Predictive current control system of PMSM based on LADRC

Wang Xiaopeng Zhao Jun Wang Bohui Li Baomin

(School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China)

Abstract：For a permanent magnet synchronous motor (PMSM) model predictive current control (MPCC) system, when the speed loop adopts proportional-integral (PI) control, speed regulation is easily affected by motor parameters, resulting in the inability to balance the system robustness and dynamic performance. A PMSM optimal control strategy combining linear active disturbance rejection control (LADRC) and two-vector MPCC (TV-MPCC) is proposed. Firstly, a mathematical model of a PMSM is presented, and the PMSM TV-MPCC model is developed in the synchronous rotation coordinate system. Secondly, a first-order LADRC controller composed of a linear extended state observer and linear state error feedback is designed to reduce the complexity of parameter tuning while linearly simplifying the traditional active disturbance rejection control (ADRC) structure. Finally, the conventional PI speed regulator in the motor speed control system is replaced by the designed LADRC controller. The simulation results show that the speed control system using LADRC can effectively deal with the changes in motor parameters and has better robustness and dynamic performance than PI control and similar methods. The system has a fast motor speed response, small overshoot, strong anti-interference, and no steady-state error, and the total harmonic distortion is reduced.

Key words：permanent magnet synchronous motor(PMSM); two-vector model predictive current control; linear active disturbance rejection control; speed control system

Permanent magnet synchronous motors (PMSMs) are widely used in servo and high-performance speed regulation systems due to their high efficiency, high power, and low loss[1-2]. Vector control and direct torque control are two typical control strategies for a PMSM. However, as vector control, its closed-loop performance is restricted due to the conflict between the linearity of the proportional-integral (PI) controller adopted in its current loop and the rotation speed loop and nonlinearity of the PMSM. For the direct torque control, the hysteresis loop controller adopted in its inner loop results in an increased ripple of the electromagnetic torque and three-phase current. In recent years, with the rapid development of microprocessors, predictive control has become a novel nonlinear control strategy for PMSMs, owing to its fast dynamic response and high control precision. Model predictive control (MPC) has become a high-performance control scheme because of its merits such as fast dynamic response, simplicity, and no requirement for the parameter tuning of the current loop[3-6]. MPC mainly includes model predictive torque control and MPCC. Compared with the former control, MPCC has a value function whose control variable only involves current. Without considering the weight of different control variables, it can achieve the fast tracking of current overshoot, which is highly favorable for actual control. To further improve the static performance of the PMSM control system, domestic and foreign scholars have performed a substantial amount of research on the two-vector model predictive current control (TV-MPCC) strategies of PMSMs[7-9].

APMSM is a typical nonlinear multi-variable strongly coupled system. Its predictive current control system generally adopts a PI controller as its rotation speed loop controller. However, due to the complexity of the PMSM’s rotation speed control model, a mathematical model is not precise enough. Moreover, PI control is sensitive to the parameter changes of the PMSM model, resulting in the poor dynamic performance and robustness of the speed regulation system. In response to the inherent defects of the conventional PI, Han[10]proposed active disturbance rejection control (ADRC), which is characterized by simple regulation, high accuracy, real-time estimation, and compensation of internal and external disturbances without the need for an accurate and controlled mathematical model system. In recent years, ADRC has been widely applied to the PMSM’s control system. Zhou et al.[11]suggested that a speed control system with ADRC showed better dynamic and static performance than the speed control system with PI control in the vector control of PMSMs. Hezzi et al.[12]investigated the speed tracking capability and elastic control performance of the ADRC technique under different operating conditions based on a five-phase PMSM for electric vehicles. Accordingly, ADRC can ensure higher dynamic performance and robustness of a system than PI control. Zhang et al.[13]proposed an extended state observer (ESO)-based predictive control strategy for a PMSM’s active disturbance rejection model to reduce the ripple of electromagnetic torque, which demonstrates a satisfactory performance of the rotation speed tracking and disturbance rejection. However, ADRC cannot be widely applied due to its complex parameter calibration and nonlinear function. Accordingly, Gao[14]simplified nonlinear ADRC into linear ADRC (LADRC), which has been applied to power systems[15].

This paper attempts to design a one-order LADRC controller that consists of the following parts: linear extended state observer (LESO) and linear state error feedback (LSEF)[16]. The innovative LADRC controller provides intuitive parameter calibration and significantly reduces computational effort, achieving the same high performance as nonlinear ADRC, making it more suitable for PMSM speed loop control system[17-18]. The proposed control strategy can increase the disturbance rejection capacity and robustness of the PMSM’s predictive current control system.

1 PMSM Mathematical Model

Based on the magnetic field orientation theory, ignoring the hysteresis loss of the PMSM, the state equation of the stator current for the table-posted PMSM in the synchronous rotation coordinate system is

(1)

whereidandiqare the stator current,udanduqare the stator voltage, respectively;Ld=Lq=Lsdenote the stator’s winding inductance;Rsis the stator resistance;ψfis the permanent magnet flux linkage; andωris the rotor electrical angular velocity.

The mechanical motion equation of the motor is

(2)

whereTLandTeare the load torque and electromagnetic torque, respectively;Jis the rotational inertia; andFis the rotor viscous friction coefficient. Inωr=pωm,ωmrefers to the rotor mechanical angular velocity, andpis the number of pole pairs.

The electromagnetic torque equation is

Te=1.5pψfiq

(3)

2 PMSM TV-MPCC

A two-level three-phase voltage source inverter may generate eight basic switch states: Six valid voltage vectorsU1(001),U2(010),U3(011),U4(100),U5(101),U6(110) and two zero vectorsU0(000) andU0(111). They can determine the limited number of output current possibilities for the next sampling interval. A traditional MPCC only selects one optimal voltage vectorVopt1as the output voltage. However, the electromagnetic torque and stator current demonstrate obvious ripples. With TV-MPCC, another voltage vector is selected within one control period to determine the second optimal voltage vectorVopt2. The optimal voltage vectorVopt1and the eight voltage vectors are combined and pre-allocated to the action time of two voltage vectors in each combination. Therefore, the selected voltage vector is accurate. It can effectively improve the current tracking accuracy and obtain good static performance. TV-MPCC aims to minimize the error between the prediction current value and the set current value by calculating the value function. Withd-qaxis currents as the control quantities, the value function can be established as follows:

(4)

For state Eq.(1), the Eulerian method is adopted to obtain the discretizedd-qphase current prediction equation:

(5)

wheretg1andtg2are the action times ofVopt1andVopt2within a control cycleTS, respectively,tg1=Ts-tg2;i′d1(k),i′d2(k),i′q1(k) andi′q2(k) denote the current slope ofdandqaxis components with the actions ofVopt1andVopt2. The calculation is as follows:

(6)

(7)

wheren=1,2，udnanduqndenote thedandqaxis voltages ofVopt1andVopt2, respectively. Combining the equation above, the action time ofVopt1is obtained:

(8)

tg2=TS-tg1

(9)

When the motor is started, there may be circumstances where the calculation results oftg1andtg2do not fall within 0 andTs. At this time,tg1andtg2should be reallocated. There are mainly two circumstances: iftg1<0, then,tg1=0; iftg1>Ts, then,tg1=Ts. The structure block diagram of the PMSM TV-MPCC is shown in Fig.1.

Fig.1 Structure block diagram of the PMSM TV-MPCC

3 Design of the LADRC Controller

3.1 LADRC rotation speed loop controller

For the surface-mount PMSM (id=0) with the givenLd=Lq, the following expression is obtained from Eqs.(2) and (3):

(10)

The disturbance is set as

(11)

Eq.(10) can be rewritten as

(12)

Fig.2 Structure diagram of the designed one-order LADRC

3.2 LESO

LESO is the core part of the LADRC controller. It is mainly applied to observe the actual value of the disturbance actions inside and outside the system and compensate for the feedback to eliminate the influence of disturbances and improve the performance of the control system.

The equation of the controlled object is set as follows:

(13)

whereyandudenotes the output and input;tis the system’s time-varying state; andbis the gain of the controlled variable. The true value ofbis hard to estimate in real systems. Therefore, Eq.(13) can be written as

(14)

wheref(y,ω,t) is the total internal and external disturbances.

(15)

(16)

Substitute the parameters forA,B,LandCinto Eq.(16), and the LESO equation can be obtained as follows:

(17)

3.3 LSEF

The traditional proportional-integral-derivative (PID) is a simple weighted sum of the error’s proportion, integral, and differential. As the LESO can achieve real-time estimation and compensate for internal and external disturbances, there is no need to employ traditional PID integration to eliminate residual errors under constant disturbances. Similar to a PI controller, the LSEF is mainly used to control the reference input, error feedback, and offset disturbance. It converts a complex mathematical model into an integral series model, thus eliminating the negative influence of the integrator. The control law can be expressed as

(18)

wherekpis the gain coefficient. Supposing thatωcis the bandwidth of the LADRC controller, the high-performance tracking of the signal can be achieved whenkp=ωc.

3.4 Parameter calibration strategy

In a rotation speed loop’s one-order LADRC controller, only three parameters need to be calibrated: bandwidthωcof the controller, bandwidthωoof the observer, and compensation coefficientb0.ωcdetermines the controller’s response speed. Moreover, the greater theωc, the better the control effect. However, excessiveness may cause system divergence.ωodetermines the tracking speed of the LESO, where the greater theωo, the faster the LESO’s estimation of disturbance. However, the system is still sensitive to noise. Therefore, a balance between the observer’s parameters and the system’s sensitivity to noise must be found.b0represents the characteristics of the controlled object and can be directly exported based on those characteristics.

But the king was left raging in his empty palace, and he called together his army, and got ready his ships of war, in order that he might go after the vessel and bring back what had been taken away

Gao[14]proposed a strategy that simplifies the LADRC parameter adjustment to the bandwidth parameter adjustment problem. Therefore, for most engineering projects, the relationship betweenωoandωccan be set asωo≈(3～5)ωc. In simulations, appropriate parameters can be found through multiple comparative analyses using cut-and-try methods.

4 Simulation Results and Analysis

To verify the correctness and effectiveness of the proposed control strategy, the performance test of the PMSM system with the TV-MPCC as the current loop and the LADRC controller as the speed loop was performed on the MATLAB/Simulink platform, and the results were compared with those of the TV-MPCC PMSM where the speed loop adopts a PI controller. The general block diagram of the PMSM’s control system is shown in Fig.3. The selected main parameters of the PMSM are shown in Tab.1. The system sampling cycleTsis 50 μs, and the base frequency of both systems is 133.33 Hz. The LADRC parameters are as follows:ωo=1 950,ωc=480, andb0=1 734.5.

Fig.3 General block diagram of the PMSM’s control system

Tab.1 Selected main parameters of the PMSM

The PI parameters were adjusted at the same simulation conditions so that the two TV-MPCC PMSM systems that adopt PI and LADRC achieve the same dynamic performance as much as possible. Accordingly,kp=0.20 andki=0.014 can be obtained. Fig.4 shows the system response of the TV-MPCC PMSM with PI.

4.1 Simulation results with the PMSM parameter match

To solidly and comparatively analyze the dynamic performance of the two systems, the PI parameters were

(a)

(b)

(c)

adjusted so that the two TV-MPCC PMSM systems using PI and LADRC can achieve the same anti-interference performance as much as possible. Accordingly,kp= 0.90 andki= 0.70 can be obtained. The simulation conditions are as follows: When the motor operated at a stable speed of 2 000 r/min without load, 8 N·m load disturbance was abruptly added at 0.1 s. Figs.4 and 5 show the corresponding response curves of the two systems.

Figs.4 and 5 show that when the rotation speed loop has the same anti-load disturbance capacity, the fluctuation of the rotation speed wave and the overshoot are great with the PI controller during the initial startup period of the motor. It takes a long transition time whenTs= 16 ms. However, as the LADRC controller was adopted, the overshoot was smaller with the transition time (Ts=8 ms), and the response was faster. Figs.4(b) and 5(b) show the waves of the three-phase current (ia,ib, andic) under the control of the PI and LADRC, respectively, in a stable state after 0.16 s. A fast Fourier transform (FFT) analysis was conducted foria, and one cycle after 0.16 s was selected. As shown in Figs.4(c) and 5(c), the THDs controlled by the PI and LADRC are 3.61% and 3.32%, respectively.

The robustness of the LADRC and PI controllers are compared in Figs.5 and 6. When the speed loop basically has the same dynamic performance, the rotation speed controlled by PI was reduced by 3.35% as an abrupt 8 N·m load was added at 0.1 s, as shown in Fig.6(a). At the same time, when the rated rotation speed was resumed, there was a 2.4% steady-state error. In Fig.5(a), however, the rotation speed is reduced by only 1.62%. Furthermore, when the rated rotation speed is resumed, there is no steady-state error. Figs.5(b) and 6(b) show the wave of three-phase current (ia,ib, andic), respectively, under the control of the LADRC and PI in a stable state after 0.16 s. The FFT analysis was conducted foria, and one cycle after 0.16 s was selected.

(a)

(b)

(c)

(a)

(b)

(c)

The THDs controlled by the PI and LADRC were 4.17% and 3.32%, respectively.

To further test the application of the designed LADRC in the high-performance PMSM speed regulation system, the control strategy was simulated in comparison with that presented in Ref.[13], in which the control strategy speed loop employs the conventional active disturbance rejection controller under the same simulation conditions. The PMSM system response of Ref.[13] is shown in Fig.7.

(a)

(b)

(c)

Comparing Figs.5(a) and 7(a), the proposed control strategy demonstrates better transient response features and disturbance rejection control than that of Ref.[13]. Figs.5(b) and 7(b) show the three-phase currents (ia,ib, andic) of the proposed control strategy and those in Ref. [13] in a steady state after 0.16 s, respectively. The FFT analysis also demonstrates that the THD of the phase current is small. Tab.2 presents the control system performance of the speed loop with different control strategies.

Tab.2 Control system performances of the rotation speed loop adopting different control strategies

4.2 Simulation results with the PMSM parameter mismatch

During the actual running of the motor, due to factors such as temperature rise and magnetic saturation, the PMSM parameters may deviate. To verify the effectiveness of the proposed LADRC controller in the case of mismatched motor parameters, under the same conditions as in Section 4.1, the actual values of the inductance, rotor flux linkage, and stator resistance in the control strategy are set to be 1.5, 1.2, and 0.8 times their nominal values, respectively. Fig.8 shows the response of the PI-based TV-MPCC PMSM system, wherekp=0.90 andki=0.70. Fig.9 presents the corresponding response whenkp=0.20 andki=0.014.

Fig.10 shows the corresponding system response of the LADRC-based TV-MPCC PMSM system in the case of a motor parameter mismatch. Comparing Figs.8 and 9, the motor speed response time of the proposed control strategy is short, the steady-state error is small, and the simulation is close to the nominal value of the motor parameters. Based on the FFT analysis onia(see Fig.10(c)), the THD of the phase current is small. Hence, the system under the proposed control strategy has good performance, can effectively deal with the changes in motor parameters, and has good robustness when the motor parameters do not match.

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

(c)

5 Conclusions

1) A control strategy employing the LADRC controller as the speed loop of the PMSM predictive current control system is proposed, which effectively overcomes the contradiction between PI control robustness and dynamic performance. The proposed LADRC controller is of simple design with intuitive parameters without reliance on precise mathematical models. Moreover, it improves the disturbance rejection capacity and robustness of the PMSM’s predictive current control system.

2) Based on the simulation and comparative analysis results of two TV-MPCC PMSM systems based on PI and LADRC, the comprehensive performance of the rotation speed loop based on the LADRC is better than that of the PI controller in terms of its fast response speed, low startup overshoot, high disturbance rejection performance, and non-steady-state error. The LADRC achieves more satisfactory results than the PI controller in controlling speed and current.

3) The proposed control strategy can be applied in the fields of aerospace, wind power systems, new energy vehicles, home appliances, and elevator control. However, due to the addition of the LADRC controller in the proposed control strategy, the computational complexity of the sampling period is increased. Nonetheless, the parameter setting of the LADRC controller is relatively vague, and further research is required.