Abstract:
A system and method controls a rotor angular speed of an induction motor by first sensing an operation condition of the induction motor to produce measured signals, which are transformed by applying a state transformation to an induction motor model to produce a transformed induction motor model. Transformed state estimates of the transformed induction motor model are produced based on the measured signals. An inverse of the state transformation is applied to the transformed state estimates to produce state estimates of the induction motor model, which are then used to determine control input voltages for the induction motor, based on the state estimates, to control the rotor angular speed of the induction motor.

Description:
RELATED APPLICATION 
     This U.S. patent application is related to U.S. Ser. No. 14/612,460 (MERL-2783) co-filed herewith 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 electric motors, and more particularly to sensorless angular speed control of an induction motor. 
     BACKGROUND OF THE INVENTION 
     Adjustable speed motor drives for induction motors are widely used in industrial applications due to their low maintenance cost and high performance. However, the control of induction motors is challenging due to highly nonlinear dynamics. Among various means, vector (field oriented) control appears to be a good solution and has evolved as a mature technology. Speed sensorless motor drives for electric motors are advantageous in practice by avoiding measuring the motor speed. 
     The prior art describes speed sensorless control technologies including a voltage model-based direct integration approach, an adaptive observer approach, and an extended Kalman filter approach, etc. The voltage model-based direct integration suffers from accumulation error due to inaccurate measurement. 
       FIGS. 1B and 2B  show a prior art speed sensorless motor drive for an induction motor  104 . 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 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 a speed 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 motor torque, to produce a difference signal  122 . The current control block  103  determines the stator voltage signals  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 voltages signals, 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  131  phase currents  125  as input signals, and outputs estimated or measured stator currents  115  and  121 , an estimated rotor flux amplitude  112 , and an estimated rotor speed signal  118  to produce the difference signals  113 ,  116 ,  119 , and  122 . The signal  119  is used for speed control  102 . 
     The performance of the prior art sensorless speed motor drives relies heavily on the performance of the flux and speed estimators  106  and  107 . 
       FIGS. 2A and 2B  show prior art estimator methods based on stator currents and voltages signals  211 , which are measured by sensors of a sensing induction motor  202  and are assumed in balanced three-phases and in orthogonal stationary frame, and an induction motor model  201 . A Clarke transformation  203  is first applied to transform the induction motor model  201  and the sensed signals  211  so that quantities (including variables in the induction motor model and measured signals) in balanced three-phases are converted into quantities in balanced two-phases. 
     Balanced two-phases quantities, as a result of Clarke transformation of balanced three-phases quantities, are still in orthogonal stationary frame, and thus called quantities in balanced two-phases orthogonal stationary frame. Some prior art further applies Park transformation to the quantities in balanced two-phases orthogonal stationary frame which converts the quantities into quantities in balanced two-phases orthogonal rotating frame. 
     A block  204  represents an estimator, which is designed on the basis of the induction motor model, as a result of applying Clarke, or Clarke and Park transformations, to produce estimates of stator currents, rotor flux, and rotor speed signals, which are referred here as state coordinates. Note that both Clarke and Park transformations are not state transformation, and thus state variables in the induction motor model bear the same physical meanings after Clarke and Park transformations are applied. This imposes limitations on choices of estimators, and thus leads to unsatisfactory estimation performance. For instance, the voltage model-based direct integration suffers accumulation error due to inaccurate measurement. Adaptive observer and extended Kalman filter approaches yield slow speed tracking performance because the speed is treated as an unknown parameter and its identification is slow. 
     This fact is elaborated by  FIG. 2B , where block  222  represents the unnecessary assumption that the rotor speed is an unknown parameter, and a speed estimator  223  produces the rotor speed estimate based on outputs of block  221  and the assumption  222 . 
     Overall, most prior art speed sensorless motor drives produce limited speed tracking performance because estimator design is performed in fixed state coordinates and under unnecessary assumption (for instance parameter assumption). Performing estimator design for a system with fixed state coordinates fails to exploit the freedom of state transformations, which may simplify the induction motor model thus admit high performance estimators. 
     SUMMARY OF THE INVENTION 
     The embodiments of the invention provide a speed sensorless control system and method applicable to motor drives of variable speed induction motors. The embodiments use state transformations of a model of the induction motor to simplify the method. 
     This invention is based on the realization that a high bandwidth speed sensorless control system is difficult to achieve because the induction motor model in original coordinates is highly coupled, and not in any structure admitting a simple observer design unless certain assumptions are imposed, for instance, treating the rotor speed as an unknown parameter, as in the prior art. 
     This invention teaches that the state transformation, or change of coordinates, can be introduced to put the induction motor model into certain structures, and thus the induction motor model in new coordinates is partially decoupled. The structured induction motor model typically simplifies the observer design, and leads to high performance in estimation. 
     The invention further teaches the determination of observer gains to enforce fast convergence of estimation error dynamics. In one embodiment, applying a state transformation to the induction motor model gives a transformed induction motor model such that estimation error dynamics of the rotor flux and stator current in the d-axis are partially decoupled from the rest estimation errors. By enforcing the fast convergence of the estimation error of the rotor flux and stator current in the d-axis, the rest estimation errors dynamics are simplified, and thus the observer gain selection is relatively simple. 
     In the prior art, the observer gain is based on error dynamics, which are nonlinear, thus the design is complicated, and stability cannot be guaranteed. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1A and 1B  are block diagrams of a prior art sensorless speed motor drive; 
         FIGS. 2A and 2B  are block diagram of prior art estimator methods based on stator currents and voltages signals; 
         FIG. 3  is a block diagram of a method for estimating a state of an induction motor according to embodiments of the invention; 
         FIGS. 4A and 4B  are block diagrams of decomposing a transformed induction motor model into multiple subsystems according to embodiments of the invention; 
         FIG. 5  is a block diagram of decomposing a transformed induction motor model into multiple subsystems according to embodiments of the invention; 
         FIG. 6A  is a block diagram of one embodiment of the sequential design based on the decomposition of the transformed induction motor model as shown in  FIG. 5 ; and 
         FIG. 6B  is a block diagram of another embodiment of the sequential design based on the decomposition of the transformed induction motor model as shown in  FIG. 5 . 
     
    
    
     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. 
     To facilitate the detailed description of the embodiments of the invention for a speed sensorless control system and method for induction motors, the following notations are defined. Assume ζ is a dummy variable, then ζ denotes a measured variable, {circumflex over (ζ)} denotes an estimate of the variable, and {tilde over (ζ)}=ζ−{circumflex over (ζ)} is an estimation error. 
     
       
         
               
             
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Notations 
               
             
          
           
               
                   
                 Notation 
                 Description 
               
               
                   
                   
               
             
          
           
               
                   
                 Φ dr , Φ qr   
                 a. 
                 rotor fluxes in d- and q-axes 
               
               
                   
                 ω 
                 b. 
                 rotor angular speed 
               
               
                   
                 ξ 
                 c. 
                 angle of a rotating frame 
               
               
                   
                 T l   
                 d. 
                 load torque 
               
               
                   
                 J 
                 e. 
                 inertia of a rotor and a load 
               
               
                   
                 L s , L m , L r   
                 f. 
                 stator, mutual, and rotor inductances 
               
               
                   
                 R s , R r   
                 g. 
                 stator and rotor resistances 
               
               
                   
                 ω 1   
                 h. 
                 angular speed of a rotating frame 
               
               
                   
                 i ds , i qs   
                 i. 
                 stator currents in d- and q-axis 
               
               
                   
                 u ds , u qs   
                 j. 
                 stator voltages in d- and q-axes 
               
               
                   
                   
               
               
                   
                 σ 
                 k.  
                 
                   
                     
                       
                         
                           
                             
                               L 
                               s 
                             
                             ⁢ 
                             
                               L 
                               r 
                             
                           
                           - 
                           
                             L 
                             m 
                             2 
                           
                         
                         
                           
                             L 
                             s 
                           
                           ⁢ 
                           
                             L 
                             r 
                           
                         
                       
                     
                   
                 
               
               
                   
                   
               
             
          
         
       
     
     Induction Motor Model 
     A model of the induction motor including stator currents, flux and angular speed as its states. This choice of states define a set of state coordinates, called the original state coordinates, can be expressed by the equations in the following induction motor model 
                         i   .     ds     =         -   γ     ⁢           ⁢     i   ds       +       ω   1     ⁢     i   qs       +     β   ⁡     (       α   ⁢           ⁢     Φ   dr       +     ω   ⁢           ⁢     Φ   qr         )       +       u   ds         L   s     ⁢   σ           ⁢     
     ⁢         i   .     qs     =         -   γ     ⁢           ⁢     i   qs       -       ω   1     ⁢     i   ds       +     β   ⁡     (       α   ⁢           ⁢     Φ   qr       -     ω   ⁢           ⁢     Φ   dr         )       +       u   qs         L   s     ⁢   σ           ⁢     
     ⁢           Φ   .     dr     =         -   α     ⁢           ⁢     Φ   dr       +       (       ω   1     -   ω     )     ⁢     Φ   qr       +     α   ⁢           ⁢     L   m     ⁢     i   ds           ,     
     ⁢         Φ   .     qr     =         -   α     ⁢           ⁢     Φ   qr       -       (       ω   1     -   ω     )     ⁢     Φ   dr       +     α   ⁢           ⁢     L   m     ⁢     i   qs           ,     
     ⁢       ω   .     =       μ   ⁡     (         Φ   dr     ⁢     i   qs       -       Φ   qr     ⁢     i   ds         )       -       T   l     J         ,     
     ⁢     y   =       [           i   ds           i   qs           ]     T       ,             (   1   )               
where y represents sensed signals, ω I  is the angular speed of a reference frame, and
 
     
       
         
           
             
               γ 
               = 
               
                 
                   1 
                   
                     
                       L 
                       s 
                     
                     ⁢ 
                     σ 
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       R 
                       s 
                     
                     + 
                     
                       
                         
                           L 
                           m 
                           2 
                         
                         
                           L 
                           r 
                         
                       
                       ⁢ 
                       
                         
                           R 
                           r 
                         
                         
                           L 
                           r 
                         
                       
                     
                   
                   ) 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               α 
               = 
               
                 
                   R 
                   r 
                 
                 
                   L 
                   r 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               β 
               = 
               
                 
                   1 
                   
                     
                       L 
                       s 
                     
                     ⁢ 
                     σ 
                   
                 
                 ⁢ 
                 
                   
                     L 
                     m 
                   
                   
                     L 
                     r 
                   
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               μ 
               = 
               
                 
                   2 
                   3 
                 
                 ⁢ 
                 
                   
                     
                       L 
                       m 
                     
                     
                       
                         
                           L 
                           s 
                         
                         ⁢ 
                         
                           L 
                           r 
                         
                       
                       - 
                       
                         L 
                         m 
                         2 
                       
                     
                   
                   . 
                 
               
             
           
         
       
     
     Note that the induction motor model ( 1 ) is in an orthognal rotating frame with a rotation speed of ω 1 ; and quantities i ds , i qs , Φ dr , Φ qr , ω are referred as balanced two-phase quantities in orthognal rotating frame, i.e. both Clarke and Park transformations have been applied to arrive at the model ( 1 ). 
     When ω 1 =0, the equations in the model ( 1 ) reduced to 
                         i   .     ds     =         -   γ     ⁢           ⁢     i   ds       +     β   ⁡     (       α   ⁢           ⁢     Φ   dr       +     ω   ⁢           ⁢     Φ   qr         )       +       u   ds         L   s     ⁢   σ           ⁢     
     ⁢         i   .     qs     =         -   γ     ⁢           ⁢     i   qs       +     β   ⁡     (       α   ⁢           ⁢     Φ   qr       -     ω   ⁢           ⁢     Φ   dr         )       +       u   qs         L   s     ⁢   σ           ⁢     
     ⁢           Φ   .     dr     =         -   α     ⁢           ⁢     Φ   dr       -     ωΦ   qr     +     α   ⁢           ⁢     L   m     ⁢     i   ds           ,     
     ⁢         Φ   .     qr     =         -   α     ⁢           ⁢     Φ   qr       +     ωΦ   dr     +     α   ⁢           ⁢     L   m     ⁢     i   qs           ,     
     ⁢       ω   .     =       μ   ⁡     (         Φ   dr     ⁢     i   qs       -       Φ   qr     ⁢     i   ds         )       -       T   l     J         ,     
     ⁢     y   =       [           i   ds           i   qs           ]     T       ,             (   2   )               
which represents the induction model without applying a Park transformation. Park transformation are known to those of ordinary skill in the art, and thus not repeated here. In another words, the induction motor model ( 1 ) is in orthognal stationary frame, and quantities i ds , i qs , Φ dr , Φ qr , ω are referred as balanced two-phase quantities in orthognal stationary frame, i.e. Clarke transformation has been applied to arrive at the model ( 1 ).
 
     Conventional estimator designs are usually based on the model according to Equations (1) or (2), which have the same state coordinates denoted by (i ds , i qs , Φ dr , Φ qr , ω) T . A direct application of existing estimator designs, e.g., sliding mode observer, high gain observer, and a Luenberger observer to the model of Equations (1) or (2) produce an unsatisfactory estimation of stator currents, rotor flux, and the rotor speed due to highly coupled nonlinear terms in the left hand side of differential Equations (1) or (2). For instance, the term ωΦ qr  in the right hand side of the differential Equation defining i ds , i.e., 
     
       
         
           
             
               
                 i 
                 . 
               
               ds 
             
             = 
             
               
                 
                   - 
                   γ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   i 
                   ds 
                 
               
               + 
               
                 β 
                 ⁡ 
                 
                   ( 
                   
                     
                       α 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Φ 
                         dr 
                       
                     
                     + 
                     
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Φ 
                         qr 
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   
                     u 
                     ds 
                   
                   
                     
                       L 
                       s 
                     
                     ⁢ 
                     σ 
                   
                 
                 . 
               
             
           
         
       
     
     The induction motor model in Equations (1) or (2) is highly coupled because of the fact that the right hand side of each differential Equation in (1) or (2) depends on almost all state variables. This invention realizes that such a tight coupling poses significant difficulty in design of speed sensorless control motor drives, including controller and estimator design, to achieve high-bandwidth speed control loop. Performing estimator design on the basis of the completely unstructured induction motor model in the original state coordinates, i.e., in Equations (1) or (2), is challenging and ineffective. 
     This invention realizes introduction of state transformations to represent the induction motor model under different state coordinates might partially break up coupling among state variables, and the resultant induction motor model after applying a state transformation, named after a transformed induction motor model, bears certain structures, which admit simple estimator design. The invention provides a method and system and embodiments for controlling an angular speed of the induction motor by introducing state transformations. 
     As shown in  FIG. 3 , a state transformation  301 , which defines a different set of state coordinates and facilities systematic estimator design, is incorporated into the estimator design flow. The state transformation  301  can be performed on the induction models of Equations (1) or (2), i.e., A direct—quadrature—zero (Park) transformation  203  may or may not be present. Applying the state transformation to the induction motor  201  gives a transformed induction motor model, on the basis of which the estimator design  302  is performed. 
     In one embodiment, the state transformation can be
 
 z ( x )=└ i   ds   i   qs αΦ dr +ωΦ qr αΦ qr −ωΦ dr ω┘.  (3)
 
where z=(z 1 ,z 2 ,z 3 ,z 4 ,z 5 ) T , and T is a transpose operator. One can verify that the state transformation is globally defined and has the inverse transformation
 
                 x   ⁡     (   z   )       =     [           z   1           z   2               α   ⁢           ⁢     z   3       -       z   4     ⁢     z   5         η               α   ⁢           ⁢     z   4       +       z   3     ⁢     z   5         η           z   5           ]       ,         
with η=α 2 +z 5   2 . The transformed induction motor model is written as
 
 ż=f   z ( z )+ g   z   1 ( z ) T   l   +g   z   2   u,  
 
 y=Cz,   (4)
 
where g z   2 =g x   2 , and
 
     
       
         
           
             
               
                 
                   f 
                   z 
                 
                 ⁡ 
                 
                   ( 
                   z 
                   ) 
                 
               
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                 [ 
                 
                   
                     
                       
                         
                           
                             - 
                             γ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             z 
                             1 
                           
                         
                         + 
                         
                           
                             ω 
                             1 
                           
                           ⁢ 
                           
                             z 
                             2 
                           
                         
                         + 
                         
                           β 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             z 
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             - 
                             
                               ω 
                               1 
                             
                           
                           ⁢ 
                           
                             z 
                             1 
                           
                         
                         - 
                         
                           γ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             z 
                             2 
                           
                         
                         + 
                         
                           β 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             z 
                             4 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             κ 
                             3 
                           
                           ⁡ 
                           
                             ( 
                             z 
                             ) 
                           
                         
                         
                           η 
                           2 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             κ 
                             4 
                           
                           ⁡ 
                           
                             ( 
                             z 
                             ) 
                           
                         
                         
                           η 
                           2 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             κ 
                             5 
                           
                           ⁡ 
                           
                             ( 
                             z 
                             ) 
                           
                         
                         η 
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   g 
                   z 
                   2 
                 
                 ⁡ 
                 
                   ( 
                   z 
                   ) 
                 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         
                           - 
                           
                             
                               ( 
                               
                                 
                                   α 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     z 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     z 
                                     3 
                                   
                                   ⁢ 
                                   
                                     z 
                                     5 
                                   
                                 
                               
                               ) 
                             
                             
                               η 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               J 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               
                                 α 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   z 
                                   3 
                                 
                               
                               - 
                               
                                 
                                   z 
                                   4 
                                 
                                 ⁢ 
                                 
                                   z 
                                   5 
                                 
                               
                             
                             ) 
                           
                           
                             η 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             J 
                           
                         
                       
                     
                     
                       
                         
                           - 
                           
                             1 
                             J 
                           
                         
                       
                     
                   
                   ] 
                 
                 . 
               
             
           
         
       
     
     The terms κ i ,3≦i≦5 are given by 
     
       
         
           
             
               
                 κ 
                 3 
               
               = 
               
                 
                   
                     α 
                     6 
                   
                   ⁢ 
                   
                     L 
                     m 
                   
                   ⁢ 
                   
                     z 
                     1 
                   
                 
                 + 
                 
                   
                     α 
                     5 
                   
                   ⁢ 
                   
                     L 
                     m 
                   
                   ⁢ 
                   
                     z 
                     2 
                   
                   ⁢ 
                   
                     z 
                     5 
                   
                 
                 + 
                 
                   2 
                   ⁢ 
                   
                       
                   
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                     α 
                     4 
                   
                   ⁢ 
                   
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                     m 
                   
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                     1 
                   
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                     z 
                     5 
                     2 
                   
                 
                 + 
                 
                   2 
                   ⁢ 
                   
                       
                   
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                     3 
                   
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                     m 
                   
                   ⁢ 
                   
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                     5 
                     3 
                   
                 
                 + 
                 
                   
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                     m 
                   
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             , 
             
               
 
             
             ⁢ 
             
               
                 κ 
                 4 
               
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                     6 
                   
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                     m 
                   
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                     2 
                   
                 
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                   ⁢ 
                   
                       
                   
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                     3 
                   
                   ⁢ 
                   
                     L 
                     m 
                   
                   ⁢ 
                   
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       FIGS. 4A and 4B  show the steps of methods and the structure of a system for controlling the angular speed of the the induction motor according to embodiments of the invention. The method and system can be implemented in a microcontroller, field programmable gate array (FPGA), digital signal processor (DSP), or custom logic. 
     In  FIG. 4A , a block  402  measures stator voltages and currents of the induction motor during operation. Both the stator voltages and currents are transformed using block  403  according to the state transformation to new coordinates, and fed into estimators of subsystems  430  to produces transformed state estimates of the transformed induction motor model. An inverse state transformation using block  440  is applied to the transformed state estimatess of the transformed induction motor model to produce the state estimate  450  of the induction motor model. Then, the controller  101  determines a control command to control the angular rotor speed of the induction motor  105  based on the state estimate  450 . 
       FIG. 4B  shows the steps for designing the estimators of subsystems on the basis of the induction motor model  401 . A state transformation  403  is applied to the model  401  of the induction motor  402 , see  FIG. 4A , to obtain a transformed induction motor model  405 . The transformed model is decomposed  410  into a set of subsystems  415  using Equations (4)  406 . Estimators of subsystems  430  are designed by applying a sequential state estimator design technique  420 . That is, the states of previous subsystems are known for subsequent subsystems, as described in detail below. 
       FIG. 5  shows one embodiment of decomposition where the transformed induction motor model  405 , represented by Equation (4), is decomposed into a set of three subsystems  502 ,  503 , and  504 . The states of the three subsystem  415  are, for example, respectively
 
Σ 1 :( z   1   ,z   3 ),
 
Σ 2 :( z   2   ,z   4 ), and
 
Σ 3   : z   5 .
 
     By verifying certain assumptions, for example, all states z are bounded, and subsystems Σ 1  and Σ 2  have certain structures, various systematic estimator design techniques such as a high gain observer or a finite time convergent observer of the states can be applied to produce state estimates {circumflex over (z)} 1 , {circumflex over (z)} 2 . The resultant estimators for subsystems Σ 1  and Σ 2  guarantees that estimation errors, i.e., a difference between the true state z 1 , z 2  and its estimate {circumflex over (z)} 1 ,{circumflex over (z)} 2 , are bounded or convergent to zero. 
       FIG. 6A  shows one embodiment of sequential estimator design based on the decomposition of the transformed induction motor model ( 4 ) according to  FIG. 5 . A state estimator  601  for subsystem Σ 1    502  is designed on the basis of the sensed stator current and voltage signals  211  and the model of subsystem Σ 1  to produce the state estimate {circumflex over (z)} 1    611  of the state z 1  of subsystem Σ 1 . A state estimator  602  for subsystem Σ 2  is designed on the basis of the sensed stator current and voltage signals  211 , estimated state  611 , and the model of subsystem Σ 2    503  to produce the state estimate {circumflex over (z)} 2    612  of the state z 2  of subsystem Σ 2 . A state estimator  603  is designed on the basis of stator current and voltage signals  211 , estimated states  611  and  612 , and the model of subsystem Σ 3    504 , to produce the state estimate {circumflex over (z)} 3    613 , of the state z 3  of subsystem Σ 3 . 
     Note that while designing the state estimator  601 , state variables z 2  and z 3  appearing in the model of Σ 1  are treated as bounded uncertainties. Similarly, while designing the state estimator  602 , state variable z 3  appearing in the model of Σ 2  is treated as bounded uncertainties, on the other hand, state variable z 1  appearing in the model of Σ 2  is treated as known and replaced by {circumflex over (z)} 1 ; while the design the state estimator  603 , both state variables {circumflex over (z)} 1  and {circumflex over (z)} 2  are treated as known and replaced by {circumflex over (z)} 1  and {circumflex over (z)} 2  respectively. 
     As an example, a high gain observer technique can be applied to design estimators  601  and  602 . While designing estimators using high gain observer technique, one can treat 
               κ   3       η   2           
as uncertainties bounded by L 1 &gt;0, and design the estimator  601  for the subsystem Σ 1  as follows
 
                   z     ^   .       1     =       [               -   γ     ⁢           ⁢       z   ^     1       +     β   ⁢           ⁢       z   ^     3                 0         ]     +       [           l   1               l   3           ]     ⁢     (       z   1     -       z   ^     1       )           ,         
where l 3 &gt;&gt;l 1 &gt;&gt;0 depend on the bound of uncertainties.
 
     Similarly, 
               κ   4       η   2           
can be treated as uncertainties bounded by L 2 , and the estimator  602  for subsystem Σ 2  takes the following expression
 
                   z     ^   .       2     =       [               -   γ     ⁢           ⁢       z   ^     2       +     β   ⁢           ⁢       z   ^     4                 0         ]     +       [           l   2               l   4           ]     ⁢     (       z   2     -       z   ^     2       )           ,         
where l 4 &gt;&gt;l 2 &gt;&gt;0 depend on L 2 . Similarly, with z 1  treated as known and replaced by {circumflex over (z)} 1 , the estimator  602  for subsystem Σ 2  can also be taken as follows
 
                   z     ^   .       2     =       [               -   γ     ⁢           ⁢       z   ^     2       +     β   ⁢           ⁢       z   ^     4                       κ   ^     4         η   ^     2             ]     +       [           l   2               l   4           ]     ⁢     (       z   2     -       z   ^     2       )           ,         
where
 
{circumflex over (η)}=α 2   +{circumflex over (z)}   5   2 ,
 
{circumflex over (κ)} 4 =κ( z   1   ,z   2   ,{circumflex over (z)}   3   ,{circumflex over (z)}   4   ,{circumflex over (z)}   5 ).
 
     Another embodiment of estimators  601  and  602  can be obtained by applying finite time convergent observer design techniques for both subsystems. For instance, a finite time convergent observer for Σ 2  is 
                   z     ^   .       2     =       [               -   γ     ⁢           ⁢       z   ^     2       +     β   ⁢           ⁢       z   ^     4                 0         ]     +       [           l   2               l   4           ]     ⁢           ⁢   sign   ⁢           ⁢     {     (       z   2     -       z   ^     2       )     }           ,         
where sign{ε} is an operator given by
 
     
       
         
           
             
               sign 
               ⁢ 
               
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                 ɛ 
                 } 
               
             
             = 
             
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                         , 
                       
                     
                     
                       
                         
                           
                             if 
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                           0 
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           - 
                           1 
                         
                         , 
                       
                     
                     
                       otherwise 
                     
                   
                 
                 . 
               
             
           
         
       
     
     One embodiment of estimator  603  has the following form 
                   z     ^   .       5     =         l   51     ⁡     (       z   1     -       z   ^     1       )       +       l   52     ⁡     (       z   2     -       z   ^     2       )       -       μ       z   ^     5       ⁡     [           ρ   1     ⁡     (   t   )       ⁢   α     +         ρ   2     ⁡     (   t   )       ⁢       z   ^     5         ]           ,         
where l 51  and l 52  are estimator gains, and
 
ρ 1 ( t )=2μα( z   1   {circumflex over (z)}   4   −z   2   {circumflex over (z)}   3 ), and
 
ρ 2 ( t )=2μ( z   1   {circumflex over (z)}   3   +z   2   {circumflex over (z)}   4 )
 
     If the sign of rotor rotation is known, another embodiment of estimator  603  is 
                   z       _   ^     .       5     =         l   51     ⁡     [           Φ   ^     qr     ⁡     (       z   1     -       z   ^     1       )       -         Φ   ^     dr     ⁡     (       z   2     -       z   ^     2       )         ]       -       1       z     _   ^       5       ⁡     [           ρ   1     ⁡     (   t   )       ⁢           z     _   ^       5     -     α   2           +         ρ   2     ⁡     (   t   )       ⁢     (         z     _   ^       5     -     α   2       )         ]           ,         
where l 51  and l 52  are constant, and
 
 {circumflex over (z)}   5 =√{square root over ( {circumflex over (z)}   5 )}sign( z   5 ).
 
       FIG. 6B  shows another embodiment of sequential design based on the decomposition of the transformed induction motor model according to  FIG. 5 . A state estimator  604  for subsystems Σ 1  and Σ 2  is designed on the basis of the sensed stator current and voltage signals  211  and the models of subsystems Σ 1  and Σ 2 , denoted by block  502  and  503 , to produce the state estimate {circumflex over (z)} l  and {circumflex over (z)} 2    614 , of the state z 1  and z 2 , respectively. The state estimator  605  for subsystem Σ 3  is designed on the basis of the sensed stator current and voltage signals, the estimated state  614 , and the model of subsystem Σ 3  block  504  to produce the state estimate {circumflex over (z)} 3    613  of the state z 2 , see  FIG. 6A . 
     In one embodiment, the estimator  604  for subsystems Σ 1  and Σ 2  is 
                       [             z     ^   .       1                 z     ^   .       2           ]     =       [               -   γ     ⁢           ⁢     z   1       +     β   ⁢           ⁢       z   ^     3                       -   γ     ⁢           ⁢     z   2       +     β   ⁢           ⁢       z   ^     4                       κ   ^     3         η   ^     2                     κ   ^     4         η   ^     2             ]     +       [           θ   ⁢           ⁢     I   2           0           0           θ   2     ⁢     I   2             ]     ⁢     S     -   1       ⁢       C   _     ⁡     (     y   -     y   ^       )             ,     
     ⁢       y   ^     =       [             z   ^     1             z   ^     2           ]     T               (   5   )               
where {circumflex over (z)} 1  and {circumflex over (z)} 2  are estimates of z 1  and z 2 , respectively,
 
                   κ   ^     3     =       κ   3     ⁡     (       z   1     ,     z   2     ,       z   ^     3     ,       z   ^     4     ,       z   ^     5       )         ,     
     ⁢       I   2     =     [         1       0           0       1         ]       ,     
     ⁢       C   _     =     [           I   2             0           0         ]       ,         
and S is a matrix determined by solving
 
 S+A   T   S+SA=CC   T  
 
with
 
     
       
         
           
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               . 
             
           
         
       
     
     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.