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 statedependent Brownian perturbations of size ε, and fast periodic sampling with period δ; 0 < ε, δ 1. Thus, state samples (measurements) are taken every δ 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 ε, δ, can be approximated, as ε, δ 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 δ 0 at the same rate as, or faster than, ε 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 ε, δ 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.

Asymptotic analysis of dynamical systems driven by Poisson random measures with periodic sampling

“…The behavior of dynamical systems perturbed by a small white noise has been extensively studied in [FW12], and references therein. Getting motivation from the asymptotic analysis presented in chapter 2 of [FW12], we have explored this asymptotic behavior for general nonlinear systems in the presence of fast periodic sampling and small state-dependent white noise, see, [DP22b];…”