Discontinuous piecewise‐linear discrete‐time dynamics – maps with gaps in electronic systems

Abstract

PurposeMany important electronic systems are modelled by discrete-time equations with nonlinearities that are discontinuous and piecewise-linear, often arising as a result of quantization.Approximations based on linearization -the standard engineering response to nonlinearity -are often quite unhelpful in these systems, because of the form of the nonlinearity. Certain methods and results have been developed over a number of years for the analysis of discontinuous piecewise-linear discrete-time dynamics. The aim of this tutorial paper is to review that body of knowledge, and to show how it can be applied to representative electronic systems. ApproachThe paper uses an important electronic circuit -the sigma-delta modulator -as a central example, and considers the dynamical behaviour exhibited by this circuit and related circuits. FindingsThe circuits under investigation exhibit complex forms of behaviour that can be explained by the application of methods of nonlinear discrete-time dynamics. ValueThis paper is intended to provide a brief introduction to the body of research that exists into the behaviour of nonlinear discrete-time circuits and systems with discontinuous piecewise-linear nonlinearities.3

Predicting the unstable period-1 orbit (UPO-1) of DC–DC converters by extracting harmonic content

“…Power electronic converters are typically piecewise smooth dynamical systems (Tse and Bernardo, 2002; Aroudi et al ., 2010; Feely, 2011). Nonlinear dynamic behaviors of these systems have received much attention (Chen et al ., 2008; Xie et al ., 2013; Wang et al ., 2017), and it is commonly accepted that the performances of converters deteriorate when nonlinear phenomena such as bifurcations and chaos occur with the variation of their working conditions.…”