Necessary and sufficient stability condition for time-delay systems arising from Legendre approximation

TL;DR

This paper extends a stability criterion for time-delay systems using Legendre polynomial projections, reducing the required approximation order and enabling efficient stability analysis with lower computational effort.

Contribution

It introduces a Legendre polynomial-based approach to stability analysis, improving the efficiency and accuracy of the necessary and sufficient stability condition for time-delay systems.

Findings

Reduced approximation order needed for stability determination

Able to identify stable regions at low orders

Faster convergence compared to previous methods

Abstract

Recently, sufficient conditions of stability or instability for time-delay systems have been proven to be necessary. In this way, a remarkable necessary and sufficient condition has then been developed by Gomez et al. It is presented as a simple test of positive definiteness of a matrix issued from the Lyapunov matrix. In this paper, an extension of this result is presented. Without going into details, the uniform discretization of the state has been replaced by projections on the first Legendre polynomials. Like Gomez et al., based on convergence arguments, the necessity is obtained in finite order, which can be calculated analytically. Compared to them, by relying on the fast convergence rate of Legendre approximation, the required order to ensure stability has been reduced. Thanks to this major modification, as shown in the example section, it is possible the find stable regions for low orders and unstable ones for even smaller orders.