## 1 Introduction  
Urban rail transit's ATO (Automatic Train Operation system) is a direct unit responsible for the train's safe operation and control, and its advanced control algorithm is the key to ensure the train's efficiency, punctuality, control accuracy, comfort, and energy conservation [1]. In the case of the changeable operation environment and the system state changing, ATO control algorithm with high control accuracy, fast response speed, and less braking times is the core to ensure the tracking operation control of multiple trains under moving block [2]. In the process of the development of ATO, there are many different train operation control algorithms, and some achievements have been made, such as PID control [3], fuzzy control [4], adaptive control [5], iterative learning control [6], genetic algorithm control [7], predictive control [8], etc. However, the response speed of PID control is slow, which cannot meet the real-time requirements of train control. The adjustment of fuzzy control parameters and control rules is time-consuming and labor-consuming and needs experienced engineers to design, so it is not easy to implement in the project. Additionally, in the design process of the controller, the quasi parameters of the model need to be known in advance, but the parameters of the train operation model change with the changes of the train load, performance, and operating environment. The authors of [9] introduced an adaptive control to better overcome the control inaccuracy caused by the uncertainty of model parameters. However, there are certain limitations in the way of dealing with the additional resistance of the route. When facing the line slope change point, the switching of control input is too frequent, which affects the service life of the train braking system equipment. In order to overcome the problem of frequent switching of control input caused by adaptive control, [10] introduced an auxiliary system to control variable slope points smoothly. Although the adaptive control algorithm can make the system stable, when the route disturbance is large, the train operation state cannot converge quickly. The authors of [11] applied iterative learning control to realized train precise control. After repeated learning many times, the train can finally overcome the interference of the operation environment, but the real-time performance of continuous iteration is poor. The authors of [12] applied parameter estimation to realized on-line identification of train model parameters, and a fuzzy predictive control algorithm is designed, which can better deal with the uncertainty and external disturbance of model parameters. However, the problem of control accuracy degradation due to the limit value, delay, and the fault of many links such as an actuator, speed sensor, and interface equipment are not considered. In [13], a robust adaptive and fault-tolerant control scheme for high-speed trains is established by constructing a Lyapunov function with virtual parameter estimation error to ensure the safe and reliable operation of trains in normal conditions or in case of traction or braking failure. However, only the faults of the traction or braking mechanism are considered in the literature, and the control problems caused by input saturation and speed delay are not considered. In [14], an iterative learning control algorithm considering speed constraints is proposed to realize train operating curve tracking under the condition of variable parameters, but it does not involve the safe train operating control under input saturation and actuator traction or braking failure. In [15], a train adaptive iterative learning control method with speed delay and input saturation is designed, and the Lyapunov-Krasovskii function method is used to solve the problem of speed delay. In [16], an adaptive fault-tolerant ATO control method considering input saturation and speed delay is proposed, and the convergence of speed tracking is guaranteed. However, ref.