Global bifurcation phenomena in discontinuous piecewise-smooth maps in theory and applications for power converter systems

点击翻译按钮获取中文摘要

In recent years, the interest in power converters increased because they are inherent parts of renewable energy sources (solar panels and wind turbines) and electric cars. There are two main methods for controlling power converters. The first method, pulse width modulation, uses a fixed switching frequency and has been the mainstream for many years. Here, a sufficiently high frequency is required to maintain good signal quality. The second method, called hysteresis control and becoming increasingly popular nowadays, uses a variable (adaptive) switching frequency that can be significantly lower. This results in lower energy losses and increased efficiency. However, designing these converters is challenging because of the complicated bifurcation phenomena that occur under parameter variation. From the mathematical point of view, power converters with either control technique belong to the scope of the piecewise-smooth (PWS) systems theory. This theory describes and predicts the behavior of systems operating in different regimes which appear in many areas of engineering, social and life sciences. Over the last three decades, many unusual phenomena caused by sudden changes in the dynamics occurring in such systems have been discovered and explained. However, there is a major gap in the PWS systems theory. For models in discrete time, many results are obtained under the assumption that the function governing the dynamics is piecewise smooth but continuous (i.e., for continuous PWS maps), while the dynamics of models with a discontinuous system function (discontinuous PWS maps) is largely inaccessible to the existing theory. The main difficulty here is that all bifurcation phenomena in such maps are necessarily global and possess no normal forms. This is a significant problem for applications in power electronics, since in general, models of power converters with hysteresis control are given by discontinuous PWS maps. The goal of the proposed project is to contribute to the solution of this problem. For discontinuous maps, we propose a novel approach how to overcome the difficulty related to missing normal forms. We will investigate bifurcation phenomena involving three major types of dynamics: periodic, chaotic and related to closed invariant curves. Specifically, we will focus our research on organizing principles of complicated and heavily affected by multistability bifurcation structures related to periodic dynamics; on border collision bifurcations of chaotic attractors specific for discontinuous maps and discovered only recently; and on border collision bifurcations of closed invariant curves in discontinuous maps barely investigated so far. On the theoretical side, we will explain bifurcation phenomena in discontinuous maps that are still inaccessible for the existing PWS systems theory. On the practical side, our results will support the development of power converters important for industrial applications.