# Time-Delay Vibration Reduction Control of 3-DOF Vehicle Model with Vehicle Seat
## 4 Suspension Time-Delay Control Parameter Optimization
### The Establishment Method of Objective Function
Due to the inconsistency of the units and magnitudes of each performance index, it will be divided by the corresponding passive suspension performance index value, and the fitness function obtained is as follows  
\[\min(J)=\mu_{1}\frac{\sigma_{1}}{\sigma_{1 pars}}+\mu_{2}\frac{\sigma_{2}}{ \sigma_{2 pars}}+\mu_{3}\frac{\sigma_{3}}{\sigma_{3 pars}}+\mu_{4}\frac{\sigma_{4 }}{\sigma_{4 pars}} \tag{15}\]where the weighting coefficients \(\mu_{1}\),\(\mu_{2}\),\(\mu_{3}\), and \(\mu_{4}\) are used as the weight coefficients of the evaluation function to measure the relative importance of each sub-objective function. In this paper, we take \(\mu_{1}\)= 0.3, \(\mu_{2}\)= 0.3, \(\mu_{3}\)= 0.2, \(\mu_{4}\)= 0.2 and \(\sigma_{1\text{pas}}\), \(\sigma_{2\text{pas}}\), \(\sigma_{3\text{pas}}\), and \(\sigma_{4\text{pas}}\) to represent the corresponding performance of the passive suspension. \(\sigma_{1}\) is the root mean square value of the occupant's vertical acceleration:  
\(\sigma_{1}=\sqrt{\frac{\sum\limits_{k=1}^{n}\left(\tilde{x}_{pk}\right)}{n}}\)  
\(\sigma_{2}\) is the root mean square value of body acceleration:  
\(\sigma_{2}=\sqrt{\frac{\sum\limits_{k=1}^{n}\left(\tilde{x}_{sk}\right)}{n}}\)  
\(\sigma_{3}\) is the root mean square value of suspension dynamic deflection:  
\(\sigma_{3}=\sqrt{\frac{\sum\limits_{k=1}^{n}\left(x_{sk}-x_{uk}\right)}{n}}\)  
\(\sigma_{4}\) is the root mean square value of tire dynamic load:  
\(\sigma_{4}=\sqrt{\frac{\sum\limits_{k=1}^{n}k_{t}\left(x_{uk}-x_{r}\right)}{n}}\)  
where \(\tilde{x}_{pk}\) is the vibration acceleration of the occupant at time \(t_{\text{k}}\), \(x_{sk}\) is the vertical displacement of vehicle vibration at time \(t_{\text{k}}\), and \(x_{uk}\) is the vertical displacement of wheel vibration at time \(t_{\text{k}}\). Optimization Constraints  
For the suspension dynamic deflection constraint, in order to prevent frequent collision with the buffer block when driving on bad roads, the dynamic stroke of the suspension must meet the inequality of Equation (16) within the safe stroke range:  
\[\left|x_{s}(t)-x_{u}(t)\right|\leq z_{\text{max}} \tag{16}\]  
where the maximum travel value of the suspension is represented by \(z_{\text{max}}\). For tire grounding, in order to prevent the tire from falling off the ground during driving and cause driving safety problems, it is necessary to meet the inequality of Equation (17):  
\[\left|k_{2}(x_{u}(t)-x_{r}(t))\right|\leq F \tag{17}\]  
where the static load of the tire is represented by the calculation formula, which is as follows:  
\[F=(m_{p}+m_{s}+m_{u})g \tag{18}\]  
The formula of the mathematical model of the optimized objective function is as follows:  
\[\left\{ \begin{array}{l}\min(J)=\mu_{1}\sigma_{1}{}^{2}+\mu_{2}\sigma_{2}{}^{ 2}+\mu_{3}\sigma_{3}{}^{2}\\ s.t.\\ \sigma_{1}<\sigma_{1\text{pas}}\\ \sigma_{2}<\sigma_{2\text{pas}}\\ \left|x_{s}(t)-x_{u}(t)\right|\leq z_{\text{max}}\\ \left|k_{2}(x_{u}(t)-x_{r}(t))\right|\leq F\\ \tau>0\end{array} \right. \tag{19}\]