## 1 Introduction
### Literature Review  
More sophisticated control methods are necessary to enhance the performance of PMSMs [3]. Currently, prominent PMSM control methods include constant voltage-tofrequency ratio control, magnetic field orientation control, and direct torque control [8]. Position sensorless control technology for PMSMs simplifies the control system by employing suitable control algorithms to estimate rotor speed and position [9]. Key position sensorless control techniques encompass the Sliding Mode Observer method, the Model Reference Adaptive System method, and the Extended Kalman Filter method. The model-referenced adaptive speed control was proposed, utilizing an ultra-twisted sliding mode observer to estimate and compensate for aggregate perturbations [10]. This approach updates the error compensation term online using adaptive rates, leading to improved dynamic performance and robustness in speed tracking control for PMSMs, as demonstrated through both experiments and simulations. Additionally, [PERSON] proposed a bi-level fuzzy exponential convergence rate fractional order sliding mode control, which enhances the control performance and robustness of motor drives, as evidenced by simulation results [11]. [PERSON] designed a super-helical sliding mode observer that mitigates the chattering issue typically associated with sliding mode observers while enhancing system robustness. This approach effectively addresses the jitter problem of traditional sensor systems [12]. [PERSON] introduced a sliding mode control method for fast terminal sliding mode observers in permanent magnet synchronous motor drive systems, effectively reducing system dependence on accurate models and improving disturbance immunity [13]. A robust current control method with disturbance cross-coupling compensation was proposed by [PERSON], employing a discrete-time backstepping perturbation observer combined with a sliding mode controller to improve system robustness and stability, as indicated by simulation results [14]. [PERSON] suggested a control strategy that combines a robust two-degree-of-freedom controller and an extended sliding mode parameter observer. This strategy utilizes iterative learning control and a series structure to suppress unmodeled perturbations and periodic components, with the ESMPO enhancing parameter identification to improve anti-interference capabilities [15]. [PERSON] proposed the Model Predictive Control method for driving the current loop, employing Higher Order Sliding Mode Control for the outer speed loop and a Super twisting Algorithm to suppress jitter, thereby effectively enhancing motor performance [16]. [PERSON] proposed a comprehensive electromagnetic model to address elliptical and corrugated deviations. In their research, a three-pole active magnetic bearing with a simple structure was employed to dampen vibrations and was integrated into a permanent magnet synchronous motor system. By utilizing the iterative sliding mode control in the design of the control system, the motor's vibration is effectively mitigated [17]. To mitigate pulsations, [PERSON] devised a robust insertion repetition controller with phase compensation within the outer speed loop of the cascade PI control system in a permanent magnet synchronous motor. Additionally, a third-order Butterworth filter with phase correction is proposed to enhance the system's resilience against low-frequency periodic disturbances and ensure its high-frequency stability [18]. Applying a barrier function-based adaptive strategy to first-order disturbed systems significantly enhances system robustness while maintaining effective performance [19]. [PERSON] proposed a uniform reaching phase strategy within adaptive sliding mode control, effectively ensuring uniform reaching of the sliding phase within a predetermined time while adapting to the perturbation norm [20].