A Lyapunov-Based Small-Gain Theorem for Fixed-Time Stability

TL;DR

This paper develops a Lyapunov-based small-gain theorem to ensure fixed-time stability in interconnected systems, extending existing stability analysis methods to fixed-time guarantees.

Contribution

It introduces a novel small-gain approach for fixed-time stability using Lyapunov functions, filling a gap in interconnected system stability analysis.

Findings

Theorem establishing FxTS for interconnected systems under small-gain conditions.

Application to fixed-time feedback optimization without time-scale separation.

Numerical examples demonstrating the effectiveness of the proposed method.

Abstract

This paper introduces a novel Lyapunov-based small-gain methodology for establishing fixed-time stability (FxTS) guarantees in interconnected dynamical systems. Specifically, we consider interconnections in which each subsystem admits an individual fixed-time input-to-state stability (ISS) Lyapunov function that certifies FxT-ISS. We then show that if a nonlinear small-gain condition is satisfied, then the entire interconnected system is FxTS. Our results are analogous to existing Lyapunov-based small-gain theorems developed for asymptotic and finite-time stability, thereby filling an important gap in the stability analysis of interconnected dynamical systems. The proposed theoretical tools are further illustrated through analytical and numerical examples, including an application to fixed-time feedback optimization of dynamical systems without time-scale separation between the plant and the controller.