Adaptive Partial Feedback Linearization of Im

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Procesdlngs of Ihe 291h Confere nceon Deddon end ControlHonolulu, Hawall December1990 FP=8-12:OOADAPTIVE PARTIAL FEEDBACK LINEARIZATIONO F INDUCTION MOTORS

RICCARDO ARINO,* SERGEI ERESADA," PAOLOALIGI'*Seconda Universith di Roma, Dipartimento di Ingegneria Elettronica

Via 0.Raimondo 00173Roma ITALIA.**KievPolytechnical Institute, Department of Electrical EngineeringProspect Pobedy, 37 Kiev 252056 USSR.

Abstract. A nonlinear adaptive sta te feedback input-outputlinearizing control is designed for a fifth order model of an in-duction motor which includes both electrical and mechanicaldynamics under the assumptions of linear magnetic circuits.The control algorithm contains a nonlinear identification schemewhich asymptotically tracks the true values of the load torqueand rotor resistance which are assumed to be constant but un-known. Once those parameters are identified, the two controlgoals of regulating rotor speed and rotor flux amplitude are de-coupled. Full state measurements are required.

1. INTRODUCTIONIn the last decade significant advances have been made in thetheory of nonlinear state feedback control (see [I] for a com-prehensive introduction to nonlinear geometric control): in par-ticular feedback linearization and input-output decoupling tech-niques have proved useful in applications [2]. More recently theproblems of feedback linearization and input-output lineariza-tion have been generalized allowing for some parameters not tobe known [3],[4],[5].n this paper we address the problem ofadaptive speed regulation for induction motors with load torqueand rotor resistance being unknown but constant parameters.Non adaptive input-output decoupling controls were presentedin (9],[lO],[ll]sing geometric techniques. We develop an adap-tive version of the controller presented in [ll], assuming thatload torque and rotor resistance are unknown parameters. Thepaper is organized as follows. In Section 2 a fifth-order statespace model of an induction motor, which includes both elec-trical and mechanical dynamics, is given. In Section 3 previ-ous control schemes are reviewed and it is shown that field ori-ented control can be viewed as a feedback transformation whichachieves asymptotic input-output decoupling and linearization.In Section 4, ollowing the results presented in (41, we developan adaptive version of the exact decoupling and linearizing con-trol given in (111 which covers the more realistic situation inwhich the load torque and the rotor resistance are not known.We present a second order nonlinear identification scheme whichasymptotically tracks the correct value of load torque and, whenelectric torque is different han zero, the correct value of rotor re-sistance as well. The adaptive state-feedback linearizing controlachieves full decoupling in speed and rotor flux magnitude regu-lation as soon as the identification scheme has converged to thetrue parameter values. The contribution of the paper is to showhow the theory of adaptive feedback linearization leads directlyto the design of a nonlinear adaptive control algorithm whichhas some advantages over the classical scheme of field orientedcontrol: with a comparable complexity, two critical parametersare identified and exact decoupling is achieved.

2. INDUCTION MOTOR MODELThe reader is referred to [12] nd [13] or the general theory ofelectric machines and induction motors and to (81 for related

control problems. The symbols used and their meaning are col-lected in the Appendix. An induction motor is made by threestator windings and three rotor windings. Krause [14] ntro-duced a two phase equivalent machine representation with tworotor windings and two stator windings. The dynamics of aninduction motor under the assumptions of equal mutual induc-tances and linear magnetic circuit are given by the fifth-ordermodel

where i, $',U , denote current, flux linkage and stator voltageinput to the machine; the subscripts s and T stand for statorand rotor; (a , b ) denote the components of a vector with respectto a fixed stator reference frame andu = L s - - M 2 * - (Ma&+2L:Rs)L , J - UL ,From now on we will drop the'subscripts T and s since we willonly use rotor fluxes $ r b ) and stator currents ( i s a , i s b ) . Let

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

P = ( ~ 1 7 ~ 2 ) ~(TL- TLN, r -RrN)T (2.2)be th e unknown parameter deviations from the nominal valuesTLN nd R,N of load torque TL nd rotor resistance R,.TL istypically unknown whereas R, may have a range of variations of3150% around i ts nominal value (see [8], ag. 224) due to rotorheating. Let U = (ua , be the control vector. Let

p = - , y = - . - - M2RrN +5 , - YE,e aL r ' UL, UL? Ureparametrization of the induction motor model, where a, , 7,p are known parameters depending on the nominal valueR,N.System (2.1) an be rewritten in compact form as

2? = f ( z) + U a g a + U b g b +Plfl +p'2f2(z) , (2.3)where the vector fields f , a , g b , f ~ ,2 are

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3. INDUCTION MOTOR CONTROL3.1 Field Oriented ControlA classical control technique for induction motors is the fieldoriented control. First introduced by Blaschke [6], [7] in 1971,it involves the transformation of the vectors ( i a , i b ) , ( $ a ,$ b ) inthe fixed stator frame (a , ) into vectors in a frame ( d , ) whichrotat e along with the flux vector ( $ , , $ b ) ; if one defines*= arctan4

$a

the transformations are

We now reinterpret field oriented control as a state feedbacktransformation (involving sta te space change of coordinate andnonlinear stat e feedback ) to a control system of simpler struc-ture. If we define the sta te space change of coordinates

w = wd'd = J4Z. + $

*alC'bp = arctan-

. $ais + * b i b, * n i b - * b i nZq =

2 1 = I*IMI

and the state feedback

(:;) = 441 ( b $;) li 2$d- nywi , - ~ M X V d

the closed loop system (2 .1) , (3 .5) n new coordinates becomes

In other words system (2.1) is transformed into (3 .6) by thefeedback transformation (3.4), (3.5). System (3 .6) has a simplerstructure : flux amplitude dynamics are linear

and can be independently controlled by V d for instance via a PIcontroller, as proposed in [8]

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

and can be independently controlled by v q , for instance by twonested loops of PI controllers, as proposed in [8]

vq = - k q ~ ( T - T r e f ) - K q ~ ( T ( ~ ) - T r e j ( ~ ) ) d 71;T r e f = - k q 3 ( W - W r e f ) - q41 U(.) - W r e f ) T

(3 .10)If w and y!Jd are defined as outputs, field oriented control achi-eves asymptotic input-output linearization and decoupling viathe nonlinear state feedback (3 .5) , (3.8), 3.10) : PI controllersare used to counteract parameters variations.During flux transient the nonlinearity * d i p in (3 .6) makes thefirst four equations in (3 .6) still nonlinear and coupled. Fluxtransients occur when the motor has to be operated above thenominal speed. In this case flux weakening, (for instance $ref =k)s required in order to keep applied voltage within inverterceiling limits ([8], .217) and the speed transients of the closedloop system (3 .6) , (3 .8) , (3 .10) are difficult to evaluate and maybe unsatisfactory. It should also be mentioned that flux mea-surements, which are required in (3 .5) , are difficult to obtain(see [15], [16]), even though flux observers from stator currentsand rotor speed measurements have been determined [17].

T = p $ d i q .

w../

3.2 Input-Output DecouplingAs shown in [ l l] ,one can improve field oriented control byachieving exact input-output decoupling and linearization viaa nonlinear state feedback control which is not more complexthan (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 fieldf ( z) = (f i (r) , ' . fn(z))

Iteratively we define L id = Lf(L?-')+).The outputs to be controlled are w and $2 + $:. Let us definethe change of coordinates

y5 = arctan ( 2 ) +31which is one to one in R = {z E R5 $: +$; # 0) but it is ontoonly for y3 > 0, -90 5 y5 5 90. The inverse transformation isdefined in as

w = Y1$a =6 OS ~5

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

4 3 - npw= w, =-rN M $sib - baa(3.19)r $ t ;

np I$P'RrN T-

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

$b = fi in Y5ia =

(3.12)(cosy5 (*) - s iny , (yz +y) ) where v = ( V a , V b ) T is the new input vector. Substituting the

i b = -&(siny5 (*) t 1cosy5 (yz +y)). state feedback (3.20) in (3.13) the closed loop dynamics become,in y-coordinatesThe dynamics of the induction motor wi th nominal parameters Y l = YZ

$2 = vare given in new coordinates byYl = Yz Y3 = Y4 (3.21)

$4 = VbY2 = LZf41+ Lg,LfdlUa + LgbLfdlUb Rr N 1Y5 = npyl + -----(Jyz + T L N ) .np Y3Equations (3.18) represents the dynamics which have been madeunobservable from the outputs by the state feedback control(3.20).In order to track desired reference signals t+ej( t) and l $ l :e j ( 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

(3.13)3 = Y4

Y5 = Lf43.Y4 = Lzf42+ Lg.Lf42Ua + LgbLfhUb

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

va = - k a l ( y l - wrej(t)) - kaZ(y2 -Gre/(t)) tGref(t)vb = -kbl(Y3 - $l:ej) - cb2(Y4 - dlref) t $lref (3.22)

where (ka11k a 2 ) and (hi, kb2) are constant design parametersto be assigned in order to shape the response of the decoupled,linear second order systems

where D ( s ) s the decoupling matrix given by- 2 '. 2

(3*15)

d2 d-( wt2 - wre ) = -kal(w - ref) - az-(wt - W r e j )

Remarks1) System (3.21) is input-output decoupled; the input-outputmapping is a pair of second order linear systems. This al-

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lows for an independent regulation (or tracking) of the out- be the parameter error. Following [4] we now introduce a timeputs according to (3.23). Transient responses are now de-coupled also when flux weakening is performed. This is animprovement over the field oriented control (see also [ll]).State space change of coordinates both in the field orientedcontrol and in the decoupling control (i.e. (3.4) and (3.11))are valid in the open set R = {z E R5 $:+$: # 0); noticethat 4; + $$ = 0 is a physical singularity of the motor instarting conditions.While measurements of ( U , ,, z b ) are available, measure-ments of ($., $ 6 ) pose some problems (see [15]). As far asparameters are concerned, variation in load torque TL androtor resistance R, cause a loss of input-output decouplingand steady-state regulation errors. This calls for an adap-tive version of th e control (3.20) ,(3.22) which is given in thenext Section.Easy computations show that the induction motor model(2.1) is not feedback linearizable. The necessary and suf-ficient conditions given in [2] fail; in fact the distribution91 = span {Sa, 96 , a d f q . , a d f g b } is not involutive since thevector field [ a d f g , , d f g b ] does not belong to G'1 ( a d x Y or[ X ,Y] denotes the Lie bracket of two vector fields; one de-fine recursively a d i Y = a d x ( a d $ ' Y ) ). Following the re-sults in [18],since & = span { g a , g b } is involutive and rank91 = 4, it turns out that the largest feedback linearizablesubsystem has dimension 4 . This shows that the control(3.20),(3.22) provides the largest linearizable subsystem inthe closed loop.The state feedback control (3.20), (3.22) is essentially theone proposed in [ll]. t is made clear that the decouplingcontrol makes the angle 4 3 unobservable from the output sand that (2.1) is not feedback linearizable. Exact input-output decoupling controls for induction motors are pro-posed also in [9], [ lo] with reference to a simplified model :the mechanical dynamics in (2.1) are not considered and wis viewed as a parameter in the last four equations of (2.1).

4. ADAPTIVE INPUT-OUTPUTLINEARIZATIONIn this section we develop an adaptive version of the decouplingcontrol (3.20) under the assumptions that T L and R, are un-known constant parameters. Let us rewrite system (2.3) in they-coordinates defined by (3.11); since the Lie derivatives L f 2 Q , ,vanish, we haveL f l L f 4 1 , L f l h , L f l L f h l L f l $ 3 , L f l L f 2 h 7 L 9 a 4 3 , L g b $ 3

Y l = Yz + P l L f 1 h$2 = L Z fh + P Z L f i L f h + L gaL f Q1Ua+ L g , L f Q I U bY3 = Y4 +PZLf24Z (4.1)$4 = ~ Z f ~ 2 + P 2 ~ f 2 ~ f Q z + L , ~ L f ~ z U a + L g 6 L f Q 2 U bY5 = L f 43 + P Z L f 2 h .

Let $ ( t )= ($l ( t ) , l ; z ( t ) )T e a time varying estimate of the pa-rameters and let

varying sta te space change of coordinates depending on the pa-rameters estimate @(t)

21 = Y12 2 = YZ + 1 L l $123 = Y324 = Y4 + $ZLf26Z2 5 = YS.

In z-coordinates system (2.3) becomes

(4.3)

where

d$it-1 = - L Z f Q1 - $zL f iL f Ql - L f lQ1t

( k a l , , z ) , ( k b l ,kb2) are control parameters to be designed and21 r e f and 23 r e f are the desired values for the rotor speed andthe square of the rotor flux amplitude respectively. Since

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

e = (2 1 - 21 r e f I 2 2 3 2 3 - 3 r e f 9 Z4)T (4.7)the closed loop system becomes

11 = ez + ep1L f iQ1iz = - h e 1 - h e z + ep2L f 2 f Q163 = e4 + e PZLf2QZi4 = - h i e 3 - bze4 + e P a L f 2 L f Q z $ z L ; , Q z )i s = L f 4 3 + P z L f 2 Q 3 .

(4.8)

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I I

While the dynamics of 25 areRr J Z Z+ TL- pIi s = np21+- InP 23

the dynamics of the vector e can be rearranged as(4 .9)

(4.10)

whereK = block d iag(Ka,K b ) ,

(4 .11)

This guarantees tha t e ( t ) and $(t) re bounded and t hat e ( t ) isan Cz ignal; it follows from (4 .7) that the first four state vari-ables ( 21 ,. . zd) are bounded. We are guaranteed to avoid thesingularities 23 = 0 and & = - R r ~ or the decouplimg -matrix,and therefore for the control (4 .5) as well, if t he initia l condi-tions ( e ( O) , p ( 0 ) ) re in S = { ( e , P )E R6 e TPe +e Tre P 5 K } ,the largest set entirely contained in { ( e , e p )E R6 ep i < R, +a l , e 3 > 02 - g r e f } , where a1 > 0 and ruz > 0 are arbitrary.Since W(z ,&) is continuous, contains only bounded functionsof 25 (sine and cosine), and (21,22,23,24 ,h are bounded, it fol-lows that W( z , f i )s bounded and therefore d and 6 are boundedas well; since, according to (4 .9) , s is bounded for ( e ,e P )E S , tfollows that I = & [ K e +W ( s , & ) e p ] is bounded as well. Now,since e is a bounded f? signal with bounded derivative 6, byBarbalat lemma ([19], p. 211) it follows that

pz l4t)ll = 0 (4 .18)i.e. zero steady-state regulation error is achieved. Since e isbounded aswell, e is uniformly continuous and (4 .18) implies byBarbalat lemma again that& l W = 0 (4.10)therefore it must be

2L rf i h =- M($aia + $bib) - $: + $':)) Equation (4.20) implies, from (4 .11), that

i.e.lim e p l ( t )= 0t-m (4.21)W ( z , h ) s called the regressor matriz and is a function of the

s-variables (and therefore of th e 2-variables).Let P = block diag ( P n , P b )be the positive definite symmetricsolution to the Liapunov equation

and, since limt-oo T( t )= TL, whenever Tt + o, i.e. in anyphysical situation,lim e p , ( t )= 0 (4.22)

tha t is parameter convergence s achieved. The difficulty n iden-tifying rotor resistance under no-load condition is a common

t-m

(4'12)~ PP K = - Qwith Q = block diag ( Q n ,Q b ) , Q . and Qb positive definite sym-metric matrices. Consider the quadratic function

v = e T P e + e T r e p (4.13)where I' is a positive definite symmetric matrixi The time deriva-tive of V is

If we now define -e, = - r - l W T p e (4 .15)dt-f i = r-lWTpe (4 .16)dt

which defines the dynamics of the parameter estimate $(t), nduse (4 .12), equation (4 .14) becomes

or equivalently

problem ([20]) and it is related to physical reasons. If the motoris unloaded, when speed and rotor flux regulation is achieved,the slip frequency in (3.19) is zero so that the flux vector rotatesat speed npw and we have Rrir,+ = 0 , Rrirg, = 0; it followsthat rotor currents are zero and therefore rotor resistance is notidentifiable in steady-state. It is proposed in [20] to track a si-nusoidal reference signal for $ +$ so that rotor currents aredifferent 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

6 = ICe + W e pdp = - r - ' W T P e .

If the initial conditions (e(O), e p ( 0 ) )E S we havelim lle(t)ll = 0lim lep, ( t) l = 0 .t-m

t- m

(4.23)

(4.24)dV- -eT&..d t (4.17)

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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. CONCLUSIONSIn this paper it is shown how the theory of input-output decou-pling and its adaptive versions ead to th e designof a satisfactorycontroller for a detailed nonlinear model of an induction motordeduced from basic physical principles. The control io adaptivewith respect to two parameters which cannot be measured and isbased on a converging dentification algorithm. T he main draw-back of the proposed control is the need of flux measurements.However nonlinear flux observers from stator currents and rotorspeed measurements have been determined [17]. Preliminarysimulations show that a good performance is maintained whenflux signals are provided by the observers to the adaptive controlalgorithm. This is a direction of further investigation. Anotherdirection of research is the real implementation of the controlin order to verify the influence of sampling rate, truncation er-rors in digital implementation, measurement noise, simplifyingmodeling assumptions, unmodeled dynamics and saturations .

ACKNOWLEDGEMENTWe would like to thank Prof. A. Bellini for providing us thedata of the motor and for useful discussions.This work was suppor ted in part by Minister0 della UniversitAe della Ricerca Scientificae Tecnologica (fondi 40%).

APPENDIX.List of Symbols

R, = stator resistanceR, = rotor resistancei , = stator current$, = stator flux linkageir = rotor current$, = rotor flux linkageU = voltage inputw = angular sp e d

np= number of pole pairs6= angle of rotation

L, = stator inductanceL, = rotor inductanceM = mutual inductance

J = rotor inertiaTL load torqueT = electric motor torque

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Adaptive Partial Feedback Linearization of Im

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