Global Adaptative Output Feedback Control of Induction Motor With

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 5, MAY 1999 967

Global Adaptive Output Feedback Controlof Induction Motors with Uncertain

Rotor ResistanceRiccardo Marino, Sergei Peresada, and Patrizio Tomei

Abstract— The authors design a global adaptive output feed-back control for a fifth-order model of induction motors, whichguarantees asymptotic tracking of smooth speed references on thebasis of speed and stator current measurements, for any initialcondition and for any unknown constant value of torque loadand rotor resistance.

The proposed seventh-order nonlinear compensator generatesestimates both for the unknown parameters (torque load androtor resistance) and for the unmeasured state variables (rotorfluxes); they converge to the corresponding true values underpersistency of excitation which actually holds in typical operating

conditions. In these cases the rotor flux modulus asymptoticallytracks desired smooth reference signals which allows the motorto operate within saturation limits (so that modeling assumptionsare met). As in field-oriented control, the control algorithmgenerates references for the magnetizing flux component and forthe torque component of stator current which lead to significantsimplifications for current-fed motors. Simulations show that theproposed controller is suitable for high dynamic performanceapplications. This is confirmed by experiments since the controlis robust with respect to modeling errors, sensors and actuatorsnoise, control discretization, and simplification. Experimentalcomparisons with a classical indirect field-oriented control ex-hibit a significant improvement of power efficiency when rotorresistance differs from its nominal value.

Index Terms— Adaptive nonlinear control, induction motor,

online parameter estimation, output feedback.

I. INTRODUCTION

INDUCTION motors are more reliable and less expensive

than those permanent magnet switched reluctance and d.c.

motors which are currently employed for high dynamic per-

formance applications; this is due to their simpler construction

since they have no brushes, no commutator, no permanent

magnet, and no windings in squirrel cage rotors.

On the other hand, the control of induction motors is rather

difficult. It is a highly coupled and nonlinear multivariable

problem with two control inputs (stator voltages) and two

output variables (rotor speed and flux modulus), required totrack desired reference signals. Since flux sensors are not

available, it is an output feedback problem in which the outputs

to be controlled do not coincide with measured outputs (speed

Manuscript received May 16, 1997; revised June 30, 1998. Recommendedby Associate Editor, J.-B. Pomet. This work was supported in part by MURST.

R. Marino and P. Tomei are with the Dipartimento di Ingegneria Elettronica,Universita di Roma “Tor Vergata,” 00133 Roma, Italy.

S. Peresada is with the Department of Electrical Engineering, Kiev Poly-technical Institute, Kiev 252056, Ukraine.

Publisher Item Identifier S 0018-9286(99)02825-1.

and stator currents). Moreover, there are uncertain critical

parameters: in addition to load torque, which is typically

unknown in all electric drives, in induction motors rotor

resistance is also largely uncertain since it may vary up to

100% during operations, due to rotor heating. In comparison,

the control of currently employed high dynamic performance

motors is much simpler; this is in fact one of the reasons for

their widespread use. However, the availability of low cost

powerful digital signal processors (requiring only 40 ns to ex-

ecute a simple instruction) and advances in power electronicsmake complex control algorithms now implementable even

for medium- and small-size induction motors which could

replace currently used motors, provided that high dynamic

performance in positioning and speed tracking along with

high-power efficiency are achieved; this motivates current

research efforts in induction motor control design.

Assuming that all state variables (including rotor fluxes) are

available from measurements and all parameters (including ro-

tor resistance) are known, the problem of controlling induction

motors was solved by the so-called field-oriented control [1],

[2] and, more recently, by the input–output linearizing control

[3] (see also [4] for a different feedback linearizing control and

[5] for a variable structure approach). As clarified in [6], bothfield-oriented and input–output linearizing controls make use

of nonlinear state-space change of coordinates and nonlinear

state feedback transformations to achieve different objectives.

While input–output linearization guarantees exact decoupling

so that speed and flux modulus can be independently controlled

with linear dynamics, field-oriented control achieves the same

properties only asymptotically, provided that the reference

for the flux modulus is constant. This is a disadvantage

especially in low power motors since at high speed, when the

flux modulus reference has to be lowered (flux weakening)

to operate the motor within the linear magnetic region and

without voltage saturation or varied for torque maximization

(see [7]), the speed tracking dynamics are perturbed.

Advances in nonlinear adaptive control (see for instance

[8]) allowed us to design in [6] an input–output state feedback

linearizing control which is adaptive with respect to both

load torque and rotor resistance. It is shown in [6] that

load torque and rotor resistance estimation errors converge to

zero in physical operating conditions. However, flux sensors

are typically not available in induction motors since they

reduce reliability and imply additional costs and technological

difficulties. Therefore, the design of state feedback controls is

0018–9286/99$10.00 © 1999 IEEE



968 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 5, MAY 1999

only a first step toward the solution of the output feedback

control of induction motors which points out two facts: high

tracking performance is achieved while rotor resistance is

estimated online; the resulting control is highly nonlinear but

implementable with available microprocessors. The currently

open problem, which is solved in this paper, is whether the

same results may be obtained without flux measurements.

Provided that rotor resistance is known, flux observers have

been studied in [9]–[11] and exponentially convergent flux

estimators are designed in [12] and [13]. They may be rather

sensitive with respect to rotor resistance variations. Exponen-

tially convergent flux observers have been tested in closed loop

in experiments and simulations providing flux estimates both

to field-oriented [14], [2], [15] and input–output linearizing

algorithms [16]–[20]. An independent line of research was

focused on the design of rotor resistance estimators starting

with [21] (see [22]–[28]), in which simplifying assumptions

are made including linear approximations and steady state

operations. More recently, an online nonlinear exponentially

convergent rotor resistance estimator and an adaptive flux

observer have been obtained in [29] which shows that bothrotor fluxes and rotor resistance may be estimated on line from

speed and stator current measurements. An output feedback

algorithm consisting of a flux observer which is adaptive

with respect to rotor resistance together with a state feedback

controller which is adaptive with respect to torque load is given

in [30]; however, the closed-loop behavior is only illustrated

by simulations since its stability is not proved in [30].

In fact, as far as nonlinear systems are concerned, replacing

state measurements by exponentially convergent state esti-

mates in global tracking controls does not yield, in general,

global output feedback tracking controls. This property, which

holds for linear systems, has to be independently established

for nonlinear ones. Therefore the output feedback control prob-lem for induction motors was addressed as such in [31]–[36],

assuming known rotor resistance. In [31] speed and flux

tracking are achieved by output feedback provided that the

initial conditions lie in a predetermined region. In the series

of papers [33]–[36], torque and flux tracking are globally

obtained by output feedback using a passivity approach. As

far as current-fed induction motors are concerned, an output

feedback controller is presented in [37] which is free of

singularities and experimentally gives high performance both

in speed and flux tracking: however, it is shown in [37]

that errors in rotor resistance reduce power efficiency since

higher currents are needed. Hence, it is proposed in [37] to

update the rotor resistance value used by the controller bya rotor resistance estimator; nevertheless, since no stability

proof has been obtained in [37] for the closed-loop system

with continuously updated rotor resistance, the design of a

global output feedback control which is adaptive with respect

to rotor resistance, is still an open problem. Very recently, an

adaptive output feedback position tracking controller is pro-

posed in [38]; position tracking is achieved despite unknown

inertia, load, and rotor resistance. The controller exhibits

a singularity when the magnitude of the estimated rotor

flux is zero and does not provide converging estimates for

rotor resistance and rotor flux so that the rotor flux tracking

objective (which is crucial to improve power efficiency) is not

fulfilled.

The contribution of this paper is to design a global output

feedback control for induction motors which is adaptive with

respect to both load torque and rotor resistance and guarantees

asymptotic tracking of smooth speed references for any initial

condition of the closed-loop motor. The resulting control is

a seventh-order dynamic compensator which provides voltage

inputs on the basis of rotor speed and stator currents measure-

ments. The dynamic control algorithm generates estimates for

load torque, rotor resistance, and rotor fluxes that converge to

the corresponding true values under persistency of excitation,

which actually holds in physical operating conditions. The con-

trol also generates as internal signals stator current estimates

and an estimate for the rotor flux angle. When persistency

of excitation condition is satisfied the rotor flux modulus

asymptotically tracks desired smooth reference signals, so that

the motor operates within saturation limits at higher speed

and the modeling assumptions (linear magnetic circuits) are

met. As in classical field-oriented control, reference signals

both for the direct and the quadrature components of statorcurrents (in a frame attached to the estimated rotating flux

vector) are generated which are responsible for flux modulus

and speed tracking, respectively. Direct and quadrature current

errors are forced to zero so that simplified controls may be used

in current-fed machines relying on high gain current loops.

Simulations and experiments show converging estimates

both for rotor flux and for unknown parameters within 2 s

and very precise tracking of speed and flux modulus reference

signals, which are typically required in high-performance

applications in spite of sharp load torque variations and rotor

resistance uncertainties. Experiments confirm that since the

proposed control algorithm achieves rotor resistance estima-

tion and flux modulus tracking, an improved power efficiencyis obtained in comparison with a classical control scheme such

as the indirect field-oriented control.

II. ADAPTIVE OUTPUT FEEDBACK CONTROL DESIGN

A. Problem Statement

Assuming linear magnetic circuits, i.e., no magnetic satura-

tion, the dynamics of a balanced induction motor in a fixed

reference frame attached to the stator are given by the fifth-

order model (see for instance [6] for its derivation and [14]

and [39] for modeling assumptions)

(1)

in which the state variables are rotor speed , rotor fluxes

and stator currents ; the control inputs are



MARINO et al.: GLOBAL ADAPTIVE OUTPUT FEEDBACK 971

We replace the unknown variables (recall that

are not measured) by the new error variables

(24)

so that the error equations (21) and (22) are expressed in new

coordinates as

(25)

The advantage of using the unknown variables instead

of relies on the fact that their dynamics no longer

depend on [compare (25) with (21)]. We now define some

of the yet undetermined terms in (17), (18), (22) as ( and

in (22) are still to be chosen)

(26)

with positive design parameter and estimates of

the unknown error variables

defined by (24). Note that the dynamics of

are yet to be defined. If converge to ,

they allow us to recover the rotor flux vector since

are known variables and is a known parameter. Hence, the

new variables may be viewed as rotor flux estimates.

Substituting (26) in (25), we obtain

(27)

We compute from (3), (17), (25), and (26) the dynamics of the

stator current tracking errors

(28)

with




(29)

In order to determine , the dynamics of the estima-

tion variables , and the feedback control inputs, we consider the function

(30)

in which are positive parameters and

are estimation errors. We define

with and to guarantee that is

chosen so that with the minimum value of

(assumed to be known). Alternatively, we may choose

and with the maximum value of (assumed to

be known). From (27) and (30), the time derivative of (30)

results in

(31)

We now consider the function

(32)

From (31), (27), and (28), its time derivative is

(33)

The terms , the feedback controls , and the

dynamics of are now chosen in order to force

to be negative semidefinite. Choosing the yet undetermined

terms in (22)

(34)




we obtain

(35)

We finally define

(36)

(37)

so that (35) becomes

(38)

Equations (36), (37), (29), (22), (26), (34), (17), (18), and (2)

define a seventh-order dynamic feedback compensator, whose

state variables are , which generates

the control signals on the basis of the measurements

, the reference signals and their time deriva-

tives . In order to guarantee that for

every we modify the dynamics (37) according to

(39)

where is the smooth projection algorithm given in

[40] and defined in our case by

if

if and

otherwise

in which with the minimum

(known) value of and such that . The

initial condition in (39) is chosen so that . The

projection algorithm has the following properties:1) ;

2) is Lipschitz continuous;

3) ;

4) .

Property 4) implies that substituting (39) instead of (37) in

(35), we obtain instead of (38) the inequality

(40)

From (32) and (40), it follows that

are bounded for every . Their

bounds depend on the initial errors. Therefore,according to (36), and, consequently,

are bounded for every

. Hence, are bounded and therefore

are uniformly continuous. On the other hand,

integrating (40) we have

which implies by Barbalat’s lemma (see for instance [41] and

[8]) that

This shows that asymptotic speed tracking is achieved from

any initial condition provided that in (39). Moreover,

current estimation errors and current tracking errors

asymptotically tend to zero.




D. Parameter Convergence

We now analyze under which conditions also

tend asymptotically to zero.

The error equations are

(41)

Whenever projection does not occur, i.e., , (41)

may be rewritten as

(42)

with and

the expressions shown at the bottom of the page.

Making the change of coordinates

with , from (42) we obtain

Since the structure of and is such that

setting

we finally have




The radially unbounded function (32) may be written as

, whose time derivative is

. Since is skew-symmetric and recall-

ing the structure of and , we obtain , with

. From now on, using

the same arguments adopted in the proof of the persistency of

excitation Lemma B.2.3 in [8, p. 367] we can establish that

if persistency of excitation conditions are satisfied, i.e., thereexist two positive constants and such that

(43)

then the equilibrium point of system (42) is uni-

formly asymptotically stable and all trajectories tend asymp-

totically to zero.

If projection occurs, it is easy to see that the arguments in

the proof of [8, Lemma B.2.3] are still valid and, therefore,

we can conclude that if (43) holds then the equilibrium

point of system (41)is uniformly asymptotically stable and all trajectories tend

asymptotically to zero provided that . In summary,

if (43) is satisfied then, in addition to asymptotic speed

tracking:

1) since both and tend to zero, from (24)

it follows that rotor flux traking is achieved and, in

addition, the rotor flux vector is asymptotically oriented

with respect to the frame, i.e.,

2) since tend to zero, both rotor resistance and

torque load are asymptotically estimated, i.e.,

3) since tend to zero, the rotor flux vector is

asymptotically estimated: defining, according to (24)

(44)

we have in fact

III. CONTROL IMPLEMENTATION AND PERFORMANCE

A. Control Implementation

Let us first summarize the seventh-order dynamic feedback

control algorithm designed in Section II

(45)

with given in (26), (34) and

, given in (29).

The dynamic compensator (45) contains eight control pa-

rameters whose role may be eval-

uated by examining both the closed-loop error equations (41)and the corresponding function (32) with time derivative (40).

The parameters determine the rate of decay of

in (40) and directly affect [see (41)] the dynamics of

speed tracking error , current estimation errors and

current tracking errors , respectively. The parameter

determines the influence of speed tracking errors on the

other errors and is typically chosen much smaller than one.

The parameters and may be separately tuned using

(16), (9), and (13) so that has the desired transients.

The parameters and are the adaptation gains for

and , respectively. The smaller they are chosen, the




slower the adaptations for and result. The parameter

is a weighting factor in the function

(32); the choice of depends on the interval of variation of

the uncertain parameter . The parameter should be the

last parameter to be tuned and should be chosen sufficiently

large so that the current error dynamics are much faster than

speed error dynamics while voltages are within saturation

limits. Of course, different tuning strategies should be followed

depending on sensor noise features, discretization strategies,

and other implementation issues.

For induction motors which allow for high gain current

loops the control may be greatly simplified as follows:

(46)

with obtained by setting in (17), (18),(22), (26), (34), (36), and (39)

(47)

B. Control Performance

We tested the proposed controller (45) both by simulations

and by experiments [using its simplified version (46), (47)] for

a three-phase single pole pair 0.6-kW induction motor (OE-

MER 7-80/C), whose parameters are listed in the Appendix

(see [29] for experimental and computed static speed-torque

characteristics). Since flux and torque measurements are not

available during the experiments, we first tested by simula-

tions how a typical flux modulus reference (including flux

weakening) is tracked and the role of torque in persistency of

excitation condition (43) when the controller (45) is used. We

then illustrate by experiments the robustness of the simplified

controller (46), (47) with respect to sensors noise, inaccura-

cies on mechanical and electrical parameters, control signals

distortions generated by the power inverter, control discretiza-

tion and truncation errors, inverter unmodeled dynamics, and

unmodeled saturation effects of the magnetic circuits. Finally,

we compare performances achieved by the proposed control

with those given by a classical indirect field-oriented control.

The proposed control algorithm (45) has been tested first

by simulation with the control parameters (all values are in SI

units):

. Recall that may be chosen

negative provided that is negative. All initial conditionsof the motor and of the controller are set to zero excepting

Wb and s , which is 50% greater

than the true parameter value s . The references

for speed and flux modulus along with the applied torque are

reported in Fig. 1. The flux reference starts from 0.01 Wb at

and grows up to the rated constant value 1.16 Wb;

field weakening starts at s. The speed reference is

zero until s and grows up to the constant value 100

rad/s; at s the speed is required to go up to the value

200 rad/s, while the reference for the flux is reduced to 0.5

Wb. A constant load torque (5.8 Nm, the rated value) which

is unknown to the controller is applied at s and reduced

to 0.5 Nm at s. Fig. 2 shows the time histories of speed,flux modulus, load torque estimate and , the estimate

of . The speed tracks tightly the reference even

though load torque sharply changes (at s and s),

since the load torque estimate quickly recovers the applied

unknown value. Also the estimate of converges within 1 s

to the true value. Note that the higher the torque is (see

Fig. 3), the larger the convergence rate for is. Persistency of

excitation condition (43) has been checked to hold. The flux

tracks its reference: there is, however, a coupling with speed

tracking at s and at s when speed is perturbed

by an unknown load torque. Fig. 3 shows the time histories of

torque, direct component of current estimation error, phase-

current, and phase- voltage. Currents and voltages are withinsaturation limits and reduced overshoots are noticed in torque

when unknown load torques are applied at s and

s. The current estimation errors tend rapidly to zero

after perturbations due to speed tracking errors.

The simplified control algorithm (46), (47) was then tested

experimentally with the control parameters values:

; the gains of the PI controllers (46) are

chosen so that a unit step reference is tracked with a settling

time of about 2.5 ms; all initial conditions of the controller

are set equal to zero excepting . The following typical




Fig. 1. Reference signals and load torque in simulations.

Fig. 2. Speed, flux modulus, and parameter estimates in simulations.

operating conditions were experimentally tested: the unloaded

motor is required to reach the rated speed 100 rad/s with

acceleration 1000 rad/s in 140 ms starting from 0.5 s; during

the initial time interval s, the motor flux modulus

is driven from the initial value 10 Wb to its rated value

1.16 Wb, with flux speed 3.87 Wb/s; both speed and flux

reference signals (given in Fig. 4) are twice differentiable with

bounded second-order derivatives (the bounds are rad/s

and 38.7 Wb/s , respectively); after start-up a constant load

torque, equal to the rated value (5.8 Nm) is applied. Speed

measurements are provided by an optical incremental encoder

with 2000 lines per revolution, while current measurements

are filtered by low pass filters with cut off frequency equal

to 2.6 kHz and then converted by 12 bit A/D converters




Fig. 3. Torque, ~

i

d

; i

a

; and u

a

in simulations.

Fig. 4. Reference signals in experiments.

with conversion time of 50 s. A 32-bit DSP (AT&T 32C)

performs data acquisition, implements the control law using

an improved Euler integration algorithm with a sampling time

equal to 0.5 ms, and generates reference voltages for the power

inverter with symmetrical PWM and switching frequency of

15 kHz. The DSP is hosted by a PC which programs the DSP,generates smooth speed and flux modulus reference signals,

generates torque commands for a current controlled d.c. motor

(which is connected to the induction motor), and stores and

displays experimental data. We performed two experiments:

in the first one underestimates the correct value , i.e.,

; while in the second one is overestimated, i.e.,

. The closed-loop performance in the two cases

is documented in Figs. 5 and 6, respectively, in which speed

error , current estimation errors , estimated flux

modulus with given by (44), cur-

rents , voltage , and the normalized estimate

are given. In both cases speed errors are compatible with a

high-performance drive; estimated flux modulus converges to

the reference value and converges to 1 within 2 s; the

estimation of flux modulus and of depends on the torque

level (which may be evaluated from ). Finally, we performed

for comparison the same two experiments by using the indirectfield-oriented control (FOC), which is implemented by (46)

with

(48)

where is used in the last equation instead of the usual




Fig. 5(a). Experimental ~ ! ( t ) ; (

^

2

d

( t ) +

^

2

q

( t ) )

1 = 2

; i

q

( t ) and i

d

( t ) with initial underestimated rotor resistance.

Fig. 5(b). Experimental u

q

; ^ ( t ) = ;

~

i

q

( t ) and ~

i

d

( t ) with initial underestimated rotor resistance.

estimated flux modulus. Comparing (48) with (47) we note that

(48) may be viewed as a simplification of (47) and therefore

the same tuning can be used ( with

and ) which guarantees satisfactory speed

tracking when in (48). The flux is estimated by the

observer (which is converging outside the magnetic saturation

region)




Fig. 6(a). Experimental ~ ! ( t ) ; (

^

2

d

( t ) +

^

2

q

( t ) )

1 = 2

; i

q

( t ) and i

d

( t ) with initial overestimated rotor resistance.

Fig. 6(b). Experimental u

q

; ^ ( t ) = ;

~

i

q

( t ) and ~

i

d

( t ) with initial overestimated rotor resistance.

which makes use of the true value . The performance

achieved by FOC in the two cases are reported in Figs. 7 and

8; while the speed error is still satisfactory, the flux modulus

goes above the reference rated value when and

below when . In both cases higher currents

(when compared with the corresponding in Figs. 5 and 6)

are required to produce the rated torque: this is due to magnetic

saturation when and to low flux modulus when

. Experiments show that the controller proposed

in this paper gives improved transient performance and a




Fig. 7. Experimental ~ ! ( t ) ; (

^

2

d

( t ) +

^

2

q

( t ) )

1 = 2

; i

q

( t ) and i

d

( t ) with FOC and underestimated rotor resistance.

Fig. 8. Experimental ~ ! ( t ) ; (

^

2

d

( t ) +

^

2

q

( t ) )

1 = 2

; i

q

( t ) and i

d

( t ) with FOC and overestimated rotor resistance.

power efficiency which is bigger than the one obtained by

the indirect field-oriented control (46), (48) with an inaccurate

rotor resistance estimate.

IV. CONCLUSION

We have designed for the fifth-order model (1) of an

induction motor with constant load torque a seventh-order

nonlinear adaptive control (45) which, on the basis of rotor

speed and stator currents measurements, guarantees asymptotic

tracking of smooth speed references for any initial condition

and for any unknown constant value of load torque and rotor

resistance. Under persistency of excitation condition (43), we

have shown that smooth rotor flux modulus reference signals




are asymptotically tracked while the rotor flux vector tends to

coincide with the -axis of the frame, achieving the

so-called field orientation; both unknown parameters (rotor

resistance and load torque) and the rotor flux vector are

asymptotically estimated. The control algorithm structure leads

to a straightforward simplification for current-fed motors given

by (46), (47).

Simulations of typical operating conditions show accurate

tracking of speed and flux modulus smooth reference signals

usually required in high-performance applications in spite

of load torque perturbations and rotor resistance variations,

converging estimates of uncertain parameters and unmeasured

state variables within 2 s. The simplified version (46), (47) of

the controller was experimentally tested and compared with the

classical indirect field-oriented control (46), (48); simulation

results were confirmed showing robustness with respect to

sensor noise, unmodeled dynamics and saturation effects,

inverter distortions, parameter inaccuracies, control discretiza-

tion, and simplification. Moreover, improved power efficiency

and transient performance are documented in comparison with

the classical indirect field-oriented control algorithm whenrotor resistance differs from its nominal value.

APPENDIX

MOTOR PARAMETERS

Rated power 600 W.

Rated speed 1000 rev/min.

Rated torque 5.8 Nm.

Rated frequency 16.7 Hz.

Excitaton current 2 A.

Rated current 4 A.

Stator resistance

Rotor resistanceMutual inductance H.

Rotor inductance H.

Stator inductance H.

Motor-load inertia kg m .

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Riccardo Marino was born in Ferrara, Italy, in1956. He received the degree in nuclear engineeringand the M.S. degree in systems engineering fromthe University of Rome “La Sapienza,” in 1979and in 1981, respectively, and the D.Sc. degree insystem science and mathematics from WashingtonUniversity, St. Louis, MO, in 1982.

Since 1984 he has been with the Department of

Electronic Engineering at the University of Rome“Tor Vergata” where he is currently a Professor of systems theory. He is the author, with P. Tomei, of

Nonlinear Control Design (Englewood Cliffs, NJ: Prentice-Hall, 1995). Hisresearch interests include theory and applications of nonlinear control.

Sergei Peresada was born in Donetsk, Ukraine, onJanuary 14, 1952. He received the Diploma of Elec-trical Engineer from Donetsk Polytechnical Institutein 1974 and the Candidate of Sciences degree inelectrical engineering from the Kiev PolytechnicalInstitute, Ukraine, 1983.

From 1974 to 1977 he was a Research Engineer inthe Department of Electrical Engineering, Donetsk Polytechnical Institute. Since 1977 he has beenwith the Department of Electrical Engineering, Kiev

Polytechnical Institute, where he currently is anAssociate Professor. From 1985 to 1986 he was a Visiting Professor in theDepartment of Electrical and Computer Engineering, University of Illinois,Urbana-Champaign. His research interests include applications of modern con-trol theory (nonlinear control, adaptation, VSS control) in electromechanicalsystems, model development, and control of electrical drives and internalcombustion engines.

Patrizio Tomei was born in Rome, Italy, on June 21,1954. He received the “dottore” degree in electronicengineering in 1980 and the “dottore di ricerca”degree in 1987, both from the University of Rome“La Sapienza.”

He currently is an Associate Professor of Adap-tive Systems at the University of Rome “Tor Ver-gata.” He is coauthor of the book Nonlinear Control

Design (Englewood Cliffs, NJ: Prentice-Hall, 1995)with R. Marino. His research interests include adap-tive control, nonlinear control, robotics, and control

of electrical machines.

Global Adaptative Output Feedback Control of Induction Motor With

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