Stability and Performance Analysis on Self-dual Cones

TL;DR

This paper investigates the stability and performance of cone-preserving dynamical systems, providing novel eigenvalue criteria, conditions for H-infinity norm bounds, and insights into the H-infinity norm's attainment at zero frequency.

Contribution

It introduces new eigenvalue criteria and conditions for stability and H-infinity norm bounds in cone-preserving systems, extending results even to positive systems.

Findings

Eigenvalue criterion for cone-preserving Sylvester equations

Conditions equivalent to H-infinity norm bounds

H-infinity norm is attained at zero frequency

Abstract

In this paper, we consider nonsymmetric solutions to certain Lyapunov and Riccati equations and inequalities with coefficient matrices corresponding to cone-preserving dynamical systems. Most results presented here appear to be novel even in the special case of positive systems. First, we provide a simple eigenvalue criterion for a Sylvester equation to admit a cone-preserving solution. For a single system preserving a self-dual cone, this reduces to stability. Further, we provide a set of conditions equivalent to testing a given H-infinity norm bound, as in the bounded real lemma. These feature the stability of a coefficient matrix similar to the Hamiltonian, a solution to two conic inequalities, and a stabilizing cone-preserving solution to a nonsymmetric Riccati equation. Finally, we show that the H-infinity norm is attained at zero frequency.