## 1 Introduction  
Permanent magnet synchronous motor (PMSM) servo systems can have many excellent features, which are simple construction, high dependability, and low noise [1; 2]. Thus, PMSM servo systems are commonly applied in industrial manufacturing, electric vehicles, and other application domains where strict and efficient velocity control is needed [3; 4]. But it is difficult to formulate a control scheme for the operation of the PMSM drive system because the PMSM is subjected to several disturbances which include the load torque and parameter uncertainties [5; 6], which will significantly affect the speed control of the PMSM. Thus, the results achieved by the traditional linear control methods like the one presented above, the PI control strategy, are low primarily owing to the fact that a nonlinear system has been controlled directly with a linear PI controller [7]. As the related digital signal processing technology and power electronics technology have advanced in the recent years, more and more new nonlinear control methods based on the modern control theory have been applied into the control system of the PMSM to overcome some of the problems coming from the traditional PI controllers, which include active disturbance rejection control [8; 9], model predictive control [10; 11], intelligence control [12; 13], and sliding mode control (SMC) [14; 15; 16]. In contrast to the other control strategies, SMC is a special nonlinear control, which is widely used in all modern motion control systems. Thus, SMC has been proven to be less complex, dependable, and significantly robust. However, this results in the problem of limiting external interference and the variation of parameters by expanding the sign-function switching gain of the sliding mode. On one hand, the increase in gain can enhance the robustness of SMC; however, on the other hand, the increase in gain will lead to the chattering problem in the system. To enhance the control performance of the conventional SMC control and decrease the chattering effect on the system, many advanced SMC methods have been developed [17; 18; 19]. Designing SMC using sliding mode reaching laws has become a prevalent method due to its simplicity and computational efficiency, leading to extensive research in recent years [17; 20]. The purpose of the reaching law is to ensure that state trajectories reach a specified sliding surface within a finite time. However, conventional sliding mode reaching laws often suffer from slow reaching, regardless of the system's proximity to the sliding surface [21]. This issue can adversely affect dynamic performance and disturbance rejection capability in practical applications. To accelerate the reaching process of this method, a terminal sliding mode reaching law proposed in [22] aims to accelerate convergence and reduce chattering, but it slows down when the system state is far from the sliding-mode surface. An adaptive sliding mode controller based on a novel sliding mode reaching law, designed for the speed control of PMSM in [23], effectively reduces chattering and shortens convergence time. Moreover, a sliding mode control method based on a new sliding mode reaching law (NSMRL) is proposed, incorporating the system state variable and a power term bounded by the absolute value of the switching function. This allows the reaching law to be expressed in two forms, effectively suppressing inherent chattering and increasing the speed at which the system state reaches the sliding-mode surface [24]. Since disturbance observers can compensate for external disturbances, they can improve the control quality of the system, as indicated in [25]. By designing a time-varying switching gain and a sliding surface, an improved SMC based on a nonlinear disturbance observer (DOB) is developed to counteract disturbances and ensure closed-loop stability.