Systems & Control Letters 23 (1994) 315 323 315 North-Holland

State observers are unnecessary for induction motor control

Gerardo Espinosa UmversMad Nacwnal Autonoma de Mexico, DEPFI, P 0 Box 70-256, D F Mexico

Romeo Ortega Untversltb de Technologw de Compi~gne, URA CNRS 817, BP 649, 60206 Complbgne cedex, France

Received 13 May 1993 Rewsed 6 September 1993

Abstract Induction motors constitute a theoretically interesting and practically important class of nonhnear systems, which are evolving into a benchmark example for nonhnear control They are described by a fifth-order nonhnear differential equation with two mputs, and only three state variables avadable for measurement A lot of research m the field has been devoted to the design of observers whmh, combined with a suitable control strategy, would yield stable behavlour Our contnbut lon in this paper is to show that the control objectwe can be achmved without reconstructmn of the motor state Specifically, we present a globally stable nonhnear dynamic output feedback controller for torque tracking of reduction motors which does not rely on state reconstruction ideas Another important feature of our sheme ~s that the control law is globally defined, even in startup This stems from the fact that we do not aim at hneanzmg the system dynamics, but instead exploit the energy dissipation properties of the motor model For the sake of dlustratmn we present the result for a model described in the stator frame (ab model), but the theory apphes as well to models expressed in a rotating frame (dq model) We also show how, as a particular case of torque tracking, we can solve the rotor speed tracking problem

Keywords Induction motor, observers, nonlinear systems, stabilization

1. Problem formulation

We consider in this paper the well-known ab model of a three-phase induction motor [14],

DYe + C(x)x + Rx = Q, (1.1)

where the state vector is given by the a and b components of the stator currents (tsa, zsb) and the rotor fluxes (fir°, frb) respectively, and the rotor speed (o9), that is

X~--Oso,Z~b, ~ro, frb,CO]T ~[xl,X2,X~,X,,X~] v,

w i t h

I ° 0 D& 0 ~ 2 , C(x ) ~-~ 0 - x 5 ~ 2 ,

o o Lr,l - - fI (x) 0

Correspondence to Dr R Ortega, Unlverslt6 de Compl6gne, URA CNRS 817, BP 649, 60206, Compl6gne cedex, France

0167-6911/94/$07 00 © 1994 Elsevier Scmnce B V All rights reserved SSDI 0 1 6 7 - 6 9 1 1 ( 9 3 ) E 0 1 3 0 - 9

"~16 6 Esptnosa, R Orteqa lnduttum motor ~ontrol

e A

LrtY;'J2 MRf J2 0 Lr

MR~ ~2 R~ J2 0 L~ Lr

0 0 L~B

Q A

LrUl Lru2

0

0

0

- L r ~

f l ( x ) ~ - M x 4 ] = x3 j 2 ~ 0 - t r ~ L ~ - - - )'~- Mx3 ] M J 2 x4 ' 1 ' Lr ' aL E

where ~'2 IS the 2 x 2 Identity matrix, Rs,Rf,L~, Lr are the stator and rotor resistances and inductances, respectively, M is the mutual inductance, u~, u 2 are the stator voltages, and TL lS the load torque

The control problems we solve m this paper may be formulated as follows

Torque tracking problem. Consider the induction motor model (1.1) with control inputs the stator voltages u~, u2 and regulated output the generated torque

z~ a M T = h ( x ) = £ ( x 2 x 3 -- XlX4). (1_2)

Assume' (A1) Stator currents xl , x2 and rotor speed x5 are available for measurement. (A2) Motor parameters are exactly known (A3) The desired torque Ta IS a smooth bounded dlfferentiable function with a known bounded first-order

derivative and the load torque TL is smooth and bounded. Under these conditions, design a control law that will ensure

h m ( T - T o ) = 0

with all internal signals bounded

(1.3)

Rotor speed tracking problem. For the problem formulation above assume (A1), (A2) and the following (AY) The desired rotor speed X5d IS a smooth bounded dlfferentiable function with known bounded first-

and second-order derivatives (A4) Load torque can be linearly parametrized as

TL = p r o (1 41

where p e r q is an unknown constant vector with q~,q~ bounded measurable signals. Further, we assume known an upperbound K v on the norm of the parameter vector, 1.e, II P II ~< K r where II " II stands for the Euclidean norm

Under these conditions, design a control law that will ensure

hm (x5 -- XSd) = 0

with all internal signals bounded

(1.5t

Remark 1.1. The model (1.1) coincides with [8, eq. (16)], with the exception of the mechanical damping constant B > 0, which is set to zero in [-8]. As will become clear later, the existence of nonzero mechanical damping will be essential to solve the rotor speed tracking problem The need for this assumpnon stems from the Inablhty to add damping to the mechanical system with partial state measurement See [11] for further d~scusslon on this point, and [10] for physical motivation of the chosen motor representation.

G Espmosa, R Ortega / Inductzon motor control 317

Remark 1.2. It is worth pointing out that from the point of view of modern applications of induction motors (e.g. electric vehicles, robotics), the regulated output of interest is torque. We refer the reader to [10, 11] for comments regarding the assumptions and further motivation for these problems

Remark 1.3. It has been shown in [10] that the induction motor model with zero load torque TL defines a passive mapping [3] from stator voltages u~,u2 to stator currents This property follows immediately taking the time derivative of the motor total energy x r D x along the trajectories of (1 1), and using the positivlty of D, R and the skew-symmetry of C(x). We will find it convenient below to decompose this passive system into the feedback lnterconnection of two passive subsystems This decomposition reveals the interaction of the subsystems and will allow us to solve our stabilization problem

2. Literature review and paper contributions

A brief review of the hterature follows. The problem of torque regulation assuming full state measurement was studied using linearization techniques by Deluca [2] for a model neglecting the mechanical dynamics, i e x5 = const Marino et al. I-8] propose an adaptive version of the feedback linearlzatlon scheme of [7] to address the speed regulation problem with unknown rotor resistance Rr and unknown constant load torque, but assuming measurable state. Kanellakopoulos et al [6] established local stabIhty of a scheme designed using backstepping, which IS a recent Lyapunov-based stabilization technique, for the velocity control problem with flux observer and known motor parameters.

All the schemes referred above suffer from the drawback that the control laws are not globally defined. The set of singularities, besides being difficult to analyse, is defined by the design methodology The physical rationahzatlon of these singularities is usually an a posterlorl step, leaving the designer with a pious hope that it will not contradict physical operation. This seems to stem from the fact that both techniques, feedback hnearlzatlon and backstepping, neglect the physical structure of the system. In contrast with these ap- proaches, an energy shaping plus damping injection controller design methodology that exploits the energy dissipation properties of the system has been proposed in 1-9] i This methodology is used in [10] to derive the first globally defined globally stable solution to the torque regulation problem with partial state feedback and unknown constant load. The result was established for constant desired torques which satisfy an upper bound determined by the motor mechanical damping An adaptive scheme to handle uncertain rotor resistance IS also presented in that paper, but requires full measurement of the state The result of [10] Is extended In [11] to handle time-varying torque references, and to remove the upper-bound condition described above. The key idea introduced in [11-] is a procedure to inject damping to the mechanical subsystem. In [1] a local solution to the problem of position control of robots with induction motor drives is presented.

The schemes of 1-10, 11] are derived on the basis of a model where the variables are expressed in a synchronously rotating frame (dq model). In [5] it is shown that, for the case of full state measurement, we can rotate back the coordinates of the control signals of the scheme of 1-11] to solve the torque tracking problem for a motor model given in the fixed stator frame (ab model). This 'coordlnate-lnvarlance' feature of the controller of [11] is not surprising since the design procedure of [9] is based on the input-output (energy dissipation) properties of the motor, which are 'coordinate-independent'

It is not known at this point if the coordinate rotation used in 1-5] applies also for the observer-based scheme of [11]. In any case, it is clear that it will lead to a highly complex controller dynamics One of the contributions of the present work is to provide a very simple 'observerless' solution to the torque tracking problem above To this end, we make the fundamental observation that the mechanical part of the induction motor dynamics defines a passive feedback around the electrical subsystem, which in turn is also passwe Therefore, we can apply the energy shaping procedure only to the electrical part and treat the effect of the

1 The methodology is a natural extension of the highly successful passivity-based controller designs used m robotics, see e g [12]

"~18 G Espmo,~a R Orteqa , lndmtton motor tontrol

mechamcs as a passive perturbation. Interestingly enough thts can be done measuring only stator varmbles The main stumbhng block of this procedure is that we cannot reject addttlonal damping to the mechamca} subsystem Th~s dependence on the system's natural damping limits the achievable performance of the deszgn

3. Motor model revisited

In th~s section we explain the rationale of the controller that we propose To thts end, we find ~t convenient to express the induction motor model as a feedback interconnectlon of passive systems as follows. Model (1 l) can be rewritten m an alternative form as

J~s + Bx~ = T - TL, (3 2)

where

oo I ¢ °1 ° J2 ' MJz - ~z x5,

Ro(xs)~=

L r a T J 2 MR,. J2 L~

Mar :: R, -- MJ2x5 L~ -~ J2

x ~ X2

X3

X4

Lru~]

LrU21 .

Notice that to obtain a skew-symmetrac 2 Ce(XS) we have added and substracted MJ2x5 in the third and fourth rows of (3 1). This has in its turn added a term - MJ2x5 m the (2, 1)-entry of the 'damping matrix' R,(x~). As shown below, we will be able to recover the posaivlty of this matrix using control signals that compensate this term (3.1), (3.2) define two operators in a feedback mterconnectmn

1 ~ t - T+ TL), s~d S,2 ( - T + TL)+ - xS - j s + dr'

where TL is v~ewed as an external disturbance Notice that

MRr l ) L~ayJ2 T ~ J2[

Cl&2m~' MR, j 2 Rr l > 0

- L - -T

[ul 1 S 1 v ~ u2

- - X 5

with 2.~,,{ ] the mimmum elgenvalue, Now, takmg the derivative of the total energy of the electrical subsystem x~D,x~ along the trajectories of (3,D, we can show that Sa as passive [3] Specifically, it satisfies the diss~patxon inequality

2 As discussed m [12] a representation that enjoys this property reveals the workless forces m the system Since these forces do not affect

the system stablhty there is no need to cancel them

G Espmosa, R Orteoa / Inductmn motor control 319

for all t _ 0 and s o m e c 2 ~ . On the other hand, the system z~ 2 IS strictly posttlve real for all J _ 0, B > 0; thus It is also passive. Mot iva ted by the wel l -known fact that feedback mterconnect lon of passive subsystems is also pasmve, we then propose to design an energy shaping plus damping injection control ler for the electrical subsystem only, and to t reat the mechanical subsystem as a 'passive disturbance ' . This is in stark contras t to the approach of [10], where the controller alms at shaping the energy of the overall system. Interest ingly enough, the energy shaping of the electrical subsystem and the damping rejection can be carried out using only measurable varmbles as shown below

4. Main results

Proposition 4.1 Let the controller be defined as

(Torque tracking). Consider the reduction motor model (1.1), (1.2) under assumptwns (A 1)-(A3).

ut M - < , L~2dJ LX2dJ LT/L .J- X2d)J

where

[ Xld] Lr

= z, / ~ > 0 ,

I MflJ2 x4d J

wzth z = [Zl, z2] T the two-dzmenstonal controller state solutmn of

for some mztzal condition 3 z(O) = [cos a, sin a] x, aeR, and K(x5) given as

M 2 K(xs ) = --4-e-e x 2, 0 < e < R r .

Under these conditions,

llm ( T - To) = 0

wzth all internal si#nals bounded.

(4 1)

(4.2)

(4.3)

(4 4)

Proposition 4.2 (Speed tracking). Consider the induction motor model (1 1), (1.2) under assumptions (A1), (A2), (A3'), (A4) Let the controller be defined by (4 1)-(4.4), wtth desired torque

Td = pT qb + JYcsd + nxsd (4.5)

is updated wlth a #radient update law with projection

/~ = Proj {/~, - g~b(x5 - Xsd)}, i0(0) = poel~ q, y > 0 (4 6)

where Proj { , } is a smooth projection 4 that keeps the estimates mszde a sphere of radius K r Under these conditions,

hm (x5 - xsa) = 0 t ~ 3

with all mternal stgnals bounded

3 The motivation for this particular choice of initial condmons wdl become clear m the proof of the proposmon 4 See e g [13] for an example of such a projection

~20 G Esptnoscl, R Orteqa , Imlu~tlcm nlotor tontrol

Remark 4.1. Not ice that the control lers of the p ropos i t ions above are nonhnea r dynamic feedback~ of the measurab le var iables x l , x2, i s , where the cont ro l le r states are z and/~ As will become clear in the proofs, there are two types of terms in (4.1) the first three r igh t -hand terms achieve the energy shaping of the closed loop, while the last r igh t -hand terms Inject the required damping to achieve asympto t i c s tablhty

R e m a r k 4.2. The cont ro l (4 1) requires the knowledge of "~ld, X2d, which in turn implies avai labi l i ty of 7~a This hampers the Inclusion into the definit ion of To of the actual ro tor speed x~ wi thout accelera t ion measurement Also, this explains the need for our a s sumpt ion of known ~ in the speed t racking problem formula t ion

5. Proof of main results

5 1 ProoJ o f ProposltlOn 4 1

The proof proceeds along the following steps First, we define a change of coordinates (an error signal) for the electrical variables xe such that convergence to zero in these coordinates (with bounded states) imphes the torque tracking objective Then, we derive the dynamic equations in the new coordinates and prove that, if there is no fimte escape time, then convergence to zero is achieved. Finally, we prove the existence of solutions for the whole positive real axis 5

Step 1 Let us define an error signal x,&[x1,3~2,)~3,)~':I-]T~Xe - - Xed, where x~a is given by (4.2) Now, by direct substitution we can prove that the solutions to (4.3) may be written in the form

zl = cos0, z 2 = sin0,

where 0 is the solution of

0 - Rr f12 M 2 T d + x 5, 0(0) = a

Using the expression above, replacing (4 2) in (1.2), and after some lengthy but straightforward calculations we establish the key property of

h(x~a) - Td

On the other hand, it IS easy to show that

with

M T T - Td = h(x) - h(x~e) = ~;-. ( ~ W;~ + 2~r~ Wxea)

ZLr

w,[ 0 J2] - J 2 0

Therefore, noting that W IS a unitary matnx, the bound below follows immediately 6

M I T - Td[ ~< ~ ( l l ; o q [ z + 2 I1-~oll Hxoall)- (5 1)

From (5.1) we conclude that asymptotic torque tracking will be achieved if we can ensure that .~o --* 0 as t--* cc.~ with bounded X~d

s It is worth mentioning that the only techmcally revolved part of the proof, which imposes some of the restrictions m our problem formulatmn, ~s th~s last step

Notice that h(-) is not globally Ltpschltz

G Espmosa, R Ortega / lnductwn motor control 321

Step 2 Writing (3.1) m terms of ~ we get

De:~, + C~(xs)Y~¢ + R~(xs)'2~ = ~,

where

~ ~-Q~ - [Djca + Ce(xs)xd + Re(XS)Xd]

which can be further expressed as

MRr "~1" L~ul -- LrO'Xld + M x s x 4 d -- LrO'yXld + ~ - - r X3d

M R r ~2 L r u 2 - - Lr0"X2d -- MXsX3d -- Lr0"yX2d "b - - X4d

Lr MRr R,

MR~ Rr

Direct replacement of (4.1) shows that

Now, from (4 2) and (4.3) we have

[x3dl ~ ( Rr /~2 Fx3dl, X4dJ ~ Td + x s / LX4dJ

Lr

LX2d I = ~ + LX4d j

(5 2)

(5.3)

Thus, after some lengthy but simple calculations we can prove that ~3 = ~4 = 0. This leads finally to the error equation

D ~ + C~(xs)~ + Rcs(xs):re = O,

where

[ 1 L, EaT+K(xs)]J2 - ½ M J ~ x s - ~ 2

R~s(Xs) ~= 1 MRr R~ ~ " - z M J z x 5 Jz ~ 2

Lr

(5.4)

In view of(5 4) we say that the total energy of the subsystem Z1 m a closed loop with the control law has been shaped to match the function

H d ~ ½ -T ~ X e Dexe

with external forces R,(xs)~o The terms (5.2) inject the required damping into the error system, since it is easy to verify that with the choice of K(xs) given in (4.4) we have

/~m,n {Res(XS)} ~-~ ¢Y > 0

Now, let us temporarily assume that xs cannot escape to infinity in finite t~me Under this con&tmn, we can evaluate the derivative of Hd along the trajectones of (5 4) to get

H~ = - ~ R . ( ~ , ) ~ c .

322 G E w m o s . . R Orteqa lndut tton motor ~ ontrol

Thus, we can determine positive constants m~, p~, which are independent of t, such that

The only remaining difficulty IS thus to prove that there is no finite escape time behaviour. Step 3 To this end, nohce that the dynamics of the closed-loop system are fully described by (5.4), (3.2) and

(1 2). Notice that this set of differential equations is locally Llpschltz In the state and, under the assumptions on the reference, is also cont inuous in t. Therefore, there exists a time interval [0, tl ) where the solutions exist and are unique. Now, for all t~[O, tl ), (5.5) holds This, together with the boundedness of II Xed II (4.2), proves that on the time interval [-0, tl ), the r ight-hand side of(3 2) is bounded by a constant ~1 independent of t~ Therefore, its solutions cannot grow faster than an exponential on that interval Consequently, x5 remains bounded for all t~ [0, tl ], and II-~e II cannot escape to infinity on [0, tl ]. Since m~, p~, 9q are Independent of t j we can repeat this argument for a new Initial condit ion at tl and define the solution cont inuat ion in a new time interval [-t 1,2tl ] This allows us to extend the time interval of the existence of solutions to the whole real axis, and conclude the proof []

5 2 Proo[ of Proposmon 4 2

The first and second steps of this p roof follow exactly the same arguments as above In the third step we have the additional difficulty that now Td lS not a bounded external reference but is given as (4 5) To ensure (3 2) has a bounded right-hand side we included the projection opera tor in (4.6) and the assumptions of bounded XSd,-;CSd, 49 F r o m here we conclude that x5 cannot grow faster than an exponential on [0, tl), and proceeding as above we can extend the solutions to the whole real axis to conclude that ~e is bounded and ~e--* 0 as t ~ :c

We now proceed to prove asymptot ic speed tracking To this end, we replace (4.5) in (3 2) to get

J-'~5 + B-,~5 = T - Td + /~T49, (5 6)

where we have defined the errors ~5 & x5 - Xsd a n d / 5 - -a/~ - p. Let us evaluate the derivative of the quadratic function

1 1

along the traJectories of (5.6) and (4.6) to get

I 2 = - BYs 2 + -~5(T-- Td) (5.7)

T - - Ta bounded ensures via (5 7) that Y5 is bounded. F rom here, using (5.6), we conclude that also xs is bounded, and consequently ~5 is uniformly continuous. Now, integrating (5 7) and using the fact that T ~ Td as t--, oc, we conclude that -~5 is square integrable. The proof is completed recalling that a uniformly cont inuous square integrable function converges to zero [ ]

6. Concluding remarks

We have illustrated in this paper how by using energy shaping (passivity-based) design techniques we can obviate the need of state observers for the control of induction motors. This has been a long-standing problem, attracting the attention of many researchers, in both the practical and in the theoretical camps We believe that this new apphcat lon of the technique shows its potential and versatility, as well as underscores the importance of incorporat ing physical understanding into the controller design

Our main motivat ion in carrying out this research was to address the combined parameter adapta t ion and state observation problem, which to the best of our knowledge is essentially open As is well known, the

G Espmosa, R Ortega / Induction motor control 323

difficulty in that case stems from the fact that joint observation of states and parameters leads to bllinear estimation problems It is our belief that 'observerless' schemes such as the one presented here provide a first step towards its solution, since the bdlnearity is removed. Even though in this work we managed to solve a (restricted) load torque adaptation task, we were not able to address the parameter estimation problem. Some s~mple calculations show that, if we replace the parameter Rr by an estimate, we stdl get a linear m the parameters error equation. However, the measurements needed to design the update law include the rotor flux Current research is under way to remove th~s &fficulty.

As a concluding remark we would hke to stress that the proposed controller can be obtained from a 'coordinate rotation' of the scheme in [11-1 plus a new damping injection mechanism to remove the need to measure (or observe) rotor flux Consequently, as explained in [10], it is consistent wtth the field orientation idea extensively used in applications. Also, as seen from (5.3) it leads to steady-state balanced operation

References

[1] C Canudas, R Ortega and S Seleme, Robot motion control using induction motor drives, in Proc IEEE ICRA "93, Atlanta, GA, May 1993

[2] A Deluca, Design of an exact nonhnear controller for induction motors, IEEE Trans Automat Control 34 (1989) 1304-1307 [3] C Desoer and M Vldyasagar, Feedback Systems Input Output Propertws (Academic Press, New York, 1975) [4] G Espmosa, Nonhnear control of reduction motors, UNAM Ph D Thesis, under preparaUon [5] G Espmosa and R Ortega, Control of induction motor models m a fixed reference frame, UTC Int Report, Nov 1992 [6] I Kanellakopoulos, P Krem and F Dlsllvestro, Nonhnear flux-observer-based control of reduction motors, in Proc ACC,

Chicago, IL, 1992 [7] Z Krzeminskl, Nonhnear control of inductton motor, in Proc lOth IFAC World Conor, Munich (1987) 349-354 [8] R Manno, S Peresada and P Vahgl, Adaptwe partial feedback hneanzatlon of reduction motors, IEEE Trans Automat Control

38 (1993) 208-221 [9] R_ Ortega and G Esplnosa, A controller design methodology for systems with physical structures- application to induction motors,

in Proc IEEE CDC, Brighton, UK, Dec 1991 [10] R Ortega and G Espmosa, Torque regulation for reduction motors, m Proc ECC '92, Grenoble, June 1992, also to appear in

Automatwa [11] R Ortega, C Canudas and S Seleme, Nonhnear control ofmduction motors torque tracking with unknown load disturbances, in

Proc ACC, Chicago, June 1992, also to appear at" IEEE TAC [12] R Ortega and M Spong, Adaptive motion control of ngld robots_ a tutonal, Automatlca 25 (1989) 877-888 [13] J Pomet and L_ Praly, Adaptwe nonhnear regulation estimation from the Lyapunov equation, IEEE Trans Automat Control 37

(1992) 729-740 [14] S Seely, Electromechamcal Energy Conversion (McGraw-Hill, New York, 1962)