Robust output regulation of 2 x 2 hyperbolic systems part I: Control law and Input-to-State Stability

We consider the problem of output feedback regulationfor a linear first-order hyperbolic system with collocatedinput and output in presence of a general class of disturbancesand noise. The proposed control law is designed through abackstepping approach incorporating an integral action. Toensure robustness to delays, the controller only cancels partof the boundary reflection by means of a tunable parameter.This also enables a trade-off between disturbance and noisesensitivity.We show that the boundary condition of the obtainedtarget system can be transformed into a Neutral DifferentialEquation (NDE) and that this latter system is Input-to-StateStable (ISS). This proves the boundedness of the controlledoutput for the target system. This extends previous worksconsidering an integral action for this kind of system [16], andconstitutes an important step towards practical implementationof such controllers. Applications and practical considerations,in particular regarding the system's sensitivity functions arederived in a companion paper.