模式矩阵对结构化扰动的秩弹性及其应用
Rank Resilience of Pattern Matrices against Structured Perturbations with Applications
AI总结：
本文研究了模式矩阵对结构化扰动的秩弹性，提出了判定矩形模式矩阵在给定扰动下失去满秩或任意指定秩的组合必要与充分条件，并将其应用于结构化系统的可控性等弹性分析。
摘要：
在结构化系统理论中，模式矩阵是指其元素要么固定为零，要么可以取任意数值的矩阵。模式矩阵的（一般）秩是指其几乎所有实现的秩。各种系统性质的弹性与相应模式矩阵的秩弹性密切相关。然而，现有文献主要通过关注改变模式矩阵零-非零结构的扰动来探讨后者，这对应于链路的增加/删除。在本文中，我们考虑模式矩阵对结构化扰动的秩弹性，该扰动可以任意改变一组指定元素的值，对应于链路权重的变化。我们说一个模式矩阵在结构上对扰动模式具有秩 $r$ 弹性，如果该模式矩阵的几乎所有实现在扰动模式的任意复值实现下都具有不小于 $r$ 的秩。我们建立了该概念中的一般性质。然后，我们给出了矩形模式矩阵在给定扰动模式下失去满秩的组合必要条件和充分条件。我们还将它们推广，得到了失去任意指定秩的一个充分条件和一个必要条件。最后，我们展示了我们的结果可以应用于结构化（描述符）系统各种性质的弹性分析，包括可控性和输入状态可观测性，以及刻画零结构固定模。
英文摘要：
In structured system theory, a pattern matrix is a matrix with entries either fixed to zero or free to take arbitrary numbers. The (generic) rank of a pattern matrix is the rank of almost all its realizations. The resilience of various system properties is closely tied to the rank resilience of the corresponding pattern matrices. Yet, existing literature predominantly explores the latter aspect by focusing on perturbations that change the zero-nonzero structure of pattern matrices, corresponding to link additions/deletions. In this paper, we consider the rank resilience of pattern matrices against structured perturbations that can arbitrarily alter the values of a prescribed set of entries, corresponding to link weight variations. We say a pattern matrix is structurally rank $r$ resilient against a perturbation pattern if almost all realizations of this pattern matrix have a rank not less than $r$ under arbitrary complex-valued realizations of the perturbation pattern. We establish a generic property in this concept. We then present combinatorial necessary and sufficient conditions for a rectangular pattern matrix to lose full rank under given perturbation patterns. We also generalize them to obtain a sufficient condition and a necessary one for losing an arbitrarily prescribed rank. We finally show our results can be applied to the resilience analysis of various properties of structured (descriptor) systems, including controllability and input-state observability, as well as characterizing zero structurally fixed modes.
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