Bao-jun QU, Qing-xin YANG, Yong-jian LI, Jing-song LI,Chang-geng ZHANG,Hong-qiang LI

(1State Key Laboratory of EFEAR, Hebei University of Technology, Tianjin 300130, China) (2State Key Laboratory of Control and Simulation of Power System and Generation Equipments, Tsinghua University, Beijing 100084, China) (3School of Mechanical Engineering, Shandong University of Technology, Zibo 255049, China)

Abstract: This paper explores the current flaw that drivers feel clumsy when turning the steering wheel quickly. The article analyzes the poor dynamic performance of weak magnetic control, based on the permanent magnet synchronous motor (PMSM) speed regulation for electric power steering system (EPS), and presents a comprehensive flux-weakening method based on feed forward and fuzzy PI. This method achieves a smooth transition from the adjustment of the direct-axis current to the adjustment of the quadrature-axis current by the flux-weakening control, and broadens the flux-weakening depth of the feed forward control. Finally, the validity of the method is verified by simulation analysis and hardware-in-the-loop test.

Key words: Electric power steering system (EPS), Fuzzy PI, Feed forward, Flux-weakening control

1 Introduction

EPS requires that the torque output of the auxiliary motor in various steering conditions can quickly and accurately track the target torque without steering hysteresis, which is different from position or speed servo system commonly used in industry. Since the PMSM cannot continue to rise when it reaches a certain rotational speed, it cannot provide the large rotational speed required to quickly operate the steering wheel [1, 2]. As a result, the steering feel is particularly heavy when the steering wheel is quickly operated. To solve this problem, the EPS system usually adopts a flux-weakening control algorithm, which includes feed forward control and feedback control [3-5]. The article [6] presents a weak feedback flux-weakening control algorithm. But the motor has a certain jitter when switching from the flux-weakening control to vector control. The article [7] presents two weak magnetic control methods, which belong to open-loop control and have poor system stability. The article [8] presents a six-step operation algorithm control to realize the dynamic performance of the current control, but the current will oscillate at the limit speed. So, this paper analyzes the cause of the poor dynamic performance of traditional flux-weakening control, and presents the EPS control system based on fuzzy PI flux-weakening control.

2 Analysis on flux-weakening control for pmsm in eps

The EPS control system controls the PMSM based on the preset curves of assist currentiqref(the target current ofq-axis), which is calculated by the preset steering torque signal and the vehicle speed signal. So that the output assist characteristic of EPS system satisfies the steering performance requirement of the vehicle.

In Fig.1 the block diagram of the electric drive system is shown. The control scheme is based on vector control. Each time moment currentsia,ib,icand voltagesVa,Vb,Vcare measured and then transformed ind-qreference frame. The transformations used in the control scheme have DC-like characteristics which are simple and efficient. Due to their simplicity, the computational time required is not important.

Fig.1 EPS control system based on flux-weakening control

The DC side supply voltage of the EPS is generally 12 V, but the rated current of the motor is up to tens of amps, even hundreds of amps. Compared with the phase voltage, the ratio of the voltage drop of the stator resistance is large. So the influence of the stator resistance cannot be ignored. Surface mounted PMSM used in this paper is considered that the armature inductances of the d-axis and q-axis are approximately equal. Therefore, the voltage and electromagnetic torque equations of the PMSM in thed-qreference frame are as follows:

(1)

(2)

Where,ud,uqisd- andq-axis components of stator voltage;id,iqisd- andq-axis components of stator current;Lisd- andq-axis components of armature inductance;Rsis resistance of armature winding;ωris rotor (electrical) angular velocity;Ψfis flux linkage of the PMSM;Pnis pole pair number of the motor;Teis electromagnetic torque caused by the flux linkage.

It can be observed from Eq.(2) that the electromagnetic torque of the PMSM is linear with theiq, and the torque control can achieve the same control quality as the DC motor. Since the EPS system requires less power, but it has higher requirements on the capacity of overload and the response performance of torque. So the maximum torque current ratio control is used in the constant torque area, and the flux-weakening control is used in the constant power area.

When the PMSM is running, the amplitudes of the phase voltageUsand phase currentisare limited by the limit voltageUlimand the limit currentIlimthat the controller can output:

(3)

(4)

(5)

It can be observed from Eq.(5) that the voltage constraint of the motor running is a cluster of circles on thed-qcurrent plane. The radius and center of these circles vary with speed. So the running area and current optimal control track of PMSM are shown in the Fig.2.

Fig.2 The running area and current optimal control trajectory of PMSM

Below the base speed,id=0 is used to ensure that the motor operates in region of the MTPA (Maximum Torque Per Ampere). The running track of the stator current is mainly limited by the current limit circle, that is, the current is running along the MTPA curve (line “bao”).

Above the base speed, the flux-weakening control is used to ensure maximum torque obtained from the intersection of the voltage and current limit circles, such as point c,d. Therefore, In order to ensure that the torque-current ratio is maximized, thed-qcurrents should be as follows according to the speed:

(1) When the motor is running at a lower speed, it enters the flux-weakening area I. For example, the speed of motor is 2 000 r/min;d-qcurrents use the point a, the intersection of the voltage limit circle and the q current. And in the case of low torque and no load, the d current is zero, as in the constant torque region. So the best track is cao.

(2) When the motor is running at a higher speed, it enters the flux-weakening area II. For example, the speed of motor is 3 000 r/min; but there is no intersection of the voltage limit circle and the q current. The optimum track remains on the voltage limit circle MTPV (Maximum Torque Per Voltage, arc “de”), and a negative d current component is used even during no-load operation, such as point e.

3 Flux-weakening controller based on fuzzy PI control algorithm

3.1 Traditional flux-weakening control

Fig.3 The schematic diagram of voltage negative feedback flux weakening control

3.2 A novel flux-weakening control strategy based on feed forward control and fuzzy PI control

As shown in Fig.4, when the speed is above the base speed, the motor runs in the flux-weakening area, the feed forward control module gives the corresponding d currentidffaccording to the current speed and theqcurrent command to improve the dynamic response capability of the flux-weakening controller. The fuzzy PI controller is designed mainly to eliminate the deviation of thedcurrent caused by the change of motor parameters.

Fig.4 Flux-weakening controller based on feed forward control and fuzzy PI

According to the current working state of the motor, feed forward controller generates directly the required flux-weakening current and does not need to wait until the value of a system variable have changed. So it has good dynamic response performance and does not affect the stability of the original system. In order to improve the dynamic response performance, an improved feed forward controller is proposed. Thed-qreference currents given by the controller are directly calculated by the formula according to the optimal current track. The following equations are the optimal control tracks of the stator current when the motor runs in different states.

(6)

(7)

(8)

(9)

When the motor is working in the constant torque area, the motor current vector should run along the MTPA curve, and then thed-qreference currents should be calculated according to the Eq.(6). When the motor is working in flux-weakening area I, if the reference current vector is located in the current limit circle, thed-qreference currents should be calculated according to Eq.(7). Otherwise, the reference currents should be recalculated according to Eq.(8). When the motor is working in flux-weakening area II, thed-qreference currents should run along the MTPV curve, and be calculated by Eq.(9).

Fuzzy adaptive controller is used to adjust the PI parameters according to the real-time state of the motor. Its input is the erroreof the reference voltage vector outputted by the current regulator andUlim，and the rate of change of the error Δe. When the motor is running, the fuzzy controller deals with the input variables by fuzzy logic and then outputs the proportion correction coefficientΔKpand the integral correction coefficient ΔKi[11]. According to the two coefficients and the original PI parameters coefficients, two parameters of PI controller can be obtained as follows:

(6)

The principle of the fuzzy control rule should be formulated in this paper as follows: When the voltage erroreis larger, the adjustment of the fuzzy output variable should be based on reducing the overshoot and eliminating the steady-error. The fuzzy domain where the linguistic variablese,Δe, ΔKpand ΔKiare located is divided into 7 fuzzy sets [-3, -2, -1, 0, 1, 2, 3], and the corresponding fuzzy language values are NB, NM, NS, ZO, PS, PM, PB. According to the above PI parameter adjustment principle and the actual debugging experience, this paper establishes the fuzzy control rule table of variables ΔKpand ΔKioutputted by fuzzy PI flux-weakening controller, as shown in the table.

Fuzzy adaptive PI controller can not only provide the flux-weakening current needed in steady state, but also ensure the output torque of motor.

4 Experimental results

In order to verify the feasibility of the proposed flux-weakening control strategy under EPS, the simulation analysis and experimental verification are carried out according to the motor parameters in Table 3.

Table 1 The fuzzy control rule table of variables

Table 2 The fuzzy control rule table of variables

Table 3 Motor parameters

4.1 Simulation analysis

The simulation model is built in Matlab/Simulink, including mechanical steering system model, controller model and PMSM model. The step signal is used as the input of the steering wheel torque and the given speed is 60 km/h. The simulation waveform shown in Fig.5 is obtained.

Fig.5 Torque simulation waveforms

As can be seen from the figure, compared with the traditional flux-weakening control, the new field-weakening control proposed in this paper makes that the response time of the PMSM output torque is faster, and the overshoot obviously is smaller, and the amplitude is accelerated to a smoother level. So the jitter problem of steering wheel can also be improved.

4.2 Hardware-in-the-loop test

In order to verify the effect of the flux-weakening control strategy proposed in this paper, it was verified on the EPS hardware-in-the-loop test bench，which is mainly made up of mechanical steering systems, input motor, load motor, PMSM, EPS controller, and test system based on CAN signal, etc., as shown in Fig.6.

Fig.6 Hardware-in-the-loop test bench

Fig.7 shows the response curve of thed-qcurrents under the traditional flux-weakening control when the motor speed changes rapidly. As shown in Fig.7 (b), there is a large overshoot in the response process of the motor speed when the motor speed reaches 2 600 r/min from standstill within 0.08 s. As shown in Fig.7(a), when the speed of the motor changes rapidly, the motor has entered the flux-weakening area, but the field-weakening controller has not yet given an appropriate reference current for the change of speed, so that the current regulator is saturated, and the q current is violently oscillated. The oscillation of the q current will cause the output torque of the motor to fluctuate greatly, and eventually affect the driver’s comfort of steering control.

Fig.8 shows the response curve of thed-qcurrents under the new flux-weakening control when the motor speed change rapidly. The experimental conditions are basically the same as those in Fig.7. It can be seen from Fig.8 (a) that when the speed of the motor changes rapidly, thed-qcurrent of the motor can accurately track its target value. In the dynamic process, the two-axis current not only responds quickly, but also has a smooth transition without large fluctuations.

Fig.7 D-q current response curves under traditional flux-weakening control

Fig.8 D-q current response curves under new flux-weakening control

5 Conclusion

This paper presents the EPS control system based on fuzzy PI flux-weakening control by analyzing the cause of the poor dynamic performance of traditional flux-weakening control. The simulation analysis and the hardware-in-the-loop test prove that the control strategy can improve the motor speed response to effectively track the input of speed when the driver quickly manipulates the steering wheel, and has a certain improvement effect on the jitter caused when the driver’s quick steering.