## 1 Introduction  
Permanent magnet synchronous motors (PMSMs) adopt the field-oriented control (FOC) which have the advantages of small size, simple structure, high efficiency, large power density, and wide speed range. Therefore, PMSM can be widely used in aerospace, household appliances, and medical equipment fields [1]. To achieve efficient control of PMSM, a Hall sensor, optical encoder, and resolver are usually used to monitor the rotor position in time. However, this control scheme is highly dependent on the sensor, which increases the control cost. If the connection cable is too long, interference may be introduced into the system. In addition, working in high temperatures, low temperatures dirty air, and other environments will reduce the reliability of the system [2, 3]. However, PMSM plays an important role in high-performance control places, so sensorless control technology came into being. The sensorless control technology discards the position sensor. By detecting the electrical signal in the motor winding, a certain control algorithm is used to estimate the rotor position and speed. Sensorless control represents the development trend of the three-phase PMSM control system. The core of sensorless control is the estimation of rotor position. Therefore, selecting an appropriate scheme to accurately estimate the real-time position of the PMSM rotor is an important part of sensorless control. The sensorless rotor position acquisition of the PMSM mainly depends on the back electromotive force (EMF) estimation of the system [4]. As an effective position estimation method, the sliding-mode observer (SMO) has the advantages of a simple algorithm and strong robustness. In recent years, the SMO has been widely used in sensorless high-performance control of PMSM [5, 6]. However, the switching function used by the traditional SMO has the problems of phase delay and speed chattering. Therefore, reducing phase delay and speed chattering is a hot issue in SMO control. In Ref. [7], a new design of SMO was proposed to improve control accuracy. An observer is built according to the back EMF model after the back EMF equivalent signal is obtained. In this way, not only are low-pass filter and phase compensation modules eliminated but also estimation accuracy is improved. In Ref. [8], an SMO based on an adaptive super-twisting algorithm is proposed, in which the nonlinearity of the voltage source inverter (VSI) is considered. The algorithm reduces the error by compensating for the voltage distortion. Some schemes propose using the sigmoid function as a switching function to improve response speed and reduce chattering [9]. In addition, considering the double closed-loop module for processing the estimated parameters, the sliding mode controller (SMC) can replace the classical proportional-integral (PI) control algorithm. The SMC is insensitive to disturbances and parameters. In addition, it also has a fast response speed. Therefore, the SMC has been widely used in motor speed regulation systems and works with other control schemes [10; 11; 12]. In Ref. [13], a hybrid control strategy combining fuzzy SMC with load torque observer was proposed to accurately estimate load torque and suppress chattering. In Ref. [14], a fuzzy sliding mode speed controller for PMSM is proposed. The scheme combines the continuous terminal SMO, which can reach the balance point in a limited time to ensure the continuity of control and fast-tracking. Moreover, to improve the speed tracking accuracy, a neural network SMC scheme is proposed to optimize the control system [15]. Active disturbance rejection control (ADRC) has the advantages of strong anti-interference ability, weak model dependence, and a simple design process. Therefore, the ADRC has been successfully applied to motor drive systems. Extended state observer (ESO) is the core of ADRC, which was first proposed at the end of the 20 th century [16].