Lyapunov-Krasovskii Functionals of Robust Type for the Stability Analysis in Time-Delay Systems

TL;DR

This paper introduces a new class of Lyapunov-Krasovskii functionals of robust type, which enhance the robustness bounds in stability analysis of time-delay systems by relating to algebraic Riccati equations.

Contribution

It proposes an alternative to complete-type functionals, improving robustness bounds for linear systems with delays using algebraic Riccati equations.

Findings

Derived linear bounds on admissible perturbations.

Proven existence of robust functionals for stable systems.

Improved robustness bounds over previous complete-type functionals.

Abstract

Inspired by the widespread concept of Lyapunov-Krasovskii functionals of complete type, this article proposes an alternative class of functionals, termed Lyapunov-Krasovskii functionals of robust type. Their construction aims at improving deducible robustness bounds of linear systems with a constant delay. These refer to bounds on nonlinear or uncertain terms that can be added to the system without compromising the proof of stability. The defining equation of complete-type functionals relies on the template of a Lyapunov equation. In contrast, the proposed functionals are related to an algebraic Riccati equation. The article proves properties that make these functionals suitable tools for the stability analysis via Lyapunov arguments. The derived linear bounds on the norm of admissible perturbations mirror bounds from the small gain theorem or the complex stability radius. More general sector-based absolute stability bounds can also be addressed. Existence of the functionals is proven via the Kalman-Yakubovich-Popov lemma combined with a splitting approach. In particular, for any asymptotically stable nominal system, there exists a Lyapunov-Krasovskii functional of robust type that proves a nonzero bound on admissible perturbations. This robustness bound significantly improves results from complete-type functionals.