## 1 Introduction  
With the ongoing development of rare earth materials and advancements in power electronics technology, synchronous motors have gradually surpassed induction motors in terms of torque density and operating efficiency [1; 2]. Among these, permanent magnet synchronous motors (PMSMs) are widely used in servo control applications due to their high efficiency, reliability, power density, and ease of control [3; 4]. However, the traditional control methods for PMSMs often fall short due to their complex nature as high-order systems with significant coupling and numerous adjustable parameters. These methods typically exhibit slow response speeds, excessive overshoot, and suboptimal control performance, which fail to satisfy the accuracy and response demands of position servo systems. Consequently, improving the high-performance control of PMSMs has become a significant focus for researchers in motor control theory and related disciplines [5]. Active disturbance rejection control (ADRC) offers a promising solution by estimating both the internal and external disturbances within the system. This approach is characterized by a robust series configuration that demonstrates low dependence on precise modeling [6; 7]. After more than 20 years of development, many scholars have provided summaries and researched the related thoughts, theories, and advancements in this field. Professor [PERSON] introduced active disturbance rejection control (ADRC) through unique analyses of proportional-integral-derivative (PID) control [8]. He elaborated on the evolution from PID control to ADRC, highlighting the advantages of active disturbance rejection control [9]. Professor [PERSON] further discussed the characteristics of and development trends for ADRC, focusing specifically on disturbance suppression [10]. Active disturbance rejection control operates on the fundamental principle of PID deviation control, allowing it to function without the need for precise modeling of the control object. This approach effectively addresses the challenge of balancing rapid response with overshoot. ADRC is characterized by high control accuracy, fast response times, and strong stability. However, it includes numerous adjustable parameters that can lead to increased interdependency among them. It is important to recognize that the performance index of the state observer significantly affects both the precision and stability of the controller [11; 12; 13]. Numerous scholars have researched active disturbance rejection control (ADRC) and its enhancements to control strategies to address the challenge of disturbance suppression in permanent magnet synchronous motors. The fundamental issues associated with ADRC primarily encompass three areas: parameter tuning and optimization, performance analysis, and stability/convergence proof [14]. Furthermore, improvements to ADRC can be classified into two categories: one involves enhancing ADRC through the introduction of additional tools, while the other focuses on refining various components within the ADRC framework, such as the tracking differentiator (TD), the extended state observer (ESO), and the nonlinear state error feedback law (NLSEF) [14]. In reference [15], the ESO is improved based on error principles to enhance the nonlinear function. Reference [16] describes enhancing the ESO structure by integrating first-order active disturbance rejection, which facilitates better disturbance compensation in the system through the addition of supplementary terms. Additionally, reference [17] proposes a modified third-order extended state observer. The performance of the controller can be significantly improved by enhancing its tracking capabilities and disturbance rejection through the implementation of the proportional-integral disturbance update law. Reference [18] addresses the challenge of the measurement noise resulting from the bandwidth of the extended state observer (ESO) in an analysis of an enhanced active disturbance rejection controller.