# Enhancement of Yaw Moment Control for Drivers with Excessive Steering in Emergency Lane Changes
## 4 Yaw Moment Control Design
### Control System Modeling with Polytopic LPV  
We use the polytopic LPV modeling technique to design the control input for the speed-varying error system (37). In the \((q_{1},q_{2})\) parameter space shown in Figure 4, we consider the following affine matrices  
\[\widetilde{\mathbf{A}}_{v}(q_{1},q_{2})= \begin{bmatrix}0&-1\\ \frac{-C_{f}l/+C_{r}l}{l_{z}}&0\end{bmatrix} +q_{1} \begin{bmatrix}-\frac{C_{f} +C_{r}}{m}&0\\ 0&-\frac{C_{f}l_{f}^{2}+C_{r}l_{z}^{2}}{l_{z}}\end{bmatrix} +q_{2} \begin{bmatrix} 0&\frac{-C_{f}l/+C_{r}l}{m}\\ 0&0\end{bmatrix}  \tag{47a}\] \[\widetilde{\mathbf{B}}_{v}(q_{1},q_{2})= \begin{bmatrix}0\\ \frac{C_{f}l_{f}}{l_{z}}\end{bmatrix} +q_{1} \begin{bmatrix}\frac{C_{f}}{m}\\ 0\end{bmatrix} \] (47b) \[\widetilde{\mathbf{B}}_{w}(q_{1},q_{2})=[\widetilde{\mathbf{B}}_{v}(q_{1},q_{ 2})\ \mathbf{B}_{m}]= \begin{bmatrix}0&0\\ \frac{C_{f}l_{f}}{l_{z}}&\frac{1}{l_{z}}\end{bmatrix} +q_{1} \begin{bmatrix} \frac{C_{f}}{m}&0\\ 0&0\end{bmatrix} \] (47c) and a curve segment \[C_{V}:\ q_{2}=q_{1}^{2},\ q_{1}\in \begin{bmatrix}V_{\max}^{-1},V_{\min}^{-1} \end{bmatrix}  \tag{48}\]  
It can be seen that the affine matrices (47) on the curve segment (48) indeed are the matrices \(\mathbf{A}_{v}(V_{0})\), \(\mathbf{B}_{v}(V_{0})\) and \(\mathbf{B}_{w}(V_{0})\) in (1) and (37), i.e.,  
\[\mathbf{A}_{v}(V_{0})=\widetilde{\mathbf{A}}_{v}(q_{1},q_{1}^{2}),\ \mathbf{B}_{v}(V_{0})= \widetilde{\mathbf{B}}_{v}(q_{1},q_{1}^{2}),\ \mathbf{B}_{w}(V_{0})= \widetilde{\mathbf{B}}_{w}(q_{1},q_{1}^{2}),\ q_{1}=V_{0}^{-1}\in \begin{bmatrix} V_{\max}^{-1},V_{\min}^{-1}\end{bmatrix}  \tag{49}\]  
A triangular polytope \(\Theta\) framed by the lines \(L_{1}\), \(L_{2}\), and \(L_{3}\) as depicted in Figure 4 is chosen to contain the curve segment \(C_{V}\) (see [6] and reference therein), where \(L_{1}\) is the line passing through the two points \(a\big{(}V_{\max}^{-1},V_{\max}^{-2}\big{)}\) and \(b\big{(}V_{\min}^{-1},V_{\min}^{-2}\big{)}\), while \(L_{2}\) and \(L_{3}\) are the tangent lines to the curve segment \(C_{V}\) at the points \(a\big{(}V_{\max}^{-1},V_{\max}^{-2}\big{)}\) and \(b\big{(}V_{\min}^{-1},V_{\min}^{-2}\big{)}\), respectively. The intersection of lines \(L_{2}\) and \(L_{3}\) is the point \(c\big{(}(V_{\max}^{-1}+V_{\min}^{-1})/2,V_{\max}^{-1}V_{\min}^{-1}\big{)}\). Thus, the triangular polytope \(\Theta\) containing the curve segment \(C_{V}\) has its vertices \(a\big{(}V_{\max}^{-1},V_{\max}^{-2}\big{)}\), \(b\big{(}V_{\min}^{-1},V_{\min}^{-2}\big{)}\), and \(c\big{(}(V_{\max}^{-1}+V_{\min}^{-1})/2,V_{\max}^{-1}V_{\min}^{-1}\big{)}\). Based on the concept of polytopic LPV, there exists a speed-parameter-dependent coefficient \(\eta_{i}(V_{0})\geq 0\), \(i=1,2,3\) such that \(\sum_{i=1}^{3}\eta_{i}(V_{0})=1\) and the following convex combinations hold:  
\[ \begin{split}\mathbf{A}_{v}(V_{0})=\sum_{i=1}^{3}\eta_{i}(V_{0}) \mathbf{A}_{vi},\ \mathbf{B}_{v}(V_{0})=\sum_{i=1}^{3}\eta_{i}(V_{0})\mathbf{B}_{vi},\ \ \ \mathbf{B}_{w}(V_{0})=\sum_{i=1}^{3}\eta_{i} \ (V_{0})\mathbf{B}_{w},\\ \forall V_{0}\in\big{[}V_{\min},V_{\max}\big{]}\end{split}  \tag{50}\]  
Figure 4: The polytope \(\Theta\) and curve segment \(C_{V}\) in the parameter \((q_{1},q_{2})\) space.