2016 12th World Congress on Intelligent Control and Automation (WCICA) June 12-15, 2016, Guilin, China Model Predictive Torque Control of PMSM Systems Based on Sliding Mode Control Shuyuan, Qingfang, Guofei Abstract-Based on sliding mode control (SMC) method, a is model predictive torque control (MPTC) strategy proposed for permanent magnet synchronous motor (PMSM) systems. The mathematical model of PMSM system is built firstly. Then the SMC and MPTC are respectively designed, with the former being used to make the motor speed track its reference quickly and accurately while the latter being used to reduce the torque and flux ripples. Compared with conventional PI-based MPTC PMSM system, the designed system possesses better dynamical response behavior and stronger robustness in the presence of variation of load torque. The simulation results validate the feasibility and effectiveness of the proposed scheme. I. INTRODUCTION P ermanent magnet synchronous motors (PMSMs) nowadays are widely used in industry applications due to their high efficiency and high power/torque density. As for high performance PMSM systems, different control strategies are presented in the past. Of all the strategies, model predictive torque control (MPTC) is an emerging control concept and has received significant attention from the motor drives community [1]-[6]. MPTC has several merits such as easy inclusion of nonlinearities and constraints [2]. Besides, its algorithm is simple and its implementation is easy. MPTC is competitive comparing with field-oriented control (FOC) and direct torque control (DTC) in [7].1 So far, a conventional PI-based MPTC motor system consists of two loops, one being the outer-loop and the other the inner-loop. PI and MPTC acted as the outer and inner loops, respectively [8]-[11]. In general, PI is used to regulate the rotor speed and generate the reference torque and its parameters commonly adopt a fixed gain. It may perform well under certain operating conditions, but degrade dynamic performance under other operating conditions when external disturbances arise. Sliding mode control (SMC) has been applied into general design method for wide spectrum of system types. The most eminent feature of SMC is its completely insensitive to parametric uncertainty and external disturbances during sliding mode [12]-[14]. Function fal(·) is proposed in active disturbance rejection control Shuyuan is with Graduate Management Team Department, Engineering University of Armed Police Force, Xi’an, Shanxi, 710078, China (e-mail: 379467329@qq.com). Qingfang is with Department of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070 China (e-mail: tengqf@mail.lzjtu.cn). Guofei is with Department of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070 China. (e-mail: 251661756@qq.com). This work was supported by National Natural Science Foundation of China under Grant No.61463025. 978-1-4673-8414-8/16/$31.00 ©2016 IEEE 2677 (ADRC) by Han [15]-[17], and is widely used in system control because of its engineering application merits [18]. In order to improve the robustness of speed regulator in MPTC system, a sliding mode controller, which adopts nonlinear function fal(·), is designed as speed regulator in this paper. For PMSM drive system, in order to strength dynamic performance and system robustness, a SMC-based MPTC strategy is proposed in our research. Section 2 of the paper presents modeling of PMSM system. In Section 3, the SMC-based MPTC PMSM system is designed. The numerical simulation analysis and conclusion are reported in Section 4 and Section 5, respectively. II. MATHEMATICAL MODEL OF PMSM DRIVE SYSTEM The schematic diagram of PMSM fed by three-phase voltage source inverter (VSI) is shown in Fig. 1. dcu 1T 4T 3T 1D 5T 3D 6T 4D 2T 6D 5D 2D Fig.1 PMSM system fed by VSI As far as Fig.1 is concerned, three-phase stator voltages in abc-system are given by: − = − (2 S a dc S b S c ) au , a b u u u u = 1 3 1 3 1 u u 3 bu and = c − ( S a dc + 2 S b − S c ) (1) + − − c a b S S S ) dc 2 ( cu are the stator voltages in the where abc-system dcu is the DC bus voltage of power unit, and Sii= a, bc the upper power switch state of one of three legs. Si =1 or Si=0 when the upper power switch is on or off as shown in Fig.1. The different combination of Sa, Sb, Sc may form eight switching states, which produce corresponding eight voltage space vectors as shown in Fig. 2. Consider a surface-mounted PMSM. The mathematical model in -system can be expressed as follows 

+ ) ψω θ r sin p ( f r − ) ψω θ r cos p ( f r R i s R i s = u − R i s = u − R i s ) ) + u + u (2) (3) i d t d i d t d = = − − ( 1 L ( 1 L ψ d t d ψ d t d ψ , β , and ,u uα stator stator voltages, ψ are the and axis stator ,i iα β , where currents, in the -system, respectively. L is the stator winding ψ the inductance, rθ and rωare the measured permanent magnet flux, and rotor angular displacement and speed, respectively. p is pole pairs. sR the stator winding resistance, fluxes f (01 0) 3V 2V (11 0) 4V (011) 0V (0 0 0) 7V (111) 1V (1 0 0) 5V (0 01) 6V (1 01) Fig.2 The layout of voltage space vectors of VSI Stator voltage and current expressed in -system can be obtained using the following Clark transformation A. SMC design Take into account the disturbance impacting on PMSM system, (5) can be rewritten as bellow, − ω + = − T e T L B m r (6) d t ( ) J ω d r t d Where d(t) is the practical disturbance. Then ω rd t d = ∗ − T e J T L − f t ( ) (7) eT ∗ is the reference value of motor torque, f(t) Where can be regarded as the total disturbance which is expressed as ω + B m r ∗ T e − d t ( ) T e f t ( ) = − J It can be observed that f(t) is a multivariable function of both the internal and external disturbance. It is reasonable to assume that f(t) is bounded, therefore the following relation is satisfied: l≤ t ( ) f where l is a positive real constant. ∗ω . Definite speed error: Let the desired speed be r ∗= r ω ω ω e − r Select sliding variable as following eω= s The differentiation of (9) can be obtained: = s ω ∗ rd t d ∗ T e − − J T L + f t ( ) (8) (9) (10) The sliding mode controller is designed as + αδ ) Jkfal e ( , ω ∗ = T e + , Where fal x αδ is defined as: , , ( ω ∗ d r t d J ) T L (11) T abc- 1 2 = 3 0 − 1 2 3 2 − − 1 2 3 2 (4) ( fal x , ) αδ , α x = δ − 1 sign , ( ) x ≤ δ > δ x x ⋅ α , x By Newton’s law, the torque equation can be written as (5) − ω B m r T L T e = − J ω d r t d where J, Te, TL and Bm are the inertia of moment electromagnetic torque, load torque and viscous friction coefficient, respectively. III. DESIGN OF MPTC FOR PMSM SYSTEMS BASED ON SLIDING MODE CONTROL The objective of SMC-based MPTC strategy is that PMSM system not only can work reliably and its speed and torque can be controlled to achieve satisfactory dynamic performance but also can be strongly robust to internal and external disturbance. The block diagram of SMC-based MPTC system proposed is shown in Fig.3. ( ) , , The fal x αδ is nonlinear function during 0 1α< < . small error with large gain, and large error with small fal x αδ , which gainis the main characteristic of meets the demand of engineering rule to improve performance of system [18]. ( ) , , Select the Lyapunov function as V 21 s= 2 Then V ss = = e ω ω ∗ rd t d ∗ T e − T L − J + f t ( ) (12) 2678 

kω r Fig.3 SMC-based model predictive torque control for PMSM fed by VSI eT ∗ in (12) with (11). When eω δ> , the (13) Replace following result can be obtained + + = − ≤ − V + 1 + 1 α α k e ω k e ω ( k e ω = − α − t ( ) e f ω l e ω ) l e ω If e ω 1 α c + l > k satisfied: , the following relation could be where c is a positive real constant. < − V c + l k 1 α c < 0 (14) The above result illustrates that SMC speed regulator is stable. The speed tracking error could be limited in following convergence area as follows min. { } = T g * i e { ∈ V s t V . . i 0 − T k e V 1 + 1 + + 1 k s ψ − ψ } k * 1 s V V 6 7 (16) 1 * s eT and * kT + and e ψ are reference values for torque and where k +ψ predictions for stator flux, respectively. torque and stator flux at (k+1)th instant, respectively. k1 is the weighting factor. Vi as shown in Fig. 2 are eight voltage space vectors generated by VSI with respect to the different switches states. 2) Predictive model for stator currents 1 s According to (2), the predictions of the stator currents at the next sampling instant are expressed as the following ψω θ k r ψω θ k r T s L T s L ) ) R i k s R i k s cos ( ( sin 17 i k i k i k i k = + − + + = + − − + u u + 1 + 1 k k k r k r f f e ω 1 α c + l ≤ k (15) 1ki + α and 1ki + where are predicted values of stator currents at β sT is the sampling period. (k+1)th instant, k > t ( ) in parameter adjust to Therefore, + and then schedule parameterαsuch c f k guarantee that the requirement of anti-disturbance ability can be satisfied. B. Model predictive torque control order 1) Basic principle of MPTC The basic idea of MPTC is to predict the future behavior of the variables over a time frame based on the model of the system. In fact, MPTC is an extension of DTC, as it replaces the look-up table of DTC with an online optimization process in the control of machine torque and flux. Different from the employment of hysteresis comparators and switching table in DTC, the principle of vector selection in MPTC is based on evaluating a defined cost function [1]. The selected voltage vector from switching table in DTC is not necessarily the best one in terms of reducing torque and flux ripples. For eight voltage space vectors generated by VSI as shown, it is easy to evaluate the effect of each voltage vector and select the one to minimize the cost function in MPTC. For MPTC, the minimum cost function is such chosen that both torque and flux at the end of the cycle are as close as possible to their reference values. Its definition is 1ki + β 1ki + α and After obtaining , both the torque and flux at the (k+1)th instant can be estimated according to the following Section 3.2.3. 3) Torque and flux estimators According to (3), the predictions of the flux-linkage at the (k+1)th instant can be expressed as following: + 1 ψ ψ k ψ ψ k = = k k + 1 + + T u ( k s T u ( k s − − R i k s R i k s ) ) (18) The predictions of the magnitudes of stator flux linkage instant the (k+1)th torque at and electromagnetic respectively are + 1 ψ k s = + 1 T k e = . p 1 5 2 2 + 1 + 1 k ) ) + ( ψ k ( −ψ ( ψ ) + 1 + 1 k + i k 1 ψ k + i k 1 (19) (20) Substituting (17) and (18) into (20), the predictive torque can be estimated. IV. SIMULATION AND ANALYSIS In order to validate the effectiveness of proposed control scheme, the designed control system from Fig.3 is 2679 

b b implemented in Matlab/Simulink/Simscape platform. The parameters of PMSM are given in Table 1. TABLE 1 PARAMETERS OF PMSM Symbol Quantity Nominal phase resistance Stator-winding inductance Rotor magnetic flux Number of pole pairs DC bus voltage Rated speed Value 2.875 0.0085H 0.175Wb 1 450V 3000rpm Rs L f p udc nN J Tn Bm increased to 2 Nm at 0.2 seconds. The sampling period sT is 10us, and value 1k in (16) is selected to be 33. The ψ is 0.175Wb. The parameters of reference stator flux * s SMC in Fig.3 are k =0.16, δ =0.01, α = 0.7 For the SMC-based MPTC PMSM system, in order to verify its strong robustness, two systems are compared, which correspond to the PI-based MPTC and SMC-based MPTC PMSM systems, respectively. Except their distinct outer-loop controllers (i.e. PI and SMC), two systems have completely structures and parameters. For comparison purpose, the parameters of PI for PI-based MPTC PMSM system are adjusted as follows, identical MPTC = K p 0.1, K I = 0.1 So that PI-based MPTC system has almost identical transient response as SMC-based one. Moment of inertia 0.0008Kg.m2 Rated torque Viscous friction coefficient 3Nm 0 In the simulation, their reference speeds n* are set to 1000 rpm and their load torques of 1Nm at the start are 1200 1000 800 600 400 200 ) m p r ( d e e p S 0 0 6 5 4 3 2 1 0 -1 0 ) m N ( e u q r o T 0.2 0.15 0.1 0.05 ) b W ( 0 -0.05 -0.1 -0.15 1010 1000 990 980 0.19 0.2 0.21 0.1 0.2 0.3 0.4 0.5 Time(s) (a) Rotor speed response ) m p r ( d e e p S 1200 1000 800 600 400 200 0 0 1010 1000 990 980 0.19 0.2 0.21 0.1 0.2 0.3 0.4 0.5 Time(s) (a) Rotor speed response 6 5 4 3 2 1 0 ) m N ( e u q r o T 0.1 0.2 0.3 0.4 0.5 Time(s) (b) Torque response -1 0 0.1 0.2 0.3 Time(s) 0.4 0.5 (b) Torque response 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 ) b W ( -0.2 -0.2 -0.1 0 ( W b 0.1 0.2 (c) Trajectory of stator flux linkage -0.2 -0.2 -0.1 0 (Wb) 0.1 0.2 (c) Trajectory of stator flux linkage 2680 

) A ( t n e r r u c 30 20 10 0 -10 -20 -30 0 ia ib ic 0.1 0.2 0.3 Time(s) 0.4 0.5 (d) Stator current ia,ib and ic 30 20 ) 10 A ( t n 0 e r r u c -10 -20 -30 0 ia ib ic 0.1 0.2 0.3 0.4 0.5 Time(s) (d) Stator currents ia,ib and ic Fig.4 Dynamic responses of PI-based MPTC scheme Fig.5 Dynamic responses of SMC-based MPTC for PMSM system scheme for PMSM system Drives,” IEEE Transactions on Power Electronics, vol.31, no.2, pp.1381-1390, 2016. [7] F. Niu, B. Wang, A. S. Babel, K. Li and E. G. Strangas, “Comparative Evaluation of Direct Torque Control Strategies for Permanent Magnet Synchronous Machines,” IEEE Transactions on Power Electronics, vol.31, no.2, pp.1408-1424, 2016. [8] F. Wang, Z. Zhang; S. A. Davari, R. Fotouhi, D. A. Khaburi, J. Rodríguez and R. Kennel, “An Encoderless Predictive Torque Control for an Induction Machine with a Revised Prediction .IEEE Transactions on Industrial Model and EFOSMO,” Electronics, vol.61, no.12, pp. 6635-6644, 2014. [9] C. A. Rojas, J. Rodriguez, F. Villarroel, J. R. Espinoza, C. A. Silva and M. Trincado, “Predictive torque and flux control without weighting factors,” IEEE Transactions on Industrial Electronics, vol.60, no.2, pp. 681-690, 2013. [10] F. Wang, Z. Zhang and D. Alireza, “An experimental assessment of finite-state predictive Torque control for electrical drives by considering different online optimization methods,” Control Engineering Practice, vol.31, pp. 1-8, 2014. [11] M. Preindl and S. Bolognani, “Model predictive direct torque control with finite control set for PMSM drive systems, Part 1: Maximum torque per ampere operation,” IEEE Transactions on Industrial Informatics, vol.9, no.4, pp. 1912-1921, 2013. [12] Q. TENG, J. BAI, J. ZHU and Y.GUO, “Sensorless model predictive torque control using sliding-mode model reference adaptive system observer for permanent magnet synchronous motor drive systems,” Control Theory & Applications, vol.32, no.2, pp.150-161, 2015. [13] Q. Xu, “Digital Sliding Mode Prediction Control of Piezoelectric Micro/Nanopositioning System,” IEEE Transactions on Control Systems Technology, vol.23, no.1, pp.297-304, 2015. [14] X. Zhang, L. Sun, K. Zhao and L Sun, “Nonlinear Speed Control for PMSM System Using Sliding-Mode Control and Disturbance IEEE Transactions on Power Compensation Techniques,” Electronics, vol.28, no.28, pp.1358-1365, 2013. [15] J. Q. Han, “Auto Disturbances Rejection Control Techn-iqu e,” Frontier Science, vol.1, pp. 24-31, 2007. [16] J. Q. Han, “From PID to active disturbance rejection control,” transactions on Industrial Electronics, vol.56, no.3, IEEE pp.900-906,2009. [17] J. Q. Han, “From PID technique to active disturbance rejection control technique,” Control Engineering of China, vol.9, no.3, pp. 13-18, 2002. [18] Q. Zhong, Y. Zhang, J. Yang and J. Wu, “Non-linear auto-disturbance rejection control of parallel active power filters,” IET Control Theory & Applications, vol.3, no.7, pp. 907-916, 2009. their dynamical Figs.4 and 5 show responses. Comparing Fig.4(a) with Fig.5(a), it can be observed that, for SMC-based MPTC PMSM system, its speed decrease is less than PI-based one’s after the change of external load, and it also can recover to its reference value more quickly. Therefore, for SMC-based MPTC PMSM system, its capability of accommodating the change of load disturbance is superior to PI-based one’s. V. CONCLUSION In this paper, a SMC-based MPTC strategy is developed for PMSM system. Then the SMC is designed for tracking its reference speed quickly and accurately while the MPTC for torque and flux ripple reduction. The simulation result illustrated that, the speed and torque of PMSM could be regulated in a satisfactory manner. 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