# An Optimized Fractional-Order PID Horizontal Vibration Control Approach for a High-Speed Elevator
## 4 Numerical Results and Validation
### Results and Discussion
Figure 11: Comparison of active control force. Figure 10: Comparison of vibration spectrum diagram. vibration compared to the uncontrolled situation, and the inhibition effect is related to the control cost. Using the same optimization algorithm to optimize the conventional PID controller, a Pareto solution set like the FOPID controller is obtained (Figure 12). The two curves are shown below. Using the controller C2 designed above as the comparison object, the control effects between PID and FOPID are compared and analyzed. The control cost of the designed FOPID controller C2 is 77. Select a point on the PID curve where the control cost is similar and bring the corresponding parameters and PID controller into the model for simulation analysis. Figure 13 compares the horizontal acceleration under the control of two types of controllers, respectively. From the curve in Figure 13, compared to the PID controller, the horizontal acceleration controlled by the FOPID controller is smaller. Figure 14 compares the displacement at the car frame's bottom relative to the centerline under two control situations. The two controllers have almost the same control effect on the bottom displacement relative to the centerline. Figure 15 shows the bottom offset between the car body and the car frame under two control situations. From Figure 15, the relative deviation of FOPID is slightly smaller than that of PID. Figure 16 \begin{table}
\begin{tabular}{c c c c c} \hline  & **Uncontrolled** & **C1** & **C2** & **C3** \\ \hline \(J_{1}\) & \(4.817\times 10^{4}\) & \(9.616\times 10^{3}\) & \(1.519\times 10^{4}\) & \(2.784\times 10^{4}\) \\ \(J_{2}\) & 0 & 106.3202 & 77.7123 & 47.9846 \\ Relative displacement of & & & & \\ car frame bottom (\(X_{d}\)) & 0.0020 & 0.0019 & 0.0019 & 0.0019 \\ Relative offset (\(X_{e}\)) & \(8.5845\times 10^{-4}\) & \(1.4406\times 10^{-4}\) & \(2.3772\times 10^{-4}\) & \(4.5487\times 10^{-4}\) \\ Horizontal acceleration & & & & \\ of the car body (\(\tilde{x_{2}}\)) & 0.3933 & 0.0785 & 0.1240 & 0.2273 \\ Control force (\(F_{1}\)) & 0 & 647.8452 & 464.9079 & 283.5612 \\ Control force (\(F_{2}\)) & 0 & 75.9603 & 10.6733 & 19.4114 \\ Control force (\(F_{3}\)) & 0 & 573.9356 & 431.7551 & 271.3025 \\ Control force (\(F_{4}\)) & 0 & 14.4725 & 10.3735 & 19.6114 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of corresponding indicators. Figure 12: Comparison of Pareto front between PID and FOPID. To sum up, both controllers can suppress elevator vibration, but when the output force is basically equal, the parameters such as elevator acceleration and car offset under the same conditions, the parameters such as elevator acceleration and car offset under the same conditions are different. Figure 16: Comparison of active control force of PID and FOPID. Figure 15: Comparison of bottom offset of PID and FOPID. optimized FOPID controller control are superior to optimized PID control. As shown in Table 3, comparing the RMS of system performance indicators, control cost, horizontal acceleration, and other parameters under the two control conditions, compared to the PID controller, \(J_{1}\) of FOPID controller decreased by 24.9%, the horizontal acceleration at the bottom of the car body decreased by 24.97%, the relative displacement at the bottom of the car frame decreased by 13.63%, and the relative bottom displacement between the car body and frame decreased by 27.71%. It can be found that FOPID is superior to PID control.