Fluctuation analysis for a class of nonlinear systems with fast periodic sampling and small state-dependent white noise

TL;DR

This paper studies the behavior of nonlinear systems with small state-dependent noise and fast periodic sampling, showing how the stochastic process can be approximated by an ODE and analyzing fluctuations with a limiting SDE.

Contribution

It introduces a novel approximation framework for nonlinear sampled systems with noise, deriving the limiting SDE that captures the interplay between sampling and stochastic perturbations.

Findings

The stochastic process converges to an ODE as noise and sampling intervals go to zero.

Fluctuations around the ODE are characterized by a limiting SDE depending on sampling and noise rates.

Effective drift terms are computed to describe the combined effects of noise and sampling.

Abstract

We consider a nonlinear differential equation under the combined influence of small state-dependent Brownian perturbations of size $\varepsilon$, and fast periodic sampling with period $\delta$; $0<\varepsilon, \delta \ll 1$. Thus, state samples (measurements) are taken every $\delta$ time units, and the instantaneous rate of change of the state depends on its current value as well as its most recent sample. We show that the resulting stochastic process indexed by $\varepsilon,\delta$, can be approximated, as $\varepsilon,\delta \searrow 0$, by an ordinary differential equation (ODE) with vector field obtained by replacing the most recent sample by the current value of the state. We next analyze the fluctuations of the stochastic process about the limiting ODE. Our main result asserts that, for the case when $\delta \searrow 0$ at the same rate as, or faster than, $\varepsilon \searrow 0$, the rescaled fluctuations can be approximated in a suitable strong (pathwise) sense by a limiting stochastic differential equation (SDE). This SDE varies depending on the exact rates at which $\varepsilon,\delta \searrow 0$. The key contribution here involves computing the effective drift term capturing the interplay between noise and sampling in the limiting SDE. The results essentially provide a first-order perturbation expansion, together with error estimates, for the stochastic process of interest. Connections with the performance analysis of feedback control systems with sampling are discussed and illustrated numerically through a simple example.