# Adaptive Cooperative Control of Multiple Urban Rail Trains with Position Output Constraints
## 3 Control Law Design
### Adaptive Cooperative Control Design of Multiple Trains with POC
: _For a set of \(n\) urban rail trains system (4) operating on the same route, an adaptive cooperative controller (5) with POC is used, in which each train can track the target speed-distance curve during train operation, the separation distance between two neighboring trains is always kept within the safe range of \((l_{1},l_{2})\), and the speed of each train is able to reach the desired speed. That is, the closed-loop dynamic system satisfies the control objective._  
**Proof of Theorem 1. ** The symmetrical Barrier Lyapunov function for the POC is chosen as follows:  
\[V_{i,1}(\widetilde{x}_{i,1})=\frac{1}{2}\alpha_{i}\sum_{i=1}^{n}\log\frac{k_{i}^{ 2}}{k_{i}^{2}-\widetilde{x}_{i,1}^{2}} \tag{11}\]  
According to Lemma 1, for \(k_{i}>0\), if \(|\widetilde{x}_{i,1}(0)|<k_{i}\) holds, \(\forall t\in[0,\infty)\), thus [23; 24]  
\[\log\frac{k_{i}^{2}}{k_{i}^{2}-\widetilde{x}_{i,1}^{2}}<\frac{\widetilde{x}_{ i,1}^{2}}{k_{i}^{2}-\widetilde{x}_{i,1}^{2}} \tag{12}\]  
The differential of \(V_{i,1}\) is obtained as follows:  
\[\dot{V}_{i,1}<\alpha_{i}\sum_{i=1}^{n}\frac{\widetilde{x}_{i,1}\dot{\widetilde {x}}_{i,1}}{k_{i}^{2}-\widetilde{x}_{i,1}^{2}}=\alpha_{i}\sum_{i=1}^{n}\frac{ \widetilde{x}_{i,1}\widetilde{x}_{i,2}}{k_{i}^{2}-\widetilde{x}_{i,1}^{2}} \tag{13}\]  
The Lyapunov function is chosen as follows:  
\[V_{i}=V_{i,1}+\frac{1}{2}\sum_{i=1}^{n}\widetilde{x}_{i,2}\widetilde{x}_{i,2} +\frac{1}{2}\sum_{i=1}^{n}\left(\frac{\widetilde{f}_{i,1}^{2}}{\lambda_{i,1} }+\frac{\widetilde{a}_{i,2}^{2}}{\lambda_{i,2}}+\frac{\widetilde{a}_{i,3}^{2} }{\lambda_{i,3}}\right) \tag{14}\]  
The differential of \(V_{i}\) is obtained as follows:  
\[\dot{V}_{i} = \alpha_{i}\sum_{i=1}^{n}\frac{\widetilde{x}_{i,1}\widetilde{x}_{ i,2}}{k_{i}^{2}-\widetilde{x}_{i,1}^{2}}+\sum_{i=1}^{n}\widetilde{x}_{i,2} \dot{\widetilde{x}}_{i,2}+\sum_{i=1}^{n}\left(\frac{\widetilde{f}_{i,1}^{2}}{ \lambda_{i,1}}+\frac{\widetilde{a}_{i,1}\dot{\widetilde{a}}_{i,1}}{\lambda_{i,2}}+\frac{\widetilde{a}_{i,2}\dot{\widetilde{a}}_{i,2}}{\lambda_{i,3}}\right) \tag{15}\] \[= \alpha_{i}\sum_{i=1}^{n}\frac{\widetilde{x}_{i,1}\widetilde{x}_{ i,2}}{k_{i}^{2}-\widetilde{x}_{i,1}^{2}}+\sum_{i=1}^{n}\widetilde{x}_{i,2} \left[\dot{\widetilde{x}}_{i,2}-\dot{x}_{i,2d}\right]+\sum_{i=1}^{n}\left(\frac {\widetilde{f}_{i,1}^{2}}{\lambda_{i,1}}+\frac{\widetilde{a}_{i,1}\dot{ \widetilde{a}}_{i,1}}{\lambda_{i,2}}+\frac{\widetilde{a}_{i,2}\dot{\widetilde {a}}_{i,2}}{\lambda_{i,3}}\right)\] \[= \alpha_{i}\sum_{i=1}^{n}\frac{\widetilde{x}_{i,1}\widetilde{x}_{ i,2}}{k_{i}^{2}-\widetilde{x}_{i,1}^{2}}+\sum_{i=1}^{n}\widetilde{x}_{i,2}\left[ \widetilde{f}_{i}+\widetilde{a}_{i,1}x_{i,2}+\widetilde{a}_{i,2}x_{i,2}^{2}- \beta_{i}\widetilde{x}_{i,2}-\alpha_{i}\frac{\widetilde{x}_{i,1}}{k_{i}^{2}- \widetilde{x}_{i,1}^{2}}\right]\] \[-\sum_{i=1}^{n}\left(\widetilde{x}_{i,2}\widetilde{f}_{i}+ \widetilde{x}_{i,2}\widetilde{a}_{i,1}x_{i,2}+\widetilde{x}_{i,2}\widetilde{a }_{i,2}x_{i,2}^{2}\right)\] \[= -\beta_{i}\sum_{i=1}^{n}\widetilde{x}_{i,2}^{2}\]  
From \(\beta_{i}>0\), we can know that \(\dot{V}_{i}\leq 0\). Then, \(\dot{V}_{i}\) is semi-negative definite, so \(V_{i}\) is a not-increasing state, and its numerical boundary value is \(V_{i}(0)\), i.e., \(V_{i}\) is bounded, which further ensures the boundedness of \(\widetilde{x}_{i,1}\), \(\widetilde{x}_{i,2}\), \(\widetilde{f}_{i}\), \(\widetilde{a}_{i,1}\), and \(\widetilde{a}_{i,2}\). Therefore, the control input \(u_{i,1}\) is bounded. It is further guaranteed that \(\dot{V}_{i}\) is bounded by differentiating both sides of Equation (15), so \(\dot{V}_{i}\) is consistently continuous. According to Barbalat Lemma, it is known that \(\lim\limits_{t\rightarrow\infty}\dot{V}_{i}(t)=0\). According to the linear control theory, it is known that the designed controller can guarantee the asymptotic speed and position tracking of each train, and achieve the tracking of the target speed-distance curve.