# Adaptive Cooperative Control of Multiple Urban Rail Trains with Position Output Constraints
## 4 Simulation Results
From Figures 10 and 11, it can be seen that the speed tracking effect is great and each train can automatically regulate the control input conferring to the data of the neighboring trains, as shown in Figure 12. In the section with sudden change of desired speed and complex route in Figure 10, that is, the train runs in 52 s to 248 s, the speed tracking error of each train in the multiple train cooperative operation system shown in Figure 11 fluctuates, but the speed error of the preceding trains does not affect the speed control of the tracking trains. The speed tracking error of each train decreases with the propagation of train number, that is, the fluctuation of speed tracking error of the fifth train is the smallest, which ensures the stability of multiple trains cooperative operation. In other acceleration and deceleration sections, the speed error of each train is significantly reduced. In particular, in the cruise section, the speed error of each train is almost zero, which has good control accuracy. As can be seen from Figure 12, when the trains are just started, the controller applies a large control input to overcome the external resistance. When the train runs normally, the control input remains stable. When the train continues to run for 52 s to 248 s, the controller adjusts the control input in order to maintain the control accuracy. The control inputs of other sections are relatively stable, even in acceleration and deceleration sections. The fluctuation is very small to meet the comfort of passengers. Thus, it can be seen that in the multiple trains single-value learning adaptive cooperative control system with POC, the trains can regulate their own operation status conferring to the operation data of the neighboring trains, which usefully ensures the safety of multiple trains cooperative operation, reduces the complexity of the controller, and provides a reference for the development and application of the new generation of MTCC system. Figure 8: Desired position and position tracking curves for single-value learning. Figure 9: Position tracking error curves for single-value learning. Figure 11: Speed tracking error curves for single-value learning. Figure 10: Desired speed and speed tracking curves for single-valued learning. In addition, in order to demonstrate the performance and engineering practicality advantages of the controller proposed in this paper, the PID control algorithm is extensively applied to train operation control systems (such as the Beijing Metro Yizhuang Line) as selected in this paper, and the PID controller structure is chosen in the form of reference [27], with the PID controller parameters set as \(K_{p,1}=0.05\), \(K_{i,1}=0.01\), \(K_{d,1}=0.03\), \(K_{p,2}=0.15\), \(K_{i,2}=0.03\), \(K_{d,2}=0.013\), \(K_{p,3}=0.13\), \(K_{i,3}=0.035\), \(K_{d,3}=0.014\), \(K_{p,A}=0.11\), \(K_{i,4}=0.032\), \(K_{d,4}=0.012\), \(K_{p,5}=0.1\), \(K_{i,5}=0.03\), \(K_{d,5}=0.02\). Figures 13 and 14 give the position tracking errors and speed tracking errors under the PID controller. It can be seen from Figure 13 that the position tracking error of PID controller is distributed in the range of [\(-1.41\), \(1.58\)], which is relatively large as a whole. Due to the lack of adaptive mechanism of PID controller, the ability to deal with the change of train running resistance and external environment was not strong, and the degree of position tracking error decreasing in turn with the propagation of train number was not obvious. The overall position error is much larger than \(0.2\) m proposed in this paper, and there is continuous fluctuation. It can be seen from Figure 14 that the speed error of the PID controller varies widely, among which the speed error of the first train is the largest, and the speed error of the fifth train is improved, but there were overall fluctuations.