## 4 Numerical Simulation Results  
In this section, controllers for the suspension of a quarter car are designed, and the results of the simulations that have been tried on the system are given. The control algorithms developed and applied for the system are LQR, PSO-based FLC, and PSO-based fuzzy-LQR methods. Active control of the system was performed using these methods, and the obtained results were presented with graphs and tables. The main objective in controlling the car suspension system is to minimize the impact of the disturbance caused by the road entrance on passenger comfort. The initial parameters of the system are determined as follows: Matlab/Simulink package program was used in this study. The physical parameters of the car suspension system are m\({}_{\text{b}}\) = 300 kg, m\({}_{\text{w}}\) = 60 kg, k\({}_{\text{s}}\) = 16,000 N/m, b\({}_{\text{s}}\) = 1000 Ns/m, k\({}_{\text{t}}\) = 190,00  
This study presents the graphs resulting from the passive state of the suspension system and the active control of the LQR, PSO-based FLC, and PSO-based fuzzy-LQR methods. Figures 10 and 11 show the graphs obtained as a result of the motion of the car body and the acceleration of the car body, which are the parameters affecting passenger comfort, and the passive state and active control with the LQR, PSO-based fuzzy, and PSO-based fuzzy-LQR control methods. By looking at Figure 10, the plot of the vertical deflection of the car body as a function of time, it is clear that all the methods used significantly reduce the vibration amplitude and the seat time compared to the passive suspension system. It can be seen that the smallest change in vibration amplitude is in the PSO-based fuzzy-LQR control (0.018-0 m range) and the highest amplitude is in the LQR control (0.038-0.005 m). The acceleration acting on the car due to the poor road entrances can reach dimensions that disturb the passengers. Therefore, reducing car acceleration amplitudes is important for car ride  
Figure 11: The response to the body acceleration using the methods described. Figure 10: The response to the body motion using the methods described. comfort. By examining the acceleration curve of the car body, it can be seen that the PSO-based fuzzy-LQR control type (in the range of 1-1 m/s\({}^{2}\)) attenuates the acceleration amplitudes much more successfully. Similarly, the highest amplitude is observed for the LQR control type (about 2.1-2.4 m). It is also observed that the LQR control type causes undesirable amplitude increases. The suspension deflection and tire deflection resulting from the methods used are shown in Figures 12 and 13. Another criterion for the success of the methods is whether the methods used in car suspension systems cause the problem of suspension constriction. The suspension deflection curve in Figure 12 shows that the suspension range has reached zero, i.e., the initial position, for all three controller types and thus does not narrow. It can be seen that the PSO-based fuzzy-LQR (in the range of about 0.018-0 m) occurs at a much lower amplitude in the control mode. Similarly, the highest amplitude is observed in LQR control (about  
Figure 12: The response to the suspension deflection using the methods described. Figure 13: The response to the tire deflection using the methods described. 0.033-0.005 m). The graph of the tire deflection in Figure 13 shows that the tire deflection occurs in all three types of controllers. It can be seen that the PSO-based fuzzy-LQR (about \(-0.0012\)-\(0.0018\) m) occurs at a much lower amplitude in the control type. Similarly, it can be seen that the highest tire deviation occurs in the LQR control type (about \(0.0018\)-\(0.003\) m). Figure 14 shows the control forces that result from active control. In car vibration reduction studies, the fact that the control force required by a controller is low is accepted as an indicator of the successful operation of that controller.