Robust Control of Permanent Magnet Synchronous Motors

Transcription

1 IEEE/CAA JOURNA OF AUTOMATICA SINICA VO. 2 NO. 2 APRI Robust Control of Permanent Magnet Synchronous Motors Zhenhua Deng and Xiaohong Nian Abstract In this paper permanent magnet synchronous motors (PMSMs) are investigated. According to the feature of PMSMs a novel state equation of PMSMs is obtained by choosing suitable state variables. Based on the state equation robust controllers are designed via interval matrix and PI control idea. In terms of bilinear matrix inequations sufficient conditions for the existence of the robust controller are derived. In order to reduce the conservation and the dependence on parameter the control inputs of PMSMs are divided into two parts a feedforward control input and a feedback control input and relevant sufficient conditions for the existence of the controller are obtained. Because of the suitable choice of state variables the proposed control strategies can cope with the load uncertainty and have robustness for disturbance. Finally simulations are carried out via Matlab/Simulink soft to verify the effectiveness of the proposed control strategies. The performance of the proposed control strategies are demonstrated by the simulation results. Index Terms Bilinear matrix inequality (BMI) interval matrix permanent magnet synchronous motor (PMSM) robust control. I. INTRODUCTION PERMANENT magnet synchronous motors (PMSMs) have been widely used in the motion control field transmission control field servo system and so on thanks to theirs features of small volume high power low noise and high torque-to-current ratio. Compared with other motors the PMSMs have many advantages but theirs mathematical model has the characteristics of high order nonlinear strong coupling and multi-variables. Therefore it is difficult to design theirs controller. Because the PMSMs has an irreplaceable role in engineering application huge economic benefit will be achieved and good motion/traction performance will be obtained if it has been well controlled. An increasing number of scholars came up with all kind of control strategies in this field in recent decades of years. For example in order to deal with the Manuscript received April ; accepted June This work was supported by National Natural Science Foundation of China ( U113418) Ph. D. Programs Foundation of Ministry of Education of China ( ) and Science Foundation of Innovation Research Groups of National Natural Science Foundation of China (613213). Recommended by Associate Editor Zhiyong Geng. Citation: Zhenhua Deng Xiaohong Nian. Robust control of permanent magnet synchronous motors. IEEE/CAA Journal of Automatica Sinica 215 2(2): Zhenhua Deng is with the School of Information Science and Engineering Central South University Changsha 414 China and also with the Key ab of System and Control Academy of Mathematics and System Science Chinese Academy of Sciences Beijing 119 China ( deng dzh@163.com). Xiaohong Nian is with the School of Information Science and Engineering Central South University Changsha 414 China ( xhnian@csu.edu.cn). disturbance of parameter and load torque robust control 1 2 was developed. Adaptive control 3 4 has been researched so that the controller can adapt to the change of parameter and/or load. In 5-6 some performance indexes were optimized when predictive control was used. In order to get good output performance even the motor s parameters change with time model reference adaptive control 7 was introduced in this field. Sensorless controller 8 and observer-based 9 controller were designed to reduce the cost of controller and minimize the volume of controller. Fuzzy controller 1 11 was widely used to handle the structured and unstructured uncertainty. In order to deal with the disturbances slide mode controller 12 has been researched. Internal model control was utilized to improve the robustness of the whole system 13. In 14 torque feedforward controller was established to achieve good dynamic performance. Besides many other control strategies were developed such as near optimal control 15 and neural 1 13 network control 16. Recently hybrid control methods have been widely adopted. Interval matrix is often used to cope with systems parameters are unknown but vary (or locate) in a certain range Its mainly idea is an interval matrix whose elements change (or locate) in a specific range can be expressed by several matrices which can be further handled by other methods. For a given PMSM restricted by supply power operation environment protection equipment and so on current and speed are and must be in a specific range likes rated range. Therefore interval matrix can be introduced to design the controller of PMSMs. Besides although many control strategies have been developed in fact PI control method is still a main control strategy in engineering application because it is easy to implement and has a good control performance i.e. many advance control strategies have not been widely adopted in engineering because it is not easy to apply or its cost is higher than that of the conventional PI control method. In order to get a easy realized controller which can deal with load uncertainty in this paper robust controllers are developed via interval matrix and PI control idea. This paper is organized as follows. In Section II the mathematical model of PMSMs and lemmas are presented. Robust control strategies are developed in Section III. To verify the effectiveness of the proposed methods in section IV simulations are carried out via Matlab/Simulink soft. In Section V some conclusions are obtained. II. MODE DESCRIPTION AND EMMAS According to 19-2 the mathematical model of PMSMs can be expressed as

2 144 IEEE/CAA JOURNA OF AUTOMATICA SINICA VO. 2 NO. 2 APRI 215 di d di q Rs i d + ωi q + 1 u d ωi d Rs i q 1 ωψ r + 1 u q dω n2 p J ( )i d i q + n2 p J ψ ri q np J T l B f J ω are d- and q-axis inductance; R s is stator resistances; ψ r is permanent magnet flux linkage; n p is the number of pole pairs; B f is frictional coefficient; J is moment inertia; i d and i q are d- and q-axis stator currents; ω is rotor angular velocity; u d and u q are d- and q axis voltages and T l is load torque. Usually the model expressed as (1) could not accurately describe the actual model of PMSMs because of the existence of disturbance. For example resistance will increase with the rise of temperature. In other words disturbances must be considered in the model to make the model more accurately present the actual model. Thus the following model is established if disturbances are considered. di d di q Rs i d + ωi q + 1 u d + d d ωi d Rs i q 1 ωψ r + 1 u q + d q dω n2 p J ( )i d i q + n2 p J ψ ri q np J T l B f J ω + d ω (2) d d d q and d ω represent the total disturbances acting on d-axis q-axis and motion equations respectively which may be caused by the environment or the change of parameter. And in fact the disturbances and the load change very slowly i.e. the following equations usually are satisfied. dd d dd q dd ω dt l (1). (3) emma For any scalar ε > X and Y are real matrices with appropriate dimensions so that the following inequality is obtained. X T Y + Y T X 1 ε XT X + εy T Y. emma Given symmetric matrix ( ) S11 (x) S S(x) 12 (x) S12(x) T S 22 (x) and S 11 (x) R K K the following conditions are equivalent: 1) S(x) < ; 2) S 11 (x) < S 22 (x) S12(x)S T 11 1 (x)s 12(x) < ; 3) S 22 (x) < S 11 (x) S 12 (x)s22 1 (x)st 12(x) <. If matrices A m a m ij n n and A M a M ij n n satisfy a m ij am ij for all 1 i j n A m A M {a ij : a m ij a ij a M ij i i j n} is established. Assume A Rn n and A A m A M then A is called interval matrix. emma For a given interval matrices A A m A M and A R n n A can be rewritten as follows. A A + E G A 1 2 (AM + A m ) H h ij n n 1 2 (AM A m ). Elements of matrix A m A M are made up of the low bound and the upper bound of the elements in matrix A respectively. Obviously each element of matrix H is nonnegative. { R n2 n 2 diag { χ 11 χ 1n χ n1 χ nn } χ ij 1 i j 1 n} E h 11 e 1 h 1n e 1 hn1 e n hnn e n G h 11 e 1 h 1n e n h n1 e 1 h nn e n T n n 2. emma For any given interval matrix A A m A M matrix A 1 2 (AM + A m ) and H h ij n n 1 2 (AM A m ) A can be represented as A A + n ij1 e if ij e T j f ij h ij. Notations. I is identity matrix with appropriate dimension and e i is the ith column of unit matrix with appropriate dimension. X < Y X and Y are symmetric matrices means that the matrix Y X is positive definite. III. ROBUST CONTRO Although many advance control strategies have been developed in the PMSMs control field PI controller is still currently main control strategy in practical application because it is easy to implement and can get good control performance. Therefore in this section robust controllers of PMSMs will be developed via PI control idea. A. Robust Controller According to the character of PMSMs the state variables and control inputs can be chosen as X di i d di q d ω ω d(ω ω ) T U (4) T u d u q ω is the desired speed of the system and is a constant. According to (2) (4) the following state equation is obtained. Ẋ A(t)X + B U (5) 1 Rs A(t) a ij 5 5 ω n 2 p J ( )i q ω i q Rs ( i d + ψr 1 n 2 p J (( )i d + ψ r ) B J T 1 B sd 1. ) For a given PMSM which is constrained by the voltage source switch frequency protect facilities operating environment and so on current and speed are and must be in a certain

3 DENG AND NIAN: ROBUST CONTRO OF PERMANENT MAGNET SYNCHRONOUS MOTORS 145 range such as rated range. In other words current is not larger than rated current and speed is not higher than rated speed. Hence the following assumptions are reasonable. i m d i d i M d i m q i q i M q ω m ω ω M (6) i m d im d im q i M q ω m and ω M are the lower bound and the upper bound of i d i q and ω respectively. Based on condition (6) A(t) is viewed as interval matrix and according to emma 3 A(t) can be rewritten as A(t) A + E (t)m (t) (7) 1 Rs a c 23 a c 25 A c a c 32 Rs a c 35 1 a c 52 a c 53 B J a c 23 (ωm +ω m ) 2 a c 25 2( sd (i M q + i m q ) ) a c 32 (ω M +ω m ) a35 c sd (i M d +im d ) + ψr 2 2 a c 52 n2 p 2J ( )(i M q + i m q ) ( ) a c 53 n2 p J ( ) (im d +im d ) 2 + ψ r H h ij 5 5 sd (ω M ω m ) 2 n 2 p 2J (i M q i m q ) (ω M ω m ) 2 2 (i M q i m q ) sd (i M d im d ) 2 n 2 p 2J (i M d im d ) The following theorem can be obtained. Theorem 1. The PMSMs presented as (2) can be stabilized and the rotor angular speed can track the desired speed under the conditions (3) and (6) if there exist a symmetric and positive-definite matrix P 1 R 5 5 matrix K R 2 5 and positive scalar ε such that the following BMI holds: A T P 1 + P 1 A + K T B T P 1 + P 1 BK + εm T P M 1 E < (8) E T P 1 εi And the robust control law is U KX. Proof. The control law is designed as Choose yapunov function as follows. U KX. (9) V 1 X T P 1 X. (1) Taking the derivative of yapunov function (1) with respect to t along system (5) with (7) yields V 1 X T (A T P 1 + P 1 A + K T B T P 1 + P 1 BK)X+ X T M T T (t)e T P 1 X + X T P 1 E (t)mx. (11) Equation (11) can be written as follows via emma 1 and (t) T (t) I. V 1 X T Q 1 X (12) Q 1 A T P 1 +P 1 A +K T B T P 1 +P 1 BK +εm T M + P 1EE T P 1 ε. If Q 1 < the time derivative of yapunov function V 1 is negative which means the system can be stabilized by the controller and the PMSMs speed can follow the desired speed according to yapunov stability theory. And Q 1 < can be expressed as (8) by emma 2. There are six interval variable in matrix A(t) i.e. a 23 a 25 a 32 a 35 a 52 and a 53. The matrix A(t) can be rewritten as follows via emma 4 to reduce the conservative of BMI. A c (t) A c + e 2 f c 23e T 3 + e 2 f c 25e T 5 + e 3 f c 32e T 2 + e 3 f c 35e T 5 + e 5 f c 52e T 2 + e 5 f c 53e T 3 (13) f 23 h 23 f 25 h 25 f 32 h 32 f 35 h 35 f 52 h 52 and f 53 h 53. The following theorem can be obtained. Theorem 2. The PMSMs presented as (2) can be stabilized and the rotor angular speed can track the desired speed under the conditions (3) and (6) if there exist symmetric and positivedefinite matrix P 2 R 5 5 matrix K R 2 5 real scalar ε 23 > ε 25 > ε 32 > ε 35 > ε 52 > and ε 53 > such that the following BMI holds: AT P 2 + P c2 A + K T B T P 2 + Ξ P 2 BK + Ξ c2 1 < (14) Ξ T 2 Ξ 3 Ξ 1 ε 23 (h 23 ) 2 e 3 e T 3 + ε 25 (h 25 ) 2 e 5 e T 5 + ε 32 (h 32 ) 2 e 2 e T 2 + ε 35 (h 35 ) 2 e 5 e T 5 + ε 52 (h 52 ) 2 e 2 e T 2 + ε 53 (h 53 ) 2 e 3 e T 3 Ξ 2 P c2 e 2 P c2 e 2 P c2 e 3 P c2 e 3 P c2 e 5 P c2 e 5 Ξ 3 diag { ε 23 ε 25 ε 32 ε 35 ε 52 ε 53 } and the robust control law is U KX. Proof. yapunov function is designed as V 2 X T P 2 X. (15) Taking the derivative of yapunov function (15) with respect to t along system (5) with (13) yields V 2 X T (A T P 2 + P 2 A + K T B T P 2 + P 2 BK)X+ X T (e 2 f 23 e T 3 + e 2 f 25 e T 5 + e 3 f 32 e T 2 + e 3 f 35 e T 5 + e 5 f 52 e T 2 + e 5 f 53 e T 3 ) T P 2 X + X T P 2 (e 2 f 23 e T 3 + e 2 f 25 e T 5 + e 3 f 32 e T 2 + e 3 f 35 e T 5 + e 5 f 52 e T 2 + e 5 f 53 e T 3 )X. (16) According to emma 1 and f ij T f ij (h ij ) 2 the following inequality can be obtained. V 2 X T Q 2 X. (17)

4 146 IEEE/CAA JOURNA OF AUTOMATICA SINICA VO. 2 NO. 2 APRI 215 Q 2 A T P 2 + P 2 A + K T B T P 2 + P 2 BK + P 2e 2 e T 2 P 2 ε 23 + ε 23 (h 23 ) 2 e 3 e T 3 + P 2e 2 e T 2 P 2 + ε 25 (h 25 ) 2 e 5 e T 5 + ε 25 P 2 e 3 e T 3 P 2 + ε 32 (h 32 ) 2 e 2 e T 2 + P 2e 3 e T 3 P 2 + ε 32 ε 35 ε 35 (h 35 ) 2 e 5 e T 5 + P 2e 5 e T 5 P 2 ε 52 + ε 52 (h 52 ) 2 e 2 e T 2 + P 2 e 5 e T 5 P 2 ε 53 + ε 53 (h 53 ) 2 e 3 e T 3. If the inequality Q 2 < holds it stands the time derivative of yapunov function V 2 is negative which means the system can be stabilized by the controller and the PMSMs speed can follow the set point according to yapunov stability theory. Q 2 < can be written as (14) via emma 2. B. Robust Controller with Feedforward Compensation In order to further reduce the conservative and the dependence on parameter of system the control inputs can be divided into two parts i.e. ud u q udf u qf + udc u qc (18) u df i q ω u qf i d ω which can be viewed as feedforward compensator. And let U cf u dc u qc T. (19) According to (3) (4) (18) and (19) the system can be described as A cf (t) a cf ij Ẋ A cf (t)x + BU cf (2) Rs Rs ψr 1 a cf 52 a cf 53 B J a cf 52 n2 p J ( )i q a cf 53 n2 p J (( )i d + ψ r ). Condition (6) can be simplified as i m d i d i M d i m q i q i M q. (21) Similar to previous A cf (t) can be rewritten as A cf (t) A cf + E cf cf (t)m cf cf (t) cf (22) 1 Rs A cf Rs ψr 1 a cf 52 a cf 53 B J a cf 52 n2 p 2J ( )(i M q + i m q ) a cf 53 n2 p J (( ) (im d + im d ) 2 H cf ij ψ r ) n2 p 2J (i M q i m q ) 53 n2 p 2J (i M d i m d ) E cf 11 e 1 15 e 1 51 e 5 55 e 5 M cf 11 e 1 15 e 1 51 e 5 55 e 5 cf { cf (t) R cf (t) diag { χ cf 11 (t) χcf 15 (t) χ cf 51 (t) χcf 55 (t) } χ cf ij (t) 1. The following theorem can be obtained. Theorem 3. The PMSMs presented as (2) can be stabilized and the rotor angular speed can track the desired speed under the conditions (3) and (21) if there exist a symmetric and positive-definite matrix P cf1 R 5 5 matrix K cf R 2 5 and positive scalar λ such that the following BMI holds: A T cf P cf1 + P cf1 A cf + εmcf T M cf + Kcf T P BT P cf1 + P cf1 BK cf1 E cf cf < Ecf T P cf1 λi (23) udf and the robust control law is U + u K cf Xdτ. qf Proof. The proof is similar with previous. Here we omit it. Similar to previous A cf (t) can be rewritten as f cf 52 hcf 52 A cf (t) A cf + e 5 f cf 52 et 2 + e 5 f cf 53 et 3 (24) and f cf 53 hcf 53. The following theorem can be obtained. Theorem 4. The PMSMs presented as (2) can be stabilized and the rotor angular speed can track the desired speed under the conditions (3) and (21) if there exist symmetric and positive-definite matrix P cf2 R 5 5 matrix K cf R 2 5

5 DENG AND NIAN: ROBUST CONTRO OF PERMANENT MAGNET SYNCHRONOUS MOTORS 147 real scalar λ 52 > and λ 53 > such that the following BMI holds: A T cf P cf2 + P cf2 A cf + Kcf T Ξ BT P cf2 + P cf2 BK cf + Ξ cf2 cf1 < Ξ T cf2 Ξ cf3 (25) Ξ cf1 λ 52 ( 52 )2 e 2 e T 2 + λ 53 ( 53 )2 e 3 e T 3 Ξ cf2 Pc2 e 5 P c2 e 5 Ξcf3 diag { } λ 52 λ 53. And the udf robust control law is U + u K cf Xdτ. qf Proof. The proof is similar with previous. Here we omit it. Remark 1. The robust controller without feedforward compensator is U KX and it can be expressed as ud u P I d u cc u q u P q I + d q u P I d k c 11i d + u P I k c 12i d dτ d k c 13i q + k c 14(ω ω ) + q k c 24(ω ω ) + q k c 21i d + k c 25(ω ω )dτ k c 22i d dτ + k c 23i q. (26) k c 15(ω ω )dτ And kij c is the i-th row and j-th column element of matrix K. It is obvious that u P d I and u P q I can be viewed as PI controller and d ucc q act as feedback compensating controller. The control inputs of robust controller with feedforward compensator can be written as ud udf u P I + dc u cc u q u qf u P qc I + dc (27) qc u P I dc k cf 11 i d + k cf 12 i ddτ dc k cf 13 i q + k cf 14 (ω ω ) + u P I qc k cf 24 (ω ω ) + qc k cf 21 i d + k cf 25 (ω ω ) k cf 22 i ddτ + k cf 23 i q. k cf 15 (ω ω )dτ And k cf ij is the i-th row and j-th column element of matrix K cf. It is obvious that u P dc I and u P qc I can be viewed as PI controller and dc ucc qc act as feedback compensating controller. IV. SIMUATION In this section simulations will be done to verify the effectiveness of proposed control methods by Matlab/Simulink soft. Refer to 6 the parameters of the PMSM are given as: P rated 75 W ω rated 314 rad/s i rated 4.71 A T rated 2 N m.4 V.4 H R s 1.74 Ω ψ r.1167 Wb n p 4 B f N m s/rad and J kg/m 2. According to the parameter of PMSM current and speed of the PMSM satisfy i d 3 A i q 4 A ω 35 rad/s. The PWM switching frequency is 1 KHz and the DC-link voltage is 3 V. Solve inequality (8) with the help of Matlab soft the following equations can be obtained. K P ε It means the motor can be stabilized and the rotator angular speed can track the desired speed if Theorem 1 is correct and the robust control law (Method 1) are u d 7i d 1 i d dτ u q 2i q 7(ω ω ) 25 (ω ω )dτ. Only need conditions i d 3 A i q 4 A can the following equations be got via Theorem 3 with the help of Matlab soft. K cf P cf λ If Theorem 3 is effective the PMSM can be stabilized and the rotor angular speed can track the desired speed by the control law (proposed Method 2) which is described as follows. u d.4i q ω 7i d 1 i d dτ u q.4i d ω 2i q 7(ω ω ) 25 (ω ω )dτ. Next simulations are carried out via Matlab/Simulink soft to demonstrate the judgment of Theorems 1 and 3. In the

6 148 IEEE/CAA JOURNA OF AUTOMATICA SINICA VO. 2 NO. 2 APRI 215 practical application the desired speed may be needed to change to satisfy the requirements and the load torque is often influenced by the environment. Besides the parameters of motor are also with time. For example with the rise of temperature the resistance value will increase. In order to comprehensive test the proposed controllers three cases will be considered in the simulation. Case 1. 1) The PMSM s parameters are normal. 2) The desired angular speed: 157 rad/s 314 rad/s (.3 s) 157 rad/s (.7 s). 3) The load is constant and unknown. Case 2. 1) The PMSM s parameters are normal. 2) The desired angular speed ω : 314 rad/s. 3) The initial load is constant and unknown and the load suffer unit step disturbance in.4 s and.7 s. Case 3. 1) The PMSM s parameters change with time. 2) The desired angular speed: 157 rad/s 314 rad/s (.3 s) 157 rad/s (.7 s). 3) The load is constant and unknown. The simulation results are depicted as Figs It is obvious from these figures that the current and the speed all are in the given range which mean the required conditions of the relevant theorems are satisfied. Figs. 1 3 shows the simulation results of proposed method 1 under three cases. The simulation results of proposed Method 2 under three cases can be seen in Figs In the figures ω ω ω stands the speed error; i a and u an are the phase a current and the line to the neutral voltage respectively. Figs. 1 and 4 show that the speed of PMSM can quickly track the desired speed and its change under the control of proposed Methods 1 and 2 respectively. These reflect the two methods have good tracking performance. it is obvious from Figs. 2 and 5 that the proposed control Methods 1 and 2 can make the system effectively resist the load disturbance. Figs. 3 and 4 reflect that although the parameters of PMSM change with time the two proposed control methods can make the speed of motor track the desired speed which comprehensively verify the control performance of the two methods. Comparing Figs. 1 3 and Figs. 4 6 we can conclude that Fig. 2. The simulation results of proposed Method 1 under Case 2. (a) Speed and current (b) Change of parameters Fig. 3. The simulation results of proposed Method 1 under Case 3. Fig. 1. The simulation results of proposed Method 1 under Case 1. the two control methods have similar control performance. Moreover from these cases we know that the load information is not needed in the two proposed controllers which means the two proposed control strategies can deal with the load uncertainty.

7 DENG AND NIAN: ROBUST CONTRO OF PERMANENT MAGNET SYNCHRONOUS MOTORS 149 Fig. 4. The simulation results of proposed Method 2 under Case 1. (b) Change of parameters Fig. 6 The simulation results of proposed Method 2 under Case 3. bilinear matrix inequations sufficient conditions for the existence of the robust controllers have been derived. The proposed controllers can deal with load uncertainty and resist disturbances. In order to demonstrate the effectiveness of proposed control methods computer simulations have been carried out via Matlab/Simulink. Simulation results have explained the proposed control methods have good control performance. REFERENCES 1 Yang S S Zhong Y S. Robust speed tracking of permanent magnet synchronous motor servo systems by equivalent disturbance attenuation. IET Control Theory Applications 27 1(3): Fig. 5. The simulation results of proposed Method 2 under Case 2. V. CONCUSIONS In this paper a novel state equation of PMSMs has been obtained via the suitable choice of state variables. Based on the state equation robust controllers of PMSMs have been developed via interval matrix and PI control idea. In terms of 2 Corradini M Ippoliti G onghi S Orlando G. A quasisliding mode approach for robust control and speed estimation of PM synchronous motors. IEEE Transactions on Industrial Electronics (2): Choi H H Vu N T T Jung J W. Digital implementation of an adaptive speed regulator for a PMSM. IEEE Transactions on Power Electronics (1): Underwood S J Husain I. Online parameter estimation and adaptive control of permanent-magnet synchronous machines. IEEE Transactions on Industrial Electronics 21 57(7): Zhu H Xiao X i Y D. Torque ripple reduction of the torque predictive control scheme for permanent-magnet synchronous motors. IEEE Transactions on Industrial Electronics (2): iu H X i S H. Speed control for PMSM servo system using predictive functional control and extended state observer. IEEE Transactions on Industrial Electronics (2): Jin H Z ee J M. An RMRAC current regulator for permanentmagnet synchronous motor based on statistical model interpretation. IEEE Transactions on Industrial Electronics 29 56(1): (a) Speed and current 8 Hamida M A eon J D Glumineau A Boisliveau R. An adaptive interconnected observer for sensorless control of PM synchronous motors with online parameter identification. IEEE Transactions on Industrial Electronics 213 6(2):

8 15 IEEE/CAA JOURNA OF AUTOMATICA SINICA VO. 2 NO. 2 APRI Yue X Vilathgamuwa D M Tseng K J. Observer-based robust adaptive control of PMSM with initial rotor position uncertainty. IEEE Transactions on Industry Applications 23 39(3): Chow. Design of optimal digital controller for stable super-high-speed permanent-magnet synchronous motor. IEE Proceedings Electric Power Applications (2): Choi H H Jung J W. Discrete-time fuzzy speed regulator design for PM synchronous motor. IEEE Transactions on Industrial Electronics 213 6(2): Chen B S Yuan Y Chen W Ni G Chen M. Electric Drive Automatic Control System: Motion Control System (Third Edition). Beijing China: China Machine Press 23 (in Chinese) 11 Chaou H Sicard P. Adaptive fuzzy logic control of permanent magnet synchronous machines with nonlinear friction. IEEE Transactions on Industrial Electronics (2): Khargonekar P Petersen I Zhou K M. Robust stabilization of uncertain linear systems: quadratic stabilizability and H control theory. IEEE Transactions on Automatic Control (3): eu V Q Choi H H Jung J W. Fuzzy sliding mode speed controller for PM synchronous motors with a load torque observer. IEEE Transactions on Power Electronics (3): iu G H Cheng Zhao W X Jiang Y Qu. Internal model control of permanent magnet synchronous motor using support vector machine generalized inverse. IEEE Transactions on Industrial Informatics 213 9(2): Boyd S Ghaoui Balakrishnan V. inear Matrix Inequalities in System and Control Theory. Philadelphia PA: SIAM Mao W J Chu J. Quadratic stability and stabilization of dynamic interval systems. IEEE Transactions on Automatic Control 23 48(6): Cheng B Tesch T R. Torque feedforward control technique for permanent-magnet synchronous motors. IEEE Transactions on Industrial Electronics 21 57(3): Do T D Choi H H Jung J W. SDRE-based near optimal control system design for PM synchronous motor. IEEE Transactions on Industrial Electronics (11): Zhenhua Deng received the B. Sc. degree in Dalian Maritime University Dalian China and the M. Sc. degree in Central South University China in 211 and in 214 respectively. His main research interests include control of electrical machines and distributed optimization. Corresponding author of this paper. 16 Wai R J. Hybrid fuzzy neural-network control for nonlinear motor-toggle servomechanism. IEEE Transactions on Control Systems Technology 22 1(4): u J G Chen Y Q. Robust stability and stabilization of fractional-order interval systems with the fractional order α: the α 1 case. IEEE Transactions on Automatic Control 21 55(1): Guo. H output feedback control for delay systems with nonlinear and parametric uncertainties. IEE Proceedings Control Theory and Applications (3): Zhao Ham C Han Q Wu T X Zheng Sundaram K Kapat J Xiaohong Nian received the B. Sc. degree the M.Sc. degree and the Ph. D. degree from Northwest Normal University Shudong University and Peking University in and 1999 respectively. He was a research fellow in the Institute of Zhuzhou Electric ocomotive from 24 to 28. Currently he is a professor the dean of the Faculty of Control Engineering of Central South University and an editor of Converter Technology & Electric Traction. His research interests include coordinated control and optimization of complicated multi-agent system converter technology and drive control control of tractive power supply system in high-speed train.