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

功率变换器系统的理论和应用中不连续分段平滑映射中的全局分岔现象

• 批准号: 529252663
• 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
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/529252663?language=en
• 关键词: Global; bifurcation; phenomena; discontinuous; piecewise

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.

近年来，对功率转换器的兴趣增加，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。存在两种用于控制功率转换器的主要方法。第一种方法，脉宽调制，使用固定的开关频率，多年来一直是主流。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，称为滞后控制，现在越来越流行，使用可变（自适应）开关频率，可以显着降低。这导致更低的能量损失和更高的效率。然而，设计这些转换器是具有挑战性的，因为复杂的分岔现象下发生的参数变化。从数学的观点来看，具有任一控制技术的功率变换器都属于分段平滑（PWS）系统理论的范围。该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，许多不寻常的现象所造成的突然变化的动力学发生在这样的系统已被发现和解释。然而，在PWS系统理论中存在着一个重大的空白。对于离散时间模型，许多结果是在假设控制动态的函数是分段光滑但连续的（即，对于连续PWS映射），而具有不连续系统函数（不连续PWS映射）的模型的动力学在很大程度上是现有理论无法达到的。这里的主要困难是，所有的分歧现象，在这样的地图必然是全球性的，并拥有没有正规型。这是一个重要的问题，在电力电子应用中，因为在一般情况下，模型的功率转换器与滞后控制的不连续PWS地图。拟议项目的目标是促进解决这一问题。对于不连续映射，我们提出了一种新的方法，如何克服与丢失的正规形有关的困难。我们将研究涉及三种主要类型的动力学分岔现象：周期，混沌和相关的封闭不变曲线。具体来说，我们将集中研究与周期动力学相关的复杂和严重受多稳定性分岔结构影响的组织原理;不连续映射的混沌吸引子的边界碰撞分岔，最近才发现;以及迄今为止几乎没有研究过的不连续映射中闭合不变曲线的边界碰撞分岔。在理论方面，我们将解释在不连续映射的分歧现象，仍然无法访问现有的PWS系统理论。在实践方面，我们的研究结果将支持重要的工业应用电源转换器的发展。