Input / Output Stability of a Damped String Equation coupled with Ordinary Differential System

TL;DR

This paper investigates the input/output stability of a coupled system involving a damped string equation and an ODE, deriving new stability criteria using classical and modern control theory tools, with numerical validation.

Contribution

It introduces robust stability criteria for the coupled system based on the Small-Gain theorem and Quadratic Separation, utilizing Legendre polynomial projections.

Findings

Exact stability results via pole analysis

Robust stability criteria derived from Small-Gain and Quadratic Separation

Numerical comparisons showing the effectiveness of proposed criteria

Abstract

The input/output stability of an interconnected system composed of an ordinary differential equation and a damped string equation is studied. Issued from the literature on time-delay systems, an exact stability result is firstly derived using pole locations. Then, based on the Small-Gain theorem and on the Quadratic Separation framework, some robust stability criteria are provided. The latter follows from a projection of the infinite dimensional state on an orthogonal basis of Legendre polynomials. Numerical examples comparing these results with the ones in the literature are proposed and a comparison of its efficiency is made.