Adaptive Variable Structure Rotor Flux Observer For An Induction Motor

318 ADAPTIVE VARIABLE STRUCTURE ROTOR FLUX OBSERVER FOR AN INDUCTION MOTOR

G. Garcia Soto, E. Mendes, A. Razek Laboratoire de GCnie Electrique de Paris - CNRS - SUPELEC - UniversitCs Paris XI et Paris VI, France.

Abstract.- This paper deals with robust flux observation of an induction motor using non-linear observer based on sliding mode technique. The nonlinear observer is developed and is used to perform the direct field oriented control of an induction machine. By employing Lyapunov’s theorem, resistance adaptation schemes are designed to increase the observer process accuracy. Computer simulations and experimental tests have been carried out. List of symbols

(a,P) stator fixed frame axes vmP stator voltage vector isae stator current vector +mp rotor flux vector electrical rotor frequency R,, R, stator and rotor resistances per phase Td motor drive torque p number of pole pairs CT total leakage factor L, stator self inductance. Lm=(1-o)L,magnetising inductance * denotes estimated values

~

The direct field oriented control (DFOC) is often used in high performance induction motor drives [I]. Such control method needs the knowledge of the rotor flux. In order to avoid expensive sensors, the rotor flux is not measured but estimated. Unfortunately, the estimation approach is very sensitive to the motor parameter variations. Many works have been concemed with rotor flux estimation for vector controlled induction motors. A first kind of estimators is based on the linear observation theory. The observer structure and its gains allow to achieve reduced sensitivity to rotor resistance variations as is shown in [2]. However, they are sensitive to stator resistance variations, which cause steady state and dynamic errors. A second kind of observers is based on the extended Kalman filter (EKF) [3][4]. These solutions are very complex and the estimation accuracy depends also on the stator resistance knowledge. Several works (see for example [5]-[7]) have shown a promising solution using the Luenberger observer theory and Lyapunov’s technique. The Lyapunov’s analysis is used to design the

parameters adaptation laws to eliminate the influence of the stator resistance uncertainties. The sliding-mode techniques offer an interesting solution because of their robustness to parameter uncertainties and of their fast response [SI. Some papers have presented the application of the sliding-mode method to the observation process in vector controlled induction motor [9]-[13]. However, the parameter sensitivity is not completely eliminated. This paper presents a sliding-mode observer with resistance adaptation schemes. The main contribution of this work can be divided into two parts. In the first part, a non-linear sliding mode rotor flux observer is studied and implemented. The main feature lies in its fast convergence rate. The second part deals with resistance adaptation mechanisms in order to improve the rotor flux estimation accuracy. The main property of this adaptive flux observer is its robustness against thermal variations (insensitive to stator and rotor resistances variations). Computer simulations and experimental tests are presented to highlight the effectiveness of the proposed observer. INDUCTION MOTOR MODEL The electromagnetic model of a three phases squirrel cage induction motor in the stator reference frame can be expressed as [l] :

(3) Considering the shaft speed dynamics very slower than the electromagnetic ones, we can express the linear time-variant motor model as : (4) i ( t ) = A ( o , ) x ( t ) + Bu(t) with

Power Electronics and Variable Speed Drives, 21-23 September 1998, Conference Publicafion No. 456 0 IEE 1998

31 9

v

= si {fi - 61sign(sl))+ s2 I f 2 - a2sign(s2))< 0 The resulted conditions to ensure the attractiveness of S (convergence to zero) can be resumed as :

6, >Ifiland 62 > I f 2 1

(7)

SLIDING-MODE ROTOR FLUX OBSERVER Determination of rotor corrector gains A, The full order observer with switching corrector gains is defined as [l 11:

The determination of the rotor corrector gains A, can be achieved using the equivalent control method [ 101. Since the stator current dynamics are much more greater than the rotor flux ones, the system is locally stable when S = 0, S = 0 . In these conditions we obtain :

where The equivalent expression of the switching functions is given by : Assuming that the model is a perfect image of the system, the estimation error equation is : Then, substituting (8) in the rotor flux dynamic error equation, we obtain the following expression :

The rotor corrector gains A, are easily obtained by imposing the dynamics of the reduced order model as :

with

Determination of stator corrector gains A,

where

p=["l

0

"1

with

q1> O

and q2 > O

92

The resulting rotor corrector gains Ar are The stator corrector gains Asare determined in order to ensure the convergence of the estimated stator currents to real ones (convergence of S to zero) by applying the Lyapunov's method [9]. The Lyapunov candidate function Vis defined as :

1 v = -9s 2 Forcing the time derivative of V to be strictly negative, we ensure that the estimated variable converge to the real one. Then we can determine the stator corrector gains As.Letting :

The switching function To avoid high switching frequency of the switching function (to decrease the chattering effect) a suitable hysteresis band A has been introduced as in figure 1. ,sign(s) 1-L

Lr-:

-____

I

A

T

-1

Figure 1 : the switching function

320 observer. For this, a speed control has been simulated using the field oriented control (FOC) method and the studied observer for the rotor flux estimation. As mentioned before, the FOC needs the accurate knowledge of the modulus and phase of the rotor flux vector. Then, the observer has to reconstruct precisely not only the rotor flux modulus but also its phase to obtain an accurate estimated motor torque. Figure 3 shows the simulation study of the observer robustness, the motor speed reference is taken 30 rpm and the load torque is set to 50% of the motor rated value. At the beginning of these simulations, the motor and observer rotor resistances are set to their nominal values (4 ohms). At time t=1.5 s, the motor rotor resistance is changed to 2 ohms, and at time t=3 s, the motor rotor resistance is again changed to 6 ohms. Figures 3a and 3b show the motor and estimated rotor flux magnitudes, and the motor and estimated electromagnetic torque respectively for different values of parameter q of the observer. The conclusion of the presented simulations below is that we can not obtain simultaneously a good estimation of both rotor flux magnitude and motor electromagnetic torque. For example, with q=15, the estimation of the rotor flux magnitude is almost insensitive to the rotor resistance variations, while the estimation of the motor torque is not satisfactory. Moreover, if fast convergence rate of the observer is desired (high q value), the observer is very sensitive to the rotor resistance variations. This is why a rotor resistance adaptation scheme is needed. Although, the observer is robust to stator resistance variations in the medium and high speeds ranges, it is interesting to adapt this parameter for the very low speed region operation.

Simulation study To study the behaviour of the sliding mode observer, simulations have been carried out. Response time of the observer. The first simulation, figure 2, shows the behaviour of the rotor flux dynamic error for different values of the parameter q,=q2=q. At the beginning of this simulation the states values of the motor are set to : (b,,=0.5 Wb, $,,=O.O Wb, i,,=0.07 A and i,,=O A while all observer states initial values are set to zero.

[SI

Figure 2 : simulated rotor flux error response The rotor time constant of the used induction motor is L,&= 0.1 s. Then, with q=10, the response time of the observer (0.3 s) is almost the same of the one obtained with an open loop estimator (3L,/RJ. On the other hand with q=70, the response time is much more smaller and the dynamic behaviour is very satisfactory. This feature is the main advantage of the sliding mode observer. Robustness to rotor resistance variations. In order to show the sensitivity of the non-linear observer to the rotor resistance variations, simulations have been carried out for different values of parameter q of the

6

E 5

s

E

.s5 4 -0

g61 3 2 1

time$]

4

5

1

timed

4

5

(a) motor and estimated torque responses (b) motor and estimated rotor flux magnitude responses Figure 3 : simulation study of the observer robustness to the rotor resistance variations for different values of q.

321 ROTOR AND STATOR RESISTANCES ADAPTATION MECHANISMS Taking into account the uncertainties on the resistances, the resulting system error becomes :

one in the motor, we can see that it converges to the real motor value in about 8 s. At time a = 10 s., the estimated stator resistance is forced to 13 ohms (+45% of the real value in the motor), then it returns to the real value in the motor in about 3 s, while the rotor flux magnitude is not much affected by the transients of the estimated resistances. To verify the efficiency of the proposed method the same tests as those shown on figures 4a and 4b have been done experimentally (figures 5a and 5b). For details of experimental system see appendix. A very good correlation between simulated and experimental results is obtained. real value

with

The time derivative of V becomes :

0

V = q(h - 61sign(s,)} + %(f2 - 82sign(s2)}

1.2

To ensure, for stability, that stays always strictly negative, the perturbation terms :

s'- ( ' s - i s ) .

04

. 's

and ~----(i~-k) ['r-k)

4

-

--

E:::; 3 0.6 w

0.0

16

20

"

-m

___

-

0.2 0.4

8 12 time [SI

esthrated

" " " '

OLS

are compensated using adaptive resistance schemes. For this, the candidate Lyapunov's function V, is chosen as : Figures 4 : Simulated rotor flux sliding-mode observer with stator and rotor resistances adaptation schemes. 16

,

where h , , ~are , positive constants. Then, the time derivative of v, becomes :

2h

' 0

4

8 12 time [SI

16

20

(a) experimental :rotor and stator resistances adaptation The resulting resistances adaptation mechanisms are :

These adaptation mechanisms have been simulated and implemented experimentally. The speed reference is 50 rpm, the load torque is set to 20% of the rated motor value, and the gain of the observer is q = 70. Figures 4a and 4b show the simulated behaviour of the resistances adaptation mechanisms and estimated rotor flux magnitude. At time t=2 s, the estimated rotor resistance is forced to a value 200% greater than the real

0.2 4

8 12 time [SI

16

20

(b) experimental : estimated rotor flux magnitude Figures 5 : Experimental rotor flux sliding-mode observer with stator and rotor resistances adaptation schemes.

322 FIELD WEAKENING RANGE OPERATION Tests have also been done in the field weakening range. With the used experimental system (see appendix), the field weakening range is reached for speeds greater than 400 rpm. Then, the speed reference has been chosen to vary between 100 rpm and 500 rpm. Figures 6a and 6b show the corresponding simulation test. We can see the excellent behaviour of the rotor flux reconstruction. The same test has been carried out experimentally (figures 7a and 7b). Again, the behaviour of the rotor flux is very satisfactory.

COMPARISON OF THE PROPOSED OBSERVER WITH OTHER TECHNIQUES The proposed Sliding Mode Observer (SMO) has been compared with other solutions : 0 an Extended Kalman Filter such in [3]; 0 an adaptive Luenberger observer (ALO) such in [5]. For the comparisons, the observers gains were determined to obtain the best steady-state and dynamic performances accounting to observation noise level and stability of the algorithms. Execution time Table 1 shows the needed execution time of each observer, when they were implemented on the experimental DSP-card. Table1 -Execution time of the observer algorithms

5

o'oO

IO time [SI

15

20

24 ps 26 ps 40 ps

SMO ALO [ 5 ] EKF [3]

(a) rotor flux magnitude

Rotor flux error response

tine [SI

(b) shaft speed Figure 6 : Simulated speed and flux responses.

The first advantage of an observer with respect to an estimator (which does not use corrector gains) is that one can set the observers corrector gains to permit more fast convergence. In our case, a flux estimator has a response time of 300 ms. (in our case the rotor time constant is approximately 0.1 s). Table 2 and figure 8 give a comparison of the rotor flux error response of the three compared observers. For this, the initial state conditions of the motor model were set as: +,=0.5 [Wb], +lp=-O.l [Wb], isa=1.7 [A] and isp=O [A]. All initial observer state conditions were set to zero. Table 2 -Rotor flux error response 45 ms 130 ms 200 ms

SMO EKF [3] ALO [5]

time [SI

(a) rotor flux magnitude

-g

-20

=g' -40 0

5

10

15

20

time [SI

(b) shaft speed Figure 7 : Experimental speed and flux responses.

0,O

0,l

42 0,3 tinr:Is1

0,4

0,5

Figure 8 : Simulated rotor flux error response

323 APPENDIX :EXPERIMENTAL SYSTEM The used induction machine is a 1.1 kW, 2201380 V, 50 Hz, 1500 rpm, 3.4 A, 7 Nm, closed rotor cage slots. The nominal values of its parameters (at 25OC) are Rs = 8 ohms, Rr = 4 ohms, Ls = 0.47 H and CT = 0.12. The induction motor load is a powder-brake, and the inertia of the entire drive is J = 0.06 kg.m2. The induction machine is fed through a 300 V, 10 A, 13 kHz, MOSFET inverter controlled by the Pulse Width Modulation (PWM) method. Since the rated induction machine flux is (@)rated = 1.22 Wb and the DC link voltage of the inverter is only 300 V, the base motor speed is 400 r.p.m (before the field weakening operation). The entire machine control, including the PWM, is implemented on a Digital Signal Processor (DSP) card using the DSP32C RISC-microprocessor from AT&T. The use of this card allows the processing of the control with the following sampling time periods: currents control Ti = 76.6 ps, flux observation and control To= 153 ps, and speed control T, = 10 ms. The induction machine stator voltages are not measured. They are estimated using the inverter duty cycles, the DC link voltage and the average values model of the inverter. The numerical simulations are carried out using exactly the same control programs written in C language. The simulation of the induction machine takes into account the magnetic non-linearities [14]. The equations of the system have been discretised by the Euler method (first order).

CONCLUSIONS The different obtained results show that the developed adaptive non-linear rotor flux observer used with the direct field-oriented control method gives very good results for the induction machine control. The advantages of the proposed adaptive observer may be resumed as follows : high convergence rate of the rotor flux components estimation, insensitivity to resistances variations due to an adaptation mechanism of both stator and rotor resistances and low computational time.

Atkinson D.J., Acamley P.P., Finch J.W., "Estimation of rotor resistance in induction motors", IEE Proc. Electronic Power Applications, 1996, Vol. 143, NO. 3, pp. 87-94. Wade S., Dunnigan M.W., Williams B.W., "Improving the accuracy of the rotor resistance estimate for vector-controlled induction machines IEE Proc. Electric Power Applications, 1997, Vol. 144, NO. 5, pp. 285-294. H. Kubota, "DSP-Based speed Adaptive Flux Observer of Induction Motor", IEEE Trans. Industry Applications, Vol. 29, No 2, MarcWApril 1993. G. Yang, C. Tung-Hai, "Adaptive-Speed Identification Scheme for a Vector-Controlled Speed sensorless Inverter-Induction Motor Drive", IEEE Trans. on Industry Applications , Vol. 29, No 4, JulylAugust 1993. H. Kubota, "Speed-Sensorless Field Oriented Control of InductionMotor with Rotor Resistance Adaptation", IEEE Trans. on Industry Applications, Vol. 30, No5, October 1994. H. BQhler, "Rtglage par mode de glissement", Presses polytechniques romandes, 1986. J.H.E. Stoline, J.K. Hedrick and E.A. Misawa, "Nonlinear state estimation using sliding observers", Proceeding of 2Sth Conference on Decision and Control, pp. 332-339, Athens, Greece. December 1986. V. I. Utkin, "Sliding mode control design principals and applications to electrical drives", IEEE Trans. on Industrial Electronics, Vol. 40, No. 1, pp. 23-36, February 1993. C. Malarge, "Sur l'observation non lindaire de flux rotorique des machines asynchrones en rtgime de magnetisation variable", Technical report, University of Orsay, Paris XI, 1995. J. Hemandez, "Sur la synthbe de lois de commande non-lin6aires avec observateur : application a la robotique et a l'Clectrotechnique", PhD thesis, University of Orsay, Paris XI, 1994. C. C. Chan, H.Q. Wang, "New scheme of slidingmode control for high performance induction motor drives", IEE Proc. on Electric Power Applications, Vol. 143, No 3, pp. 177-185, May 'I,

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