# Practical Approach for Developing Lateral Motion Control of Autonomous Lane Change System
## 2 Controller Design in Cylinder Domain
### Interpolation Based Gain-Scheduling  
Now, we choose the following vertex \(\mathbf{V}\) among the several candidates by considering a bound of the time-varying parameter:  
\[\mathbf{V}= \begin{bmatrix}v_{1}&v_{2}&v_{3}&v_{4}\end{bmatrix} = \begin{bmatrix} \underline{\theta}_{1}&0&\overline{\theta}_{1}&0\\ 0&\underline{\theta}_{2}&0&\overline{\theta}_{2}\end{bmatrix} . \tag{18}\]  
Then, the varying parameter vector, \(\theta\in\mathbb{R}^{2}\) is rewritten in the following polytopic form  
\[\theta(x_{1},L_{w})=\mathbf{V}\boldsymbol{\eta}(\theta) \tag{19}\]  
where \(\boldsymbol{\eta}(\theta)= \begin{bmatrix}\eta_{1}(\theta)&\eta_{2}(\theta)& \eta_{3}(\theta)&\eta_{4}(\theta)\end{bmatrix} ^{T}\in\mathbb{R}^{4}\), \(\eta_{i}\geq 0\), \(\sum_{i=1}^{4}\eta_{i}(\theta)=1\), is the convex interpolation parameter vector. Note that the convex interpolation parameter vector \(\xi(\theta)\) is uniquely determined by the given time-varying parameter vector \(\theta\) if the vertex \(\mathbf{V}\) is invertible in (19). However, the given vertex in (18) is not invertible and one cannot determine the interpolation parameter vector. Therefore, we introduce the internal division constraints to make the given vertex invertible. The defined varying parameters satisfy the trigonometric constraints; thus, the parameters always exist on the surface of the cylinder as follows:  
\[\theta_{1}^{2}+\theta_{2}^{2}=\cos^{2}\left(\frac{2\pi}{L_{w}}x_{1}\right)+ \sin^{2}\left(\frac{2\pi}{L_{w}}x_{1}\right)=1. \tag{20}\]  
Furthermore, the ranges of the vertices are equal in a 2D space. Considering the same range of varying parameter and \(\sum_{i=1}^{4}\eta_{i}(\theta)=1\) lead to the following equation:  
\[\eta_{1}(\theta)+\eta_{3}(\theta)=\eta_{2}(\theta)+\eta_{4}(\theta)=0.5. \tag{21}\]  
In addition, \(\theta_{1}\) is the internal division of \(\overline{\theta}_{1}\) and \(\underline{\theta}_{1}\) by \(\eta_{1}\) and \(\eta_{3}\). Thus, the varying parameter can be represented as follows:  
\[ \begin{split}\theta_{1}&=\frac{\eta_{1}\underline{ \theta}_{1}+\eta_{3}\overline{\theta}_{1}}{\eta_{1}+\eta_{3}}\\ &=\eta_{1}2\underline{\theta}_{1}+\eta_{3}2\overline{\theta}_{1}. \end{split}  \tag{22}\]  
We can also obtain the equation of \(\theta_{2}\) as \(\theta_{2}=\eta_{2}2\underline{\theta}_{2}+\eta_{4}2\overline{\theta}_{2}\). As a result, constraints (21) and (22) give the following expanded vertices matrix \(\hat{\mathbf{V}}\) and the expanded parameter vector \(\hat{\Theta}\) can be defined as follows:  
\[\hat{\mathbf{V}}= \begin{bmatrix}2\mathbf{V}\\ M^{2\times 4}\end{bmatrix} = \begin{bmatrix}2\underline{\theta}_{1}&0&2\overline{ \theta}_{1}&0\\ 0&2\underline{\theta}_{2}&0&2\overline{\theta}_{2}\\ 1&0&1&0\\ 0&1&0&1\end{bmatrix} , \tag{23}\] \[\hat{\Theta}= \begin{bmatrix}\Theta\\ s^{2\times 1}\end{bmatrix} = \begin{bmatrix}\theta_{1}&\theta_{2}&0.5&0.5 \end{bmatrix} ^{T},\]  
where \(M\in\mathbb{R}^{2\times 4}\) is the expanded matrix for \(\hat{\mathbf{V}}\), and \(s\in\mathbb{R}^{2}\) is the expanded vector for \(\hat{\Theta}\). Now, the vertex in (23) is invertible and the convex interpolation parameter is uniquely determined as \(\eta(\theta)=\hat{\mathbf{V}}^{-1}\hat{\Theta}\). Using the convex interpolation parameters, \(\eta_{i}(\theta)\), one can represent the system matrix as the following parameter-dependent matrix \(A_{\xi}(\theta)\):  
\[A_{\xi}(\theta)=\sum_{i=1}^{4}\eta_{i}(\theta)A_{\xi}^{(i)}, \tag{24}\]  
where  
\[A_{\xi}^{(1)}=A_{\xi}(0)+\underline{\theta}_{1}\hat{A}_{\xi}^{(1)}, \tag{25}\] \[A_{\xi}^{(2)}=A_{\xi}(0)+\underline{\theta}_{2}\hat{A}_{\xi}^{(2 )},\] \[A_{\xi}^{(3)}=A_{\xi}(0)+\overline{\theta}_{1}\hat{A}_{\xi}^{(1)},\] \[A_{\xi}^{(4)}=A_{\xi}(0)+\overline{\theta}_{2}\hat{A}_{\xi}^{(2 )},\]  
with  
\[A_{\xi}(0)= \begin{bmatrix}0&\sigma&0&0&0\\ -\sigma&0&0&0&0\\ 0&0&a_{22}&a_{23}&a_{24}\\ 0&0&0&0&1\\ 0&0&a_{42}&a_{43}&a_{44}\end{bmatrix} ,\] \[\hat{A}_{\xi}^{(1)}=