplify the structure of the EKF estimator. Because a simplified conventional induction motor model implicitly assumes speed as a parameter, the corresponding EKF necessarily performs as an adaptive estimator, which generally yields slowly convergent estimation error dynamics. 
     We reduce the effect of this limitation by considering the EKF estimator based on the full induction motor model of equation (2). 
     With the sampling rate of the filter denoted by T s  and the forward difference method, the system of equation (2) is discretized as
 
 x   k   =T   s   f ( x   k-1   ,u   k-1 )+ x   k-1   =f ′( x   k-1   ,u   k-1 ).  (7)
 
     Linearization of the dynamics is expressed as 
     
       
         
           
             
               A 
               = 
               
                 
                   
                     ∂ 
                     f 
                   
                   
                     ∂ 
                     x 
                   
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           - 
                           
                             a 
                             3 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               a 
                               1 
                             
                           
                           ⁢ 
                           
                             x 
                             5 
                           
                         
                       
                       
                         
                           
                             a 
                             1 
                           
                           ⁢ 
                           
                             x 
                             4 
                           
                         
                       
                       
                         
                           
                             a 
                             1 
                           
                           ⁢ 
                           
                             x 
                             3 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               a 
                               1 
                             
                           
                           ⁢ 
                           
                             x 
                             2 
                           
                         
                       
                     
                     
                       
                         
                           
                             a 
                             6 
                           
                           ⁢ 
                           
                             x 
                             5 
                           
                         
                       
                       
                         
                           - 
                           
                             a 
                             7 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           a 
                           5 
                         
                       
                       
                         
                           
                             a 
                             6 
                           
                           ⁢ 
                           
                             x 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           
                             - 
                             
                               a 
                               6 
                             
                           
                           ⁢ 
                           
                             x 
                             4 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           - 
                           
                             a 
                             7 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               a 
                               6 
                             
                           
                           ⁢ 
                           
                             x 
                             1 
                           
                         
                       
                       
                         
                           a 
                           5 
                         
                       
                     
                     
                       
                         
                           
                             - 
                             
                               a 
                               9 
                             
                           
                           ⁢ 
                           
                             x 
                             5 
                           
                         
                       
                       
                         
                           a 
                           10 
                         
                       
                       
                         0 
                       
                       
                         
                           - 
                           
                             a 
                             8 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               a 
                               9 
                             
                           
                           ⁢ 
                           
                             x 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           
                             a 
                             9 
                           
                           ⁢ 
                           
                             x 
                             4 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           a 
                           10 
                         
                       
                       
                         
                           
                             a 
                             9 
                           
                           ⁢ 
                           
                             x 
                             1 
                           
                         
                       
                       
                         
                           - 
                           
                             a 
                             8 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
           
         
       
     
     hence, A k =T s A+I with the identity matrix I. 
     Predictions 
     A predicted state estimate is
 
 {circumflex over (x)}   k|k-1   =f ′( {circumflex over (x)}   k-1|k-1   ,u   k|k-1 ), and  (8)
 
     a predicted covariance estimate is
 
 P   k|k-1   =A   k-1   P   k-1|k-1   A   k-1   T   +Q   k-1 ,  (9)
 
     Update 
     A measurement residual is
 
 {tilde over (y)}   k   =y   k   −C{tilde over (x)}   k|k-1 ,  (10)
 
     a residual covariance is
 
 S   k   =C   k   P   k|k-1   C   k   T   +R   k ,  (11)
 
     a near-optimal Kalman gain is
 
 K   k   =P   k|k-1   C   k   T   S   k   −1 ,  (12)
 
     an updated state estimate is
 
 {circumflex over (x)}   k|k   ={circumflex over (x)}   k|k-1   +K   k   {tilde over (y)}   k ,  (13)
 
     an updated covariance estimate is
 
 P   k|k =( I−K   k   C   k ) P   k|k-1 , and  (14)
 
     for the general observer form 
     
       
         
           
             
               
                 
                   L 
                   = 
                   
                     
                       
                         K 
                         k 
                       
                       Ts 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Integrated Control Design 
     Assumptions and General Treatment 
     We consider the general case where a system is subject to parametric uncertainties θε   p , and uncertain nonlinearities {tilde over (d)}. The following practical assumption can be used in a speed control step described below. 
     Assumptions 
     Both parametric and nonlinear uncertainties are bounded, i.e., 
                       θ   ∈     Ω   θ       ⁢     =   Δ     ⁢     {       θ   ⁢     :     ⁢     θ   min       ≤   θ   ≤     θ   max       }       ,   and           (   16   )                     d   ~     ∈     Ω   d       ⁢     =   Δ     ⁢     {         d   ~     ⁢     :     ⁢          d   ~            ≤     δ   d       }       ,           (   17   )               
where {tilde over (d)}=d−{circumflex over (d)}, θ min =[θ 1min , . . . , θ pmin ] T , θ max =[θ 1max , . . . , θ pmax ] T  are known constants. The operation ≦ for two vectors is performed componentwise, and δ d  is a known bounding function.
 
     Parametric uncertainties are treated by adaptation and projection. Let {circumflex over (θ)} denote the estimate of θ and {tilde over (θ)} so that an estimation error is {tilde over (θ)}=θ−{circumflex over (θ)}). In view of equation (16), the following adaptation law with discontinuous projection modification can be used
 
{circumflex over (θ)}=Proj {circumflex over (θ)} (Γτ),  (18)
 
where Γ&gt;0 is a diagonal matrix, τε   p  is a vector of adaptation functions synthesized later. The projection mapping
 
Proj {circumflex over (θ)} (•)=[Proj {circumflex over (θ)}     1   (• 1 ), . . . ,Proj {circumflex over (θ)}     p   (• p )] T  
 
is taken as follows
 
     
       
         
           
             
               
                 
                   
                     
                       Proj 
                       
                         
                           θ 
                           ^ 
                         
                         i 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         • 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             0 
                           
                           
                             
                               
                                 if 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     θ 
                                     ^ 
                                   
                                   i 
                                 
                               
                               = 
                               
                                 
                                   
                                     θ 
                                     imax 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   and 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     • 
                                     i 
                                   
                                 
                                 &gt; 
                                 0 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               
                                 if 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     θ 
                                     ^ 
                                   
                                   i 
                                 
                               
                               = 
                               
                                 
                                   
                                     θ 
                                     imin 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   and 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     • 
                                     i 
                                   
                                 
                                 &lt; 
                                 0 
                               
                             
                           
                         
                         
                           
                             
                               • 
                               i 
                             
                           
                           
                             otherwise 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     It can be shown that for any τ, the adaption law in equations (18)-(19) ensures equation (16) and
 
{tilde over (θ)} T (Γ −1 Proj {circumflex over (θ)} (Γτ)−τ)≦0,∀τ.  (20)
 
     For the induction motor case, parametric uncertainties can be due to the slowly time-varying resistance and load torque, and uncertain nonlinearities can be result from the state estimation error {tilde over (x)}=x−{circumflex over (x)}. 
     When restricted to the induction motor case, state estimation error {tilde over (x)} is bounded as 
                         x   ~     ∈     Ω   x       ⁢     =   Δ     ⁢     {         x   ~     ⁢     :     ⁢       x   ~     min       ≤     x   ~     ≤       x   ~     max       }       ,           (   21   )               
where {tilde over (x)} min =[{tilde over (x)} 1min , . . . , {tilde over (x)} 5min ] T  and {tilde over (x)} max =[{tilde over (x)} 1max , . . . , {tilde over (x)} 5max ] T  Similarly, the boundedness of the state estimation error is guaranteed by equations (4)-(5).
 
     Adaptive Backstepping Induction Motor Control 
       FIG. 2A  shows the overall architecture of the speed sensorless control method and system.  FIG. 2B  shows an alternative embodiment with a combined flux and speed estimator  209 . The control design is performed using backstepping technique, and includes four steps: 
     Step 1—Speed Control; 
     Step 2—Torque Control; 
     Step 3—Flux Outer Loop Control; and 
     Step 4—Flux Inner Loop Control. 
     The first step is the speed control loop outputs an electromagnetic torque as a first virtual control signal. The second step is the torque control loop which regulates the electromagnetic torque. The third step is the flux control outer loop, and the fourth step is the flux control inner loops, which regulate the magnetic flux magnitude. State estimation errors are explicitly taken into account within all control design steps under the assumption described above. 
     The reference numerals for  FIGS. 2A and 2B  are:
           201  Speed control (step 1)     202  Torque control (step 2)     203  Flux control outer loop (step 3)     204  Flux control inner loop (step 4)     104  Clarke Transformation/Power electronics     105  Induction motor     106  Flux estimator     107  Speed estimator     208  Control transformation     111  desired angular rotor speed     112  estimated angular rotor speed     113  estimated angular rotor speed error     214  desired torque     215  estimated torque     216  estimated torque error     117  desired flux modulus     118  estimated flux modulus     119  estimated flux error     124  measured voltage inputs     125  current measurement     126  estimated flux     127  input voltage command.     209  flux/speed estimator     220  flux loop virtual control     221  estimated flux loop control     222  flux loop virtual control error     223 - 224  combined voltage inputs       

     The components of the motor drive can be implemented as hardware circuits or other electronic components. 
     Comparison with Prior Art 
     Before describing the structure of the controller according to the embodiments of this invention, details of the differences with the prior art controller are noted. The prior art controller focuses on the current  123  produced by the current control  103  for the transformation power electronics  104 , which is not required in the embodiment shown in  FIG. 2A . Also, the prior art has a relatively simple flux control  102 , compared to the flux control of  FIG. 2A  with an inner  224  and outer  203  loop. The torque control loop  202  and control transformation  208  of  FIG. 2A  are missing from  FIG. 1 . 
     It is also noted that the embodiments use a number of “virtual” control signals, e.g., v 1 , v 2 , v 3 , and v 4 . The reason the controls are virtual is that the signal do not represent real physical values, but rather intermediate values. 
     Step 1—Speed Control 
       FIG. 3  shows the structure and operation of the speed control loop  201 . The loop includes model compensation block  310  receiving the angular speed ω d (t), a linear stabilizing feedback block  320 , a speed dynamics block, and two adders  340  and  350 . 
     We define a speed tracking error as a difference  350  z 1 ={circumflex over (x)} 1 −ω d , and determine its time derivative as 
     
       
         
           
             
               
                 
                   
                     
                       z 
                       . 
                     
                     1 
                   
                   = 
                   
                     
                       
                         - 
                         
                           a 
                           3 
                         
                       
                       ⁢ 
                       
                         z 
                         1 
                       
                     
                     + 
                     
                       
                         a 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               x 
                               3 
                             
                             ⁢ 
                             
                               
                                 x 
                                 ^ 
                               
                               4 
                             
                           
                           - 
                           
                             
                               x 
                               2 
                             
                             ⁢ 
                             
                               
                                 x 
                                 ^ 
                               
                               5 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         O 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               x 
                               ~ 
                             
                             2 
                           
                           , 
                           
                             
                               x 
                               ~ 
                             
                             2 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         ω 
                         . 
                       
                       d 
                     
                     - 
                     
                       
                         a 
                         3 
                       
                       ⁢ 
                       
                         ω 
                         d 
                       
                     
                     - 
                     
                       
                         
                           a 
                           ^ 
                         
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     We define a first virtual control signal v 1 , and define a difference between v 1  and an estimated electromagnetic torque η 1 =a 1 (x 3 {circumflex over (x)} 4 −x 2 {circumflex over (x)} 5 ) as
 
 z   2   =a   1 ( x   3   {circumflex over (x)}   4   −x   2   {circumflex over (x)}   5 )− v   1 .  (23)
 
     Thus, the speed tracking error dynamics  330  can be rewritten as
 
 ż   1   =−a   3   z   1   +v   1   +z   2   +O   1 ( {tilde over (x)}   2   ,{tilde over (x)}   2 )−{dot over (ω)} d   −a   3 ω d   −â   2 .   (24)
 
     The first virtual control signal v 1  has two parts
 
 v   1   =v   1a   +v   1s ,  (25)
 
     where v 1a  is the feedforward model compensation given by
 
 v   1a   =−O   1 ( {tilde over (x)}   2   ,{tilde over (x)}   2 )+{dot over (ω)} d   +a   3 ω d   +â   2 ,  (26)
 
     and v 1s  is an the feedback from the linear stabilization control  320 
 
 v   1s   =−k   1s1   z   1 .  (27)
 
     Thus, equation (24) can be rewritten as follows
 
 ż   1   =−k   1   z   1   +z   2   ,k   1   =a   3   +k   1s1 .  (28)
 
     Overall the first virtual control signal v 1  is
 
 v   1   =−k   1s1   z   1   −O   1 ( {tilde over (x)}   2   ,{tilde over (x)}   2 )+{dot over (ω)} d   +a   3 ω d   +â   2 ,  (29)
 
and its derivative is determined as follows
 
                         v   .     1     =         k     1   ⁢   s   ⁢           ⁢   1       ⁢     k   1     ⁢     z   1       +       ω   ¨     d     +       a   3     ⁢     ω   d       +       a   ^     2     +       (         l   11     ⁢     a   7       +       l   11     ⁢     l   21       +       l   12     ⁢     l   31         )     ⁢       x   ~     2       +       (         l   12     ⁢     a   7       +       l   11     ⁢     l   22       +       l   12     ⁢     l   32         )     ⁢       x   ~     3       -       (         l   11     ⁢     a   5       -       l   12     ⁢     a   6     ⁢       x   ^     1         )     ⁢       x   ~     4       -       (         l   12     ⁢     a   5       +       l   11     ⁢     a   6     ⁢       x   ^     1         )     ⁢       x   ~     5       -       (         l   11     ⁢     a   6     ⁢     x   5       -       l   12     ⁢     a   6     ⁢     x   4         )     ⁢       x   ~     1       -       k     1   ⁢   s   ⁢           ⁢   1       ⁢     z   2           ,           (   30   )               
which is used in the torque control step.
 
     Step 2—Torque Control 
       FIG. 4A  shows the torque control loop  202  with model compensation  410 , adaptation  420 , nonlinear feedback  430 , and torque dynamics  450 . Three adders  460 ,  470  and  480  are used. 
     According to equations (23), (30), and (3), we have
 
 ż   2   =v   2 +ψ 2   +d   2 .  (31)
 
where v 2 =a 21 bu 1 +a 22 bu 2  with a 21 =−a 1 {circumflex over (x)} 5 , a 22 =a 1 {circumflex over (x)} 4 , and
 
     
       
         
           
             
               
                 
                   
                     
                       ψ 
                       2 
                     
                     = 
                     
                       
                         
                           - 
                           
                             ( 
                             
                               
                                 a 
                                 7 
                               
                               + 
                               
                                 a 
                                 8 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           v 
                           1 
                         
                       
                       - 
                       
                         
                           a 
                           1 
                         
                         ⁢ 
                         
                           a 
                           6 
                         
                         ⁢ 
                         
                           
                             
                               x 
                               ^ 
                             
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   x 
                                   ^ 
                                 
                                 4 
                                 2 
                               
                               + 
                               
                                 
                                   x 
                                   ^ 
                                 
                                 5 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       - 
                       
                         
                           a 
                           1 
                         
                         ⁢ 
                         
                           a 
                           9 
                         
                         ⁢ 
                         
                           
                             
                               x 
                               ^ 
                             
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   5 
                                 
                                 ⁢ 
                                 
                                   x 
                                   3 
                                 
                               
                               + 
                               
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   4 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           a 
                           1 
                         
                         ⁢ 
                         
                           
                             a 
                             10 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   x 
                                   3 
                                 
                               
                               - 
                               
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   3 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           a 
                           1 
                         
                         ⁢ 
                         
                           x 
                           3 
                         
                         ⁢ 
                         
                           
                             O 
                             4 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   x 
                                   ~ 
                                 
                                 2 
                               
                               , 
                               
                                 
                                   x 
                                   ~ 
                                 
                                 3 
                               
                             
                             ) 
                           
                         
                       
                       - 
                       
                         
                           a 
                           1 
                         
                         ⁢ 
                         
                           x 
                           2 
                         
                         ⁢ 
                         
                           
                             O 
                             5 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   x 
                                   ~ 
                                 
                                 2 
                               
                               , 
                               
                                 
                                   x 
                                   ~ 
                                 
                                 3 
                               
                             
                             ) 
                           
                         
                       
                       - 
                       
                         
                           k 
                           
                             1 
                             ⁢ 
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                         ⁢ 
                         
                           k 
                           1 
                         
                         ⁢ 
                         
                           z 
                           1 
                         
                       
                       - 
                       
                         
                           ω 
                           ¨ 
                         
                         d 
                       
                       - 
                       
                         
                           a 
                           3 
                         
                         ⁢ 
                         
                           
                             ω 
                             . 
                           
                           d 
                         
                       
                       - 
                       
                         
                           a 
                           
                             ^ 
                             . 
                           
                         
                         2 
                       
                       - 
                       
                         
                           ( 
                           
                             
                               
                                 l 
                                 11 
                               
                               ⁢ 
                               
                                 a 
                                 7 
                               
                             
                             + 
                             
                               
                                 l 
                                 11 
                               
                               ⁢ 
                               
                                 l 
                                 21 
                               
                             
                             + 
                             
                               
                                 l 
                                 12 
                               
                               ⁢ 
                               
                                 l 
                                 31 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           2 
                         
                       
                       - 
                       
                         
                           ( 
                           
                             
                               
                                 l 
                                 12 
                               
                               ⁢ 
                               
                                 a 
                                 7 
                               
                             
                             + 
                             
                               
                                 l 
                                 11 
                               
                               ⁢ 
                               
                                 l 
                                 22 
                               
                             
                             + 
                             
                               
                                 l 
                                 12 
                               
                               ⁢ 
                               
                                 l 
                                 32 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           3 
                         
                       
                       + 
                       
                         
                           k 
                           
                             1 
                             ⁢ 
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                         ⁢ 
                         
                           z 
                           2 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       d 
                       2 
                     
                     = 
                     
                       
                         
                           a 
                           1 
                         
                         ⁢ 
                         
                           
                             a 
                             5 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   5 
                                 
                                 ⁢ 
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   4 
                                 
                               
                               - 
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   4 
                                 
                                 ⁢ 
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   5 
                                 
                               
                             
                             ) 
                           
                         
                       
                       - 
                       
                         
                           a 
                           1 
                         
                         ⁢ 
                         
                           a 
                           6 
                         
                         ⁢ 
                         
                           
                             x 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   4 
                                 
                                 ⁢ 
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   4 
                                 
                               
                               + 
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   5 
                                 
                                 ⁢ 
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   5 
                                 
                               
                             
                             ) 
                           
                         
                       
                       - 
                       
                         
                           a 
                           1 
                         
                         ⁢ 
                         
                           a 
                           6 
                         
                         ⁢ 
                         
                           
                             
                               x 
                               ~ 
                             
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   x 
                                   ^ 
                                 
                                 4 
                                 2 
                               
                               + 
                               
                                 
                                   x 
                                   ^ 
                                 
                                 5 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           ( 
                           
                             
                               
                                 l 
                                 11 
                               
                               ⁢ 
                               
                                 a 
                                 5 
                               
                             
                             - 
                             
                               
                                 l 
                                 12 
                               
                               ⁢ 
                               
                                 a 
                                 6 
                               
                               ⁢ 
                               
                                 
                                   x 
                                   ^ 
                                 
                                 1 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           4 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             
                               
                                 l 
                                 12 
                               
                               ⁢ 
                               
                                 a 
                                 5 
                               
                             
                             + 
                             
                               
                                 l 
                                 11 
                               
                               ⁢ 
                               
                                 a 
                                 6 
                               
                               ⁢ 
                               
                                 
                                   x 
                                   ^ 
                                 
                                 1 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           5 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             
                               
                                 l 
                                 11 
                               
                               ⁢ 
                               
                                 a 
                                 6 
                               
                               ⁢ 
                               
                                 x 
                                 5 
                               
                             
                             - 
                             
                               
                                 l 
                                 12 
                               
                               ⁢ 
                               
                                 a 
                                 6 
                               
                               ⁢ 
                               
                                 x 
                                 4 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             
                               x 
                               ~ 
                             
                             1 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     Note that d 2  denotes the disturbance due to state estimation errors. Although d 2  is typically time varying, one can estimate its slowly time-varying component, which can result in improved performance. In this case, the adaption law  420  for identifying d 2  can be
 
τ 22   =z   2  
 
   2 =Proj(Γ 22 τ 22 )  (33)
 
with {circumflex over (d)} 2  representing the estimation of d 2 .
 
     We denote d 2 ={circumflex over (d)} 2 +{tilde over (d)} 2 . The second virtual control signal v 2  is rewritten as
 
 v   2   =v   2a   +v   2s ,  (34)
 
where the feedforward model compensation term is
 
 v   2a =ψ 2   −{circumflex over (d)}   2   −z   1 ,  (35)
 
and the feedback stabilization term v 2s  has a nominal stabilization term v 2s1  and a control term v 2s2  
 
 v   2s   =v   2s1   +v   2s2  
 
 v   2s1   =−k   2s1   z   2  
 
 k   2   =a   7   +a   8   +k   2s1   (36)
 
     Thus, equation (31) can be rearranged as
 
 ż   2   =k   2   z   2   +v   2s2   +{tilde over (d)}   2   −z   1 .  (37)
 
     Given the above assumption, {tilde over (d)} 2  is bounded. Thus, there exists v 2s2  such that the following conditions hold
 
 z   2 ( v   2s2   +{tilde over (d)}   2 )≦ε 2 , and
 
 z   2   v   2s2 ≦0,
 
where ε 2  is an arbitrarily small threshold parameter. One example of v 2s2  that satisfies above conditions is
 
                       v     2   ⁢           ⁢   s   ⁢           ⁢   2       =       -     1     4   ⁢           ⁢     ɛ   2           ⁢     h   2   2     ⁢     z   2         ,           (   38   )               
where h 2 =δ d     2    and δ d     2    is the bound of {tilde over (d)} 2 .
 
     The first and second virtual control signals v 1  and v 2  ensure that the speed z 1  and torque z 2  are bounded. We define a positive definite function 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       
                         s 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     = 
                     
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           z 
                           1 
                           2 
                         
                       
                       + 
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           z 
                           2 
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     and its time derivative 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             V 
                             . 
                           
                           
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               - 
                               
                                 k 
                                 1 
                               
                             
                             ⁢ 
                             
                               z 
                               1 
                               2 
                             
                           
                           + 
                           
                             
                               z 
                               1 
                             
                             ⁢ 
                             
                               z 
                               2 
                             
                           
                           + 
                           
                             
                               z 
                               2 
                             
                             ⁢ 
                             
                               
                                 z 
                                 . 
                               
                               2 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               - 
                               
                                 k 
                                 1 
                               
                             
                             ⁢ 
                             
                               z 
                               1 
                               2 
                             
                           
                           - 
                           
                             
                               k 
                               2 
                             
                             ⁢ 
                             
                               z 
                               2 
                               2 
                             
                           
                           + 
                           
                             
                               z 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   v 
                                   
                                     2 
                                     ⁢ 
                                     s 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     d 
                                     ~ 
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         ≤ 
                           
                         ⁢ 
                         
                           
                             
                               - 
                               
                                 k 
                                 1 
                               
                             
                             ⁢ 
                             
                               z 
                               1 
                               2 
                             
                           
                           - 
                           
                             
                               k 
                               2 
                             
                             ⁢ 
                             
                               z 
                               2 
                               2 
                             
                           
                           + 
                           
                             ɛ 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         ≤ 
                           
                         ⁢ 
                         
                           
                             
                               - 
                               
                                 λ 
                                 2 
                               
                             
                             ⁢ 
                             
                               V 
                               
                                 s 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               ɛ 
                               2 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     Hence, V s2  is bounded by 
                       V     s   ⁢           ⁢   2       ≤         exp   ⁡     (       -     λ   2       ⁢   t     )       ⁢       V     s   ⁢           ⁢   2       ⁡     (   0   )         +         ɛ   2       λ   2       ⁡     [     1   -     exp   ⁡     (       -     λ   2       ⁢   t     )         ]           ,     
     ⁢       where   ⁢           ⁢     λ   2       =     2   ⁢           ⁢     min   ⁡     (       k   1     ,     k   2       )           ,           (   41   )               
where the function min returns a minimum.
 
     If the state estimator design can guarantee finite time convergence, i.e., {tilde over (d)} 2 =0 after some t 2 , then it can be shown that from Barbalat&#39;s lemma and equation (41) with {tilde over (d)} 2 =0, the tracking error approaches zero over time, i.e., z 1 , z 2 →0, as t→∞. 
     Step 3—Flux Outer Loop Control 
       FIG. 4B  shows the structure and operation of the flux control outer loop with model compensation  415 , linear stabilizing feedback  425 , dynamics  445  and two adders  445  and  455 . 
     Flux tracking control also uses backstepping. We define a flux module tracking error as
 
 z   3   ={circumflex over (x)}   4   2   +{circumflex over (x)}   5   2 −φ d   2 , and  (42)
 
denote ψ d (t)=φ d   2 (t), and its time derivative
 
 ż   3 =2 a   10 ( x   2   {circumflex over (x)}   4   +x   3   {circumflex over (x)}   5 )+ψ 3 ,  (43)
 
where
 
     
       
         
           
               
             ⁢ 
             
               
                 ψ 
                 3 
               
               = 
               
                 
                   
                     - 
                     2 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       a 
                       8 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             x 
                             ^ 
                           
                           4 
                           2 
                         
                         + 
                         
                           
                             x 
                             ^ 
                           
                           5 
                           2 
                         
                       
                       ) 
                     
                   
                 
                 - 
                 
                   2 
                   ⁢ 
                   
                     
                       a 
                       10 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             
                               x 
                               ~ 
                             
                             2 
                           
                           ⁢ 
                           
                             
                               x 
                               ^ 
                             
                             4 
                           
                         
                         + 
                         
                           
                             
                               x 
                               ~ 
                             
                             3 
                           
                           ⁢ 
                           
                             
                               x 
                               ^ 
                             
                             5 
                           
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   2 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     l 
                     41 
                   
                   ⁢ 
                   
                     
                       x 
                       ~ 
                     
                     2 
                   
                   ⁢ 
                   
                     
                       x 
                       ^ 
                     
                     4 
                   
                 
                 + 
                 
                   2 
                   ⁢ 
                   
                     l 
                     42 
                   
                   ⁢ 
                   
                     
                       x 
                       ~ 
                     
                     3 
                   
                   ⁢ 
                   
                     
                       x 
                       ^ 
                     
                     4 
                   
                 
                 + 
                 
                   2 
                   ⁢ 
                   
                     l 
                     51 
                   
                   ⁢ 
                   
                     
                       x 
                       ~ 
                     
                     2 
                   
                   ⁢ 
                   
                     
                       x 
                       ^ 
                     
                     5 
                   
                 
                 + 
                 
                   2 
                   ⁢ 
                   
                     l 
                     52 
                   
                   ⁢ 
                   
                     
                       x 
                       ~ 
                     
                     3 
                   
                   ⁢ 
                   
                     
                       x 
                       ^ 
                     
                     5 
                   
                 
                 - 
                 
                   
                     
                       ψ 
                       . 
                     
                     d 
                   
                   . 
                 
               
             
           
         
       
     
     Note that ψ 3  depends on accessible signals. We use a third virtual control signal v 3 , and a state z 4  to denote the discrepancy between v 3  and
 
η 2 =2 a   10 ( x   2   {circumflex over (x)}   4   +x   3   {circumflex over (x)}   5 ), i.e.,
 
 z   4 =2 a   10 ( x   2   {circumflex over (x)}   4   +x   3   {circumflex over (x)}   5 )− v   3 .  (44)
 
     Similar to the rewriting of the first virtual control signal v 1 , the second virtual control signal v 3  is rewritten as
 
 v   3   =v   3a   +v   3s ,  (45)
 
where the feedforward model compensation is v 3a =−ψ 3  and the feedback stabilization term v 3s =−k 3s1 z 3 . Thus, we have
 
 ż   3   =−k   3   z   3   +z   4 ,  (46)
 
where k 3 =k 3s1 .
 
     Step 4—Flux Inner Loop Control 
       FIG. 4C  shows the structure and operation of the flux control inner loop with model compensation  461 , adaptation  462 , nonlinear feedback  465 , linear stabilizing feedback, dynamics and three adders  466 ,  467  and  468 . 
     The derivative of the virtual control signal discrepancy is
 
 ż   4   =v   4 +ψ 4   +d   4 ,  (47)
 
where v 4 =a 21 bu 1 +a 22 bu 2  with a 41 =2a 10 {circumflex over (x)} 4  and a 42 =2a 10 {circumflex over (x)} 5 , and
 
     
       
         
           
               
             ⁢ 
             
               
                 ψ 
                 4 
               
               = 
               
                 
                   2 
                   ⁢ 
                   
                     a 
                     10 
                   
                   ⁢ 
                   
                     
                       a 
                       5 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             x 
                             ^ 
                           
                           4 
                           2 
                         
                         + 
                         
                           
                             x 
                             ^ 
                           
                           5 
                           2 
                         
                       
                       ) 
                     
                   
                 
                 - 
                 
                   2 
                   ⁢ 
                   
                     a 
                     10 
                   
                   ⁢ 
                   
                     
                       a 
                       7 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             
                               x 
                               ^ 
                             
                             2 
                           
                           ⁢ 
                           
                             
                               x 
                               ^ 
                             
                             4 
                           
                         
                         + 
                         
                           
                             
                               x 
                               ^ 
                             
                             3 
                           
                           ⁢ 
                           
                             
                               x 
                               ^ 
                             
                             5 
                           
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   2 
                   ⁢ 
                   
                     
                       a 
                       10 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             l 
                             21 
                           
                           ⁢ 
                           
                             
                               x 
                               ~ 
                             
                             2 
                           
                           ⁢ 
                           
                             
                               x 
                               ^ 
                             
                             4 
                           
                         
                         + 
                         
                           
                             l 
                             22 
                           
                           ⁢ 
                           
                             
                               x 
                               ~ 
                             
                             3 
                           
                           ⁢ 
                           
                             
                               x 
                               ^ 
                             
                             4 
                           
                         
                         + 
                         
                           
                             l 
                             31 
                           
                           ⁢ 
                           
                             
                               x 
                               ~ 
                             
                             2 
                           
                           ⁢ 
                           
                             
                               x 
                               ^ 
                             
                             5 
                           
                         
                         + 
                         
                           
                             l 
                             32 
                           
                           ⁢ 
                           
                             
                               x 
                               ~ 
                             
                             3 
                           
                           ⁢ 
                           
                             
                               x 
                               ^ 
                             
                             5 
                           
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   2 
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           l 
                           41 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           4 
                         
                       
                       + 
                       
                         
                           l 
                           51 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           5 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           - 
                           
                             ( 
                             
                               
                                 a 
                                 7 
                               
                               + 
                               
                                 l 
                                 21 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           2 
                         
                       
                       - 
                       
                         
                           l 
                           22 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           3 
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   2 
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           l 
                           42 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           4 
                         
                       
                       + 
                       
                         
                           l 
                           52 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           5 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           - 
                           
                             l 
                             31 
                           
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           2 
                         
                       
                       - 
                       
                         
                           ( 
                           
                             
                               a 
                               7 
                             
                             + 
                             
                               l 
                               32 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           3 
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   2 
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           l 
                           41 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           2 
                         
                       
                       + 
                       
                         
                           l 
                           42 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           3 
                         
                       
                       + 
                       
                         
                           a 
                           10 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           2 
                         
                       
                       - 
                       
                         2 
                         ⁢ 
                         
                           a 
                           8 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           4 
                         
                       
                     
                     ) 
                   
                   × 
                   
                     ( 
                     
                       
                         
                           - 
                           
                             a 
                             8 
                           
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           4 
                         
                       
                       - 
                       
                         
                           a 
                           9 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           1 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           5 
                         
                       
                       + 
                       
                         
                           a 
                           10 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           2 
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   2 
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           l 
                           41 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           2 
                         
                       
                       + 
                       
                         
                           l 
                           42 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           3 
                         
                       
                       + 
                       
                         
                           a 
                           10 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           2 
                         
                       
                       - 
                       
                         2 
                         ⁢ 
                         
                           a 
                           8 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           4 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           l 
                           41 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           2 
                         
                       
                       + 
                       
                         
                           l 
                           42 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           3 
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   2 
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           l 
                           51 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           2 
                         
                       
                       + 
                       
                         
                           l 
                           52 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           3 
                         
                       
                       + 
                       
                         
                           a 
                           10 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           3 
                         
                       
                       - 
                       
                         2 
                         ⁢ 
                         
                           a 
                           8 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           5 
                         
                       
                     
                     ) 
                   
                   × 
                   
                     ( 
                     
                       
                         
                           - 
                           
                             a 
                             8 
                           
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           5 
                         
                       
                       + 
                       
                         
                           a 
                           9 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           1 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           4 
                         
                       
                       + 
                       
                         
                           a 
                           10 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           3 
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   2 
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           l 
                           51 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           2 
                         
                       
                       + 
                       
                         
                           l 
                           52 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           3 
                         
                       
                       + 
                       
                         
                           a 
                           10 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           3 
                         
                       
                       - 
                       
                         2 
                         ⁢ 
                         
                           a 
                           8 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           5 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           l 
                           51 
                         
                         ⁢ 
                         
                           
                             x 
                             ~ 
                           
                           2 
                         
                       
                       + 
                       
                         
                           l 
                           52 
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           3 
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     k 
                     3 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         
                           - 
                           
                             k 
                             3 
                           
                         
                         ⁢ 
                         
                           z 
                           3 
                         
                       
                       + 
                       
                         z 
                         4 
                       
                     
                     ) 
                   
                 
                 - 
                 
                   
                     ψ 
                     ¨ 
                   
                   d 
                 
               
             
           
         
       
       
         
           
             
               d 
               4 
             
             = 
             
               
                 
                   ( 
                   
                     
                       2 
                       ⁢ 
                       
                         a 
                         5 
                       
                       ⁢ 
                       
                         l 
                         41 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         4 
                       
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         a 
                         6 
                       
                       ⁢ 
                       
                         l 
                         42 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         4 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         1 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         a 
                         5 
                       
                       ⁢ 
                       
                         l 
                         51 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         5 
                       
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         a 
                         6 
                       
                       ⁢ 
                       
                         l 
                         52 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         5 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         1 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   
                     x 
                     ~ 
                   
                   4 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       2 
                       ⁢ 
                       
                         l 
                         41 
                       
                       ⁢ 
                       
                         a 
                         6 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         1 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         4 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         l 
                         51 
                       
                       ⁢ 
                       
                         a 
                         6 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         1 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         5 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         l 
                         42 
                       
                       ⁢ 
                       
                         a 
                         5 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         4 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         l 
                         52 
                       
                       ⁢ 
                       
                         a 
                         5 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         5 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   
                     x 
                     ~ 
                   
                   5 
                 
               
               + 
               
                 2 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         l 
                         41 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         4 
                       
                     
                     + 
                     
                       
                         l 
                         51 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         5 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   a 
                   6 
                 
                 ⁢ 
                 
                   
                     x 
                     ~ 
                   
                   1 
                 
                 ⁢ 
                 
                   x 
                   5 
                 
               
               - 
               
                 2 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         l 
                         42 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         4 
                       
                     
                     + 
                     
                       
                         l 
                         52 
                       
                       ⁢ 
                       
                         
                           x 
                           ^ 
                         
                         5 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   a 
                   6 
                 
                 ⁢ 
                 
                   
                     x 
                     ~ 
                   
                   1 
                 
                 ⁢ 
                 
                   x 
                   4 
                 
               
             
           
         
       
     
     A function of accessible signals is ψ 4 , and d 4  represents the disturbance result from the state estimation error. Although d 4  is time-varying, one can estimate its slowly time-varying component by adaptation and a discontinuous projection
 
τ 44   =z   4  
 
   4 =Proj(Γ 44 τ 44 ),  (48)
 
where {circumflex over (d)} 4  is the estimate of d 4 .
 
     We denote the fourth virtual control v 4  as
 
 v   4   =v   4a   +v   4s ,  (49)
 
where the feedforward model compensation is
 
 v   4a =−ψ 4   −{circumflex over (d)}   4   −z   3 .  (50)
 
and the feedback term includes a nominal stabilization term v 4s1  and a control term v 4s2  given by
 
 v   4s   =v   4s1   +v   4s2  
 
 v   4s1   =−k   4   z   4 .  (51)
 
     We define {tilde over (d)} 4 =d 4 −{circumflex over (d)} 4 . Thus,
 
 ż   4   =−k   4   z   4   +v   4s2   +{tilde over (d)}   4   −z   3 .  (52)
 
     Given the above assumption, d 4  is bounded. Also, the adaptive law equation (48) ensures the boundedness of {circumflex over (d)} 4 . Consequently, we know {tilde over (d)} 4  is bounded. Hence, v 4s2  can be determined such that
 
 z   4 ( v   4s2   +{tilde over (d)}   4 )≦ε 4 , and
 
 z   4   v   4s2 ≦0,
 
where ε 4  is an arbitrarily small design parameter. One example of v 4s2  satisfying above inequalities is
 
                       v     4   ⁢   s   ⁢           ⁢   2       =       -     1     4   ⁢           ⁢     ɛ   4           ⁢     h   4   2     ⁢     z   4         ,           (   53   )               
where h 4 =δ d     4    and δ d     4    is the bound of {tilde over (d)} 4 .
 
     Third and fourth virtual control signals v 3  and v 4  guarantee the boundedness of z 3  and z 4 . Taking a Lyapunov function candidate 
                       V     s   ⁢           ⁢   4       =         1   2     ⁢     z   3   2       +       1   2     ⁢     z   4   2           ,           (   54   )               
and determining its time derivative, we can derive
 
                       V     s   ⁢           ⁢   4       ≤         exp   ⁡     (       -     λ   4       ⁢   t     )       ⁢       V     s   ⁢           ⁢   4       ⁡     (   0   )         +         ɛ   4       λ   4       ⁡     [     1   -     exp   ⁡     (       -     λ   4       ⁢   t     )         ]           ,           (   55   )               
with λ 4 =2 min(k 3 , k 4 ). This implies the boundedness of z 3  and z 4 . If we assume that z 3  and z 4  dynamics are only subject to parametric uncertainties, i.e., {tilde over (d)} 4 =0), after some finite time t 4 , the zero solution of z 3  and z 4  dynamics is asymptotically stable, i.e., z 3 , z 4 →0, as t→∞.
 
     The control input can be determined by 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             1 
                           
                         
                       
                       
                         
                           
                             u 
                             2 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         
                           
                             1 
                             b 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       - 
                                       
                                         a 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         x 
                                         ^ 
                                       
                                       5 
                                     
                                   
                                 
                                 
                                   
                                     
                                       a 
                                       1 
                                     
                                     ⁢ 
                                     
                                       
                                         x 
                                         ^ 
                                       
                                       4 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       a 
                                       10 
                                     
                                     ⁢ 
                                     
                                       
                                         x 
                                         ^ 
                                       
                                       4 
                                     
                                   
                                 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       a 
                                       10 
                                     
                                     ⁢ 
                                     
                                       
                                         x 
                                         ^ 
                                       
                                       5 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   v 
                                   
                                     2 
                                     ⁢ 
                                     a 
                                   
                                 
                                 + 
                                 
                                   v 
                                   
                                     2 
                                     ⁢ 
                                     s 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   v 
                                   
                                     4 
                                     ⁢ 
                                     a 
                                   
                                 
                                 + 
                                 
                                   v 
                                   
                                     4 
                                     ⁢ 
                                     s 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
           
         
       
     
     For the control input voltage to has unique solution, we have 
                                    -     a   1       ⁢       x   ^     5               a   1     ⁢       x   ^     4                 2   ⁢     a   10     ⁢       x   ^     4             2   ⁢     a   10     ⁢       x   ^     5                  =         -   2     ⁢     a   1     ⁢       a   10     ⁡     (         x   ^     4   2     +       x   ^     5   2       )         ≠   0       ,           (   57   )               
which means that the flux modulus estimate should not be zero.
 
     Flux and Speed Estimator 
       FIG. 5  shows the flux and speed estimator  209  of  FIG. 2B  in greater detail. One embodiment of the flux and speed estimator is an EKF  515  designed on the basis of the induction motor model  510 , which comprises stator and flux dynamics  512  and speed dynamics  514 . The EKF  515  estimate the flux and speed according to measured signals: stator voltages  124  and stator currents  125 . 
     Although the invention has been described by way of examples of 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 appended claims to cover all such variations and modifications as come within the true spirit and scope of the invention.