变差递减线性定常系统
Variation diminishing linear time-invariant systems
摘要：
本文研究了$k$ -正线性时不变系统的变差递减性质，该系统将输入具有$k-1$符号变化映射到输出至多具有相同的变差.我们刻画了有限维系统的Toeplitz和Hankel算子的这一性质。我们的主要结果是，这些运营商有一个优势的逼近形式的串联或并联互连的$k$一阶正系统。这通过将LTI系统的$k$阳性表示为$k$复合LTI系统的外部阳性（即$1$阳性）来显示。我们的表征概括了公知的外部正系统（$k=1$）和完全正系统（$k=\infty$ ;也称为松弛系统）的性质。
英文摘要：
This paper studies the variation diminishing property of $k$-positive linear time-invariant (LTI) systems, which map inputs with $k-1$ sign changes to outputs with at most the same variation. We characterize this property for the Toeplitz and Hankel operators of finite-dimensional systems. Our main result is that these operators have a dominant approximation in the form of series or parallel interconnections of $k$ first order positive systems. This is shown by expressing the $k$-positivity of a LTI system as the external positivity (that is, $1$-positivity) of $k$ compound LTI systems. Our characterization generalizes well known properties of externally positive systems ($k=1$) and totally positive systems ($k=\infty$; also known as relaxation systems).
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