# Risk-Sensitive Rear-Wheel Steering Control Method Based on the Risk Potential Field
## 2 Design of Rear-Wheel Steering Control Based on the Risk Potential Field
### Feedforward Controller  
Equations (13) and (14) are Laplace transformed and solved for the rear steering angle (\(\delta_{r}\)), assuming a steady state. Therefore, the feedforward term of the desired rear steering angle (\(\delta_{r}\dot{\ }_{\ \  
Furthermore, the feedback term of the desired rear-wheel steering angle (\(\delta_{r}^{*}\_{-fb}\)) can be defined by applying the error state feedback to the error state equation as follows:  
\[\delta_{r}^{*}\_{-fb}\ =\ -K_{\beta}\Delta\beta-K_{r}\Delta r, \tag{19}\]  
where \(K_{\beta}\) and \(K_{r}\) indicate the state feedback gains. The LQ control theory is used to design the regulator, and the state feedback gain is determined to minimize this evaluation function.  
\[J_{LQ}\ =\ \int_{0}^{\infty}\left[\left(\frac{\Delta\beta}{\beta_{\max}} \right)^{2}+\left(\frac{\Delta r}{r_{\max}}\right)^{2}+\left(\frac{\delta_{r }^{*}\_{-fb}}{\delta_{\max}}\right)^{2}\right]dt, \tag{20}\]  
where the denominator of each term in the evaluation function indicates the tolerance of the error of the state quantity and the limit of the control input. In this paper, the control parameters are set as follows: \(\beta_{\max}\) = 0.1 rad, \(r_{\max}\) = 0.1 rad/s, and \(\delta_{r}\) = 0.1 rad. The stability of the model matching the control of the active rear wheel steering control is guaranteed, as the weighting coefficients of the performance index are set in the way that the poles of the feedback-controlled vehicle are located on the left side of the complex plane. Regarding the system stability, the closed loop poles can be determined by calculating the eigenvalues of the closed loop matrix \(A-BK\):  
\[\dot{x}\ =\ Ax+Bu\ =\ (A-BK)x, \tag{21}\]  
where the feedback gain matrix \(K\) is calculated by the LQ control theory to minimize the cost function:  
\[J_{LQ}\ =\ \int_{0}^{\infty}x^{T}Qx+u^{T}Rudt, \tag{22}\]  
The Riccati equation for finding the feedback gain is expressed as follows:  
\[PA+A^{T}P-PBR^{-1}B^{T}P+Q\ =\ 0, \tag{23}\]  
Using \(P\), which satisfies Equation (23), the feedback gain is calculated as follows:  
\[K\ =\ R^{-1}B^{T}P. \tag{24}\]  
Here, all elements of each matrix are represented as follows:  
\[A\ =\ \left[\begin{array}{cc}A_{11}&A_{12}\\ A_{21}&A_{22}\end{array}\right],\ B\ =\ \left[\begin{array}{cc}B_{1}\\ B_{2}\end{array}\right],\ x\ =\ \left[\begin{array}{cc}\Delta\beta\\ \Delta r\end{array}\right],\ u\ =\ \left[\delta_{r-fb}^{*}\right]\] \[K\ =\ \left[\begin{array}{cc}K_{\beta}&K_{r}\end{array}\right],\ Q \ =\ \left[\begin{array}{cc}\frac{1}{\beta_{\max}}&0\\ 0&\frac{1}{r_{\max}}\end{array}\right],\ R\ =\ \left[\frac{1}{\delta_{\max} ^{2}}\right]\]  
Moreover, in the control part, the constraint of the control input (rear-wheel steering) is under a hard constraint (\(\delta_{r}\leq 3.0\) deg). In addition, for securing the vehicle stability during collision avoidance, as the physical limit of the vehicle motion, each tire force should not exceed its friction limit as shown in the following expression:  
\[\sqrt{F_{x}{}^{2}+F_{y}{}^{2}}\leq\mu F_{z}, \tag{25}\]  
where \(F_{x}\) and \(F_{y}\) indicate the tire longitudinal and lateral forces, respectively, \(\mu\) denotes the road friction coefficient, and \(F_{z}\) indicates the tire vertical load.  
The closed loop system poles with respect to the change of the velocity can be calculated as the following complex plane (Figure 4).