## 1 Introduction  
The permanent magnet synchronous motor (PMSM) has been widely used in modern industrial fields because of its advantages of having a small size, simple design, high efficiency, and high power density, and it is used in many applications such as numerically controlled machine tools, robots, new energy vehicles, and other fields [1]. In the operation process of PMSM, there are factors such as current coupling, magnetic field saturation, and external or internal unknown interference, which make the motor to become a highly nonlinear system [2]. Therefore, obtaining high-performance control algorithms has become the key to PMSM applications. In recent years, with the development of modern control theory and power electronics technology, many researchers have developed a large number of PMSM control strategies and control algorithms. The main control strategies are magnetic field directional control (FOC), direct torque control (DTC), maximum torque per ampere (MPTA), etc. The control algorithms are used for the specific implementation to design the controllers in each control strategy. At present, the most widely used control strategy is the FOC, because it can effectively realize the decoupling of the stator current [3]. After the development and application of the FOC strategy, the main problem that restricts the control performance of PMSM becomes how to solve the impact of the external or internal unknown disturbances. The internal disturbance of PMSM mainly includes motor parameter variation [4], back electromotive force, and torque ripple [5], and the external disturbance mainly refers to load torque disturbance [6]. In order to solve the disturbance problem, researchers have proposed different control algorithms, such as the PID control algorithm, sliding mode control algorithm, backsteppingcontrol algorithm, adaptive control algorithm, robust control algorithm, neural network control algorithm, and fuzzy control algorithm [7; 8; 9; 10; 11; 12]. Among them, the PID control algorithm is the longest and most widely used control algorithm. The control structure of the PID control algorithm is simple and easy to use, and the integral loop itself can suppress special types of disturbances. However, the PID control algorithm is a linear algorithm, and it also has the shortcomings of a linear algorithm, that is, there is a contradiction between the overshoot of the system dynamic response and the settling time [13]. In addition, the response speed of the traditional PID controller to suppress disturbances is dependent on its parameters, and although there are many tuning algorithms that have improved the performance of traditional PID controllers, it is difficult for traditional PID controllers to ensure zero steady-state error control under time-varying parameter disturbance [14]. Among the remaining control algorithms listed above, backstepping algorithm and sliding mode algorithm are the control algorithms that researchers pay most attention to [15]. The backstepping algorithm is widely recognized because it is a control algorithm with self-stabilization and wide relaxation matching conditions. Under the semi-accurate conditions of the PMSM model, the backstepping algorithm is a control algorithm that is closest to the essence of the PMSM motor model. In addition, the q-axis model of the PMSM is a second-order model, and the backstepping method can connect the second-order models like a bridge and ensure that each order state is asymptotically stable in the global scope [16; 17]. The sliding mode control algorithm has a simple structure, and more importantly, it has a strong disturbance rejection capability [18]. However, the sliding mode surface of the traditional sliding mode control algorithm is mostly designed as a linear sliding mode surface, which makes the sliding mode control algorithm more sensitive to model mismatch and unknown disturbances [19].