# Adaptive Cooperative Control of Multiple Urban Rail Trains with Position Output Constraints
## 2 Preliminaries System Model
During the actual train operation, the basic operating resistance coefficients \(a_{i,0}\), \(a_{i,1}\), \(a_{i,2}\) and additional resistance \(f_{i,s}\), \(f_{i,c}\), \(f_{i,t}\) change with different trains, weather conditions, outside environments, and other factors, these parameters are extremely difficult to get precisely, resulting in the parameters uncertainties of the train dynamics model. Among them, \(|a_{i,0}|\leq a_{i,0}^{+}\), \(|a_{i,1}|\leq a_{i,1}^{+}\), \(|a_{i,2}|\leq a_{i,2}^{+}\), \(|f_{i,s}|\leq f_{i,s}^{+}\), \(|f_{i,c}|\leq f_{i,c}^{+}\), and \(|f_{i,t}|\leq f_{i,t}^{+}\), \(a_{i,0}^{+}\), \(a_{i,1}^{+}\), \(a_{i,2}^{+}\), \(f_{i,s}^{+}\), \(f_{i,c}^{+}\), and \(f_{i,t}^{+}\) are the unknown upper bounds of the time-varying function, respectively. Thus, in this paper, an adaptive control scheme is designed to identify uncertain parameters to ensure the control performance of multiple trains cooperative control system. Definition \(f_{i}=a_{i,0}+f_{i,s}+f_{i,c}+f_{i,l}\), then the system model Equation (3) can be rewritten as follows:  
\[\left\{ \begin{array}{l}\dot{x}_{i,1}(t)=x_{i,2}(t)\\ \dot{x}_{i,2}(t)=u_{i}(t)-f_{i}-a_{i,1}x_{i,2}(t)-a_{i,2}x_{i,2}^{2}(t)\end{array} \right. \tag{4}\]  
In order to ensure that the trains accurately track the operation curve and the requirements of station fixing stop in the MTCC operation system, the higher and lesser bounds of the train position output bounds need to be strictly restricted. It is defined as follows. **Lemma 1** ([23]). : _For the train operation control system with position error \(\widetilde{x}_{i,1}\) and speed error \(\widetilde{x}_{i,2}\), there exist continuously differentiable and positive definite functions \(V_{i,1}\) and \(V_{i,2}\). The desired position outputs \(x_{i,1d}\), and \(k_{i}>0\) is the parameter limiting the value of the train position output. Define a position error of \(\widetilde{x}_{i,1}=x_{i,1}-x_{i,1d}\), such that_  
1. _When_ \(\widetilde{x}_{i,1}\rightarrow-k_{i}\) _or_ \(\widetilde{x}_{i,1}\to k_{i}\)_,_ \(V_{i,1}(\widetilde{x}_{i,1})\rightarrow+\infty\)_,_
2. \(\chi_{1}(\|\widetilde{x}_{i,2}\|)\leq V_{i,2}(\widetilde{x}_{i,2})\leq\chi_{2 }(\|\widetilde{x}_{i,2}\|)\)_, where_ \(\chi_{1}\) _and_ \(\chi_{2}\) _are_ \(K_{\infty}\) _class functions._  
_Set \(|\widetilde{x}_{i,1}(0)|<k_{i}\), take \(V_{i}=V_{i,1}(\widetilde{x}_{i,1})+V_{i,2}(\widetilde{x}_{i,2})\), if \(\dot{V}_{i}\leq 0\) can satisfy, then \(|\widetilde{x}_{i,1}(t)|<k_{i}\), \(\forall t\in[0,\infty)\)._  
The design objectives of multiple trains distributed adaptive cooperative control are as follows. * The position errors of all trains in the multiple trains cooperative operation are limited to a preset range, that is \(|\widetilde{x}_{i,1}(t)|<k_{i}\), and each train can accurately track the desired speed and distance curve. * The separation distance between each train and neighboring trains in the MTCC system is kept within a predefined safety range. * In the process of multiple trains tracking and cooperative operation, the speed of each train approaches the desired speed.