## 1 Introduction
For the problems of sliding mode control, the current research is generally divided into two categories: The first category mainly uses some classic control design tools, such as using the Riccati inequation method to design a new type of sliding surface [20], to study the robust stability of various systems under mismatched uncertainties. The sliding mode controller designed based on the above method must require that these model mismatch and unknown disturbances must be bounded by the H\({}_{2}\) norm, that is to say, these unmatched disturbances must belong to the self-decaying disappearance type. Taking PMSM as an example, the load torque and the mismatched parameters disturbance therein may have nonzero steady-state values and thus do not have bounded H\({}_{2}\) norms [21]. The second method is to add an integral module to the traditional sliding mode controller to construct an integral sliding mode controller [22]. The design idea of the integral sliding mode control algorithm is to design a high-frequency switching gain to force the state to achieve an integral sliding surface. When matching the unknown disturbance action, the integral action of the sliding surface drives the state to the desired equilibrium state. Compared with the first type of mismatch uncertainty, the integral sliding mode control algorithm has received more attention from researchers because of its simplicity and robustness. Of course, the integral action will often bring some adverse effects to the control system, such as a large overshoot and long transition time [23]. In [24; 25], the authors proposed nonsingular integral terminal sliding mode control, which guarantees fast transient convergence both at a distance from and at a close range to the equilibrium point. The above two types of SMC methods use a robust way to suppress the influence of the model mismatch and unknown disturbances, which means that the attenuation of the model mismatch and unknown disturbances is at the expense of the performance of the PMSM servo control system. In addition, the torque jitter of the sliding mode control algorithm itself is still an urgent problem to be solved [26]. The main research contents of this paper are as follows: Based on _[PERSON]_'s second stability theory, a backstepping sliding mode controller and a backstepping integral sliding mode controller are designed, and the control performance of the two control algorithms in the face of model mismatch and unknown disturbances is compared. Aiming at the deficiencies of the two types of sliding mode control algorithms, a nonlinear disturbance observer is designed and a backstepping sliding mode composite controller based on the nonlinear disturbance observer is constructed. The design idea is to offset or weaken the negative influence of the model mismatch and unknown disturbances in the PMSM servo system on the dynamic response of the system through the nonlinear disturbance observer. The second chapter of this paper establishes the basic mathematical model of PMSM and describes the feedback system. Section 3 designs two types of sliding mode controllers and nonlinear disturbance observers, and proves the stability of the back-step sliding mode composite control system based on nonlinear disturbance observers. Section 4 gives the simulation experiments. Section 5 is the conclusion of this article.