Improved control strategy for PMSM based on fuzzy sliding mode control and sliding-mode observer

ZHAO Feng， LUO Wen， GAO Fengyang， YU Jiale

(School of Automation and Electrical Engineering， Lanzhou Jiaotong University， Lanzhou 730070， China)

Abstract： Aimed at the problems of large torque ripple, obvious chattering and poor estimation accuracy of back-EMFs in traditional permanent magnet synchronous motor (PMSM) control system with sliding mode observer (SMO), an improved control strategy for PMSM based on a fuzzy sliding mode control (FSMC) and a two-stage filter sliding mode observer (TFSMO) is proposed. Firstly, a novel reaching law (NRL) used in the speed loop based on hyperbolic sine function is studied, and fuzzy control ideal is shown to achieve the self-turning of the parameter for the reaching law, thus a fuzzy integral sliding mode controller based on the novel reaching law is designed in speed loop. Then the suppression effect upon chattering caused by the novel reaching law is analyzed strictly by discrete equation. Secondly, in order to restrain the high frequency components and measurement noise in back-EMFs, a two-stage filter structure based on a variable cut-off frequency low-pass filter (VCF-LPF) and a modified back-EMF observer (MBO) is conceived, and the rotor position is compensated reasonably. As a result, a TFSMO is designed. The stability of the proposed control strategy is proved by Lyapunov Criterion. The simulation and experiment results show that, compared with traditional SMO, the controller suggested above can obtain very nice system respond when the motor starts and is subjected to external disturbances, and effectively improve the problems about torque ripple, chattering and the estimation accuracy of back-EMF.

Key words： permanent magnet synchronous motor (PMSM); novel reaching law (NRL); fuzzy sliding mode control (FSMC); two-stage filter sliding mode observer (TFSMO)

0 Introduction

In recent years, because of the high mechanical strength and good reliability of permanent magnet synchronous motor (PMSM), new urban rail transportation technology using PMSM has been strongly supported and encouraged in China. In the meantime, it is necessary to improve performance of PMSM control system. The traditional sliding mode control (SMC) in speed loop used in PMSM has some problems, such as large torque ripple, obvious chattering and poor anti-disturbance ability. At the same time, the PMSM control system based on the traditional sliding mode observer (SMO) uses a fixed cut-off frequency low-pass filter, which can not eliminate the measurement noise of back-EMFs. In order to improve the performance of PMSM control system, sensorless control strategy for PMSM has been widely studied by scholars all over the world[1-6].

A novel reaching law is proposed to suppress the sliding mode chattering. However, the control structure of the system is so simple that the speed response and torque response of PMSM would have certain overshoot and jitter when the PMSM is subjected to external disturbances[7-8]. SMC and maximum torque per ampere (MTPA) are combined together to suppress the starting current harmonic rate of PMSM effectively[9]. A double sliding mode direct torque control strategy based on PLL is proposed to restrain flux ripple of traditional direct torque control (DTC) for PMSM. However, the accuracy of rotor position compensation is poor while the observed speed has obvious chattering and overshoot[10]. In the filtering section of the SMO, two low-pass filters (LPF) are used to improve the estimation accuracy of the rotor position and speed. However, owing to the fixed cut-off frequency of the filters, the adaptive ability of LPF is poor when the high-frequency components of the back-EMFs change[11]. A piecewise exponential function is proposed to replace the symbolic function in traditional SMO. The proposed function is one of the effective methods to weaken the sliding mode chattering. The current response of the system is effectively improved. However, due to the poor adaptive ability, the torque of system still has obvious overshoot and pulsation when the motor starts and is subjected to external disturbances[12]. Genetic algorithm is used to improve the current loop control accuracy and ensure the stable operation of the motor in the whole range[13]. The design of motor parameter identification system based on mode reference adaptive system (MRAS)[14-15]effectively reduces the influence of the changes of motor parameters on the performance of control system.

In order to solve the problems such as simple filtering structure, high harmonic content of back-EMFs and poor adaptive ability in the above literatures, taking the PMSM vector control withid=0 as background, this paper designs an integral sliding mode structure based on a novel reaching law (NRL) in the speed loop, and realizes self-tuning of the proposed reaching law parameters through fuzzy control method. In the SMO, a variable cut-off frequency low-pass filter (VCF-LPF) and a modified back-EMF observer (MBO) are combined together to suppress the high frequency components and measurement noise in back-EMFs, and the rotor position estimation is compensated reasonably. Thus, the structure of improved control strategy based on fuzzy sliding mode controller (FSMC) and two-stage filter sliding mode observer (TFSMO) are achieved. Finally, the superiority of the improved control strategy is verified by simulations in MATLAB/Simulink and experiments.

1 Fuzzy sliding mode speed controller designed with novel reaching law

1.1 PMSM model

In the rotorα-βcoordinates, the model of a surface mounted PMSM can be expressed as

(1)

whereuαanduβrepresentα-βaxis voltages;iαandiβrepresentα-βaxis currents;prepresents differential operator;Ris stator resistance;Lsis stator inductance;eαandeβrepresentα-βaxis back-EMFs whose expression can be given by

(2)

whereωeis electrical angular velocity;Ψfis flux linkage of permanent magnets;θeis rotor angle.

Torque equation and motion equation ind-qaxis can be expressed as

(3)

whereωis rotor angular velocity;Bis viscous friction coefficient;Jis rotational inertia;TLis load torque;npis polar logarithm.

1.2 NRL

NRL based on hyperbolic sine function is achieved to accommodate the variations of system states. This reaching law is given by

(4)

wheresis sliding mode surface;x1is state variable of system;αis positive.

In this NRL, it can be seen that if system state is so far from the sliding mode surface (sinh(α|x1|)>1) thatεsinh(α|x1|)>ε. Thus the proposed reaching law will have a faster reaching speed. One hand, when system state is close to the sliding mode surface,x1approximates to zero to suppress the sliding mode chattering. Another hand, while the sliding mode surface is close to zero, according to Eq.(4), a discrete expression of the proposed reaching law is given as

s(n+1)-s(n)=-εsinh(α|x1|)sgn(s(n))T-

qs(n)T,

(5)

whereTis sampling period.

Assuming that the trajectory of system approaches the sliding mode surface in a finite time froms>0, which manifestss(n)=0+, then the equation in the next period can be obtained as

s(n+1)≈-εsinh(α|x1|)T.

(6)

On the contrary, assuming that the trajectory of system approaches the sliding mode surface in a finite time froms<0, which manifestss(n)=0-, thus the equation in the next period can be obtained as

s(n+1)≈εsinh(α|x1|)T.

(7)

Hence, according to Eqs.(6) and (7), the width of discrete sliding mode band can be obtained by

Δ≈εsinh(α|x1|)T.

(8)

Δ1≈ε1T.

(9)

It can be found that the bandwidth is a constant in Eq.(9). That indicates the system state might form a chattering betweenε1Tand -ε1Tand can not approach the equilibrium point. However, the bandwidth of the proposed reaching law in Eq.(8) would decrease withx1when the trajectory of system is on the sliding mode surface. The proposed reaching law has a better chattering suppression and reaching speed. The system trajectory of the exponential reaching law and the novel reaching law are shown in Fig.1.

(a)

1.3 Speed controller based on SMC and NRL

The system state variables are defined as

(10)

whereω*is reference rotor angular velocity;ωis actual rotor angular velocity.

In order to improve the dynamic performance of system, an integral sliding mode structure which can reduce the interference of high frequency disturbances is used. The sliding mode surface is designed as

s=x1+cx2,

(11)

wherecis positive.

Taking the time derivative of Eq.(10), it can be obtained by

(12)

Taking the time derivative of Eq.(11), it can be obtained by

(13)

According to Eqs.(4), (12) and (13), the input of sliding mode controller is

iq=(εsinh(α|s|)f(s)+qs+cx1+

(14)

Then, in order to prove the stability of the proposed reaching law, the Lyapunov function is chosen as

(15)

The derivative of Eq.(15) is obtained by

-ε|s|sinh(α|x1|)-ks2≤0.

(16)

According to Lyapunov theory, the proposed novel reaching law is stable.

1.4 Fuzzy control

According to the principle of fuzzy control, the fuzzy sets are defined as

s={PB,PS,ZO,NS,NB};

ε={PS,PM,PB}.

The membership functions of inputs are shown in Fig.2. And the fuzzy control rules are shown in Table 1. Finally, centroid method is used to achieve defuzzification.

(a)

Table 1 Fuzzy control rules

2 Two-stage filter sliding mode observer design

2.1 Sliding mode observer

Eq.(1) can be transformed into a state equation of stator currents inα-βaxis as

(17)

Following Eq.(17), SMO is constructed by

(18)

Furthermore, the error equation of estimated currents can be obtained by subtracting Eq.(17) from Eq.(18) as

(19)

In Ref.[12], a piecewise exponential function was proposed to replace the symbolic function in traditional SMO. As a result, based on the piecewise exponential function, the two-stage filter sliding mode observer in this paper is designed. The piecewise exponential function expression is given by

(20)

Therefore, the sliding mode rule is designed as

(21)

When system state of SMO reaches the equilibrium point, the estimated back-EMFs can be expressed as

(22)

2.2 Two-stage filter structure

In the operation of PMSM, motor speed might be changed because of the external disturbances. This would influence high frequency components in estimated back-EMFs. Because the cutting-off frequency of low-pass filter in traditional SMO is a constant, the filter has a poor adaptive ability when faced with the changes of frequency components in estimated back-EMFs, thus a VCF-LPF is designed. The schematic diagram of control system for PMSM based on traditional SMO is shown in Fig.3.

Fig.3 Schematic diagram of control system for PMSM based on traditional SMO

In order to improve the adaptive ability of LPF, cut-off frequency is related to the rotor speed. The variable cut-off frequency of VCF-LPF is designed as

(23)

wherekfandkeare positive, andkecan improve the filtering ability of VCF-LPF at low speed.

Therefore, the VCF-LPF can be constructed as

(24)

The estimated value of the back-EMFs by filtering with VCF-LPF is

(25)

After filtering by VCF-LPF, the high frequency components in estimated back-EMFs are effectively reduced. However, there still are many disturbances and measurement noise in estimated back-EMFs, thereby a MBO is proposed to achieve two-stage filter.

The model of back-EMFs can be expressed as

(26)

Taking the time derivative of Eq.(26), it can be obtained as

(27)

According to Eq.(27), the MBO is constructed as

(28)

The error equation of estimated back-EMFs can be obtained by subtracting Eq.(27) from Eq.(28), that is

(29)

In order to prove the stability of the proposed two-stage filter structure, the Lyapunov function is chosen as

(30)

The derivative of Eq.(30) is expressed as

(31)

According to Lyapunov theory, the proposed two-stage filter structure is stable.

2.3 Estimation and compensation of rotor position

The estimated rotor position can be calculated with the trigonometric function as

(32)

Because low-pass filter is used in the filtering process, the estimated component of the back-EMFs will have phase delay. Therefore, it is necessary to improve the accuracy of rotor position estimation by angle compensation. The compensation expression is obtained by

(33)

The estimated electrical rotor angular velocity expression by the derivative of Eq.(33) can be obtained as

(34)

The schematic diagram of two-stage filter sliding mode observer is shown in Fig.4.

Fig.4 Schematic diagram of SMO based on two-stage filter

3 Results and discussion

3.1 Simulation results

To demonstrate the effectiveness and feasibility of the proposed control strategy, the involved simulations are carried out in MATLAB/Simulink based on the traditional SMO method and the proposed control method in one PMSM system. The vector control based onid=0 is used. The parameters of PMSM are shown in Table 2, and the schematic diagram of control system for PMSM based on proposed method is shown in Fig.5. The main Simulink elements are from the libraries of Simulink-Math Operations and Simscape.

Table 2 Parameters of the PMSM

Fig.5 Schematic diagram of system in proposed control strategy

The simulation parameters of traditional SMO method arekps=1,kis=0.2,kpc=8,kic=2.06,ωc=20 000,ks=200. The simulation parameters of the proposed method arekpc=8,kic=2.06,q=500,ks=200,kf=8,ke=3 000,Kl=1 000, andεis self-tuning parameter of fuzzy control method.

In order to test the start-up performance and anti-disturbance performance of the proposed control strategy, the initial reference speed is 1 000 r/min, the reference speed is suddenly added to 1 200 r/min att=0.07 s, and load torque is suddenly added to 5 N·m att=0.14 s.

From Figs.6 and 7, it can be seen that the THD of the alpha-axis back-EMF in Fig.6(b) reduces from 9.25% to 3.88%. It is meant that there are a lot of harmonic disturbances in back-EMFs after filtering by VCF-LPF, and the proposed two-stage structure can suppress harmonics effectively.

(a)

Compared with traditional SMO, the speed response of proposed control strategy is more stable than that of traditional SMO, and the chattering is effectively suppressed (Fig.8). The speed of the proposed control strategy can be stabilized at a given speed quickly when starting, and the speed fluctuation is reduced when dealing with external disturbances.

(a)

(a)

Fig.9 show the speed error response of traditional SMO method and proposed method.

(a)

It can be found that the speed error of traditional SMO in steady state fluctuates sharply between -8 r/min and 10 r/min, and the chattering is very obvious. The speed error variation range of the proposed control strategy in steady state is between -2 rad/min and 3 rad/min, and the chattering is weakened effectively when dealing with external disturbances. Therefore, the speed estimation accuracy and robustness of the proposed control strategy are better than that of the traditional SMO.

Fig.10(a) shows that the torque response of traditional SMO has an obvious ripple in steady state, especially in starting and suffering from external disturbances. From Fig.10(b), the torque response of proposed control strategy has no obvious ripple and satisfying robustness. The starting performance and anti-disturbance performance of the proposed control strategy are better than that of the traditional SMO. As shown in Fig.11(a), the distortion of the three-phase currents in the traditional SMO is very serious, and there is a long restoring stability time when PMSM starts and is disturbed.

(a)

(a)

As shown in Fig.11(b), the three-phase currents of the proposed control strategy can approach sinusoidal wave fast, and the dynamic performance is better than that of the traditional SMO when the reference speed and load torque change suddenly.

3.2 Experiment results

The experiments platform is constructed by DSP TMS320F28335 processor of TI company. To prove the proposed method, the experimental system for back-EMF inαaxis and three-phase current of PMSM is constructed. The photo of experimental devices is shown in Fig.12. The structure diagram of experimental bench is shown in Fig.13.

Fig.12 Photo of experimental devices

Fig.13 Structure diagram of experimental bench

The experimental parameters of proposed controller and observer are:q=500,ks=200,kf=8,ke=3 155,Kl=1 022. The experimental parameters have a little difference from the simulation’s. In this experiment， the reference speed for PMSM is given as 1 000 r/min, and the reference load torque is given as 4 N·m.

Firstly, the experimental results for back-EMF inαaxis with the proposed two-stage filter observer are shown in Fig.14. It is obvious that the back-EMF inαaxis after second filter is more similar to sinusoidal wave than that after first filter.

(a)

Secondly, the experimental results for three-phase current with traditional SMO and the proposed control strategy are shown in Fig.15. From Fig.15(a), it can be seen that the current with traditional SMO has obvious distortion. From Fig.15(b), the current with the proposed control strategy is more similar to sinusoidal wave than that with traditional SMO.

4 Conclusions

In this paper, the improved control method based on the FSMC and the TFSMO is suggested. Firstly, in speed loop of control system, the NRL and fuzzy control thought are combined together to achieve self-turning of reaching law parameter. In order to reduce the disturbances of sliding mode chattering and high frequency noise, an integral sliding mode structure is designed.

Secondly, to solve the problems of the bad adaptive ability and the poor filtering ability for measurement noise of LPF, whose cut-off frequency is a constant, VCF-LPF and MBO are used in SMO to achieve two-stage filter.

Meanwhile, the chattering suppression of NRL and the stability of MBO is analyzed strictly.

Finally, compared with traditional SMO, the effectiveness and feasibility of the proposed strategy are demonstrated by simulations in MATLAB/Simulink and experiments.