# Improved Sliding Mode Control for Permanent Magnet Synchronous Motor Speed Regulation System
## 1 Introduction  
Permanent magnet synchronous motors (PMSMs) have been widely used as servo motors and have become a good choice in servo drives in high-dynamic and high-precision applications due to several distinct advantages such as fast response time, high power density, high acceleration, robustness, etc. [1; 2; 3]. Usually, PMSMs use the strategy of the field-oriented control (FOC), a cascade control structure with an inner current loop and an outer speed loop, in their control systems. The proportional integral (PI) control, which offers simple and easy implementation, is widely used in PMSM systems in both the inner and outer loops [4; 5]. In applications, unknown random disturbances and parametric uncertainties always affect PMSM systems and induce ripples. These ripples induce periodic oscillations in the PMSM's speed, and the speed oscillations cause a degradation of the servo system's robustness and tracking performance, as well as undesirable mechanical vibrations [6; 7; 8]. Conventional proportional-integral control is not always effective in improving the performance of PMSM systems. A number of direct ways to address the chattering phenomenon from a theoretical viewpoint have been developed recently [9]. These methods, which deal with the issue of pulsating torque, can be categorized into two types. The first is the improvement of PMSM optimization, such as the use of a skewing magnet, a fractional number of slots per pole, winding optimization, etc. [10; 11; 12]. The second method is through the control system [13; 14]. As the controller is a necessary part in PMSM systems, the latter requires no additional hardware and can be applied more easily. The main indexes in the control systems, settling time, rising time, and overshoot, are very important and indicate the performance of the system [15; 16]. As a nonlinear, strong coupling and multi-variable system, the PMSM is very sensitive to internal parameter perturbation and external disturbance. Therefore, the traditional linear control method cannot solve the dynamic response performance of PMSMs for its fixed proportional gain and integral time constant [17; 18]. Due to the complexity of algorithm and the real time requirement of application, it is difficult to achieve for many nonlinear control strategies in a PMSM system. Based on the advantages of strong robustness and simplicity of both tuning and implementation, sliding mode control (SMC) can realize high performance by frequently switching the system structure to restrain system parameter variations and external disturbances [19; 20]. SMC is applied widely in servo systems due to its simple implementation and insensitivity to the matched parameter perturbation and disturbance; however, its inherent chattering phenomenon impedes SMC implementation. Hence, many methods have been proposed to reduce or eliminate the effect of chattering. Since an unsuitably large switching gain leads to large chattering [6], one effective method is to add a human-like self-learning capability to the control system and select a suitable switching gain for the SMC law [21; 22]. Additionally, Reference [23] presents a general regularization procedure, which adds an appropriate penalization of the total variation, is proposed. A criterion, which bases on a speed of convergence of the costs, is quantified. Similarly, a more simply way is proposed in Reference [24]. It is an adaptive exponential reaching law, and devotes to the chattering phenomenon in the sliding mode control system. According the distance between the current states and the sliding switching surface, the adaptive exponential reaching law changes the reaching speed dynamically. Moreover, in Reference [25], a novel sliding mode method, the unidirectional auxiliary surfaces are utilized to design a SMC controller, is proposed to reduce the chattering phenomenon. With different unidirectional auxiliary surfaces, the chattering phenomenon and robustness of the SMC controller can be adjusted according to requirements.