Method and system for controlling an angualr speed of an induction motor

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.

Claims

1. A method for controlling a rotor angular speed of an induction motor, comprising steps of: sensing an operation condition of the induction motor to produce measured signals; transforming the measured signal by applying a state transformation to an induction motor model to produce a transformed induction motor model; producing transformed state estimates of the transformed induction motor model based on the measured signals, wherein the transformed state estimates of the transformed induction motor model are produced by a set of estimators of a set of subsystems of the transformed induction motor model, wherein the set of the subsystems are determined by applying the state transformation to the induction motor model to obtain the transformed induction motor model; decomposing the transformed induction motor model into the set of the subsystems; and designing a state estimator of each subsystem by treating states of previous subsystems as known; applying an inverse of the state transformation to the transformed state estimates to produce state estimates of the induction motor model; determining control input voltages to the induction motor based on the state estimates; and applying the control input voltages to the induction motor to control the rotor angular speed.

2. The method of claim 1, wherein the measured signals are stator voltages and currents of the induction motor.

3. The method of claim 1, wherein states of the set of the subsystems are estimated sequentially so the states for a previous subsystems are known for subsequent subsystems.

4. The method of claim 1, wherein a particular state transformation is, where i_ds,i_qs,φ_dr,φ_qr,ω denote a stator current in a d-axis, a state current in q-axis, a rotor flux in the d-axis, a rotor flux in a q-axis, and the rotor angular speed, respectively, and α is predetermined constant.

5. The method of claim 4, wherein the set of the subsystems comprises a subsystem with states (i_ds,αφ_dr+ωφ_qr), a subsystem with states (i_qs,αφ_qr-ωφ_dr), and a subsystem with states ω.

6. The method of claim 4, where the set of the subsystems comprises a subsystem with states (i_ds,αφ_dr+ωφ_qr,i_qs,αφ_qr-ωφ_dr), and a subsystem with states ω.

7. The method of claim 4, wherein a particular state transformation is
z=[i_ds,i_qs,βφ_dr+i_ds,βφ_qr+i_qs,ω], where z denotes new coordinates, and β is a predetermined constant.

8. The method of claim 4, where the set of the subsystems comprise a subsystem with states (i_ds,i_qs,βφ_dr+i_ds,βφ_qr+i_qs), and a subsystem with states ω.

9. The method of claim 1, wherein a high gain observer is used for each subsystem.

10. The method of claim 1, wherein a finite time convergent observer is used for each subsystem.

11. A system for controlling a rotor angular speed of an induction motor, comprising: a sensor configured to sense an operation condition of the induction motor to produce measured signals; a transformation block configured to transform the measured signal by applying a state transformation to an induction motor model to produce a transformed induction motor model; means for producing transformed state estimates of the transformed induction motor model based on the measured signals, and applying an inverse of the state transformation to the transformed state estimates to produce state estimates of the induction motor model, wherein the transformed state estimates of the transformed induction motor model are produced by a set of estimators of a set of subsystems of the transformed induction motor model, wherein the set of the subsystems are determined by applying the state transformation to the induction motor model to obtain the transformed induction motor model; decomposing the transformed induction motor model into the set of the subsystems; and designing a state estimator of each subsystem by treating states of previous subsystems as known; means for determining control input voltages to the induction motor based on the state estimates, and applying the control input voltages to the induction motor to control the rotor angular speed.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIGS. 1A and 1B are block diagrams of a prior art sensorless speed motor drive;

(2) FIGS. 2A and 2B are block diagram of prior art estimator methods based on stator currents and voltages signals;

(3) FIG. 3 is a block diagram of a method for estimating a state of an induction motor according to embodiments of the invention;

(4) FIGS. 4A and 4B are block diagrams of decomposing a transformed induction motor model into multiple subsystems according to embodiments of the invention;

(5) FIG. 5 is a block diagram of decomposing a transformed induction motor model into multiple subsystems according to embodiments of the invention;

(6) 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

(7) 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

(8) The embodiments of the invention provide a method and system for controlling an angular rotor speed of an induction motor.

(9) 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.

(10) TABLE-US-00001 TABLE 1 Notations Notation Description Φ.sub.dr, Φ.sub.qr a. rotor fluxes in d- and q-axes ω b. rotor angular speed ξ c. angle of a rotating frame T.sub.l d. load torque J e. inertia of a rotor and a load L.sub.s, L.sub.m, L.sub.r f. stator, mutual, and rotor inductances R.sub.s, R.sub.r g. stator and rotor resistances ω.sub.1 h. angular speed of a rotating frame i.sub.ds, i.sub.qs i. stator currents in d- and q-axis u.sub.ds, u.sub.qs j. stator voltages in d- and q-axes σ k. $\frac{L_s{L}_r-L_m^2}{L_s{L}_r}$

(11) Induction Motor Model

(12) 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

(13) $\begin{array}{cc}{\stackrel{.}{i}}_{\mathrm{ds}}=-\gamma i_{\mathrm{ds}}+{\omega }_1{i}_{\mathrm{qs}}+\beta \left(\alpha {\Phi }_{\mathrm{dr}}+\omega {\Phi }_{\mathrm{qr}}\right)+\frac{u_{\mathrm{ds}}}{L_s\sigma }{\stackrel{.}{i}}_{\mathrm{qs}}=-\gamma i_{\mathrm{qs}}-{\omega }_1{i}_{\mathrm{ds}}+\beta \left(\alpha {\Phi }_{\mathrm{qr}}-\omega {\Phi }_{\mathrm{dr}}\right)+\frac{u_{\mathrm{qs}}}{L_s\sigma }{\stackrel{.}{\Phi }}_{\mathrm{dr}}=-\alpha {\Phi }_{\mathrm{dr}}+\left({\omega }_1-\omega \right){\Phi }_{\mathrm{qr}}+\alpha L_m{i}_{\mathrm{ds}},{\stackrel{.}{\Phi }}_{\mathrm{qr}}=-\alpha {\Phi }_{\mathrm{qr}}-\left({\omega }_1-\omega \right){\Phi }_{\mathrm{dr}}+\alpha L_m{i}_{\mathrm{qs}},\stackrel{.}{\omega }=\mu \left({\Phi }_{\mathrm{dr}}{i}_{\mathrm{qs}}-{\Phi }_{\mathrm{qr}}{i}_{\mathrm{ds}}\right)-\frac{T_l}{J},y={\left[\begin{array}{cc}{i}_{\mathrm{ds}}& i_{\mathrm{qs}}\end{array}\right]}^T,& \left(1\right)\end{array}$
where y represents sensed signals, ω.sub.I is the angular speed of a reference frame, and

(14) $\gamma =\frac{1}{L_s\sigma }\left(R_s+\frac{L_m^2}{L_r}\frac{R_r}{L_r}\right),\alpha =\frac{R_r}{L_r},\beta =\frac{1}{L_s\sigma }\frac{L_m}{L_r},\mu =\frac{2}{3}\frac{L_m}{L_s{L}_r-L_m^2}.$

(15) Note that the induction motor model (1) is in an orthognal rotating frame with a rotation speed of ω.sub.1; and quantities i.sub.ds, i.sub.qs, Φ.sub.dr, Φ.sub.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).

(16) When ω.sub.1=0, the equations in the model (1) reduced to

(17) $\begin{array}{cc}{\stackrel{.}{i}}_{\mathrm{ds}}=-\gamma i_{\mathrm{ds}}+\beta \left(\alpha {\Phi }_{\mathrm{dr}}+\omega {\Phi }_{\mathrm{qr}}\right)+\frac{u_{\mathrm{ds}}}{L_s\sigma }{\stackrel{.}{i}}_{\mathrm{qs}}=-\gamma i_{\mathrm{qs}}+\beta \left(\alpha {\Phi }_{\mathrm{qr}}-\omega {\Phi }_{\mathrm{dr}}\right)+\frac{u_{\mathrm{qs}}}{L_s\sigma }{\stackrel{.}{\Phi }}_{\mathrm{dr}}=-\alpha {\Phi }_{\mathrm{dr}}-{\mathrm{\omega \Phi }}_{\mathrm{qr}}+\alpha L_m{i}_{\mathrm{ds}},{\stackrel{.}{\Phi }}_{\mathrm{qr}}=-\alpha {\Phi }_{\mathrm{qr}}+{\mathrm{\omega \Phi }}_{\mathrm{dr}}+\alpha L_m{i}_{\mathrm{qs}},\stackrel{.}{\omega }=\mu \left({\Phi }_{\mathrm{dr}}{i}_{\mathrm{qs}}-{\Phi }_{\mathrm{qr}}{i}_{\mathrm{ds}}\right)-\frac{T_l}{J},y={\left[\begin{array}{cc}{i}_{\mathrm{ds}}& i_{\mathrm{qs}}\end{array}\right]}^T,& \left(2\right)\end{array}$
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.sub.ds, i.sub.qs, Φ.sub.dr, Φ.sub.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).

(18) Conventional estimator designs are usually based on the model according to Equations (1) or (2), which have the same state coordinates denoted by (i.sub.ds, i.sub.qs, Φ.sub.dr, Φ.sub.qr, ω).sup.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.sub.qr in the right hand side of the differential Equation defining i.sub.ds, i.e.,

(19) ${\stackrel{.}{i}}_{\mathrm{ds}}=-\gamma i_{\mathrm{ds}}+\beta \left(\alpha {\Phi }_{\mathrm{dr}}+\omega {\Phi }_{\mathrm{qr}}\right)+\frac{u_{\mathrm{ds}}}{L_s\sigma }.$

(20) 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.

(21) 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.

(22) 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.

(23) In one embodiment, the state transformation can be
z(x)=└i.sub.dsi.sub.qsαΦ.sub.dr+ωΦ.sub.qrαΦ.sub.qr−ωΦ.sub.drω┘.  (3)
where z=(z.sub.1,z.sub.2,z.sub.3,z.sub.4,z.sub.5).sup.T, and T is a transpose operator. One can verify that the state transformation is globally defined and has the inverse transformation

(24) $x\left(z\right)=\left[\begin{array}{ccccc}{z}_1& z_2& \frac{\alpha z_3-z_4{z}_5}{\eta }& \frac{\alpha z_4+z_3{z}_5}{\eta }& z_5\end{array}\right],$
with η=α.sup.2+z.sub.5.sup.2. The transformed induction motor model is written as
ż=f.sub.z(z)+g.sub.z.sup.1(z)T.sub.l+g.sub.z.sup.2u,
y=Cz,  (4)
where g.sub.z.sup.2=g.sub.x.sup.2, and

(25) $f_z\left(z\right)=\left[\begin{array}{c}-\gamma z_1+{\omega }_1{z}_2+\beta z_3\\ -{\omega }_1{z}_1-\gamma z_2+\beta z_4\\ \frac{{\kappa }_3\left(z\right)}{{\eta }^2}\\ \frac{{\kappa }_4\left(z\right)}{{\eta }^2}\\ \frac{{\kappa }_5\left(z\right)}{\eta }\end{array}\right],g_z^2\left(z\right)=\left[\begin{array}{c}0\\ 0\\ -\frac{\left(\alpha z_4+z_3{z}_5\right)}{\eta J}\\ \frac{\left(\alpha z_3-z_4{z}_5\right)}{\eta J}\\ -\frac{1}{J}\end{array}\right].$

(26) The terms κ.sub.i,3≦i≦5 are given by

(27) ${\kappa }_3={\alpha }^6{L}_m{z}_1+{\alpha }^5{L}_m{z}_2{z}_5+2{\alpha }^4{L}_m{z}_1{z}_5^2+2{\alpha }^3{L}_m{z}_2{z}_5^3+{\alpha }^2{L}_m{z}_1{z}_5^4+\alpha L_m{z}_2{z}_5^5-{\alpha }^2\mu z_1{z}_4^2+{\alpha }^2\mu z_2{z}_3{z}_4-2\alpha \mu z_1{z}_3{z}_4{z}_5+\alpha \mu z_2{z}_3^2{z}_5-\alpha \mu z_2{z}_4^2{z}_5-\mu z_1{z}_3^2{z}_5^2-\mu z_2{z}_3{z}_4{z}_5^2-{\alpha }^5{z}_3-{\alpha }^4{z}_4{z}_5-2{\alpha }^3{z}_3{z}_5^2-2{\alpha }^2{z}_4{z}_5^3-\alpha z_3{z}_5^4-z_4{z}_5^5,{\kappa }_4={\alpha }^6{L}_m{z}_2-{\alpha }^5{L}_m{z}_1{z}_5+2{\alpha }^4{L}_m{z}_2{z}_5^2-2{\alpha }^3{L}_m{z}_1{z}_5^3+{\alpha }^2{L}_m{z}_2{z}_5^4-\alpha L_m{z}_1{z}_5^5+{\alpha }^2\mu z_1{z}_3{z}_4-{\alpha }^2\mu z_2{z}_3^2+\alpha \mu z_1{z}_3^2{z}_5-\alpha \mu z_1{z}_4^2{z}_5+2\alpha \mu z_2{z}_3{z}_4{z}_5-\mu z_1{z}_3{z}_4{z}_5^2-\mu z_2{z}_4^2{z}_5^2-{\alpha }^5{z}_4+{\alpha }^4{z}_3{z}_5-2{\alpha }^3{z}_4{z}_5^2+2{\alpha }^2{z}_3{z}_5^3-\alpha z_4{z}_5^4+z_3{z}_5^5,{\kappa }_5=-\mu \left[\alpha \left(z_1{z}_4-z_2{z}_3\right)+\left(z_1{z}_3+z_2{z}_4\right)z_5\right].$

(28) 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.

(29) 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.

(30) 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.

(31) 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.sub.1:(z.sub.1,z.sub.3),
Σ.sub.2:(z.sub.2,z.sub.4), and
Σ.sub.3: z.sub.5.

(32) By verifying certain assumptions, for example, all states z are bounded, and subsystems Σ.sub.1 and Σ.sub.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)}.sup.1, {circumflex over (z)}.sup.2. The resultant estimators for subsystems Σ.sub.1 and Σ.sub.2 guarantees that estimation errors, i.e., a difference between the true state z.sup.1, z.sup.2 and its estimate {circumflex over (z)}.sup.1,{circumflex over (z)}.sup.2, are bounded or convergent to zero.

(33) 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.sub.1 502 is designed on the basis of the sensed stator current and voltage signals 211 and the model of subsystem Σ.sub.1 to produce the state estimate {circumflex over (z)}.sup.1 611 of the state z.sup.1 of subsystem Σ.sub.1. A state estimator 602 for subsystem Σ.sub.2 is designed on the basis of the sensed stator current and voltage signals 211, estimated state 611, and the model of subsystem Σ.sub.2 503 to produce the state estimate {circumflex over (z)}.sup.2 612 of the state z.sup.2 of subsystem Σ.sub.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.sub.3 504, to produce the state estimate {circumflex over (z)}.sup.3 613, of the state z.sup.3 of subsystem Σ.sub.3.

(34) Note that while designing the state estimator 601, state variables z.sup.2 and z.sup.3 appearing in the model of Σ.sub.1 are treated as bounded uncertainties. Similarly, while designing the state estimator 602, state variable z.sup.3 appearing in the model of Σ.sub.2 is treated as bounded uncertainties, on the other hand, state variable z.sup.1 appearing in the model of Σ.sub.2 is treated as known and replaced by {circumflex over (z)}.sup.1; while the design the state estimator 603, both state variables {circumflex over (z)}.sup.1 and {circumflex over (z)}.sup.2 are treated as known and replaced by {circumflex over (z)}.sup.1 and {circumflex over (z)}.sup.2 respectively.

(35) 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

(36) $\frac{{\kappa }_3}{{\eta }^2}$
as uncertainties bounded by L.sub.1>0, and design the estimator 601 for the subsystem Σ.sub.1 as follows

(37) 0${\stackrel{\stackrel{.}{^}}{z}}^1=\left[\begin{array}{c}-\gamma {\hat{z}}_1+\beta {\hat{z}}_3\\ 0\end{array}\right]+\left[\begin{array}{c}{l}_1\\ l_3\end{array}\right]\left(z_1-{\hat{z}}_1\right),$
where l.sub.3>>l.sub.1>>0 depend on the bound of uncertainties.

(38) Similarly,

(39) $\frac{{\kappa }_4}{{\eta }^2}$
can be treated as uncertainties bounded by L.sub.2, and the estimator 602 for subsystem Σ.sub.2 takes the following expression

(40) ${\stackrel{\stackrel{.}{^}}{z}}^2=\left[\begin{array}{c}-\gamma {\hat{z}}_2+\beta {\hat{z}}_4\\ 0\end{array}\right]+\left[\begin{array}{c}{l}_2\\ l_4\end{array}\right]\left(z_2-{\hat{z}}_2\right),$
where l.sub.4>>l.sub.2>>0 depend on L.sub.2. Similarly, with z.sup.1 treated as known and replaced by {circumflex over (z)}.sup.1, the estimator 602 for subsystem Σ.sub.2 can also be taken as follows

(41) ${\stackrel{\stackrel{.}{^}}{z}}^2=\left[\begin{array}{c}-\gamma {\hat{z}}_2+\beta {\hat{z}}_4\\ \frac{{\hat{\kappa }}_4}{{\hat{\eta }}^2}\end{array}\right]+\left[\begin{array}{c}{l}_2\\ l_4\end{array}\right]\left(z_2-{\hat{z}}_2\right),$
where
{circumflex over (η)}=α.sup.2+{circumflex over (z)}.sub.5.sup.2,
{circumflex over (κ)}.sub.4=κ(z.sub.1,z.sub.2,{circumflex over (z)}.sub.3,{circumflex over (z)}.sub.4,{circumflex over (z)}.sub.5).

(42) 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.sub.2 is

(43) ${\stackrel{\stackrel{.}{^}}{z}}^2=\left[\begin{array}{c}-\gamma {\hat{z}}_2+\beta {\hat{z}}_4\\ 0\end{array}\right]+\left[\begin{array}{c}{l}_2\\ l_4\end{array}\right]\mathrm{sign}\left\{\left(z_2-{\hat{z}}_2\right)\right\},$
where sign{ε} is an operator given by

(44) $\mathrm{sign}\left\{\mathrm{.Math.}\right\}=\{\begin{array}{cc}1,& \mathrm{if}\mathrm{.Math.}>0,\\ -1,& \mathrm{otherwise}\end{array}.$

(45) One embodiment of estimator 603 has the following form

(46) ${\stackrel{\stackrel{.}{^}}{z}}_5=l_{51}\left(z_1-{\hat{z}}_1\right)+l_{52}\left(z_2-{\hat{z}}_2\right)-\frac{\mu }{{\hat{z}}_5}\left[{\rho }_1\left(t\right)\alpha +{\rho }_2\left(t\right){\hat{z}}_5\right],$
where l.sub.51 and l.sub.52 are estimator gains, and
ρ.sub.1(t)=2μα(z.sub.1{circumflex over (z)}.sub.4−z.sub.2{circumflex over (z)}.sub.3), and
ρ.sub.2(t)=2μ(z.sub.1{circumflex over (z)}.sub.3+z.sub.2{circumflex over (z)}.sub.4)

(47) If the sign of rotor rotation is known, another embodiment of estimator 603 is

(48) ${\stackrel{\stackrel{.}{\widehat{\_}}}{z}}_5=l_{51}\left[{\hat{\Phi }}_{\mathrm{qr}}\left(z_1-{\hat{z}}_1\right)-{\hat{\Phi }}_{\mathrm{dr}}\left(z_2-{\hat{z}}_2\right)\right]-\frac{1}{{\stackrel{\widehat{\_}}{z}}_5}\left[{\rho }_1\left(t\right)\sqrt{{\stackrel{\widehat{\_}}{z}}_5-{\alpha }^2}+{\rho }_2\left(t\right)\left({\stackrel{\widehat{\_}}{z}}_5-{\alpha }^2\right)\right],$
where l.sub.51 and l.sub.52 are constant, and
{circumflex over (z)}.sub.5=√{square root over ({circumflex over (z)}.sub.5)}sign(z.sub.5).

(49) 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.sub.1 and Σ.sub.2 is designed on the basis of the sensed stator current and voltage signals 211 and the models of subsystems Σ.sub.1 and Σ.sub.2, denoted by block 502 and 503, to produce the state estimate {circumflex over (z)}.sup.l and {circumflex over (z)}.sup.2 614, of the state z.sup.1 and z.sup.2, respectively. The state estimator 605 for subsystem Σ.sub.3 is designed on the basis of the sensed stator current and voltage signals, the estimated state 614, and the model of subsystem Σ.sub.3 block 504 to produce the state estimate {circumflex over (z)}.sup.3 613 of the state z.sup.2, see FIG. 6A.

(50) In one embodiment, the estimator 604 for subsystems Σ.sub.1 and Σ.sub.2 is

(51) $\begin{array}{cc}\left[\begin{array}{c}{\stackrel{\stackrel{.}{^}}{z}}^1\\ {\stackrel{\stackrel{.}{^}}{z}}^2\end{array}\right]=\left[\begin{array}{c}-\gamma z_1+\beta {\hat{z}}_3\\ -\gamma z_2+\beta {\hat{z}}_4\\ \frac{{\hat{\kappa }}_3}{{\hat{\eta }}^2}\\ \frac{{\hat{\kappa }}_4}{{\hat{\eta }}^2}\end{array}\right]+\left[\begin{array}{cc}\theta I_2& 0\\ 0& {\theta }^2{I}_2\end{array}\right]S^{-1}\stackrel{\_}{C}\left(y-\hat{y}\right),\hat{y}={\left[\begin{array}{cc}{\hat{z}}_1& {\hat{z}}_2\end{array}\right]}^T& \left(5\right)\end{array}$
where {circumflex over (z)}.sup.1 and {circumflex over (z)}.sup.2 are estimates of z.sub.1 and z.sub.2, respectively,

(52) ${\hat{\kappa }}_3={\kappa }_3\left(z_1,z_2,{\hat{z}}_3,{\hat{z}}_4,{\hat{z}}_5\right),I_2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\stackrel{\_}{C}=\left[\begin{array}{c}{I}_2\\ 0\\ 0\end{array}\right],$
and S is a matrix determined by solving
S+A.sup.TS+SA=CC.sup.T
with

(53) 0$A=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right].$

(54) 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.