An Improved Sliding Mode Control Method to Increase the Speed Stability of Permanent Magnet Synchronous Motors

Abstract

A permanent magnet synchronous motor (PMSM) plays an important role in the operation performance of an electric vehicle. In order to increase the speed stability of PMSMs, an improved sliding mode control (SMC) method is proposed in this paper. Firstly, the control process of a PMSM is divided into parts, which are the torque closed-loop control and speed closed-loop control. Secondly, an integrated control method with proportional integral (PI) and inverter switching frequency is adopted in order to increase the torque performance of the PMSM. Lastly, compared with the traditional SMC method, an improved SMC method with saturation function is proposed in order to decrease the speed fluctuation of the PMSM from 7.7% to 5.9%, thereby increasing the speed stability of the PMSM. The results of the experimental test indicate that both the speed stability and starting performance of the PMSM are increased by the improved SMC method.

1. Introduction

Compared with the traditional induction motor, permanent magnet synchronous motors (PMSMs) are widely used in electric vehicle drive systems and aerospace systems, mainly due to advantages such as their simple rotor structure, high speed, high torque, and so on. However, due to the high stability requirements of the driving motor for electric vehicles, especially for aerospace systems, it is very necessary to develop a better control method to reduce the speed fluctuation of the driving motor, so as to improve the stability of its speed.

In order to improve the operation stability of PMSMs, previous studies [1,2] proposed a fuzzy auto disturbance rejection speed control method, and the feasibility of this control method was verified by MATLAB/Simulink simulation software. In one study [1], the simulation results indicated that the disturbance rejection control method eliminated overshoots of the PMSM’s speed and resulted in a faster response time (about 0.6 s) during the starting process of the PMSM. In another study [2], the stabilization problem of T–S fuzzy system was investigated by the integral sliding-mode control method, and the global asymptotic stabilization criteria was established from the perspective of the $H_{\infty }$ performance index. Several studies [3,4,5] proposed one kind of SMC method with the variable structure adaptive reference model, which limits the upper and lower bounds of torque of the motor; the validity of this method to decrease the speed fluctuation of the motor was verified by theoretical analysis. The experimental results of one study [3] show that the induction motor had a good and fast speed response (about 1.2 s), which was similar to the simulation results, but the detailed speed stability was not analyzed. For another study [5], the stator currents of a six-phase induction motor based on the discrete sliding model control method was investigated in detail, including the presence of uncertainties, perturbations, and unmeasurable rotor currents. Furthermore, some other control methods, such as the wavelet analysis, cross-coupling current control, neural network control, and so on, were also utilized in the speed stability control of the motor [6,7,8,9]. If the speed response of a PMSM is decreased to less than 0.5 s, and the speed stability of the PMSM is directly analyzed by data (not the internal performance, such as the stator current, $H_{\infty }$ performance index, and so on), research on the PMSM will be further specified.

In the above references, although the SMC method showed advantages in the form of high anti-jamming capability and a simple structure, the fluctuation is still its disadvantage, especially in the control operation of the motor [10,11,12]. Therefore, some other improved SMC methods have been proposed for the operation control of the motor [13,14]. However, these references either put the focus on the simulation research or experimental test investigation by only one method. If the starting performance and speed stability of a PMSM are both analyzed by the simulation and experimental test, especially the performance comparison by different SMC methods, then the unity of theory and practice for PMSM’s stable operation can be achieved.

In this paper, an improved SMC method based on the saturation function is proposed in order to improve the speed stability of a PMSM. Firstly, the control process of a PMSM is divided into a torque closed-loop control and speed closed-loop control. Secondly, via simulation and hardware experiments, the optimal proportional integral (PI) parameters of the torque closed-loop control of the PMSM are selected. Finally, an improved SMC method based on the saturation function is proposed for the speed closed-loop and stability control of the PMSM. The simulation and experimental results indicate that the improved SMC method based on saturation function can further improve the speed stability of the PMSM, and the starting performance of the PMSM is also increased.

2. Control Model and Equation of the PMSM

2.1. Control Model of the PMSM

The structure of SMC model of the PMSM is shown in Figure 1, which consists of a torque closed-loop control and speed closed-loop control [5,7,14]. Generally, the SMC steps of the PMSM are as follows. Firstly, the parameter setting of the torque closed-loop control is preliminarily determined by the PI control method. Secondly, based on the SMC method of symbolic function and saturation function, the speed closed-loop control of the PMSM is researched. Finally, the above control methods are simulated and verified by experimental testing.

2.2. Torque Closed-Loop Control of PMSM

The motor involved in this paper is an interior-type PMSM, where the permanent magnets are embedded in the rotor core of the PMSM, as illustrated in Figure 2. From Figure 2 it can be concluded that the magnetization direction of adjacent permanent magnets in the rotor core is opposite. For the interior-type PMSM, its mathematical equations of torque closed-loop control are described as follows.

If neglecting the electromagnetic losses of rotor core and stator core [15], the PMSM’s KVL equations in the d-q coordinate system (i.e., the synchronous rotation coordinate system) can be expressed as follows:

$$\left\{\begin{array}{l}{u}_d={Ri}_d+L_d\frac{d}{dt}{i}_d-{\mathsf{\omega }}_e{L}_q{i}_q\hfill \\ u_q={Ri}_q+L_q\frac{d}{dt}{i}_q+{\mathsf{\omega }}_e\left(L_d{i}_d+{\mathsf{\psi }}_f\right)\hfill \end{array}\right.$$

(1)

where $u_d$ and $u_q$ are the voltages, $i_d$ and $i_q$ are the currents, $L_d$ and $L_q$ are the inductances, $R$ is the resistance, ${\mathsf{\omega }}_e$ is the electrical angle, and ${\mathsf{\psi }}_f$ is the flux linkage that is generated by the permanent magnets.

Ignoring the coupling terms (${\mathsf{\omega }}_e{L}_q{i}_q$ and ${\mathsf{\omega }}_e{L}_d{i}_d$) and dynamic item (${\mathsf{\omega }}_e{\mathsf{\psi }}_f$) of the d-q coordinate system, which will also be considered in the simulation process and experimental test, then Equation (1) can be simplified as follows:

$$\left\{\begin{array}{l}{u}_d={Ri}_d+L_d\frac{d}{dt}{i}_d\hfill \\ u_q={Ri}_q+L_q\frac{d}{dt}{i}_q\hfill \end{array}\right.$$

(2)

Under the assumption of zero initial conditions, and after the Laplace transform of Equation (2), then the transfer function relationship between the current and voltage can be obtained as follows:

$$\left\{\begin{array}{l}{G}_d\left(s\right)=\frac{i_d\left(s\right)}{u_d\left(s\right)}=\frac{1}{L_{d}s+R}\hfill \\ G_q\left(s\right)=\frac{i_q\left(s\right)}{u_q\left(s\right)}=\frac{1}{L_{q}s+R}\hfill \end{array}\right.$$

(3)

Then, based on the classical PI parameter tuning method of the torque closed-loop control of the motor [16], and considering the switching cycle of the inverter, the open-loop transfer function of the q-axis voltage of the PMSM can be written as follows:

$${\left.G_q\left(s\right)=\frac{k_{pq}\left({\mathsf{\tau }}_{i}s+1\right)}{{\mathsf{\tau }}_{i}s}\frac{k_{pwm}}{T_{pwm}s+1}\frac{1}{L_{q}s+R}\right|}_{{\mathsf{\tau }}_i=k_{pq}/k_{iq}}$$

(4)

In Equation (4), $k_{pq}$ and $k_{iq}$ are the proportional coefficient and integral coefficient of PI parameter tuning, respectively, $k_{pwm}$ is the magnification of the inverter, $T_{pwm}$ is the switching cycle of the inverter, and $\frac{1}{L_{q}s+R}$ is the transfer function.

With the classic automatic control theory [17], the closed loop transfer function of q-axis voltage Equation (4) can be written as follows:

$${\mathsf{\Phi }}_q\left(s\right)=\frac{k_{pq}{k}_{pwm}}{L_q{T}_{pwm}{s}^2+L_{q}s+k_{pq}{k}_{pwm}}=\frac{k_g}{s^2+\frac{1}{T_{pwm}}s+k_g}\left|{}_{k_g=\frac{k_{pq}{k}_{pwm}}{L_q{T}_{pwm}}}\right.$$

(5)

Comparing Equation (5) and the classical second-order closed-loop transfer function $\mathsf{\Phi }\left(s\right)=\frac{{\mathsf{\omega }}^2}{s^2+2\mathsf{\zeta }\mathsf{\omega }s+{\mathsf{\omega }}^2}$, the PI parameter tuning results of the closed-loop transfer function of q-axis voltage Equation (5) can be written as follows:

$$k_{pq}=\frac{L_q}{4{\mathsf{\zeta }}^2{T}_{pwm}{k}_{pwm}}$$

(6)

$$k_{iq}=\frac{k_{pq}}{{\mathsf{\tau }}_i}=\frac{R}{4{\mathsf{\zeta }}^2{T}_{pwm}{k}_{pwm}}$$

(7)

According to the classical motion control theory, if the damping coefficient is 0.707, then the adjustment time (which also means the time that the motion system enters the steady state) is shortest, and the overshoot is smaller. Therefore, it is assumed that the amplification factor of the inverter is $k_{pwm}=1$, and the damping coefficient is $\mathsf{\zeta }=0.707$, then the PI parameter tuning result of Equations (6) and (7) can be simplified as follows:

$$\left\{\begin{array}{c}{k}_{pq}=\frac{L_q}{2T_{pwm}}\\ k_{iq}=\frac{R}{2T_{pwm}}\end{array}\right.$$

(8)

Similarly, from Equation (3) it can be concluded that the d-axis voltage equation is consistent with the q-axis voltage equation; therefore, the PI parameter tuning result of the closed-loop transfer function of the d-axis voltage equation can be written as follows:

$$\left\{\begin{array}{c}{k}_{pd}=\frac{L_d}{2T_{pwm}}\\ k_{id}=\frac{R}{2T_{pwm}}\end{array}\right.$$

(9)

2.3. Speed Closed-Loop Control of the PMSM

According to the principle of mechanical motion [18,19], the motion equation of the PMSM can be written as follows:

$$J\frac{d{\mathsf{\omega }}_m}{dt}=T_e-T_L$$

(10)

$$T_e=\frac{3}{2}{P}_n{i}_q{\mathsf{\psi }}_f$$

(11)

where $J$ is the moment of inertia, ${\mathsf{\omega }}_m$ is the mechanical angular speed, $T_e$ is the electromagnetic torque, and $T_L$ is the load torque.

Adopting the method of d-axis current $i_d=0$, then by considering Equations (1), (10), and (11), the equations for the q-axis current and mechanical angular speed of the PMSM can be expressed as follows:

$$\left\{\begin{array}{l}\frac{d{i}_q}{dt}=\frac{1}{L_q}\left(-R{i}_q-P_n{\mathsf{\psi }}_f{\mathsf{\omega }}_m+u_q\right)\\ \frac{d{\mathsf{\omega }}_m}{dt}=\frac{1}{J}\left(-T_L+\frac{3}{2}{P}_n{\mathsf{\psi }}_f{i}_q\right)\end{array}\right.$$

(12)

Under the no-load condition of the PMSM, a transformation on the angular velocity variable of Equation (12) can be conducted as follows:

$$\left\{\begin{array}{l}{x}_1={\mathsf{\omega }}_{ref}-{\mathsf{\omega }}_m\\ {\stackrel{•}{x}}_1={\stackrel{•}{\mathsf{\omega }}}_{ref}-{\stackrel{•}{\mathsf{\omega }}}_m=-{\stackrel{•}{\mathsf{\omega }}}_m=-\frac{3}{2J}{P}_n{\mathsf{\psi }}_f{i}_q\\ {\stackrel{••}{x}}_1=-{\stackrel{••}{\mathsf{\omega }}}_m=-\frac{3}{2J}{P}_n{\mathsf{\psi }}_f{\stackrel{•}{i}}_q\end{array}\right.$$

(13)

where ${\mathsf{\omega }}_{ref}$ is the set value of mechanical angular speed of the PMSM, ${\mathsf{\omega }}_m$ is its actual mechanical angular speed, and the symbol ‘$\stackrel{•}{ }$’ represents a derivation.

This paper proposes an improved SMC method with a saturation function for the speed closed-loop control of the PMSM; the control process is described as follows.

The sliding mode surface of the SMC method is defined as follows:

$$s=c{x}_1-{\stackrel{•}{x}}_1=-\mathsf{\epsilon }\mathrm{sgn}\left(s\right)-gs$$

(14)

where $c$ is the coefficient of the sliding mode surface, $\mathsf{\epsilon }$ is the constant velocity approach rate, $\mathrm{sgn}\left(s\right)$ is the symbolic function, and $g$ is the exponential approach rate. Next, the derivative of Equation (14) is considered, which is illustrated as follows:

$$\stackrel{•}{s}=c{\stackrel{•}{x}}_1-{\stackrel{••}{x}}_1=-c\frac{3}{2J}{P}_n{\mathsf{\psi }}_f{i}_q+\frac{3}{2J}{P}_n{\mathsf{\psi }}_f{\stackrel{•}{i}}_q=-\mathsf{\epsilon }\mathrm{sgn}\left(s\right)-gs$$

(15)

By transforming Equation (15) into an integral mode, the expression of q-axis current control can be obtained as follows:

$$i_q^{\ast }=\frac{2J}{3P_n{\mathsf{\psi }}_f}{\displaystyle {\int }_0^t\left(c\frac{3}{2J}{P}_n{\mathsf{\psi }}_f{i}_q-\mathsf{\epsilon }\mathrm{sgn}\left(s\right)-gs\right)dt}$$

(16)

The so-called improved SMC method with saturation function is a means to replace the symbolic function $\mathrm{sgn}\left(s\right)$ of Equation (16) with a saturation function $\mathrm{sat}\left(s\right)$, as described in Equation (17).

$$i_q^{\ast }=\frac{2J}{3P_n{\mathsf{\psi }}_f}{\displaystyle {\int }_0^t\left(c\frac{3}{2J}{P}_n{\mathsf{\psi }}_f{i}_q-\mathsf{\epsilon }\mathrm{sat}\left(s\right)-gs\right)dt}$$

(17)

Figure 3 shows the comparison between symbolic function $\mathrm{sgn}\left(s\right)$ and saturation function $\mathrm{sat}\left(s\right)$. For the saturation function $\mathrm{sat}\left(s\right)$, when the input variables are beyond the boundary layer Δ, the sliding mode switching control is adopted, otherwise the linearized feedback control is adopted. Compared with the symbolic function $\mathrm{sgn}\left(s\right)$, the saturation function $\mathrm{sat}\left(s\right)$ can reduce the speed fluctuation of the controlled object (PMSM in this paper) in order to improve its stability. Especially, the value of boundary layer Δ can be optimized to further improve the speed stability of the PMSM.

3. Simulation and Analysis

3.1. The Stability of the Torque Closed-Loop Control of the PMSM

According to the control method of the PI parameter tuning that described in Section 2.1 of this paper, the torque closed-loop control of the PMSM is simulated and analyzed in this section. The main characteristic parameters of the PMSM involved in this paper is shown in

Table 1. The main characteristic parameters of the PMSM.

Item	Value	Unit
Number of phases	3	-
Rated power	1.5	kW
Rated voltage	220	V
Rated current	4.5	A
Phase resistance	2.92	Ω
d-axis inductance	0.0089	H
q-axis inductance	0.0122	H

.

Based on the frequency domain analysis method of automatic control theory [20,21]; Equations (5), (8), and (9) for the above PMSM; and the characteristic parameters of Table 1, the simulation and analysis results of the torque closed-loop control of the PMSM are shown in

Table 2. The simulation and analysis results of the d-axis current of the torque closed-loop control.

${\mathit{T}}_{\mathit{p}\mathit{w}\mathit{m}}$	${\mathit{k}}_{\mathit{p}\mathbf{d}}$	${\mathit{k}}_{\mathit{i}\mathit{d}}$	$\mathit{G}\mathit{M}$	$\mathit{P}\mathit{M}$	Stability
0.01 s	0.5	14.6	∞dB	96.1°	Yes
0.01 s	0.5	145.7	∞dB	66.4°	Yes
0.001 s	4.5	1457	∞dB	65.7°	Yes
0.001 s	4.5	145.7	∞dB	109°	Yes

and

Table 3. The simulation and analysis results of the q-axis current of the torque closed-loop control.

${\mathit{T}}_{\mathit{p}\mathit{w}\mathit{m}}$	${\mathit{k}}_{\mathit{p}\mathbf{q}}$	${\mathit{k}}_{\mathit{i}\mathit{q}}$	$\mathit{G}\mathit{M}$	$\mathit{P}\mathit{M}$	Stability
0.01 s	0.6	14.6	∞dB	97.7°	Yes
0.01 s	0.6	145.7	∞dB	65.3°	Yes
0.001 s	6.2	1457	∞dB	65.6°	Yes
0.001 s	6.2	145.7	∞dB	95.1°	Yes

. In Table 2 and Table 3, the symbol $T_{pwm}$ is the switching cycle of the inverter, $k_{pd}$ is the proportional coefficient, $k_{id}$ is the integral coefficient, $GM$ is the gain margin, and $PM$ is the phase margin.

Table 2 and Table 3 indicate that the d-axis current and q-axis current of the PMSM remain stable (which means that the torque closed-loop control of the PMSM remains stable), although the switching cycle $T_{pwm}$, proportional coefficients ($k_{pd}$ and $k_{pq}$), integral coefficients ($k_{id}$ and $k_{iq}$), and the phase margin $PM$ are different.

However, under the different control parameters, Table 2 and Table 3 cannot reflect the rise time, stability time, overshoot time, and adjustment time of the torque of the PMSM. Therefore, it is necessary to conduct a hardware experimental test, which is based on the preliminary simulation results in Table 2 and Table 3, and obtain the optimal control parameters of the torque closed-loop control of the PMSM.

The optimal control parameters of the torque closed-loop control of the PMSM will lay the premise and foundation for the next step of the speed closed-loop control of the PMSM.

3.2. The Simulation and Analysis of the Speed Closed-Loop Control of the PMSM

Given the inertia of $J=0.00104\begin{array}{c}\end{array}\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2$ and the external interference signal of $d\left(t\right)=10\sin \left(3.14t\right)$, the speed fluctuation curve of the PMSM by different SMC methods is shown in Figure 4. The comparison result shows that the SMC method with the saturation function $\mathrm{sat}\left(s\right)$ can enhance the speed stability of the PMSM. In particular, as the boundary layer Δ of the saturation function $\mathrm{sat}\left(s\right)$ decreases, the speed stability of the PMSM is further enhanced.

Figure 5 shows the fluctuation curves of q-axis current $i_q^{\ast }$ (shown in Figure 1), which is transmitted by the above two SMC methods of speed closed-loop control. Based on the SMC method with saturation function $\mathrm{sat}\left(s\right)$, it can be concluded that there was barely any fluctuation in the q-axis current $i_q^{\ast }$ curve; however, curve fluctuation occurs via the SMC method with symbolic function $\mathrm{sgn}\left(s\right)$. According to the vector control theory of PMSMs, the fluctuation of $i_q^{\ast }$ will have a negative effect on the speed stability of the PMSM. In addition, the phase angle of q-axis current $i_q^{\ast }$ can be changed by the boundary layer Δ; this is the key to further improve the speed stability of the PMSM (see Figure 4).

4. Experimental Verification

In order to verify the correctness of the above simulation and analysis results, an experimental test platform was built, as shown in Figure 6. The controller can compile and download the MATLAB/Simulink program, and then complete the torque closed-loop control and speed closed-loop control of the current and speed signals of the PMSM.

Figure 7 shows the q-axis current response result of the torque closed-loop control of the PMSM. For the torque closed-loop control process, the set value of the q-axis current of the PMSM is 0.3A, and the PMSM is still in the locked rotor state. From Figure 7 it can be concluded that the best q-axis current response performance occurs when the proportional coefficient of the d-axis current is $k_{pd}=4.5$, the proportional coefficient of the q-axis current is $k_{pq}=6.2$, the integral coefficient of the d-axis current and q-axis current is $k_{id}=k_{ip}=1457$, and the switching cycle of the inverter is $T_{pwm}=0.001$.

Figure 8 shows the speed fluctuation curve of the PMSM, where the PMSM is controlled by different SMC methods. For the same sliding surface coefficient $c=400$, constant velocity approach rate $\mathsf{\epsilon }=100$, and exponential approach rate $g=300$, the SMC method with saturation function $\mathrm{sat}\left(s\right)$ can limit the speed fluctuation of the PMSM to 5.9%, whereas the speed fluctuation of the PMSM by the SMC method with symbolic function $\mathrm{sgn}\left(s\right)$ is 7.7%. The simulation result (see Figure 4) and experimental result (see Figure 8) indicate that the SMC method with saturation function $\mathrm{sat}\left(s\right)$ can improve the speed stability of the PMSM.

Figure 9 shows the starting performance of the PMSM by different SMC methods. Under the same sliding mode surface coefficient $c=400$, constant velocity approach rate $\mathsf{\epsilon }=100$, and exponential approach rate $g=300$, the two SMC methods can make the PMSM have almost the same stability time (about 0.15 s). However, Figure 9 also shows that the SMC method with saturation function $\mathrm{sat}\left(s\right)$ can reduce the overshoot time of the speed of the PMSM. The main reason for this phenomenon is that the SMC method with saturation function $\mathrm{sat}\left(s\right)$ has the advantage of adjusting the boundary layer Δ.

5. Discussion

In considering both a simulation analysis and experimental test, this paper adopts two SMC methods to investigate the speed stability of a PMSM. Although the results indicate that the SMC method with saturation function $\mathrm{sat}\left(s\right)$ has more advantages in terms of ensuring the speed stability of the PMSM, some other aspects need to be further investigated. Firstly, the high-speed stability of the PMSM needs to be investigated, as this paper focuses on the medium- (or low-) speed operation process. Secondly, the overload stability and robustness performance of the PMSM need to be further researched, since this paper only tests the PMSM in the rated state and without abnormal load interference.

6. Conclusions

In this paper, the SMC method and PI control method are combined in order to complete the theoretical simulation of a double closed-loop control of a PMSM, and the correctness of the theoretical simulation is verified by an experimental test. The results of the theoretical simulation and experimental test show that compared with the traditional SMC method with symbolic function $\mathrm{sgn}\left(s\right)$, the speed fluctuation of the PMSM can be decreased from 7.7% to 5.9% by the improved SMC method with the saturation function $\mathrm{sat}\left(s\right)$, and the stability time of the speed of the PMSM can be shortened to 0.15 s (compared with 0.6 s from a previous study [1] and 1.2 s from an earlier study [3]). Therefore, the SMC method with saturation function $\mathrm{sat}\left(s\right)$ can improve the speed stability and starting performance of a PMSM.