Discrete-time k-positive linear systems

TL;DR

This paper introduces and analyzes discrete-time linear systems that preserve vectors with limited sign variations, extending the concept of positive systems to a broader class with convergence properties.

Contribution

It defines k-positive systems that generalize positive systems, analyzes their properties using sign-regular matrices, and studies their behavior on parallelotopes.

Findings

Almost all solutions converge to vectors with limited sign variations.

Properties are characterized using sign-regular matrix theory.

System operation on k-dimensional parallelotopes is examined.

Abstract

Positive systems play an important role in systems and control theory and have found many applications in multi-agent systems, neural networks, systems biology, and more. Positive systems map the nonnegative orthant to itself (and also the nonpositive orthant to itself). In other words, they map the set of vectors with zero sign variations to itself. In this note, discrete-time linear systems that map the set of vectors with up to $k-1$ sign variations to itself are introduced. For the special case $k=1$ these reduce to discrete-time positive linear systems. Properties of these systems are analyzed using tools from the theory of sign-regular matrices. In particular, it is shown that almost every solution of such systems converges to the set of vectors with up to $k-1$ sign variations. Also, the operation of such systems on $k$-dimensional parallelotopes are studied.