# Sliding Mode Control of Servo Feed System Based on Fuzzy Reaching Law
## 4 Simulation Analysis  
In order to verify the control performance of the fuzzy convergence law sliding mode control method proposed in this paper, a flexible body model of the servo feed system is established and model simulation comparison experiments are conducted. The structure of the fuzzy convergence law sliding mode control system is shown in Figure 5.  
The main parameters of the flexible body model of the servo-feed system are shown in Table 5.  
The smooth reference trajectory helps to suppress the vibration, and the cubic acceleration trajectory is chosen as the reference command for the table position. The maximum displacement is 0.065 m, the maximum velocity is 0.1 m/s, and the maximum acceleration is 1 m/s\({}^{2}\), as shown in Figure 6.  
Figure 4: Fuzzy rule 3D diagram. Figure 5: Block diagram of control system structure. \begin{table}
\begin{tabular}{c c} \hline \hline Parameter & Numeric Value \\ \hline \(m_{1}(\mathrm{kg})\) & 50 \\ \(m_{2}(\mathrm{kg})\) & 80 \\ \(k(\mathrm{N/m})\) & \(5.994\times 10^{7}\) \\ \(c(\mathrm{N/m\cdot s^{-1}})\) & \(4.596\times 10^{3}\) \\ \(b_{1}(\mathrm{N/m\cdot s^{-1}})\) & 0.0694 \\ \(b_{2}(\mathrm{N/m\cdot s^{-1}})\) & \(1.382\times 10^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Main parameters of the flexible body model. The flexible body model of the servo feed system is added to the controller designed in this paper, and the given reference trajectory is tracked and compared with the sliding model control and PID control to verify the feasibility and control effect of the fuzzy convergence law sliding mode control algorithm. The simulation model built by MATLAB /Simulink simulation module is shown in Figure 7.  
The fuzzy convergence law sliding mode controller is implemented by programming the S-function. The system inputs are the reference command \(\mathrm{x}\mathrm{r}\), the ball screw rotational equivalent displacement \(\mathrm{x}_{1}\), the table linear displacement \(\mathrm{x}_{2}\), and the convergence law parameters \(\varepsilon\) and \(h\). The outputs are the slide surface function \(\sigma\), the slide surface derivative \(\mathrm{d}\sigma\), and the voltage signal \(\mathrm{u}\mathrm{t}\) for controlling the torque. \(\sigma\) and \(\mathrm{d}\sigma\) are used as inputs to the fuzzy controller to adjust the parameters \(\varepsilon\) and \(h\) in the exponential convergence law adaptively through fuzzy rules to reduce the chattering phenomenon of the system. Take the coefficients \(\lambda 1\), \(\lambda 2\), \(\lambda 3\) and \(\lambda 4\) of the sliding mode function as \(30\), \(1\), \(30\) and \(0.1\), respectively, without considering the interference term, so that \(\mathrm{d}1=\mathrm{d}2=0\), then the simulation results of dynamic regulation of parameters \(\varepsilon\) and h are shown in Figure 8.  
Figure 6: Position reference trajectory of the working table. Figure 7: Sliding mode control simulation model based on fuzzy convergence law. The fuzzy controller feeds the adjusted parameters \(h\) and \(\varepsilon\) back to the fuzzy convergence law sliding mode controller in real time for the voltage signal adjustment. The voltage signal is passed through the AC servo motor output to control the torque signal of the load. The voltage signal outputs the torque signal of the control load through the AC servo motor, so as to accurately locate the table. The simulation results of the position trajectory tracking of the table are shown in Figure 9.  
As can be seen from Figure 9, the table generates significant tracking errors when it is in the speed rise and fall phase. In order to compare the performance of the controller more intuitively, the tracking error and control voltage signal of the platform are simulated, and the simulation results are shown in Figures 10 and 11. Figure 8: Adaptive adjustment process of convergence law parameters \(h\) and \(\varepsilon\). Figure 9: Trajectory tracking of the table.