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

Michael Tang, Miroslav Krstic, Jorge Poveda

TL;DR

This work introduces a Lyapunov-based nonlinear small-gain theorem to certify fixed-time stability (FxTS) for interconnections of FxT ISS subsystems, relaxing the need for time-scale separation. By allowing gains from either the plus or inverse classes of $\\mathcal{K}_\\infty$, and constructing a max-form, scaled FxTS Lyapunov function, the authors derive a global FxTS result under a mild small-gain condition. The approach is demonstrated through a second-order interconnection with cross-couplings and a fixed-time feedback optimization problem without time-scale separation, underscoring both theoretical and practical relevance. Appendices supply supporting lemmas to ensure the FxTS properties persist under various gain structures, enabling potential extensions to large-scale networks and hybrid systems.

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.

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

TL;DR

This work introduces a Lyapunov-based nonlinear small-gain theorem to certify fixed-time stability (FxTS) for interconnections of FxT ISS subsystems, relaxing the need for time-scale separation. By allowing gains from either the plus or inverse classes of , and constructing a max-form, scaled FxTS Lyapunov function, the authors derive a global FxTS result under a mild small-gain condition. The approach is demonstrated through a second-order interconnection with cross-couplings and a fixed-time feedback optimization problem without time-scale separation, underscoring both theoretical and practical relevance. Appendices supply supporting lemmas to ensure the FxTS properties persist under various gain structures, enabling potential extensions to large-scale networks and hybrid systems.

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.

Paper Structure

This paper contains 11 sections, 9 theorems, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Fixed-Time stability
  5. Fixed-time input-to-state stability
  6. Main Results
  7. Analysis
  8. Second-Order example
  9. Fixed-Time Feedback Optimization without Timescale Separation
  10. Conclusion
  11. Appendix

Key Result

Lemma 1

Given $\underline{p}\le p\le \overline{p}$, the following holds for all $x\ge 0$.$\square$

Figures (3)

  • Figure 1: A block diagram depicting the interconnection \ref{['sysnu']}.
  • Figure 2: Trajectories of system \ref{['ex0plot']} with varying $x(0),y(0)$.
  • Figure 3: Trajectories of system \ref{['accplant']} with \ref{['ex_fdbk']} and varying initial conditions, where the parameters are given by \ref{['accplant_param']} and \ref{['quadcost_param']}.

Theorems & Definitions (25)

  • Lemma 1
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • ...and 15 more