Properties of Eventually Positive Linear Input-Output Systems

TL;DR

This paper studies systems whose trajectories become nonnegative after some finite time, extending the concept to input-output systems and providing methods to analyze their stability and energy properties.

Contribution

It introduces the concept of eventually positive systems, computes invariant cones and Lyapunov functions, and extends these ideas to input-output systems while preserving key properties.

Findings

Computed forward-invariant cones for eventually positive systems

Extended the notion to input-output systems maintaining classical properties

Provided linear programming methods for induced norm computation

Abstract

In this paper, we consider the systems with trajectories originating in the nonnegative orthant becoming nonnegative after some finite time transient. First we consider dynamical systems (i.e., fully observable systems with no inputs), which we call eventually positive. We compute forward-invariant cones and Lyapunov functions for these systems. We then extend the notion of eventually positive systems to the input-output system case. Our extension is performed in such a manner, that some valuable properties of classical internally positive input-output systems are preserved. For example, their induced norms can be computed using linear programming and the energy functions have nonnegative derivatives.