Observer design for linear switched differential-algebraic equations

动力系统包括一个潜在物理现象的数学模型，它有两个基本组成部分：将系统与其环境互连的外部信号和根据模型描述而演变的内部状态。外部信号通常可以进一步分解为输入和输出。与任何动态系统相关的基本问题之一是构造一个观测器，该观测器利用外部信号的可用信息来估计内部状态。这个项目的目的是开发动态系统的观测器，该系统被建模为切换微分-代数方程(DAE)。研究这类特殊系统的动机有两个：1)与常微分方程组(ODE)不同，DAE既包括微分方程组，也包括代数方程。实际上，每个系统的模型首先也包含代数方程，因此使用DAE(而不是简化的ODE)作为起点是很自然的。2)可能的结构变化(如电路中的开关或一般物理系统中的元件故障)可以在切换系统的框架内建模。作为拟议项目的一个应用，考虑(国家)电网，它是被建模为DAE的大型电路。然后，将使用观察者来监测通过传输线的能量流动，并可以防止超载。电网中的突然结构变化是常见的，必须加以考虑；例如，由于恶劣的天气条件导致电线跳闸，或者当整个风力发电场在太强的风中被关闭时，风力涡轮机的发电量突然下降。因此，拟议项目所获得的理论结果可能应用于改进电网的监测工具。

A dynamical system comprises a mathematical model of an underlying physical phenomenon.It has two basic components: external signals which interconnects the system with its environment and the internal state that evolves according to the model description. The external signals can usually further be split into inputs and outputs.One of the basic problem associated with any dynamical system is that of constructing an observer which uses the available information of the external signals to estimate the internal state. The purpose of this project is to develop observers for dynamical systems modeled as switched differential-algebraic equations (DAEs). The motivation to study this particular system class is twofold: 1) In contrast to ordinary differential equations (ODEs), DAEs include differential as well as algebraic equations. Practically every system's model also contains algebraic equations in the first place so it is natural to use DAEs (instead of the simplified ODEs) as a starting point. 2) Possible structural changes (like switches in electrical circuits or component faults in general physical system) can be modeled within the framework of switched systems. As an application of the proposed project, consider for example (national) electrical grids, which are large electrical circuits modeled as DAEs. An observer would then be used to monitor the energy flows through the transmission lines and could prevent overloading. Sudden structural changes in electrical grids are common and have to be taken into account; examples are: tripping of power lines due to harsh weather conditions, or a sudden drop in the energy production by wind turbines when whole wind parks are switched off in the presence of too strong winds. Hence a possible application of the theoretical results obtained by the proposed project could be improved monitoring tools for electrical grids.