Deterministic and Stochastic Fixed-time Stability of Discrete-time Autonomous Systems

TL;DR

This paper establishes Lyapunov-based conditions for deterministic and stochastic fixed-time stability of nonlinear discrete-time systems, including perturbations and noise, with bounds on settling times verified through examples.

Contribution

It introduces new Lyapunov criteria for fixed-time stability in both deterministic and stochastic discrete-time systems, extending existing results to perturbed and noisy scenarios.

Findings

Lyapunov conditions for fixed-time stability are derived.

Fixed upper bounds on settling times are established.

Simulation examples verify theoretical results.

Abstract

This paper studies deterministic and stochastic fixed-time stability of autonomous nonlinear discrete-time (DT) systems. Lyapunov conditions are first presented under which the fixed-time stability of deterministic DT system is certified. Extensions to systems under deterministic perturbations as well as stochastic noise are then considered. For the former, the sensitivity to perturbations for fixed-time stable DT systems is analyzed, and it is shown that fixed-time attractiveness is resulted from the presented Lyapunov conditions. For the latter, sufficient Lyapunov conditions for fixed-time stability in probability of nonlinear stochastic DT systems are presented. The fixed upper bound of the settling-time function is derived for both fixed-time stable and fixed-time attractive systems, and the stochastic settling-time function fixed upper bound is derived for stochastic DT systems. Illustrative examples are given along with simulation results to verify the introduced results.