A Lyapunov-like Characterization of Predefined-Time Stability

TL;DR

This paper develops Lyapunov-like conditions to characterize and analyze predefined-time stability in dynamical systems, providing a unified framework that includes controller design and stability analysis for uncertain systems.

Contribution

It introduces a general Lyapunov-like theorem for predefined-time stability and extends it to uncertain systems and controller design, unifying previous results.

Findings

Lyapunov-like conditions characterize predefined-time stability.

The framework applies to uncertain and controlled systems.

Simulation demonstrates effective settling time estimation.

Abstract

This technical note studies Lyapunov-like conditions to ensure a class of dynamical systems to exhibit predefined-time stability. The origin of a dynamical system is predefined-time stable if it is fixed-time stable and an upper bound of the settling-time function can be arbitrarily chosen a priori through a suitable selection of the system parameters. We show that the studied Lyapunov-like conditions allow to demonstrate equivalence between previous Lyapunov theorems for predefined-time stability for autonomous systems. Moreover, the obtained Lyapunov-like theorem is extended for analyzing the property of predefined-time ultimate boundedness with predefined bound, which is useful when analyzing uncertain dynamical systems. Therefore, the proposed results constitute a general framework for analyzing predefined-time stability, and they also unify a broad class of systems which present the predefined-time stability property. On the other hand, the proposed framework is used to design robust controllers for affine control systems, which induce predefined-time stability (predefined-time ultimate boundedness of the solutions) w.r.t. to some desired manifold. A simulation example is presented to show the behavior of a developed controller, especially regarding the settling time estimation.