Optimal Energy Reference Observer-Based Torque Tracking : A Nonlinear Feedback Control in the ap Frame

S.I. Seleme Jr.' DEE - Centro de CiGncias Tecno16gicas/FEJ - UDESC

89223-100 Joinville - SC Brasil

Abstract This paper presents, as the main contribution, an observer-based control feedback for the torque track- ing of induction motors, conceived in terms of optimal state reference trajectories. This control scheme ex- tends the result presented in [7] by the inclusion of a state observer and by the use of the stationnary CYP frame model for the feedback design. Inherent passiv- ity and dissipativity properties of the model are used. ,

1 Introduction In a previous work, we have presented a control ap- proach for the torque tracking of induction motors [7] which inherits both the dynamical performance of the Euler-Lagrange based schemes proposed in [l, 5, 41 and the power efficiency features of the minimization of the magnetic energy proposed in [6]. The feedback design is conceived on the dq rotating frame aligned with the rotor flux.

It is well known that the main problem of control schemes based on the dq (synchronous frame) model is the recovery of the flux angular position which implies the use of state observers.

Nevertheless, the conception of an optimal energy ref- erence torque tracking approach for the a@ (station- nary) frame is not only possible but equivalent to the dq one given in [7].

Therefore, the Section 2 of this paper is reserved to the motor models derived from the system Lagrangian function which are related by rotation transformations known also as the Blondel-Park Extended llansforma- tions (BPE) [3].

"This work waa partially supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnol6gico - CNPq, Brazil

Section 3 presents some relevant properties of the sys- tem. Section 4 gives the general reference trajectories, expressed in terms of the desired torque and the optimal slip frequency.'I'he derivation of optimal state reference trajectories, function of the desired torque and sub- ject to an optimization criterion (minimum magnetic energy, minimum losses, etc.), is obtained exactly the same way as in [7].

Section 5 presents the main result of this paper con- sisting on an observer-based torque tracking controller designed in the CUP frame. The torque tracking error converges to zero globally and exponentially.

Section 6 presents conclusions about our contribution and the related open problems.

2 The motor models Ortega and Espinosa 151 have derived an energy based model for the induction motor coupled with the driven mechanical system which is obtained from the La- grangian function described in terms of the following generalized coordinates :

where the last term corresponds to the angular position of the rotor with respect to the stator and is given by

(Y = / W . d t + (YO, (2.2)

and wr corresponds to the rotor speed. represents the stator and rotor

current components with respect to two different fraines - one fxed with the stator for the stator current arid the other turning with the rotor for the rotor currents - arid the rotor speed :

The 5 x 1 vector

0-7803-2559-1195 $400 0 1995 IEEE 4446

Note that w, = &. The generalized forces are given by

where Vab := [v,, Vb]* and

A¶ :=

72 := diag{R, b } ,

E := [O, o ,o , 0, --7-L]*.

The Lagrangian function is given by

1 L(qab1 i a b ) = 2&Dab((?ab)iabi

where

This model corresponds to the general rotating machine representation also known as primitive machine or ma- chine of Kron (see [2], Vol. l, Chap. 5).

There are several possible choices of factorization to obtain the second term of the equation above. One which achieves the skew-symmet ry property rendering

(2.4)

Z T [ f i a b - 2C,b]Z = 0 t/% E R5, (2.12)

is given by

(2.7) cosva sinva

-sinva cosva V(va) :=

and where v is the number of pole pairs of the machine, R, and R, are the stator and rotor resistances, L,,L, and L,, are the stator, rotor and mutual inductances, respectively. Z2 is the (2 x 2 ) identity matrix and 3 2 is the ( 2 x 2 ) skew-symmetric matrix

The a b model obtained applying the Euler-Lagrange equations is given by

which can also be expressed as

D a b ( a ) k a b + Cab(at x a b ) r a b = Qabt (2.10)

with the output torque given by

Tab := c T (a1 xab)Zab, (2.11)

where

Remark. Note that the Lagrangian is a function of L ( a , z,b). Physically, it represents the “kinetic ener& of the whole system which, in this case, is the sum of the magnetic energy stored in the coupling fields of the machine and the kinetic energy of the turning masses. (2.6)

2.1 The rotation transformation The motor model can be expressed, as well, in differ- ent reference frames. Typically, it is represented in one single frame, which can be obtained from the ab model given in (2.10), by a rotation transformation of the type :

~ ( a , e ) := diag {u(e), u(e - va), I ) , (2.13)

where U(.) is the 2 x 2 rotating matrix defined as in (2.8) and 0 defines the angular position of the frame. By making 0 = 0 we obtain the stationary reference model also known as a@ model (see 121, Vol. 1, Chap. 5); the choice of 0 equal to the rotor speed gives a reference frame turning with the rotor and by making e equal to the angular position of the input voltage we obtain the dq model. In this case the derivative 0 = w, is the electrical frequency applied to the machine. Then, by the following transformations,

:= T ( a , O ) Z a b , Q := T(al 6)Qab , (2.14)

the dq model can be obtained. The dq reference frame turns with the primary frequency, w,, which is one of the control inputs. The dq frame angular position is 0, which is given as :

@ = B o + wa d t . (2.15) I 2.2 The dq model

The dq model can be derived from the ab model (2.10) by applying the transformations defined by (2.13)- (2.14). This transformation gives

Dk + [ A f ( ~ 3 ~ x ) + 721 x = M [ :t ] + 6 , (2.16)

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where V, := [ u I , u2IT are the stator input voltages in the dq frame and 213 is the primary frequency. For more details on this model, see [7].

3 Theap model The a@ model which is usually applied to control prob- lems is expressed in terms of the following state vari- ables :

z a p = [ ZAP, . * e 1 x z p l T : = [ G, i;1 42, $ 2 1 wr I*} (3.1)

composed by the stator current and rotor flux projec- tions in the a@ coordinate frame and the rotor speed.

This model can be derived from the a b model (2.10) by applying the transformations defined by (2.13)- (2.14) making -9 = 0, followed by another linear state transformation given by :

These transformations give

% 3 h 3 + C,p(zap)zap = MV,p - Rapzap + (} (3.3)

with V,p := [U,, uplT = Vab and the following matrices definitions :

and v is the number of pole pairs while the control in- put vector U = [ U , , up I T corresponds to the stator voltages in the stationary a@ frame and L,, L,, L,,, Rs and Rr are the motor parameters as defined for the a b model.

4 Torque tracking

The key point in the "energy shaping'' type of approach is to use the physical properties of the model in order to build a feedforward or reference model which, on one hand, performs the desired torque trajectory and, on the other hand is asymptotically attainable.

In other words, by exploiting the energy based model, time varying reference state vectors can be de- rived which satisfy some dynamical matching conditions while assuring the desired output for all time t.

Since the output torque is a bilinear function of the the states, it is not uniquely defined. Therefore, there is some degree of freedom in the choice of the desired state trajectory.

4.1 The reference a,O model

Seleme e t al. [7] have shown that there is a generic writing of the time varying reference state vector, in the dq reference frame whose degree of freedom is given by the slip frequency w, := wa -vwr. An equivalent desired trajectory can be written in termes of the coordinates in the a@ frame as :

x d = I 0

d W r

where 6 = Note that the desired speed trajectory (the fifth ele-

ment of the desired state vector) U$ must be such that some desired mechanical dynamics be achieved. It has been shown by Ortega et al. [4] that the use of speed reference trajectories in the feedback can add damping to the system and improve its dynamical performance.

(w, + w s ) dt is the desired flux position.

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The reference state variables of (4.2) satisfy the fol- lowing equation set :

d [ZS - ~ ~ + ] { ~ o p j . $ j + [Cap(Wr, x o a ) + ~ o p l x t p - = 0,

M + := ( M ~ M M ) - ' M T = [ 12 0 2 ~ 3 1 ,

for any w, # 0, where Z, is the 5 x 5 identity matrix. Note that w8 can be considered as a degree of freedom on the choice of the state trajectory.

This means that the desired states, as defined by (4.2), are reachable through two inputs, U , and up. Then, the control problem consists in rendering invari- ant the manifold defined by the chosen reference vector. The slip frequency is used to optimize the desired state trajectories with respect to an energy criterion.

This is a full order nonlinear observer which has the advantage of being less sensitive to noise in the mesured signals than reduced order observers. Also, it can be used as a filter to this noise [8].

Now, taking the quadratic function

(5.4) 1 -e= V e := - ~ , p D , p Z : p , 2

and considering

u o p , x : p E IC-,

the derivative of (5.4) along the trajectories of (5.3) is

(5.5 v e - - X@(F -eT + R&Zp > 0 , V5Zp # 0 , which ensures that ZEp 4 0, asymptotically.

As for the boundedness of uap and xzp, it will be demonstrated next that the connection of this observer with the torque tracking controller is internally stable.

5 The observer based controller in ar,B

The control scheme presented in this section can be easily represented in other reference frames, obtained through rotation transformation. The assumption of exactly know model is necessary.

5.1 State observer in a@ In order to relax the assumption of full state measure ment and to recover the control law in the stationnary frame a/3 it is necessary to estimate the rotor flux vari- ables.

Consider, for this purpose the following definitions :

zZp := [zap, 1 * * - I '$317- , 2Zp := [iop, 1 - * a , 2 3 T 5Zp := X Z p - 2Zp

where the upperscript (c) denotes an observed variable. Consider then, the following state observer:

where

F : = [ K$2 :] , K l x 2 = keZz> 0 . (5.2)

The observer error equation is obtained by putting the first four equations of the system set (3.3) into (5. l), as

5.2 Stability of the a/3 observer-based control

Consider the full system model given by (3.3)-(3.4), the reference trajectories (4.1), the state observer (5.1) and the desired torque trajectory given by :

(5.6) d 7 d = JG,d + h,d + 71, - k , ( W r -U,.) ,

which is written in terms of a reference speed trajectory given by w,d(t).

Let the a@ controller be :

Putting (5.7) in (3.3) gives the following error equa-

'Oclp; + [Cop(wrp e ) + Rap + K ] e + Se - X%zp = O , e := zap - xZp , Z Z p := xQp - ?zp , xap - wr

tion :

d5 - d

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The right hand term of (5.13) can be written as :

Let

V L s r

2Lr K1= IC122 > 0 , K2 = --w:& .

Then, the full system can be completely described by :

with

J (5.14) . .

(5.15)

1 k e = C ~ - V ~ L ? ~ ( W : ) ~ 4 ck > 1 .

L 1 d Using the same argument as in [ l ] , concerning the

speed error dynamics which cannot grow faster than an exponential, we can prove that

e5 := w, - w, .

(5.9) Remarks :

1 1 ~ 1 1 I mzlIz(0)IIe-Pzt vt E [ o , ~ ) , (5.17)

with mz,pz independent of T . This reasoning can be extended to the whole real axis and thus, proving that the convergence

G Ehq. (5.9) a) describes the error dynamia of the observer. Its convergence is exponential provided that the mechanical speed w, is bounded.

H Q. (5.9) b) describes the dynamics of the motor electrical variables error. x a p + z $ and r - - + ~ d

I &. (5.9) C) describes the dynamics of the mechan- Notice that there are posi-

is global exponential with internal signals bounded. ical speed error e5.

6 Conclusion tive constants al, a2 independent of T suchthat 1. - ~d 5 a1 + azlles(t), 6s(t)II , Vt E [O, T ) .

As it can be seen from the ab model in Section 2, ob- tained from the system Lagrangian, the workless forces explicitly given in this model are the key point in the

The stability proof follows the same rationale df the Theorem 2 in [l]. Consider the quadratic function :

1 2

(5.10)

with its derivative along the solutions of (5.9) a) and b), for t E (0, T ) , given as :

VT = s ~ ~ p D , p Z i p 1 -eT + -ETDapE,

VT := --tTQ(w,,w$)z . (5.11)

with the extended vector z := [ E T , Z$IT and the matrix

Knowing that (Rap+diag{211Z2, 0)) > 0, the matrix Q(w,d) is positive definite if and only if

(Rap + diag(2k1Z2,O) > ;L ' T (wr)(R,p d + F)-lL(w,d) . (5.13)

design of the control law and the state observer. These workless forces which do not have to be com-

pensated by the controller, stem from the passivity of the system while the dissipativity which add natural damping to the control, is due to the coil losses and the mechanical friction. These properties are preserved under the Blonde1 Park Extended transformation. This allows the use of simpler models which, as far as the control low is concerned, preserve the same structure and dynamical characteristics.

The observer-based torque tracking control scheme for the ap frame, based on the optimal energy trajec- tory, has been developed. The torque tracking is glob- ally exponentially attained with internal stability.

The criteria of optimization of the reference state tra- ject'ory are chosen in terms of the control goals. They can be the motor magnetic energy, the losses, the input current norm or other convex state function.

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The desired slip frequency is used as a degree of free- dom in the construction of the optimal trajectories. This is possible because the torque is a bilinear func- tion of the states and can be obtained through different combinations of them. The optimization is not revisited in this paper since it is exactly the same as in [7].

The inclusion of the model uncertainties and the pa- rameters estimation combined with the optimal refer- ence trajectory control is still an open problem.

References [l] C. Canudas de Wit, R Ortega, and S. I. Se-

leme. Robot motion control using induction mo- tor drives. In Proc. of the IEEE Int. Conf. on Robotics and Automation, 1992. Also submitted to the IEEE Trans. on Robotics and Automation.

[2] J. Chatelain. Machines Electrigues. Dunod, Paris, 1983.

[3] X. 2. Liu, G. C. Verghese, J. H. Lang, and M. K. Onder. Generalizing the blondel-park transforma- tion of electrical machines : Necessary and suffi- cient conditions. IEEE Trans. on Cicuits and Sys- tems, 36(8):1058-1067, 1989.

[4] R. Ortega, C. Canudas, and S. I. Seleme. Nonlin- ear control of induction motors: Torque tracking with unknown load disturbance. IEEE Trans. on Automatic Control, 38(11):1675-1680, 1993. Also appeared in Proc. of the American Control Con- ference 92.

[5] R Ortega and G. Espinosa. Torque regulation of induction motors. Automutica, 29(3):621-633, 1993.

[SI S. I. Seleme Jr., C. Canudas de Wit, and T. Brandt. Minimum energy torque control of in- duction motors via nonlinear feedback. In 3rd In- ternational Workshop on Microcomputer Control of Electric Drives, pages Kl-K11, 1991.

[I S. I. Seleme Jr., M. Petersson, and C. Canudas de Wit. The torque tracking of induction motors via magnetic energy optimization. 1994. Submitted to the Proc. of the IEEE Conference on Decision and Control’94 and to the IEEE Trans. on Automatic Control.

[8] G. C. Verghese and S. R. Sanders. Observers for flux estimation in induction machines. IEEE Trans. on Industrial Electronics, 35(1):85-94, 1988.

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