Modeling and Analysis of Higher-Order Switched Linear Systems

开关线性动力学出现在电力电子，分布式电力系统，多控制器方案等；经典地，它们是由同阶的状态空间方程来建模的。然而，这样的表示可能是不必要的限制或复杂的：第一原理模型通常是高阶的，并且通常这些模式通常并不真正共享一个公共的全局状态空间。例如，考虑分布式电力系统，其中连接到源的负载具有不同复杂性的动态：状态空间的变化取决于实际连接的负载。使用全局状态变量对这样的系统建模会导致比必要的更复杂的模型，减少模块化，因此主要是为了满足优先定义的结构属性。我正在开发一个新的框架其中动力模态由高阶常系数线性微分方程系统描述。系统轨迹在由开关信号决定的时间间隔上满足这些方程。“胶合条件”，即涉及系统轨迹及其在切换时刻前后的导数的代数方程，规定了是否可以将分段限制连接起来形成可接受的轨迹。这个框架是基于多项式代数的，因此有利于使用计算机代数技术进行建模、分析和控制。它在用于系统建模的变量和方程的数量上是有效的，完全模块化，并与分层建模完美集成。我的研究的最终目的是促进建模，仿真和设计环境的创建基于一个健全的数学方法与有效的仿真技术相结合。在这个框架下，我获得了关于李亚普诺夫稳定性的令人鼓舞的结果，但仍有许多工作要做。我建议在这里研究3个方面：-脉冲现象的检测：粘合条件可能隐含地指定某些轨迹不能在切换时顺利连接。这可能意味着系统变量值的瞬时激增，这可能导致组件故障。我的目标是开发代数测试来确定这种情况何时可能发生。这些测试可用于从描述系统的方程中自动检测脉冲行为的存在，因此对于在计算机辅助设计环境中实现我的框架非常有用。-耗散开关系统：我想将我的框架扩展到开放系统，并建模与能量交换相关的动态和环境之间的相互作用。这是研究基于耗散思想的开关系统控制技术新框架的第一步。-微分变分不等式的多项式方法。在许多情况下（如电路、化学过程、遗传学、水力学等），切换依赖于代数不等式集的满足，而不是外部切换信号。这样的观点也可以有效地克服与通过切换建模转换相关的组合复杂性。我想研究如何在多项式设置中表示基于不等式的转移规则；多项式集解的适定性；稳定性的代数表征和Lyapunov泛函的计算。这是将多项式代数技术应用于动力系统描述的一个全新领域。上面描述的三个领域构成了我的方法的可靠性的具有挑战性的测试领域，并提供了在应用程序的重要方向上进一步发展它的机会。

Switched linear dynamics arise e.g. in power electronics, distributed power systems, multi-controller schemes, etc.; classically they are modeled by state space equations of the same order. However, such representations may be unnecessarily restrictive or complex: first-principles models are often of higher order, and often the modes often do not really share a common global state space. Consider e.g. distributed power systems, where the loads connected to the source have dynamics of different complexity: the state space changes depending on which loads are actually connected. Modelling such systems using a global state variable results in a more complex model than necessary, reduces modularity, and thus is done mainly to satisfy a priori-defined structural properties. I am developing a new framework where the dynamical modes are described by systems of higher-order linear constant coefficient differential equations. The system trajectories satisfy these equations on time-intervals determined by a switching signal. "Gluing conditions", i.e. algebraic equations involving the system trajectories and their derivatives before and after the switching instant, specify whether the piecewise restrictions can be concatenated to form an admissible trajectory. This framework is based on polynomial algebra and is thus conducive to the use of computer algebra techniques for modelling, analysis, and control. It is efficient in the number of variables and equations used to model a system, completely modular, and integrates perfectly with hierarchical modelling. The final aim of my research is contributing to the creation of a modelling, simulation and design environment based on a sound mathematical methodology aligned with efficient simulation techniques.In this framework I obtained encouraging results on Lyapunov stability, but much work remains to be done. I propose here to investigate 3 areas: - detection of impulsive phenomena:Gluing conditions may implicitly specify that certain trajectories cannot be concatenated smoothly at switching. This may imply instantaneous surges in the values of the system variables, which may lead to component breakdown. I aim at developing algebraic tests to ascertain when such situations may occur. These tests could be used to detect automatically the presence of impulsive behavior from the equations describing the systems, and thus would be useful for implementing my framework in a computer-aided design environment. - dissipative switched systems:I want to extend my framework to open systems and to modelling the interaction between dynamics and environment associated with energy exchange. This is a first step towards the investigation in this new framework of control techniques based on dissipation ideas for switched systems. - polynomial methods for differential variational inequalities. In many situations (e.g. in circuits, chemical processes, genetics, hydraulics, etc.) switching depends on the satisfaction of sets of algebraic inequalities, rather than on an external switching signal. Such a point of view can also efficiently overcome the combinatorial complexity associated with modelling transitions via switches. I want to investigate how to represent inequality-based transition rules in a polynomial setting; the well-posedness of solutions in a polynomial setting; the algebraic characterization of stability and the computation of Lyapunov functionals. This is a completely new area of application of polynomial algebraic techniques to the description of dynamical systems. The three areas described above constitute challenging test fields for the soundness of my approach, and offer the opportunity for developing it further in directions important for applications.