Robust Speed and Position Control of Permanent Magnet Synchronous Motor Using Sliding Mode Controller with Fuzzy Inference

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1 Preprint of the paper presented on 8 th European Conference on Power Electronics and Applications. EPE 99, , Lausanne, Switzerland. DOI: Robust Speed and Position Control of Permanent Magnet Synchronous Motor Using Sliding Mode Controller with Fuzzy Inference Stefan Brock, Jan Deskur, Krzysztof Zawirski POZNAN UNIVERSITY OF TECHNOLOGY ul. Piotrowo 3A Poznan, POLAND phone , fax {Stefan.Brock, Jan.Deskur, Krzysztof.Zawirski}@put.poznan.pl Acknowledgements This work was support partially by a grant from State Committee for Science Research (Poland), contract No 8 T1A Keywords Adaptive control, Emerging technologies, Fuzzy logic, Permanent magnet motors, Sliding mode control. Abstract In the paper a problem of robust speed and position control of permanent magnet synchronous motor is discussed. Basic idea of the proposed approach consists in statement that gained robustness of speed control guarantees robust position control even if position controller has simple structure. The proposed robust speed control algorithm bases on modified sliding mode controller equipped with a fuzzy adaptive mechanism. Presented computer simulation study and experimental results show clearly the advantages of elaborated control method. Introduction High dynamic performance of servo motor drives is required today in many automatically controlled machines providing high quality and high productivity in industrial technology. Recently permanent magnet synchronous motor (PMSM) without reduction gear has been often used in this high performance application such as robots and machine tools. It is well known that the control performance of the PMSM is very sensitive to load disturbances and variations of its parameters when the motor shaft is directly coupled with external load. These disturbances are the dominant harmful factor in the PMSM control system. This is the reason that robustness against parameter variation and load disturbances in speed and position control system is of particular importance. The sliding mode control (SMC) is one of the methods which enables to solve this problem [4,5,11,12.13]. A number of research on the SMC application to speed and position control have been reported [4,7,8,9]. However the SMC still creates some problems in practical applications such as chattering and steady state control error. In the ideal system the switching frequency can be very high and the state slides smoothly on the reference trajectory. In the real system, such as digital control system or system with presence of high frequency periodical distributions at its input, the switching rate and control accuracy are limited [3]. According to this the system state chatters around the sliding line and undesirable steady state control error is caused. In the drive system the chattering causes harmful effects such as torque pulsation, current harmonics,

2 acoustic noise and excitation of unmodelled dynamics. Therefore to obtain a chattering free sliding mode control has become an open question for researchers. One of the methods of chattering reduction bases on introduction of integrator to the SMC [1,2,6,7,12]. The additional component of control signal, which is the integral of control error, leads to elimination of steady state error and reduction of chattering effect. This basic idea recommended in reported works was modified in the paper and proposed in form of original SMC structure equipped with some fuzzy adaptation mechanism. The robustness of the speed control considered in the paper requires as follows: 1) The system has fast response for speed reference input change insensitive to load parameter variation without chattering and overshoot. 2) The system has low sensitivity to external load disturbances - transient speed deviation under rapid change of load torque is small with fast recovery. 3) There is not steady state control error during external load torque disturbances. The change of load torque and its moment of inertia is assumed as a load parameter variation. These disturbances have strong influence on speed control loop so robust speed control is of particular importance. Robustness of speed controller allows to design ordinary position controller with simple structure, which robustness is guaranteed by robust speed controller. The control method was investigated by means of computer simulation technique. The laboratory system was constructed with application of single-chip microcontroller SAB 8C166. The simulation and experimental results clearly demonstrate effectiveness of proposed control method. Control strategy The general structure of the proposed controller is shown in fig.1. The ordinary SMC, which contains only error calculation block, switching block or equivalent control block [11] is equipped with additional modules, which are : integral block and fuzzy control module. Special role plays the integral block. In a number of reported research works it was proved that the introduction of the integral block improves properties of SMC [1,2,6,7,12]. The calculation of integral of output signal of switching block guarantees elimination of steady control error and effectively reduces chattering. This idea is a modification of concept presented in [1] but in opposition to that concept in proposed controller the integrator operates continuously and the signal of switching block has not direct influence on output signal. Fuzzy control module fuzzification inference defuzzification e ė en dn ez ep dz dp dndz dp en S S B ez S S S ep B S S K s =K s_min *ms +K s_max *mb w ref - + w e T TPs Z -1 FIR 2 ė + + s T S 1 S Big K S 1 T F óó Limiter I ref Error calculation block Functional block

3 Fig.1 : General structure of proposed controller In ordinary SMC the switching block has ideal switching (relay) characteristic according to the formula (1) U = K sgn( s) (1) s s where s is a generalise control error described by (2), K S is a switching gain. d s e dt e æ = + = ç e + T d s è dt e ö l l ø (2) where T s = 1 l represents time constant of sliding control system. In many practical applications of SMC a boundary layer characteristic is introduced by application a sat function [1] as it is shown in formula (3). æ s U s = Ks sat ç è s big ö ø (3) where s big is a half of boundary layer. Both characteristics of switching block were analysed in the paper. Simulation results proved that the modified SMC has relatively large speed control error caused by step change of load torque. Significant improvement of this properties was achieved by introducing a fuzzy adaptive mechanism, which role is tuning controller parameters. Fuzzy control rules, fuzzy input membership functions and defuzzification method are shown in fig.1. After testing many variants of fuzzy inference mechanism the simple one was selected, which tunes switching gain K S on the base of two input signals. These signals are error(e) and error derivative (é). As it is visible in fig.1 for each input signal only three linguistic values: N - negative, P - positive and Z - zero, have been assumed. The fuzzy control rule can be formulated as follows: if the control error and the change of error are positive (P) then the switching gain should be big (B), if the control error and the change of error are negative (N) then the switching gain should be big (B), otherwise the switching gain should be small (S). Defuzzification procedure uses two singletons, which are extreme values of switching gains (KS) : maximum K Smax and minimum K Smin. Value of switching gain K S is calculated according formula KS = KS_ min ms + KS_ max m B (4) where ms and mb are highest of proper clipped fuzzy sets being result of firing fuzzy rules [11]. Simulation study Such mathematical model of PMSM was assumed, which is described in number of papers [6,8,13]. Applying description in rotating d-q frame and field oriented control of stator currents it can be assumed that current in d-axis ( i d ) is controlled to be zero. Then the electromagnetic torque ( T e ) depends only on current in q-axis ( i q ). Investigations basing on computer simulation technique were carried on in two stages. In the first the torque control system was represented by a first order lag system with the time constant T m =.25 ms.

4 Transient responses on the step change of speed reference signal under various values of moment of inertia (J min - J max = 6 J min ) were analysed. The investigations confirmed well known disadvantages of ordinary SMC, which are a high level of ripples of current reference signal and due to it a low accuracy of speed control. Introducing the integral block into the proposed controller but with ideal switching characteristic (without boundary layer) gives a structure of controller called Sliding Mode - Integral (SMI), which effectively reduces chattering and speed control dynamic error. Transients of this controller are shown in fig.2.a. Complete elimination of chattering is achieved by introducing to the SMI controller a boundary layer. This controller called Soft Sliding Mode - Integral (SSMI) leads to the transients very similar to SMI but without ripples (fig.2.b). Complete elimination of overshoot for SSMI controller can be obtained by increasing time constant of integrator (T F ) and time constant of sliding control (T S ) but this enlarges the speed control error cased by step change of the load torque. This relatively large dynamic speed error is some disadvantaged of the controllers SMI and SSMI. S M I a ) S S M I b ) J m in J m in Speed [rad/s] - - J m ax I ref [%] J m in J m in Tim e [m s ] Tim e [m s ] Fig.2 : Simulation transient responses on step change of speed reference signal with controllers : SMI (a) and SSMI (b) under various moment of inertia. Model of the torque control system - simplified. T S = 1 ms. Elimination of this disadvantage was an aim of the second stage of investigations. In this stage the more accurate model of the drive was elaborated. In the model such phenomena were taken into account as: non-linear characteristic of drive mechanical losses, irregularity of the motor torque as a function of rotational angle, errors in speed measurements, delay of.5-1 ms caused by sampling analogue reference input signal of the torque control system, increased time constant of torque control system T m =.5 ms, calculation of error derivative by means of FIR 2 filter.

5 5 Introduction to the model all additional phenomena led to remarkable decrease of control quality. Transients for the accurate model are shown in fig.3. In this figure all transients of step responses were obtained for values of time constant T F and T S bigger than 3 ms. Speed transients for controller SMI and SSMI are similar but for SSMI a smaller value of chattering is observed. Regular pulsation of the I qref comes from irregularity of the motor torque. The controller equipped with fuzzy adaptive mechanism was called Fuzzy tuned Soft Sliding Mode - Integral (FSSMI). This controller has a much smaller dynamic deviation of speed caused by step change of the load torque, what is presented in fig.3.c. The effect of fuzzy adaptive mechanism is clearly visible for small value of moment of inertia. The FSSMI controller has been selected for experimental investigations. S M I a ) S S M I b ) F S S M I c ) J m in Speed [rad/s] - J m in - J m in I ref [%] J m in J m in J m in T im e [m s ] Tim e [m s ] T im e [m s ] Fig.3 : Simulation transient responses on step change of speed reference signal with controllers : SMI (a), SSMI (b) and FSSMI (c) under various moment of inertia. Model of the torque control system - accurate. T S = 35 ms. Experimental results Fig. 4 shows the block diagram of investigated laboratory system, which consists of PMSM, PWM inverter and microprocessor control system. Parameters of PMSM are listed in Table 1. To exam the control performance under various values of load inertia a set of additional wheels with well calibrated moment of inertia was used to mount on the motor shaft. Investigated robust speed controller was realised by means of a single-chip microcontroller SAB 8C166. Speed feedback signal came from a pulse encoder additionally mounted on the motor shaft. The encoder had high resolution given 1. pulses per rotation. The speed measurement based on M/T method [9] with some modification, which

6 6 led to constant sampling time - equals.2 ms. This measurement procedure used internal counters and timers of the microcontrollers. Desktop PC RS-232C Counter Microcontroller w SAB8C166 D/A I ref A/D DigitAx Current field-oriented control i s PWM Inverter q PMSM q Fig.4 : Block diagram of laboratory system Table I. Specifications of PMSM Rated torque 2.3 Nm Maximum torque 9.2 Nm Rated voltage 38 V Maximum current 5.8 A Rated speed 3 rpm Torque constant 1.59 Nm/A Poles 6 Moment of inertia.135 kgm 2 Moment of inertia J min.265 kgm 2 Moment of inertia J max.159 kgm 2 Fig.5 shows transient response on step change of speed reference under various values of moment of inertia. The proposed controller leads to required equal wave forms of speed change, without overshoot and chattering and with limited derivative of the torque. The shape of speed wave forms depends on inclination of sliding trajectory, what can be done by change of time constant T S. This effect, shown in fig.6, is independent of value of moment of inertia. Faster speed change is limited by delay existing in current (torque) control system. The small ripples of speed and current reference signal for small value of moment of inertia, visible in fig.5 and 6, come from some asymmetry of motor construction and are observed even for the constant torque reference signal. In fig.7. step responses of position control are presented. Robust speed controller leads to robust position control even if very simple proportional controller is used (fig.7). Presented transients confirm robustness of position control and show significant improvement of position control due to influence of fuzzy adaptive mechanism of speed controller.

7 7 J=J m in J=3*J m in J=6*J m in Speed [rad/s] I ref [%] Tim e [m s] Time [ms] Time [ms] Fig.5 : Experimental transient responses on step change of speed reference signal with proposed controller under various moment of inertia - FSSMI.

8 8 T s = 3 5 m s T s = 7 m s Speed [rad/s] - Speed [rad/s] I ref [%] 2 I ref [%] 2 J m i n J m i n T i m e [ m s ] T i m e [ m s ] Fig.6 : Modification of experimental transient responses on step change of speed reference signal by changing inclination of sliding trajectory ( T S = 35, 7 ms ) - FSSMI.

9 9 J min a) J max b) J max c) angle [rad] error [rad] speed [rad/s] time [ms] time [ms] time [ms] Fig.7 : Experimental transient responses on step change change of position reference signal 21 rad with fuzzy module (FSSMI) for minimum (a) and maximum (b) value of moment of inertia and (c) without fuzzy module (SSMI) for maximum value of moment of inertia Conclusions The paper deals with the robust speed and position control of PMSM using modified sliding mode speed controller with the integrator and fuzzy inference for switching gain adaptation. The robust speed control leads to robust position control, insensitive to load parameters variation, such as moment of inertia and load torque. The chattering and steady state control error have been successfully eliminated from speed control. The effectiveness of the proposed control system was proved by simulation and experimental results. References 1. A. Bartoszewicz. Integral compensation of disturbance in sliding mode control systems, Proc. of 3rd Conference - Control in Power Electronics and Electrical Drives, Lodz (POLAND) November 1997, Vol S. Brock, J. Deskur. Combining of PI servodrive controller with sliding-mode servodrive controller by means of fuzzy logic, Proc. of 3rd Conference - Control in Power Electronics and Electrical Drives, Lodz (POLAND) November 1997, Vol. 1. (in Polish). 3. S. Brock, J. Deskur, K.Zawirski. Modified sliding-mode speed controller for servo drives, Proc. of the IEEE International Symposium on Industrial Electronics ISIE 99, Bled, Slovenia, July F.J. Chang, S.H. Twu, S. Chang. Tracking control of DC motor via an improved chattering alleviation control, IEEE Trans. Ind. Electron., Vol. 39, No. 1 February 1992.

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