Internally Hankel $k$-positive systems

TL;DR

This paper introduces internally Hankel $k$-positive systems, extending the concept of internal positivity, and provides conditions for their verification and applications in bounding system response overshoot and undershoot.

Contribution

It defines internally Hankel $k$-positive systems as a new class, extending internal positivity, with tractable verification conditions and geometric criteria for realizations.

Findings

Derived conditions for $k>1$ cases via internal positivity of compound systems.

Established a new positive realization problem framework.

Applied results to bound overshoot and undershoot in LTI system responses.

Abstract

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.