# Optimal Sliding Mode Control for an Active Suspension System Based on a Genetic Algorithm
## 5 Simulations and Analysis  
In order to verify the control performance of the OSMC controller optimized by the GA, the models of the passive suspension system and the active suspension system with SMC controller were also built in MATLAB/Simulink. The model of the whole system is shown in Figure 8, and the related parameters of the whole system used in simulations are listed in Table 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Symbol** & **Value** & **Unit** & **Symbol** & **Value** & **Unit** \\ \hline \(m_{u}\) & 42 & kg & \(c_{2}\) & 8.2 & \\ \(m_{s}\) & 343 & kg & \(c_{3}\) & 0.0148 & \\ \(c_{s}\) & 1.2 & \(\frac{\text{kN}}{\text{s}/\text{m}}\) & \(\epsilon_{e}\) & 0.06 & \\ \(k_{t}\) & 200 & kN/m & \(k_{e}\) & 29.63 & \\ \(k_{s}\) & 20 & kN/m & \(\delta_{1}\) & 20000 & \\ \(c_{1}\) & 2.538 & \(\backslash\) & \(\delta_{2}\) & 100 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The actual parameters of the quarter suspension system. Figure 7: The relationship between the fitness function value and the generation. The values of \(c_{1},c_{2},c_{3},\varepsilon_{e},k_{e}\) were obtained through the GA proposed in Section 4.3 based on the other values in Table 3. The values of \(\delta_{1}\) and \(\delta_{2}\) are cited from Reference [35], which is obtained based on the analytic hierarchy process method. For the passive suspension system, its parameters \((m_{u},m_{s},k_{t},k_{s},c_{s})\) are the same as the quarter active suspension system. For the active suspension system with an SMC controller, the related values of the SMC controller were \(c_{1}=-3,c_{2}=3,c_{3}=1,\varepsilon_{e}=35,k_{e}=0.01\), which are quoted from Reference [36]. The optimal parameters \((c_{1},c_{2},c_{3},\varepsilon_{e},k_{e})\) of Table 3 were put into the OSMC controller. In other words, the state equations of the three suspension systems are the same, the only difference is the control strategy. Based on the above conditions, the simulations based on MATLAB/Simulink were carried out, and the simulation results are listed from Figures 9-12. Figure 8: The whole simulation model of three suspension systems. Figure 9: Comparison of the three suspension systems’ vertical acceleration. Figure 11: Vehicle body displacements of the three suspension systems. Figure 12: Performance index J of the three suspension systems. Figure 10: Different wheel hops of the three suspension systems. Based on the curves shown from Figures 9-12, the indicators of the active suspension system with an OSMC controller, SMC controller, and passive suspension system can be clearly analyzed. In the aspects of vertical acceleration \(\tilde{z}_{s}\), the active suspension systems excelled over the passive system, and the OSMC controller performed significantly better than the SMC controller in the active suspension system. For a better comparison of the indexes of the three suspension systems, the root mean square (RMS) values of indicators are listed in Table 4. As shown in Table 4, the vertical acceleration \(\tilde{z}_{s}\) of the OSMC controller decreased 17.28% and 8.1% more than the SMC controller and passive systems, respectively. The comprehensive performance index \(J\) of OSMC controller decreased 12.73% and 33.4% more than the SMC controller and passive systems, respectively. The vertical acceleration of the sprung mass is the most important riding comfort performance index [7; 8; 9]. As a result, the OSMC controller performed better than others in the aspect of riding comfort. As can be seen from Figures 10 and 11 and Table 4, the differences of \(z_{u}-z_{q}\) and \(z_{s}-z_{u}\) among the three suspension systems was significantly small, which indicated that the vehicle stability and the suspension rattle space were almost the same. According to the above analyses, we can come to the conclusion that the  
As shown in Figure 13, the RMS of the vertical acceleration \(\ddot{z}_{s}\) decreased 19.34% compared with the optimum results in Reference [7] based on the OSMC controller.