Synthesis And Digital Implementation Of A Reduced Order Rotor Flux Observer For Im Drive

Synthesis and Digital Implementation of a Reduced Order Rotor Flux Observer for IM Drive Alfonso Damiano, Gianluca Gatto, Ignazio Marongiu, Alessandro Pisano University of Cagliari Department of Electrical and Electronic Engineering P.zza d’Armi, 09123 Cagliari, Italy. { alfio, gatto, marongiu, pisano) @ elettrol .unica.it

Abstract -This paper deals with the rotor flux detection in IM drives by means of reduced-order observers. The basic structure discussed by Verghese et al.[l] is slightly modified, and a novel observer class is proposed. A detailed stability analysis by Lyapunov approach is conducted, and the effect of rotor resistance variation is investigated. The discrete time implementation of the observer model is considered, and the Z-transform method is used to ensure the stability of the observer whichever is the adopted sampling period. Experimental investigations are conducted on a direct field oriented (DFO) IM drive to test the on-line behaviour of the proposed observer class.

1.

INTRODUCTION

Due to its ruggedness, maintenance free operation and many other advantages, induction motors (IM) have become widely used in variable-speed drives. This development is principally due to the great improvement of pPs performance, which allows implementing complex, and more efficient, control techniques. At the present time, the Direct Field Oriented (DFO) control technique is the widespread used in high performance IM drives. It allows, by means a co-ordinate transformation, to decouple the electromagnetic torque control from the rotor flux one, and, hence, to manage IM as DC motor. Its application requires the detection of rotor flux vector, which is not directly measurable. For this reason, rotor flux observers are commonly used. Being the effectiveness of the control strategy based on a right rotor flux detection, the drive performance is strictly connected to the rotor flux observer one. So, the characteristics of the observer, in terms of stability, accuracy and robustness, critically influence those of the drive. Real time simulators of the machine model are commonly used when only the steady state behaviour is prescribed. Main drawback of real time simulators is that the speed convergence of the estimated flux to the actual one depends only on the machine parameters. Nevertheless, for high-performance applications, it should be desirable to increase in some way the convergence rate. The linear control theory suggests that a corrective signal, derived from some prediction error, should be added to the simulator model [1,3]. This allows to arbitrarily setting the dynamic behavior of the observer. In particular, the reduced order observers, which estimate just the unavailable part of the state vector, are well suited for providing fast and accurate flux estimation, with increased robustness against parameter variations 0-7803-5662-4/99/$10.00 01999 IEEE

and reduced hardware (or computational) complexity. The design procedure is generally based on the assumption of perfect matching between the nominal and actual machine model. This is not the case in real operating conditions, and, for this reason, some parameter uncertainty must be taken into account when the observer performance is evaluated. Basic work in this field was that of Verghese and Sanders [I], in which an exhaustive analysis of the problem was carried out. Currently, digital processors are widely used for IM drives control. This motivates the great interest in dealing with the discrete time implementation of the observers. Here the design procedure is made in the continuous time. Then the discretization of the observer model by Ztransform method is performed, to avoid possible unstability phenomena that may occur when incautious discrete time implementation of the observer is performed. An interesting approach was proposed in [2], which consists in developing a suitable simplified discrete model of the machine, which allows performing the observer design in the discrete time domain. In [4] Bellini e Figalli proposed a reduced order observer together with an interesting procedure for setting the observer gains so as the effect of parameter variations are taken into account. In this paper, the attention is focused on the reduced order observers based on rotor vector equation. A novel observer scheme is proposed, and some issues associated with the effect of the rotor time constant variation, and with the digital implementation of the observer, are discussed. Experimental results show the good effectiveness of the proposed observer class. 11. REDUCED ORDER ROTOR FLUX OBSERVERS FOR IM DRIVES The idealized two-axis model of the IM electromagnetic dynamics can be represented by a linear time-varying system, characterised by a state vector whose components can be the stator current and rotor flux vectors, and with the rotor speed considered as a time-varying parameter. Assuming, without loss of generality, that there is one pole pair, the IM model can be represented by

-ar d dt 729

= [ - a , +wJ&, ~ +a,L,i,

ISIE’99 - Bled, Slovenia

d v, =--Lm a,. +oLS-is + r,i, L, dt dt where A,=[@, , is the rotor flux vector, &=[is, , iSblT is the stator current vector, vs=[v,, , vSblT represents the stator voltage vector, a,is the inverse value of the rotor time constant, wO(t)is the rotor speed, L,, L,, L, are the linkage, rotor and stator inductances respectively, d = 1- L i /(LsLr) is the leakage coefficient and the matrixes I and J are defined as

[ ]

J = [ o -l]

0 1

1

I= 1 0

(3)

0

Being the stator current and the rotor speed directly measurable, the reduced order observer is synthesised to detect the unavailable rotor flux components. The basic structure of this observer class has been proposed in [1], which is given by

which does not need for informations about current derivatives. Define the estimation error as

e=a,

-2,

(9)

The error dynamics turns out to be given by

[-

:r

e = I K-

[-~r,I+wJ]e

which is linear time variant, and so the stability analysis cannot be dealt with by simple eigenvalues analysis. A constant speed condition can be first assumed, and the observer gains are set according to the eigenvalues analysis. Then, by Lyapunov approach, the asymptotic stability of the time varying error system is verified. For the sake of simplicity, the observer gain matrix is set as

Error matrix has eigenvalues

-a, d~ =[-a,i+dN,+arL,is +K(G, -vs) dt

4 2

(4)

2,

where is the estimated rotor flux vector, ?, represents the estimated voltage vector and K is the observer gain matrix. The first step of the observer design is that of choosing a proper form for the estimated stator voltage vector. Then, the design procedure terminates with a proper tuning of the gain matrix coefficients k,, (i,j=1,2) which makes the error system asymptotically stable.

=

Lr (-ark io) L, - kL,

The error system is asymptotically stable if the complex pair of its eigenvalues has negative real part. To increase the convergence speed with respect to the real time simulator speed, guaranteeing at the same time asymptotic stability, the observer gain k must be chosen in accordance with

A . The Verghese-Sanders observer The predictive term is directly derived from the stator equation (2), according to

L,dd v, =-A, +oLs-is L, dt dt A

+ rsis

A rigorous approach to the stability analysis, which takes into account the effects of speed variation, requires the use of Lyapunov theory. Take the Lyapunov function candidate

v(e)=(eTe)/2

To avoid needing current derivatives, a suitable auxiliary variable z(t) can be defined in accordance with

(14)

Differentiating (14) yields

V ( e )= -a,. L, LrkLmlle1I2 so that the observer model, in terms of the z variable, turns out to be given by d = [-a,I + dflr + a,L,,,i,+ K(rsis- v.) dt

-2

X,=

[-

:I’

I K-

(7)

so that (13) is sufficient for guaranteeing the negative definiteness of the Lyapunov function derivative V ( e ) , which implies the convergence to zero of the error vector components [ 11. B. The Proposed observer

[z+oL,K~,]

The authors propose the use of a different predictive term, synthesised by means of a proper re-elaboration of 730

the IM equations. Substituting (1) into (2) yields

proposed observer feature more robustness, from this point of view. 111. EFFECT OF ROTOR RESISTANCE VARIATION

d

'

+oLs - i s dt

The voltage vector estimate

?, can be directly derived by

(16) with A, = A, , and a different observer class turns out to be correspondingly defined. Also for this observer class, an auxiliary variable has to be introduced to avoid the current derivatives Define z = Ar -KoL,i,

The observer model is given by

ir= z+aL,Ki, Put

K=kI and take the Lyapunov candidate function

This section is devoted to briefly investigate the effect of rotor resistance variation on the performance of the proposed observer. In [ l ] it was shown that, if the assumption of a perfect matching between the nominal and actual machine model is removed, then the error dynamics is modified with respect to the "ideal"case, More specifically, one has that a driving term is introduced in the error dynamics, which is autonomous and stable when the parameter variation is neglected. This causes the error does not converge to zero. On the contrary, the error converges to some residual set, which size depends on the amplitude of the parameter variations. There are two major routes that can be followed for analyzing the error behaviour. The first one consists in picking a proper Lyapunov function (e.g. the simplest Lyapunov function V(e)=eTe), aimed at showing that the error is asymptotically confined to a certain bounded region. Such an approach requires the knowledge of a-priori bounds of the parameter variations. The second approach, widely used in the literature, consists in referring to the sinusoidal steady state. A deep analysis of this problem is out from the scope of this paper. Here we are aimed to show that as faster is the error convergence speed as bigger is the steady state error of the estimated flux with respect to the actual one. Assume that the actual rotor rotor time constant a,* differs from the nominal one a,. The error dynamics can be rewritten, for the Verghese observer class and for the proposed one respectively, as

The Lypaunov function derivative can be computed as

.

e=- L,+kLi

The observer gain stability range results to be the semiinfinite interval

kc(-?,-) The error convergence to zero can be made arbitrarily fast by sufficiently increasing the observer gain. Theoretically, the Verghese observer (7),(8) and the proposed observer are able to perform comparably; it must be underlined that the practical implementation of the observer can lead to some problems, especially when high convergence speed is required, that is for k tending to LA,,, as far as the Verghese observer is concerned and for k tending to infinity for the proposed observer. In this case, one has that the Verghese observer requires some divisions for very small numbers, while the proposed observer involves some products for very high numbers. The experimental investigations will show that the

L,

[- a,I + O

J +~L, + kL, (L,

a, + Lmi,)(a,-a:)

(25)

By simple inspection of (24) and (25) it can be easily verified that a fast convergence to zero of the error unavoidably causes the effect of parameter variations to be amplified. With basis on these consideration, a right compromise have to be found, when the observer gain k is set, between the prescribed error speed decay and the consequent steady error. IV. DIGITAL IMPLEMENTATION OF THE OBSERVER The most commonly used method for discretizing continuous systems is the well known "Eulero" method. The above method allows obtain very simplified discrete models, but its use requires some care, since its convergence properties turn out to be strictly linked to the choice of the sampling period. As it was evidenced in [l], bad unstability phenomena may occur if the adopted

73 1

sampling period is not sufficiently small. Aimed at avoiding the above mentioned stability problems, the linear control theory provides more complex, and accurate, methods for deriving discrete models of continuous plants. In particular, the 2-transform method is well suited when, for some reasons, the sampling period cannot be taken sufficiently small. This is because the Z-transform method always ensures stability is always ensured whichever is the adopted period. The continuous models of the Verghese observer and of the proposed observer have the same structure, which is given by d dt

-z

(26)

= Mz +Ni, -Kv,

being K the observer gain matrix and M, N proper matrixes to be separately defined for any observer class, Assume (1 1) for the sake of simplicity. The matrixes associated to the Verghese observer class are given by

M=

Lr

Lr - kL,

(-or,I+UxJ) +a,L,+kr,

P

M=

L, + kLm

(- a,I + ~ l )

Lr

sin(bT)

and

O, = beaTsin(bT)+a(eaTc o s ( b ~ ) 1) 0, = aeaTsin(bT)-b(eaTcos(bT)- 1 )

(36)

V. EXPERIMENTAL, RESULTS The performance of the proposed observer class has been experimentally verified on a FO current regulated induction motor drive. It is constituted by 75 kVA transistors inverter with 510 V DC bus voltage and chopped brake resistance. A 15 Nm induction servomotor has been used. The ratings are given in the following Table I. TABLE I

I

Ratings of the Motor and Parameter Values 2.2 k W

RatinP meed

I Pole pair

0

Inertia constant Stator resistance I Stator leakage inductance Mutual inductance Rotor leakape inductance Rotor time constant

being T

I

The M matrix turns out to have the following structure

The particular simmetry of the M matrix allows easily MT and its integral 0. calculate the exponential matrix e More specifically, one has

(35)

Note that, at every sampling time, the discrete time observer matrixes must be real time evaluated, since they depend on the measured speed Nk]. The computational effort required by the Z-transform discretization method is slightly stronger than that corresponding to the traditional Eulero method, but this is the price that one pays for avoiding possible stability problems linked with incautious choice of the sampling period. On, the contrary, one has that the Z-transform method ensures the stability for any adopted sampling period.

T

a

(33)

being

z[k + I ] = e M T z [ k ]ON + i, [k]+ k j e M T v[k] , (30)

-7

cos(bT)

. (34)

The 2-transform method assumes that o,is and v, are piecewise constant signals with period T. This allows to define a suitable time varying discrete model, which is given by

M=[ab

[

+&-J

(28) L,L -LkL, O while, for the proposed observer class, the corresponding matrixes are expressed as

1

cos(bT) - sin(bT)

eMT

1410 mm 10.009 kg m2 10.8533 SZ 0.005 H

I

I I

0.085 H 0.005 H 194.73 ms

I

The control system is implemented using analog technique. The current loops are implemented using PI controllers and triangular carrier having 2.7 kHz frequency. The actual system is interfaced to a DSP system equipped with a TMS320C31 floating point master DSP having 40Mhz clock frequency, a fixed point slave TMS32OP14 DSP and a 128 Kword 32-bit static RAM. Moreover, the acquisition board is equipped with a parallel encoder interface channel (24-bit counter, 732

8.3MHz bandwidth) to which the 5000 ppr encoder of the servomotor has been interfaced. The board has been programmed using C language. For all the test conducted, the sampling period has been set to T=O. 1 ms.The observer gain is set to 0.55. The comparison between the evolutions of the full order observer implemented in the drive and the Vergese observer discretized by Eulero method is reported in figure 1. A square wave reference speed (=1.5 Hz, k450r-m) is used to highlight the transient phenomena due to the speed inversion. The overlapped traces 1 and 2 report the real part of the flux estimated by the full order observer and by the Verghese observer respectively, while trace 3 show the speed wave-shape. In Figure 2 the full order observer is compared with the proposed observer, discretized by Eulero method. The traces have the same structure as the previous one In Figg 3 and 4 the Verghese observer and the proposed one, both discretized by Z-Transform method, are compared with the full order observer implemented in the drive. In figure 3 the traces 1 represents the real part of the flux vector estimated by means of the full order observer , trace 2 represents the real part of the flux vector estimated by means of the Verghese observer, and trace 3 represents the actual speed. The same quantity are reported in figure 4 referring to the proposed observer . Finally, in Fig. 5 , the proposed observer flux component are reported referring to a triangular wave-shape reference speed, for evidencing the behaviour under varying speed conditions. The execution times required by the Verghese observer and by the proposed one have been evaluated during the tests. From this point of view, the two different observers perform comparably; it must be noted that, for both the observers, the Eulero discretization requires about lops, while the Z-transform discretization causes the above computation time to be doubled. VI. CONCLUSIONS

Tek ~rmpl 500 s/s

k-T+

2 Acqs

3

I _

.

4

.

.

U 500mV 13:51:01 Fig. 1: Comparison between the Vergese observed flux component and Full-order one (Eulero integration.)

. . .

. . .

. . .

U 500mV 13:54:37 Fig. 2 : Comparison between the proposed observed flux component and Full-order one (Eulero integration.)

Tek

This paper has dealt with the synthesis and digital implementation of a reduced order flux observer for IM drives. Some considerations regarding the unavoidable mismatching between the actual and nominal rotor time constant are carried out, together with the problem of a stable digital implementation of the observer. The results obtained by using the Eulero and the Ztransform discretization methods have been compared by means of experimental investigations, to evidence the necessity of using the latter method if the robustness have to be guaranteed in a wide range of sampling frequency.

500 s/s

10 Acqs

I-I-F-------kd

.

.

.

Fig. 3 Comparison between the Vergese observed flux component and Full-order one (Z-transform discretization.)

733

500 s/s

lek

I

+I-

!

c1i3zoom:

16 Acqs

1.oxvhrt

l-----i iox~brr ’

’

1

Fig.4 Comparison between the proposed observed flux component and Full-order one (Z-transform discretization.)

Fig 5: Evolution of the observed rotor flux components during a triangular speed wave-shape.

REFERENCES G.C Verghese, S . R Sanders. “Observers f o r Flux Estimation in induction Machines”, IEEE Trans. On Industrial Electronics, Vol. 35, no 1 35-94, 1988. C.P.Bottura, J.L. Silvino, P. de Resende: “A Flux Observer for induction Machines Based on a TimeVariant Discrete Model”, IEEE Trans. On Industrial Application, Vol29, no 2, 349-353, 1993. D. Luenberger: “An introduction to Observers” IEEE Trans. on Automatic Control, Vol. 16, no 6, 596-602,1911. A. Bellini, G . Figalli: “Analysis and Design of a Microcomputer-Based Observer for an Induction Machine” Automatica, Vol. 24, no 4,549-555, 1988.

734