## 2 Preliminaries and Problem Description
### Dynamic Model of MHSTs  
Employing Newtonian mechanics, a dynamic model for the \(i\)-th train has been derived. \[\left\{ \begin{array}{l}x_{i}(\ell+1)=x_{i}(\ell)+T_{s}v_{i}(\ell)\\ v_{i}(\ell+1)=v_{i}(\ell)+T_{s}(F_{i}(\ell)-f_{b,i}(\ell)-f_{a,j}(\ell))\end{array} \right. \tag{1}\]  
Within this model, the fixed sampling interval is denoted by \(T_{s}\), \(s\) denotes the moment, and \(\ell\) denotes the train number. \(F_{i}(\ell)\), \(v_{i}(\ell)\), and \(x_{i}(\ell)\) denote the control inputs, train speed, and position, respectively. Furthermore, the model incorporates resistance components, where \(f_{b,i}(\ell)\) denotes the intrinsic resistance and \(f_{a,i}(\ell)\) denotes supplementary resistance factors. \[\left\{ \begin{array}{l}f_{b,i}(\ell)=a+bv_{i}(\ell)+cv_{i}(\ell)^{2}\\ f_{a,i}(\ell)=f_{r,i}(\ell)+f_{c,i}(\ell)+f_{t,i}(\ell)\end{array} \right. \tag{2}\]  
The intrinsic and supplementary resistances are mathematically expressed through coefficients \(a\), \(b\), and \(c\), which are presently unidentified. Resistance factors related to track conditions, specifically, gradient \(f_{r,i}(\ell)\), curvature \(f_{c,i}(\ell)\), and tunnel effects \(f_{t,i}(\ell)\), are accounted for according to the train's positional context. Integrating the resistance expressions into the initial dynamic model formulation results in an overall representation involving an unknown nonlinear function \(f(\cdot)\). \[\left\{ \begin{array}{l}x_{i}(\ell+1)=x_{i}(\ell)+T_{s}v_{i}(\ell)\\ v_{i}(\ell+1)=f_{i}(v_{i}(\ell),F_{i}(\ell))\end{array} \right. \tag{3}\]  
where, \(f_{i}(\cdot)\) represents an unknown nonlinear function. The following assumptions are adopted for the operational conditions of the multi-high-speed-train (MHST) system. **Assumption 1**. : _The nonlinear function \(f_{i}(\cdot)\) with respect to \(F_{i}(\ell)\) is continuous._  
**Assumption 2**. : _Except for a finite number of instances \(\ell\), the nonlinear function \(f(\cdot)\) satisfies a generalized Lipschitz condition. Specifically, for any \(\ell_{1}\
eq\ell_{2}\) with \(\ell_{1},\ell_{2}\geq 0\), and \(F_{i}(\ell_{1})\
eq F_{i}(\ell_{2})\), the following inequality holds:_  
\[|v_{i}(\ell_{1}+1)-v_{i}(\ell_{2}+1)|\leq g\,|F_{i}(\ell_{1})-F_{i}(\ell_{2})|,\]  
_where \(g>0\) is a constant representing the generalized Lipschitz coefficient._  
**Remark 1**. : _Assumption 1 is frequently utilized in control scheme design for nonlinear systems. In contrast, Assumption 2 constrains the maximum allowable rate of change in train speed. From anenergy perspective, the bounded energy variation available from the traction network inherently restricts the magnitude of speed variations achievable by high-speed trains (HSTs)._  
**Theorem 1** ([40]). : _For the MHST (3) satisfying Assumptions 1 and 2, if \(\Delta F_{i}(\ell)\
eq 0\) holds, then there exists a parameter \(\phi_{i}(\ell)\) such that (3) can be converted into the following dynamic linearized model:_  
\[\Delta v_{i}(\ell+1)=\phi_{i}(\ell)\Delta F_{i}(\ell) \tag{4}\]  
_where \(\phi_{i}(\ell)\) is a variable called the pseudo-partial derivative (PPD), which satisfies \(|\phi_{i}(\ell)|\leq b\). \(\Delta v_{i}(\ell+1)=v_{i}(\ell+1)-v_{i}(\ell)\) and \(\Delta F_{j}(\ell)=F_{i}(\ell)-F_{i}(\ell-1)\)._  
Based on the incremental data model mentioned above, the following forward-step prediction equation can be proposed:  
\[v_{i}(\ell+1)=v_{i}(\ell)+\phi_{i}(\ell)\Delta F_{i}(\ell). \tag{5}\]  
Based on Equation (5), the following N-step forward prediction equation can be further proposed:  
\[\left\{ \begin{aligned} v_{i}(\ell+1)&=v_{i}(\ell)+ \phi_{i}(\ell)\Delta F_{i}(\ell)\\ v_{i}(\ell+2)&=v_{i}(\ell+1)+\phi_{i}(\ell+1) \Delta F_{i}(\ell+1)=v_{i}(\ell)+\phi_{i}(\ell)\Delta F_{i}(\ell)+\phi_{i}( \ell+1)\Delta F_{i}(\ell+1),\\ &\vdots\\ v_{i}(\ell+N)&=v_{i}(\ell+N-1)+\phi_{i}(\ell+N-1) \Delta F_{i}(\ell+N-1)\\ &\vdots\\ &=v_{i}(\ell)+\phi_{i}(\ell)\Delta F_{i}(\ell)+\phi_{i}(\ell+1) \Delta F_{i}(\ell+1)+\cdots\\ &\quad+\phi_{i}(\ell+N-1)\Delta F_{i}(\ell+N-1).\end{aligned} \right.