## 3 The Stability Control Strategy of the 4 WS Vehicle
### The Uncertainty Description  
The 4 WS vehicle stability control system is vulnerable to uncertainties that are unavoidable in a real control system. Hence, uncertainties must be taken into consideration to improve the robustness and robust stability of the 4 WS control system. The uncertainties are of two classes: disturbances and dynamic perturbations. The former consists of external disturbances, sensor noises, actuator noises, etc. The latter denotes the modeling uncertainty, which is the difference between the mathematical model and the actual control system. In this paper, the 4 WS vehicle modeling uncertainty, the rear wheel electrohydraulic system modeling uncertainty, the yaw rate sensor noise, and the external disturbances are regarded as the uncertainties, as shown in Figure 7. \(\Delta_{act}\) denotes the rear wheel electrohydraulic system modeling uncertainty; \(\Delta_{v}\) represents the 4 WS vehicle modeling uncertainty; \(n,f,T_{f}\) are the yaw rate sensor noise, the lateral interference force, and the yaw interference torque, respectively; \(G_{ref}\) denotes the transfer function of the reference model; \(G_{act}\) represents the transfer function of the rear wheel electrohydraulic system; \(G\) stands for the transfer function of the 4 WS vehicle model. The external disturbance signals are considered in the 4 WS vehicle model. The 4 WS vehicle model based on external disturbances can be described as follows. \[ \begin{array}{l}mV(\hat{\beta}+r)=-\left(C_{f}+C_{r}\right)\beta-\frac{l_{f} C_{f}-l_{f}C_{r}}{V}r+C_{f}\delta_{f}+C_{r}\delta_{r}+f\\ I_{z}\dot{r}=-\left(l_{f}C_{f}-l_{r}C_{r}\right)\beta-\frac{l_{f}C_{f}+l_{r}^{ 2}C_{r}}{V}r+l_{f}C_{f}\delta_{f}-l_{r}C_{r}\delta_{r}+T_{f}.\end{array}  \tag{20}\]  
Equation (20) is transformed into a state space model, which is shown below. \[\left[ \begin{array}{c}\dot{\beta}\\ \dot{r}\end{array} \right]=\left[ \begin{array}{cc}\frac{-\left(C_{f}+C_{r} \right)}{mV}&\frac{l_{f}C_{r}-l_{f}C_{f}}{mV^{2}_{r}}-1\\ \frac{l_{f}C_{r}-l_{f}C_{f}}{l_{z}}&-\frac{l_{f}C_{f}+l_{r}^{2}C_{r}}{l_{z}V} \end{array} \right]\left[ \begin{array}{c}\beta\\ r\end{array} \right]+\left[ \begin{array}{cc}\frac{C_{f}}{mV}&\frac{1}{mV}&0 &\frac{C_{r}}{mV}\\ \frac{l_{f}^{2}C_{f}+l_{r}^{2}C_{r}}{l_{z}}&0&\frac{1}{l_{z}}&-\frac{l_{r}C_{r }}{l_{z}}\end{array} \right]\left[ \begin{array}{c}\delta_{f}\\ f\\ T_{f}\\ \delta_{r}\end{array} \right]. \tag{21}\]  
The dynamic perturbations can be represented by inserting one single perturbation block \(\Delta\) into the mathematical model. As shown in Figure 7, \(\Delta_{act}\) denotes the input multiplicative perturbation in the rear wheel electrohydraulic system model, and \(\Delta_{v}\) represents the inverse additive perturbation in the 4 WS vehicle model. The linear fractional transformation (LFT) is used to transform dynamic  
Figure 7: Block diagram of the robust control considering uncertainties. perturbations into a standard configuration to represent how the model uncertainties affect the input/output relationship [37]. The dynamic perturbations can be denoted as a unified framework, as shown in Figure 8.  
\(F(M,\Delta)\) is called the LFT of interconnection transfer function of matrix M to perturbation \(\Delta\), which can be written as follows. \[ \begin{gathered} F(M,\Delta)=M_{22}+M_{21}\Delta{(I-M_{11}\Delta) }^{-1}M_{12}\\ M=\left[\begin{array}{cc}M_{11}&M_{12}\\ M_{21}&M_{22}\end{array}\right].\end{gathered}  \tag{22}\]  
Based on LFT, the relationship between input \(w\) and output \(z\) of the control system can be shown below. \[z=F(M,\Delta)w. \tag{23}\]  
The rear wheel electrohydraulic system model and the 4 WS vehicle model uncertainties can be described uniformly by Figure 8, and their interconnection transfer function matrixes are shown as follows. \[ \begin{gathered} M_{act}=\left[\begin{array}{cc}0&I\\ G_{act}&G_{act}\end{array}\right]\\ M_{0}=\left[\begin{array}{cc}-G&G\\ -G&G\end{array}\right]\end{gathered}