Randomly perturbed switching dynamics of a DC/DC converter

In this paper, we study the effect of small Brownian noise on a switching dynamical system which models a first-order dc/dc buck converter. The state vector of this system comprises a continuous component whose dynamics switch, based on the on/off configuration of the circuit, between two ordinary differential equations (ode), and a discrete component which keeps track of the on/off configurations. Assuming that the parameters and initial conditions of the unperturbed system have been tuned to yield a stable periodic orbit, we study the stochastic dynamics of this system when the forcing input in the on state is subject to small white noise fluctuations of size ϵ, 0 < ϵ ≪ 1. For the ensuing stochastic system whose dynamics switch at random times between a small noise stochastic differential equation (sde) and an ode, we prove a functional law of large numbers which states that in the limit of vanishing noise, the stochastic system converges to the underlying deterministic one on time horizons of order O(1/ϵ ^ν), 0 ≤ ν < 2/3.