Stochastic Boundedness of State Trajectories of Stable LTI Systems in the Presence of Nonvanishing Stochastic Perturbation

TL;DR

This paper investigates the stochastic boundedness of stable LTI system trajectories under persistent stochastic perturbations, providing new theoretical conditions and bounds using Lyapunov methods, with an illustrative example demonstrating effectiveness.

Contribution

It introduces new definitions and a Lyapunov-based theorem for stochastic boundedness of perturbed LTI systems, including bounds and limiting behavior analysis.

Findings

Trajectories remain stochastically bounded under certain Lipschitz conditions.

Bounds on trajectories are explicitly calculated.

Theoretical results are validated with an illustrative example.

Abstract

This paper studies stochastic boundedness of trajectories of a nonvanishing stochastically perturbed stable LTI system. First, two definitions on stochastic boundedness of stochastic processes are presented, then the boundedness is analyzed via Lyapunov theory. In this proposed theorem, it is shown that under a condition on the Lipchitz constant of the perturbation kernel, the trajectories remain stochastically bounded in the sense of the proposed definitions and the bounds are calculated. Also, the limiting behavior of the trajectories have been studied. At the end an illustrative example is presented, which shows the effectiveness of the proposed theory.