# Practical Approach for Developing Lateral Motion Control of Autonomous Lane Change System
## 2 Controller Design in Cylinder Domain
### Development of LPV System  
We can change the nonlinear dynamics into linear dynamics by considering the nonlinear bounded terms in the model as matrix coefficients. The \(\xi_{1}\) and \(\xi_{2}\) states in (10) are nonlinear for the original state \(x_{1}\). Therefore, we define these two variables as the varying parameter as \(\theta(x_{1},L_{w})=\left[\theta_{1}(x_{1},L_{w})\quad\theta_{2}(x_{1},L_{w}) \right]^{T}\):  
\[\theta_{1}(x_{1},L_{w}) =\xi_{2}, \tag{12}\] \[\theta_{2}(x_{1},L_{w}) =-\xi_{1},\]  
where \(x_{1}\) and \(L_{w}\) are available from the vision sensor. Now, the first two equations in (9) can be represented as follows:  
\[\dot{\xi}_{1} =\frac{2\pi}{L_{w}}\xi_{3}\xi_{2}=\frac{2\pi}{L_{w}}\theta_{1}\xi_ {3}, \tag{13}\] \[\dot{\xi}_{2} =-\frac{2\pi}{L_{w}}\xi_{3}\xi_{1}=\frac{2\pi}{L_{w}}\theta_{2} \xi_{3}.\]  
Note that the lateral offset error \(x_{1}\) is unbounded, and the lane width \(L_{w}\) is not zero on a highway. Nevertheless, the varying parameters are bounded because of the trigonometric function.  
\[\theta_{1}(x_{1},L_{w}) \in\left[\overline{\theta}_{1}\quad\underline{\theta}_{1}\right], \tag{14}\] \[\theta_{2}(x_{1},L_{w}) \in\left[\overline{\theta}_{2}\quad\underline{\theta}_{2}\right].\]  
Now, we introduce a small auxiliary variable for the non-linearized term between \(\xi_{1,2}\) and \(\xi_{3}\). In practice, \(\xi_{1}\) and \(\xi_{2}\) are approximately accurate because their values are calculated by the measured signal from lateral offset and width of lane. However, \(\xi_{3}\) contains uncertainty because it is obtained from an estimator or numerical derivation of the lateral offset. Therefore, we introduce auxiliary variables \(\sigma\) to \(\xi_{3}\) rather than \(\xi_{1}\) and \(\xi_{2}\). Let us define an auxiliary variable \(\sigma\) that reflects the nonlinearized term between \(\xi_{1,2}\) and \(\xi_{3}\) as follows:  
\[\dot{\xi}_{1} \simeq\frac{2\pi}{L_{w}}\left(\xi_{3}+\sigma\right)\xi_{2}, \tag{15}\] \[\dot{\xi}_{2} \simeq-\frac{2\pi}{L_{w}}\left(\xi_{3}+\sigma\right)\xi_{1}.\]  
Further, the linearized Equation (14) can be reformulated as follows using Equation (15):  
\[\dot{\xi}_{1} \simeq\frac{2\pi}{L_{w}}\left(\xi_{3}+\sigma\right)\xi_{2} \tag{16}\] \[=\frac{2\pi}{L_{w}}\xi_{3}\xi_{2}+\frac{2\pi}{L_{w}}\sigma\xi_{2}\] \[=\frac{2\pi}{L_{w}}\theta_{1}\xi_{3}+\varepsilon\xi_{2}\]  
The same can be applied for \(\dot{\xi}_{2}\).  
Therefore, the nonlinear dynamics (9) with the cylinder domain can be interpreted as a form of the LPV system with variation of lateral offset and lane width. Both parameterizing the nonlinear bounded varying parameter (12) and defining the auxiliary variable (15), result in the following LPV system:  
\[\dot{\xi}=A_{\xi}(\theta)\dot{\xi}+B_{\xi}u+B_{\xi,\varphi}\varphi, \tag{17}\]  
where\[A_{\xi}(\theta)=\begin{bmatrix}0&\sigma&\frac{2\pi}{L_{w}}\theta_{1}&0&0\\ -\sigma&0&\frac{2\pi}{L_{w}}\theta_{2}&0&0\\ 0&0&a_{22}&a_{23}&a_{24}\\ 0&0&0&0&1\\ 0&0&a_{42}&a_{43}&a_{44}\end{bmatrix},\]  
\[B_{\xi}=\begin{bmatrix}0&B\end{bmatrix}^{T},\ B_{\xi,\varphi}=\begin{bmatrix}0& B_{\varphi}\end{bmatrix}^{T}.\]  
By introducing auxiliary variable we can obtain both a reflection of the non-linearized term and controllability of the given system.