Randomly perturbed switching dynamics of a dc/dc converter

Chetan D. Pahlajani

doi:10.3934/dcdsb.2017027

Article Contents

2017, Volume 22, Issue 2: 569-584. Doi: 10.3934/dcdsb.2017027

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Randomly perturbed switching dynamics of a dc/dc converter

Chetan D. Pahlajani

Discipline of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar 382355, India

Received: December 31, 2015

Revised: April 30, 2016

Published: December 2017

The author acknowledges research support from DST SERB Project No. EMR/2015/000904.

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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 $\mathscr{O}(1/\varepsilon ^ν)$, $0 ≤ ν < 2/3$.

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Mathematics Subject Classification: Primary:60F17.

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Figure 1. The evolution of the components $x(t)$, $y(t)$ of the full state vector $z(t)$ when starting with initial condition $(x_0,1)$ where $x_0 \in (0,x_{\mathsf {ref}})$. Note that $x(t)$ is continuous and piecewise-smooth; it is smooth between corners at the switching times $t_1<s_1<t_2<s_2<\dots$. The function $y(t) \in \{0,1\}$ is piecewise-constant and right-continuous, with jumps at the switching times. The dotted vertical lines at integer times denote the periodic clock pulse which triggers the OFF $\to$ ON transition

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