Robust Speed-controlled Im Drive Using Ekf And Rls Estimators

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obust speed-controlled induction-motor F and RLS estimators F.-J. Lin

Indexing term) Induction-motor drive, RLS estimator, Kalwan Jiltei

Abstract: An induction-motor (IM) speed drive, with the application of an extended Kalman filter (EKF) and a proposed recursive least-square (RLS) estimator, is introduced. The rotor resistance of the IM is identified by the EKF, and the rotor inertia constant, the damping constant and the disturbed load torque of the IM are estimated by the proposed RLS estimator, which is composed of an RLS estimator and a torque observer. The integral proportional (IP) speed controller is on-line, designed according to the estimated rotor parameters. Then the observed disturbance torque is fed forward to increase the robustness of the induction-motor speed drive.

1

Introduction

Indirect field-oriented techniques utilising microprocessors are now widely used for the control of inductionmotor drives in high-performance applications. However, their performance strongly depends on the motor parameters, especially the rotor resistance 111. Recently, much attention has been given to the possibility of identifying the changes in motor parameters while the drive is in normal operation 121. Some authors have proposed various induction-motor drives with rotor-resistance or time-constant identification methods [2-lo], including the EKF technique [9, lo], to given better control performance. The EKF algorithm [ I l l is an optimal recursive estimation algorithm for nonlinear systems, which is very suitable for implementation in systems with measurement contaminated by noise. The current-controlled pulse-width modulated (CCPWM) voltage-source inverter (VSI) is most popular in high-performance AC servo applications. The voltage waveform of this VSI has a broad-band spectrum. This harmonic spectrum can be considered as noise input. The EKF approach is ideally suited for parameter estimation in such a system. Since the algorithm is computationally intensive, and all the steps involved require vector or matrix operations, the DSP32C digital signal processor (DSP) [12, 131, which is installed in a PC-486, is used to implement the algorithm. 0IEE, 1996 IEE Proceedings online no. 19960287 Paper received 11th July 1995 The author is with the Department of Electrical Engineering, Chung Yuan Christian University, Chung Li, Taiwan, Republic of China 186

On the other hand, to preserve the robust performance under parameter variations and external load disturbance, many studies have been made on the motor drives [ 14-21], which include feedforward control of the observed torque [19-211. An on-line tuning IP speed controller, with disturbance torque feedforward control, is proposed in this study. The tuning of the IP controller is according to the estimated rotor inertia and damping constants from the proposed RLS estimator [22], and the disturbance torque is obtained from the proposed load torque observer. The IP controller and the observer are implemented using a PC-486, and the RLS estimator is also implemented using a DSP32C. Under this co-processor structure, the decouple control of torque and flux in the indirect field-oriented mechanism is guaranteed by the estimated rotor resistance; and the robust control performance is obtained by the on-line tuning IP controller with feedforward control. The theoretic basis of the proposed controllers is derived in detail, and some simulation and experimental results are provided to demonstrate the effectiveness of the proposed control scheme. 2

Induction-motor model

The small-signal linear state space equation for the induction motor in a stationary reference frame is as follows [23]:

X=AX+BU

where

x=

A=o

(1) T

[iqs

ids

iq,

zdr]

(2)

[ (5)

igs,ids = q-axis and d-axis stator current iyr,idr = q-axis and d-axis rotor current vqr, vds = q-axis and d-axis stator voltage vqr, vdr = q-axis and d-axis rotor voltage R, = stator resistance per phase R, = rotor resistance per phase IEE Proc.-Electr. Power Appl., Vol. 143, No 3, M a y 1996

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L, = magnetising inductance per phase L, = stator inductance per phase L, = rotor inductance per phase or= rotor angular speed The discrete time equivalent model of eqn. 1 is x(k 1) = $ x ( k ) ru(k) where

+

L, G=hjo!-im

1, I] 0

0

0

+

(7)

0

(20)

k

H(X(k)) = [iqs(k) The process noise W(k) is characterised by

(21)

E { W ( k ) }= 0

(22)

E { W ( k ) W ( l ) T= ) Qdki

Q 20

(23)

The measurement noise V(k) is characterised by

r=

(1

E { V ( k ) }= 0

h

eA"ds)B

(24)

E { V ( k ) V ( l ) T=) R S k l

R20

(25)

The initial state is characterised by

where h is the sampling interval. One way to compute 4 and r is as follows [24]:

E(X(0))= xo

(26)

E ( ( X ( 0 )- XO)(X(O) - 20)) = Po

(27)

Extended Kalman filter

3

To use the Kalman filter (KF) with nonlinear plant models, such as eqn. 17, the model must be linearised about a nominal state trajectory to produce a linear perturbation model. The standard K F is then used to estimate the perturbation states. The EKF estimator can be summarised as follows [l 11: Step 1: Prediction

+ l/k) = f ( X ( k / k ) ,U @ ) ) P ( k + l / k ) = p ( / ~ ) P ( k / k ) F (+k G ) ~( k ) Q G ( k ) X(k

L

(29)

where

-L,1

0

(28)

The EKF can be used for combined state and parameter estimation by treating selected parameters as extra states and forming an augmented state vector. Whether the original state space model is linear or not, the augmented model is nonlinear because of the multiplication of states. Since R, is the parameter to be estimated, R, is augmented into X(k), and X(k) becomes

LRU, R,L,,,

=hu

-L,,L,w,

&-R,L, L,L,W,

RsL, 0

LmL,wT

& -LsX5 -L,L,w, 0

L m x j Lmx4 LsL,dT -Ls51

&

-

Lsx5 - L , x ~

0

1 -

X ( k ) = [ 2 q s ( k ) 2 d s ( k ) z q r ( k ) Z d r ( k ) % ( k ) IT (16) = [ z i ( k ) % ( k ) 2 3 ( k ) 24(k) 2 5 ( k ) ] Considering the inherent stochastic characteristic of PWM, treating the fundamental component as the deterministic input U(k) and all the high-order harmonics as white gaussian noise W(k), and considering the measurement noise V(k), the dynamic behaviour of a three-phase induction motor can be modelled as

X(k

+ 1) = f ( X ( k ) U, ( k ) )+ G W ( k ) Y ( k )= H ( X ( k ) )+ V ( k )

(17) (18)

where

f ( X ( k ) U@)) , =

hn

I

(&-R,L, ) z I - L ~ , w ,s 2 + L , , , z 3 z ~ - L , , L , W r m 4 + L r p * ~ L2,w,z1+(&--R5

L,)m2+L7,LL,w, z 3 + L , r , z 4 z 8 + L r p d ,

R b L r r , ~ i + L T r b L X3Zw + (,

~

-

~

,

~

~

)

~

~

+

L

L

c

L

r

W

~

1

~

4

-

~

where klk denotes a prediction at time k based on data up to time k. Similarly, (k + 1)lk denotes a prediction at time k + 1 based on data up to time k. The block diagram of the EKF estimator is shown in Fig. 1. The estimated R, will be used in the indirect field-oriented mechanism.

~

r

~

~

y

r

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187

Y(k+l) HP(S)

I

=

where P = number of poles iis = torque current command generated from the speed controller = flux current command J = total mechanical inertia constant B = total damping constant. In the current-controlled PWM inverter of the indirect field-oriented induction-motor drive, the current commands in the synchronous reference frame, denoted by i;, and i;,, must be transformed into the phase domain to yield the reference currents. The unit vector (cos@,+ jsino,) used in the transformation matrix is generated and by using the measured rotor angular velocity CO, the following estimated slip angular velocity wx1:

ii5

Fig.1 EKF block diagram 4 Modelling of field-oriented induction-motor drive

The block diagram of the indirect field-oriented induction-motor-drive system combined with an EKF estimator is shown in Fig. 2, which consists of an induction motor loaded with a DC generator, a ramp comparison current-controlled PWM voltage source inverter, a field-orientation mechanism, a coordinate translator and a speed-control loop. The induction motor used in this drive system is a three-phase Y-connected two-pole 8OOw 60Hz 120Vi5.4A type.

Since Rr is sensitive for different operating conditions, the estimated R, from the EKF estimator will be used in eqn. 38 to guarantee a correct estimation of the slip frequency, and to preserve the decouple control characteristic. The dynamic modelling, based on measurements 1251, is applied to find the drive model off-line at the nominal case (aro= lOOOrpm, RL = On). The results are, (the scaling is 5OradislV) -

K t = 0.5756NmIA a = 0.538 b = 3.31

-

J = 6.04 x 10p3Nms2= 0.302Nmsrad/V

(39) B = 3.25 x 10p3Nms/rad = 0.1625Nm/V The estimated drive parameters will be used in the design of the proposed controller. -

5

Fig.2 Indirect field-oriented induction-motor drive bvith EKF estimator showing the system configuration

Design of IP controller

The IP controller is shown in Fig. 3. Using the nominal drive model, the transfer function of the rotor speed response to the command input of Fig. 3 can be expressed by -

Js'

KIKt

+ (B+ K,Kt)S + KIKt 2

A -

(40)

Wn

s2

+ 2CWnS + w;

where

L _ _ _ _ _ _ _ - - _ - - _ - - I

Fig.3 Indirect field-oriented induction-motor drive M.ith EKF estimator

showing the control rystem block diagram

By using the reference-frame theory and the linearisation technique, the field-oriented induction-motor drive, shown in Fig. 2, can be reasonably represented by the control system block diagram shown in Fig. 3, in which, G,(s) is an IP speed controller and Te --K 'taqs * (35)

Owing to the absence of zeros, the overshoot of the step response in eqn. 6 is avoided by setting the damping ratio 5 = 1. Then one can find the unit-step response in eqn. 40 to be LJ,(~) = 1 - eP'7Lt(l +writ) (42) For convenience in designing the IP controller quantitatively, the response time is defined as, the time required for the step response to increase from 0 to 90% of its final value. By specifying the response time as t,,, the following nonlinear equation is yielded:

0.9 = 1 - e-w"t" 188

(1

+ wnh-e)

(43)

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Solve the above nonlinear equation to obtain a,; then from eqn. 40 one can find the parameters of the IP controller as

KI = Jw:/K~

K p = ( 2 7 -~ B~) / K ,

(44)

P(k - l)C(k - 1)T

1

+ C(k

-

1

P(lc)=-[P(k-l)01

Eqn. 40 also indicates that the tracking steady-state error is zero. It follows from the above analysis that the desired tracking specifications can be completely achieved by using the simple IP controller. 6

K ( k )=

Proposed RLS estimator with torque observer

The block diagram of the proposed RLS estimator with a torque observer is shown in Fig. 4. If on-line parameter identification is available when the parameter variations occur, the IP controller can be of on-line design, according to eqn. 44, to preserve the tracking performance. Though the RLS estimator is one of the most effective methods for on-line parameter identification, it is difficult to get unbiased results [22] in this application, owing to the dynamic model of the plant being disturbed by the external load torque. As shown in Fig. 4, the proposed RLS estimator is combined with a simple torque observer to resolve the above difficulty. The torque observer uses the inverse dynamic of the motor drive to obtain the observed torque, which is denoted TLIK,. The torque current command minus this value results in the unbiased identified parameters, J and B, which denote the estimated rotor inertia constant and damping constant, through the RLS estimator. Then J A a n dB in the torque observer are replaced by J and B. By this recursive process, the identified J and B parameters and the observed load torque will quickly converge to their real values. toraue observer

(49)

l ) P ( k - 1 ) C ( k - 1)T

P ( k - 1 ) C ( k - l ) T C ( k - 1 ) P ( k - 1) 1 + C ( k - 1 ) P ( k - l)C(/k- 1)T 1 (50)

where

C ( k ) = [wT(k)&(W -%/GI

(51)

@(IC) = [-a1 (IC), b1 (IC)] (52) The value of the forgetting factor a should be restricted to 0 < a 2 1. After O(k) is obtained, the estimated values of J and B can be easily determined from eqn. 46. The convergence of the RLS algorithm with the torque observer will be shown by simulation results. The observed torque is also fed forward through a weighting factor W, to realise a robust speed control system, as shown in Fig. 4. Ideally, the value of W is set at 1. For the system possessing nonlinearities, e.g. limiter, dead-time element etc., the weighting factor should be selected at less than 1 to preserve stability. 7

Design and simulation results

7. I

Simulation of EKF estimator

The sampling interval used in the simulation is 2ms. The process noise covariance and measurement noise covariance are set as

The simulation result is shown in Fig. 5. Between 0 and 1s, the estimated rotor resistance quickly converges to its real value, which is 1.3Q. Beginning at 1s, the real rotor resistance is varied according to the following equation: R, = 1.3 (t - l ) f l (54)

+

where t is the simulation time. The estimated rotor resistance can trace the variance of the real value as shown in Fig. 5.

Fig. 4

Proposed controller with the RLS estimator

The Z transform of the plant model with ZOH, when TL is zero, is as follows: o.5[

1

where a1

= - exp(-T,B/J)

bl = ( l y , / B ) ( l ~ e x p ( - T s ~ / J ) )

(46) and T, is the sampling interval. The system model can be written as

w,(k

+ 1) = -a1w,(k) + b l i & ( k )

(47)

From the above equation, a discrete RLS estimator for use in estimating system parameters can be written as [22]: O ( k ) = O ( k - 1)+ K ( k ) [ w , ( k )- C ( F ) O ( k- l)]

(48)

O

0

0.2

I

0.4

' 0.6

, 0.8

~

1 1.2 time,s

' 1.4

1.6

~ 1.8

~ 2

Fig. 5 Simulation result ofthe EKF estimator

a rotor resistor b estimated rotor resistor

7.2 Design of IP controller Using the parameters listed in eqn. 39 and setting t,, at 0.3s, the parameter of the IP speed controller can be obtained from eqn. 44 as

K p = 13.25

K I = 87.7

IEE Proc.-Electr. Power Appl., Vol. 143, No. 3, May 1996

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(55) 189

'

7.3 Simulation of RLS estimator with torque observer

1120

1 I

In this simulation, the load torque is set at 1Nm. In the beginning, J and B are set according to their nominal values. Then, at 2s, J is abruptly increased by a factor of three. From the simulation results shown in Fig. 6: the estimated values of J and B, after a step change in the speed command, are quickly converged to their true values under contant load disturbance. Though the value of the observed torque is influenced by the speed transient, the steady-state observed value is correct; and this observed torque transient has little effect on the desired control performance from the simulation and experimental results owing to feedforward control.

lOZO-~/ / /

'

- - - . . lo

-

,

-

,

O

1

'

'

'

'

25 3 35 4 45 5 time, s Fig. 6 Simulation se.su1t.s (j"the RLS estiniator and the torque observer. showing sotor s eed (-) and e.mmated values of'rotor speed (xi, observed torque ( a i , and ~ ( c )

0

05

1

15

2

~(6

7.4 Simulation of proposed controllers To investigate the effectiveness of the proposed robust controller, suppose that the mechanical inertia constant J is significantly changed to allow the transfer function model HJs) to be changed to Case 1 : Case 2

HpL(S)

+10.1625

= 1.51s

( J = 5 xJ) (56) ( J = 0.5 x

7) (37)

In the nominal case, the step rotor speed tracking and step load regulating responses of the drive system shown in Fig. 4, without and with the proposed on-line IP controller design and feedforward control, are shown in Figs. 7 and 8 by curves a and d, respectively. The results are identical for the tracking conditions, and the regulating performance is improved. Suppose that the mechanical inertia constant J is changed according to eqns, 56 and 57; then, the rotor speed responses due to a step command change and a step load torque change, without and with the proposed online IP controller design and feedforward control, are also compared in Figs. 7 and 8. Significant performance improvements both in the tracking and regulating responses by the proposed controller are observed from the results. 8

Implementation and experimental results

The block diagram of the co-processor computer control system for the indirect field-oriented induction motor servo drive is shown in Fig. 9. The robust speed control algorithm is realised in a 486DX-66 and the EKF and RLS algorithms are realised in a DSP32C. 190

-60' 0

"

0.2

04

1

06

0.8

'

1 1.2 tirne,s

,

14

1

1.6

'

1.8

2

Fig.8 Simulation results of proposed controller showing load regulation Nominal case with IP controller only b Case 1 with IP controller only c Case 2 with IP controller only d Nominal case with proposed controllers e Case 1 with proposed controllers f Case 2 with proposed controllers N

The control intervals are all selected at 5ms. To reduce the calculation burden of the CPU and to increase the accuracy of the three-phase command current, the coordinate transformation in the field-oriented mechanism is implemented by an AD2S100 AC vector processor. The control interval of the vector processor is selected at 0.2ms. The pure differentiator, which is shown in Fig. 4, may amplify the high-frequency noise, so the operating stability of the closed-loop controlled drive will be greatly affected. Thus in practical implementation, a filter is used as an alternative. It is designed to behave as a pure differentiator for the main low-frequency dynamic signal, but it becomes a lowpass filter for high-frequency signals. Some experimental results are provided in the following to show the effectiveness of the estimators and the proposed controllers. The estimated rotor resistance obtained from the EKF estimator, which converges to its real value, 1.3!2, within 0.3s, is shown in Fig. 10. This estimated value, after suitable filtering, is used online in the indirect field-oriented mechanism. To avoid the numerical problem of dividing by zero during the implementation for the parameters J and B, the parameters a and b (expressed in eqn. 37) are estimated instead of J and B. Moreover, the load resistance of the DC generator is set at 118R during load torque observation. The estimated values of a, b and the observed IEE Proc.-Electr. Powes Appl., Vol. 143, No. 3, May I996

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L

Fig.9 DSP-bused computer-control drive .system with vector processor

load torque are shown in Figs. 11 and 12, respectively. The values of U , b and observed load torque all converge to their real value within 0.5s. The measured rotor-speed responses, owing to step command change and step load resistance change, without and with online IP controller design and feedforward control at the nominal condition, are compared in Figs. 13-16. The results show that, while the tracking responses of these two cases are almost identical, the regulating response is significantly improved by augmenting the proposed controllers. Now let the drive be operated at 2000 rpm, the drive rotor speed tracking and regulating responses, without and with on-line IP controller design and feedforward control at another operating condition, are shown in Figs. 17-20. Better control performance yielded by the proposed controller is obvious from the results.

command

,

1

Fig. 13 M e a w e d rotor speed r a p o n w at the nominal case owing to step command change with an IP controller

i

T

i

44rpm

+-----,

jI

0 25s

1

Fig.14 Measured rotor speed reAponses at the nominal case owing to step load resistance change with an IP controller

rotor resistance

command

i

lOOOrpm

I

~

-

0 25s Fig. 15 Measured rotor speed rerponses at the nominal c u x owing to

Fig. 10 Estimated rotor resistance

I

...

+ t*

b r 3 31

-

step command change with on-line IP contioller design and feedforward control

E

L

+

0 25; Fig.11

Experimental r e d 3 of the RLS estimator and the torque observer showing the valne~of the a and b parumeters

Fig.16 Measured rotor speed res onhe5 at the nominal case owin step load resistance change with on-fne IP controller design andjee&fi ward control

9 i

Fig.12 Experimental resultr oj the RLS estimator and the torque observer showing observed loud torque

Conclusions

The EKF and the proposed RLS estimators are successfully implemented in this study for the speed control of an indirect field-oriented induction-motor drive.

IEE Proc.-Elect?. Power Appl., Vol. 143, No. 3, May 1996

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191

First, an EKF estimator was implemented to estimate the rotor resistance. The estimated rotor resistance was used to estimate the slip frequency. Next, an IP speed controller was quantitatively designed, according to the estimated nominal drive model and the prescribed speed-tracking specifications. The proposed RLS estimator was implemented to estimate the rotor parameters which are used to design the IP controller on-line. The proposed RLS estimator was combined with a torque observer to obtain unbiased results. Then the observed torque was fed forward to obtain robust control performance.

f

- command

T

t

00- T

speed

I / 2000rpm

rpm

+

-c

4 : : ’’ ’ ! ’ I



b

---

+. .

i 0 25s I Fi .I7 Mearured rotor speed iesponses at on operatiii condition

of

2080 rpnt owing to ~ t e ptommand change wtlz an IP conti ol&

f

(R~=Ofl-+24Ofl)

4--+

0 25s

60rpm

Fi . I 8 Measured rotor speed response, at an opercitrizg condition of 20%0 rpnt m i n g to step load reiistarzce change ~ 1 t an h IP controllei

1 command

+

t

t

7 . . ”‘ ” ’ + t : :” !-

it 2000rpm

L

0 25s

Fi . I 9 Measured rotor .speed responses at an operating condition oJ

2080 rprn owing to step command change with orz-line IP controiier design and feedforward control

17rpm

f

L

+-----c

0 25s

Fi .20 Measured rotor s eed iesponses at iin operating condition of 20%0 rpm owing to ste ~ o J ‘ r xc m “n c e change irirh on-line IP controller design ani~fc.edfor,ycirrontrol

10 I

192

References

KRTSHMAN, R., and DORAN, F.C.: ‘Study of parameter sensitivity in high-performance inverter-fed induction motor drive systems’, I E E E Trans., 1992, IA-23, pp. 623-635

KRISHMATU’,R., and BHARADWAJ, AS.: ‘A review of parameter sensitivity and adaptation in indirect vector controlled induction motor drive systems’, IEEE Power Electronics Specialist Conference Record, 1990, pp. 560-566 MATSUO, T.; and LIPO, T.A.: ‘A rotor parameter identification scheme for vector-controlled induction motor drives’, IEEE Trans.: 1985, IA-21, pp. 624-632 OHNISHI, K., UEDA, Y., and MIYACHI, K.: ‘Model reference adaptive system against rotor resistance variation in induction motor drive’, IEEE Trans., 1986, I C 3 3 , pp. 217-223 SUIMOTO, H., and TAMAI, S.: ‘Secondary resistance identification of an induction motor applied model reference adaptive system and its characteristics’, I E E E Trans., 1987, TA-23, pp. 296303 KOWALSKA, T.O.: ‘Application of extended Luenberger observer for flux and rotor time-constant estimation in induction motor drives’, I E E Proc. D,1989, 136, pp. 324-330 CHAN. C.C., and WANG, H.: ‘An effective method for rotor resistance identification for high-performance induction motor vector control’, IEEE Trans., 1990, 1IG37, pp. 477482 HUNG. K.T.. and LORENZ, R.D.: ‘A rotor flux error-based, adaptive tunmg approach for feedforward field oriented induction machine drives’; Proceedings of IEEE IAS Annual Meeting, 1990, pp. 589-594 ZAI: L.C.. and LIPO, L.A.: ‘An extended Kalman filter approach to rotor time constant measurement in PWM induction motor drives‘. Proceedings of IEEE IAS Annual Meeting, 1987, pp. 177-183 10 ATKINSON, D.J., ACARNLEY, P.P., and FINCH, J.W.: ‘Observers for induction motor state and parameter estimation’, I E E E Trans., 1991, IA-27, pp. 1119-1127 11 MENDEL, J.M.: ‘Lessons in digital estimation theory’ (PrenticeHall, Englewood Cliffs, NJ, 1987) 12 FUCCIO, M.L.: ‘The DSP32C : AT & T’s second-generation floating-point digital signal processor’, IEEE Micro, 1988, 8, (12), pp. 3 0 4 7 13 YEH. H.G.: ‘Real-time implementation of a narrow-band Kalman filter with a floating-point processor DSP32’, IEEE Trans., 1990, IC37, pp. 13-18 14 HO. Y.Y.; and SEN, P.C.: ‘A microcontroller-based induction motor drive system using variable structure strategy with decoupling’, I E E E Trans., 1990, I C 3 7 , pp. 227-235 15 HSIA. T.C.: ‘A new technique for robust control of servo system’, IEEE Trans., 1989, I C 3 6 , pp. 1-7 16 OHISHI, K., NAKAO, M., OHNISHI, K., and MIYACHI, K.: ’Microprocessor-controlled DC motor for load-insensitive position servo system’, I E E E Trans., 1991, I C 3 8 , pp. 21-25 17 LIAW, C.M., and LIN, F.J.: ‘A discrete adaptive induction position servo drive‘, IEEE Trans., 1993, EC-8, pp. 350-356 18 LIN, F.J., and LIAW, C.M.: ‘Control of induction field-oriented induction motor drives considering the effects of dead-time and parameter variarions’, I E E E Trans., 1993, I W O , pp. 486495 19 MATSUI. N., MAKINO, T., and SATOH, H.: ‘Autocompensatioii of torque ripple of direct drive motor by torque observer’, IEEE T ~ N I ~1993, s . , IA-29, pp. 187-194 20 IWASAKI, M.; and MATSUI, N.: ‘Robust speed control of IM with torque feedforward control’, IEEE Trans., 1993, IG-lO, pp. 553-560 21 KO, J.S.. LEE, J.H., and YOUN, M.J.: ‘Robust digital position control of brushless DC motor with adaptive load torque observer’, I E E Proc. Electr Power Appl., 1994, 141, pp. 63-70 22 LAUDAU, J.D.: ‘System identification and control design’ (Prentice-Hall, Englewood Cliffs, NJ, 1990) 23 BOSE, B.K.: ‘Power electronics and AC drives’ (Prentice--Hall, Englewood Cliffs, NJ, 1986) 24 ASTROM, K.J., and WITTEHMARK, B.: ‘Computer controlled systems‘ (Prentice-Hall, Englewood Cliffs, NJ, 1990) 25 LIAW, C.M.. OUYANG, M., and PAN, C.T.: ‘Reduced order parameter estimation for continuous system from sampled data’, Trans. ASME, J. Dyn. Syst. Meas. Control, 1990, 11, pp. 305-308

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