## 4 Performance Results  
The accuracy of the developed FTDO (12) and controller (27) is validated for a vehicle platoon subject to matched and mismatched disturbances through a numerical example in this section. The proposed FTDO-based sliding mode controller is applied to a vehicle platoon that includes a leader vehicle (5) and six follower vehicles (4). The disturbances are given by \(\varpi_{i1}(t)=0.5 sin(t)Q\) and \(\varpi_{i2}(t)=1.5 sin(3t)Q\), with \(Q=e^{-(t-5-0.2i)^{2}}\). The acceleration profile of the leader vehicle is given by [16]  
\[a_{0}(t)=\left\{ \begin{array}{ll}0.5t&\text{m/s${}^{2}$,}\quad 2s<t\leq 5s\\ -1&\text{m/s${}^{2}$,}\quad 10s<t<12s\\ 0&\text{m/s${}^{2}$,}\quad\quad\text{otherwise.}\end{array} \right.\]  
The parameters of the FTDO and controller are \(\lambda_{0}=8,\lambda_{1}=6,\lambda_{2}=5,\lambda_{3}=4\), \(\lambda_{4}=3,\varrho_{0}=8,\varrho_{1}=6,\varrho_{2}=5,\varrho_{3}=4,\varrho _{4}=1\), \(L_{i}=0.07\), \(k_{1}=1,k_{2}=0.05,\gamma=0.9\), \(\ell_{i}=50\), \(h_{i}=1\) s, \(\delta_{i}=10\) m, for \(i\in\{1,2,\dots,6\}\)[16,36]. By utilizing the proposed FTDO (12) and FTDO-based sliding mode controller (27), a numerical example is executed on the vehicle platoon (4) and (5) under the topology shown in Figure 1. The results are illustrated in Figures 4 and 5. As shown in Figure 4a, the mismatched disturbance error exhibits significant fluctuations around the 5 s mark, but rapidly converges to zero after 21 s, indicating that the method proposed in this paper can effectively estimate mismatched disturbances. From Figure 4b, we observe that \(\tilde{\varpi}_{i2}\) exhibits minor fluctuations at 5 s but rapidly converges to zero, which implies that the matched disturbance is accurately estimated within finite time. The profiles of the position are given in Figure 5a. As there are no crossed and overlapped profiles in Figure 5b, collisions between neighboring vehicles are avoided. The curves of the velocity and velocity error are given in Figure 5b and Figure 5c, respectively. It is evident that velocity synchronization is realized among all follower vehicles and the leader. As shown in Figure 5d, the amplitude of \(e_{i}\) does not increase along the platoon, indicating that strong string stability is achieved. The numerical example results demonstrate that the vehicle platoon can achieve the control goals by the designed FTDO-based sliding mode controller. The efficacy of the proposed FTDO and controller is verified. To demonstrate the superior performance of the proposed FTDO-based sliding mode controller, the results of the disturbance observer (DO)-based sliding mode controller in [38] are presented in Figures 6 and 7 for comparison. From Figure 6a, we can see that \(\delta_{i1}\) fluctuates slightly at 5 s, but quickly converges to zero after 24 s, indicating that mismatched  
Figure 4: The estimation results of the proposed FTDO. (**a**) Estimation error of mismatched disturbances \(\varpi_{i1}\). (**b**) Estimation error of matched disturbances \(\varpi_{i2}\). Figure 5: The control results of the proposed FTDO-based sliding mode controller. (**a**) Position of \(i\)th vehicle. (**b**) Velocity of \(i\)th vehicle. (**c**) Velocity error \(e_{\eta_{i}}\). (**d**) Spacing error \(e_{i}(t)\). disturbances are effectively estimated. As shown in Figure 6b, \(\bar{\omega}_{i2}\) fluctuates slightly but converges to zero quickly, which illustrates matched disturbances are effectively estimated by DO. From Figure 7a, we can obtain that the collisions between neighboring vehicles are avoided. The velocity and velocity error curves in Figure 7b,c show that the vehicle platoon achieves velocity synchronization. Figure 7d shows that \(e_{i}\) decreases along the platoon, which indicates that strong string stability is achieved. Figure 6: The estimation results of the DO. (**a**) Estimation error of mismatched disturbances \(\bar{\omega}_{i1}\). (**b**) Estimation error of matched disturbances \(\bar{\omega}_{i2}\). Figure 7: The control results of the DO-based controller.