## 4 String Stability Analysis  
By taking the Laplace transform of the governing Equation (11), the spacing error transfer function of the \(i\)-th vehicle is obtained as  
\[G_{i}(s)=\frac{E_{i}(s)}{E_{i-1}(s)}=\frac{k_{a}s^{2}\mathrm{e}^{-\gamma_{2}s}+k _{v}\mathrm{se}^{-\mathrm{n}s}+k_{r}\mathrm{e}^{-\tau_{1}s}}{Ts^{3}+s^{2}+k_{a} s^{2}\mathrm{e}^{-\gamma_{2}s}+(hk_{r}+k_{v})\mathrm{se}^{-\tau_{1}s}+k_{r} \mathrm{e}^{-\tau_{1}s}} \tag{22}\]  
where \(E_{i}(s)\) is Laplace transform of the spacing error \(e_{i}(t)\). \(G_{i}(s)\) describes how the spacing error propagates upstream in the vehicular platoon. For a string-stable platoon, when the leader vehicle performs a sinusoidal acceleration maneuver at the frequency \(\omega\), there must exist [23]  
\[|G_{i}(\iota\cdot\omega)|\leq 1 \tag{23}\]  
**Theorem 1**. : _For any scalar \(\omega>0\), the closed-loop platoon dynamics described by (12) are string-stable if one of the following conditions is satisfied._  
\[\Delta<0\ \wedge\beta_{6}>0\ \wedge\beta_{2}>0 \tag{24a}\] \[\Delta\geq 0\ \wedge\beta_{6}>0\ \wedge\beta_{4}>0\ \wedge\beta_{2}>0\] (24b) _where_ \[\left\{ \begin{array}{l}\Delta=\beta_{4}^{2}-4 Tk_{r}\beta_{6}\beta_{2},\\ \beta_{6}=T-2k_{a}\tau_{2},\\ \beta_{4}=1-2 Thk_{r}-2 TK_{v}-2k_{a}-2(Tk_{r}+hk_{r}+k_{v})\tau_{1}\\ -2 hk_{r}k_{a}|\tau_{2}-\tau_{1}|,\\ \beta_{2}=h^{2}k_{r}+2 hk_{v}-2.\end{array} \right.\] _With respect to the string stability conditions (_24_), the bound of the time headway is deduced by_ \[h>h_{\mathrm{A,min}}=\frac{2(T+\tau_{1})}{1-2k_{a}-2 Tk_{r}\tau_{1}} \tag{25}\]  
**Proof. ** We define \(G_{i}(\iota\cdot\omega)=\frac{N}{N+M}\), where  
\[\left\{ \begin{array}{l}N=[-k_{a}\omega^{2}\cos(\tau_{2}\omega)+k_{v}\omega \sin(\tau_{1}\omega)+k_{r}\cos(\tau_{1}\omega)]\\ +\iota\cdot[k_{a}\omega^{2}\sin(\tau_{2}\omega)+k_{v}\omega\cos(\tau_{1} \omega)-k_{r}\sin(\tau_{1}\omega)]\\ M=[-\omega^{2}+hk_{r}\omega\sin(\tau_{1}\omega)]+\iota\cdot[-T\omega^{3}+hk_{r} \omega\cos(\tau_{1}\omega)]\end{array} \right.\]  
When the vehicular platoon is string stable as (23), there exists  
\[ \begin{array}{l}|M+N|^{2}-|N|^{2}=T^{2}\omega^{6}-2 Tk_{a}\omega^{5}\sin(\tau _{2}\omega)+\omega^{4}-2T(hk_{r}+k_{v})\omega^{4}\cos(\tau_{1}\omega)\\ +2k_{a}\omega^{4}\cos(\tau_{2}\omega)+2(Tk_{r}-hk_{r}-k_{v})\omega^{3}\sin( \tau_{1}\omega)\\ +2 hk_{r}k_{a}\omega^{3}\sin(\tau_{2}-\tau_{1})\omega+(h^{2}k_{r}^{2}+2 hk_{r}k_ {v})\omega^{2}\\ -2k_{r}\omega^{2}\cos(\tau_{1}\omega)\geq 0\end{array}  \tag{26}\]  
According to the facts that \(\pm\sin(\tau_{k}\omega)\geq-\tau_{k}\omega\) and \(\pm\cos(\tau_{k}\omega)\geq-1\), \(k=1,2\), it is obtained that  
\[|M+N|^{2}-|N|^{2}\geq T\beta_{6}\omega^{6}+\beta_{4}\omega^{4}+k_{r}\beta_{2} \omega^{2}\geq 0 \tag{27}\]  
If one of the conditions (24) is satisfied, the inequality in (27) is fulfilled for \(\omega>0\). Then, the bound of the time headway is deduced by (24b). When (24b) is satisfied, \(\beta_{2}>0\) and \(\beta_{4}>0\) can be rewritten as  
\[k_{v}>\frac{1}{h}-\frac{hk_{r}}{2} \tag{28a}\]\[k_{v}<\frac{1-2k_{a}-2 Tk_{r}\tau_{1}-2 hk_{r}(T+\tau_{1}+k_{a}|\tau_{2}-\tau_{1}|)} {2(T+\tau_{1})} \tag{28b}\]  
By combining (28a) and (28b), we obtain  
\[(1-2k_{a}-2 Tk_{r}\tau_{1})h-2(T+\tau_{1})-[Tk_{r}+k_{r}\tau_{1}+2k_{r}k_{a}\Big{|} \tau_{2}-\tau_{1}\Big{|}]h^{2}>0 \tag{29}\]  
Due to that \(\,Tk_{r}+k_{r}\tau_{1}+2k_{r}k_{a}|\tau_{2}-\tau_{1}|>0\), (29) is reduced to  
\[(1-2k_{a}-2 Tk_{r}\tau_{1})h-2(T+\tau_{1})>0 \tag{30}\]  
After some simplifications, the bound of the time headway is given by  
\[h>h_{\text{A,min}}=\frac{2(T+\tau_{1})}{1-2k_{a}-2 Tk_{r}\tau_{1}} \tag{31}\]  
This completes the proof of Theorem 1. \(\square\)  
**Remark 3**. : _Theorem 1 provides a sufficient condition to guarantee the string stability of the vehicular platoon with sensing delay and communication delay, no matter how large the frequency/oof the leader vehicle's maneuver is.