## 4 String Stability Analysis
Moreover, Theorem 1 reveals that both the sensing delay and communication delay have a negative impact on the string stability of the platoon and restrict the available ranges of selecting the controller gains \(k_{r},k_{v},k_{a}\)._  
**Remark 4**. : _Theorem 1 deduces the minimum time headway of the vehicular platoon for \(\omega>0\) when the string stability conditions are satisfied. By taking into account the objective of enhancing road capacity, it is desirable to set the time headway as close as possible \(toh_{\text{A,min}}\). By doing so, the platoon can achieve a high level of efficiency in utilizing the available road space while maintaining the string stability._  
In practice, the leader vehicle is unlikely to maneuver with high frequency due to the physical and mechanical constraints. Most of the energy of the spacing errors is in the region of low frequency, so this region is most critical for analyzing the string stability. Next, we only consider the low-frequency maneuver of the leader vehicle, aiming at providing a more practical string stability criterion. **Theorem 2**. : _For the low frequency \(\omega\) satisfying \(\sin(\tau_{k}\omega)>0\) and \(\cos(\tau_{k}\omega)>0\), \(k=1,2\), the closed-loop platoon dynamics described by (12) are string-stable if one of the following conditions is satisfied._  
\[\overline{\Delta}<0\ \wedge\overline{\beta}_{6}>0\ \wedge\beta_{2}>0 \tag{32a}\] \[\overline{\Delta}\geq 0\ \wedge\overline{\beta}_{6}>0\ \wedge\overline{\beta}_{4}>0\ \wedge\beta_{2}>0 \tag{32b}\]  
_where_  
\[\left\{ \begin{array}{l}\overline{\Delta}=\overline{\beta}_{4}^{2}-4k_{r} \overline{\beta}_{6}\beta_{2},\\ \overline{\beta}_{6}=T^{2}-2 Tk_{a}\tau_{2}-k_{a}\tau_{2}^{2},\\ \overline{\beta}_{4}=1-2 Thk_{r}-2 Tk_{v}+2k_{a}-2(hk_{r}+k_{v})\tau_{1}+\theta,\\ \theta=\left\{\begin{array}{c}0,\\ 2 hk_{r}k_{a}(\tau_{1}-\tau_{2}),\ \tau_{1}>\tau_{2}\end{array}\right..\end{array} \right.\]  
_With respect to the string stability conditions (32), the bound of the time headway is given by_  
\[h>h_{\text{L,min}}=\frac{2(T+\tau_{1})}{1+2k_{a}} \tag{33}\]Proof. : According to the assumption of Theorem 2, there exist that \(\sin(\tau_{k}\omega)>0,-\sin(\tau_{k}\omega)>-\tau_{k}\omega\), \(\cos(\tau_{k}\omega)=1-2\sin^{2}(0.5\tau_{k}\omega)>1-0.5\tau_{k}^{2}\omega^{2}\), and \(-\cos(\tau_{k}\omega)>-1,k=1,2\). Thus, the inequality in (26) can be simplified as  
\[\left|M+N\right|^{2}-\left|N\right|^{2}\geq\overline{\beta}_{6}\omega^{6}+ \overline{\beta}_{4}\omega^{4}+k_{r}\beta_{2}\omega^{2}\geq 0 \tag{34}\]  
When the inequality (34) is established, the string stability conditions for the low frequency \(\omega\) can be obtained as (32a) and (32b). Then, for low-frequency maneuver, the bound of the time headway (33) can be deduced from (32b). It is assumed that \(\overline{\Delta}\geq 0\) and \(\overline{\beta}_{6}>0\). \(\overline{\beta}_{4}>0\) and \(\beta_{2}>0\) can be rewritten as  
\[k_{\
u}>\frac{1}{h}-\frac{hk_{r}}{2} \tag{35a}\] \[k_{\
u}<\frac{1-2 Thk_{r}+2k_{a}-2 hk_{r}\tau_{1}+\theta}{2(T+\tau_{1})} \tag{35b}\]  
Since that \(\theta\) is different for the situations that \(\tau_{1}<\tau_{2}\) and \(\tau_{1}>\tau_{2}\), these two situations are analyzed as follows, respectively. Firstly, we consider the situation that \(\tau_{1}<\tau_{2}\), i.e., \(\theta=0\). By combining (35a) and (35b), one achieves  
\[h(1+2k_{a})-2(T+\tau_{1})-h^{2}k_{r}(T+\tau_{1})>0 \tag{36}\]  
Due to that \(T+\tau_{1}>0\), the bound of the time headway for low frequency (33) can be obtained by reorganizing (36). Secondly, by considering the other situation that \(\tau_{1}>\tau_{2}\), i.e., \(\theta=2 hk_{r}k_{a}(\tau_{1}-\tau_{2})\), we combine (35a) and (35b) to achieve  
\[h(1+2k_{a})-2(T+\tau_{1})-h^{2}k_{r}[\tau_{1}+T-2k_{a}(\tau_{1}-\tau_{2})]>0 \tag{37}\]  
By assuming that \(T>(2k_{a}-1)\tau_{1}-2k_{a}\tau_{2}\), the bound of the time headway can still be represented by (33), which is the same as the result of the situation that \(\tau_{1}<\tau_{2}\).