A Sensorless Robust Vector Control of Induction Motor

Drives

O. Barambones, A.J. Garrido and F.J. MasedaDpto. Ingeniera de Sistemas y Automatica E.U.I.T.I Bilbao.

Universidad del Pas Vasco.Plaza de la Casilla, 48012 Bilbao (Spain)

Tel: +34 946014459; Fax: +34 944441625; E-mail: ispbacao@ehu.es

Abstract: In this paper, an indirect eld-orientedinduction motor drive with a sliding-mode controlleris presented. The design includes rotor speed esti-mation from measured stator terminal voltages andcurrents. The estimated speed is used as feedback inan indirect vector control system achieving the speedcontrol without the use of shaft mounted transducers.Stability analysis based on Lyapunov theory is alsopresented, to guarantee the closed loop stability. Thehigh performance of the proposed control scheme un-der load disturbances and parameter uncertainties isalso demonstrated via simulation examples.

Key-Words: Vector control. Sliding mode contro-ller. Induction drive. Speed sensorless.

1 Introduction

Indirect eld-oriented techniques utilizing micropro-cessors are now widely used for the control of induc-tion motor servo drive in high-performance applica-tions. With the eld-oriented techniques (Leonhard1996, Bose 2001, Vas 1994), the decoupling of torqueand ux control commands of the induction motor isguaranteed, and the induction motor can be contro-lled linearly as a separated excited D.C. motor. Ho-wever, the control performance of the resulting linearsystem is still inuenced by the uncertainties, whichusually are composed of unpredictable parameter va-riations, external load disturbances, unmodelled andnonlinear dynamics. Therefore, many studies havebeen made on the motor drives in order to preserve theperformance under these parameter variations and ex-ternal load disturbance, such as nonlinear control, op-timal control, variable structure system control, adap-tive control and neural control (Lin 1993, Ortega et.al. 1993, Marino et. al. 1998).

In the past decade, the variable structure controlstrategy using the sliding-mode has been focussed onmany studies and research for the control of the ACservo drive system (Sabanovic & Izosimov 1981, Park

& Kim 1991, Chern et al. 1998, Benchaib & Ed-wards 2000). The sliding-mode control can oer manygood properties, such as good performance againstunmodelled dynamics, insensitivity to parameter va-riations, external disturbance rejection and fast dyna-mic response (Utkin 1993). These advantages of thesliding-mode control may be employed in the positionand speed control of an AC servo system.

On the other hand, in indirect eld-oriented con-trol of induction motors, a knowledge of rotor speed isrequired in order to orient the injected stator currentvector and to establish speed loop feedback control.Tachogenerators or digital shaft-position encoders areusually used to detect the rotor speed of motors.These speed sensors lower the system reliability andrequire special attention to noise. In addition, forsome special applications such as very high-speed mo-tor drives, there exist diculties in mounting thesespeed sensors.

Recently, many research has been carried on thedesign of speed sensorless control schemes, (Shauder1992, Kubota 1993, Peng & Fukao 1994, Zamora 1998,Huang 1998). In these schemes the speed is obtai-ned based on the measurement of stator voltages andcurrents. However the estimation is ussually complexand heavily dependent on machine parameters. The-refore, although sensorless vector-controlled drives arecommercially available at this time, the parameter un-certainties imposes a challenge in the control perfor-mance.

This paper presents a new sensorless vector controlscheme consisting on the one hand of a speed estima-tion algorithm which overcomes the necessity of thespeed sensor and on the other hand of a novel variablestructure control law with an integral sliding surfacethat compensates the uncertainties that are presentin the system.

The closed loop stability of the proposed schemeis demonstrated using the Lyapunov stability theory,and the exponential convergence of the controlledspeed is provided.

1

This report is organized as follows. The rotorspeed estimation is introduced in Section 2. Then,the proposed variable structure robust speed controlis presented in Section 3. In the Section 4, some simu-lation results are presented. Finally some concludingremarks are stated in the last Section.

2 Calculation of the motorspeed

Many schemes (Abbondanti 1975) based on simpliedmotor models have been devised to sense the speed ofthe induction motor from measured terminal quanti-ties for control purposes. In order to obtain an ac-curate dynamic representation of the motor speed, itis necessary to base the calculation on the coupledcircuit equations of the motor.

Since the motor voltages and currents are measu-red in a stationary frame of reference, it is also con-venient to express these equations in that stationaryframe.From the stator voltage equations in the stationaryframe it is obtained (Bose 2001):

dr =LrLm

vds LrLm

(Rs + Lsd

d t)ids (1)

qr =LrLm

vqs LrLm

(Rs + Lsd

d t)iqs (2)

where is the ux linkage; L is the inductance; vis the voltage; R is the resistance; i is the currentand = 1 L2m/(LrLs) is the motor leakage coe-cient. The subscripts r and s denotes the rotor andstator values respectively refereed to the stator, andthe subscripts d and q denote the dq-axis componentsin the stationary reference frame.The rotor ux equations in the stationary frame are(Bose 2001):

dr =LmTr

ids wrqr 1Tr

dr (3)

qr =LmTr

iqs + wrdr 1Tr

qr (4)

where wr is the rotor electrical speed and Tr = Lr/Rris the rotor time constant.

The angle e of the rotor ux vector (r) in rela-tion to the d-axis of the stationary frame is dened asfollows:

e = arctan(

qrdr

)(5)

being its derivative:

e = we =drqr qrdr

2dr + 2qr(6)

Substituting the equations (3) and (4) in the equa-tion (6) it is obtained:

we = wr LmTr

(driqs qrids

2dr + 2qr

)(7)

Then Substituting the equations (6) in the equa-tion (7), and nding wr we obtain:

wr =12r

[drqr qrdr Lm

Tr(driqs qrids)

]

(8)where 2r =

2dr +

2qr.

Therefore, given a complete knowledge of the mo-tor parameters, the instantaneous speed wr can becalculated from the previous equation, where the sta-tor measured current and voltages, and the rotor uxestimated obtained from a rotor ux observer basedon equations (1) and (2) are employed.

3 Variable structure robustspeed control

In general, the mechanical equation of an inductionmotor can be written as:

Jwm + Bwm + TL = Te (9)

where J and B are the inertia constant and the vis-cous friction coecient of the induction motor systemrespectively; TL is the external load; wm is the rotormechanical speed in angular frequency, which is re-lated to the rotor electrical speed by wm = 2wr/pwhere p is the pole numbers and Te denotes the gene-rated torque of an induction motor, dened as (Bose2001):

Te =3p4

LmLr

(edrieqs eqrieds) (10)

where edr and eqr are the rotor-ux linkages, with the

subscript e denoting that the quantity is refereed tothe synchronously rotating reference frame; ieqs andieds are the stator currents, and p is the pole numbers.

The relation between the synchronously rotatingreference frame and the stationary reference frame isperformed by the so-called reverse Parks transforma-tion: xaxb

xc

=

sin(e) cos(e)sin(e 2/3) cos(e 2/3)

sin(e + 2/3) cos(e + 2/3)

[

xdxq

]

(11)where e is the angle position between the d-axix ofthe synchronously rotating and the stationary refe-rence frames, and it is assumed that the quantitiesare balanced.

2

Using the eld-orientation control principle (Bose2001) the current component ieds is aligned in the di-rection of the rotor ux vector r, and the currentcomponent ieqs is aligned in the direction perpendicu-lar to it. At this condition, it is satised that:

eqr = 0, edr = |r| (12)

Therefore, taking into account the previous re-sults, the equation of induction motor torque (10) issimplied to:

Te =3p4

LmLr

edrieqs = KT i

eqs (13)

where KT is the torque constant, and is dened asfollows:

KT =3p4

LmLr

e

dr (14)

where e

dr denotes the command rotor ux.

With the above mentioned proper eld orienta-tion, the dynamic of the rotor ux is given by (Bose2001):

dedrd t

+edrTr

=LmTr

ieds (15)

Then, the mechanical equation (9) becomes:

wm + awm + f = b ieqs (16)

where the parameter are dened as:

a =B

J, b =

KTJ

, f =TLJ

; (17)

Now, we are going to consider the previous me-chanical equation (16) with uncertainties as follows:

wm = (a +a)wm (f +f) + (b +b)ieqs (18)where the terms a, b and f represents the un-certainties of the terms a, b and f respectively.

Let us dene dene the tracking speed error asfollows:

e(t) = wm(t) wm(t) (19)where wm is the rotor speed command.

Taking the derivative of the previous equationwith respect to time yields:

e(t) = wm wm = a e(t) + u(t) + d(t) (20)where the following terms have been collected in thesignal u(t),

u(t) = b ieqs(t) awm(t) f(t) wm(t) (21)and the uncertainty terms have been collected in thesignal d(t),

d(t) = awm(t)f(t) +b ieqs(t) (22)

Now, we are going to dene the sliding variableS(t) with an integral component as:

S(t) = e(t) t0

(k a)e() d (23)

where k is a constant gain.

In order to obtain the speed trajectory tracking,the following assumption should be formulated:

(A 1) The gain k must be chosen so that the term(k a) is strictly negative, therefore k < 0.Then the sliding surface is dened as:

S(t) = e(t) t0

(k a)e() d = 0 (24)

The variable structure speed controller is designed as:

u(t) = k e(t) sgn(S) (25)where the k is the gain dened previously, is theswitching gain, S is the sliding variable dened in eqn.(23) and sgn() is the signum function.

In order to obtain the speed trajectory tracking,the following assumption should be formulated:

(A 2) The gain must be chosen so that |d(t)|for all time.

Theorem 1 Consider the induction motor given byequation (18). Then, if assumptions (A 1) and (A 2)are verified, the control law (25) leads the rotor me-chanical speed wm(t) so that the speed tracking errore(t) = wm(t) wm(t) tends to zero as the time tendsto infinity.

The proof of this theorem will be carried out usingthe Lyapunov stability theory.

Proof : Dene the Lyapunov function candidate:

V (t) =12S(t)S(t) (26)

Its time derivative is calculated as:

V (t) = S(t)S(t)=S [e (k a)e]=S [(a e + u + d) (k e a e)]=S [u + d k e]=S [k e sgn(S) + d k e]=S [d sgn(S)]( |d|)|S|0 (27)

It should be noted that the eqns. (23),(20) and(25), and the assumption (A 2) have been used in theproof.

3

dqe abc CurrentController

VSCController

Limiter

PWMInverter

IM

wrEstimator

iabc

we

e

vabc

FieldWeakening

idsCalculation

iqsids

edr

wr

wr

iqs

+

wr

e

iabc

Pulses

Figure 1: Block diagram of the proposed sliding-mode eld oriented control

Using the Lyapunovs direct method, since V (t)is clearly positive-denite, V (t) is negative deniteand V (t) tends to innity as S(t) tends to innity,then the equilibrium at the origin S(t) = 0 is globallyasymptotically stable. Therefore S(t) tends to zeroas the time t tends to innity. Moreover, all trajecto-ries starting o the sliding surface S = 0 must reachit in nite time and then will remain on this surface.This systems behavior once on the sliding surface isusually called sliding mode (Utkin 1993).

When the sliding mode occurs on the sliding sur-face (24), then S(t) = S(t) = 0, and therefore thedynamic behavior of the tracking problem (20) is equi-valently governed by the following equation:

S(t) = 0 e(t) = (k a)e(t) (28)

Then, under assumption (A 1), the tracking errore(t) converges to zero exponentially.

It should be noted that, a typical motion undersliding mode control consists of a reaching phase du-ring which trajectories starting o the sliding surfaceS = 0 move toward it and reach it in nite time, follo-wed by sliding phase during which the motion will beconned to this surface and the system tracking errorwill be represented by the reduced-order model (28),where the tracking error tends to zero.

Finally, the torque current command, iqs(t), canbe obtained directly substituting eqn. (25) in eqn.(21):

iqs(t) =1b[k e sgn(S) + awm + wm + f ] (29)

Therefore, the proposed variable structure speedcontrol resolves the speed tracking problem for the in-duction motor, with some uncertainties in mechanicalparameters and load torque.

4 Simulation Results

In this section we will study the speed regulation per-formance of the proposed sliding-mode eld orientedcontrol versus reference and load torque variations bymeans of simulation examples.

The block diagram of the proposed robust controlscheme is presented in gure 1.

The block VSC Controller represent the propo-sed sliding-mode controller, and it is implemented byequations (23), (29). The block limiter limits thecurrent applied to the motor windings so that it re-mains within the limit value, and it is implementedby a saturation function. The block dqe abc ma-kes the conversion between the synchronously rotatingand stationary reference frames, and is implementedby equation (11). The block Current Controller con-sists of a three hysteresis-band current PWM control,which is basically an instantaneous feedback currentcontrol method of PWM where the actual current(iabc) continually tracks the command current (iabc)within a hysteresis band. The block PWM Inverteris a six IGBT-diode bridge inverter with 780 V DCvoltage source. The block Field Weakening gives theux command based on rotor speed, so that the PWMcontroller does not saturate. The block ieds Calcula-tion provides the current reference ieds from the rotorux reference through the equation (15).

4

The block wr Estimator represent the proposedrotor speed and synchronous speed estimator, and isimplemented by the equations (8) and (6) respecti-vely. The block IM represents the induction motor.

The induction motor used in this case study isa 50 HP, 460 V, four pole, 60 Hz motor having thefollowing parameters: Rs = 0.087, Rr = 0.228,Ls = 35.5mH, Lr = 35.5mH, and Lm = 34.7mH.

The system has the following mechanical parame-ters: J = 1.662 kg.m2 and B = 0.1N.m.s. It is as-sumed that there is an uncertainty around 20 % inthe system parameters, that will be overcome by theproposed sliding control.

The following values have been chosen for the con-troller parameters, k = 100, = 30.

In this example the motor starts from a standstillstate and we want the rotor speed to follow a speedcommand that starts from zero and accelerates untilthe rotor speed is 90 rad/s. The system starts with aninitial load torque TL = 50N.m, and at time t = 1 sthe load torque steps from TL = 50N.m to TL =100N.m.

Figure 2 shows the desired rotor speed (dashedline) and the real rotor speed (solid line). As it maybe observed, the rotor speed track the desired speedin spite of system uncertainties. Moreover, the speedtracking is not aected by the load torque change atthe time t = 1 s, because when the sliding surface isreached (sliding mode) the system becomes insensitiveto the boundary external disturbances.

Figure 3 shows the current of one stator winding.This gure shows that in the initial state, the currentsignal presents a high value because it is necessary ahigh torque to increment the rotor speed. In the cons-tant speed region, the motor torque only has to com-pensate the friction and the load torque and so, thecurrent is lower. Finally, at time t = 1 s the currentincreases because the load torque has been increased.

Figure 4 shows the motor torque. As in the caseof the current (g. 3), the motor torque has a highinitial value speed acceleration zone, then the valuedecreases in a constant region and nally increasesdue to the load torque increment. In this gure itmay be seen that in the motor torque appears theso-called chattering phenomenon, however this highfrequency changes in the torque will be ltered by themechanical system inertia.

5 Conclusions

In this paper a sensorless sliding mode vector con-trol has been presented. The rotor speed estimator is

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 220

0

20

40

60

80

100

Time (s)

wm*

, w

m (ra

d/s)

wm*

wm

Figure 2: Reference and real rotor speed signals(rad/s)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2500

400

300

200

100

0

100

200

300

400

500

Time (s)

i sa (A

)

Figure 3: Stator Current isa (A)

based on stator voltage equations and rotor ux equa-tions in the stationary reference frame. It is propo-sed a variable structure control which has an integralsliding surface to relax the requirement of the acce-leration signal, that is usual in conventional slidingmode speed control techniques. Due to the nature ofthe sliding control this control scheme is robust underuncertainties caused by parameter error or by chan-ges in the load torque. The closed loop stability ofthe presented design has been proved thought Lyapu-nov stability theory. Finally, by means of simulationexamples, it has been shown that the proposed con-trol scheme performs reasonably well in practice, andthat the speed tracking objective is achieved underuncertainties in the parameters and load torque.

5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2100

50

0

50

100

150

200

250

300

350

Time (s)

T e (N

*m)

Figure 4: Motor torque (N.m)

Acknowledgments

The authors are grateful to the Basque Country Uni-versity and to MCYT for partial support of this workthrough the research projects 1/UPV 00146.363-E-13992/2001 and DPI2000-0244 respectively.

References

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[2] Benchaib, A. and Edwards, C., 2000, Nonli-near sliding mode control of an induction motor,Int. J. of Adaptive Control and Signal Procesing,14, 201-221.

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[4] Chern, T.L., Chang, J. and Tsai, K.L.,1998,Integral variable structure control based adaptivespeed estimator and resistance identier for aninduction motor. Int. J. of Control, 69, 31-47.

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[10] Ortega, R., Canudas, C. and Seleme, I.S.,1993, Nonlinear Control of Induction Motors:Torque Tracking with Unknown Load Disturban-ces, IEEE Tran. on Automat. Contr., 38, 1675-1680.

[11] Park M.H. and Kim, K.S., 1991, Chatteringreduction in the position contol of induction mo-tor using the sliding mode, IEEE Trans. PowerElectron., 6 317-325.

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[14] Schauder C., 1992, Adaptive Speed Identica-tion for Vector Control of Induction Motors wit-hout Rotational Transducers, IEEE Trans. In-dus. Applica., 28, 1054-1061.

[15] Utkin V.I., 1993, Sliding mode control de-sign principles and applications to electric drives,IEEE Trans. Indus. Electro., 40, 26-36.

[16] Vas, P., 1994, Vector Control of AC Machines.Oxford Science Publications, Oxford.

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