## 3 The Stability Control Strategy of the 4 WS Vehicle
### The H2/H\(\infty\) Robust Control Strategy  
In order to compensate the robustness against the unavoidable uncertainties in the real 4 WS control system, the feedback controller was designed based on the mixed H2/H\(\infty\) control theory. The H\(\infty\) controller was formulated as a minimization problem of the infinitive norm of a closed-loop transfer function matrix. The H2 theory was studied to improve system performance. As a generalization of the H\(\infty\) control problem, the H2/H\(\infty\) control theory was used to find an optimal controller that can improve the robustness and performance of the 4 WS control system. The H2/H\(\infty\) standard design problem is shown in Figure 10, where \(w\) is the input signal; \(u\) is the feedback input signal; \(z_{\infty}\) and \(z_{2}\) are the outputs of the H2/H\(\infty\) controller indexes; \(y\) is the output signal; \(G(s)\) is the generalized controlled object; \(K(s)\) is the feedback controller; \(\Delta\) represents the uncertainties. \(G(s)\) is a linear, time-invariant system, which can be described by the state space model, as shown below. \[\left\{ \begin{array}{l}\dot{x}=Ax+B_{1}w+B_{2}u\\ z_{\infty}=C_{1}x+D_{11}w+D_{12}u\\ z_{2}=C_{2}x+D_{21}w+D_{22}u\\ y=C_{y}x+D_{y1}w+D_{y2}u\end{array} \right. \tag{26}\]  
Figure 10: The standard H2/H\(\infty\) design problem. Figure 9: Linear fractional transformation (LFT) representation of 4 WS vehicle stability control system with dynamic perturbations. where \(A,B_{1},B_{2},C_{1},D_{11},D_{21},C_{2},D_{21},D_{22},C_{y},D_{y1},D_{y2}\) are the parameter matrixes of the state space. Equation (26) is represented as follows. \[\left[ \begin{array}{c}z_{\infty}\\ z_{2}\\ y\end{array} \right]=G(s)\left[ \begin{array}{c}w\\ u\end{array} \right]=\left[ \begin{array}{cc}G_{11}(s)&G_{12}(s)\\ G_{21}(s)&G_{22}(s)\end{array} \right]\left[ \begin{array}{c}w\\ u\end{array} \right]. \tag{27}\]  
Suppose \(G(s)\) meets the following requirements:  
1. \((A,B_{1})\) is stable, and \((C_{1},A)\) can be measured. 2. \((A,B_{2})\) is stable, and \((C_{2},A)\) can be measured. Based on the assumptions mentioned above, the H2/Heo control theory mainly takes advantage of the output signal to design a suitable controller and obtains the appropriate feedback control input shown below. \[u=K(s)y. \tag{28}\]  
The designed H2/H\(\infty\) controller can make the closed loop system meet different characteristics with respect to the following requirements. 1. The closed-loop system is internally stable. In other words, all eigenvalues of the state matrix of the closed-loop system are in the left half open plane. 2. The H2/H\(\infty\) norm of \(T_{zw}\) is minimized or less than a given value, as shown below. \[\left\{ \begin{array}{c}min||T_{zw}(s)||_{\infty}\\ ||T_{z_{\infty}w}(s)||_{\infty}<\gamma\end{array} \right..\] (29)  
The feedback controller \(K(s)\) can be expressed by state space, which is shown below. \[\left\{ \begin{array}{c}\dot{x}_{K}=A_{K}x_{K}+B_{K}y\\ u=C_{K}x_{K}+D_{K}y\end{array} \right. \tag{30}\]  
where \(A_{K},B_{K},C_{K},D_{K}\) are the parameter matrixes of the feedback controller \(K(s)\); \(x_{K}\) is the state vector of \(K(s)\). By applying Equation (30) into Equation (27), the closed-loop system can be expressed as follows. \[\left[ \begin{array}{c}\dot{\varepsilon}\\ z_{\infty}\\ z_{2}\end{array} \right]=\left[ \begin{array}{cc}A_{C}&B_{C}\\ C_{C1}&D_{C11}\\ C_{C2}&D_{C21}\end{array} \right]\left[ \begin{array}{c}\varepsilon\\ w\end{array} \right] \tag{31}\]  
where  
\[\varepsilon=\left[ \begin{array}{c}x\\ x_{K}\end{array} \right]\]  
\[A_{C}=\left[ \begin{array}{cc}A+B_{2}D_{K}C_{y}&B_{2}C_{K}\\ B_{K}C_{y}&A_{K}\end{array} \right]\]  
\[B_{C}=\left[ \begin{array}{cc}B_{1}+B_{2}D_{K}D_{y1}\\ B_{K}D_{y1}\end{array} \right]\]  
\[C_{C1}=\left[ \begin{array}{cc}C_{1}+D_{12}D_{K}C_{y}&D_{12}C_{K}\\ C_{C2}=\left[\begin{array}{cc}C_{2}+D_{22}D_{K}C_{y}&D_{22}C_{K}\end{array} \right]\end{array}