## 1 Introduction  
Permanent magnet synchronous motors (PMSM) have been widely used in the fields of electric vehicle drive system, robot, aerospace, intelligent manufacture, and other fields because of the advantages of excellent efficiency, high power density, low inertia, high torque-current ratio, no excitation loss, et al. [1]. In order to ensure the smooth running of the motor, it is very important to adopt efficient and stable control strategies for each part of the three-phase permanent magnet synchronous motor system, such as speed control [2], current control [3], and pulse width modulation [4]. Besides, sensor-less control also needs observer strategy or other control methods [5]. Among them, speed control is particularly important. The control of current loop can be simply divided into current control in static coordinate system and current control in synchronous rotating coordinate system. For the current control in synchronous rotation coordinates, hysteresis current control and PI current control are commonly used at present. Hysteresis current control can meet the requirements of motor operation, but the electromagnetic torque fluctuates greatly during the whole starting process, and PI regulator has good dynamic performance and anti-disturbance ability, which can meet the needs of actual motor control performance [6]. The pulse width demodulation technology of three-phase voltage source inverter mainly includestwo-level space vector pulse width modulation (SVPWM) and sinusoidal pulse width modulation (SPWM). SVPWM has better harmonic elimination effect and torque ripple resistance than SPWM [7]. In recent years, many control technologies are used in the research of speed control for PMSM, which can be divided into linear control and nonlinear control [8]. As is known to us all, many linear control methods based on transfer functions (the most popular one is PI control [9]) are sensitive to the system model accuracy, which is highly susceptible to external disturbances and internal parameter mismatch. Therefore, many scholars draw attention on nonlinear control design for PMSM, such as robust control [10], adaptive control [11], and sliding mode control [12]. Among these speed control strategies for PMSM, sliding mode control with the merits of simple structure and fast response has become one of the most popular methods. In reference [13], a nonlinear speed-control algorithm for the PMSM control systems using sliding-mode control (SMC) is proposed to optimize the dynamic performance of speed regulation system. In which, the new sliding-mode reaching law (NSMRL) using the system state variable and the power term of sliding surface function can be expressed in two different forms during the reaching. This method can not only effectively suppress the inherent chattering, but also increases the velocity of the system state reaching to the sliding-mode surface. A new adaptive terminal sliding mode reaching law (ATSMRL) is proposed with continuous fast terminal sliding mode control (CFTSMC) in reference [14]. It aims to enhance the speed control performance of the permanent magnet synchronous motor (PMSM) with internal and external disturbances. In reference [15], a novel sliding mode control (NSMC) strategy is developed to improve the robustness, disturbance rejection, and dynamic response performance of permanent magnet synchronous motor (PMSM) speed servo system. The integral of speed error is introduced into sliding mode surface to avoid the requirements of acceleration signal and reduce the steady-state error of the system. It can be found in the existing literature that many scholars focus on the design of new sliding mode reaching law and novel sliding mode surface. One cannot deny that these literatures have achieved good control performance. However, sometimes it is hard to know how to get a new reaching law or a new sliding mode surface, that is, it requires more professional and profound knowledge of sliding-mode control theories.