# Sliding Mode Control of Servo Feed System Based on Fuzzy Reaching Law
## 1 Introduction  
CNC machine tool is a kind of high precision and high efficiency machining equipment, and its servo feed system is one of the most important control systems in machine tools. The system enables precise control of parameters such as feed speed, position and force during machine tool machining by using components such as sensors, actuators and controllers, resulting in higher accuracy and stability of the machine tool machining process [1]. With the continuous development of the aviation industry, automotive industry and mold manufacturing, the performance of CNC machine tools has put forward higher requirements. Typically, the drive model of a servo feed system can be equated to a rigid model. Although the rigid body model of the system is simple, it can only reflect the dynamic characteristics of the system during low-speed operation. When the servo feed system operates at high speed, the screw drive components generate axial vibration and torsional deformation, causing structural vibration [2]. The generation of structural vibrations then limits the achievable bandwidth and is a decisive factor in the tracking and positioning accuracy in the feed system [3]. In real systems, both parameter uncertainty and external perturbations are important factors that affect high accuracy positioning and tracking performance [4]. In addition, external disturbances dominate the nonlinear friction and cutting forces [5]. Structural vibrations, external disturbances and parameter uncertainties of the servo-feed system are the main factors that lead to the inability of the classical controller to meet the requirements of high precision control [6]. In response to these challenges, a range of advanced control strategies and advanced technologies are required to develop faster and higher tracking performance servo-feed systems [7]. The literature [8, 9] investigated various control issues of servo position systems, including their stability, output regulation capability, and interference immunity. To address the system uncertainty, [PERSON] et al. [10] proposed a nonlinear sliding mode controller with a feedforward compensator, which led to a significant improvement in the tracking performance of the feed drive system. Bymeans of Lyapunov stability theory, the stability of the system is analyzed and confirmed, and its convergence to the sliding surface is ensured. For the high-precision position control of electro-hydraulic servo pump control system, [PERSON] et al. [11] proposed a compensation control algorithm based on fuzzy control theory on the basis of classical PID control algorithm for reducing the influence of system by load disturbance and enhancing the anti-interference capability of the system. [PERSON] et al. [12] proposed a composite nonlinear feedback adaptive integral sliding mode controller with convergence law (CNF-AISMRL) for fast and accurate control of a servo position control system subject to external disturbances. [PERSON] et al. [13] applied an autoturbation controller (ADRC) to motor position control of a servo-driven feed system to suppress vibration and disturbances, and used a simple PI controller to achieve accurate position control of the motion table. [PERSON] et al. [14] proposed a sliding mode control method based on extended Kalman filter and traceless Kalman filter online state estimation to ensure the tracking accuracy at high speed feed in the presence of noise, friction and uncertainty factors in the ball screw feed system, but the isokinetic convergence law it uses leads to jitter problems. The sliding mode control algorithm has a fast response to the system, is less affected by internal and external disturbances, and has good robustness [15], and is widely used to control servo-feed systems [16]. However, chattering is easy to occur in the controlled system because of the switching control behavior near the sliding mode surface. The presence of chattering makes it difficult to improve the performance of the control system and limits the application of sliding mode control [17].