# A Generalized LMI Formulation for Input-Output Analysis of Linear Systems of ODEs Coupled with PDEs

## TL;DR

This paper develops a unified LMI-based framework for analyzing input-output properties of coupled PDE-ODE systems without discretization, extending the KYP Lemma to infinite-dimensional systems and demonstrating computational efficiency.

## Contribution

It generalizes the KYP Lemma to coupled PDE-ODE systems using a boundary-condition-free representation and PQRS parameterization, enabling finite-dimensional LMI analysis without discretization.

## Key findings

- LMI conditions can verify system passivity and L2-gain.
- The method is less conservative than existing discretization-based approaches.
- Computational complexity is lower than traditional PDE analysis methods.

## Abstract

In this paper, we consider input-output properties of linear systems consisting of PDEs on a finite domain coupled with ODEs through the boundary conditions of the PDE. This framework can be used to represent e.g. a lumped mass fixed to a beam or a system with delay. This work generalizes the sufficiency proof of the KYP Lemma for ODEs to coupled ODE-PDE systems using a recently developed concept of fundamental state and the associated boundary-condition-free representation. The conditions of the generalized KYP are tested using the PQRS positive matrix parameterization of operators resulting in a finite-dimensional LMI, feasibility of which implies prima facie provable passivity or L2-gain of the system. No discretization or approximation is involved at any step and we use numerical examples to demonstrate that the bounds obtained are not conservative in any significant sense and that computational complexity is lower than existing methods involving finite-dimensional projection of PDEs.