New Adaptive Flux Observer Of Induction Motor For Wide Speed Range Motor Drives

NEW ADAPTIVE FLUX OBSERVER OF INDUCTION MOTOR FOR WIDE SPEED RANGE MOTOR DRIVES Hisao Kubota,Member, Kouki Matsuse,Senior Member, and Takayoshi Nakano Department of Electrical Engineering Meiji University Fuji Electric Co. R & D Higashimita, Tama-ku, Kawasaki 214 Fuji, Hino, Tokyo 191 JAPAN JAPAN Abstract where A Flux Observer of an induction motor

w j t h a parameter adaptive scheme will be proposed. The parameters identified adaptively are stator and rotor resistance which vary with motor temperature. A stability of the proposed adaptive flux observer will be proved by the L y a p u n o v ' s t h e o r e m . F u r t h e r m o r e , robustness of the induction motor drive system with the proposed flux observer will be shown. A n adaptive scheme can be applied also to estimate motor speed without speed sensors.

iqsjT : stator current : rotor flux v,=[vd, vqSjT : stator voltage k l = - { R s / ( aL,) + (1-0) / ( O r r ) }I=arxlI k z = M / ( aL,L,) { (1/r,)I-~IJ}=arx21+aixzJ Azl= (M/rr)I=arzlI Az2=- ( l/rr) I+orJ=arzzI+aizzJ Bi=l/ ( ULs)I=biI is=[ida

Introduction The indirect field oriented control of an induction motor is widely used. This method has a disadvantage which is sensitivity to the n i o l or parameter variation. Especially, resist ancc varies widely with the motor temperature. On the other h a n d , the direct field oriented control method is robust against the motor parameter variation because the measured motor flux is fed back to the reference. The disadvantage of this method is that the installation of a flux sensor is necessary. Therefore, flux estimation from the terminal variables is usually used instead of the flux mcasurernent. However, flux estimation is also sc'nsitive to the motor parameter variation. Recently, flux observers have been proposed for flux estimationcl)-ce). The sensitivity of the flux observers to the rotor resistance variation is much lower than that of' thc conventional flux estimator. However, t t i c sensitivity of these observers cannot be c-ornplclely zero. The sensitivity becomes high, cspecially in a low speed range. To solve these problems, we have proposed ;I rotor flux observer with parameter adaptive scheme('). Parameters identified adaptively a r c the stator and rotor resistance which are clrpcndent upon the motor temperature. In this p
A n induction motor is described by the f o l lowing statc equations:

&=jT

#r=[@dr

I=

[ y]

J=

[ -;]

The full order state observer which estimates the stator current and the rotor flux is written by following equations.

[

is

4ir

]=[

All AlZ Azl

I[ I+[ ;

a22

is

]I_

4ir

+G(L-L) where means the estimated values. The poles of the observer are made proportional to those of the induction motor. Then the gain matrix G is calculated as follows ~

where c=-(aLsLr)/M, k is the proportional constant Influence of Parameter Variation on Flux Estimation Stator and rotor resistance vary with the motor temperature. Therefore, it is hard to use the exact value in the flux observer for these parameters. Influence of the parameter variations on the flux estimation will be investigated in this section. Figure 1 shows the actual rotor fluxes and the estimated ones when the estimated motor parameters (stator and rotor resistance) are incorrect. Ratings of the tested induction

087942-6004/90/1100-0921$01.00 0 1990 IEEE

R,

=

1.2~-

'dr

'qr

.5

-

-$?2 rl Ir,

.5

.4

.4 I

.3

.2 .I

Ll

-$2

*3

x

-2

;;J

.I

0

0

0

2

2

-.I

2

-.l

-.2

-.2

-.3

-.3

-.4

-.4

-.5

-.5

(a) Influence of Stator Resistance

(b) Influence of Rotor Time Constant

Figure 1. Influence of Parameter Variation on State Estimation

Figure 2 . Observer Based Direct Field Oriented Controller

Figure 3. Indirect Field Oriented Controller

motor are shown in Table 1. These results were calculated under the condition that the motor speed was 30 [rpml with 2 0 [Nml load torque and the proportional constant k was 1.0. It is evident that both o f the stator and rotor resistance exert great influences on the flux estimation. Although the m u t u a l i n d u c t a n c e a l s o varies with the motor condition, it can be treated as a function of the motor flux. Therefore, the variation of the mutual inductance is not taken into consideration in this paper.

Table 1. Ratings of Tested Motor 3.7[kWl, 2OO[Vl, 15[Al 50[Hzl, 1420[rpml, 4[polesl

rect. Stator resistance variation does not affect the indirect field oriented controller. However, it has influence on the direct field oriented controller with the flux observer. In a low speed range the influence is especially significant, which creates a problem. In Figure 4(b), the only rotor resistance is incorrect. Rotor resistance variation also has a great influence o n the direct field oriented controller in a low speed range. The influence on the indirect field oriented controller is much more significant than that on the direct field oriented controller. In Figure 4(c), the both parameters are incorrect in the same degree. In this case, the influence will be restrained with a proper proportional constant. However, it is difficult to find the proper constant, because the degree of the incorrectness for the stator and rotor resistance varies widely. Therefore, some compensation for the influence of the parameter variation is necessary, especially in a low speed range.

Influence of Parameter Variation on Field Oriented Induction Motor Drive System Figures 2 and 3 s h o w a d i r e c t f i e l d oriented controller and a n indirect field oriented controller, respectively. The most important performance for motor drives is to produce the desired torque. Therefore, influences of the parameter variation on the produced torque o f two kinds of induction motor drive systems are investigated. Figure 4 shows the ratio of the produced torque to the desired value. These results were calculated under the condition that the load torque was 20 [Nm]. Figure 4(a) is the results of the case that the only stator resistance is incor922

1-

--- - - - - - - - - - - - - ---

----_____

-k=0.5

-k=l.O

o.6

t

6

cl,

0 '

(a) R,

0'

2 0'0 3d0 Rotor Speed [rpml

100

=

-k=1.5 Indirect Field Orientation

2 ob

ibo

3 do

1

J 100

Rotor Speed [rpml

(b) i,

1.2RS

=

0 . 8 ~ ~

(c) R,

=

1.2R,,

I I 200 300 Rotor Speed Irpml

fr

=

0 . 8 ~ ~

Figure 4. Ratio of Produced Torque to Desired Torque

P r o o f o f Stability of Flux Observer with Parameter Adaption

Rotor Flux Observer with Parameter Adaption

Adaptive Adjustment Scheme A stability of the proposed flux observer with the parameter adaption scheme is proved by the Lyapunov's theorem. For the simplification of the proof, the induction motor and the observer are expressed by the following equations with complex variables and k=1.0.

We propose the addition of a parameter adaptive scheme to the flux observer described by Eq.(2) in order to solve the problem ment.i oned above. We propose the following update law:

(3)

= A x + bv,

(4)

x

=

(A+AA)x+ bv,

where eida=ide-ids,eiqs=iqs-iss a l , d z : arbitrary positive gain where A A is an error matrix caused by the Darameter variation. An error equation can be expressed by the following equation:

'I'he parameters are updated only in a powering operation. Figure 5 shows a block diagram of the proposed flux observer with the parameter adaptive scheme.

e=Ae-AAx

where e = x - x Now we define a following Lyapunov function candidate:

is I

(7)

I

Computing the time derivative of V and using (3),(4)gives following equation:

Eqs.

V = e'(A'+A)e

Pigure 5. Block Diagram of Proposed Flux Observer

923

(9)

Equation (9) is negative-semi definite, bec a u s e the matrix A is negative definite. Therefore the flux observer with the parameter adaptive scheme. The stability in the case of k f 1.0 can be proved similarly.

.G

0. 2

L.-

0

:

3

I

6

9

12 TIME

15

(sec)

(a) 30[rpml, 20[Nml 1.

CLX

0

a 3 6 9 12 15

TIME

.2

0

0

3

6

9

12

(sec)

TIME

(c) 300[rpml, 2uLNml

15

(sec)

(d) 30[rpmI, 2lNml

Figure 6. S i m u l a t i o n R e s u l t s of Parameter Adaption (k=1.0)

0

3

6

9

(e) 30Lrpm1, 10+5sin(2st)[Nm]

Figure 6 shows the simulation results of the parameter adaption. Initial values of the estimated stator resistance and rotor time constant were 1.2 and 0 . 8 times as much as actual values, respectively. The arbitrary gains h ;i and k were set to following values: ;i

15

12

TIME ( s e c )

Simulation Results and Discussion

1=0.01

1.0

h 2=0.1

'

T*=1[PU]

k =1.0 Figures 6(a),6(b) were calculated under the condition that the induction motor was driven at a low speed (30 [rpml, 3 [rpml) with an approximate rating load torque (20 [Nm]). Both the estimated stator resistance and rotor time c o n s t a n t c o n v e r g e quickly to the actual values. In the case of a higher speed condition, the convergence behavior for the stator resistance identification deteriorates slightly as shown in Fig. 6(c). Furthermore, the lighter the load torque is, the slower the convergence speed becomes as shown in Fig. 6(d). Figure 6(e) is an example of a transient state case. The convergence behavior of the parameter a d a p t i o n is s t i l l good w i t h a sinusoidal load. Figure 7 shows the characteristics of the produced torque when an induction motor is driven by the direct field oriented controller with proposed flux observer. We find that the proposed induction motor drive system can produce the desired torque without respect to the parameter variation.

-E 0. --

a,

0. 7

6

?i

a

0. .. 5

T*=O . 5 [PU]

-a E 0.3 -0.2-. U

0. 4

p-i

0.1

'

+

0

T*=0.25[PU] T*=O.1 [PU] I

160

I

200 300 R o t o r Speed [rpml

Figure 7 . Produced Torque Characteristics for Proposed Method 924

Furthermore, we investigated the effect the p r o p o r t i o n a l c o n s t a n t k o n t h e parameter adaption. Figure 8 shows the simulation results of the parameter adaption calculatcd under the same conditions as those of b i g . 6(a) except the proportional constant k. 'I'hc smaller the proportional constant is, the Vaster the convergence speed of the parameter ~ r r o rbecomes. Therefore, we had better select ;I small proportional constant to identify the parameters. oI'

Application of Adaptive Scheme to Speed Estimation

The other adaptive scheme is for identification of the motor speed. Using this scheme, a speed sensor can be eliminated from a field oriented induction motor control system. A stability of the proposed adaptive schemes has been proved by the Lyapunov's theorem and validity have been verified b y simulation results.

> I

The removal of a speed sensor is required Vor n cheap drive system or a drive system in hostile environments. The flux observer with t h c parameter adaptive scheme can be applied l o the speed estimation when the motor speed i s regarded as a parameter. In this case, the I'ollowing adaptive scheme is used.

4-c

0 - c

I

0

3

6

9

15

12

TIME ( s e c ) (a) k=0.5

=

urp

+

(10)

urx

wlic~rc K P and KI are arbitrary positive gain. A stability of this observer also can be proved by the Lyapunov's theorem. In this case, we define a Lyapunov function candidate ;Is ~ollows:

o

!

I

0

3

6

9

12 TIME

15

(sec)

(b) k=1.5 Figure 8. S i m u l a t i o n R e s u l t s o f Parameter Adaption ( k f 1 . 0 , 30[rpml, 20[Nml) '1%~

time derivative of V becomes V

=

lOrpm/d i v

e'(A'+A)e

I * q n a t ion ( 1 2 ) is negative-semi definite, so l t r r I'lux observer with the speed adaptive s r h c v r c is stable.

1

Figure 9 shows a simulation result of a stcp response. The estimated speed was f o t l back in stead of the actual speed. An ex( ~ ( ~ I L c nresponse t was obtained and validity of 1 quation ( 1 0 ) was verified. 5pc'ctf

1 depS/,

-Actual Reference Speed

1

---

II

Conclusion

c

Two kinds of parameter adaptive schemes l o r '
I/

Estimated Speed :

-

5Omsec/div

Figure 9. Simulation Results of Speed Step Response without Speed Sensor

925

2.

Appendix

Symbols 3.

Stator Resistance Rotor Resistance Stator Self Inductance L, Rotor Self Inductance LXM Mutual Inductance U Leakage Coefficient Rotor Time Constant rr Motor Angular Velocity & &* Angular Velocity Reference Slip Angular Frequency Oe in* Field Current Reference it* Torque Current Reference Produced Torque T Torque Reference T* RS RX-

4.

5.

6.

Acknowledgment

The authors would like to thank Mr. M. Ozaki of Meiji University for his assistance. References 7.

1.

A.Bellini, G.Figalli, and G.Ulivi " A Microprocessor-Based State Observer for the Feedback Control of Induction Motor Drives" Proc. of European Conference on Power Electronics and Applications, pp. 3.45-3.50, 1985

926

G.Verghese, and S.R.Sanders "Observers f o r F l u x Estimation in Induction Machines" I E E E T r a n s . o n Industrial Electronics vo1.35 No.1 pp.85-94, 1988 H . K o b a y a s h i , M.Koizumi, H.Hashimoto, S.Kondo, and F.Harashima " A New Controller for I n d u c t i o n M o t o r s using F l u x Observer" IEEE Pesc '88 Record pp.10631068, 1988 Y.Hori, and T.Umeno "Implementation of Robust Flux Observer Based Field Orientation (FOFO) Controller for Induction Machines" 1989 IEEE IAS Annual Meeting pp.523-528 G.Franceschini, C.Tassoni, and A.Vagati "Flux Estimation for Induction ServoMotors" Proc. of 1990 International Power Electronics C o n f e r e n c e ( I P E C - T o k y o ) pp.1227-1234 S.Sangwongwanich, T.Yonemoto, T. Furuhashi, and S.Okuma "Design of Sliding Observer for Robust Estimation of Rotor Flux of Induction Motors" Proc. of 1990 International Power Electronics Conference (IPEC-Tokyo) pp.1235-1242 H. Kubota, and K.Matsuse "Flux Observer of Induction Motor with Parameter Adaptation for Wide Speed Range Motor Drives" P r o c . o f 1 9 9 0 I n t e r n a t i o n a l Power E l e c t r o n i c s C o n f e r e n c e (IPEC-Tokyo) pp.1213-1218