# A Polishing Processes Optimization Method for Ring-Pendulum Double-Sided Polisher
## 4 Simulation and Experiment Verification
### Simulation  
We set step, sine, and square wave with amplitudes of 10 rpm and periods of 16 s as input signals. We separately added noise and load disturbances to the system output to validate the proposed method's disturbance rejection capability. The initial system states for the STRMPC and MPC were defined as \(x_{0}=[-10,-3.5]\). We used the integral of the time-weighted absolute error (ITAE) performance index to evaluate the control performance. (1) Load disturbance  
To verify the system disturbance rejection performance with load disturbance, we superimposed a 1 rpm load disturbance onto the system output at 16 s and computed the ITAE indices. In Figure 5, \(x_{1}\) is presented on the horizontal axis as the system state and \(x_{2}\) is presented on the vertical axis as the derivative of the system state. The green patches in Figure 5 represent the single-tube terminal constraint set \(\mathbb{Z}(\alpha,s)\), while the gray patches denote the nominal MPC reachable set \(\mathbb{X}_{f}\). The convergence results shows that the actual system state \(x(k)\) follows \(\overline{x}(k)\) and \(\hat{x}(k)\), then eventually converges within the single-tube constraint set. Table 2 presents the ITAE indices obtained from the simulations. The STRMPC method achieves an ITAE index of 1.9203 in square wave tracking, which is approximately 89.87% lower than the MPC method and 94.63% lower than the no controller method. Similarly, for sine wave tracking and step response, the ITAE index of the STRMPC method is also 83.87% and 30.79% lower than the MPC method and 91.87% and 90.13% lower than the no controller method, respectively. The STRMPC method achieves higher control effectiveness compared to MPC and no controller. Compared with the results of the no controller and MPC method, the proposed STRMPC method has the smallest ITAE index, which indicates that the STRMPC method has better control performance. (2) Noise disturbance  
The random white noise disturbance with the amplitude range of [\(-\)0.5, 0.5] is set as the noise interference. From Figure 9, the STRMPC method is able to maintain \(x(k)\) in a terminal constraint set. Figures 10-12 demonstrate that the proposed STRMPC method is more effective in suppressing noise disturbance compared to the MPC method. Based on the simulation results in Table 3, it can be observed that the STRMPC method demonstrates higher control efficiency compared to the other methods. In square wave tracking, the STRMPC method has an ITAE index of 4.3713, which is approximately 83.29% lower than the MPC method and 84.86% lower than the no controller method. Similarly, for sine wave tracking and step response, the ITAE index of the STRMPC method is lower than the other two methods. It shows that the proposed method has the lowest index value, which also demonstrates its stability and tracking accuracy. To further verify the noise suppression capability of the proposed method, we expanded the noise range to \([-1,\,1]\). The simulation results are presented below. From Figure 13 it can be seen that the system actual states remain within the terminal constraint set. Therefore, this simulation result indicates that increasing noise interference has a rare effect on the system stability of the STRMPC method. Figure 11: Sine wave tracking with \(\pm\)0.5 noise disturbance. Figure 10: Step response with \(\pm\)0.5 noise disturbance. Figure 9: Convergence of system state and terminal constraint set with \(\pm\) 0.5 noise disturbance. Figure 14-16 show the ability of the STRMPC method to effectively suppress the noise disturbance. The ITAE indices in Table 4 show that the STRMPC has a lowest value, indicating higher control accuracy compared to the other methods. The results demonstrate that the proposed method effectively suppresses interference, even in the presence of increased noise interference. Figure 14: Step response with \(\pm 1\) noise disturbance. Figure 12: Square wave tracking with \(\pm 0.5\) noise disturbance.