内部Hankel $k$ -正系统
Internally Hankel $k$-positive systems
摘要：
最近，外Hankel $k$ -正LTI系统和自治$k$ -正LTI系统被定义，并且它们的性质和应用开始在全正性和变差递减算子的框架下被研究.在这项工作中，这两个系统类被归入一类新的内部Hankel $k$ -正系统，我们定义为状态空间LTI系统的$k$ -正可控性和可观性运营商。我们证明了内部Hankel $k$ -正性是著名的内部正性性质（$k=1$）的自然扩张，并导出了证明第一$k$复合系统内部正性形式的情形$k> 1$的易处理的条件.由于这些条件定义了一个新的正实现问题，我们还讨论了几何条件时，最小内部Hankel $k$ -正实现存在。最后，我们使用我们的研究结果，建立一个新的框架，在一般的LTI系统的阶跃响应的过冲和下冲的数量的边界。
英文摘要：
The classes of externally Hankel $k$-positive LTI systems and autonomous $k$-positive systems have recently been defined, and their properties and applications began to be explored using the framework of total positivity and variation diminishing operators. In this work, these two system classes are subsumed under a new class of internally Hankel $k$-positive systems, which we define as state-space LTI systems with $k$-positive controllability and observability operators. We show that internal Hankel $k$-positivity is a natural extension of the celebrated property of internal positivity ($k=1$), and we derive tractable conditions for verifying the cases $k> 1$ in the form of internal positivity of the first $k$ compound systems. As these conditions define a new positive realization problem, we also discuss geometric conditions for when a minimal internally Hankel $k$-positive realization exists. Finally, we use our results to establish a new framework for bounding the number of over- and undershoots in the step response of general LTI systems.
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