# Cooperative Roundabout Control Strategy for Connected and Autonomous Vehicles
## 2 Methodology
### Vehicle Platoon Control
As shown in Figure 3, an angular coordinate \(\theta_{n},\ \omega_{n}\) is used to describe the position and angular speed of vehicle \(n\) (i.e., vehicle matched with vehicle slot \(n\)). Each vehicle is modeled as a double integrator. The control inputs for vehicle \(n\) in the platoon are assumed to depend only on its speed error \(\omega_{n}-\omega_{des}\) and the relative headway errors between itself and its immediate neighbors (i.e., its predecessor and follower). Denote the positions of the preceding and following vehicles of vehicle \(n\) as \(\theta_{n}^{p}\) and \(\theta_{n}^{f}\), respectively. The desired angular headway is \(\Delta=2\pi/N\). The vehicle dynamics can be described as  
\[\frac{d\theta_{n}(t)}{dt}=\omega_{n} \tag{10a}\] \[\frac{d\omega_{n}(t)}{dt}=-k_{v}(\omega_{n}-\omega_{des})+k_{d}\left(\theta_{n }^{p}-\theta_{n}-\Delta\right)-k_{d}\left(\theta_{n}-\theta_{n}^{f}-\Delta\right) \tag{10b}\]  
where \(k_{v}\) and \(k_{d}\) are positive constants. To facilitate analysis, a coordinate change is considered using the initial position of slot \(1\), \(\theta_{1}(0)\), as a reference point:  
\[v(\Delta t_{des}) =v_{des} \tag{9b}\]0 \[v(\Delta t_{des}) =v_{des} \tag{9b}\]1  
To describe the dynamics of the circular platoon, define \(\widetilde{\theta}\)=\(\left[\widetilde{\theta}_{1},\ \widetilde{\theta}_{2},\ \ldots,\ \widetilde{\theta}_{N}\right]^{T}\), \(\widetilde{\omega}\)=\(\left[\widetilde{\omega}_{1},\widetilde{\omega}_{2},\ldots,\ \widetilde{\omega}_{N}\right]^{T}\). Substituting (11a-b) into (10a-b), the following equation is obtained:  
\[ \begin{bmatrix}\dot{\widetilde{\theta}}\\ \dot{\widetilde{\omega}}\end{bmatrix} = \begin{bmatrix}\mathbf{0}&I\\ -k_{d}T&-k_{v}I\end{bmatrix}\begin{bmatrix}\widetilde{\theta}\\ \dot{\widetilde{\omega}}\end{bmatrix} , \tag{12}\]  
\[T= \begin{bmatrix}2&-1&0&\ldots&0&-1\\ -1&2&-1&0&\ldots&0\\ 0&-1&2&-1&\cdots&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&\ldots&0&-1&2&-1\\ -1&0&\ldots&0&-1&2\end{bmatrix} . \tag{13}\]  
Equation (12) indicates that the circular platoon to be controlled is a linear system. The following lemma [18] illustrates the non-negativity of \(T\)'s eigenvalues. **Lemma 1** ([18]). : _Given an \(n\times n\) circulant matrix \(C\) with the form_  
\[\mathbf{C}= \begin{bmatrix}c_{0}&c_{n-1}&\ldots&c_{2}&c_{1}\\ c_{1}&c_{0}&c_{n-1}&\ldots&c_{2}\\ \vdots&c_{1}&c_{0}&\ddots&\vdots\\ c_{n-2}&\ldots&\ddots&\ddots&c_{n-1}\\ c_{n-1}&c_{n-2}&\ldots&c_{1}&c_{0}\end{bmatrix} , \tag{14}\]the normalized eigenvectors are the Fourier modes, namely,_  
\[v_{j}=\frac{1}{\sqrt{n}}\Big{(}1,\;\kappa^{j},\;\kappa^{2j},\;\ldots,\;\kappa^{(n -1)j}\Big{)},\;j=0,\;1,\;\ldots,\;n-1, \tag{15}\]  
_where \(\kappa=exp\Big{(}\frac{2\pi i}{n}\Big{)}\) and \(i\) is the imaginary unit. The corresponding eigenvalues are then given by_  
\[\lambda_{j}=c_{0}+c_{n-1}\kappa^{j}+c_{n-2}\kappa^{2j}+\ldots+c_{1}\kappa^{(n -1)j},\;j=0,\;1,\;\ldots,\;n-1 \tag{16}\]  
Note that \(\mathbf{T}\) is a \(N\times N\) circulant matrix with \(c_{0}=2,\;c_{1}=c_{N-1}=-1,\;c_{i}=0\;(i=2,\;3,\;\ldots,\;N-2)\). According to Lemma 1, the \(j^{th}\) eigenvalue of \(\mathbf{T}\) is  
\[\lambda_{Tj}=2-\kappa^{j}-\kappa^{(N-1)j},\;j=0,\;1,\;\ldots,\;N-1, \tag{17}\]  
where \(\kappa=\exp\Big{(}\frac{2\pi i}{N}\Big{)}\). Note that \(\kappa^{(N-1)j}=-\kappa^{j}\). Therefore,  
\[\lambda_{Tj}=2-\kappa^{j}-\kappa^{-j}=2-2\cos\!\left(\frac{2\pi j}{N}\right) \geq 0,\;j=0,\;1,\;\ldots,\;N-1, \tag{18}\]  
**Theorem 1**[19]. : _The equilibrium states for a general linear time-invariant system \(\dot{x}=Ax\) are stable if and only if each eigenvalue \(\lambda_{A}\) of \(A\) satisfies \(\Re(\lambda_{A})\leq 0\) and \(\Re(\lambda_{A})<0\) if \(\lambda_{A}\) is defective, where \(\Re(\lambda_{A})\) is the real part of \(\lambda_{A}\)._  
From Theorem 1 (Theorem 5.4 of [19]), the eigenvalues of the system can be described by (12) as follows:  
\[