# Study on the Multi-Mode Optimal Control of Four-Wheel Steering Vehicle
## 2 Vehicle Model
### Ideal Reference Model  
The ideal steering characteristics of the active four-wheel steering vehicle are to ensure that the system has the same steering sensitivity as a conventional front-wheel steering vehicle, but also to ensure that the body has a good trajectory and attitude. That is, the steady-state gain of the yaw rate is required to be the same as that of the traditional front-wheel steering vehicle, and the sideslip angle is reduced to 0 as far as possible.  
Differential formulas of vehicle motion for linear 2-DOF front-wheel steering:  
\[\left\{\begin{array}{l}\left(k_{f}+k_{r}\right)\beta+\frac{1}{u}\left(ak_{f }+bk_{r}\right)\omega_{r}-k_{f}\delta_{f}=mu\left(\hat{\beta}+\omega_{r} \right)\\ \left(ak_{f}+bk_{r}\right)\beta+\frac{1}{u}\left(a^{2}k_{f}+b^{2}k_{r}\right) \omega_{r}-ak_{f}\delta_{f}=I_{Z}\dot{\omega}_{r}\end{array}\right. \tag{3}\]  
Let the initial condition be 0, and the transfer function of the yaw rate is:  
\[\frac{\gamma_{d}}{\delta_{f}}(s)=\frac{muk_{f}s+Lk_{r}k_{r}}{muI_{Z}s^{2}- \begin{bmatrix}I_{Z}\left(k_{f}+k_{r}\right)+m\left(a^{2}k_{f}k_{r}\right) \end{bmatrix}s+\frac{k_{f}+k_{r}}{u}L^{2}+\begin{bmatrix}\frac{mm^{2}\left(bk_ {r}-ak_{f}\right)}{k_{f}k_{r}L^{2}}\end{bmatrix}} \tag{4}\]  
It can be seen from Formula (3) that when the sideslip angle tends to 0, the yaw rate is the first-order lag response of the front wheel angle, so Formula (4) can be written as the following standard form:  
\[\frac{\gamma_{d}}{\delta_{f}}(s)=G_{\gamma d}\frac{1}{1+T_{e}s} \tag{5}\]  
where  
\[G_{\gamma d}=\frac{u}{(1+Ku^{2})L} \tag{6}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Units** \\ \hline m & 1880 & kg \\ L & 2.8 & m \\ a & 1.365 & m \\ b & 1.435 & m \\ I\({}_{Z}\) & 4044 & kg\(\cdot\)m\({}^{2}\) \\ k\({}_{f}\) & \(-\)28,000 & N/rad \\ k\({}_{r}\) & \(-\)52,000 & N/rad \\ \hline \hline \end{tabular}
\end{table}
Table 1: Key parameters of the model.  
\[K=\frac{m}{L^{2}}\left(\frac{a}{k_{r}}-\frac{b}{k_{f}}\right) \tag{7}\]  
\[T_{e}=\frac{1}{\sqrt{P}} \tag{8}\]  
\[P=\frac{L^{2}k_{f}k_{r}}{mI_{Z}u^{2}}\left(1+Ku^{2}\right) \tag{9}\]  
Formula (5) is \(K\) for the front wheel steering vehicle stability factor, \(T_{e}\) for the first-order inertia link-time constant, then the ideal reference model transfer function is as follows:  
\[x_{d}(s)=\begin{bmatrix}\beta_{d}\\ \gamma_{d}\end{bmatrix}=\begin{bmatrix}G_{\beta d}\\ G_{\lambda d}\frac{1}{1+T_{e}s}\end{bmatrix}\delta_{f}(s) \tag{10}\]  
The ideal active four-wheel steering vehicle can avoid the influence of front-wheel angle change and vehicle speed on yaw rate steady-state gain and lateral acceleration gain, and reduce the driving difficulty and overcome high-speed driving fatigue. For vehicle steering, it is best to keep the direction of the vehicle body consistent with the direction of the centroid velocity. The result is that the vehicle body can maintain a good attitude at any time in the steering process, so the sideslip angle is set to 0.  
\[\frac{\beta}{\delta_{f}}(s)=G_{\beta d}=0 \tag{11}\]  
When defining \(x_{d}=\begin{bmatrix}\beta_{d}&\gamma_{d}\end{bmatrix}^{T}\) as the state variable of the reference model, \(U_{d}=\begin{bmatrix}\delta_{f}\end{bmatrix}\) is the input of the reference model, and the state space equation of the ideal reference model is:  
\[x_{d}=A_{d}x_{d}+B_{d}U_{d} \tag{12}\]  
where  
\[A_{d}=\begin{bmatrix}0&0\\ 0&-\frac{1}{T_{e}}\end{bmatrix};\;B_{d}=\begin{bmatrix}0\\ \frac{G_{\gamma d}}{T_{e}}\end{bmatrix}\]