Balanced truncation of $k$-positive systems

TL;DR

This paper studies balanced truncation of discrete-time Hankel $k$-positive systems, revealing conditions under which the truncated system remains positive and relates to total positivity, bridging known properties of positive systems.

Contribution

It establishes that truncating a $k$-positive system to order $k$ or less yields an $ ext{infinity}$-positive system, connecting positivity properties with balanced truncation.

Findings

Truncated systems of order $k$ are Hankel totally positive.

Balanced truncation preserves positivity under certain conditions.

Provides a class of systems with guaranteed minimal internally positive truncations.

Abstract

This paper considers balanced truncation of discrete-time Hankel $k$-positive systems, characterized by Hankel matrices whose minors up to order $k$ are nonnegative. Our main result shows that if the truncated system has order $k$ or less, then it is Hankel totally positive ($\infty$-positive), meaning that it is a sum of first order lags. This result can be understood as a bridge between two known results: the property that the first-order truncation of a positive system is positive ($k=1$), and the property that balanced truncation preserves state-space symmetry. It provides a broad class of systems where balanced truncation is guaranteed to result in a minimal internally positive system.