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Properties of Eventually Positive Linear Input-Output Systems

Aivar Sootla

TL;DR

This paper addresses extending positivity concepts to systems whose trajectories become nonnegative after a transient, and to their input-output extensions. It develops a control-theoretic framework based on invariant cones, spectral characterizations, and Lyapunov functions to analyze eventual positivity, and shows that energy functions and induced norms retain key positive-system properties, with linear-programming based certificates for norms. For IO systems, it provides necessary and sufficient conditions for internal eventual positivity and derives LP-based criteria for norm computations, supported by an illustrative example. The work broadens the toolbox for stability and model-reduction tasks in systems that are not strictly positive but eventually exhibit positivity, with potential extensions to nonlinear dynamics and Koopman-theoretic formulations.

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.

Properties of Eventually Positive Linear Input-Output Systems

TL;DR

This paper addresses extending positivity concepts to systems whose trajectories become nonnegative after a transient, and to their input-output extensions. It develops a control-theoretic framework based on invariant cones, spectral characterizations, and Lyapunov functions to analyze eventual positivity, and shows that energy functions and induced norms retain key positive-system properties, with linear-programming based certificates for norms. For IO systems, it provides necessary and sufficient conditions for internal eventual positivity and derives LP-based criteria for norm computations, supported by an illustrative example. The work broadens the toolbox for stability and model-reduction tasks in systems that are not strictly positive but eventually exhibit positivity, with potential extensions to nonlinear dynamics and Koopman-theoretic formulations.

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.

Paper Structure

This paper contains 12 sections, 11 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. (Eventually) Positive Systems
  5. Results
  6. Eventual Positivity and Positivity
  7. Lyapunov Functions
  8. Eventually Positive Input-Output Systems
  9. Energy Functions
  10. Induced Norms
  11. Illustrative Example
  12. Conclusion

Key Result

Proposition 1

The system $\dot x = A x$ is ${\mathcal{K}}$-positive if and only if $\lambda \in {\mathcal{K}}^\ast$, $x \in \partial {\mathcal{K}}$, $\lambda^T x = 0$ implies that $\lambda^T A x \ge 0$.

Figures (1)

  • Figure 1: Trajectories of the dynamical system $\dot x = A x$ with initial conditions $x_{0 1} = (1~~0~~0)^T$ (left panel) and $x_{0 2} = (0~~0~~1)^T$ (right panel).

Theorems & Definitions (16)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Proposition 2: Theorem 3 in rantzer2015ejc
  • Definition 3
  • Proposition 3
  • Remark 1
  • Theorem 1
  • Proposition 4
  • Theorem 2
  • ...and 6 more