Randomly perturbed switching dynamics of a DC/DC converter

TL;DR

This paper analyzes how small random perturbations affect the switching behavior of a DC/DC buck converter modeled as a stochastic hybrid system, demonstrating convergence to deterministic dynamics over specific time scales.

Contribution

It provides a rigorous proof of a functional law of large numbers for a stochastic hybrid system modeling a buck converter under small noise perturbations.

Findings

Stochastic system converges to deterministic system as noise vanishes.

Convergence holds over time horizons of order 1/ε^ν for 0 ≤ ν < 2/3.

Small noise causes the switching dynamics to approximate the deterministic orbit.

Abstract

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 $\varepsilon$, $0<\varepsilon \ll 1$. 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 $\mathscr{O}(1/\varepsilon^\nu)$, $0 \le \nu < 2/3$.