Computing Input-Output Properties of Coupled PDE systems

TL;DR

This paper introduces an LMI-based method to analyze input-output properties of coupled PDE systems, extending state-space theory and positive-real lemmas to infinite dimensions without discretization.

Contribution

It develops a novel LMI-based approach for coupled PDEs that avoids discretization and provides provable, non-conservative bounds on system properties.

Findings

Bounds are not significantly conservative

Method is computationally feasible on desktop computers

Applicable to systems with up to 20 coupled PDEs

Abstract

In this paper, we propose an LMI-based approach to analyze input-output properties of coupled linear PDE systems. This work expands on a newly developed state-space theory for coupled PDEs and extends the positive-real and bounded-real lemmas to infinite dimensional systems. We show that conditions for passivity and bounded L2 gain can be expressed as linear operator inequalities on RxL2. A method to convert these operator inequalities to LMIs by using parameterization of the operator variables is proposed. This method does not rely on discretization and as such, the properties obtained are prima facie provable. We use numerical examples to demonstrate that the bounds obtained are not conservative in any significant sense and that the bounds are computable on desktop computers for systems consisting of up to 20 coupled PDEs.