# A Precise Modeling Method for PMSM Servo System of Spacecraft
## 1 Introduction
In order to compensate for the nonlinearity of backlash, a state feedback controller is designed to reduce the impact, eliminate the limit cycle oscillation and improve the accuracy of position control. A linear and a nonlinear method of ESO for a two-mass system with ADRC speed control loop of the motor side are proposed in [13]. The performance of the two approaches, such as damping of the torsional vibration and influence of backlash width, is compared. Ref. [14] presents an adaptive model predictive control to eliminate the effects of clearance and coulomb dry friction, which is proved to be correct in servo position tracking simulation. In [15], the influence of the backlash and friction is analyzed by a description function method and simulation. A comprehensive two-mass full-closed-loop system incorporating both backlash and friction is also constructed. The friction models of real systems are usually discontinuous or piecewise continuous, which is not conducive to the actual realization of the control system. Some continuous friction models have been proposed [16], such as Stribeck friction model, LuGre friction model, etc. For nonlinear servo systems with friction compensation, a continuous friction model is adopted to obtain the friction dynamics [17]. In order to suppress the dynamic disturbance, an adaptive parameter estimation method is proposed to identify the friction model parameters, which improves the system's disturbance resistance and tracking accuracy. The servo system of continuous friction model [18] establishes a nonlinear parameterized friction model and uses the nonlinear least squares method to estimate parameters of the model, such as viscosity coefficient and the Coulomb coefficient. Based on the established friction model, a backstepping controller is designed. Ref. [19] proposes linearization control for servo systems with friction based on the LuGre model and gives a high-gain closed-loop speed control method to identify the dynamic parameters of the LuGre model. An experimental friction model for the PMSM servo system is proposed in [20]. A control system with maximum bandwidth is designed using the proposed friction model and analysis results, which improves the dynamic response of the system. Ref. [21] derives a modified LuGre friction model to compensate for dynamic friction behaviors. A discontinuous projection mapping is employed to enhance the stability of parameter adaptation during the learning process. In [22], a new static compensation method combined with a Stribeck model of fuzzy low-pass filter is proposed to study the friction compensation of the PMSM servo system. servo system, which is to solve the friction degradation problem of speed and position tracking accuracy in PMSM servo systems. Due to the special application scenario of the research object, the relevant literature is less published, or focuses on a certain direction of the electrical or mechanical system. The above research has studied the modeling mechanism and action rules of nonlinear factors such as elasticity, backlash, and friction in servo systems from different perspectives, and has improved the performance to a certain extent. However, these modeling methods did not consider the relationship between elasticity and resonances, the interaction between elasticity and backlash, and the influence of the external environment on friction [23; 24]. Furthermore, it is not possible to fully and accurately characterize the PMSM servo system through the study of single and partially coupled factors. In this regard, this paper proposes a modeling method that considers elasticity, backlash, and friction characteristics. The elastic dead zone model and the improved LuGre friction model considering environmental effects are organically combined to form a high-precision PMSM servo system model through simulation analysis and experimental verification. The main contributions of this paper are as follows:  
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