## 2 Active Suspension Systems and Control Systems
### Simulation Analysis  
In the optimization process of GA, to guarantee the control effect of BA and reduce the number of iterations, the weighting coefficient of BA is set to 1. We took the optimization results of the literature [14] as references and expanded the range of weighting coefficients \(q_{1}\) and \(q_{2}\) intervals as larger as possible to ensure the reliability of the optimization search results. For GA, the main parameters are the population size, the number of interactions, the crossover fraction, and the mutation fraction. We set these parameters of GA using the values set in the literature [16; 17] as references. As can be seen from Figure 5a, the value of the fitness function has reached the optimum at the 12 th iterations. Therefore, it is sufficient to set the generations and termination generation to 20. Furthermore, the selection of crossover fraction and the genetic fraction is crucial to the performance of genetic algorithms. The larger the crossover fraction, the faster new individuals are generated. However, when the crossover fraction is too large, there is a possibility of the genetic pattern being destroyed, making the structure of individuals with high fitness destroyed. Additionally, if the crossover fraction is too small, the search process is slow, and the search time is prolonged. For the mutation fraction, if its value is too small, it is not easy to generate new individuals. If the mutation fraction is too large, the randomness of the genetic algorithm search is too large. Therefore, we used the values in the literature [16; 17] as references, and the two parameters were fine-tuned to simulation. Finally, the crossover fraction is set to 0.4 and the mutation fraction to 0.05. The parameters of the GA are shown in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model Parameter** & **Value** & **Model Parameter** & **Value** \\ \hline Search interval of \(q_{1}\) & \((0,\,2\times 10^{5})\) & Crossover fraction & 0.4 \\ Search interval of \(q_{2}\) & \((0,\,2\times 10^{4})\) & Generations & 20 \\ Population size & 100 & Termination generation & 20 \\ Elite count & 10 & Fitness function deviation value & \(1\times 10^{-4}\) \\ Mutation fraction & 0.05 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of GA.  
Figure 5: Optimization results of GA: (**a**) the variation in fitness function value; (**b**) optimal solution of weighting coefficients. The variation in fitness function value is shown in Figure 5a. The optimal solution of weighting coefficients is shown in Figure 5b. From the simulation results in Figure 5, it can be seen that the fitness function value is optimized with the population evolving and finally converges to 2.42819. The weighting coefficients' optimal solution is: \(q_{1}=68680\), \(q_{2}=9259\), \(q_{3}=1\). Simulations are performed when the road input is the A-class to D-class composite road, simulation speed is 20 m/s, and simulation time is 40 s. The time domain curves of \(1/4\) vehicle model's BA, SWS, and DTD are obtained, as shown in Figure 6a-d. In Figure 6, the solid lines are the results of a passive suspension, and the dot-dash lines are those for active control. Compared with passive suspension, the vibration range of each performance index is reduced after the optimization and control. To quantitatively describe the improvement effect, the RMS values and improvement rate of each performance index are calculated, as shown in Table 3.  
Figure 6: Comparison curves of time domain response for two suspensions (passive suspension and active suspension based on GA-LQR) at four road classes: (**a**) A-class road; (**b**) B-class road; (**c**) C-class road; (**d**) D-class road.