Switched Differential Algebraic Equations
Switched Differential Algebraic Equations
复制标题
交换微分代数方程
DOI：
10.1007/978-1-4471-2885-4_6
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发表时间：
2012
影响因子：
15.9
通讯作者：
Stephan Trenn
中科院分区：
文献类型：
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作者：
Stephan Trenn
在本章中，带开关的电路被建模为开关微分代数方程（开关DAE），即每个模式都由形式为Ex′=Ax+Bu的DAE描述，其中E通常是奇异矩阵，u是输入。所产生的时间变化来自电路中存在的开关的动作，但也可以由电路中发生的故障引起。一般来说，开关或组件故障会引起某些状态变量的跳跃，并且通常会基于物理参数定义额外的跳跃映射。然而，事实证明，作为切换DAE的公式已经隐式地定义了这些跳跃，不必给出额外的跳跃映射。事实上，一个简单的方法来计算这些跳跃将提出的一致性投影仪。结果表明，一般的开关微分方程不仅可以有跳跃的解决方案，但也狄拉克脉冲和/或其衍生物。为了捕捉这种脉冲行为，分段光滑分布的空间被用作底层解空间。有了这个基本的解空间，就有可能证明开关DAE解的存在性和唯一性（包括开关引起的跳跃的唯一性）。借助一致性投影仪，我们给出了开关（或故障）是否能引起解的跳跃甚至狄拉克脉冲的条件。此外，开关DAE的稳定性进行了研究，再次一致性投影机发挥了重要作用。
In this chapter, an electrical circuit with switches is modelled as a switched differential algebraic equation (switched DAE), i.e. each mode is described by a DAE of the form Ex′=Ax+Bu where E is, in general, a singular matrix and u is the input. The resulting time-variance follows from the action of the switches present in the circuit, but can also be induced by faults occurring in the circuit. In general, switches or component faults induce jumps in certain state-variables, and it is common to define additional jump-maps based on physical arguments. However, it turns out that the formulation as a switched DAE already implicitly defines these jumps, no additional jump-map must be given. In fact, an easy way to calculate these jumps will be presented in terms of the consistency projectors. It turns out that general switched DAEs can have not only jumps in the solutions but also Dirac impulses and/or their derivatives. In order to capture this impulsive behaviour, the space of piecewise-smooth distributions is used as an underlying solution space. With this underlying solution space it is possible to show existence and uniqueness of solutions of switched DAEs (including the uniqueness of the jumps induced by the switches). With the help of the consistency projector a condition is formulated whether a switch (or fault) can induce jumps or even Dirac impulses in the solutions. Furthermore, stability of the switched DAE is studied; again the consistency projectors play an important role.