# Exploring Safety-Stability Tradeoffs in Cooperative CAV Platoon Controls with Bidirectional Impacts
## 3 Methodology
### String Stability Analysis  
This section derives the string stability of the bidirectional control model. String stability represents the ability of one vehicle to withstand small perturbations and progress to the steady state where vehicles travel with an identical gap and speed in homogenous traffic [46]. This research adopts the linear stability analysis method described in [21; 32; 47; 48] to analyze string stability. We first derive the stability condition for the generic model, which adopts both spacing and velocity difference information in Equation (2). We then derive the stability condition for the specific models that only use one type of information. Considering a platoon with \(N\) vehicles, in the steady state, each vehicle can be represented as  
\[\overline{x}_{n}(t)=(N-n)h_{0}+\overline{v}t,\ \ \ \ \ \ \ \ \ \ \ n=1,2\ldots,N \tag{13}\]  
where \(h_{0}\) denotes the average headway of adjacent vehicles in the steady state in homogenous traffic, \(\overline{v}\) represents the velocity of vehicles in the steady state, and \(\overline{x}_{n}(t)\) is the location of vehicle \(n\) at time \(t\). We assume that a small perturbation affects the steady state solution of vehicle \(n\) at time \(t\). We denote the perturbation by \(y_{n}(t)\) that has a linear Fourier-mode expansion,  
\[y_{n}(t)=ce^{i\alpha_{k}n+zt}=x_{n}(t)-\overline{x}_{n}(t),\ y_{n}(t)\to 0,\ \alpha_{k}= \frac{2\pi k}{N} \tag{14}\]  
where \(c\) is a constant and \(\alpha_{k}=\frac{2\pi k}{N}(k=0,1,\ldots,N-1)\)Taking the second derivative of both sides of Equation (14), we obtain  
\[\ddot{y}_{n}(t+t_{d})=\ddot{x}_{n}(t+t_{d})-(\overline{x}_{n}(t+t_{d}))^{\prime \prime}=\ddot{x}_{n}(t+t_{d})=\frac{dv_{n}(t+t_{d})}{dt} \tag{15}\]  
Based on Equation (2), we rewrite Equation (15) as  
\[\ddot{y}_{n}(t+t_{d})=f_{n}(s_{n}(t),\;v_{n}(t),\;\Delta v_{n}(t),\;s_{n+1}(t), \;\Delta v_{n+1}(t)) \tag{16}\]  
By linearizing Equation (16), we can derive the following equation:  
\[ \begin{split}\ddot{y}_{n}(t+t_{d})=f_{s_{n}}&(y_ {n-1}(t)-y_{n}(t))+f_{v_{n}}\dot{y}_{n}(t)+f_{\Delta v_{n}}\big{(}\dot{y}_{n-1 }(t)-\dot{y}_{n}(t)\big{)}\\ &+f_{s_{n+1}}(y_{n}(t)-y_{n+1}(t))+f_{\Delta v_{n+1}}\big{(}y_{n }(t)-y_{n+1}(t)\big{)}\end{split}  \tag{17}\]  
where \(f_{s_{n}}=\left.\frac{\partial f_{s}}{\partial s_{n}}\right|_{(\overline{x}, \;\overline{y})}>0\), \(f_{\bar{v}_{n}}=\left.\frac{\partial f_{s}}{\partial s_{n}}\right|_{(\overline {x},\;\overline{y})}<0\), \(f_{\Delta v_{n}}=\left.\frac{\partial f_{s}}{\partial\Delta v_{n}}\right|_{( \overline{x},\;\overline{y})}<0\), \(f_{s_{n+1}}=\left.\frac{\partial f_{s}}{\partial s_{n+1}}\right|_{(\overline {x},\;\overline{y})}\) and \(f_{\Delta v_{n+1}}=\left.\frac{\partial f_{s}}{\partial\Delta v_{n+1}}\right|_ {(\overline{x},\;\overline{y})}\). We rewrite Equation (17) and substitute \(y_{n}(t)=ce^{ia_{k}n+zt}\) and \(\dot{y}_{n}(t)=ce^{ia_{k}n+zt}\) into Equation (17). Simplifying the resulting equation, we can obtain  
\[ \begin{split}\big{(}e^{t_{d}z}-1\big{)}&\Big{[}ze^ {t_{d}z}-f_{v_{n}}+\big{(}e^{-ia_{k}}-1\big{)}\big{(}f_{\Delta v_{n}}+e^{ia_{ k}}f_{\Delta v_{n+1}}\big{)}\Big{]}\\ &=t_{d}\cdot\big{(}e^{-ia_{k}}-1\big{)}\big{(}f_{s_{n}}+e^{ia_{ k}}f_{s_{n+1}}\big{)}\end{split}  \tag{18}\]  
We expand \(z\) in a power series solution, where \(z=z_{1}(ia_{k})+z_{2}(ia_{k})^{2}+\cdots\) and \(e^{t_{d}z}=1+t_{d}z+\frac{t_{d}^{2}z^{2}}{2}+\cdots\). We can insert this solution into Equation (18) to derive the first-order and second-order terms of coefficients in expression of z, given, respectively, the following:  
\[z_{1}=\frac{f_{s_{n}}+f_{s_{n+1}}}{f_{v_{n}}} \tag{19}\]  
\[z_{2}=\frac{z_{1}^{2}-z_{1}\big{(}f_{\Delta v_{n}}+f_{\Delta v_{n+1}}\big{)}- \frac{1}{2}\big{(}f_{s_{n}}-f_{s_{n+1}}\big{)}}{f_{v_{n}}}-\frac{1}{2}(z_{1}^{ 2}t_{d}) \tag{20}\]  
The platoon is string stable if \(z_{2}>0\). The string stability condition is derived as follows, assuming no time delays (\(t_{d}=0\)):  
\[