Adaptive error feedback regulation problem of a first-order hyperbolic PDE system with unknown exosystem

Abstract

This paper studies the output regulation problem for a first-order hyperbolic PDE system with disturbances generated by an unknown finite-dimensional exosystem. The main challenges arise from unbounded control and observation operators, as well as non-collocated input–output configuration. We first introduce a coordinate transformation that simplifies the system dynamics. Next, based on the transformed system, we design an observer and apply an adaptive internal model principle to estimate the unknown harmonic frequencies of the exosystem. We present a controller that achieves exponentially stable output regulation for the resulting closed-loop system. Finally, the effectiveness of the controller is demonstrated through numerical simulations which demonstrate effective parameter tracking, $g(0,t)$ (the regulated output) achieves accurate tracking of ${\Phi }_{ref}\left(t\right)$ (the reference signal), and the solution remain uniformly bounded of the $g$-part in closed-loop system.