Adaptive Partial Feedback Linearization Of Im

Procesdlngs of Ihe 291h Conference on Deddon end Control Honolulu, Hawall December 1990

FP=8-12:OO ADAPTIVE PARTIAL FEEDBACK LINEARIZATION OF INDUCTION MOTORS RICCARDO MARINO,* SERGEIPERESADA," PAOLO VALIGI' *Seconda Universith di Roma, Dipartimento di Ingegneria Elettronica Via 0. Raimondo 00173 Roma ITALIA. **KievPolytechnical Institute, Department of Electrical Engineering Prospect Pobedy, 37 Kiev 252056 USSR.

Abstract. A nonlinear adaptive state feedback input-output linearizing control is designed for a fifth order model of an induction motor which includes both electrical and mechanical dynamics under the assumptions of linear magnetic circuits. The control algorithm contains a nonlinear identification scheme which asymptotically tracks the true values of the load torque and rotor resistance which are assumed to be constant but unknown. Once those parameters are identified, the two control goals of regulating rotor speed and rotor flux amplitude are decoupled. Full state measurements are required.

control problems. The symbols used and their meaning are collected in the Appendix. An induction motor is made by three stator windings and three rotor windings. Krause [14]introduced a two phase equivalent machine representation with two rotor windings and two stator windings. The dynamics of an induction motor under the assumptions of equal mutual inductances and linear magnetic circuit are given by the fifth-order model

1. INTRODUCTION

In the last decade significant advances have been made in the theory of nonlinear state feedback control (see [I] for a comprehensive introduction to nonlinear geometric control): in particular feedback linearization and input-output decoupling techniques have proved useful in applications [2]. More recently the problems of feedback linearization and input-output linearization have been generalized allowing for some parameters not to be known [3],[4],[5]. In this paper we address the problem of adaptive speed regulation for induction motors with load torque and rotor resistance being unknown but constant parameters. Non adaptive input-output decoupling controls were presented in (9],[lO],[ll] using geometric techniques. We develop an adaptive version of the controller presented in [ll], assuming that load torque and rotor resistance are unknown parameters. The paper is organized as follows. In Section 2 a fifth-order state space model of an induction motor, which includes both electrical and mechanical dynamics, is given. In Section 3 previous control schemes are reviewed and it is shown that field oriented control can be viewed as a feedback transformation which achieves asymptotic input-output decoupling and linearization. In Section 4, following the results presented in (41, we develop an adaptive version of the exact decoupling and linearizing control given in (111 which covers the more realistic situation in which the load torque and the rotor resistance are not known. We present a second order nonlinear identification scheme which asymptotically tracks the correct value of load torque and, when electric torque is different than zero, the correct value of rotor resistance as well. The adaptive state-feedback linearizing control achieves full decoupling in speed and rotor flux magnitude regulation as soon as the identification scheme has converged to the true parameter values. The contribution of the paper is to show how the theory of adaptive feedback linearization leads directly to the design of a nonlinear adaptive control algorithm which has some advantages over the classical scheme of field oriented control: with a comparable complexity, two critical parameters are identified and exact decoupling is achieved.

where i, $', U , denote current, flux linkage and stator voltage input to the machine; the subscripts s and T stand for stator and rotor; ( a , b) denote the components of a vector with respect to a fixed stator reference frame and u=Ls-- M2 * +2L:Rs)

(Ma&

L,J

UL,

-

From now on we will drop the'subscripts T and s since we will $ r b ) and stator currents ( i s a , i s b ) . Let only use rotor fluxes

x = (w,$a,$b,i.a,ib)T be the state vector and let

P =(

~ 1 7 ~= 2 (TL ) ~ -

TLN,Rr - RrN)T

Lr '

p = - , yM =-.-UL,

M2RrN UL?

+

5,

~

U

-

YE,

be a

reparametrization of the induction motor model, where a,p, 7, p are known parameters depending on the nominal value R,N. System (2.1)can be rewritten in compact form as 2?

= f(z)

+

Uaga

+ U b g b +Plfl

where the vector fields f , g a , g b ,

+p'2f2(z),

f ~ f2, are

2. INDUCTION MOTOR MODEL

The reader is referred to [12]and [13]for the general theory of electric machines and induction motors and to (81 for related

CH2917-3/90/0000-3313$1.00 @ 1990IEEE

(2.2)

be the unknown parameter deviations from the nominal values TLN and R,N of load torque TL and rotor resistance R,. TL is typically unknown whereas R, may have a range of variations of 3150% around its nominal value (see [8],pag. 224) due to rotor heating. Let U = ( u a , be the control vector. Let

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(2.3)

In other words system (2.1) is transformed into (3.6) by the feedback transformation (3.4), (3.5). System (3.6) has a simpler structure : flux amplitude dynamics are linear 3. INDUCTION MOTOR CONTROL 3.1 Field Oriented Control

A classical control technique for induction motors is the field oriented control. First introduced by Blaschke [6], [7] in 1971, it involves the transformation of the vectors ( i a , i b ) , ( $ a , $ b ) in the fixed stator frame ( a , b) into vectors in a frame ( d , q ) which rotate along with the flux vector ( $ , , $ b ) ; if one defines

and can be independently controlled by controller, as proposed in [8]

Vd

for instance via a P I

*

p = arctan 4 $a

When the flux amplitude ?+!Id is regulated to the constant reference value $d r e f , rotor speed dynamics are also linear

the transformations are

We now reinterpret field oriented control as a state feedback transformation (involving state space change of coordinate and nonlinear state feedback ) to a control system of simpler structure. If we define the state space change of coordinates

and can be independently controlled by v q , for instance by two nested loops of P I controllers, as proposed in [8] vq Tref

w=w d'd

+ $:

= J4Z.

lC'b

p = arctan *a

.

21 ,

Zq

= =

$ais +*bib

I*I * n i b -*bin

MI

and the state feedback

(:;)

= 441

(

$b

(

$;)

-l

i2 -nywi, - ~ M + XVd $d

the closed loop system (2.1),(3.5) in new coordinates becomes

=-kq~(T-Tref)-Kq~

= -kq3(W

- W r e f ) - kq4

1;

( T ( ~ ) - T r e j ( ~ ) ) d 7

1

(U(.)

- W r e f )d T

T = p$diq. (3.10) If w and y!Jd are defined as outputs, field oriented control achieves asymptotic input-output linearization and decoupling via the nonlinear state feedback (3.5), (3.8), (3.10) : PI controllers are used to counteract parameters variations. During flux transient the nonlinearity * d i p in (3.6) makes the first four equations in (3.6) still nonlinear and coupled. Flux transients occur when the motor has to be operated above the nominal speed. In this case flux weakening, (for instance $ref = k) is required in order to keep applied voltage within inverter w../ ceiling limits ([8],p.217) and the speed transients of the closed loop system (3.6), (3.8), (3.10) are difficult to evaluate and may be unsatisfactory. It should also be mentioned that flux measurements, which are required in (3.5), are difficult to obtain (see [15], [16]), even though flux observers from stator currents and rotor speed measurements have been determined [17]. 3.2 Input-Output Decoupling As shown in [ll], one can improve field oriented control by achieving exact input-output decoupling and linearization via a nonlinear state feedback control which is not more complex than (3.5).

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We will use the following notation for the directional (or Lie) derivative of state function 4(z) : R" --f R along a vector field f ( z ) = (fi(r), ' . 3 fn(z))

Iteratively we define L i d = Lf(L?-')+). The outputs to be controlled are w and the change of coordinates

$2 + $:.

Let us define

The difference between flux angular speed & and rotor speed npw is usually called slip speed, ws, which can be expressed, recalling the expression of a, as

y5

= arctan

(2)2

43

- npw = w, = RrNM $sib - $baa

np

+

6COS ~5

=

= fisin Y5

ia = ib

=

(3.12)

(*) -&(siny5 (*) (cosy5

- isiny,

(yz

t 1 cos y5 (yz

y)) + y)). +

where v = ( V a , V b ) T is the new input vector. Substituting the state feedback (3.20) in (3.13) the closed loop dynamics become, in y-coordinates

The dynamics of the induction motor with nominal parameters are given in new coordinates by Yl

I$P'

represents the electric torque. The input-output linearizing feedback for system (3.13) is given bv

w = Y1 $'b

(3.19)

- RrN T

which is one to one in R = { z E R5: $: $; # 0) but it is onto only for y3 > 0, -90 5 y5 5 90. The inverse transformation is defined in as

$a

$: t $;

Lr

+31

Y l = YZ

= Yz

+ Lg,LfdlUa + LgbLfdlUb

Y2

= LZf41

Y3

= Y4

Y4

= Lzf42

Y5

= Lf43.

(3.13)

+ Lg.Lf42Ua + LgbLfhUb

$2

= va

Y3 $4

= Y4 = Vb

Y5

= npyl

(3.21) RrN 1 + -----(Jyz np Y3

The first four equations in (3.13) can be rewritten as (3.14)

Equations (3.18) represents the dynamics which have been made unobservable from the outputs by the state feedback control (3.20). In order to track desired reference signals t+ej(t) and l$l:ej(t) for the speed y1 = w and the square of the flux modulus y3 = $: t $!, the input signals vQ and Vb in (3.20) we designed as

where D ( s ) is the decoupling matrix given by

va

(3*15)

+TLN).

= -kal(yl

vb = -kbl(Y3

- wrej(t)) - kaZ(y2 - Gre/(t)) t Gref(t) - 2 '. 2 - l$l:ej) - Icb2(Y4 - Idlref) t l$lref

(3.22)

where (ka11 k a 2 ) and ( h i , kb2) are constant design parameters to be assigned in order to shape the response of the decoupled, linear second order systems d2 -(w dt2

d

- wre j ) = -kal(w - uref) - kaz-(w dt

-Wrej)

Remarks 1) System (3.21) is input-output decoupled; the input-output mapping is a pair of second order linear systems. This al3315

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lows for an independent regulation (or tracking) of the outputs according to (3.23). Transient responses are now decoupled also when flux weakening is performed. This is an improvement over the field oriented control (see also [ll]).

be the parameter error. Following [4] we now introduce a time varying state space change of coordinates depending on the parameters estimate @(t)

State space change of coordinates both in the field oriented control and in the decoupling control (i.e. (3.4) and (3.11)) are valid in the open set R = {z E R5: $:+$: # 0); notice that 4; $$ = 0 is a physical singularity of the motor in starting conditions.

+

21

= Y1

22

= YZ

23

= Y3

24

= Y4

+i 1 L

fl

$1

(4.3)

+ $ZLf26Z

2 5 = YS. In z-coordinates system (2.3) becomes

While measurements of ( U , i,, z b ) are available, measurements of ($., $ 6 ) pose some problems (see [15]). As far as parameters are concerned, variation in load torque TL and rotor resistance R, cause a loss of input-output decoupling and steady-state regulation errors. This calls for an adaptive version of the control (3.20),(3.22) which is given in the next Section. Easy computations show that the induction motor model (2.1) is not feedback linearizable. The necessary and sufficient conditions given in [2] fail; in fact the distribution 91 = span {Sa, 96, a d f q . , adfgb} is not involutive since the vector field [ a d f g , ,adfgb] does not belong to G'1 ( a d x Y or [ X ,Y] denotes the Lie bracket of two vector fields; one define recursively a d i Y = a d x ( a d $ ' Y ) ). Following the results in [18],since & = span { g a , g b } is involutive and rank 91 = 4, it turns out that the largest feedback linearizable subsystem has dimension 4. This shows that the control (3.20),(3.22) provides the largest linearizable subsystem in the closed loop. The state feedback control (3.20), (3.22) is essentially the one proposed in [ll]. It is made clear that the decoupling

where

control makes the angle 4 3 unobservable from the outputs and that (2.1) is not feedback linearizable. Exact inputoutput decoupling controls for induction motors are proposed also in [9], [lo] with reference to a simplified model : the mechanical dynamics in (2.1) are not considered and w is viewed as a parameter in the last four equations of (2.1).

d$i

t-1

= -LZfQ1 - $ z L f i L f Q l - -LflQ1 dt

4. ADAPTIVE INPUT-OUTPUT

LINEARIZATION In this section we develop an adaptive version of the decoupling control (3.20) under the assumptions that TL and R, are unknown constant parameters. Let us rewrite system (2.3) in the y-coordinates defined by (3.11); since the Lie derivatives L f 2 Q , , LflLf41,

L f l h ,

L f l L f h l

Lfl$3,

L f l L f 2 h 7

L9a43,

( k a l ,k , z ) , (kbl,kb2) are control parameters to be designed and 21 r e f and 23 r e f are the desired values for the rotor speed and the square of the rotor flux amplitude respectively. Since

Lgb$3

vanish, we have

+

the decoupling matrix is singular not only when ($: $,") = 0 as in the nonadaptive case but also when j Z ( t ) = -R,N; this additional singularity has to be taken into account in the design of the adaptive algorithm. Defining the regulation error

Yl

= Yz + P l L f 1 h

$2 Y3

= LZfh+ P Z L f i L f h = Y4 +PZLf24Z

$4

=~Zf~2+P2~f2~fQz+L,~Lf~zUa+Lg6LfQ2Ub

Y5

= Lf43 + P Z L f 2 h .

+ LgaLfQ1Ua+ L g , L f Q I U b (4.1)

e = ( 2 1 - 21 r e f I 2 2 3 2 3 - 23 r e f 9 Z4)T

(4.7)

the closed loop system becomes

Let $(t) = ($l(t),l;z(t))Tbe a time varying estimate of the parameters and let

+

11 = ez ep1LfiQ1 iz = - h e 1 - h e z ep2Lf2LfQ1 63 = e4 ePZLf2QZ i4 = - h i e 3 - kbze4 ePa( L f 2 L f Q z+ $ z L ; , Q z ) is = L f 4 3 + P z L f 2 Q 3 .

+

+

+

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(4.8)

I

I

While the dynamics of

i s = np21

This guarantees that e ( t ) and $(t)are bounded and that e ( t ) is an Cz signal; it follows from (4.7) that the first four state variables (21,. . .,zd) are bounded. We are guaranteed to avoid the singularities 23 = 0 and & = - R r ~for the decouplimg -matrix, and therefore for the control (4.5) as well, if the initial conditions (e(O),e p ( 0 ) )are in S = { ( e ,e P ) E R6: eTPe+eTreP 5 K } , the largest set entirely contained in { ( e , e p ) E R6 : epi < R, a l , e 3 > 02 - 2 g r e f } , where a1 > 0 and ruz > 0 are arbitrary. Since W ( z ,&) is continuous, contains only bounded functions of 25 (sine and cosine), and (21,22,23,24, h ) are bounded, it follows that W ( z , f i )is bounded and therefore d and 6 are bounded as well; since, according to (4.9), i s is bounded for ( e , e P )E S , it follows that I = & [ K e W ( s , & ) e p ] is bounded as well. Now, since e is a bounded f? signal with bounded derivative 6, by Barbalat lemma ([19], p. 211) it follows that

are

25

Rr J Z Z+ TL - epI +nP

I

23

(4.9)

the dynamics of the vector e can be rearranged as

+

(4.10)

+

where

pzIl4t)ll

K = block diag(Ka, K b ) ,

=0

(4.18)

i.e. zero steady-state regulation error is achieved. Since e is bounded as well, e is uniformly continuous and (4.18) implies by Barbalat lemma again that

&I l W = 0

(4.10)

therefore it must be (4.11) 2 L f i h = - (M($aia Lr

+ $bib) - ($: + $':))

Equation (4.20) implies, from (4.11), that

i.e. lim e p l ( t )= 0

(4.21)

t-m

W ( z , h ) is called the regressor matriz and is a function of the s-variables (and therefore of the 2-variables). Let P = block diag ( P n , P b )be the positive definite symmetric solution to the Liapunov equation

K ~ +PP K = - Q

(4'12)

with Q = block diag ( Q n ,Qb), Q . and Qb positive definite symmetric matrices. Consider the quadratic function

v = eTPe + eTrep

(4.13)

where I' is a positive definite symmetric matrixi The time derivative of V is

and, since limt-oo T ( t ) = TL, whenever Tt physical situation, lim e p , ( t ) = 0

+

t-m

o,

i.e. in any

(4.22)

that is parameter convergence is achieved. The difficulty in identifying rotor resistance under no-load condition is a common problem ([20]) and it is related to physical reasons. If the motor is unloaded, when speed and rotor flux regulation is achieved, the slip frequency in (3.19) is zero so that the flux vector rotates at speed npw and we have Rrir,+ = 0 , Rrirg, = 0; it follows that rotor currents are zero and therefore rotor resistance is not identifiable in steady-state. It is proposed in [20] to track a sinusoidal reference signal for $: $: so that rotor currents are different than zero and rotor resistance can be identified.

+

In summary we have shown that the adaptive feedback control (4.5),(4.16) gives the closed loop system

If we now define

de, = - r - l W T p e dt

6 = ICe + W e p

(4.15)

dp = - r - ' W T P e .

or equivalently

' f i = r-lWTpe

(4.16) dt which defines the dynamics of the parameter estimate $(t),and use (4.12), equation (4.14) becomes

dV = -eT&.. dt

(4.23)

If the initial conditions (e(O), e p ( 0 ) )E S we have lim lle(t)ll = 0

t-m

lim lep,(t)l = 0 .

(4.17)

t-m

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(4.24)

Moreover if

TL# 0 we also have

References. (4.25)

From (4.18) and (4.19) it follows that in any case we have

(4.26)

6. CONCLUSIONS

In this paper it is shown how the theory of input-output decoupling and its adaptive versions lead to the design of a satisfactory controller for a detailed nonlinear model of an induction motor deduced from basic physical principles. The control io adaptive with respect to two parameters which cannot be measured and is based on a converging identification algorithm. The main drawback of the proposed control is the need of flux measurements. However nonlinear flux observers from stator currents and rotor speed measurements have been determined [17]. Preliminary simulations show that a good performance is maintained when flux signals are provided by the observers to the adaptive control algorithm. This is a direction of further investigation. Another direction of research is the real implementation of the control in order to verify the influence of sampling rate, truncation errors in digital implementation, measurement noise, simplifying modeling assumptions, unmodeled dynamics and saturations. ACKNOWLEDGEMENT We would like to thank Prof. A. Bellini for providing us the data of the motor and for useful discussions. This work was supported in part by Minister0 della UniversitA e della Ricerca Scientificae Tecnologica (fondi 40%).

APPENDIX. List of Symbols

R, = stator resistance R, = rotor resistance i , = stator current $, = stator flux linkage ir = rotor current $, = rotor flux linkage U = voltage input w = angular s p e d np = number of pole pairs 6 = angle of rotation L, = stator inductance L, = rotor inductance M = mutual inductance J = rotor inertia TL= load torque T = electric motor torque

A. Isidori, Nonlinear Control Systems, (2nd edition), Communications and Control Engineering Series, Springer, Berlin, 1989. B. Jakubczyk, W. Respondek, ’’ On linearization of control systems.”, Bull. Acad. Pol. Sci., Ser. Sci. Math., Vol. 28, 9-10, pp. 517-522, 1980. D.G. Taylor, P.V. Kokotovic, R. Marino, I. Kanellakopoulos, ’’ Adaptive regulation of nonlinear systems with unmodeled dynamics”, IEEE Trans on Automatic Control, Vol. 34, pp. 405-412, 1989. R.Marino, I. Kanellakopoulos, P.V. Kokotovic, ”Adaptive tracking for feedback linearizable SISO systems.”, Proc. 28th CDC Conference, Tampa FL, pp. 1002-1007, 1989. S.S. Sastry, A. Isidori, “Adaptive control of linearizable systems.”, IEEE Trans. on Automatic Control, Vol. 35, pp. 1123-1131, 1990. F. Blaschke, ” Das Prinzip der Feldorientierung, die Grundlage fur die transvector Regelung von Asynchronmaschienen”, Siemens Zeitschrift 45, p. 757-760, 1971. F. Blaschke, ”The Principle of field orientation applied to the new transvector closed loop control system for rotating field machines”, Siemens Rev., Vol. 39, pp. 217-220, 1972.

W. Leonhard, Control of Electrical Drives, Springer Verlag, Berlin, 1985. A. De Luca, G. Ulivi, ”Dynamic decoupling of voltage frequency controlled induction motors”, 8th Int. Conf. on Analysis and Optimization of Systems, INRIA, Antibes, pp. 127-137, 1988. A. De Luca, G.Ulivi, ”Design of exact nonlinear controller for induction motors”, IEEE Trans. on Automatic Control, vol. AC-34, no.12, pp. 1304-1307, December 1989. 2. Kneminski, Nonlinear control of induction motor”, 10th IFAC World Congress, Munich, pp. 349-354, 1987. A.E. Fitzgerald, C.Kingsley Jr, S.D.Umans, Electric Machinery, Mc. Graw-Hill, 1983. P.C. Krause, Analysis of Electric Machinery, Mc Graw Hill, 1986. P.C.Krause, C.H. Thomas, ”Simulation of symmetrical induction machinery”, IEEE Trans. on Power Apparatus and System, Vol. PAS-84,no. 11, pp. 1038-1053, Nov. 1965. R.Gabrie1, W.Leonhard, ”Microprocessor control of induction motors”, Proc. IEEE/ IAS Int. Semiconductor Power Converter Conf., Orlando, pp. 385-396, 1982. W. Leonhard, ” Microcomputer control of high dynamic performance AC-drives : a survey”, Automatica, Vol. 22, no. 1, pp.1-19, 1986. G.C. Verghese, S.R. Sanders, ”Observers for flux estimation in induction machines”, IEEE Trans. on Industrial Electronics, Vol. 35, no. 1, pp. 85-94, February 1988. R. Marino, ”On the largest feedback linearizable subsystem”, Systems and Control Letters, Vol. 6, pp. 345-351, January 1986. V.H. Popov, Hyperstability of Control System, Springer, Berlin, 1973. H. Sugimoto, S.Tamai, ”Secondary resistance identification of an induction-motor applied model reference adaptive system and its characteristics”, IEEE Trans on Ind. Appl., Vol. IA-23, no. 2, pp. 296-303, March 1987.

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