Stability verification for monotone systems using homotopy algorithms

TL;DR

This paper presents a numerical method using homotopy algorithms to verify stability regions in monotone systems, particularly for large-scale systems, by computing order intervals within the region of attraction.

Contribution

It introduces a novel numerical approach to find points indicating stability and verifies generalized small-gain conditions in monotone systems.

Findings

Method effectively computes regions of attraction.

Enables numerical verification of stability conditions.

Applicable to large-scale systems with monotone dynamics.

Abstract

A monotone self-mapping of the nonnegative orthant induces a monotone discrete-time dynamical system which evolves on the same orthant. If with respect to this system the origin is attractive then there must exists points whose image under the monotone map is strictly smaller than the original point, in the component-wise partial ordering. Here it is shown how such points can be found numerically, leading to a recipe to compute order intervals that are contained in the region of attraction and where the monotone map acts essentially as a contraction. An important application is the numerical verification of so-called generalized small-gain conditions that appear in the stability theory of large-scale systems.