# An Optimized Fractional-Order PID Horizontal Vibration Control Approach for a High-Speed Elevator
## 4 Numerical Results and Validation
### Results and Discussion  
According to the above process, the parameters of the controller are optimized by using MOGA. Kp, Ki, Kd, the integral order \(\lambda\), and differential order \(\mu\) are chosen as variables which are all nonnegative. The upper bounds for \(\lambda\) and \(\mu\) are 1. The upper bounds for Kp, Ki, and Kd change in [1\(e^{3}\),1\(e^{4}\)]. The population size is 400, and the maximum algebra is 50. The function tolerance is \(1\times 10^{-4}\). The frontier of the Pareto solution obtained through the optimization is shown in Figure 6. The horizontal and vertical axes represent the two objective functions of the optimization algorithm. The horizontal axis represents system performance, while the vertical axis represents control cost. The curve in the Figure 6 consists of all feasible solutions obtained through optimization, namely the Pareto solution set, with each point representing a set of feasible controller parameters. Select three points  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Value** \\ \hline \(m_{c}\) & kg & 1100 \\ \(m_{f}\) & kg & 2400 \\ \(l_{c}\) & kg\(\cdot\)m\({}^{2}\) & 1600 \\ \(l_{f}\) & kg\(\cdot\)m\({}^{2}\) & 8600 \\ \(k_{1}\) & N/m & 120,000 \\ \(k_{2}\) & N/m & 170,000 \\ \(c_{1}\) & N\(\cdot\)s/m & 2000 \\ \\(C_{1}\), \(C_{2}\), \(C_{3}\) on this curve. Three sets of controller parameters are brought into Simulink for simulation, and their differences in time domain response can be analyzed. Figure 7 shows a comparison of the bottom horizontal acceleration of the car body with different controllers (\(C_{1}\), \(C_{2}\), \(C_{3}\)) and without control. According to the curve in Figure 7, compared to the case without control, the three groups of controllers can reduce the horizontal acceleration response. In the curve shown in Figure 6, the point near the vertical axis has better system performance and greater control costs. The controller parameters can be reasonably selected based on actual needs. Figure 5: Parameter interval optimization process. Figure 6: Pareto solution front. Figure 8 compares the relative centerline displacement of the car frame bottom under three sets of controllers and the situation without a controller. The image curve shows that the controller has almost no effect on the relative displacement because the relative displacement occupies a small weight in the system performance \(J_{1}\). The controller does not increase the displacement, which meets the control expectations. Figure 9 compares the bottom offset between the car body and frame under the control of three sets of controllers and without control. The curve results show that under the control of three sets of controllers (C1/C2/C3), the relative offset of the system decreases compared to the case without controllers. Figure 10 compares the magnitude of vibration without a controller and controlled by optimized FOPID controller C2. The vibration of the elevator is mainly caused by the excitation of the guide rail, and the energy of the guide rail excitation is mainly concentrated in the low-frequency region. Figure 10 shows that the optimized FOPID controller can effectively reduce the vibration amplitude of the elevator compared to no controller. Figure 11 compares the output forces of three group controllers (C1/C2/C3). The output force corresponding to the controller is proportional to the control cost. Figure 7: System acceleration comparison. Figure 8: Comparison of relative displacement of car frame bottom. Figure 9: Comparison of relative offset between car body and car frame. As shown in Table 2, the RMS values of system performance indicators, control cost, horizontal acceleration, and other parameters are compared. Taking controller C2 as an example, compared with those without a controller \(J_{1}\) has been reduced by 68.46%, and the horizontal acceleration of the main control target car bottom has been reduced by 68.47%. The results show that the designed FOPID controller effectively suppresses elevator performance.