# 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  
This paper presents a new optimization method. When the complex excitation is known, the linear function is used to make the complex excitation equivalent in the discrete time interval. This means dividing continuous time into small periods of time. The linear function \(g(x)=l_{a}x+l_{b}\) is equivalent to the complex excitation in each time interval. When the time interval is finely divided, the linear function can approximate the original complex excitation. In the continuous time interval, the amplitude of both is the same at each discrete time point. The optimization problem of complex excitation is transformed into the optimization problem of the normal force in the discrete time interval. This simplifies complex issues. At the same time, the external excitation is directly introduced into the solution process of time-delay control parameter optimization in the objective function. The quantitative relationship among the time-domain vibration response, time-delay control parameters, and external excitation are established. Assume that \(f(x)\) is a complex excitation, and take the linear equivalent function \(g(x)=l_{a}x+l_{b}\). Among them, \(l_{a}\) and \(l_{b}\) are the equivalent parameters. In a small time interval of \([t_{k},t_{k+1}]\), then \(\left\{ \begin{array}{l}f(x_{t_{k}})=l_{a}x_{t_{k+1}}+l_{b}\\ f(x_{t_{k+1}})=l_{a}x_{t_{k+1}}+l_{b}\end{array} \right..\) Of these, \(f(x)\) is known. It is only necessary to solve for the equivalent parameters of \(l_{a}\) and \(l_{b}\). Then, equivalent complex excitation \(g(x)=\sum\limits_{k=1}^{n}l_{ak}x+l_{bk}\) is brought into Equations (1) to (3). It can solve the vibration response of the system at each time point \(t_{k}\) by solving the dynamic equation of the system:  
\[\left\{ \begin{array}{l}x_{p}=\left[x_{p1},x_{p2},\cdots x_{pk}\cdots,x_{pn} \right]\\ \dot{x}_{p}=\left[\dot{x}_{p1},\dot{x}_{p2},\cdots\dot{x}_{pk}\cdots,\dot{x}_{ pn}\right]\\ x_{s}=\left[x_{s1},x_{s2},\cdots x_{sk}\cdots,x_{sn}\right]\\ \dot{x}_{s}=\left[\dot{x}_{s1},\dot{x}_{s2},\cdots\dot{x}_{sk}\cdots,\dot{x}_{ sn}\right]\\ x_{u}=\left[x_{u1},x_{u2},\cdots x_{uk}\cdots,x_{un}\right]\\ \dot{x}_{u}=\left[\dot{x}_{u1},\dot{x}_{u2},\cdots\dot{x}_{uk}\cdots,\dot{x}_{ un}\right]\end{array} \right.\quad k\in[1,2,\cdots,n] \tag{13}\]  
where \(x_{pk}\), \(\dot{x}_{pk}\) are the vibration displacement and velocity of the occupant at \(t_{k}\); \(x_{sk}\), \(\dot{x}_{sk}\) are the vibration displacement and velocity of the body mass at \(t_{k}\); and \(x_{uk}\dot{x}_{uk}\) are the vibration displacement and velocity of the unsprung mass at \(t_{k}\). The vibration response of Equation (7) is brought into Equations (1)-(3) to obtain the vibration acceleration of the occupant, the body mass, and the unsprung vibration at time \(t_{k}\). \[\left\{\begin{array}{l}\ddot{x}_{pn}=-\left(c_{1}\big{(}\dot{x}_{p}-\dot{x}_ {s}\big{)}+k_{1}\big{(}x_{p}-x_{s}\big{)}\right)/m_{p}\\ \ddot{x}_{sn}=-\left( \begin{array}{c}-c_{1}\big{(}\dot{x}_{p}-\dot{x}_{s} \big{)}-k_{1}\big{(}x_{p}-x_{s}\big{)}\\ +c_{2}\big{(}\dot{x}_{s}-\dot{x}_{u}\big{)}+k_{2}\big{(}x_{s}-x_{u}\big{)}+f \\ \ddot{x}_{un}=-\big{(}-c_{2}\big{(}\dot{x}_{s}-\dot{x}_{u}\big{)}-k_{2}\big{(}x _{s}-x_{u}\big{)}+k_{3}\big{(}x_{u}-x_{r}\big{)}-f\big{)}/m_{u}\end{array} \right. \tag{14}\]  
According to the vehicle's ride comfort requirements, this paper takes the occupant vertical acceleration, body acceleration, suspension dynamic stroke, and tire dynamic load weighting as the optimization objective function and meets the ride comfort requirements to the greatest extent as the optimization objective.