Finite-dimensional output stabilization for a class of linear distributed parameter systems -- a small-gain approach

TL;DR

This paper introduces a small-gain method for analyzing the stability of linear diffusion-reaction systems controlled by finite-dimensional observers, addressing spillover effects and providing a way to determine the necessary subsystem dimension.

Contribution

It develops a novel small-gain framework for stability analysis of infinite-dimensional systems with finite-dimensional controllers, including spillover effect considerations.

Findings

Effective stability conditions derived for controlled systems

Method to compute the minimal dimension of the slow subsystem

Simulation results validate theoretical predictions

Abstract

A small-gain approach is proposed to analyze closed-loop stability of linear diffusion-reaction systems under finite-dimensional observer-based state feedback control. For this, the decomposition of the infinite-dimensional system into a finite-dimensional slow subsystem used for design and an infinite-dimensional residual fast subsystem is considered. The effect of observer spillover in terms of a particular (dynamic) interconnection of the subsystems is thoroughly analyzed for in-domain and boundary control as well as sensing. This leads to the application of a small-gain theorem for interconnected systems based on input-to-output stability and unbounded observability properties. Moreover, an approach is presented for the computation of the required dimension of the slow subsystem used for controller design. Simulation scenarios for both scalar and coupled linear diffusion-reaction systems are used to underline the theoretical assessment and to give insight into the resulting properties of the interconnected systems.