基于 ESO 的 PMSM 系统自抗扰 FCS-MPC 策略

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].

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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.

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1 Mathematical model of PMSM

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

(1)

where id, iq and ud, uq are d, q axis currents and voltages of stators; Rs is stator resistor; Ls is stator inductance; ed, eq is back EMF; ωr is rotor velocity; ωe is electrical angular velocity; P is pole pairs; ψf is rotor permanent magnetic flux; TL is load torque; J is moment of inertia; and B is 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.

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=[id iq],

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

And h in Eq.(2) is extended to the new state variable v2, where q is present, and q(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)

2.3.2 Control of delay compensation

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

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(5)

where are the estimated values of id, iq; are the estimated values of counter electromotive items.

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

(6)

where are the estimated values of ψd, ψq. The estimated value and position can be obtained as

(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.

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)

where ωm is speed setpoint value, w is the transition variable, parameters b1>0, a1>0. a1 is close to 1 in usual. The appropriate increase of b1 can increase the response speed and reduce the tracking error. The ideal response output of ωm can 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)

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

f(t)=-Bωr/J-TL/J.

(11)

Let x2(t)=f(x1), and x2(t) is extended variable, that is The first order system constructed by Eq.(10) is extended to

(12)

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

其中用来表示拟合优度的为调整后的判定系数R方为0.975，这说明有97.5%的影响因素可以由此模型来加以解释，相应的各自变量的系数显著性概率值全都小于0.05，所有的T值均大于临界值T（27-3-1）=2.069，表示结果在α=0.05的水平上是显著的。此外，该回归模型的DW值为1.4，说明随机误差项不存在一阶序列相关；AIC值为-0.63较小意味着滞后阶数较为合适。总体表明所建立的回归模型比较好。

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(13)

where b2>0, b3>0, a2>0. a2 is close to 1 in usual. The astringency of ESO is easy to be influenced by parameter b2. The extended state observer constructed by Eq.(13) can achieve accurate state variable estimation of the expansion Eq.(12) only when b2-b3a2>0, that is z1(t)→x1(t), z2(t)→x2(t).

2.2.3 First-order active-disturbance rejection controller

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The error feedback control rate u0(t) shown in Fig.2 is used to feed forward the interference signal. It is designed as

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where b0=1.5ψfP/J, y2 is the input of speed controller, iq is the output of speed controller.

(14)

where parameters b4>0, a3>0. a3 is close to 1 in usual. A small ripple of control quantity u can be got by selecting the proper value of b4.

u=[ud uq],

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)

where is=[id iq] is current vector; us=[ud uq] 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

In Eq.(16), the ideal voltage vector is obtained when i.e.,

(17)

Let uk(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

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(19)

The method can be summarized as follows: (1) is calculated by Eq.(17). (2) Judging the sector k (k=1,2,…,6) that located by Eq.(18). (3) Comparing e0 with euk by Eqs.(18) and (19). If e0>euk, the best voltage vector is euk, otherwise, the best voltage vector is e0.

where β1>0, β2>0, β3>0, and arsh(·) is a smooth continuous nonlinear function.

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The actual output voltage of the controller lags the current change in the actual digital system control[22]. To eliminate the negative effect of the system control performance by the delay, is calculated according to Eq.(36) in Ref.[14], and then the voltage is compensated at k+1 by appling the second-order Euler discretization method, which compensates the delay of the system.

(20)

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

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.

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.

Simulation analyses are as follows.

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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.

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.

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.

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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