482 Tian-Jun Fu, Wen-Fang Xie / ISA Transactions 44, (2005) 481–490 Various SMC techniques for induction motors model of induction motor is given in Section 2 andhave been proposed in many literatures. The lin- SVM techniques in induction motor drives are dis-earization SMC techniques were suggested in cussed in Section 3. Details of sliding-mode con-Refs. ͓2,6,7͔. Linear reference models or input- troller design are given in Section 4, while theoutput linearization techniques were used in the simulation results are presented in Section 5. Fi-control of the nonlinear systems. A fuzzy SMC nally, some concluding remarks are given in Sec-method was developed in Ref. ͓3͔. SMC acts in a tion 6.transient state to enhance the stability, while fuzzytechnique functions in the steady state to reducechattering. In Refs. ͓8–10͔, the Lyapunov direct 2. Dynamic model of induction motormethod is used to ensure the reaching and sustain-ing of the sliding mode. These SMC methods re- A three-phase induction motor with squirrel-sult in a good transient performance, sound distur- cage rotor is considered in the paper. Assumingbance rejection, and strong robustness in a control that three-phase ac voltages are balanced and sta-system. However, the chattering is a problem in tor windings are uniformly distributed and basedSMC and causes the torque, ﬂux, and current on the well-known two-phase equivalent motorripple in the systems. In Ref. ͓9͔, sliding-mode representation, the nonsaturated symmetrical in-concepts were used to implement pulse width duction motor can be described in the ﬁxed coor-modulation ͑PWM͒. This implementation method dinate system ͑␣ , ␤͒ by a set of ﬁfth-order nonlin-is simple and efﬁcient by means of power inverter ear differential equations with respect to rotorsince both implementation of SMC and PWM im- velocity , the components of rotor magnetic ﬂuxply high-frequency switching. However, this ␣ , ␤, and of stator current i␣ , i␤ ͓4͔:method causes severe ripple in the torque signaldue to the irregular logic control signals for in- d␣ Rr Lm = − ␣ − ␤ + Rr i␣ ,verter. To overcome this problem, an rms torque- dt Lr Lrripple equation was developed in Ref. ͓11͔ tominimize torque ripple. In Ref. ͓12͔, a direct d␤ Rr Lmtorque control ͑DTC͒ is combined with space vec- = − ␤ + ␣ + Rr i␤ , dt Lr Lrtor modulation ͑SVM͒ techniques to improve thetorque, ﬂux, and current steady-state wave formsthrough ripple reduction. With the development of microprocessors, the di␣ = Lr 2 − dt LsLr − Lm ͩ Lm d␣ Lr dt − R si ␣ + u ␣ , ͪ ͩ ͪSVM technique has become one of the most im-portant PWM methods for voltage source inverter di␤ Lr Lm d␤ = 2 − − R si ␤ + u ␤ ,͑VSI͒. It uses the space vector concept to compute dt LsLr − Lm Lr dtthe duty cycle of the switches. It simpliﬁes thedigital implementation of PWM modulations. An d Paptitude for easy digital implementation and wide = ͑ T − T L͒ , dt Jlinear modulation range for output line-to-linevoltages are the notable features of SVM ͓13,14͔. 3P LmThus SVM becomes a potential technique to re- T= ͑i − i ͒ , ͑1͒ 2 Lr ␤ ␣ ␣ ␤duce the ripple in the torque signal. This paper presents a new sliding-mode control- where is the electrical rotor angle velocity; ler for torque regulation of induction motors. This = ͓␣␤͔T, i = ͓i␣i␤͔T, and u = ͓u␣u␤͔T are rotornovel control method integrates the speed sensor- ﬂux, stator current, and stator voltage in ͑␣ , ␤͒less SMC with the SVM technique. It replaces the coordinate, respectively; T and TL are the torquePWM component in the conventional SMC with of motor and load torque; J is the inertia of thethe SVM so that the torque ripple of induction rotor; P is the number of pole pairs. Rr and Rs aremotors is effectively reduced while the robustness rotor and stator resistances, Lr and Ls are rotor andis ensured at the same time. stator inductances, and Lm is the mutual induc- The paper is organized as follows. The dynamic tance.
Tian-Jun Fu, Wen-Fang Xie / ISA Transactions 44, (2005) 481–490 483 Fig. 1. Three-phase two-level PWM inverter.3. SVM techniques in induction motor drives Fig. 2. Space vectors. The SVM technique is the more preferablescheme to the PWM voltage source inverter since sponding to the rotation frequency of the vectorit gives a large linear control range, less harmonic ͓14͔. In order to reduce the number of switchingdistortion, and fast transient response ͓13,14͔. A actions and make full use of active turn-on timescheme of a three-phase two-level PWM inverter for space vectors, the vector us is commonly splitwith a star-connection load is shown in Fig. 1. into two nearest adjacent voltage vectors and zero In Fig. 1, uLi, i = 1,2,3, are pole voltages; ua , ub, vectors U0 and U7 in an arbitrary sector. For ex-and uc are phase voltages; uo is neutral point volt- ample, during one sampling interval, vector us inage; Vdc is the dc link voltage of PWM. Their sector I can be expressed asrelationships are T0 T1 T2 T7 1 u s͑ t ͒ = U0 + U1 + U2 + U7 , ͑3͒ uLi = ± Vdc, i = 1,2,3, TS TS TS TS 2 where TS is the sampling time, and TS − T1 − T2 1 = T0 + T7 ജ 0, T0 ജ 0, and T7 ജ 0. uo = ± Vdc , 6 The required time T1 to spend in active state U1 is given by the fraction of U1 mapped by the de- ua = uL1 − uo ;ub = uL2 − uo ;uc = uL3 − uo . ͑2͒ composition of the required space vector uS onto the U1 axis, shown in Fig. 2 as U1X. Therefore The SVM principle is based on the switchingbetween two adjacent active vectors and two zero ͉U1X͉vectors during one switching period ͓13͔. From T1 = T ͑4͒Fig. 1, the output voltages of the inverter can be ͉ U 1͉ Scomposed by eight switch states U0 , U1 , … , U7, and similarlycorresponding to the switch statesS0͑000͒ , S1͑100͒ , … , S7͑111͒, respectively. These ͉U2X͉vectors can be plotted on the complex plane ͑␣ , ␤͒ T2 = T . ͑5͒as shown in Fig. 2. ͉ U 2͉ S The rotating voltage vector within the six sec- From Fig. 2, the amplitude of vector U1X andtors can be approximated by sampling the vector U2X are obtained in terms of ͉us͉ and ,and switching between different inverter statesduring the sampling period. This will produce an ͉ u S͉ ͉U2X͉ ͉U1X͉ = = . ͑6͒ sin͑2 3͒ ր sin sin͑ 3 − ͒րapproximation of the sampled rotating space vec-tor. By continuously sampling the rotating vectorand high-frequency switching, the output of the Based on the above equations, the required timeinverter will be a series of pulses that have a domi- period spending in each of the active and zeronant fundamental sine-wave component, corre- states are given by
484 Tian-Jun Fu, Wen-Fang Xie / ISA Transactions 44, (2005) 481–490 t5 Ά · 3Tz Tm 4 + 2 + Tm+1 , sector = I,III,V; m = 1,3,5, respectively 3Tz Tn = + + Tm , sector = II,IV,VI; , 4 2 m = 2,4,6 and n = 3,5,1, respectively Fig. 3. Pulse command signal pattern. 3Tz t6 = + Tm + Tn . ͑8͒ 4 ͉uS͉sin͑ 3 − ͒ ր T1 = TS , ͉U1͉sin͑2 3͒ ր 4. Sliding-mode controller design ͉uS͉sin T2 = TS , ͉U2͉sin͑2 3͒ ր The objective of SMC design is to make the modulus of the rotor ﬂux vector r, and torque T T z = T 0 + T 7 = T s − ͑ T 1 + T 2͒ . ͑7͒ track to their reference value r and T*, respec- * tively. The pulse command signals pattern for the in-verter for Sector I can be constructed in Fig. 3. 4.1. Selection of the sliding surfaces Similarly, according to the vector sequence andtiming during a sampling interval given in Table 1, The transient dynamic response of the system isother ﬁve pulse command signal patterns, associ- dependent on the selection of the sliding surfaces.ated with sector II, sector III, …, sector VI can be The selection of the sliding surfaces is not unique.obtained. Hence the required time periods in a According to Ref. ͓15͔, the higher-order slidingsampling interval can be given as modes can be selected; however, it demands more information in implementation. Considering the Tz t1 = , SMC design for an induction motor supplied 4 through an inverter ͑Fig. 1͒, two sliding surfaces Ά · Tz Tm are deﬁned as 4 + 2 , sector = I,III,V; ˆ S1 = T* − T , ͑9͒ m = 1,3,5, respectively t2 = Tz Tn , 4 + 2 , sector = II,IV,VI; d * ˆ * ˆ S 2 = C ͑ r − r͒ + ͑ r − r͒ . ͑10͒ n = 3,5,1, respectively dt Tz Tm + Tn The positive constant C determines the convergent t3 = + , speed of rotor ﬂux. T* and r are the reference * 4 2 ˆ torque and reference rotor ﬂux, respectively. T and 3Tz Tm + Tn ˆ r are the estimated torque and rotor ﬂux, and t4 = + , 4 2 ˆ ͱ ˆ2 ˆ2 ˆ ˆ r = ␣ + ␤, where ␣ and ␤ are the estimated rotor ﬂux in ͑␣ , ␤͒ coordinate. Once the system isTable 1 driven into sliding surfaces, the system behaviorTime duration for selected vectors. will be determined by S1 = 0 and S2 = 0 in Eqs. ͑9͒ U0 U ma U na U7 Un Um U0 and ͑10͒. The objective of control design is to force the system into sliding surfaces so that theTz / 4 Tm / 2 Tn / 2 Tz / 2 Tn / 2 Tm / 2 Tz / 4 torque and rotor ﬂux signals will follow the re-a Um and Un are two adjacent voltage vectors. spective reference signals.
Tian-Jun Fu, Wen-Fang Xie / ISA Transactions 44, (2005) 481–490 4854.2. Invariant transformation of sliding surfaces transformation of a discontinuity surface has no effect upon the equivalent control value on the In order to simplify the design process, the time manifolds S = 0 or q = 0.derivative of sliding surfaces’ function S can bedecoupled with respect to two phase stator voltage 4.3. Selection of the control lawvectors u = ͓u␣u␤͔T. Projection of the systems mo-tion in the subspaces S1 and S2 can be written as The direct method of Lyapunov is used for the stability analysis. Considering the Lyapunov func- dS tion candidate = 0.5STS ജ 0, its time derivative is = F + Au, ͑11͒ dt = ST͑F + Au͒ . ˙ ͑16͒where F = ͓f 1 f 2͔T, u = ͓u␣u␤͔T, and S = ͓S1S2͔T. Select the control law as Functions f 1 , f 2, and matrix A can be obtained u␣ = − k1 sgn͑q1͒ − k2q1 , ͑17a͒as follows by differentiating structure switchingfunction ͑9͒ and ͑10͒ and substituting correspond- u␤ = − k1 sgn͑q2͒ − k2q2 , ͑17b͒ing relations from the mathematical model, where ˙ f 1 = T* + ͩ 3P 1 ˙ ˆ˙ ˆ 2 Rr r r ˆ2 ˆ ͪˆ · + L s r · + ␥ T , sgn͑q͒ = ͭ + 1,q Ͼ 0 − 1,q Ͻ 0 , ͮ ͑12͒ and k1,k2 2 ˆ T are positive constants. ˙* ¨* ˆ f 2 = C r + r + R rR s r − Rr ˆ 3P ˆ Theorem: Consider the induction motor ͑1͒, with r the developed sliding mode controller ͑17a͒ and − ͩ ͪ 2 3P 2 2 Rr ˆ T2 ˆ3 r + ͩ 2Rr Lr ˙ ˆ − C , ͪ ͑13͒ ͑17b͒ and stable sliding surfaces ͑9͒ and ͑10͒. If k1 , k2 are chosen so that ͑k1 + k2͉qi͉͒ Ͼ max͑f *͒, where i = 1, 2, the reaching condition of sliding i ˙ surface = STS Ͻ 0 is satisﬁed, and control system ͫ ͬ ˙ ˆ ˆ a 1 ␤ − a 1 ␣ will be stabilized. A= , ͑14͒ Proof: From the time derivative of Lyapunov ˆ ˆ a 2 ␣ a 2 ␤ function ͑16͒, the following equation can be de- rived:where = 1 / ͑LsLr − Lm͒, ␥ = LrRs + LsRr, a1 2 = ST͑F + Au͒ = ͑q1 f * − k1͉q1͉ − k2q2͒ ˙ 1 1 ˆ= ͑3P / 2͒Lm and a2 = −͑1 / r͒RrLm; is the es- ˆtimated rotor angle velocity. + ͑ q 2 f * − k 1͉ q 2͉ − k 2q 2͒ , 2 2 From Eqs. ͑12͒ and ͑13͒, it is noted that func- ͑18͒tions f 1 and f 2 do not depend on either u␣ or u␤.Therefore the transformed sliding surfaces, q where ͓f * f *͔ = ͑A−1F͒T. 1 2= ͓q1q2͔T, are introduced to simplify the design From Eq. ͑18͒, it is noted that if one choosesprocess and to construct the candidate Lyapunov ͑k1 + k2͉qi͉͒ Ͼ max͑f *͒, where i = 1 , 2, the time de- ifunction in the next subsection. Sliding surfaces q rivative of Lyapunov function Ͻ 0. Thus the ori- ˙and S are related by an invariant transformation: gin in the space q ͑and in the space S as well͒ is asymptotically stable, and the reaching condition q = ATS. ͑15͒ of sliding surface is guaranteed. The torque T andˆ Remark 1: According to Ref. ͓4͔, the purpose of ˆ rotor ﬂux r will approach to the reference torqueinvariant transformation is to choose the easiest and reference rotor ﬂux, respectively.implementation of the SMC technique from the Remark 2: From Eqs. ͑17a͒ and ͑17b͒, it is ob-entire set of feasible techniques. A linear invariant served that the control command u␣ is used to
486 Tian-Jun Fu, Wen-Fang Xie / ISA Transactions 44, (2005) 481–490 Fig. 4. The block diagram of SMC with SVM. Fig. 5. The block diagram of PI with SVM.force sliding mode occurring on the manifold q1 ing mode control method ͑SMC with SVM͒.= 0, while u␤ is used to force sliding mode occur- Meanwhile, the proposed control method has beenring on the manifold q2 = 0. The sliding mode oc- compared with the conventional SMC ͓9͔ andcurring on the manifold q = 0 is equivalent to its classical PI control method ͓16͔. The sliding-modeoccurrence on the manifold S = 0 ͓4͔. After the observer discussed in Ref. ͓5͔ is adopted to esti-sliding mode arises on the intersection of both sur- mate the rotor ﬂux and the torque of an induction ˆ * ˆ motor without using speed sensors. This observerfaces S1 = T* − T = 0 and S2 = C͑r − r͒ + ͑d / dt͒ has been proved to have good convergence and * ˆ ˆ ˆϫ͑r − r͒ = 0, then T = T* and r = *. Therefore a r asymptotic stability ͓9͔. The block diagrams ofcomplete decoupled control of torque and ﬂux is torque control of the induction motor are shown inachieved. Fig. 4 ͑SMC with SVM͒, Fig. 5 ͑PI with SVM͒, Remark 3: It is well known that sliding-mode and Fig. 6 ͑conventional SMC͒.techniques generate undesirable chattering and * * In Fig. 4, u␣ and u␤ are control signals, derivedcause the torque, ﬂux, and current ripple in the from the control law ͑17a͒ and ͑17b͒, andsystem. However, in the new control system, due ͱ͑u␣͒2 + ͑u␤͒2 = ͉us͉, = a tan͉͑u␤͉ / ͉u␣͉͒ ͑see Fig. * * * *to the SVM technique giving a large linear control 2͒. In Fig. 5, the parameters of the PI controllerrange and the regular logic control signals for in- are tuned by trial and error to achieve the “best”verter ͓13͔, which means less harmonic distortion, control performance. In Fig. 6, the inverter logicthe chattering can be effectively reduced. control signals are obtained through the SMC method while they are calculated by using SVM5. Simulations techniques in the proposed method ͑Fig. 4͒. This turns out to be the major difference between the In this section, simulation results are presented conventional SMC and the proposed SMC methodto show the performance of the proposed new slid- with SVM. Fig. 6. The block diagram of conventional SMC.
Tian-Jun Fu, Wen-Fang Xie / ISA Transactions 44, (2005) 481–490 487Table 2Induction motor nominal parameters.Ls= 590 H P=1Lr= 590 H J = 4.33e − 4 N m s2Lm= 555 H B = 0.04 N m s / radRs= 0.0106 ⍀ rated voltage= 24 VRr= 0.0118 ⍀ The simulations are implemented by using A Matlab S function is developed toMatlab/Simulink.implement the SVM block. A 10-kHz ﬁxedswitching frequency for the inverter is used. ForSMC with SVM, parameters k1 and k2 are selectedas k1 = 0.1 and k2 = 0.3. The nominal parameters of the test inductionmotor are listed in Table 2.5.1. Simulation results of stator current, rotortorque and rotor ﬂux Figs. 7–9 show the stator current i␣, torque re- Fig. 8. Rotor ﬂux responses. ͑a͒ PI with SVM, ͑b͒ SMC, ͑c͒ SMC with SVM. sponses, and rotor ﬂux responses when the refer- ence torque signal is a rectangular wave with fre- quency 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 con- trol scheme has the most oscillation and biggest overshoot, while SMC with SVM has the least os- cillation and no overshoot. Due to the sudden change of stator current, two disturbances appear at 0.2 and 0.4 s in Figs. 8͑a͒ and 8͑c͒. However, noFig. 7. Stator current i␣. ͑a͒ PI with SVM, ͑b͒ SMC, ͑c͒ disturbances are found in Fig. 8͑b͒. This demon-SMC with SVM. strates the fact of the strong robustness of the con-
488 Tian-Jun Fu, Wen-Fang Xie / ISA Transactions 44, (2005) 481–490Fig. 9. Torque responses. ͑a͒ PI with SVM, ͑b͒ SMC, ͑c͒ Fig. 10. Torque responses with a sine-wave reference sig-SMC with SVM. nal. ͑a͒ PI with SVM, ͑b͒ SMC, ͑c͒ SMC with SVM.ventional SMC since it is used in the observer, the 5.2. Torque trackingcontroller, and even the PWM. Fig. 9 and Table 3 In order to test the torque tracking convergenceshow that, among three control methods, SMC to various reference torque signals, different kindswith SVM has the best torque tracking perfor- of waves are selected as the reference torque sig-mance with signiﬁcant reduced torque ripple. The nals. Figs. 10 and 11 show torque responses of thesimulation results demonstrate that the new con- three control methods when the reference torquetrol approach can achieve the exact decoupling of signals are sine wave and piecewise wave, respec-the motor torque and rotor ﬂux, and shows satis- tively.factory dynamic performance. From Figs. 10 and 11, it is noted that the pro- posed new control method exhibits high accuracyTable 3 in torque tracking when the reference torque sig-Comparison of the three control methods. nal is changed to different signals. Mean-square error Controllers of output torque Torque ripple 5.3. Load disturbances PI with SVM 0.637% ±12% SMC 0.284% ±8% To test the robustness of the developed controlSMC with SVM 0.004% ±0.85% method, the external load disturbance has been in- troduced to the proposed control system. Fig. 12
Tian-Jun Fu, Wen-Fang Xie / ISA Transactions 44, (2005) 481–490 489Fig. 11. Torque responses with a piecewise wave referencesignal. ͑a͒ PI with SVM, ͑b͒ SMC, ͑c͒ SMC with SVM.shows torque and speed responses of three controlmethods when external load disturbance is a band-limited white noise with 6.25e − 5 noise power. From Fig. 12, it is demonstrated that the torqueresponse of the proposed new control system isinsensitive to external load perturbation. Althoughthe speed has small oscillation because of the dis-turbance, the new control system is stable, andstrong robust. Fig. 12. Torque and speed responses with disturbance. ͑a͒6. Conclusions PI with SVM, ͑b͒ SMC, ͑c͒ SMC with SVM. In this paper, a novel SMC approach integrating ventional SMC method, this new scheme has lowwith the SVM technique for an induction motor torque ripple, low current distortion, and high-has been presented. Complete decoupled control performance dynamic characteristics. Moreover,of torque and ﬂux is obtained and signiﬁcant this new control scheme can achieve high accu-torque ripple reduction is achieved. Comparing racy in torque tracking to various reference torquewith the classical PI control method and the con- signals and shows very strong robustness to exter-
490 Tian-Jun Fu, Wen-Fang Xie / ISA Transactions 44, (2005) 481–490nal load disturbances. Therefore the proposed space-vector modulation in a high-performance sen-novel control method is simple, accurate, and ro- sorless AC drive. IEEE Trans. Ind. Appl. 40, 170–176bust. ͑2004͒. ͓13͔ Holtz, J., Pulsewidth modulation for electronic power conversion. Proc. IEEE 82, 1194–1213 ͑1994͒.Acknowledgment ͓14͔ Zhou, K. and Wang, D., Relationship between space- vector modulation and three-phase carrier-based The project was supported by the Faculty Re- PWM: A comprehensive analysis. IEEE Trans. Ind.search Development Program from Concordia Electron. 49, 186–196 ͑2002͒.University. The authors would like to thank the ͓15͔ Perruquetti, W., et al., Sliding Mode Control in Engi-reviewers for comments and suggestions. neering. Marcel Dekker, Inc., New York, 2002. ͓16͔ Tursini, M., Petrella, R., and Parasiliti, F., AdaptiveReferences sliding-mode observer for speed-sensorless control of induction motors. IEEE Trans. Ind. Appl. 36, 1380– ͓1͔ Rodic, M. and Jezernik, K., Speed-sensorless sliding- 1387 ͑2000͒. mode torque control of an induction motor. IEEE Trans. Ind. Electron. 49, 87–95 ͑2002͒. Tian-Jun Fu received the B.S. ͓2͔ Chen, F. and Dunnigan, M. W., Sliding-mode torque degree in electrical engineering and ﬂux control of an induction machine. IEE Proc.: from Shenyang University of Electr. Power Appl. 150, 227–236 ͑2003͒. Technology, China, in 1988. He ͓3͔ Barrero, F., Gonzalez, A., Torralba, A., Galvan, E., had been working as a senior en- and Franquelo, L. G., Speed control of induction mo- gineer and project manager in tors using a novel fuzzy sliding-mode structure. IEEE several electric motor companies in China from 1988 to 2002. He is Trans. Fuzzy Syst. 10, 375–383 ͑2002͒. currently working toward the ͓4͔ Utkin, Vadim I., Sliding Modes in Control and Opti- M.A.Sc. degree in mechanical mization. Springer-Verlag, Berlin, 1992. and industrial engineering at Con- ͓5͔ Utkin, Vadim I., Sliding mode control design prin- cordia University, Canada. His re- ciples and applications to electric drives. IEEE Trans. search interests include control Ind. Electron. 40, 23–36 ͑1993͒. theory applications, electrical ma- ͓6͔ Shieh, Hsin-Jang and Shyu, Kuo-Kai, Nonlinear chine drives, power electronics, and hybrid electric vehicle control. sliding-mode torque control with adaptive backstep- ping approach for induction motor drive. IEEE Trans. Ind. Electron. 46, 380–389 ͑1999͒. Wen-Fang Xie is an assistant ͓7͔ Benchaib, A., Rachid, A., and Audrezet, E., Sliding professor with the Department of mode input-output linearization and ﬁeld orientation Mechanical and Industrial Engi- for real-time control of induction motors. IEEE Trans. neering at Concordia University, Canada. She was an Industrial Power Electron. 14, 3–13 ͑1999͒. Research Fellowship holder from ͓8͔ Soto, Rogelio and Yeung, Kai S., Sliding-mode con- Natural Sciences and Engineering trol of induction motor without ﬂux measurement. Research Council of Canada and IEEE Trans. Ind. Appl. 31, 744–750 ͑1995͒. served as a senior research engi- ͓9͔ Yan, Zhang, Jin, Changxi, and Utkin, V. I., Sensorless neer in InCoreTec, Inc. Canada sliding-mode control of induction motors. IEEE Trans. before she joined Concordia Uni- Ind. Electron. 47, 1286–1297 ͑2000͒. versity. She had worked as a re-͓10͔ Benchaib, A., Rachid, A., and Audrezet, E., Real-time search fellow in Nanyang Techno- sliding-mode observer and control of an induction mo- logical University, Singapore from 1999 to 2001. She received her Ph.D. from The Hong Kong tor. IEEE Trans. Ind. Electron. 46, 128–137 ͑1999͒. Polytechnic University in 1999 and her Masters degree from͓11͔ Kang, Jun-Koo and Sul, Seung-Ki, New direct torque Beijing University of Aeronautics and Astronautics in 1991. Her control of induction motor for minimum torque ripple research interests include nonlinear control in mechatronics, artiﬁ- and constant switching frequency. IEEE Trans. Ind. cial intelligent control, induction motor control, advanced process Appl. 35, 1076–1082 ͑1999͒. control, image processing, and pattern recognition.͓12͔ Lascu, C. and Trzynadlowski, A. M., Combining the principles of sliding mode, direct torque control, and