A novel sliding-mode control of induction motor using space vector modulation technique

ISA TRANSACTIONS® ISA Transactions 44 共2005兲 481–490

A novel sliding-mode control of induction motor using space vector modulation technique Tian-Jun Fu* Wen-Fang Xie† Department of Mechanical & Industrial Engineering, Concordia University, Montreal, Quebec H3G 1M8, Canada

共Received 20 September 2004; accepted 27 March 2005兲

Abstract This paper presents a novel sliding-mode control method for torque control of induction motors. The control principle is based on sliding-mode control combined with space vector modulation technique. The sliding-mode control contributes to the robustness of induction motor drives, and the space vector modulation improves the torque, flux, and current steady-state performance by reducing the ripple. The Lyapunov direct method is used to ensure the reaching and sustaining of sliding mode and stability of the control system. The performance of the proposed system is compared with those of conventional sliding-mode controller and classical PI controller. Finally, computer simulation results show that the proposed control scheme provides robust dynamic characteristics with low torque ripple. © 2005 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Sliding-mode control; Induction motor; Space vector modulation

1. Introduction The induction motor is widely used in industry, mainly due to its rigidness, maintenance-free operation, and relatively low cost. In contrast to the commutation dc motor, it can be used in aggressive or volatile environments since there are no risks of corrosion or sparks. However, induction motors constitute a theoretically challenging control problem since the dynamical system is nonlinear, the electric rotor variables are not measurable, and the physical parameters are most often imprecisely known. The control of the induction motor has attracted much attention in the past few decades; especially the speed sensorless control of induction motors has been a popular area due to its low cost and strong robustness 关1兴. *E-mail address: [email protected] †

E-mail address: [email protected]

Classical PI controller is a simple method used in control of induction motor drives. However, the main drawbacks of PI controller are the sensitivity of performance to the system-parameter variations and inadequate rejection of external disturbances and load changes 关2,3兴. Sliding-mode control 共SMC兲 is a robust control since the high gain feedback control input suppresses the influence of the disturbances and uncertainties 关4兴. Due to its order reduction, good disturbance rejection, strong robustness, and simple hardware/software implementation by means of power inverter, SMC has attracted much attention in the electric drive industry, and becomes one of the prospective control methodologies for induction motor drives 关5兴. The applications of SMC to electric motors have been previously investigated by Utkin in Refs. 关4,5兴, where the author gives the basic concepts, mathematics, and design aspects of variable structure systems, as well as sliding mode as a principle operation mode.

0019-0578/2005/$ - see front matter © 2005 ISA—The Instrumentation, Systems, and Automation Society.

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Various SMC techniques for induction motors have been proposed in many literatures. The linearization SMC techniques were suggested in Refs. 关2,6,7兴. Linear reference models or inputoutput linearization techniques were used in the control of the nonlinear systems. A fuzzy SMC method was developed in Ref. 关3兴. SMC acts in a transient state to enhance the stability, while fuzzy technique functions in the steady state to reduce chattering. In Refs. 关8–10兴, the Lyapunov direct method is used to ensure the reaching and sustaining of the sliding mode. These SMC methods result in a good transient performance, sound disturbance rejection, and strong robustness in a control system. However, the chattering is a problem in SMC and causes the torque, flux, and current ripple in the systems. In Ref. 关9兴, sliding-mode concepts were used to implement pulse width modulation 共PWM兲. This implementation method is simple and efficient by means of power inverter since both implementation of SMC and PWM imply high-frequency switching. However, this method causes severe ripple in the torque signal due to the irregular logic control signals for inverter. To overcome this problem, an rms torqueripple equation was developed in Ref. 关11兴 to minimize torque ripple. In Ref. 关12兴, a direct torque control 共DTC兲 is combined with space vector modulation 共SVM兲 techniques to improve the torque, flux, and current steady-state wave forms through ripple reduction. With the development of microprocessors, the SVM technique has become one of the most important PWM methods for voltage source inverter 共VSI兲. It uses the space vector concept to compute the duty cycle of the switches. It simplifies the digital implementation of PWM modulations. An aptitude for easy digital implementation and wide linear modulation range for output line-to-line voltages are the notable features of SVM 关13,14兴. Thus SVM becomes a potential technique to reduce the ripple in the torque signal. This paper presents a new sliding-mode controller for torque regulation of induction motors. This novel control method integrates the speed sensorless SMC with the SVM technique. It replaces the PWM component in the conventional SMC with the SVM so that the torque ripple of induction motors is effectively reduced while the robustness is ensured at the same time. The paper is organized as follows. The dynamic

model of induction motor is given in Section 2 and SVM techniques in induction motor drives are discussed in Section 3. Details of sliding-mode controller design are given in Section 4, while the simulation results are presented in Section 5. Finally, some concluding remarks are given in Section 6.

2. Dynamic model of induction motor A three-phase induction motor with squirrelcage rotor is considered in the paper. Assuming that three-phase ac voltages are balanced and stator windings are uniformly distributed and based on the well-known two-phase equivalent motor representation, the nonsaturated symmetrical induction motor can be described in the fixed coordinate system 共␣ , ␤兲 by a set of fifth-order nonlinear differential equations with respect to rotor velocity ␻, the components of rotor magnetic flux ␺␣ , ␺␤, and of stator current i␣ , i␤ 关4兴:

Lm d␺␣ Rr = − ␺␣ − ␻␺␤ + Rr i␣ , Lr dt Lr Lm Rr d␺␤ = − ␺␤ + ␻␺␣ + Rr i␤ , Lr dt Lr

冉 冉

冊 冊

di␣ Lm d␺␣ Lr − R si ␣ + u ␣ , = 2 − dt LsLr − Lm Lr dt Lm d␺␤ Lr di␤ − R si ␤ + u ␤ , = 2 − dt LsLr − Lm Lr dt d␻ P = 共 T − T L兲 , dt J T=

3P Lm 共i ␺ − i ␺ 兲 , 2 Lr ␤ ␣ ␣ ␤

共1兲

where ␻ is the electrical rotor angle velocity; ␺ = 关␺␣␺␤兴T, i = 关i␣i␤兴T, and u = 关u␣u␤兴T are rotor flux, stator current, and stator voltage in 共␣ , ␤兲 coordinate, respectively; T and TL are the torque of motor and load torque; J is the inertia of the rotor; P is the number of pole pairs. Rr and Rs are rotor and stator resistances, Lr and Ls are rotor and stator inductances, and Lm is the mutual inductance.

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483

Fig. 1. Three-phase two-level PWM inverter.

3. SVM techniques in induction motor drives The SVM technique is the more preferable scheme to the PWM voltage source inverter since it gives a large linear control range, less harmonic distortion, and fast transient response 关13,14兴. A scheme of a three-phase two-level PWM inverter with a star-connection load is shown in Fig. 1. In Fig. 1, uLi, i = 1,2,3, are pole voltages; ua , ub, and uc are phase voltages; uo is neutral point voltage; Vdc is the dc link voltage of PWM. Their relationships are

1 uLi = ± Vdc, 2

i = 1,2,3,

1 uo = ± Vdc , 6 ua = uL1 − uo ;ub = uL2 − uo ;uc = uL3 − uo . 共2兲 The SVM principle is based on the switching between two adjacent active vectors and two zero vectors during one switching period 关13兴. From Fig. 1, the output voltages of the inverter can be composed by eight switch states U0 , U1 , … , U7, corresponding to the switch states S0共000兲 , S1共100兲 , … , S7共111兲, respectively. These vectors can be plotted on the complex plane 共␣ , ␤兲 as shown in Fig. 2. The rotating voltage vector within the six sectors can be approximated by sampling the vector and switching between different inverter states during the sampling period. This will produce an approximation of the sampled rotating space vector. By continuously sampling the rotating vector and high-frequency switching, the output of the inverter will be a series of pulses that have a dominant fundamental sine-wave component, corre-

Fig. 2. Space vectors.

sponding to the rotation frequency of the vector 关14兴. In order to reduce the number of switching actions and make full use of active turn-on time for space vectors, the vector us is commonly split into two nearest adjacent voltage vectors and zero vectors U0 and U7 in an arbitrary sector. For example, during one sampling interval, vector us in sector I can be expressed as

u s共 t 兲 =

T0 T1 T2 T7 U0 + U1 + U2 + U7 , TS TS TS TS

共3兲

where TS is the sampling time, and TS − T1 − T2 = T0 + T7 艌 0, T0 艌 0, and T7 艌 0. The required time T1 to spend in active state U1 is given by the fraction of U1 mapped by the decomposition of the required space vector uS onto the U1 axis, shown in Fig. 2 as U1X. Therefore

T1 =

兩U1X兩 T 兩 U 1兩 S

共4兲

T2 =

兩U2X兩 T . 兩 U 2兩 S

共5兲

and similarly

From Fig. 2, the amplitude of vector U1X and U2X are obtained in terms of 兩us兩 and ␪,

兩 u S兩

sin共2␲ 3兲

Ⲑ

=

兩U1X兩 兩U2X兩 = . sin ␪ sin共␲ 3 − ␪兲

Ⲑ

共6兲

Based on the above equations, the required time period spending in each of the active and zero states are given by

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t5

=

冦

Fig. 3. Pulse command signal pattern.

T1 =

兩uS兩sin共␲ 3 − ␪兲

T2 =

Ⲑ

兩U1兩sin共2␲ 3兲

Ⲑ

兩uS兩sin ␪

兩U2兩sin共2␲ 3兲

Ⲑ

t2 =

冦

+

Tm 2 ,

+

Tn 2 ,

TS , 共7兲

Tz , 4

sector = I,III,V; sector = II,IV,VI;

冧

m = 1,3,5, respectively 3Tz 4

+ Tm ,

sector = II,IV,VI; m = 2,4,6 and n = 3,5,1, respectively 3Tz + Tm + Tn . 4

冧

,

共8兲

n = 3,5,1, respectively t3 =

Tz Tm + Tn + , 4 2

t4 =

3Tz Tm + Tn + , 4 2

The objective of SMC design is to make the modulus of the rotor flux vector ␺r, and torque T track to their reference value ␺r* and T*, respectively. 4.1. Selection of the sliding surfaces The transient dynamic response of the system is dependent on the selection of the sliding surfaces. The selection of the sliding surfaces is not unique. According to Ref. 关15兴, the higher-order sliding modes can be selected; however, it demands more information in implementation. Considering the SMC design for an induction motor supplied through an inverter 共Fig. 1兲, two sliding surfaces are defined as

共9兲

d S2 = C共␺r* − ␺ˆ r兲 + 共␺r* − ␺ˆ r兲 . dt

共10兲

The positive constant C determines the convergent speed of rotor flux. T* and ␺r* are the reference torque and reference rotor flux, respectively. Tˆ and ␺ˆ r are the estimated torque and rotor flux, and ␺ˆ r = ␺ˆ ␣2 + ␺ˆ ␤2 , where ␺ˆ ␣ and ␺ˆ ␤ are the estimated rotor flux in 共␣ , ␤兲 coordinate. Once the system is driven into sliding surfaces, the system behavior will be determined by S1 = 0 and S2 = 0 in Eqs. 共9兲 and 共10兲. The objective of control design is to force the system into sliding surfaces so that the torque and rotor flux signals will follow the respective reference signals.

冑

U0

U ma

U na

U7

Un

Um

U0

Tz / 4

Tm / 2

Tn / 2

Tz / 2

Tn / 2

Tm / 2

Tz / 4

Um and Un are two adjacent voltage vectors.

S1 = T* − Tˆ , ,

Table 1 Time duration for selected vectors.

a

+

Tn 2

+ Tm+1 , sector = I,III,V;

4. Sliding-mode controller design

m = 1,3,5, respectively Tz 4

Tm 2

TS ,

The pulse command signals pattern for the inverter for Sector I can be constructed in Fig. 3. Similarly, according to the vector sequence and timing during a sampling interval given in Table 1, other five pulse command signal patterns, associated with sector II, sector III, …, sector VI can be obtained. Hence the required time periods in a sampling interval can be given as

Tz 4

+

t6 =

T z = T 0 + T 7 = T s − 共 T 1 + T 2兲 .

t1 =

3Tz 4

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4.2. Invariant transformation of sliding surfaces In order to simplify the design process, the time derivative of sliding surfaces’ function S can be decoupled with respect to two phase stator voltage vectors u = 关u␣u␤兴T. Projection of the systems motion in the subspaces S1 and S2 can be written as

dS = F + Au, dt

共11兲

where F = 关f 1 f 2兴T, u = 关u␣u␤兴T, and S = 关S1S2兴T. Functions f 1 , f 2, and matrix A can be obtained as follows by differentiating structure switching function 共9兲 and 共10兲 and substituting corresponding relations from the mathematical model,

冉

冊

transformation of a discontinuity surface has no effect upon the equivalent control value on the manifolds S = 0 or q = 0. 4.3. Selection of the control law The direct method of Lyapunov is used for the stability analysis. Considering the Lyapunov function candidate ␯ = 0.5STS 艌 0, its time derivative is

␯˙ = ST共F + Au兲 .

共16兲

Select the control law as

u␣ = − k1 sgn共q1兲 − k2q1 ,

共17a兲

u␤ = − k1 sgn共q2兲 − k2q2 ,

共17b兲

where

3P 1 ˙ˆ ˆ˙ f 1 = T˙* + ␺ · ␺ + ␴Ls␺ˆ r2 · ␻ˆ + ␴␥Tˆ , 2 Rr r r 共12兲

485

sgn共q兲 =

再

冎

+ 1,q ⬎ 0 , − 1,q ⬍ 0

and

k1,k2 f 2 = C␺˙ r* + ␺¨ r* + ␴RrRs␺ˆ r −

−

冉 冊 2 3P

A=

2

Rr2

冋

冉

2 Tˆ Rr ␻ˆ 3P ␺ˆ

冊

r

2Rr Tˆ2 ˙ + − C ␺ˆ , ˆ␺3 Lr r

册

a1␺ˆ ␤ − a1␺ˆ ␣ , a2␺ˆ ␣ a2␺ˆ ␤

共13兲

共14兲

2 where ␴ = 1 / 共LsLr − Lm 兲 , ␥ = L rR s + L sR r, a 1 ˆ is the es= 共3P / 2兲␴Lm and a2 = −共1 / ␺ˆ r兲␴RrLm; ␻ timated rotor angle velocity. From Eqs. 共12兲 and 共13兲, it is noted that functions f 1 and f 2 do not depend on either u␣ or u␤. Therefore the transformed sliding surfaces, q = 关q1q2兴T, are introduced to simplify the design process and to construct the candidate Lyapunov function in the next subsection. Sliding surfaces q and S are related by an invariant transformation:

q = ATS.

共15兲

Remark 1: According to Ref. 关4兴, the purpose of invariant transformation is to choose the easiest implementation of the SMC technique from the entire set of feasible techniques. A linear invariant

are positive constants. Theorem: Consider the induction motor 共1兲, with the developed sliding mode controller 共17a兲 and 共17b兲 and stable sliding surfaces 共9兲 and 共10兲. If k1 , k2 are chosen so that 共k1 + k2兩qi兩兲 ⬎ max共f *i 兲, where i = 1, 2, the reaching condition of sliding surface ␯˙ = STS˙ ⬍ 0 is satisfied, and control system will be stabilized. Proof: From the time derivative of Lyapunov function 共16兲, the following equation can be derived:

␯˙ = ST共F + Au兲 = 共q1 f *1 − k1兩q1兩 − k2q21兲 + 共q2 f *2 − k1兩q2兩 − k2q22兲 , 共18兲 where 关f *1 f *2兴 = 共A−1F兲T. From Eq. 共18兲, it is noted that if one chooses 共k1 + k2兩qi兩兲 ⬎ max共f *i 兲, where i = 1 , 2, the time derivative of Lyapunov function ␯˙ ⬍ 0. Thus the origin in the space q 共and in the space S as well兲 is asymptotically stable, and the reaching condition of sliding surface is guaranteed. The torque Tˆ and rotor flux ␺ˆ r will approach to the reference torque and reference rotor flux, respectively. Remark 2: From Eqs. 共17a兲 and 共17b兲, it is observed that the control command u␣ is used to

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Fig. 4. The block diagram of SMC with SVM.

force sliding mode occurring on the manifold q1 = 0, while u␤ is used to force sliding mode occurring on the manifold q2 = 0. The sliding mode occurring on the manifold q = 0 is equivalent to its occurrence on the manifold S = 0 关4兴. After the sliding mode arises on the intersection of both surfaces S1 = T* − Tˆ = 0 and S2 = C共␺r* − ␺ˆ r兲 + 共d / dt兲 ⫻共␺r* − ␺ˆ r兲 = 0, then Tˆ = T* and ␺ˆ r = ␺*r . Therefore a complete decoupled control of torque and flux is achieved. Remark 3: It is well known that sliding-mode techniques generate undesirable chattering and cause the torque, flux, and current ripple in the system. However, in the new control system, due to the SVM technique giving a large linear control range and the regular logic control signals for inverter 关13兴, which means less harmonic distortion, the chattering can be effectively reduced. 5. Simulations In this section, simulation results are presented to show the performance of the proposed new slid-

Fig. 5. The block diagram of PI with SVM.

ing mode control method 共SMC with SVM兲. Meanwhile, the proposed control method has been compared with the conventional SMC 关9兴 and classical PI control method 关16兴. The sliding-mode observer discussed in Ref. 关5兴 is adopted to estimate the rotor flux and the torque of an induction motor without using speed sensors. This observer has been proved to have good convergence and asymptotic stability 关9兴. The block diagrams of torque control of the induction motor are shown in Fig. 4 共SMC with SVM兲, Fig. 5 共PI with SVM兲, and Fig. 6 共conventional SMC兲. In Fig. 4, u␣* and u␤* are control signals, derived from the control law 共17a兲 and 共17b兲, and 冑共u␣* 兲2 + 共u␤* 兲2 = 兩us兩, ␪ = a tan共兩u␤* 兩 / 兩u␣* 兩兲 共see Fig. 2兲. In Fig. 5, the parameters of the PI controller are tuned by trial and error to achieve the “best” control performance. In Fig. 6, the inverter logic control signals are obtained through the SMC method while they are calculated by using SVM techniques in the proposed method 共Fig. 4兲. This turns out to be the major difference between the conventional SMC and the proposed SMC method with SVM.

Fig. 6. The block diagram of conventional SMC.

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487

Table 2 Induction motor nominal parameters. Ls= 590 ␮H Lr= 590 ␮H Lm= 555 ␮H Rs= 0.0106 ⍀ Rr= 0.0118 ⍀

P=1 J = 4.33e − 4 N m s2 B = 0.04 N m s / rad rated voltage= 24 V

The simulations are implemented by using A Matlab S function is developed to implement the SVM block. A 10-kHz fixed switching frequency for the inverter is used. For SMC with SVM, parameters k1 and k2 are selected as k1 = 0.1 and k2 = 0.3. The nominal parameters of the test induction motor are listed in Table 2. Matlab/Simulink.

5.1. Simulation results of stator current, rotor torque and rotor flux Figs. 7–9 show the stator current i␣, torque re-

Fig. 8. Rotor flux responses. 共a兲 PI with SVM, 共b兲 SMC, 共c兲 SMC with SVM.

Fig. 7. Stator current i␣. 共a兲 PI with SVM, 共b兲 SMC, 共c兲 SMC with SVM.

sponses, and rotor flux responses when the reference torque signal is a rectangular wave with frequency 2.5 Hz. Based on the simulation results shown in Fig. 9, the output torque comparison of three control methods is shown in Table 3. From Fig. 7, it is noted that the resulting current has the largest harmonic distortion for PI with SVM, and the smallest harmonic distortion for SMC with SVM. Fig. 8 shows that the estimated rotor flux tracks the reference input well in all three control methods, but PI with the SVM control scheme has the most oscillation and biggest overshoot, while SMC with SVM has the least oscillation and no overshoot. Due to the sudden change of stator current, two disturbances appear at 0.2 and 0.4 s in Figs. 8共a兲 and 8共c兲. However, no disturbances are found in Fig. 8共b兲. This demonstrates the fact of the strong robustness of the con-

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Fig. 9. Torque responses. 共a兲 PI with SVM, 共b兲 SMC, 共c兲 SMC with SVM.

Fig. 10. Torque responses with a sine-wave reference signal. 共a兲 PI with SVM, 共b兲 SMC, 共c兲 SMC with SVM.

ventional SMC since it is used in the observer, the controller, and even the PWM. Fig. 9 and Table 3 show that, among three control methods, SMC with SVM has the best torque tracking performance with significant reduced torque ripple. The simulation results demonstrate that the new control approach can achieve the exact decoupling of the motor torque and rotor flux, and shows satisfactory dynamic performance.

5.2. Torque tracking

Table 3 Comparison of the three control methods. Controllers

Mean-square error of output torque

Torque ripple

PI with SVM SMC SMC with SVM

0.637% 0.284% 0.004%

±12% ±8% ±0.85%

In order to test the torque tracking convergence to various reference torque signals, different kinds of waves are selected as the reference torque signals. Figs. 10 and 11 show torque responses of the three control methods when the reference torque signals are sine wave and piecewise wave, respectively. From Figs. 10 and 11, it is noted that the proposed new control method exhibits high accuracy in torque tracking when the reference torque signal is changed to different signals. 5.3. Load disturbances To test the robustness of the developed control method, the external load disturbance has been introduced to the proposed control system. Fig. 12

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489

Fig. 11. Torque responses with a piecewise wave reference signal. 共a兲 PI with SVM, 共b兲 SMC, 共c兲 SMC with SVM.

shows torque and speed responses of three control methods when external load disturbance is a bandlimited white noise with 6.25e − 5 noise power. From Fig. 12, it is demonstrated that the torque response of the proposed new control system is insensitive to external load perturbation. Although the speed has small oscillation because of the disturbance, the new control system is stable, and strong robust. 6. Conclusions

Fig. 12. Torque and speed responses with disturbance. 共a兲 PI with SVM, 共b兲 SMC, 共c兲 SMC with SVM.

In this paper, a novel SMC approach integrating with the SVM technique for an induction motor has been presented. Complete decoupled control of torque and flux is obtained and significant torque ripple reduction is achieved. Comparing with the classical PI control method and the con-

ventional SMC method, this new scheme has low torque ripple, low current distortion, and highperformance dynamic characteristics. Moreover, this new control scheme can achieve high accuracy in torque tracking to various reference torque signals and shows very strong robustness to exter-

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nal load disturbances. Therefore the proposed novel control method is simple, accurate, and robust. Acknowledgment The project was supported by the Faculty Research Development Program from Concordia University. The authors would like to thank the reviewers for comments and suggestions.

关13兴 关14兴

关15兴 关16兴

References 关1兴 Rodic, M. and Jezernik, K., Speed-sensorless slidingmode torque control of an induction motor. IEEE Trans. Ind. Electron. 49, 87–95 共2002兲. 关2兴 Chen, F. and Dunnigan, M. W., Sliding-mode torque and flux control of an induction machine. IEE Proc.: Electr. Power Appl. 150, 227–236 共2003兲. 关3兴 Barrero, F., Gonzalez, A., Torralba, A., Galvan, E., and Franquelo, L. G., Speed control of induction motors using a novel fuzzy sliding-mode structure. IEEE Trans. Fuzzy Syst. 10, 375–383 共2002兲. 关4兴 Utkin, Vadim I., Sliding Modes in Control and Optimization. Springer-Verlag, Berlin, 1992. 关5兴 Utkin, Vadim I., Sliding mode control design principles and applications to electric drives. IEEE Trans. Ind. Electron. 40, 23–36 共1993兲. 关6兴 Shieh, Hsin-Jang and Shyu, Kuo-Kai, Nonlinear sliding-mode torque control with adaptive backstepping approach for induction motor drive. IEEE Trans. Ind. Electron. 46, 380–389 共1999兲. 关7兴 Benchaib, A., Rachid, A., and Audrezet, E., Sliding mode input-output linearization and field orientation for real-time control of induction motors. IEEE Trans. Power Electron. 14, 3–13 共1999兲. 关8兴 Soto, Rogelio and Yeung, Kai S., Sliding-mode control of induction motor without flux measurement. IEEE Trans. Ind. Appl. 31, 744–750 共1995兲. 关9兴 Yan, Zhang, Jin, Changxi, and Utkin, V. I., Sensorless sliding-mode control of induction motors. IEEE Trans. Ind. Electron. 47, 1286–1297 共2000兲. 关10兴 Benchaib, A., Rachid, A., and Audrezet, E., Real-time sliding-mode observer and control of an induction motor. IEEE Trans. Ind. Electron. 46, 128–137 共1999兲. 关11兴 Kang, Jun-Koo and Sul, Seung-Ki, New direct torque control of induction motor for minimum torque ripple and constant switching frequency. IEEE Trans. Ind. Appl. 35, 1076–1082 共1999兲. 关12兴 Lascu, C. and Trzynadlowski, A. M., Combining the principles of sliding mode, direct torque control, and

space-vector modulation in a high-performance sensorless AC drive. IEEE Trans. Ind. Appl. 40, 170–176 共2004兲. Holtz, J., Pulsewidth modulation for electronic power conversion. Proc. IEEE 82, 1194–1213 共1994兲. Zhou, K. and Wang, D., Relationship between spacevector modulation and three-phase carrier-based PWM: A comprehensive analysis. IEEE Trans. Ind. Electron. 49, 186–196 共2002兲. Perruquetti, W., et al., Sliding Mode Control in Engineering. Marcel Dekker, Inc., New York, 2002. Tursini, M., Petrella, R., and Parasiliti, F., Adaptive sliding-mode observer for speed-sensorless control of induction motors. IEEE Trans. Ind. Appl. 36, 1380– 1387 共2000兲.

Tian-Jun Fu received the B.S. degree in electrical engineering from Shenyang University of Technology, China, in 1988. He had been working as a senior engineer and project manager in several electric motor companies in China from 1988 to 2002. He is currently working toward the M.A.Sc. degree in mechanical and industrial engineering at Concordia University, Canada. His research interests include control theory applications, electrical machine drives, power electronics, and hybrid electric vehicle control.

Wen-Fang Xie is an assistant professor with the Department of Mechanical and Industrial Engineering at Concordia University, Canada. She was an Industrial Research Fellowship holder from Natural Sciences and Engineering Research Council of Canada and served as a senior research engineer in InCoreTec, Inc. Canada before she joined Concordia University. She had worked as a research fellow in Nanyang Technological University, Singapore from 1999 to 2001. She received her Ph.D. from The Hong Kong Polytechnic University in 1999 and her Masters degree from Beijing University of Aeronautics and Astronautics in 1991. Her research interests include nonlinear control in mechatronics, artificial intelligent control, induction motor control, advanced process control, image processing, and pattern recognition.