On hyperbolic PDEs, filtered feedback control laws, and fractal-like stability crossing curves

TL;DR

This paper investigates the stability of hyperbolic PDE boundary control systems using filtered feedback laws, revealing fractal-like crossing curves and providing stability conditions considering model mismatch and filter parameters.

Contribution

It introduces a novel stability analysis framework for hyperbolic PDEs with filtered feedback, including stability charts and the fractal nature of crossing curves.

Findings

Maximum stability interval in parameter T identified

Stability conditions derived considering model mismatch

Fractal-like structure of stability crossing curves explained

Abstract

The paper addresses the boundary control of a class of hyperbolic PDEs, based on an equivalent representation in terms of an integral-difference equation. The situation is considered where direct compensation of reflection terms induces a fragile closed-loop system, in the sense of lack of strong stability. This is theoretically resolved by adding a low-pass filter to the control law, but the choice of its cut-off frequency is crucial in balancing robustness at high frequencies and performance at low frequencies. First, the maximum stability interval in parameter $T$ is determined, with $T$ the inverse of the filter's cutoff frequency. Next, model mismatch on the PDE parameters is considered and a sufficient stability condition is derived in terms of allowable mismatch and cut-off frequency, satisfied in a region in the combined parameter space with a conic shape around $T=0$. Finally, this qualitative behavior is confirmed by exact stability charts for a special case where all model mismatch is contained into one parameter. It is highlighted that the set of stability crossing curves exhibits a fractal-like structure, which is explained using a limit system with discrete delays.