## 1 Introduction  
Permanent magnet synchronous motors (PMSMs) possess the characteristics of simple structure, small volume and high power. Moreover, they have been widely applied in various types of servo control systems and stabilized platforms [1; 2; 3; 4; 5; 6]. Due to the high-precision control requirements of these systems, the speed control of a PMSM needs to have considerably high stability. Because external vibration, shock, unbalanced torque, load gravity and other disturbances are mainly transmitted to the motor rotor through the bearing, bearing friction has an obvious effect on the speed stability of PMSMs. Especially when a PMSM operates at low speed, the problem of low-speed crawling may occur. To promote the anti-disturbance performance of the servo system and solve the problem of low-speed crawling, it is of great significance to research an effective speed control algorithm of PMSMs. Bearing friction has strong nonlinearity and a series of friction models have been proposed. Among them, the LuGre model can accurately describe the dynamic and static characteristics of friction. In recent years, some control algorithms for friction compensation based on the LuGre model have been proposed. In [7], by analyzing the influence of friction heating on the friction coefficient, an improved LuGre model was established to calculate the static real-time steering torque. The authors of [8] combined the LuGre model and periodic adaptive learning control to eliminate nonlinear friction disturbance in PMSM servo systems. In [9], a nonlinear modified LuGre observer was designed to estimate friction behavior of optoelectronic tracking system. In [10], researchers modified the LuGre friction model to accurately describe the nature of the friction force in the gross sliding regime. The authors of [11] analyzed the advantages and disadvantages of the LuGremodel and some specific applications. Although these algorithms have advanced control performance, due to too many parameters in the LuGre model, it is difficult to establish the LuGre model with accurate parameters in practical engineering. A series of modified and intelligent control algorithms have been applied to disturbance suppression, including fuzzy control [12], neural networks for control [13], robust control [14], sliding mode control (SMC) [15], etc. Active disturbance rejection control (ADRC), proposed by [PERSON] [16; 17], possesses a simple structure and convenient design compared with other algorithms, and has the ability to estimate and compensate for disturbances under the condition of model uncertainty. In [18], an improved ADRC with extended state observer (ESO) and finite time stable tracking differentiator (FTSTD) was utilized in the flexion and extension motion of shoulder and elbow joints, which possess the disturbance rejection ability and robustness against parametric uncertainties. In [19], a measurement delay compensated linear ADRC was proposed to reduce the torque fluctuation of PMSMs. In [20], a hybrid control algorithm combined ADRC and adaptive fuzzy sliding mode control (AFSMC) for the optoelectronic platform to enhance the target tracking capability. In [21], a parameter tuning method based on the genetic algorithm (GA) was presented for ADRC to promote the tracking efficiency of disturbance. In these improved ADRC algorithms, the total disturbances are mainly estimated and compensated by single ESO, which increases the burden of ESO. ESO needs to select a large gain to ensure the accurate estimation of disturbances, which may cause system instability and vibration. Buffeting is not tolerated in the low-speed stability control of PMSMs. [PERSON] (2008) [16] demonstrated that partial prior knowledge of the model can help reduce the burden of ESO and improve the accuracy of disturbance estimation. In [22], the model-based control has been proven to be effective in practical applications. In [23; 24], an auxiliary system was constructed to eliminate the effect of input nonlinearities and constraints.