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

TL;DR

This paper develops a robust boundary feedback control law for 2x2 hyperbolic systems that ensures input-to-state stability despite disturbances and delays, advancing practical control strategies for such systems.

Contribution

It introduces a backstepping-based control law with integral action that handles delays and disturbances, extending previous methods to ensure ISS for hyperbolic systems.

Findings

The control law guarantees bounded output under disturbances.

The boundary condition can be transformed into a Neutral Differential Equation.

The system achieves input-to-state stability with the proposed control.

Abstract

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.