On Fixed-Time Stability for a Class of Singularly Perturbed Systems using Composite Lyapunov Functions

TL;DR

This paper develops a new Lyapunov-based approach to analyze fixed-time stability in singularly perturbed systems, enabling the design of control systems with guaranteed convergence times regardless of initial conditions.

Contribution

It extends the composite Lyapunov method to fixed-time stability in singularly perturbed systems, providing novel sufficient conditions for stability certification.

Findings

The proposed method successfully certifies fixed-time stability analytically.

Numerical examples demonstrate the effectiveness of the approach.

The approach handles interconnected systems with multiple time scales.

Abstract

Fixed-time stable dynamical systems are capable of achieving exact convergence to an equilibrium point within a fixed time that is independent of the initial conditions of the system. This property makes them highly appealing for designing control, estimation, and optimization algorithms in applications with stringent performance requirements. However, the set of tools available for analyzing the interconnection of fixed-time stable systems is rather limited compared to their asymptotic counterparts. In this paper, we address some of these limitations by exploiting the emergence of multiple time scales in nonlinear singularly perturbed dynamical systems, where the fast dynamics and the slow dynamics are fixed-time stable on their own. By extending the so-called composite Lyapunov method from asymptotic stability to the context of fixed-time stability, we provide a novel class of Lyapunov-based sufficient conditions to certify fixed-time stability in a class of singularly perturbed dynamical systems. The results are illustrated, analytically and numerically, using a fixed-time gradient flow system interconnected with a fixed-time plant and an additional high-order example.