Adaptive Sliding Mode Observer For Speed Sensor Less Control Of Im

Adaptive Sliding Mode Observer for Speed Sensorless Control of Induction Motors Francesco Parasiliti, Roberto Petrella, Marco Tursini University of L'Aquila Department of Electrical Engineering 67040 Monteluco di Roio, L'Aquila - Italy [email protected] - [email protected] - [email protected]

-

Abstract This paper presents an adaptive sliding mode observer for speed sensorless field-oriented control of induction motors. The observer detects the rotor flux components in the two-phase stationary reference frame by the motor electrical equations. The motor speed is identified by an additional relation obtained by a Lyapunov function. The analytical development of the sliding observer and the speed identification algorithm is fully explained. Experimental results are presented, based on a TMS320F240 DSP controller implementation, showing the system performance with different observer gains and the influence of the motor parameters deviations.

I. INTRODUCTION

The availability of low-cost and high performance Digital Signal Processors (DSP) and dedicated chips makes field oriented control a practical choice for a wide range of applications. Field orientation on the rotor flux is generally preferred, owing to the high dynamic and steady-state performance obtainable over all the torque-speed range. This solution needs the knowledge of both the motor speed and the rotor flux (position and amplitude). Usually, a shaft encoder or a tacho-generator is used to measure the motor speed, but the presence of these sensors increases the drive cost and encumbrance and reduces the robustness of the overall system. Owing to this, in the last decades, many research efforts have been carried out for the development of observers able to estimate both the motor speed and the rotor flux from the motor terminal quantities (currents and voltages). For a certain period, the extended Kalman filter appeared to be the unique solution for this problem, as reported in numerous papers (e.g. [ 1-21). Unfortunately, this stochastic observer has some inherent disadvantages, such as the influence of noise characteristic, the computation burden and the absence of design and tuning criteria. This has led to a renewed interest in deterministic approaches, where the structure of the standard Luenberger observer for linear system is enhanced to permit the simultaneous estimation of the rotor flux and speed. Examples of this approach are given in [3-81. In the adaptive observers [4] the speed andor other unknown parameters are identified by additional equations based on the adaptive control theory. This allow to find out the analytical conditions for stability. Among these proposals, the sliding mode observer represents an attractive choice for its being robust to disturbances, parameter deviations and system noise ([7], [8] and [lo]).

0-7803-5589-X/99/$10.00 0 1999 IEEE

In this paper an adaptive speed sensorless field-oriented control of an induction motor is presented, based on a sliding mode observer. The observer detects the rotor flux components in the two-phases stationary reference frame, using the motor voltage. equations. The motor speed is identified by a further relation obtained by a Lyapunov fbnction. The method has been implemented using the TMS320F240 fixed point DSP controller. The system performance is experimentally analysed in order to evaluate the observation errors with different observer gains and the influence of the motor parameters deviations. The paper is organised as follows. After a brief introduction of the system configuration (Section 11) the adaptive sliding mode observer for the induction motor is presented in Section 111. The conditions for a stable design of the system are discussed in Section IV. Details on the implementation of the drive system are given in SectionV, while the experimental tests and results are presented in Section VI. Finally, some concluding remarks end the paper. 11. SYSTEM CONFIGURATION

A block diagram of the considered system is shown in Fig. 1. The field-oriented controller is based on a currentcontrolled Voltage Source Inverter (VSI) structure. The control loops are arranged in the two-phase synchronously

p;q

7-

A

AV-SV

"

or

w, 2,

Adaptive Sliding Mode Observer 'SP

Fig. 1. Speed sensorless induction motor drive with adaptive sliding mode observer.

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rotating reference9ame dq aligned with the rotor flux, whilst the adaptive sliding mode observer operates in the twophases stationary referenceframe ap . The output of the speed regulator represents the q-axis reference current i:q while the flux loop generates the d-axis reference current i; . An ap to dq transformation provides the current components isq and id needed for the current regulators. The outputs of the current regulators give the reference voltages v$,viq in the dq frame. A dq to ap transformation then yields the reference voltages v f , ,v : ~in the stator frame, which are the inputs of an Adjacent Vector Space Vector Pulse Width Modulator (AV-SVPWM). Standard PI controllers (with limitation) are used for all the regulators. The adaptive sliding mode observer provides the rotorflux position p (needed for the field orientation), the flux amplitude yr (used to close the flux control loop) and the rotor speed feedback w, (used for the speed control loop). The rotor flux position and amplitude are calculated by the respective ap components as follows +r

= arc tan(

= -/,

g] .

px=Ax+Bv,

(1)

IT

where x = [is yr is the state vector, i s , v, , yr , are the stator current, voltage and rotor flux vectors respectively, and the system matrices are as follows

A , , = a l , A , , = ~ l - d l , A , ~= e l , A22=-EqlZ, B , = b , I , I=[;

'

3

J = [ l 0 -O1 ] .

,

w

d = l ,

With reference to the induction motor model (1) and considering the stator currents as the system outputs, the sliding mode observer for rotor flux estimation can be constructed as follows .

.

(3

p i = ~ + ~+Ksgn(is v , -is)

where K is a gain matrix which can be arranged in the general form

111. ADAPTIVE SLIDING MODEOBSERVER FOR

INDUCTION MOTOR The adaptive sliding mode observer for induction motor can be seen as composed by two parts (Fig.2): a sliding mode observer for rotor flux estimation and a sliding mode speed identification algorithm working in parallel. A. Rotor Flux Sliding Mode Observer

By considering the rotor speed as a system parameter, an induction motor can be described by the following state equation in the stationary reference frame a/? (the meaning of the used symbols is clarified in the nomenclature) is

IM model

The error equation from (1) and (2) which takes into account the parameter variation AA can be expressed as follows

p e = Ae+A&+Ksgn(is -is)

(3)

where

. .

e = x - x = [ei e,lT, ei=is-ts, e y

= i.v - i.y ,

-.iS

~

v

If the sliding mode is attained (i.e. the gain K is large enough) one can assume the following simplifying assumptions

A 4) ' r

e=orM,

E

V

ei

= pei = 0

(4)

V

Identification algorithm

from which equation (3) gives

0 = A12ew+ AAlljs + AA12yr- z

Fig. 2. The adaptive sliding mode observer for induction motor.

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(5)

+ AA2,is+ AA2,yr + Lz

p e , = &e,

(6)

where z = - K , sgn(is -is). From (5) and (6) the error equation for the rotor flux in sliding mode condition in obtained as follows Pe,

k,

= ( 4 2 + LA,, +

+( 4 1 +4

1

Ys+

kr

(AA,,+

Am2 = d P'O 2PE

W

By the comparison of (14) and (151, the equation for the speed identification is obtained as follows pcj, = p y [ ~sgn(jsa , - isa)- G, sgn(jsp- isp)].

kw

(16)

(7)

If the speed is a known-(measured) parameter and no other parameter variations are considered, one obtains from (7) Pew = (A22 + LA12

(15)

9

(8)

Iv. DESIGNOF THE ADAPTIVE OBSERVER A . Convergence of the Speed Identification

This relation represents the error equation of h e rotor flw observer in sliding mode condition.

Condition (13) can be used to set the observer gains in the matrix L so that the rotor speed convergence is assured. Developing the more relaxed condition

B. Rotor Speed Identification Algorithm

A' 2 -TA,, , y > o

If we consider the rotor speed as a variable parameter, the matrix AA is specialised as follows

one obtains

AA,, = o , AAl2 =-- Am' J , AAl2 = O , AA,, = AmrJ

L = -XI - y J

where

E

Aw, = C3,

-U,.

x2--E+- Y g r

We choose the candidate Lyapunov function as follows

E

(9)

V=e;e,+W

where the function W must be determined in order to assure the convergence of parameter identification according to the Lyapunov stability theory. The time derivative of V can be expressed as PV = PV, + PV2

YW, Y2E

nese conditions can be x = ( 4 - 1)E

pv, = z T l i T ~ ; ; z T

E

By the analysis of (1 3) the function W is selected as

Assuming the speed as a known parameter, the error equation of the rotor flux in sliding mode conditions is given by (8), with the system matrix equal to

4 = A,, + LA,, = -a f j p

(13) with

-a

= -a-4 - O r

p = ak - cy + mr

A

E

E

B. Stabiliry of the Rotor FIUX Observer

With this assumption the condition pV2 = 0 gives Am p w = p TL . r + v r

(20)

(12)

and A = L - d . Condition of (lo) be definite negative will be satisfied if 'pVi < 0 and pV2 = 0 . The condition pV, < 0 is satisfied choosing AT = -yAI2 , y > 0 .

Ywr

(1 1)

-id@,

P V ~ = ZA A , , - J @ r + p W

+-Y c r E

(10)

where

expressed in terms of a design

as follows

Y'4-

T

(17)

(14)

and the eigenvalues of the closed loop error system are A,2 = -a k j p

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(22)

V. DRIVESYSTEM

500

The drive system used to test the proposed adaptive sliding mode observer for induction motors is shown in Fig. 4. It consists of a single board drive unit, an induction motor and the necessary development and testing tools. The single board drive unit includes the control hardware and an integrated IGBT based Intelligent Power Module. The phase currents have been measured using a low cost technique instead of conventional Hall effect probes. The solution, integrated in the Intelligent Power Module, consists in a shunt put in series to the emitter of the lower IGBT for each leg of the inverter as shown in Fig. 5 [ 1 11. The control hardware makes use of the recent Texas Instruments TMS320F240 pC, a fixed point Digital Signal Processor (DSP) specifically developed for drive applications whose main characteristics include: > a high performance CPU core (5011s instruction cycle at 20 MHz CPU clock); > 544 words (16-bit) of data Dual Access RAM; > 16Kwords of program Flash-EEPROM; > a complete set of dedicated I/O peripherals (ADC unit, PWM unit, quadrature encoder interface).

Im 0

-500

-600

-200

-400

0

Re

5000 I

1

Scope

-3

-4

-2

-1

0

Re Fig. 3. Eigenvalues of the rotor flux closed loop error system

C. Stabiliry of the Global System Communication soffware, development and debugger tools

Introducing the convergence conditions for the speed adaptation (20) in (22) and then in (23) one obtains

+---+ ")

A,2= -q( (0,

O,'

y

7 Tj

[

w, +%E 2

(;

--

Fig. 4. Drive system

I]])

(24)

This relation demonstrates that the eigenvalues of the rotor flux error system (in sliding mode conditions) are strictly stable. Thus, the adaptive system based on the sliding mode observer plus the adaptation equation (1 6) is strictly stable. Design parameters of the adaptive system are q and y in (20) and p in (16). The first two parameters can be chosen to improve the performance of the rotor flux estimation, the third affects the dynamic response of the speed adaptation. The influence of the design parameters q and y on the placement of the eigenvalues on the complex plane is shown in Fig. 3.

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Fig. 5. Inverter with shunt resistors for current measurement

Set points and main parameters of the control scheme can be changed in real time by means of a host PC linked to the DSP through a standard RS-232 interface. The host PC is also used to run the DSP development and debugger tools. A scope is used to display in real time the variables calculated inside the control algorithm by means of a 2 channels digital to analog interface mapped on the I/O addressing space of the pC DSP. During the development of the control program, an incremental encoder has also been used to measure the actual speed and compare with the estimated one. The execution of the control algorithm has been synchronized to the PWM carrier whose period has been fixed to loops, resulting in 1OkHz switching frequency. However, the time needed to execute the whole control algorithm is less than 60ps, including the adaptive sliding mode observer which takes about 15ps. VI. EXPERIMENTAL RESULTS

.

The proposed system has been tested in order to verify the global behaviour and particularly the robustness of the adaptive sliding mode observer to parameters variations. A braking system has been used, allowing to impose the load torque in the whole range from zero to rated speed. Results are presented in per unit form. The base values assumed for scaling and the parameters of the test motor are resumed in Appendix. Fig. 6 and Fig. 7 show the transient behaviour of the system with the sliding gains kl = k2= -0.04 and the speed adaptation parameter p y = 0.086. The observer gains in matrix L are set to zero as a first test condition (parameters CASE 1). One can notice the fast convergence of the speed estimate during the transients and the capability to maintain the estimation at standstill. The set of parameters assumed in the previous tests doesn't match the design conditions (20) for the gain matrix L. In fact, we could verify that the operation over 0.7pu speed generates instability. Thus, a second set of parameters has been considered with k, = k2= -0.04, py= 0.086 and the observer gains of matrix L matching the design conditions (parameters CASE 2). with q = 0.1, y= 0 . 0 0 0 2 ~ In this case the whole speed range operation can be achieved, as demonstrated by the rated speed reversion in Fig. 8. The speed response becomes faster respect to the parameters CASE 1, as shown by the motor start-up transient in Fig. 9 (compare with Fig. 6a). The dynamic performance are satisfactory also when the rated load torque is applied, as presented in Fig. 10. The influence of the adaptation parameter ,uy is presented in Fig. 11, in the case of a speed transient from 0 to 0.7pu (the other parameters are set as in the CASE 2). According to (16) this parameter affects the rapidity of the speed adaptation. Small values reduce the adaptation rapidity. On the other hand, too large values generate responses which are not well damped and in some cases unacceptable. Moreover,

228 1

-..~ ....

5.oov

......5 . o O v.... ,* r P 2 0 . o m r

,

I....._

Chl

%=

I

A

,

Chl I

..-.....-

,

0.OOV

Fig. 7. Speed reversion from -0.6 to 0 . 6(parameters ~ CASE 1).

Fig. 8. Speed reversion from -1 to lpu (parameters T e7 kAmSIa, ,

,

f .,;-o, ,

,

0

1

,

,

, , ,

,

,I--.

,

,

,

, ,

The performance of the sensorless scheme as regard to the speed estimation error and the robustness to motor parameter variation are also tested. Fig. 13 shows the speed estimation error at no-load conditions over the whole speed range, with different choices of the observer gains. Finally, Fig. 14 shows an analysis of the system sensitivity to rotor resistance variations. Plots of the q-axis current vs. the load torque at steady-state are presented ( 0 . 5 ~speed), ~ for three different operating conditions obtained with as many (constant) values of the rotor resistance parameter used in the adaptive sliding mode observer (see Appendix). The linear behaviour demonstrates that a correct field orientation is achieved for all the cases. Moreover, the operating points are practically unaffected by the rotor resistance variations, confirming the robustness of the system.

CASE 2). ,

-_ I

VII. CONCLUSIONS In this paper an adaptive sliding mode observer for speed sensorless field-oriented control of induction motors is presented. A criteria for choosing the gains of the adaptive observer is proposed. The scheme has been implemented using one of the last generation fixed point DSP controllers, the TMS320F240. Experimental results confirm the effectiveness of the solution, whose robustness with respect to the variation of the motor parameters appears as the main feature.

Fig. 9. Speed transient from 0 to 0 . 7 (parameters ~ ~ CASE 2).

hl

- zlOO

v - 4-&,

2

Fig. 10 Speed transient from 0.1 to 0 7pu (rated load, parameters CASE 2).

Fig. 12. Phase current and estimated speed at steady state (rated load, 300rpm speed). 10 -

Fig. 1 1. Speed transient from 0 to 0.7pu with different values of parameter py.

2

4

E

2

1

v)

the estimated speed is more affected by the ripple due to the sliding mode error. Thus, this parameter must be chosen from a compromise between these requirements. The behaviour of the phase current and estimated speed at steady state operation is shown in Fig. 12. Distortion of the current due to the inverter operation is clearly evident, which causes a correspondent ripple on the speed. 2282

0 -2 0

py= 0.086, y= 0 . 0 0 0 2 ~

+ py= 0.14, y=O . 0 0 0 2 ~

-4 -6

0

300

600

900 1200 1500 1800 2100 2400 2700 3000 Speed [ W ~ I

Fig. 13. Speed error vs. operating speed at no-load

[9]

0

0.1

0.2

0.3

0.4

M. Tsuji, E. Yamada F. Parasiliti, M. Tursini, “A Digital Parameter Identification for a Vector Controlled Induction Motor,” 7th European Conference on Power Electronics and Applications (EPE ’97), Vo1.4, p.603608, Trondheim (Norway) 8-10 September 1997. [ 101 F. Parasiliti, R. Petrella, M. Tursini, “Sensorless Speed Control of a PM Synchronous Motor by Sliding Mode Observer,” IEEE International Symposium on Industrial Electronics (ISIE’97), Vo1.3, p.1106-1’111, Guimaraes, Portugal, July 7-1 1 1997. [l 11 F. Parasiliti, R. Petrella, M. Tursini, “ Low Cost Phase Current Sensing in DSP Based AC Drives,” IEEE International Symposium on . Industrial Electronics (ISIE ’99), Bled, Slovenia, July 12-16 1997.

0.5

Load toque [pu]

Fig. 14. q-axis current vs. load torque with different rotor resistance parameter.

NOMENCLATURE REFERENCES

* A

aP

reference values, observed values matrices, vectors stationary reference frame synchronously rotating reference frame

v,, ,vss ,is,, isp

afl stator voltagelcurrent

V , d , vSq, isd,isq

dq stator voltagelcurrent

wra,\vrp,w r

a;O rotor flux components

Vr

rotor flux amplitude mutual inductance statorfrotor self-inductance/resistance stator leakage coefficient reverse of the rotor time constant rotor angular speedrotor flux angle

3

G. Hennenberg, B.J. Brunsbach, Th. Klepsch “Field Oriented Control of Synchronous and Asynchronous Drives without Mechanical Sensors Using Kalman Filter”, Proc. of EPE ’91, vo1.3, p.664+67 1, Firenze, 1991. R. Kim, S.K. Sul, M.H. Park “Speed Sensorless Vector Control of Induction Motor Using Extended Kalman Filter”, IEEE Trans. Ind. Applications, Vo1.30, No.5, pp. 1225-1233, September/October 1994. T. Du, P. Vas, AF. Stronach, “Real-time DSP Implementation of an Extended Observer in a HighDynamic Performance Induction Motor Drive,” 6th European Conference on Power Electronics and Applications (EPE ’95), Vo1.3, p.45-49, Sevilla (Spain) 19-21 September 1995. H. Kubota, K. Matsuse, T. Nakano “DSP-Based Speed Adaptive Flux Observer of Induction Motor,” IEEE Trans. Ind. Applications, V01.29, No.2, pp.344-348, MarcWApril 1993. H. Kubota, K. Matsuse “Speed Sensorless Field Oriented Control of Induction Motor with Rotor Resistance Adaptation” IEEE Trans. Ind. Applications, Vo1.30, NOS, pp. 1219-1224, September/October 1994. C. Ilas, R. Magureanu “An Improved Speed Sensorless Scheme for Vector-Controlled Induction Motor Drives” ELECTROMOTION,NO.3, pp.67-67 1, 1996. S. Sangwongwanich, S. Doki, T. Yonemoto, T. Furuhashi, S. Okuma, “Design of Sliding Observers for Robust Estimation of Rotor Flux of Induction Motors,” Proc. of Int. Power Electronics Conference, pp. 12351242, Tokyo, 1990. S. Sangwongwanich, S. Doki, T. Yonemoto, T. Furuhashi, S. Okuma, “Adaptive Sliding Observers for Induction Motor Control,” Trans. of SICE, Vo1.27, N.5, pp.569-576, May 1991.

A, a

M L, ,L, ,R, ,R, 0 Or

@,?P

APPENDIX The experimentsdescribed in this paper were carried out using the following IM machine. Pole pairs Rated power Rated (base) voltage Rated (base) current Base speed Base torque R, (75 “C) R, (7511 151150 “C) L* L, M

2 225 W(at 50Hz) 117 Vrms (phase) 3.3 Arms 3000 rpm 1.5 Nm 2.64 R 2.7713.2513.66 R 65.57 mH 64.84 mH 59.52 mH

The following scaling values has been considered for displaying the experimental results.

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Flux, current, speed and position

1 p.u.-+5V