OBSERVER-BASED CONTROLLER FOR INDUCTION MOTORS

Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain

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OBSERVER-BASED CONTROLLER FOR INDUCTION MOTORS J. De Leon-Morales *, R. Alvarez-Salas **,l, J. M. Dion ** and L. Dugard** * University of Nuevo Leon, Department of Electrical Engineering, P.O. Box 148-F, 66450 San Nicolas de Los Garza, N . L. MEXICO Tel. (52) 83 76 45 14 - Fax: (52) 83 76 4 5 14 E-mail: [email protected] * * Lab. d’Automatique de Grenobk, CNRS/INPG UMR 5528, ENSIEG-BP 46, 38402 Saint Martin d’H6res Cedex FRANCE Tel. (33) 4 76 82 62 36 - FAX: (33) 4 76 82 63 88 E-mail: [email protected]

Abstract: This paper deals with the observation and control of a class of nonlinear syst,ems. A cascade observer for a class of state affine nonlinear systems is proposed. Considering an output feedback tracking controller, a stability analysis of the resulting closed-loop system is given. The proposed observed-based controller is then shown to be closed loop stable and is applied to an induct,ion motor industrial setup to show the proposed methodology. Copyright © 2002 IFAC Keywords: Induction motor, nonlinear control, nonlinear observer.

1. INTRODUCTION The use of induction motors is widespread in industry, due to their reliability, ruggedness, and low cost. However, they are difficult to control for several reasons. They are nonlinear, coupled, multivariable processes. The rotor electrical state variables are usually unavailable for measurement, and the motor parameters can vary considerably from their nominal values, which degrades the control performances. In this paper, one considers the problem of designing a control input in order to track a desired output reference when the state is not fully measurable. Several solutions have been proposed by using nonlinear techniques to design controls laws, e.g. differential geometric approach (Marino Supported by CONACYT MCxico

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et al., 1993), sliding-modes methods (Utkin et al., 1999), backstepping (Dawson et al., 1998), passivity (Ortega et al., 1998) and adaptive techniques (Marino et al., 1999). For nonlinear systems with stable zero dynamics, and assuming that all components of the state are measurable, a state feedback controller can be designed such that the state is bounded and the tracking error converges to zero. However, the vector state, in general, is not completely measurable and it should be estimated. Nonlinear observer design for a particular class of state affine nonlinear systems is considered. Moreover, the stability analysis of the closed-loop system, when the controller is a function of the estimated state, is performed for particular classes of nonlinear systems.

H2 The functions g i ( u , y , X I,...,X i ) , i = 1,..., n are globally Lipschitz w. r. t. ( X I ,..., X i ) and uniformly w. r. t. u and y .

The paper is organized as follows: an observer for a class of nonlinear systems in cascade is introduced in section 2. A feedback control for this class is given in section 3 . In section 4, a stability analysis of the closed loop system is performed. The induction motor case is considered in section 5 . In section 6, the proposed observer-based controller is applied to an industrial experimental induction motor setup. Finally, some conclusions are given.

positive constant and

n

Ti-1

r i ( u , Y )= diag{IPi,ali(u,Y),'.', a;i(u,Y)) ;=l

for i = 1, ..,n. 2. NONLINEAR OBSERVER

An observer for systems in cascade is given by

In this section, one presents a cascade observer for a class of nonlinear systems. Consider the following multi-variable nonlinear system:

where X E RN is the state vector of the system, u E R" is the control vector and y E RP is the output vector. Assume there exists a change of coordinates which transforms (1) into another system represented by:

i = 1,...,n are where M , ( u , y ) = r;'(u,y)A;;K,, the gains of the observer, AQ,= diag{ &IPz,$IPz,

..., +IPz} with 8, > O,K, is such that the matrix

= A 1 ( u , y ) X 1+ 9 1 ( U , Y , X l ) (2) X i = A ~ ( uy)Xi , gi(u, y, X i , . . . ,X i ) , i = 2, . . . , TI

x 1

+

yi = C i X i , i = 1,. . . ,n. where Xi = c o l ( x ~ ,xz,i, i , ..., x r i , i )E Rnz,C7=l ni = N , y i E RPa, C r = l p i = P , the matrices Ai E R n a X nii ,= 1 , 2 , ...,n, are

Ai(.u.,Y) =

(Ai-KiCi) is stable and Ai

From H1, the matrix

ri is nonsingular

,

Theorem 1: Assume that the system (2) satisfies assumptions H l , H2 and H3. Then, there exist 8oi > 0 , i = 1,...,n such that for all 8i > & i , the system (3) is an exponential observer for system (2). Proof: See (DeLeon et al., 2000).

3 . FEEDBACK CONTROL DESIGN where Q j i E R p i x p i , j = 1 , 2 , . . . , ri - 1 and OPi is the pi x pi null matrix. The vector fields gi E RPz, i = 1 , 2 , .. . , ri, are

[gn;i ( V i , x1,i , x 2 , i . . . x n t,i) 1 where

~i

= ( u ,y , X I , . . . , Xi-1).

The output matrix is Ci = ( IpiOPi . . . Opi where Ipi denotes the pi x pi null matrix.

Consider nonlinear system ( l ) ,suppose that the system has relative degree r and is observable. From Frobenius' Theorem, there exists a diffeomorphism [ = T ( x ) = c o l ( T ~ ( X-) Y R , T ~ ( X ) ) that transforms the original nonlinear system into the following one:

{: [r

CE

= A[?-+ B { 4 [ r , t12-r)

En-r

+ P ( [ r , [n-r)u)

= Q([r, ln-v)

Y = el

)

Moreover, it is assumed that each subsystem is observable and verifies the following assumptions:

H1 There exist positive constants cli, czi, 0 < cli < c2i < m , i = 1, ..., n; such that for all X i E Rni; 0 < cliIpi 5 C U ~ ( U , Y ) U ~ ~ ( U5, Y ) caiIp; 5 CO, j = 1,..., T i - 1.

412

(4) which is state feedback linearizable with index r , where [ = c 0 1 ( [ ~ , [ ~ - ~E ) R", Y R = (1) col(yR,y R , ...,y g - " ) , y~ is the desired reference signal, and Tl(X) = ( h ( X ) L , f h ( X ) ,..., L ; - ' h ( X ) ) , T z ( X ) = (cp,+~(X), ..., cpn(X)).The functions cpi are such that L,cpi(X) = (dcpi(X),g) = 0, for i = r + 1,..., n.

We assume the zero dynamics of the system given by In-r = Q(O,
H5 S e t Z , X E B ( 0 , p ) t h e ball centered in 0 and radius p > 0. T h e r e exist a scalar nonnegative locally Lipschitz constant f u n c t i o n E ( p ) = L i ( p ) L s ( p ) L s ( p ) s u c h that

4. ANALYSIS OF STABILITY OF THE CLOSED-LOOP SYSTEM In this section, the stability of the system (1)with the control law function of the estimated state given by the observer (3) is studied. Let us consider the estimation error equations

6 = ( A ( u y) ,

-

r-'(u,y)A;lKC)

e

+ T(u, y, e , 2)

where the measurable outputs are y = C X = cuZ(y1,y, ..., yn) = C O l ( C 1 X 1 , c,x,, ..., C n X n ) ,the estimation error e = col(e1, ..., e n ) , the state estimate Z = coZ(Z1,Z2, ..., Z n ) , and A(u,y) = diag (A1(u,;Y)- r;l(~,Y)&;KICl, '", An(? Y) - K 1 ( u , Y)A;;KnCn) T ( u , y , e , Z ) = col(gl(u,y,&) - g 1 ( u , y , z 1 - e I),..., gn(u,;Y,21,.., 2,) - gn(u,;Y,2 1 - e l , ...-GI- e n ) ) .

where ATP + P A = -&. T h e L y a p u n o v f u n c t i o n V(E) i s s u c h that a1 I l E l l

2

5 V(E) 5 a2 IIE1I2

(6)

Now, define the augmented system

2 = f ( X )+ d X ) u ( Z ) , e = (A(u,;y)- lT1(u,;y)AglKC)e + T ( u , y , e , 2) and using the following change of coordinates

Theorem 2: Consider t h e s y s t e m C N L (1) obtained via change of coordinates and a n inputo u t p u t state feedback linearizing control u depending o n t h e state estimated given b y t h e observer (3). U n d e r t h e a s s u m p t i o n s H I , H2, H3, Hd,H5 and H6, t h e d y n a m i c s of t h e whole syst e m CA i s locally asymptotically stable. More precisely, t h e d y n a m i c s of t h e tracking error 5 and t h e e s t i m a t i o n error E converge exponentially t o zero and a n e s t i m a t i o n of t h e attraction region i s given by {&. € Rn-' : 11&.(0)1I &$)} x { e E Rn : Ile(0)II fi:},

<

<

where

- G n ( u , y , X i , ...,Xn )

where ai, P i , i = I,2,3, are positive constants. L e t

Proof: For the sake of simplicity, one assumes that i = 2 . Taking the Lyapunov function candidate L(IT,€1 = W ( I r )+ V(e1, € 2 )

+rn(u,~)ril(U,~)tn)

< -vW(&-),

with Ai = {Ai - KiCi} and Gi(u,y,2 1 , ..., Zi) G i ( u , y , X 1 ,..., X i ) = r i ( u , y ) A o i ( g i ( u , y 2 , 1, ..., Zi) - g i ( u , y , X i , ..., X i ) } , i = 1,..., n. For the stability analysis of CA, the following assumptions are introduced:

H4 T h e zero d y n a m i c s stable.

En-r

= Q(O,[n-r) i s

413

Set - [ T i ( X ) - Y R ] ~ Q [ T ~-( X YR ) ] where 77 is a positive constant.

The derivative of W ( & )w. r. t . time, it follows that

dW(
< -vW(
(7)

Now, the stability domain of D of the system C A is characterized. From assumption H6, C I I ~ , C Q , P>~0, Pexist ~ and satisfy the following inequalities

From assumption H5: dm

-

- y d m + x ( p ) llell.

dt 2 Since the derivative of V is given by

(8)

v = V l ( E 1 ) + v 2 (€2) I

-

(Olh

-

a1 P1

L11 - N11) ll€lll;l

ll4t)1I25 V(f) 5 a2 Il4t)1I2

(11)

5 W(ET) 5 P 2

(12)

IlEr(t)1l2

IIET(t)1I2

Replacing (11) and (13) in (9):

Hence, for p > 0, 3 c(&) c(I9) > c(I90) and Choosing a , 81 and 02 such that the above inequality verifies V ( 4 ) = -b(O)

L(P)

a

> 0,

such that V

< -. 1

(40) - 7 ) - 2 Using (12) and (13) in ( l o ) , it follows that

114; I -.(O)V(4t))

with 19 = (191,&), where b(19)and c(19)are positive functions of 8. Then,

V ( € ( t )I) V ( E ( 0 )exp(-c(W) )

(9) This proves the attractivity of the origin of the system.

and with (6), one obtains

5. APPLICATION T O AN INDUCTION MOTOR

Replacing the above term in (8) and integrating, it follows that

The controller and observer are designed using the standard ( c I I , ~ )non linear model (Marino et al., 1993): CNL :

with y = 4 6 ) - rl 2 . Choosing c(I9) > 7, which means that the observer dynamics is faster than the system dynamics, then

A- = f ( X ) + gu

(14)

~

f ( x )=

g = [ 1z

From (9) and (10) it is concluded that L(ET,e)is Lyapunov stable.

0 000

0 -000 UL.7

414

]

T

5.2 Control design

where X = col [is,,isfi,q5r,,q5rfi,w] and u = col [us,, u,p]. The states of the system are the two phases components of the stator current and of the rotor flux and mechanical speed. The inputs are the stator voltages. It is assumed that the unknown load torque is constant (not measured) and the nominal values of the rotor resistance and the other parameters of the model are known. The variables and parameters of the motor model are given in Table 1.

A multivariable input-output linearizing state feedback is designed for the system (14). The tracking control is given by

Tab. 1: Model variables and parameters

M , L,, L , mutual, rotor, s stator currents rotor fluxes mechanical speed stator voltages rotor, stator resistances load torque inertia viscous damping coefficient pole pair number

such that the output y = col(h1,hZ) asymptotically tracks a reference signal Y E ,where hl = w , ha = q!$a d):p are considered as the outputs to be controlled and Y E = col(w,(t), IId)T(t)112) are the reference signals to be tracked for the speed and for the square of the flux norm, respectively. It is assumed that the reference signal and its derivatives are bounded for all t 2 0 and

+

5.1 Observer design The design of the cascade observer (3) for the induction motor is introduced ( i = 2). The observer of the electrical subsystem is given by (02x2

21 =

+

K W )

(&) 02x2

02x2

02x2

z1

+

( -M 712x2

02x2

-12x2

"u)

TT

u - ( 8 2 81 k k 1 4 x2 *"1(W)

O)

where ( k , ~ka2, , ka3) and ( kbal,k b 2 , k b 3 ) are constant design parameters.

) z1

Using Theorem 2 closed-loop stability is ensured.

(21 -A1) -_ \

0

6. EXPERIMENTAL RESULTS In order to illustrate the performance of the proposed scheme, some experimental results on a 4pole, 7.5 IcW, three-phase induction motor are shown. The induction motor load is simulated by a 7.5 IcW DC motor fed by an inverter with current circulation which provides four quadrants operation (see (Lubineau e t al., 2000)). The controller is tested on a wide operating domain with the following benchmark: the speed increases with constant acceleration up to nominal speed. A torque load is applied at nominal speed. Then the rotation sense is inverted. A constant torque of same sign as before is then applied to show the behavior of controller when the motor is set in energy generation operating point. Torque disturbance is maintained until speed is set back to zero and the flux reference is constant. In experiments, the observer parameters are Icl = 0.7, ICz = 0.12, 81 = 4.5, 11 = 11.0, 12 = 30 and 82 = 3. The parameters of the controller are chosen as

design parameters. Now, the observer of the mechanical subsystem is given by

with X2 = col ( w , T L ) , the output is y2 = w , = col (G,?L), lj = "JM L ,( & , i s p - & p i s a ) - $w and ( l 1 , l 2 , 82) are constant design parameters.

z,

415

Fig. 1. Rotor speed and motor torque.

Fig. 4. Estimated speed and load torque (ij = 0, w = U n o m ) . 7. CONCLUSIONS In this paper, an observer-based controller for a class of nonlinear systems is proposed and shown to be closed-loop stable. It is applied on an industrial 7.5 k W induction motor. The real-time experiments show the efficiency of the developed controller for a broad domain of operating conditions including low and high speed, as well as motor and generator behavior, with respect to standard methods. 8. REFERENCES

Fig. 2. Tracking errors.

Dawson, D. M., J. Hu and T . C. Burg (1998). Nonlinear Control of Electric Machinery. Marcel Dekker. DeLeon, J., A. Glumineau and G. Schreier (2000). Cascade nonlinear observer for induction motors. IEEE Conf. on Control Applications. Anchorage, Alaska, pp. 623 628. Lubineau, D., J. M. Dion, L. Dugard and D. Roye (2000). Design of an advanced nonlinear controller for induction motor and experimental validation on an industrial benchmark. EPJ Applied Physics 9, 165-175. Marino, R., S. Peresada and P. Tomei (1999). Global adaptive output feedback control of current-fed induction motors with unknown rotor resistance. IEEE Trans. Automat. Contr. 44(5), 967-983. Marino, R., S. Peresada and P. Valigi (1993). Adaptive input-output feedback linearizing control of induction motors. IEEE Trans. Automat. Contr. 38(2), 208-221. Ortega, R., A. Loria, P. J. Nicklasson and H. SiraRamirez (1998). Passivity- based Control of Euler-Lagrange Systems. Springer Verlag. Utkin, V., J. Guldner and J. Shi (1999). Sliding Mode Control in Electromechanical Systems. Taylor & Francis.

Fig. 3. Estimated speed and load torque. follows: kal = 3100, ka2 = 15000, ka3 = 170, kbal = 22500, k b 2 = 225000, and k b3 = 310. Figure 1 shows rotor speed and motor torque. It includes torque due to acceleration and torque due to load. Figure 2 shows tracking errors on flux and speed. The flux tracking error is given between the reference and the estimate. The speed error is less than 7% of nominal speed despite torque disturbances. Finally, the observer performance is illustrated. Figures 3 and 4 show the estimated speed and load torque, the latter one corresponds t o the case w = 0 and w = w,.

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