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

近年来，人们对电力转换器的兴趣与日俱增，因为它们是可再生能源(太阳能电池板和风力涡轮机)和电动汽车的固有部件。控制功率转换器的方法主要有两种。第一种方法是脉宽调制，使用固定的开关频率，多年来一直是主流方法。这里，需要足够高的频率来保持良好的信号质量。第二种方法被称为磁滞控制，现在越来越流行，它使用了一种可以显著降低的可变(自适应)开关频率。这导致了更低的能量损失和更高的效率。然而，这些转换器的设计是具有挑战性的，因为在参数变化的情况下会出现复杂的分叉现象。从数学的角度来看，无论采用哪种控制技术的功率变流器都属于分段光滑系统理论的范畴。这一理论描述和预测了在工程、社会和生命科学的许多领域中出现的不同制度下运行的系统的行为。在过去的三十年里，人们发现并解释了许多由这类系统中发生的动力学突然变化引起的不寻常现象。然而，在PWS系统理论中有一个重大的空白。对于离散时间的模型，许多结果是在假设控制动力学的函数是分段光滑的但连续的(即对于连续的PWS映射)的情况下得到的，而具有不连续系统函数的模型(不连续的PWS映射)的动力学在现有理论中很大程度上是不可获得的。这里的主要困难在于，这类映射中的所有分支现象都必然是全局的，并且不具有范式。这在电力电子领域的应用中是一个重要的问题，因为通常采用不连续的PWS映射来给出具有滞环控制的功率变流器的模型。拟议项目的目标是为解决这一问题作出贡献。对于不连续映射，我们提出了一种新的方法来克服与缺失范式相关的困难。我们将研究涉及三种主要动力学类型的分叉现象：周期、混沌和与闭不变曲线有关的分叉现象。具体地说，我们将集中研究与周期动力学有关的复杂且严重受多稳定分叉结构影响的混沌吸引子的组织原理；最近才发现的不连续映射特有的混沌吸引子的边界碰撞分叉；以及迄今鲜有研究的不连续映射中闭不变曲线的边界碰撞分叉。在理论方面，我们将解释不连续映射中的分叉现象，这些现象仍然是现有的PWS系统理论所不能理解的。在实际应用方面，我们的结果将支持对工业应用具有重要意义的功率转换器的开发。

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.