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Integral Quadratic Constraints on Linear Infinite-dimensional Systems for Robust Stability Analysis

Barreau Matthieu, Scherer Carsten W., Gouaisbaut Frederic, Seuret Alexandre

TL;DR

This work addresses robustness analysis for systems formed by an ODE coupled to a 1D PDE uncertainty. It introduces a projection‑based IQC framework that constructs a finite‑order filter Ψ_N from projection coefficients of the PDE state, enabling the generation of finite‑horizon IQCs with terminal cost to certify stability via LMIs. The approach generalizes IQC applicability to a broad class of linear PDEs and demonstrates its effectiveness on a time‑delay example, producing stability regions and pockets that align with, or improve upon, existing Lyapunov‑based results. The methodology offers a constructive, scalable path to quantify robustness and paves the way for extended PDE classes and performance analyses in infinite‑dimensional interconnections.

Abstract

This paper proposes a framework to assess the stability of an ordinary differential equation which is coupled to a 1D-partial differential equation (PDE). The stability theorem is based on a new result on Integral Quadratic Constraints (IQCs) and expressed in terms of two linear matrix inequalities with a moderate computational burden. The IQCs are not generated using dissipation inequalities involving the whole state of an infinite-dimensional system, but by using projection coefficients of the infinite-dimensional state. This permits to generalize our robustness result to many other PDEs. The proposed methodology is applied to a time-delay system and numerical results comparable to those in the literature are obtained.

Integral Quadratic Constraints on Linear Infinite-dimensional Systems for Robust Stability Analysis

TL;DR

This work addresses robustness analysis for systems formed by an ODE coupled to a 1D PDE uncertainty. It introduces a projection‑based IQC framework that constructs a finite‑order filter Ψ_N from projection coefficients of the PDE state, enabling the generation of finite‑horizon IQCs with terminal cost to certify stability via LMIs. The approach generalizes IQC applicability to a broad class of linear PDEs and demonstrates its effectiveness on a time‑delay example, producing stability regions and pockets that align with, or improve upon, existing Lyapunov‑based results. The methodology offers a constructive, scalable path to quantify robustness and paves the way for extended PDE classes and performance analyses in infinite‑dimensional interconnections.

Abstract

This paper proposes a framework to assess the stability of an ordinary differential equation which is coupled to a 1D-partial differential equation (PDE). The stability theorem is based on a new result on Integral Quadratic Constraints (IQCs) and expressed in terms of two linear matrix inequalities with a moderate computational burden. The IQCs are not generated using dissipation inequalities involving the whole state of an infinite-dimensional system, but by using projection coefficients of the infinite-dimensional state. This permits to generalize our robustness result to many other PDEs. The proposed methodology is applied to a time-delay system and numerical results comparable to those in the literature are obtained.

Paper Structure

This paper contains 12 sections, 3 theorems, 59 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Robust IQC Theorem on Signals
  4. The Projection Methodology
  5. Generating IQCs using Projections
  6. General idea
  7. Application to uncertainties generated by a transport equation
  8. Finite-horizon IQC with terminal cost.
  9. Finite horizon IQC with $Z = 0$.
  10. On the relation with Lyapunov functionals.
  11. Numerical Example and Discussion
  12. Conclusion & Perspectives

Key Result

Theorem 1

Let $P \in \mathbb{S}^{n_{\xi} + n}$ and a multiplier $M = M^{\top}$ be such that the finite horizon IQC with terminal cost $Z \in \mathbb{S}^{n_\xi}$ (defined in eq:finiteHorizonIQC with respect to the factorization eq:factorization) holds together with where Then system eq:problem is stable in the sense of Definition 1.

Figures (1)

  • Figure 1: Block diagram describing the interconnected system \ref{['eq:problem']}-\ref{['eq:uncertainty']}.

Theorems & Definitions (7)

  • Example 1
  • Example 2
  • Definition 1
  • Definition 2
  • Theorem 1
  • Lemma 1
  • Lemma 2