## 5 Numerical Simulations
### String Stability Verification  
The main results of the string stability conditions and the bound of the time headway derived in Theorem 1 and Theorem 2 are verified in Case 1 and Case 2, respectively. The sensing delay and communication delay are set as \(<\tau_{1},\tau_{2}>\)\(=\)\(<\) 0.01 s, 0.1 s \(>\), which satisfy the string stability conditions (24b) and (32b) with the parameters in Table 1. Meanwhile, the acceleration and deceleration of the leader vehicle are considered as the perturbations of the vehicular platoon. \begin{table}
\begin{tabular}{c c} \hline \hline
**Method** & **Simulation Time (min)** \\ \hline The proposed method & 0.05 \\ Sylvester resultant matrix–based method in [21] & 2.1 \\ ACE combined with Kronecker summation in [22] & 27.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation time. Figure 4: Performance of the vehicular platoon for the point “a”: (**a**) Spacing error. (**b**) Velocity. Figure 5: Performance of the vehicular platoon for the point “b”: (**a**) Spacing error. (**b**) Velocity. Case 1: The sinusoidal perturbations are applied as the acceleration of the leader vehicle, which are arranged as  
\[a_{0}(t)=\left\{ \begin{array}{ll}\sin(\frac{\pi}{4}t)\;\mathrm{m}/\mathrm{s}^{ 2},&20\;\mathrm{s}\leq t\leq 36\;\mathrm{s}\\ -2\sin(\frac{\pi}{4}t)\;\mathrm{m}/\mathrm{s}^{2},&80\;\mathrm{s}\leq t\leq 96\; \mathrm{s}\\ 0,&\mathrm{otherwise}.\end{array} \right. \tag{38}\]  
With the vehicular platoon and controller parameters in Table 1, the bound of the time headway (25) is \(h_{\mathrm{A,min}}=0.9127\;\mathrm{s}\). To satisfy the string stability conditions derived in Theorem 1, we set the time headway as \(h=1.5964\;\mathrm{s}>h_{\mathrm{A,min}}\). The performance of the vehicular platoon is given in Figure 6. It is seen that the spacing errors decay when propagating along the vehicle string, which illustrates the correctness of the string stability conditions proposed in Theorem 1. On the other hand, by choosing \(h=0.7764\;\mathrm{s}<h_{\mathrm{A,min}}\), the string stability of the vehicular platoon can not be guaranteed, as shown in Figure 7. Thus, the bound of the time headway (25) is valid. Case 2: To examine the main results proposed in Theorem 2, we reduce the frequency of the perturbations from \(1/4\;\mathrm{rad}/\mathrm{s}\) to \(1/16\;\mathrm{rad}/\mathrm{s}\), and hence the low-frequency perturbations are given by  
\[a_{0}(t)=\left\{ \begin{array}{ll}\sin(\frac{\pi}{16}t)\;\mathrm{m}/\mathrm{s }^{2},&20\;\mathrm{s}\leq t\leq 36\;\mathrm{s}\\ -2\sin(\frac{\pi}{16}t)\;\mathrm{m}/\mathrm{s}^{2},&80\;\mathrm{s}\leq t\leq 96\; \mathrm{s}\\ 0,&\mathrm{otherwise}.\end{array} \right. \tag{39}\]  
Compared to Case 1, the bound of the time headway for low frequency (33) can be reduced to \(h_{\mathrm{L,min}}\)\(=0.7455\;\mathrm{s}\). The time headway is set as \(h=0.7746\;\mathrm{s}>h_{\mathrm{L,min}}\) in Case 2, which is unavailable to guarantee the string stability according to Theorem 2. Then, for the vehicular platoon with the low-frequency perturbations, the spacing errors are attenuated among the followers, as shown in Figure 8. The result illustrates that a lower time headway can be used to hold the string stability for the low-frequency perturbations. After that, by choosing that \(h=0.5432\;\mathrm{s}<h_{\mathrm{L,min}}\), the vehicular platoon is not string-stable in Figure 9. Thus, the string stability conditions and the lower bound of the time headway derived in Theorem 2 are validated as effective. Figure 6: Performance of the vehicular platoon with \(h>h_{\mathrm{A,min}}\): (**a**) Spacing error. (**b**) Velocity. Figure 7: Performance of the vehicular platoon with \(h<h_{\mathrm{A,min}}\): (**a**) Spacing error. (**b**) Velocity. Case 3: In this case, the leader vehicle undergoes the \"acceleration-criuse-brake\" scenario, which is more commonly used in the real world. The acceleration and deceleration of the leader vehicle are given by  
\[a_{0}(t)=\left\{