Diffusion and robustness of boundary feedback stabilization of hyperbolic systems

TL;DR

This paper investigates the boundary feedback stabilization of hyperbolic systems, revealing that small diffusion can restore robustness against delay uncertainties, but diffusion's effect on stability can also be destabilizing.

Contribution

It demonstrates that small diffusion in hyperbolic systems can recover robustness of boundary feedback control against delay uncertainties, a novel insight in control theory.

Findings

Small diffusion restores robustness to delay uncertainties.

Diffusion can have a destabilizing effect on stability.

Analysis of interconnected transport equations with boundary sensing.

Abstract

We consider the problem of boundary feedback control of single-input-single-output (SISO) one-dimensional linear hyperbolic systems when sensing and actuation are anti-located. The main issue of the output feedback stabilization is that it requires dynamic control laws that include delayed values of the output (directly or through state observers) which may not be robust to infinitesimal uncertainties on the characteristic velocities. The purpose of this paper is to highlight some features of this problem by addressing the feedback stabilization of an unstable open-loop system which is made up of two interconnected transport equations and provided with anti-located boundary sensing and actuation. The main contribution is to show that the robustness of the control against delay uncertainties is recovered as soon as an arbitrary small diffusion is present in the system. Our analysis also reveals that the effect of diffusion on stability is far from being an obvious issue by exhibiting an alternative simple example where the presence of diffusion has a destabilizing effect instead.