Spatial adaptive iterative learning control for high-speed train with unknown speed delays

This article focuses on the high-speed train with unknown speed delays; a spatial adaptive iterative learning control (SAILC) algorithm is designed to solve the displacement-speed trajectory tracking problem by using the spatial differential operator to transform the train temporal dynamic model into a spatial model. First, parametric adaptive control is used to reduce the influence of system uncertainties. Second, a Lyapunov-Krasovskii-like spatial composite energy function (SCEF) is established, the stability of the designed controller and the convergence of the tracking error are demonstrated by verifying the differential negative definiteness and boundedness of the function. In addition, the train speed needs to be maintained within a certain range due to the speed loss when it passes through the neutral zone and overspeed protection. Therefore, the state-constrained mechanism is implanted into the train control algorithm to ensure that the train operates within the speed limits. Finally, the proposed SAILC algorithm is compared with proportional-integral-derivative (PID) and iterative learning control (ILC) algorithm; the results of the numerical simulations prove the effectiveness of the proposed method. After 20 iterations, the maximum absolute error of SAILC is 0.05 m/s, which is 4.7% of PID and 10.7% of ILC. It meets the requirements of Automatic Train Operation (ATO) that the tracking error should not be greater than 1% of the train operating speed.