Generic bifurcation structures in piecewise-smooth maps with extremely high numberof borders in theory and applications for power converter systems

具有极高边界数的分段平滑映射中的通用分叉结构在功率转换器系统的理论和应用中

• 批准号: 328158773
• 负责人: Professor Dr. Viktor Avrutin
• 金额: --
• 依托单位: Institut für Systemtheorie und Regelungstechnik (IST)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/328158773?language=en
• 关键词: Generic; bifurcation; structures; piecewise; smooth

The first steps in the development of theory of nonlinear dynamical systems have been made mainly for models with a smooth system function. However, various applications both in natural sciences and in engineering may operate in different regimes, which lead us to consider piecewise smooth (PWS) models. In such systems, the phase space of the model is subdivided into partitions where the dynamics is governed by different vector fields, separated from each other by switching manifolds. Interactions of invariant sets with these manifolds lead to numerous effects both interesting from a mathematical point of view and important for a desired behavior of the modeled system. At present, the theory of PWS systems provides a detailed description for many of such phenomena, but mainly for systems with a single switching manifold. This is a necessary but intermediate step and the goal of the proposed project is to outcome its limitations.Power electronics, dealing with switching circuits controlling the flow of electrical energy, is an established application of PWS systems theory. For a special class of power converters (DC/DC converters), the existing bifurcation theory of PWS systems is sufficient, helping to select parameter settings leading to desired mode of operation and to predict possible undesired dynamic effects. By contrast, for DC/AC converters, which are an inherent part of several applications related to renewable energy sources and electric cars, such kind of analysis is more difficult. Previously, we have shown that these systems lead to a novel class of PWS models characterized by an extremely high number of switching manifolds. The specific property of power converters leading to this class of models is that their dynamics is governed by two vastly different fixed frequencies.In the proposed continuation of the project, we will extend our research field, focusing on investigation of intriguing bifurcation phenomena occurring in models of DC/AC and AC/DC converters. We will describe the generic organizing principles of the bifurcation structures occurring in their multi-dimensional parameter spaces. Using the framework of maps with an extremely high number of switching manifolds, we will focus our research on the most challenging phenomena, such as border collision bifurcations, bubbling, and transformation of closed invariant curves with resonant and quasiperiodic dynamics. We will extend the modeling approach, making it applicable for systems with a variable high frequency. We will apply this extension to systems with hysteresis control and investigate the role of border collisions in formation of bifurcations structures occurring in DC/AC and AC/DC converters with this type of control. In this way, the proposed continuation of project will contribute to the progress of the theory of PWS dynamical systems and to development of power electronics for such applications as renewable energy sources and autonomous systems.

非线性动力系统理论发展的第一步主要是针对具有光滑系统函数的模型。然而，在自然科学和工程中的各种应用可能在不同的制度，这导致我们考虑分段光滑（PWS）模型。在这样的系统中，模型的相空间被细分为分区，其中动态由不同的矢量场控制，通过切换流形彼此分离。 不变集与这些流形的相互作用导致了许多效果，从数学的角度来看，这是有趣的，并且对于建模系统的期望行为是重要的。 目前，PWS系统的理论提供了许多这样的现象的详细描述，但主要是一个单一的开关流形的系统。这是一个必要的，但中间的步骤，提出的项目的目标是结果其局限性。电力电子，处理开关电路控制电能的流动，是PWS系统理论的一个既定的应用。 对于一类特殊的功率变换器（DC/DC变换器），PWS系统的现有分叉理论是足够的，有助于选择导致期望的操作模式的参数设置并预测可能的不期望的动态效应。相比之下，对于DC/AC转换器，其是与可再生能源和电动汽车相关的若干应用的固有部分，这种分析更困难。以前，我们已经表明，这些系统导致一类新的PWS模型，其特征在于一个非常高的开关流形。 导致此类模型的功率转换器的具体特性是，它们的动态特性由两个截然不同的固定频率决定。在该项目的拟议继续中，我们将扩展我们的研究领域，重点研究DC/AC模型中发生的有趣的分叉现象和AC/DC转换器。 我们将描述发生在其多维参数空间的分叉结构的一般组织原则。使用具有极高数量的切换流形的映射的框架，我们将把我们的研究集中在最具挑战性的现象，如边界碰撞分叉，冒泡，以及具有共振和准周期动力学的封闭不变曲线的变换。 我们将扩展建模方法，使其适用于系统的可变高频。我们将把这种扩展应用到具有滞后控制的系统中，并研究边界碰撞在DC/AC和AC/DC转换器中形成分叉结构时的作用。 通过这种方式，拟议的项目继续将有助于PWS动力系统理论的进步，并有助于可再生能源和自治系统等应用的电力电子技术的发展。