# Adaptive Cooperative Control of Multiple Urban Rail Trains with Position Output Constraints
## 2 Preliminaries System Model  
In the process of multiple trains tracking operation, each train is able to dynamically adjust its own speed based on communication with neighboring trains. In this paper, by treating each train as an agent and the information exchange between trains as the communication between agents, the tracking operation of the trains can be described by the framework of a multi-agent system. Let \(G=(V,E,A)\) be a weighted digraph of \(n\) order, where \(V=\{1,2,\cdots,n\}\) is the set of nodes, \(E\in V\times V\) is the set of edges, \(A=\left[a_{ij}\right]_{n\times n}\) is the nonnegative adjacency matrix of the digraph \(G\), representing the information interaction between trains. If \((i,j)\in E\) holds, it means that the two trains can get the status data of each other, then \(a_{ij}=1\), otherwise \(a_{ij}=0\). \(N_{i}=\{j\in V:(i,j)\in E\}\) represents the set of neighbors of node \(i\). The Laplacian matrix \(L\) corresponding to the adjacency matrix \(A\) is described as \(\left[L\right]_{ii}=\sum\limits_{j=1,j\
eq i}^{n}a_{ij}\) and \(\left[L\right]_{ij}=-a_{ij}\) for \(i\
eq j\). For the MTCC system, this paper sets that each train can communicate with its neighboring trains, then the middle train can exchange data with its preceding and following trains. The foremost train can only communicate with the one immediately behind it, and the rearmost train can only communicate with the train immediately in front of it. So, the following adjacency matrix consisting of \(n\) trains can be obtained as follows:  
\[A=\left[ \begin{array}{ccccc}0&1&0&0&\cdots\\ 1&0&1&0&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&\cdots&1&0&1\\ 0&\cdots&0&1&0\end{array} \right] \tag{1}\]  
The corresponding Laplace matrix is expressed as follows:  
\[L=\left[ \begin{array}{ccccc}1&-1&0&0&\cdots\\ -1&2&-1&0&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&\cdots&-1&2&-1\\ 0&\cdots&0&-1&1\end{array} \right] \tag{2}\]  
In addition, in order to ensure that the safety separation distance between each train and its neighboring trains is stable within a certain range, the potential function is introduced in this paper as follows. **Definition 1**. : \(U_{i,j}(\left\|x_{i,j}\right\|)\) _is a differentiable non-negative potential function of the distance \(\left\|x_{i,j}\right\|\) between train \(i\) and \(j\), \(l_{1}\) is the minimum safe separation distance between two neighboring trains, \(l_{2}\) is the maximum allowable separation distance between two neighboring trains, and \(l_{1}<l_{2}\), such that:_  
1. _When_ \(\left\|x_{i,j}\right\|\to l_{1}\) _or_ \(\left\|x_{i,j}\right\|\to l_{2}\)_,_ \(U_{i,j}(\left\|x_{i,j}\right\|)\rightarrow+\infty\)_;_
2. _When the positions of train_ \(i\) _and_ \(j\) _are at the desired position,_ \(U_{i,j}(\left\|x_{i,j}\right\|)\) _is the unique minimum value._  
Consider the cooperative operation system of multiple urban rail trains with n trains shown in Figure 1, the motion dynamics model of train \(i\) can be described as follows:  
\[\left\{ \begin{array}{l}\dot{x}_{i,1}(t)=x_{i,2}(t)\\ \dot{x}_{i,2}(t)=u_{i}(t)-(a_{i,0}+a_{i,1}x_{i,2}(t)+a_{i,2}x_{i,2}^{2}(t))-(f_ {i,s}+f_{i,c}+f_{i,t}),\;\;i=1,2,\cdots,n\end{array} \right. \tag{3}\]  
where \(x_{i,1}(t)\) and \(x_{i,2}(t)\) represent the position and speed of the \(i\)th train at time \(t\), respectively, \(u_{i}(t)\) is the cooperative control input to be designed, that is, the acceleration of the \(i\)th train, \(a_{i,0}\), \(a_{i,1}\), and \(a_{i,2}\) are the basic operating resistance coefficients for the \(i\)th train, \(f_{i,s}\) is the ramp resistance caused by the route slope, \(f_{i,c}\) is the curve resistance caused by the route curve, and \(f_{i,t}\) is the tunnel resistance caused by the route.