## 4 Numerical Simulation  
In this section, a network of four high-speed trains is modeled, and the proposed MFAPC algorithm is implemented within the MATLAB R2019b environment. The communication topology among the MHSTs is depicted in Figure 2, illustrating that information transmission is unidirectional and limited to neighboring train units.  
It has been established that the duration of the operation will be 2500 s, with a consistent sample interval of one second. The desired speed profile is presented in Figure 3. Below is a list of key parameters provided for the test: AR model order \(n_{p}=3\), \(N_{u}=3\), forward forecast step \(N=5\). \(\varepsilon=10^{-5},\delta=1,\mu=1,\lambda=1,\eta=1,\phi_{i}(1)=0.8,\phi_{i}(2 )=0.4\), \(\phi_{i}(3)=0.7,\phi_{i}(4)=0.6,F_{i}(1:3)=0,\theta_{i}(4)=[0.3,0.2,0.4]^{T}, \beta_{1}=0.3,\beta_{2}=0.4\), \(d_{1}=0.02,d_{2}=0.02\). Comprehensive information on the simulation coefficients related to running resistance is provided in refs. [15; 19]. It is important to note that these resistance coefficients are employed solely to compute the basic and additional resistances encountered during train operation. This set of coefficients is not exclusive, and alternative values may be used if necessary.  
\[a_{1} =0.2976+0.274\sin\Bigl{(}3.8\cdot 10^{-3}\ell\Bigr{)}\] \[a_{2} =2.315\cdot 10^{-2}+2.31\cdot 10^{-3}\sin\Bigl{(}3.8\cdot 10^{-3} \ell\Bigr{)}\] \[a_{3} =2.193\cdot 10^{-4}+4.42\cdot 10^{-5}\sin\Bigl{(}3.8\cdot 10^{-3} \ell\Bigr{)}\]  
The additional resistance is  
Figure 3: Reference trajectories for speed and position.  
Figure 2: Directed communication topology of multiple high-speed trains.  
\[f_{b}(k)=\begin{cases}0.01,&k\leq 490\\ 0.04,&490<k\leq 740\\ 0.01,&740<k\leq 1500\\ 0.01,&1500<k\leq 2000\\ 0.07,&2000<k\leq 2240\\ 0.02,&2240<k\leq 2500\end{cases}\]  
The effectiveness of the proposed DMFAPC scheme is evaluated by analyzing the position and velocity tracking errors. The results shown in Figures 4 and 5 indicate that the velocity and the MHSTs can be rapidly stabilized to closely follow the intended trajectory under the proposed DMFACPC algorithm. Thus, the DMFAPC algorithm then has a good control effect.  
Accordingly, a set of experiments is carried out to examine the trade-off between the false data injection (FDI) attack and the convergence characteristics of \(e(\ell)\). The outcomes corresponding to \(\bar{\beta}\) are presented in Figure 6, whereas those related to \(d_{1}\) are depicted in Figure 7. As illustrated in Figures 6 and 7, the choice of \(\bar{\beta}\) and \(d_{1}\) exerts a notable impact on both the convergence rate and the ultimate convergence bound of \(e(\ell)\). In particular, a larger \(\bar{\beta}\) or a larger \(d_{1}\) tends to undermine the stability of the system.  
Figure 4: The speed and position dynamics of the MHSTs under the implementation of the MFAPC scheme.  
Figure 5: Tracking errors in speed and position of the MHSTs.  
Furthermore, a comparative evaluation of the proposed DMFAPC scheme, the MFAC approach [33], and the conventional PI controller was performed, with the associated error trajectories presented in Figure 8. The results indicate that all three strategies successfully confine the velocity-tracking error to the prescribed interval \([-0.2,0.2]\). Nevertheless, Figure 8 clearly shows that the DMFAPC scheme exhibits faster convergence and superior steady-state accuracy compared with both the MFAC and PI algorithms.