Convergence Analysis of Backstepping‐Based Iterative Learning Control With Disturbance for PDE–PDE Cascades

This paper addresses the stabilization problem of a class of hyperbolic partial differential equation (PDE) cascades with unknown disturbance. We conduct our research under the actual traffic flow background, and the macroscopic traffic dynamics are described by Aw–Rascle–Zhang (ARZ) model, which can be transformed into the aforementioned PDE model through a series of reversible coordinate transformations. For the transformed systems, a novel backstepping‐based iterative learning control (ILC) strategy is proposed, which consists of two important parts. The first part designs backstepping controller as the basic control input to ensure the boundedness of the system states. It should be noted that this step does not handle the disturbance yet, due to the inability of the backstepping controller to directly cancel it out. The second part is to introduce ILC. By utilizing the norm of the error between the current and target states, a backstepping‐based iterative learning controller is designed to achieve uniform convergence of the states to the target trajectory, thereby eliminating the influence of external disturbance and enhancing the robustness of the systems. Rigorous Lyapunov analysis and the assistance of the composite energy function (CEF) guarantees closed‐loop stability. The simulation results serve to validate the proposed control strategy.