## 1 Introduction  
As an important power of mechanical system, permanent magnet synchronous motor (PMSM) is widely used because of the compact structure, high power density, and low noise. Especially in the field where volume is strictly limited, such as robots, aerospace, high-performance service system, PMSM is widely accepted. As we know, proportional-integral (PI) control is already used in PMSM system because of its simple implementation. However, PMSM is a nonlinear, time-varying system and it is difficult to achieve satisfactory control performance by using PI control. Therefore, it is necessary to use nonlinear control algorithm in PMSM system [1]. In recent years, many complex control algorithms have been implemented because of the development of micro controller, e.g., robust control [2; 3; 4], fuzzy control [5; 6; 7], adaptive control [8; 9; 10], sliding mode control (SMC) [11; 12; 13], backsteeping control [14; 15; 16], etc. These advanced algorithms can effectively improve control performance. Among these methods, SMC attracts more attention because of its robustness. SMC is a discontinuous control method that can force the system move along the sliding surface. Moreover, sliding motion can be designed and it is not affected by parameter change and external disturbance. In [17], a SMC speed-regulation controller consisting of a tracking controller and two synchronous controllers is designed for multi-motor system. In [18], a nonlinear speed-control algorithm based on sliding mode reaching law (SMRL) is designed for PMSM system and this algorithm can dynamically adapt to the operation status. But SMC is a feedback-basedcontrol, which only works when there is an error. So it cannot react promptly to disturbances such as load change, unmodeled dynamics, and parameters variations. As an effective method to reject disturbance, feed-forward compensation is introduced in control system. Feedforward-based control is a method to compensate disturbance, which can effectively reduce the impact of disturbance. The premise of compensation is to estimate disturbance, but most disturbances cannot be measured directly, so the disturbance observer is proposed. Disturbance observer (DOB) is proposed by [PERSON] [19] in 1987, which is an active estimation method for external disturbance and has been applied in robotic [20], aerial vehicle control [21], PMSM [22], etc. In [23], a composite sliding mode controller based on DOB is proposed for PMSM, the disturbance is compensated by DOB can take a smaller switching gain without sacrificing disturbance rejection. In [24], a disturbance observer-based sliding mode controller is designed for PMSM control system, a DOB is designed to reconstruct the external disturbance and counteract the influence of external disturbance to reduce the velocity resonance caused by disturbance. In [25], an augmented disturbance observer is proposed by embedding uncertainty/disturbance in the DOB, which can accurately estimate disturbances of different channels. Another disturbance observation technique has been widely accepted is extended state observer (ESO) which is proposed by [PERSON] [26]. ESO regards lumped disturbance, which include external disturbance and internal disturbance, as a new state variable to reconstruct a new extended state equation, and then estimate the state variable of the extended state equation. ESO does not require a model of disturbance and does not need to measure disturbances directly, so ESO is a practical disturbance observer. Disturbance feed-forward compensation based on ESO or active disturbance rejection control (ADRC) [27] has been widely applied in PMSM systems, power converter, robotic systems, etc., in [28], a second-order model of PMSM system is introduced to design a speed regulation controller; moreover, an active feed-forward compensation is designed to reduce chattering of the sliding mode speed controller and improve the robustness. In [29], a linear extended state observer (LESO) is used to achieve high precision position estimation.