Estimation of rotor resistance in induction motors

D. J . Atki nson J.W.Finch P. P. Aca r n I ey

Indexing terms: Induction motors, Estimation, Self-tuning, Vector control, Parameter estimation, Extended Kalman filter

Abstract: The evolution of high performance vector control has transformed the transient behaviour available from induction motor drives. The variation of the rotor resistance in such drives remains a problem of significant industrial interest. Parameter estimators are needed which operate in both the transient and steady-state regions of the drive. Such a parameter estimator is described, which is a novel variation on the extended Kalman filter (EKF). Execution of the EKF demands a high computational perfor- mance. The algorithm presented in the paper makes use of a model order reduction process that cuts the computational requirements to approximately one third of that demanded by the EKF. The theoretical development of the algorithm is followed by a simulation study which is used to illustrate the possible range of behaviour including the introduction of noise and modelling errors. Finally, an experimental examination of performance is presented, which shows the high standard obtained when the new estimator is applied to a practical inverter machine drive.

List of symbols

R, = stator resistance, C2 R, = rotor resistance, C2 L,y L, L, = mutual inductance, H L,, = magnetising inductance, H Le = leakage inductance, H a = turns ratio CO,

or CJ = total leakage factor V I

= Ftator self inductance, €1 = rotor self inductance, H

= stator angular frequency, radls = rotor angular frequency, radis

= two-axis voltage vector, V = two-axis current vector, A

0 IEE, 1996 IEE Proceedings online no. 19960004 Paper first received 15th March 1995 and in revised form 21st September 1995 The authors are with the Department of Electrical & Electronic Engineer- ing, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK

A(t) = continuous system matrix Wt) = continuous input matrix id$, iqS, idr, iqF = two-axis currents, A Vd$, vqs7 vdr, vqr = two-axis voltages, V t$ = sampling interval, s k = sample index Ftk) = discrete system matrix Gtk) = discrete input matrix I n = identity matrix n x n

. . . .

1 Introduction

Squirrel cage induction machines are the most widely used type of AC drive, offering robustness and econ- omy. Recent growth in the use of digital signal process- ing (DSP) for motor control has increased interest in the use of more complex control and estimation tech- niques. The search for increased drive performance and robustness using these methods has until relatively recently been hampered by the lack of high speed float- ing point hardware at a reasonable cost. The potential benefits for high performance induction motor drives are a more robust controller capable of producing fast well-damped torque dynamics over a wide rotor speed range.

Most industrial vector controlled (field orientation) drives use the feed-forward slip calculator as a flux observer to perform the necessary reference frame transformations. Sensitivity to variations in rotor time constant is a well-known major drawback of this arrangement [ 1-31. The relative simplicity of this scheme ensures its continued use. With modern micro- processors and DSP devices the basic vector control can be carried out using a fraction of the control proc- essor’s resources. This extra processing capacity can be used to operate the drive with higher control sampling rates, but this is only of benefit if the inverter is capa- ble of a matching switching frequency. An alternative approach is to use the spare capacity to run supporting algorithms which can enhance the performance of the central vector controller.

The slip calculator requires an accurate knowledge of the rotor time constant L,JRr. In many servo-applica- tions the drive is operated in the constant flux demand region [4-71, so that L, is substantially constant. This is justified in motors where the leakage flux, which is a function of the rotor current, is small compared to the magnetising flux component which is held constant. It is in this form of drive that the proposed rotor resist- ance estimator would find application. Rotor resistance can vary over a 2:l range due to cage temperature var-

87 E L ? Proc.-Efectr. Power AppL, Vof 143, iVo. 1. January 1996

iations, this effect is normally dominant. In some designs of machine skin effect variations in R, can be important, but this is normally restricted to direct mains supplied machines where deep bar or double cage designs may be used. Increasingly drives incorpo- rating inverters are specifying machine designs which exploit the lack of need for direct online starting, and have much reduced the skin effect. From a steady-state viewpoint the vector control will constrain the slip to a relatively small range thus minimising skin effect. It is unrealistic to suppose that an estimator based on stator terminal measurements could track very rapid varia- tions due to skin effect without the use of a fundamen- tally more complex nonlinear model. The algorithm described here addresses the problem of unknown or variable rotor resistance estimation without a funda- mental dependence on steady-state or transient opera- tion, or indeed on the cause of that variation.

If the motor is assumed to be operating in the steady-state condition then the estimation problem could be approached using an equivalent circuit model. Steady state measurements of voltage, current and slip could be used to solve algebraic equations based on the equivalent circuit. This approach is not appropriate to high performance drive applications where vector con- trol may be used.

A fundamental problem associated with parameter estimation on a singly fed cage induction machine is that of unmeasurable states [8, 91. Since rotor states, fluxes or currents, are not directly measurable this rules out the direct application of many simple estimator algorithms [lo-121. The use of joint state and parame- ter estimation is a possible solution and is catered for in the extended Kalman filter [13-151. Rotor resistance is redefined as a state and added to the stator and rotor currents. This new state is then estimated with rotor resistance treated as a time varying parameter [16-201. Unfortunately this causes the resulting state space model to become nonlinear because of state multiplica- tion in the equations. The resulting state estimation problem also is nonlinear and a standard Kalman filter cannot be used directly since a nonlinear estimator is required.

otor dynamic model

The Kalman filter algorithm requires a dynamic model of the induction motor. The usual circuit-based or two- axis model is given below as a matrix impedance equa- tion.

1 0 -L, 0 This model can be expressed in a more compact vector/ matrix notation

p I = A ( t ) I + B(t)V ( 3 ) The matrices A(t) and B(t) are time varying due to the presence of the rotor speed term. As eqn. 3 is continu- ous it must be discretised for use with the proposed algorithm. A forward difference approximation is used to carry out the discretisation process.

I ( k + 1) - I ( k ) P I = t s

(4) where ts is the discrete sampling interval.

Substituting eqn. 4 in eqn. 3 gives the one step ahead predictor model where I(k) is the present state and I(k+l) is the next state. The specific induction motor one step ahead predictor is therefore

(5) Examination shows that eqn. 5 can be simplified to a standard time varying linear form

where the matrices are defined as below

I ( k + 1) = I ( k ) + t , . [AI(k) + BV(k)]

I ( k + 1) = F ( k ) I ( k ) + G(k)V(k) (6)

vds ( k ) V ( k ) = [ 4 r ( k ) ]

(9)

where the leakage factor is o = 1 - Lm2/(L,Lr). The state variables can now be redefined by transfer-

ring state variables associated with the rotor currents to the output equation. The discrete model described by eqn. 6 has a fourth order state vector with stator and rotor current vector components as the state variables. Normally, the stator current vector is directly measura- ble and therefore an alternative formulation is possible. In the development that follows only the rotor current vector is considered as a state variable.

The state vector is redefined as stator and rotor sub- vectors

IEE Puoc.-Electr. Power Appl., Vol. 143, No 1, January 1996

The state equation (eqn. 6 ) is now partitioned in line

L . ' J

where the submatrices are defined as

Now consider the rotor current vector equation

Ir ( k + 1) = F21 ( k ) J s ( k ) + F 2 2 ( k ) I r ( k ) +G21 ( k ) V ( k ) (19)

Rearranging into a standard state space form gives

I r ( k + 1) = F22(k)Ir(k) + {F21(k)Is(k) + G21(k)V(k)) (20)

1s ( k + 1) = F11 ( k ) I s (IC) +F12 ( k ) I r (IC) + G11 (k)V(k) (21)

The stator current substate equation is

which can be arranged as an output equation as fol- lows

{ I s ( k + 1) - Fll(k)I.s(k) - G l ~ ( k ) V ( k ) } = F12(k)Ir(k) (22)

In the above reformulated state model (eqn. 20) the forcing term is a time varying linear combination of stator voltage and current vectors. The output vector (eqn. 22) is no longer a direct physical output but lin- ear combination of variables. This model is essentially a second order state space model. This model could be used to produce a rotor flux estimator with reduced computational cost.

2.1 Parameter estimation If a simultaneous estimate of the rotor resistance is needed then it can be defined as an auxiliary state vari- able. This overcomes the problem of trying to estimate a parameter on a model with unmeasurable states. The rotor resistance R, is redefined as a state variable and is augmented to the existing rotor current vector.

IEE Proc -Electr Power Appl , Vol 143, No 1, January 1996

' 1 0 1 The output equation can be reformulated [ r s ( k + 1) - Fi i (S ) IS (k ) - G l l ( k ) V ( k ) ]

An important point to note is that the submatrices F22(k) and F,,(k) contain the new state variable Rr(k). This means that eqns. 23 and 24 describe a nonlinear state model. It is more appropriate to use the general state space model notation which accounts for nonline- arity. The general discrete state equation notation is

where z ( k + 1) = f { z ( k ) , u ( s ) ) (25)

The general discrete output equation is

Y(k) = h { z ( k ) ) (27)

2.2 Stochastic extension The model (eqns. 25-31) serves as the basic determinis- tic core model. To use the model within a Kalman fil- ter estimator a number of stochastic terms must be added. To do this noise vectors must be added to eqns. 25 and 27 as below

z ( k + 1) = f { 4 k ) ) 4 k ) l + W ( k ) (32)

Y(k) = h { x ( k ) l + U @ ) (33 ) where w(k) is the 3x1 process noise vector and v(k) is the 2x1 measurement noise vector.

As the noise vectors w(k) and v(k) are random varia- bles then they must be described by statistical meas- ures. The mean value of the vectors are zero and the variance is described by covariance matrices. The cov- ariance matrix associated with the process noise vector is defined as

E { w ( k ) w ( k ) l = QbJ k Q 2 0 (34) and the covariance associated with the measurement noise matrix is

where Q is a 3x3 constant matrix and R is a 2x2 con- stant matrix.

The initial value of the state vector is described in terms of the mean value and covariance matrix.

E { u ( ~ ) w ( ~ ) ~ } = RS,k R > 0 (35)

89

z(0) = E{x(O)} (36)

Po = E{(x(O) - Z ( O ) ) ( X ( O ) - Z ( 0 ) ) T } (37)

3 Estimator

Having established the basic stochastic state space model it is now possible to develop the nonlinear esti- mator. This requires the application of a linear pertur- bation process to linearise the general model. The EKF then uses the linearised model to perform the estima- tion. The nonlinear state space model

L,w,(k)t, L,wT. (k ) t s L m Z d r ( k ) t s

-Lmw,(k)ts L m w r ( k ) t , LmZqr(k)ts ] (43)

oL* ULSLT.

= ULSL, ULsLr

The prediction is

q k + l / k ) = . f { ? ( k / k ) , u ( k ) }

~ ( k + i / k ) = r ( k ) p ( k / k ) r ( k ) T + Q (44)

(45) The correction is K ( k + 1) = P ( k + 1 / k ) A ( k ) [A ( k ) P( IC + 1 / k ) A (5) *+RI-'

(46) P(k + 1/k + 1) = P(k + lis) - K(k + l)A(IC)P(k + l / k )

(47)

(48) 2(k + l / k + 1) = qrc + l / k ) + K(k + l)[y(k + 1)

- h{2(k + 1/k)}]

P ( k ) = E{[Z(k ) - x ( k ) ] . [ q k ) - .(k)]*} (49)

q o ) = xo (50)

P(0) = Po (51)

The error in the estimate is described by

As can be seen from eqn. 30 the measurement quantity in the reduced order model is quite complex. The meas- urement vector y(k) is actually a combination of stator voltages, stator currents and rotor speed. The measure- ment model shown in Fig. 1 is shown expanded in Fig. 2. The physical measurements on the LHS com- bine to produce the derived measurement vector y(k) .

In the standard KF algorithm the strength of the measurement noise is specified by the R-covariance matrix. The elements of this matrix were obtained by

90

assessment of the errors in the physical measurements, i.e. the d and q axis stator currents. For the reduced order case, the assessment of the measurement covari- ance is more complex as it is a combination of the cov- ariances from each physical measurement. The R- covariance of the derived measurement is required for the reduced order EKF.

I 1 1 state

estimates

U

Fig. 7 Structure of reduced-order EKF for rotor resistance estimation

Fig. 2 Internal structure of measurement model in reduced-order EKF

Both the stator currents Is@) and the stator V(k) have measurement errors or uncertainties vxk) and vdk), respectively. These sources are modelled as zero mean Gaussian-white in keeping with normal KF assumptions.

The covariance of the voltage measurement errors is defined as

E ( W ( k ) ~ V ( j ) * } = CV&, (52)

E{.I(k).I(j)*} = C I S k , (53)

and the covariance of the current measurement is

The rotor angular velocity measurement is also assumed to be noise free for the purpose of this analy- sis which gives the derived measurement covariance as E{Y(~)Y(~)* I = cy = c , + F ~ ~ c ~ F : + G ~ ~ c ~ G T ~ (54)

The derivation of eqn. 54 is given in the Appendix.

were estimated to be have covariances of The physical voltage and current measurement errors

cv = 4.512 v2 (55)

C I = 0.0112 A2 (56)

C y = 0.0212 A2 (57) The covariance equation (eqn. 54) has a rotor speed dependency so the worst case C y was calculated in eqn. 57 which corresponds to my = 0. C Y is calculated from Cl and Cv on the assumption of white noise error sources. In practice the measurement errors are not white in character. However, the above analysis was used to give an indication of the value of the R-covari- ance matrix in the reduced-order EKF.

4 Simulation results

The theoretical development of the reduced order Kalman filter parameter estimator has been carried out

IEE Proc.-Electr. Power Appl., Vol. 143, No. 1, January 1996

in the previous Section. This has resulted in the formu- lation of an algorithm which, when implemented on a digital computer, can compute estimates of rotor resist- ance and rotor currents based on model and measure- ment information. Estimation algorithms based on the Kalman filter are very complex and their behaviour defies analytical analysis in all but very simple cases. Therefore it is imperative that digital simulation is included as part of the design process.

The motor drive model must be capable of operating in both transient- and steady-state as required for high performance induction motor drives. Ideally, in testing estimation algorithms under simulated conditions, a much more complete model would be used than the comparatively simple estimator model. This would then allow the effects of estimator model simplification to be examined in detail. However, in the case of the induc- tion motor, dynamic models which attempt to include effects beyond that handled by generalised machine theory are very complex as detailed in [21] for the inclusion of saturation effects.

Since the ultimate test of estimator performance will be carried out by experiment, using a practical inverter fed induction machine, it was decided to carry out ini- tial algorithm testing using the models developed in Section 2. These simulations were used to give a gen- eral overview of the behaviour of the algorithm under a wide range of settings and conditions. To increase the credibility of the simulation with this model it is impor- tant to check the robustness of the estimation algo- rithm when modelling errors exist, which will always be the case in a practical system. A method which will provide some indication of robustness, even if limited, is the use of random disturbance inputs. The distur- bance inputs are band limited white noise sources which are added to the deterministic two-axis voltage excitation. The band limited noise inputs serve to pro- duce a level of behaviour in the motor simulation which is not represented in the estimator model. Mod- elling errors in the form of estimatodmotor parameter mismatch can also be introduced.

The EKF settings are given below

200

5 2 O

~ l l l l l l l l " "~111111111" "1111111111111' :-200

3L

2 1 5 ; p , y<

0 .o I, 1 2 3

time, s

Fig. 3 Simulation of rotor resistance estimator with no measurement noise present

(58)

P(0) = 1013 (59)

2(0) = [0 0 0.5IT

L

R = 0.0212 (61) The P(0) and Q specifications in eqns. 59 and 60 were found to produce good results in the reduced-order algorithm. It was found that changes to Qll and QZ2 did not produce any significant improvement in the estimator performance. The measurement noise covari- ance matrix R is based on the measurement error anal- ysis of Section 3.

The simulation results are shown in Fig. 3 in which the motor is subjected to a sequence of accelerations and decelerations using open loop constant volts per hertz control. This simulation has been chosen to pro- vide a close comparison with the double speed reversal experimental results discussed later. A small switch on transient can be observed in the simulation, since for simplicity zero initial conditions for the currents are assumed. The error covariance P33, which is associated with the parameter state Rr, indicates how favourable the excitation conditions are for estimation. Examina- tion of the rotor resistance estimate shows an initial convergence to the actual value of 2.4Q. This value is held throughout the simulation period.

200 1 >. >s 0

-200

2 0.0 2'4 r 1 2 3

time, s Fig. 4 Simulation of rotor resistance estimator showing bias due to noise

To demonstrate the effects of measurement noise on estimator accuracy, band limited random noise sources were added to the stator current measurements. The effects of this noise are demonstrated in Fig. 4 which details the same motor transient conditions as in the previous noise free case. The covariance settings in the estimator remain the same as in Fig. 3. It can be seen that the rotor resistance estimate exhibits positive bias errors which are most prominent during the periods of deceleration. The independent noise sources added to the ids and iqs measurements have a bandwidth of 500Hz and a variance of 350mA2.

IEE Proc.-Electr. Power Appl.. Vol. 143, No. 1, January 1996 91

around zero speed, which is to be expected because of the reduced order algorithm's greater sensitivity to measurement errors, and this deliberate detuning.

5.3 Estimation error plotted against other parameter changes If the parameters of the motor used in the model are not accurate this can cause some degradation of esti- mator behaviour. Here the robustness of the reduced- order EKF is examined for variations in the parameters Lmag, Le and R,. These parameters correspond to the parameters associated with the equivalent circuit model of the induction motor. It is appropriate to use these parameters rather than the interdependent parameter set used in the two-axis model.

2.4 _. - - _.__ -. . -. . -. . . . . . . . ._..

CT 0.5 1.0 1 5 2.0 2.5 3 0 time, s

Fig. 7 Experimental performance of estimator with 20-lncrease in R,

The results given in Fig. 7 show the effects on Rr estimation due to changes in the stator resistance R, specified to the estimator. Here the EKF estimator is given an R, value which is 20% greater than the value in the motor. This results in a poorer estimate com- pared with that in Fig. 5 which corresponds to use of the same value of R, in the estimator as the motor. However, relatively good behaviour still results consid- ering that the initial estimate used is less than 25% of the correct value. As mentioned earlier the actual motor R, is determined from standard tests.

This parameter sensitivity can be further examined by a procedure which involves calculating the average Rr estimation error over a fixed time interval using experimental data. The results are presented in Fig. 8. In the case of the inductance parameters Lmag, Le, the reduced-order estimator exhibits instability if these parameters are varied by more than 10% of the actual motor values.

5.4 Reduction in computational requirements One significant advantage of the new reduced-order estimator is the reduced length of the algorithm. Since this is an important factor a careful estimate was pre- pared of the computational requirements for the reduced order estimator. This was done by totalling the number of multiplications and additions required for

IEE Proc.-Electr. Power Appl., Vol. 143, No. I , January 1996

the execution of a single time step by the conventional EKF [9], and repeating the process for the new tech- nique. This evaluation is independent of the rapid advances in processor speed, since this applies equally to both techniques. This yields one-third the computa- tional requirement of that required by the full-order EKF estimator.

z -0.02

-0.04

-0.08 j: 0 .6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

variation , p. U .

Fig. 8 Estimator sensitivity to other equivalent circuit parameters

6 Conclusions

Since unmeasurable rotor states occur in the cage induction motor a joint state and parameter estimation approach was adopted using an extended Kalman filter (EKF). It was established that the parameter tracking speed could be controlled by the EKF setting Q55. Higher speeds of tracking were achieved but at the expense of reduced noise immunity.

The nature of the drive conditions played a dominant role in the convergence properties of the algorithm. In periods of acceleration Rr convergence was very rapid. Deceleration gave slower convergence; this became poor during steady-state operation. Multiple speed reversal transient conditions which are commonplace in servo-applications proved to be the most favourable for estimator convergence. The effects of measurement noise on estimator accuracy are also minimised during acceleration.

The rotor resistance estimator was tested using exper- imental data gathered from a drive system. The results have a close correspondence with that of the simulation studies, but the quality of the R, estimation is a little lower, which is to be expected due in part to modelling assumptions. The experimental results confirmed the convergence properties to be most favourable during motor acceleration. These experimental results obtained on a practical induction motor drive clearly showed that a good standard of behaviour was obtained even when passing through zero speed.

The sampling rate was chosen to be 10kHz. Lower sampling rates were tried and found to produce inferior results. The requirement of rather high sampling rates is thought to be due to the linearisation process in the EKF algorithm but further investigation would be required to confirm this.

The Kalman filter algorithms all require the high pre- cision of floating point arithmetic for stable operation. There are some variants which will operate with

93

restricted word length arithmetic but these algorithms 19 WADE, S., D m N I G A N , M.W., and WILLIAMS, B.W.: ‘Parameter identification for vector controlled induction machines’. Proc. IEE Int. Conf. Control, 1994, (Coventry), Vol. are more complex. The reduced-order algorithm

described has been shown to offer significant reduction 2, pp. 1187-1192 in computational requirements, whilst exhibiting good 20 LORON, L., and LALIBERTE, G.: ‘Application of extended

Kalman fdter to parameter estimation of induction motors’. Proc. 5th European Conf. Power Electronics and Applications, 1993, performance. Its convergence properties whilst poor

under steady-state conditions, are favourable when Vol. 5. DD. 85-90 transient accelerations occur, as is the n o m in the form of servodrive most likely to require vector control.

21 B R O e , J.E., and VAS, P.: ‘The phenomenon of intersaturation in the transient operation of induction motors arising from satu- ration of main flux path’. Proc. IEE ICEM-Design and Auolica-

7

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

References

NORDIN, K.B., NOVOTNY, D.W., and ZINGER, D.S.: ‘The influence of motor parameter deviations in feedforward field ori- entation drive systems’, IEEE Trans., 1985, IA-21, pp. 1009-1015 DALAL, D., and KRISHNAN, R.: ‘Parameter compensation of indirect vector controlled induction motor drive using estimated airgap power’. Proc. IEEE IAS Annual Meeting, 1987, pp. 170- 176 GARCES, L.J.: ‘Parameter adaptation for the speed-controlled static AC drive with a squirrel-cage induction motor’, IEEE Trans., 1980, IA-16, pp. 173-178 BLASCHKE, F.: ‘The principle of field orientation as applied to the new Transvektor closed loop control system for rotating-field machines’, Siemens Rev., 34, pp. 217-220 GABRIEL, R., LEONHARD, W., and NORDBY, C.J.: ‘Field oriented control of a standard AC motor using microprocessors’, IEEE Trans., 1980, IA-16, (2), pp. 186192 GABRIEL, R., and LEONHARD. W.: ‘Microprocessor control of induction motors’. Proc. IEEE Rec. International Semiconduc- tor Power Converter Conf., 1982, pp. 385-396 HASSE, K.: ‘On the dynamics of speed control of static AC drives with squirrel-cage induction machines’. PhD thesis, D a m - stadt. Germanv

T T - .l(k + l )vv(j) G11} FINCH, J.W.,‘ACARNLEY, P.P., and ATKINSON, D. J.: ‘Prac-

tical implementation and test facility for field orientated induction motor drives’. Proc. 4th Int. IEE Conf. on Electric Machines and Drives, 1989, (London), pp. 288-292 PTKINSON, D.J., ACARNLEY, P.P., and FINCH, J.W.: Application of estimation techniques in vector-controlled induc-

+ E{-Fiivi(k)vv(j + + F i ~ v i ( k ) v v ( j ) ~ F T l

+ Fiiv1(k)vv(j)*GTl} tion motor drives’. Proc. IEE Conf. on Power Electronics and Variable Speed Drives, 1990, (London), pp. 358-363 EYKHOFF, P.: ‘System identification, parameter and state esti- mation’ (Wiley, 1974) HUNT, K.J.: ‘A survey of recursive identification algorithms’, Trans. Inst. M.C., 1986, 8, (5), pp. 273-278 IRWIN. G.W.. and ROBERTS. A.P.: ‘The Luenbereer cononical form in the state parameter es6mation of linear syzems’, Int. J . Control, 1976, 23, (6), pp. 851-864 ZAI, L.C., and LIPO, T.A.: ‘An extended Kalman fdter approach to rotor time constant measurement in PWM induction motor drives’. Proc. IEEE IAS Annual Meeting, 1987, pp. 177- 183 JAZWINSKI, A.H.: ‘Stochastic processes and filtering theory’ (Academic Press, 1970) JOHNSON, A.: ‘Process dynamics, estimation and control’, IEE Control Engineering Series 27 (Peter Peregrinus Ltd, London, 1985) SALVATORE, L., STASI, S., and TARCHIONI, L.: ‘New EKF-based algorithm for flux estimation in induction machines’, IEEE Trans., 1993, IE-40, (51, pp. 496-504 GLIELMO, L., MARINO, P., SETOLA, R., and VASCA, F.: Reduced Kalman filtering for indirect adaptive control of the

induction motor’, Int. J. Adaptive Control Signal Processing, 1994, 8, (6), pp. 527-541 DU, T., VAS, P., and STRONACH, F.: ‘Design and application of extended observers for joint state and parameter estimation in high-performance AC drives’, IEE Proc. B, 1995, 142, (2), pp. 71- 78

The covariance terms involving voltage and current vectors are put to zero as the voltage and current meas- urement errors are assumed to be uncorrellated. In the covariance terms involving vAk) and vr(k + 1) the assumption of white noise will also render these terms zero. Also the matrices Fll, Fl.lT, GI, and GllT are taken to be constant as the variations in speed band- width is considered to be much lower than the noise bandwidth.

and applying the original covariance definitions gives

94 IEE Proc.-Electr. Power Appl,, Vol. 143, No. 1, January 1996