IEGONO1: The 27th Annual Conference of the IEEE Industrial Electronics Society

A Digital Rotor Flux Observer for Robust Vector Control of the Induction Machine

Celso P. Bottura and S6rgio A. Augusto Filho

DMCSI-FEEC UKICAMP CPFlOl 13081-970 Campinas BRAZIL

cpb@turing.unicamp.br, sergioafQieee.org http://mm.dmcsi.fee.unicamp.br

Abstract- The aim of this paper is to find a solution for the rotor flux observation problem in an induction machine drive with a robust controller which performs the feedback of the slip frequency. The observer structure shall provide an accurate estimation even in the presence of either rotor resistance variation or load torque disturbances.

11. PRELIMINARIES

Consider the non-stationary discrete time model of the induction machine in dq synchronously rotating reference frame, proposed in [SI:

I . INTRODUCTION

The modern induction machine control technics have the objective of guaranteeing stability, performance and dis- turbance rejection to the electrical drives. One of the most commonly accepted technics is the vector control proposed by Blaschke [a ] , more particularly, the indirect vector con- trol. This method presents a good performance, however, its implementation requires an accurate information about the rotor flux. Generally, this information is obtained from rotor flux state estimators which: by its turn, needs the correct parameter values of the machine.

where:

1 A [ - a R s B f i- (l- ; ) k B $ c(Af -A$ + k B $ ) M k B , d . A i - k B $

The state vector is constituted by the two-component vec- tors, is = [ iq s ids ] ’ and A, = [ A q r X d r ] ‘ , which are the stator current and the rotor flux, respectively. The control input is constituted by the stator voltage vector Vs = [V,,Vd,]’, where:

Many different results have been presented in the liter- ature in order to solve this problem. As examples, we can cite the application of the adaptive and identification the- ory [6] [4], Kalman filtering [5] and, more recently, the use of robustness concepts.

In this work, we propose a robust rotor flux observer based on the discrete fourth order electrical model of the induction machine in dq synchronously rotating reference frame [8] [9]. This model aproximates the machine dy- namics over its whole operating range. Consequently, the observer equations are non-stationary. Besides that, they depend on the variation of parameters, as rotor and sta- tor resistances and leakage and mutual inductances. These above described variations are dealt as system uncertainties and are modeled as convex polyhedral sets, where each op- erating point corresponds to a combination of the extreme values of the matrices entries in the observer equation.

The synthesis of the robust observer consists in the com- putation of a static gain that makes the estimation error equation asymptotically stable for all the machine operat- ing range and parameter changes specified. The procedure adopted is the solution of the pole placement problem. The computation of the gain is done off-line, hence the imple- mentation of this robust observer dos not require any ad- ditional computational resource. On the contrary, it pro- vides some computational gain once no on-line synthesis is required.

Some simulation results are shown.

1

1

cos (wh) -sin(w h) ) A! = [ sin(wh) cos(&)

1

1

sin(wh) -(1 - cos(wh)) 1 - cos(wh) sin(wh)

cos(w, h) -sin( w, h) ) sin( w, h) cos( w, h)

sin(w,h) -(1 - cos(w,h)) 1 - cos(w,h) sin(w,h)

Bf =; [ A i = [ B,d = &

w, w, are the stator and slip frequencies, respectively. R,, R, are the stator and rotor resistances, L,, L, are the stator and rotor inductances, M is the mutual inductance and h is the sampling period. These parameters define the following constants:

M2 L T g = l - - L,L, Tr = R,

This model is completed with the torque and motion equa-

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IECONO1: The 27th Annual Conference of the IEEE Industrial Electronics Society

where wT is the rotor electrical frequency, P is the number of pairs of poles, C1 is the viscous friction, TL is the load torque, J , is the system inertia and C2 = -PC,/J,.

The vector control of the induction machine is a very accepted method when high performance of the system re- sponse is required. It is based on the decoupling of the magnetizing and torque producing components of the sta- tor current. Under this condition, the q-axis component of the rotor flux is set to zero while the daxis reaches the nominal value of the magnetizing flux, the torque equation becomes:

and a slip relation is derived:

Considering equation (4) and that A d r = Mids, the torque equation can be written as:

where the subscript z = 1 refers to the speed-flux subsystem while i = 2 refers to the stator current subsystem; z, and U , are the state and input vectors, respectively, y, is the output and w, is an external perturbation that corresponds to the load torque for the speed-flux subsystem and to the emfvoltage for the stator current subsystem. The matrices A, , B, and D, are the ones shown in equations (7) and (8) and E, are identity matrices of apropriate dimensions representing full knowledge of the states.

The speed control and the rotor flux orientation as well as the stator current regulation are achieved by the solution of a servomechanism problem resultant from the definiton of the following error equation:

zf(k + 1 ) = zf(k) + Yi(k) - y t ( k )

where y:(k) is an external reference for the output y,(k), and the definition of the augmented state vector i , ( k ) = [ z l ( k ) zf(k)]' and its dynamic equation:

Once the speed and rotor flux control are intimately re- lated to i,, and i d s control, the system is separated in two subsystems that compose a cascaded loops structure. The outer loop is constituted by the speed-flux subsystem rep- resented by the equation (7):

while the inner one is constituted by the stator current subsystem given by the equation (8):

Rewriting both subsystems with a short notation, gives:

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where v , ( k ) is the output of the augmented system, I and 0 are identity and null matrices of apropriate dimensions, respectively. Applying state feedback, ui ( k ) = - K,zi ( k ) , the closed loop state space equation becomes:

where the controller matrix gain is partitioned as K, = [Kti K22].

Note that the whole cascaded system is driven by exter- nal references for the rotor speed and the rotor flux, defined by y;(k) = [A;, A:,. u ; ( k ) l T . where A;, = 0, A;, = A,, and w : ( k ) is the desired speed. However, in the case of the stator current subsystem, the reference vector is pro- vided by the controller of the speed-flux subsystem. The input vector u l r computed in the state feedback procedure, determines the necessary values of stator current and slip frequency for the machine operating condition specified by the external reference y;. Therefore,

and

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IECONOI: The 27th Annual Conference of the IEEE Industrial Electronics Society

The stator current controller, by its turn, determines the the voltage reference that is given to the power converter:

while the stator frequency is obtained with the use of a speed sensor and the basic equation w ( k ) = w,(k) + w,*(k) .

Clearly, both subsystems are non-stationary once the model depends on the slip and rotor frequencies. The speed-flux subsystem is also non-linear because of the rela- tionship between flux and current in the torque expressions. Besides that, the system paramenters can change mainly because of variations on the temperature. The range of pa- rameter variation for the above reasons can be determined and added to the representative model of the induction ma- chine. Each extreme operating condition is represented by a specific model that defines a vertex of a polytope in the parameter space [9]. So, for any operating condition the system matrices can be derived from a convex combination of such vertices, that is, they are elements of sets defined like:

N N

where A is a generic matrix and N is the number of vertices of the polytope.

The controller structure employed here, known as inte- gral control 1121, presents a good robust performance and stability. Its behaviour under rotor resistance variation and load disturbance conditions is shown in the following simu- lation results, assuming that the actual rotor flux is known. For the induction machine described by the parameters in the appendix, the simulation starts with the rotor flux ori- ented, the rotor speed at 100rad/s and the machine param- eters at their nominal values.

At instant one second, the rotor resistance suddenly is increased in 100%. The controller reacts and the motor slip is changed to compensate the parameter variation (figure 3). However, both stator current components and conse- quently the magnetization level and the generated torque are kept constant (figure 2). At instant two seconds, a disturbance in the load torque is introduced and its value is multiplied by two. Again, the robust controller reacts on the slip frequency and also on the pcomponent of the stator current in order to generate a torque that keeps the rotor speed constant. In both cases, the orientation of the rotor flux is not affected in steady state (figure 1)

Note that the different behaviour of the speed-flux con- troller regarding the different nature of the disturbances is one of the main benefits of the slip frequency feedback. Dif- ferently from the traditional indirect vector control meth- ods where a feedforward path for the slip frequency is de- signed, the feedback of this signal allows the automatic compensation to rotor resistance variation. This is an im- portant characteristic that will be considered in the ob- server design.

0-7803-7 108-9/01/$10.00 (C)2001 IEEE 109

Fig. 1. Rotor Flux

35

3 -

Fig. 2. Stator current

I 1 5 2 2 5 T,Il,*(*)

U 5

Fig. 3. Slip frequency

IECONO1: The 27th Annual Conference of the IEEE Industrial Electronics Society

111. ROTOR FLUX OBSERVER

Consider equation (1) rewritten as the following:

iqs(k + 1) 41 -42 43 [ i d s ( k f 1) ] = [ $2 41 4 4 -$: ] [ ] Aq,(k + 1) 4 5 -46 47 -48 Aqr(k ) &r(k + 1) 46 45 48 47 A d T ( k )

where:

41 =

42 = + 43 =

4 4 =

4 5 =

46 =

47 =

4 8 =

7 1 =

7 2 =

The equation above can be partitioned as follows:

i s ( k + 1)

(14 The reduced order observer is than described by the fol- lowing equation [8] :

&-(k) = @ b b i r ( k - 1) + @ b a i s ( k - 1)

- r&(k - 1) - @ & i r ( k - I)]

Defining the estimation error as e ( k ) = & ( k ) - i T ( k ) , and representing the non-stationarity of the system by the poly- hedral sets defined in (12), the dynamic error equation is derived:

where @bb E Vabb, @ab E Vaabr L is the observer gain and ' means the matrix transposition operation.

The synthesis of the rotor flux observer constitutes the computation of the observer gain L that makes the equa- tion (16) asymptotically stable. So, it is quadratically stabilizable via linear synthesis for any @bb E v a b b and G a b E VaGb if and only if there exists a matrix P = P' > 0 such that @>P@f - P < 0. If the sets Dabb and D Q ~ ~ are polyhedral the quadratic stability condition has to be ver- ified only at the vertices, i.e., C$P@fi - P < 0, i = l...N. It is equivalent to say that there is an augmented system associated to (16), given by

~ ( k + 1) = F z ( k ) + Gr(k) (17)

which is quadratically stable, where F E % P x P , p = m f n, and the constant matrix G E WXm, are defined by

If the set DF is polyhedral, there exists a matrix W = W' > 0 such that FiWF,' - W < 0, i = l...N, and the observer gain can be evaluated by L = (W!JVcl)', where

with W1 > 0 E PXn, W2 E Pxm. The above necessary and sufficient Lyapunov condi-

tions guarantee that the poles of a> are allocated in the unity circle. However, if the matrix @bb is modified as am = f ( @ b b + 01) E Darn, the pair (Lm, Pm), solution of +,fPm@Lf - P, < 0 assigns the poles of +,f = (@bb - L m @ a b ) in a circle of ray T and centered at the point (0,a). The observer gain is computed in terms of the modified augmented matrix

Based on the above concepts, the gain L can be computed by the convex optimization problem proposed in [ll].

Generally, the correctness of the state observation de- pends on the correct values of the parameters. However, in this case, the observer is benefited by the robust control action type previously presented. Note that, under a ro- tor resistance variation scenario, the compensation action is done by slip frequency changes, independently of the sta- tor current, which is kept unaltered (see figures 2 and 3). This kind of perturbation can be made transparent for the observer if the slip used by it, wg, is computed by:

with measured currents and nominal parameter values. In this case, the observation of the rotor flux is performed as if no parameter variation had occurred.

In the presence of load torque disturbances, the control acts over the slip frequency and the stator current. As w," is

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ow /

-0 02

-OM -Oo3 t \ -005

09 091 092 093 094 095 0% 097 098 099 1 RePI

Fig. 4. Observer poles

Fig. 5. Rotor Speed

calculated with measurements of is, again the perturbation do not affect the observation.

Considering the above remarks, the observer gain is eas- ily synthesized by pole allocation, such that the observer dynamics be faster than the system one. In this case, the circle of radius 0.05 centered in point (0.95,O) of the z-plane is chosen and the poles of the equation (16) are assigned within this region. Figure 4 shows the observer poles for all possible operating points constrained by w E [0 1001 and

Some simulation results regarding the application of the flux observer are derived with the same initial conditions as before. The disturbances are also the same: +loo% step variation in the rotor resistance and in the load torque at the instants 1 and 2 seconds, respectively.

Again, the controller compensates the rotor resistance variation by acting on the slip frequency which reaches the double of its original value. The slip frequency that is cal- culated for the rotor flux observer, however, is kept con- stant (figure 6) and the estimation is not affected by the rotor resistance variation, except during a short transient time.

By its turn, the disturbance in the load torque causes a control action on both the +component of the stator current and the slip frequency. This variation in the stator current also changes the slip frequency of the observer and its output tracks the actual rotor flux.

J, E [-20 201.

IV. CONCLUSION

In this paper a solution for the observer based digital control problem of the induction machine was presented. The characteristics of the robust controller which performs the feedback of the slip frequency allowed the development of the rotor flux observer. The resultant estimation showed to be precise even for unreal changes in the rotor resistance. Very sudden variations in the load torque were also well overcame. The results under the severe conditions in which the closed loop control system was simulated proves its applicability.

0 0 5 t 1 5 2 25

r,me (S)

Fig. 6. Actual Slip and Slip for the observer

7 1 Estimated flux

06251 /I 0 625

E , , - - I , I,

2 062

' / 0 615

l

, , ,

0 5 I 1 5 2 06

Time IS)

Fig. 7. Rotor Flux

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APPENDIX Induction Motor Parameters

lHp, 220/380V, 60Hz Number of pole pairs P 2 Nominal stator current I , 4.1A Stator resistance R, 7.1R Rotor resistance Rr 5.78 R Stator inductance L, 310.5mH Rotor inductance Lr 310.5mH Mutual inductance M 284.56mH Moment of inertia J M 0.0038kg.m Viscous friction C1 0.0015Nms

REFERENCES [l] K. Hasse, On the Dynamics of Speed Control of a Statzc A C

Drive with a Squirrel-Cage Induction Machine, Ph. D. Disserta- tion, Techn. Hochschule Darmstadt, 1969.

[2] F. Blaschke, “The Principle of Field Orientation a s Applied to the New Transvector Closed Loop Control System for Rotating Field Machines”, Siemens Review, Vol. 39, pp. 217-220, 1972.

[3] K. B. Nordin, D. W. Novotny and D. S. Zinger, “The Influence of Motor Parameter Deviations in Feedforward Field Orientation Drive Systems”, IEEE Trans. on Industry Applications, vol. IA-21, no. 4, pp. 1009-1015, 1985.

[4] T. Matsuo and T . A. Lipo, “A Rotor Parameter Identification Scheme for Vector Controlled Induction Motor Drive”, Proc. of IEEE IAS Annual Meeting Conf. Rec., pp 538-545, 1984.

[.5] L. C. Zai, Application of an Extended Kalman Filter to Induc- tion Machine Parameter Estimation, Ph. D. Dissertation, Univ. of Wisconsin-Madison, 1986.

[6] H. Kubota, K. Matsuse and T. Nakano, “DSP-based speed adap- tive flux observer of induction machine”, IEEE Trans. on Industry Applications, vol. 29, no. 2, pp 344-348, 1993.

[7] G. M. Liaw and F. J. Lin, “A Robust Speed Controller for Induc- tion Motor Drives”, IEEE Trans. on Industrial Electronics, vol. 41, no. 3, pp. 308-315, 1994.

[8] C. P. Bottura, J. L. Silvino and P. Resende, “A flux observer for induction machines based on a time-variant discrete model”, IEEE Trans. on Industry Applications, vol. 29, no. 2, pp 349-354, 1993.

[9] C. P. Bottura, S. A Augusto Filho and J. L. Silvino, ‘“2 Robust Digital Vector Control for the Induction Machine”, IEE Proc. Con- trol Theory Applications, vol. 143, no. 3, pp 237-243, 1996.

[lo] C. P. Bottura and S. A Augusto Filho, “A Parameter Plane Ap- proach for State Space Induction Machine Modeling and Control”, Proc. of the 5th Electrimacs, France, pp 611-616, 1996.

[Ill A. S. Milfont, S . A. Augusto Filho and J. C. Geromel, ‘“2 Ro- bust Control Design With Regional Pole Placement for Discrete- Time Linear Systems”, Proc. of the European Control Conference- ECC95, Italy, pp 3388-3393, 1995.

[12] G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, Addison-Wesley, 1994.

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