Periodic sampled‐data adaptive controller for hyperbolic PDE–ODEs cascade systems

Abstract

This article addresses the global exponential stability of a class of hyperbolic partial differential equations (PDEs) cascaded with a set of uncertain ordinary differential equations (ODEs) under a sampled‐data boundary feedback control for a nominal continuous input. The cascade system consists of ‐dimensional coupled hyperbolic PDEs, and the ODEs enter the PDEs through the left boundary of the PDEs. We design the adaptive law based on the gradient method. We show that there exists a sufficiently small sample period that guarantees global exponential stability of the closed‐loop system. In addition, we provide easy‐to‐handle sufficient stability conditions that can be used to find an upper bound on the maximum sampling period. The results rely on a combination of Lyapunov's method and looped functionals. Finally, we validate the effectiveness of the sampled‐data‐based boundary adaptive controller design method by means of an example algorithm.