Necessary and sufficient condition for neutral-type delay systems: Polynomial approximations

TL;DR

This paper introduces a new necessary and sufficient stability test for linear neutral-type delay systems using polynomial approximations within the Lyapunov-Krasovskii framework, providing a practical and accurate method.

Contribution

It develops a novel stability criterion based on polynomial approximations of functionals, simplifying the verification process for neutral-type delay systems.

Findings

New stability test reduces to quadratic forms independent of polynomial coefficients

Chebyshev polynomial approximation quantifies functional error

Examples demonstrate effectiveness of the proposed method

Abstract

A new necessary and sufficient stability test in a tractable number of operations for linear neutral-type delay systems is introduced. It is developed in the Lyapunov-Krasovskii framework via functionals with prescribed derivatives. The necessary conditions, which stem from substituting any polynomial approximation of the functional argument, reduce to a quadratic form of monomials whose matrix is independent of the coefficients of the approximation under consideration. In the particular case of Chebyshev polynomials, the functional approximation error is quantified, leading to an estimate of the order of approximation such that the positive semi-definiteness of the functional is verified. Some examples illustrate the obtained results.