Delay-robust control design for two heterodirectional linear coupled hyperbolic PDEs

TL;DR

This paper explores a new control design paradigm for two coupled hyperbolic PDEs, emphasizing the trade-off between convergence speed and robustness to delays, especially in systems with boundary reflections.

Contribution

It demonstrates that canceling boundary reflections for systems with strong reflections reduces delay-robustness, proposing a backstepping approach that preserves some reflection for robustness.

Findings

Canceling boundary reflections yields zero delay-robustness.

Preserving some reflection ensures robustness to small delays.

Trade-off between convergence rate and delay-robustness is fundamental.

Abstract

We detail in this article the necessity of a change of paradigm for the delay-robust control of systems composed of two linear first order hyperbolic equations. One must go back to the classical trade-off between convergence rate and delay-robustness. More precisely, we prove that, for systems with strong reflections, canceling the reflection at the actuated boundary will yield zero delay-robustness. Indeed, for such systems, using a backstepping-controller, the corresponding target system should preserve a small amount of this reflection to ensure robustness to a small delay in the loop. This implies, in some cases, giving up finite time convergence.