Balanced truncation of $k$-positive systems

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.

References in corpus (3)

• Variation diminishing linear time-invariant systems
• On second-order cone positive systems
• Is my system of ODEs $k$-cooperative?

Cited by in corpus (2)

• Variation diminishing linear time-invariant systems
• Diagonal Stability of Discrete-time $k$-Positive linear Systems with Applications to Nonlinear Systems