1. Introduction
Electromechanical actuators (EMAs) consist of a motor, a reducer, a driver, a controller, sensors, etc. [
1], that can realize linear or rotary motion. EMAs are widely used in aerospace, military, robotics, transportation, equipment processing, and other fields due to their advantages of having high integration, a small size, high precision, and high efficiency [
2,
3]. Permanent magnet synchronous motors (PMSMs) are commonly used in EMAs because of their simple structure, light weight, high-power density, and good dynamic response [
4]. However, PMSMs are nonlinear multivariable strong coupling systems, and their operating performance is easily influenced by unmodeled phenomena, parameter uncertainty, and external load. An EMA servo system requires a fast response and disturbance rejection ability, but the actual work environment is complex: frequent start–stops, continuous load changing, and so on [
5,
6]. Therefore, the traditional proportional–integral–differential controller cannot meet the requirements of the high performance and high precision control of the PMSM in EMAs. Sliding mode control (SMC) is commonly used in PMSMs control in EMAs due to its simple structure, fast response, strong disturbance rejection ability, insensitivity to model uncertainty, and good robustness [
7,
8].
The SMC can improve the disturbance rejection ability and optimize the motor performance because of the structural characteristics, but it brings the chattering problem to the servo system, which seriously affects the control performance of the system [
9]. Currently, reducing the high-frequency chattering and improving the disturbance rejection ability of the SMC are the research hotspots of researchers. An improved reaching law speed sliding mode controller was proposed by Liu Yangqing [
10], and the sigmoid function was used to replace the switching function, which effectively reduced the chattering of the SMC and improved the response speed. However, the influence of external load disturbance was not considered. Based on the traditional exponential reaching law, Lin Chengmei [
11] introduced a variable exponential function and a hyperbolic tangent function to improve the reaching speed and disturbance rejection ability. Zhou Yang [
12] studied a new exponential reaching law with strong adaptive ability. The constant speed term was improved to the time variable, which solved the problem of the slow convergence of the traditional exponential reaching law system. The saturation function of the variable boundary layer was also used to replace the switching function, which reduced chattering. Yan Hongliang [
13] proposed an improved reaching law sliding mode speed controller for a time-varying disturbance environment, which improved the dynamic performance of the system and reduced chattering. Moreover, the reaching law had no other parameters, which avoided the difficulty of parameter tuning. Xiong Hualiang [
14] designed a super-twisting speed sliding mode controller to replace the q-axis PI controller, and changed the SGN function into a continuous sign function to suppress the torque and speed ripple. Zhou Caijin [
15] designed a fractional sliding mode controller for a PMSM speed control system. The fractional order reaching law was used to suppress chattering. The simulation and experimental results showed that a faster response and better stability was achieved. Based on the traditional sliding mode observer, the sigmoid function was introduced as the switching function by Wu Dinghui [
16], whereas the variable sliding mode gain method was used to reduce the high-frequency chattering. The tracking and disturbance rejection ability of the controller were verified using experiments. Although the above methods improve the response ability of the system and weaken the sliding mode chattering, they do not consider the influence of the external load disturbance and internal uncertainty of the system.
The EMA servo system should have good stability, a good tracking performance, a short settling time, and a strong disturbance rejection ability, etc. However, external load disturbances and internal uncertainties affect the stability of the EMA. Therefore, a disturbance observer should be introduced to estimate the disturbance and reduce its impact [
17]. Yong-chao Liu [
18,
19] proposed a second-order sliding mode controller based on a Hermite neural network. In the speed loop, a disturbance observer based on the HNN was used to compensate both the external disturbance and internal parameter uncertainty. At the same time, a complementary sliding mode speed controller based on a disturbance observer was also proposed. A good tracking performance and high robustness were achieved by both methods. Chen Dai [
20] designed a new current-constrained controller, and used the disturbance observer to estimate the mismatched disturbance. Lizhi Qu [
21] proposed a sliding mode speed controller based on an extended state observer for the speed tracking control of a PMSM drive system under different disturbances. Zhu Dehong [
22] studied a sliding mode controller based on a new variable exponential multi-power reaching law, and designed a new extended sliding mode disturbance observer to estimate the disturbance, which was then fed forward to the sliding mode controller. The results showed that the influence of the system disturbance was eliminated. To solve the problem of disturbance, an adaptive inversion sliding mode controller based on a nonlinear disturbance observer was designed by Lou Peibin [
23]. The switching gain was adjusted using the adaptive law to reduce the chattering of the system and improve the disturbance rejection ability. An integral sliding mode controller was designed by Li Zheng [
24] to eliminate the steady-state error. The load torque observer was designed according to the principle of the Luenberger linear observer, and the observed value of the sliding mode controller was recorded. However, there was difficulty in selecting the integral initial value, and the disturbance rejection ability was low. Furthermore, Zhao Feng [
25] proposed a new reaching law based on a piecewise function to reduce the sliding mode chattering, and observed the motor disturbance using the extended state observer (ESO). The disturbance observation value was compensated as a feedforward signal to the sliding mode controller, which effectively reduced the sliding mode chattering and improved the accuracy of the torque estimation. However, the controller’s parameter setting is complex and difficult.
In this paper, a SMC is proposed for the speed control of PMSMs to improve the overall performance of the EMA servo system. This approach aims at meeting requirements of the fast response and strong disturbance rejection ability of the PMSM in EMAs. A new reaching law is designed to reduce chattering in the SMC, where the symbol function in the traditional SMC is replaced with the symmetric S-type function. A symmetric S-type function disturbance observer (SSFDO) is designed to observe the uncertainties in PMSMs, such as the external load disturbance and internal parameter uncertainty. In order to suppress the speedy fluctuation caused by the disturbance and the chattering of the controller and to improve the disturbance rejection ability of the system, the observed value as a feedback signal is compensated to the new sliding mode controller to form a compound SMC (CSMC). The stability of the CSMC is analyzed using the Lyapunov equation. Both the simulation and experimental of the motor speed control and disturbance observations were conducted using the MATLAB/Simulink simulation and a semi-physical platform, which proves the feasibility of the combination method of a SMC and a disturbance observer.
2. Mathematical Model of PMSMs
The core design of an EMA is the PMSM control method. To simplify the analysis, it is assumed that the three-phase PMSM has a symmetrical current, and the core saturation, eddy current, and hysteresis losses can be ignored. Then, the stator voltage equation in the d–q synchronous rotating coordinate system is [
26]
the electromagnetic torque equation is [
26]
and the mechanical motion equation is [
26]
The parameters and are the voltage components and and are the current components of the d–q axis, respectively. R is the stator resistance, and and are the inductance components of the d–q axis. and are the stator flux linkage and angular velocity of the motor, respectively. is the mechanical angular velocity of the motor. is electromagnetic torque. is the pole number of the motor. is the load. is the damping coefficient. is the rotational inertia of the rotor.
If the PMSM has a surface-mounted rotor, then
. Additionally, when the the field-oriented control method with
is used, the mathematical model of the PMSM can be expressed as follows: