Boundary Control with Integral Action for a Class of Gantry Crane Systems

In this paper, we focus on the output regulation problem of a class of gantry crane systems governed by partial differential equations (PDEs) in the presence of unknown constant disturbances. We first design a preliminary state-feedback controller for the system without perturbations and establish the existence of a strict Lyapunov function under the preliminary control law. By employing the forwarding method, we then add an integral action to the preliminary controller and analyze the well-posedness of the resulting closed-loop system, demonstrating its exponential stability. Furthermore, we extend our analysis to consider the system with perturbations, solving the output regulation problem. Finally, we provide numerical simulation results to demonstrate the effectiveness of our proposed strategy.