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

Tessina H. Scholl

TL;DR

The paper develops Lyapunov-Krasovskii functionals of robust type for time-delay systems to obtain less conservative robustness bounds than complete-type functionals. It introduces a perturbation structure $(B, ext{C})$ and a perturbation-restriction in the form of $ ext{Π}$-matrices, and derives a derivative condition that reduces to an algebraic Riccati equation via a splitting approach and a Hilbert-space KYP lemma. Existence and monotonicity properties are established, leading to a small-gain–type robustness result that ties perturbation bounds to the $H_ ext{∞}$ norm of a transfer function $G(s)$. The framework accommodates sector-based absolute stability and IQC-style constraints, and an example demonstrates notably tighter bounds than those offered by complete-type LK functionals. Overall, the robust-type LK functionals provide a flexible, frequency-domain–informed path to robust stability analysis for time-delay systems with structured perturbations.

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.

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

TL;DR

The paper develops Lyapunov-Krasovskii functionals of robust type for time-delay systems to obtain less conservative robustness bounds than complete-type functionals. It introduces a perturbation structure and a perturbation-restriction in the form of -matrices, and derives a derivative condition that reduces to an algebraic Riccati equation via a splitting approach and a Hilbert-space KYP lemma. Existence and monotonicity properties are established, leading to a small-gain–type robustness result that ties perturbation bounds to the norm of a transfer function . The framework accommodates sector-based absolute stability and IQC-style constraints, and an example demonstrates notably tighter bounds than those offered by complete-type LK functionals. Overall, the robust-type LK functionals provide a flexible, frequency-domain–informed path to robust stability analysis for time-delay systems with structured perturbations.

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.
Paper Structure (24 sections, 33 theorems, 111 equations, 3 tables)

This paper contains 24 sections, 33 theorems, 111 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Problem statement and Preliminaries
  3. System class
  4. Functionals of interest
  5. Motivation: LK functionals of complete type
  6. Outline of the approach
  7. Definition of LK functionals of robust type
  8. Perturbation structure $(B,\mathcal{C})$
  9. Perturbation restriction matrices $(\Pi_{{\zeta}{\zeta}},\Pi_{{\zeta} \alpha},\Pi_{ \alpha \alpha})$
  10. Domain with an upper bound on $D_{(f+g)}^+V$
  11. Lower Bounds on $V$
  12. Existence conditions for $V$ in terms of admissible perturbation restrictions
  13. A splitting approach
  14. Existence of an LK functional of robust type
  15. Further cases of interest
  16. ...and 9 more sections

Key Result

Theorem II.1

Consider $\dot x(t)=F(x_t)$, where $F$ is Lipschitz continuous on any bounded set and $F(0_{n_{[-{h},0]}})=0_n$. If there exists a continuous functional $V:C([-{h},0],\mathbb R^n)\to \mathbb R$, upper bounded by $\exists \kappa_2\in \mathcal{K}, \forall \phi\in C:V(\phi)\leq \kappa_2(\|\phi\|_C)$, a then the zero equilibrium of $\dot x(t)=F(x_t)$ is locally asymptotically stable. Moreover, if $\Om

Theorems & Definitions (71)

  • Theorem II.1: LK theorem Hale.1993
  • Definition II.2: Complete-type LK functional Kharitonov.2013
  • Proposition II.3: Existence Kharitonov.2013
  • Proposition II.4: Lower bound on $V$ Kharitonov.2013
  • Theorem II.5: Nominal converse LK statement
  • Proposition II.6: Upper bound on $D_{(f+g)}^+V$ MelchorAguilar.2007
  • Theorem II.7: Robustness statement
  • Definition III.1: LK functional of robust type
  • Remark III.2: Solution approaches
  • Example III.3: Perturbation structure, choice
  • ...and 61 more