Active disturbance rejection control FCS-MPC strategy based on ESO of PMSM system

ZHANG Bin, WEN Xue, LI Kun-qi

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

0 Introduction

Three-phase permanent magnet synchronous motor (PMSM) has many advantages such as simple structure, small volume, small moment of inertia and large power factor. So it has been widely used in important fields, such as transportation, military, industrial, medical and aviation[1,2]. The control methods of high performance PMSM system consist of vector control (VC)[3], direct torque control (DTC)[4,5]and model predictive control[6,7]. In recent years, a kind of optimization control method finite control set-mode predictive control (FCS-MPC) has been paid wide attention. FCS-MPC can be divided into current prediction control and torque prediction control[8,11]according to the control target[9,10,12].

In traditional torque prediction control, it is necessary to adjust the weight coefficient to ensure that the system has good dynamic performance, duing to the inconsistency between the stator flux and the electromagnetic torque[13]. In contrast, current prediction control has the same magnitude with the current variable, which can avoid to design weight coefficient. The traditional FCS-MPC needs to predict current values corresponding to all the basic voltage vectors in each sampling period, which makes it have heavy computing in the actual industrial control[14]. Zhang Y C[10]proposes a fast vector selection based on FCS-MPC, which reduces the complexity of computation and algorithm. As the back electromotive force (EMF) is DC amount, the method that estimates the back EMF of the current moment by calculating the values of the first three moments and averaging them will increase system complexity. Meanwhile, external interference will lead to estimation error. In this paper, the extended state observer (ESO) is applied to observe the back EMF in real time, and a new FCS-MPC strategy is constructed to improve the control performance of the system.

FCS-MPC speed control system of PMSM requires accurate rotational speed information. The speed sensor with higher accuracy and resolution is expensive, and there are problems such as complexity of the system, noise quantization and so on. Therefore, the study of speed sensorless has drawn great attention. At present, the speed control methods of the motor system consist of high frequency injection (HFI)[15], extended kalman filter method (EKF)[16], model reference adaptive control (MRAS)[17], sliding mode (SM) variable structure[18]and extended state observer (ESO)[19]. The arc-hyperbolic sine function of many speed identification methods can make ESO have a good estimation effect, and the ESO based on the arc-hyperbolic sine function, which does not need accurate mathematical model, has strong anti-jamming capacity and rapid convergency and so on.

Speed regulator in FCS-MPC system of PMSM usually adopts PI algorithm. The control effect can be obtained by adjusting the PI parameters at different speeds and external disturbances. In order to improve the robustness of the system speed regulator, the authors proposed a new anti-disturbance rejection control strategy based on the arc-hyperbolic sine function[19-21]in recent years which has drawn wide concern. In the Ref.[20], the tracking differentiator of the arc-hyperbolic sine function is presented, in which the ideal differential signal can be obtained, and track response speed is quickly and stably. In Ref.[19], an extended state observer based on the arc-hyperbolic sine function is put forward. The observer can accurately observe the disturbance variable, and the response speed of the system is fast and stable. At the same time, the initial parameters of ESO can be used to suppress the differential amplitude. Active disturbance rejection control (ADRC), which does not require accurate mathematical models, has strong robustness and immunity. Therefore, ADRC based on arc-hyperbolic sine function is used as the speed regulator in this paper.

In order to improve the control precision, robustness and control performance of the system, the ADRC FCS-MPC strategy based on ESO is proposed for PMSM system, which can reduce the cost of sensor at the same time.

1 Mathematical model of PMSM

In this paper, taking the surface mounted PMSM system as the control object, that isLd=Lq=L, so the mathematical model in the synchronous rotating reference frame is

(1)

whereid,iqandud,uqared,qaxis currents and voltages of stators;Rsis stator resistor;Lsis stator inductance;ed,eqis back EMF;ωris rotor velocity;ωeis electrical angular velocity;Pis pole pairs;ψfis rotor permanent magnetic flux;TLis load torque;Jis moment of inertia; andBis coefficient of friction factor.

2 ADRC FCS-MPC strategy of PMSM system based on ESO

For the three-phase PMSM system, the system block diagram of active-disturbance rejection FCS-MPC based on ESO is shown in Fig.1.

Fig.1 System block diagram of active disturbance rejection FCS-MPC strategy based on ESO

2.1 Design of speed observer based on ESO

According to the principle of stator current state Eq.(1) of PMSM and ESO, the first order state space equation of the system is constructed as

(2)

where

v1=[idiq],

u=[uduq],

h=[-ed/L-eq/L],

Andhin Eq.(2) is extended to the new state variablev2, whereqis present, andq(t) is bounded. The extended equation of Eq.(2) is

(3)

Eq.(3) is observable, and the second-order ESO of Eq.(3) can be constructed as

(4)

whereβ1>0,β2>0,β3>0, and arsh(·) is a smooth continuous nonlinear function.

In the second-order ESO of Eq.(4), there are

(5)

The observed value of ESO contains the information of speed, i.e.,

(6)

(7)

2.2 Design of active disturbance rejection speed regulator based on arc-hyperbolic sine function

ADRC consists of first-order tracking differentiator (TD), ESO and state error feedback control law. The speed tracking error can be reduced through the first step tracking differential. The system state and perturbation can be estimated by ESO in real time. The disturbance can be compensated by means of the state error feedback control law according to the separation principle. The structure is shown in Fig.2.

Fig.2 Speed controller based on ADRC of PMSM system

2.2.1 Design of first order tracking differentiator

A differential equation of first order speed tracking is constructed by using arsh(·)[20]as

(8)

whereωmis speed setpoint value,wis the transition variable, parametersb1>0,a1>0.a1is close to 1 in usual. The appropriate increase ofb1can increase the response speed and reduce the tracking error. The ideal response output ofωmcan be obtained by Eq.(8).

2.2.2 Design of second-order ESO

Let

x1=ωr,y2=x1.

(9)

According to the speed differential equation constructed by Eq.(1), the first order state space equation is constructed as

(10)

whereb0=1.5ψfP/J,y2is the input of speed controller,iqis the output of speed controller.

From Eq.(10), we can see that the load torque and viscous friction coefficient are changing interference signals. The various unknown disturbances are denoted byf(t) as

f(t)=-Bωr/J-TL/J.

(11)

(12)

The second-order extended state observer of the structural system constructed by Eq.(12) is

(13)

whereb2>0,b3>0,a2>0.a2is close to 1 in usual. The astringency of ESO is easy to be influenced by parameterb2. The extended state observer constructed by Eq.(13) can achieve accurate state variable estimation of the expansion Eq.(12) only whenb2-b3a2>0, that isz1(t)→x1(t),z2(t)→x2(t).

2.2.3 First-order active-disturbance rejection controller

The error feedback control rateu0(t) shown in Fig.2 is used to feed forward the interference signal. It is designed as

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(14)

where parametersb4>0,a3>0.a3is close to 1 in usual. A small ripple of control quantityucan be got by selecting the proper value ofb4.

2.3 FCS-MPC based on fast vector selection

According to Eqs.(1) and (6), the current space state vector differential equation under the selected rotation coordinate axis is constructed as

is=(us-Rsis)/Ls-z2,

(15)

whereis=[idiq] is current vector;us=[uduq] is voltage vector.

2.3.1 Current model predictive control based on fast vector selection

The traditional FCS-MPC calculates the predicted current for each fundamental voltage vector in the rotating coordinate system first, and then calculates the corresponding minimum objective function value based on the predicted current, finally selects the voltage vector of the minimum objective function as the best output of the inverter. But it needs to forecast current for 7 times, which makes the amount of calculation relatively large. In order to obtain better control effect, the FCS-MPC strategy based on fast vector selection is presented in Ref.[10], in which the optimal voltage vector is obtained fast by summing the average to estimate the back EMF. Therefore, complexity of the algorithm and the computation is obviously reduced. However, the estimation of back EMF will increase the complexity of the computation and system, and external interference will lead to estimation error. In this paper, the Eq.(5) is used to estimate the back EMF.

In this paper, the second-order Eulerian discretization method is applied to further simplify the Eq.(10)[10], that is

(16)

where

(17)

Letuk(k=1,2,…,6) be the the base voltage vector of 6 sectors, and as long as the amplitude of the error vector is the smallest, i.e.,

(18)

When the output basic voltage vector is zero vector, the amplitude of the error vector is

(19)

2.3.2 Control of delay compensation

(20)

The flow diagram of FCS-MPC based on fast vector selection is shown in Fig.3 with considering the delay compensation.

Fig.3 FCS-MPC flow chart based on fast vector selection

3 Simulation analysis

The model of Fig.1 is structured and simulated in Matlab/Simulink environment. The simulation parameters of PMSM system are shown in Table 1. The sampling period of the system is 10 μs. In order to verify the correctness and validity of the active-disturbing FCS-MPC method of PMSM system based on ESO, two FCS-MPC systems based on PI and ADRC are constructed by appling the same speed ESO parameters, and they are compared and analyzed.

Table 1 Parameters of PMSM

3.1 Comparison of anti-load ability

Adjusting PI parameters to obtain the same transient of the two systems so that they can meet the rationality comparison. When simulation, the speed setpoint of PMSM system is 1 000 r/min, the PMSM system starts with load (1 N·m) and is loaded to the rated load (3 N·m) at 0.2 s. The response curves of two systems are shown in Figs.4 and 5, respectively.

Fig.4 Dynamic response of speed sensorless PMSM system based on PI FCS-MPC strategy

Simulation analyses are as follows.

1) ESO speed estimation. As can be seen from Figs.4 and 5(a) and (b), both the FCS-MPC PMSM systems based on PI and ADRC, the ESO speed estimates and the actual curves are almost coincident, and the speed errors are small, i.e., the speed observer can be quickly tracked and has a high recognition accuracy.

2) Robustness of ative-disturbance FCS-MPC strategy for three-phase PMSM. Figs.4 and 5(a) show that the FCS-MPC PMSM system based on ADRC has smaller torque ripple and can be restored to the reference value, which means that it has a stronger anti-load capacity comparing with the FCS-MPC PMSM system based on PI. As can be seen from Figs.4 and 5(c), the FCS-MPC PMSM system based on PI has the smaller electromagnetic torque ripple compared to the FCS-MPC PMSM system based on ADRC.

Fig.5 Dynamic response of PMSM system based on ADRC with FCS-MPC strategy

3.2 Comparison of transient response characteristics

PI parameters are adjusted so that two FCS-MPC PMSM systems based on PI and ADRC have the same anti-load change capability as possible for which can meet the rationality comparison. When simulation, the speed setpoint of PMSM system is 1 000 r/min, the PMSM system start with load (1 N·m) and is loaded to the rated load (3 N·m) at 0.2 s. The response curves of two systems are shown in Figs.6 and 7, respectively.

The following results can be obtained by simulation analysis.

1) ESO speed estimation. As can be seen from Figs.6 and 7(a) and (b), both the FCS-MPC PMSM systems based on PI and ADRC, the ESO speed estimates and the actual curves are almost coincident, and the speed errors are small, that is, the speed observer can be quickly tracked and has a high recognition accuracy.

Fig.6 Dynamic response of FCS-MPC strategy on speed sensorless PMSM system based on PI

2) Robustness of active-disturbance FCS-MPC strategy for three-phase PMSM. From Figs.6 and 7(a), the FCS-MPC PMSM system based on ADRC has smaller overshoot and shorter control period when the resistance to load change is consistent, and it has better speed transient response characteristics. As can be seen from Figs.6 and 7(c), the FCS-MPC PMSM system based on PI has the smaller electromagnetic torque ripple compared to the FCS-MPC PMSM system based on ADRC.

Above all, the control method designed in this paper can guarantee the stable operation of the system when the load changes, and it can ensure the system has strong anti-load interference ability, low electromagnetic torque ripple, and good dynamic performance. It does better than PI control in the anti-load capacity and dynamic performance.

Fig.7 Dynamic response of FCS-MPC strategy of speed sensorless PMSM system based on ADRC

4 Conclusion

In this paper, an active-disturbance-rejection FCS-MPC strategy based on ESO speed sensorless of surface mounted PMSM was proposed. ESO based on arsh(·) is designed to estimate the speed and back EMF term accurately. Compared with the PI-based FCS-MPC system, it can ensure that the system has strong anti-load interference ability, small electromagnetic torque ripple and good dynamic performance, which can achieve satisfactory torque and speed control effect. Actual results indicate the correctness and feasibility of the proposed method.

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