## 3 Stability Analysis  
Before proceeding with the stability analysis of MHSTs governed by the MFAPC framework, several fundamental assumptions and supporting lemmas are introduced to facilitate the subsequent theoretical developments. **Assumption 3**. : _For all \(j=1,2,\ldots,N\) and \(\ell=1,2,\ldots\), if \(|F_{i}(\ell)|\
eq 0\), then the sign of \(\phi_{i}(\ell)\) remains invariant. Specifically, it holds that \(\phi_{i}(\ell)>z>0\) or \(\phi_{i}(\ell)<-z<0\), where \(z\) is a small positive constant. Without loss of generality, only the case of \(\phi_{i}(\ell)>z\) is considered in this study._  
**Assumption 4**. : _The extended directed communication graph \(\overline{\mathcal{G}}\) associated with the MHST system is strongly connected. Moreover, only a subset of follower high-speed trains (HSTs) has access to the reference trajectory \(v_{0}(\ell)\)._  
**Remark 2**. : _Assumption 3 implies that the system exhibits a monotonic relationship between the control input and the output, meaning the output does not decrease as the input increases. This behavior reflects a \"quasi-linear\" characteristic of the system and is consistent with the fundamental premise of model-based control approaches, which typically require the control direction to be known or invariant. Assumption 4, on the other hand, serves as a necessary condition for achieving consensus tracking within multi-agent systems, ensuring that no high-speed train (HST) is isolated from the communication topology of the group._  
**Lemma 1** ([42]). : _If \(W(\ell)\) is an irreducible and time-varying substochastic matrix, in which the diagonal entries are positive, then there exists a set \(W\) of all possible \(W(\ell)\), such that_  
\[\|W(T)W(T-1)\cdots W(1)\|\leq\beta \tag{26}\]  
_where \(0<t<1\) and \(W(\ell)\), \(\ell=1,2\cdots P\), are chosen arbitrarily from \(W\)._  
**Theorem 2**. : _For an MHST that satisfies Assumptions 1 and 2 but is subject to an FDI attack, a distributed model-free adaptive predictive controller is used and appropriate controller parameters are chosen such that the speed tracking error is bounded, i.e., \(\lim_{k\rightarrow\infty}\|e(\ell+1)\|=\frac{T-1}{1-l}d\) when \(\lambda_{min}\ >0\) and \(\lambda>\lambda_{min}\)._  
Proof of Theorem 2.: First, we prove that \(\phi_{i}(\ell)\) is bounded. Define \(\tilde{\phi}_{i}(\ell)=\hat{\phi}_{i}(\ell)-\phi_{i}(\ell)\). Subtracting \(\phi_{i}(\ell)\) from both sides of the estimation algorithm (20) gives  
\[\tilde{\phi}_{i}(\ell)=\tilde{\phi}_{i}(\ell-1)+\frac{\eta\Delta F_{i}(\ell-1)} {\mu+\Delta F_{i}^{2}(\ell-1)}\big{(}\Delta v_{i}(\ell)-\hat{\phi}_{i}(\ell-1) \Delta F_{i}(\ell-1)\big{)}-\Delta\phi_{i}(\ell) \tag{27}\]  
Substituting Equation (4) and taking the absolute value of both sides yields:  
\[|\tilde{\phi}_{i}(\ell)|\leq\left|1-\frac{\eta\Delta F_{i}^{2}(\ell-1)}{\mu+ \Delta F_{i}^{2}(\ell-1)}\right||\tilde{\phi}_{i}(\ell-1)|+|\Delta\phi_{i}( \ell)| \tag{28}\]  
Furthermore, since \(\phi_{i}(\ell)\) is bounded, we have \(|\Delta\phi_{i}(\ell)|\leq 2b\), where \(b>0\) is the upper bound of \(\phi_{i}(\ell)\). Thus, Equation (28) can be rewritten as:  
\[|\tilde{\phi}_{i}(\ell)|\leq\left|1-\frac{\eta\Delta F_{i}^{2}(\ell-1)}{\mu+ \Delta F_{i}^{2}(\ell-1)}\right||\tilde{\phi}_{i}(\ell-1)|+2b \tag{29}\]  
We choose \(\mu>0,0<\eta<1\), thus \(\eta\Delta F_{i}^{2}(\ell-1)<\Delta F_{i}^{2}(\ell-1)<\mu+\Delta F_{i}^{2}( \ell-1)\). Then, we can obtain \(0<\delta<\big{(}\eta\Delta F_{i}^{2}(\ell-1)/\big{(}\mu+\Delta F_{i}^{2}( \ell-1)\big{)}\big{)}<1\). From Equation (29), we can have  
\[|\tilde{\phi}_{i}(\ell)| \leq(1-\delta)|\tilde{\phi}_{i}(\ell-1)|+2b \tag{30}\] \[\leq(1-\delta)^{2}|\tilde{\phi}_{i}(k-2)|+2b(1-\delta)+2b\] \[\leq\cdots\] \[\leq(1-\delta)^{k-1}|\tilde{\phi}_{i}(1)|+\frac{2b}{\delta}\]  
We can conclude that \(\tilde{\phi}_{i}(\ell)\) is bounded, which also implies that \(\phi_{i}(\ell)\) is bounded. As a direct result of Algorithms (22) and (23), the PPD prediction values \(\phi_{i}(\ell+m)\), \(m=1,2,...,N_{u}-1\) are guaranteed.