A Convex Characterization of Robust Stability for Positive and Positively Dominated Linear Systems

TL;DR

This paper presents convex conditions for robust stability of positive linear systems, simplifying stability analysis by using static gain and LMIs, with applications to wireless power control algorithms.

Contribution

It introduces convex necessary and sufficient conditions for robust stability of positive systems, leveraging the equality of structured singular value bounds for nonnegative matrices.

Findings

Structured singular value equals its convex upper bound for nonnegative matrices

Derived LMI conditions depend only on the system's static gain

Applied approach to stability testing of wireless power control algorithms

Abstract

We provide convex necessary and sufficient conditions for the robust stability of linear positively dominated systems. In particular we show that the structured singular value is always equal to its convex upper bound for nonnegative matrices and we use this result to derive necessary and sufficient Linear Matrix Inequality (LMI) conditions for robust stability that involve only the system's static gain. We show how this approach can be applied to test the robust stability of the Foschini-Miljanic algorithm for power control in wireless networks in presence of uncertain interference.