What ODE-Approximation Schemes of Time-Delay Systems Reveal about Lyapunov-Krasovskii Functionals

TL;DR

This paper introduces an ODE-based approximation approach for time-delay systems that simplifies the construction of Lyapunov-Krasovskii functionals without relying on matrix inequalities, leading to less conservative bounds.

Contribution

It presents a novel spectral method-based framework to approximate Lyapunov-Krasovskii functionals using ODEs, avoiding complex matrix inequalities and providing tighter bounds.

Findings

The approach yields less conservative bounds than existing methods.

Validation through quadrature confirms the effectiveness of the approximation.

A formula for a tight quadratic lower bound is derived and validated.

Abstract

The article proposes an approach to complete-type and related Lyapunov-Krasovskii functionals that neither requires knowledge of the delay-Lyapunov matrix function nor does it involve linear matrix inequalities. The approach is based on ordinary differential equations (ODEs) that approximate the time-delay system. The ODEs are derived via spectral methods, e.g., the Chebyshev collocation method (also called pseudospectral discretization) or the Legendre tau method. A core insight is that the Lyapunov-Krasovskii theorem resembles a theorem for Lyapunov-Rumyantsev partial stability in ODEs. For the linear approximating ODE, only a Lyapunov equation has to be solved to obtain a partial Lyapunov function. The latter approximates the Lyapunov-Krasovskii functional. Results are validated by applying Clenshaw-Curtis and Gauss quadrature to a semi-analytical result of the functional, yielding a comparable finite-dimensional approximation. In particular, the article provides a formula for a tight quadratic lower bound, which is important in applications. Examples confirm that this new bound is significantly less conservative than known results.