## 2 Problem Statement
### Closed-Loop Dynamics of the Vehicular Platoon with Sensing Delay and Communication Delay  
According to the control objective (8), the spacing error of the \(i\)-th vehicle is represented as  
\[e_{i}(t)=r_{i-1}(t)-r_{i}(t)-d-hv_{i}(t) \tag{9}\]  
With respect to the sensing delay \(\tau_{1}\) and communication delay \(\tau_{2}\), the distributed controller is given by  
\[\begin{array}{l}u_{i}(t)=k_{r}[r_{i-1}(t-\tau_{1})-r_{i}(t-\tau_{1})-d-hv_{i }(t-\tau_{1})]\\ \qquad\qquad+k_{y}[v_{i-1}(t-\tau_{1})-v_{i}(t-\tau_{1})]\\ \qquad\qquad+k_{a}[a_{i-1}(t-\tau_{2})-a_{i}(t-\tau_{2})]\end{array} \tag{10}\]  
where \(k_{r}>0\), \(k_{v}>0\), \(k_{a}>0\) are the controller gains for position, velocity, and acceleration, respectively.  
By applying the distributed controller (10) for the longitudinal vehicle dynamics (7), the governing equation with respect to the spacing error of the \(i\)-th follower is obtained as  
\[\begin{array}{l}T\ddot{e}_{i}(t)+\ddot{e}_{i}(t)+k_{a}\ddot{e}_{i}(t-\tau_ {2})+(hk_{r}+k_{v})\dot{e}_{i}(t-\tau_{1})+k_{r}e_{i}(t-\tau_{1})\\ =k_{a}\ddot{e}_{i-1}(t-\tau_{2})+k_{v}\dot{e}_{i-1}(t-\tau_{1})+k_{r}e_{i}(t- \tau_{1})\end{array} \tag{11}\]  
The lumped error vector is defined as \(\epsilon=[e_{1},\cdots,e_{N},\dot{e}_{1},\cdots,\dot{e}_{N},\ddot{e}_{1}, \cdots,\ddot{e}_{N}]^{\rm T}\). Thus, the closed-loop dynamics of the vehicular platoon can be written as  
\[\dot{e}(t)=(A\otimes I_{N})e(t)-[(BK_{1})\otimes\widetilde{\mathcal{L}}+(hBK _{2})\otimes I_{N}]e(t-\tau_{1})-[(BK_{3})\otimes\widetilde{\mathcal{L}}]e(t -\tau_{2}) \tag{12}\]  
where \(I_{N}\) denotes the \(N\)-dimensional identity matrix, and \(\otimes\) is the Kronecker product operation. \(K_{1}=[k_{r},k_{v},0]\), \(K_{2}=[0,k_{r},0]\), \(K_{3}=[0,0,k_{a}]\).  
The characteristic polynomial of (12) is calculated as \[f(s) =\det\Bigl{\{}sI_{3\text{N}}-(A\otimes I_{N})+[(BK_{1})\otimes \widetilde{\mathcal{L}}+(hBK_{2})\otimes I_{N}]\text{e}^{-\tau_{1}s}+[(BK_{3}) \otimes\widetilde{\mathcal{L}}]\text{e}^{-\tau_{2}s}\Bigr{\}} \tag{13}\] \[=\left|\begin{array}{ccc}sI_{N}&-I_{N}&0_{N}\\ 0_{N}&sI_{N}&-I_{N}\\ \frac{k_{\text{F}}\text{e}^{-\tau_{1}s}}{T}\widetilde{\mathcal{L}}&\frac{k_{ \text{F}}\text{e}^{-\tau_{1}s}}{T}\widetilde{\mathcal{L}}+\frac{h\text{k}_{ \text{F}}\text{e}^{-\tau_{1}s}}{T}I_{N}&sI_{N}+\frac{1}{1}I_{N}+\frac{k_{ \text{F}}\text{e}^{-\tau_{2}s}}{T}\widetilde{\mathcal{L}}\\ \end{array}\right|\] \[=\det(sI_{N})\text{det}\Biggl{(}\left[\begin{array}{ccc}sI_{N}&-I _{N}\\ \frac{k_{\text{F}}\text{e}^{-\tau_{1}s}}{T}\widetilde{\mathcal{L}}+\frac{h \text{k}_{\text{F}}\text{e}^{-\tau_{1}s}}{T}I_{N}&sI_{N}+\frac{1}{1}I_{N}+ \frac{k_{\text{F}}\text{e}^{-\tau_{1}s}}{T}\widetilde{\mathcal{L}}\\ \end{array}\right]-\left[\begin{array}{ccc}0_{N}\\ \frac{k_{\text{F}}\text{e}^{-\tau_{1}s}}{T}\widetilde{\mathcal{L}}\end{array} \right](sI_{N})^{-1}\bigl{[}\begin{array}{ccc}-I_{N}&0_{N}\\ -I_{N}&0_{N}\\ \end{array}\right]\Biggr{)}\] \[=\det(sI_{N})\text{det}(sI_{N})\text{det}\Bigl{[}sI_{N}+\frac{1 }{T}I_{N}+\frac{k_{\text{F}}\text{e}^{-\tau_{1}s}}{T}\widetilde{\mathcal{L}}-( k_{\text{F}}\widetilde{\mathcal{L}}+\frac{h\text{k}_{\text{F}}\text{e}^{-\tau_{1}s}}{ T}I_{N}+\frac{k_{\text{F}}\text{e}^{-\tau_{1}s}}{T}\widetilde{\mathcal{L}})\text{e}^{- \tau_{1}s}(sI_{N})^{-1}(-I_{N})\Bigr{]}\] \[=\prod\limits_{i=1}^{N}[s^{3}+\frac{1+k_{\text{F}}\text{e}^{- \tau_{2}s}}{T}s^{2}+\frac{(k_{\text{F}}+hk_{\text{F}})\text{e}^{-\tau_{1}s}}{ T}s+\frac{k_{\text{F}}\text{e}^{-\tau_{1}s}}{T}]\]  
where \(s\) is the Laplace variable, and \(\widetilde{\mathcal{L}}\) is a lower triangular matrix with all the elements of its diagonal being \(1\).  
According to (13), the closed-loop platoon dynamics (12) can be decomposed into \(N\) subsystems. The characteristic equation of each subsystem is given by  
\[f_{i}(s)=s^{3}+\frac{1+k_{a}\text{e}^{-\tau_{2}s}}{T}s^{2}+\frac{(k_{\text{F}} +hk_{\text{F}})\text{e}^{-\tau_{1}s}}{T}s+\frac{k_{\text{F}}\text{e}^{-\tau_{ 1}s}}{T} \tag{14}\]