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

Abstract

We consider a nonlinear differential equation under the combined influence of small state-dependent rownian perturbations of size ε, and fast periodic sampling with period δ; 0 < ε, δ 1. State samples (measurements) are taken every δ time units, and the instantaneous rate of change of the state depends on both the current value and most recent sample. For the resulting stochastic process indexed by ε, δ, we obtain asymptotic approximations for the mean behavior and fluctuations about the mean. The former is described by an ordinary differential equation, while the latter is governed by a stochastic differential equation (SDE).This SDE varies depending on the exact rates at which ε, δ ↘ 0. The key contribution involves computing the effective drift term capturing the interplay between noise and sampling in the limiting SDE. Connections with control systems with sampling are discussed and illustrated numerically through a simple example.