On the Existence of Block-Diagonal Solutions to Lyapunov and $\mathcal{H}_{\infty}$ Riccati Inequalities

TL;DR

This paper establishes conditions for the existence of block-diagonal solutions to Lyapunov and $\\mathcal{H}_{\infty}$ Riccati inequalities using comparison systems and linear programming, with practical numerical examples.

Contribution

It introduces a new class of comparison systems based on $\\mathcal{H}_{\infty}$ norms to determine block-diagonal solutions, providing a constructive and computational framework.

Findings

Stability of the comparison system guarantees block-diagonal solutions.

The approach reduces to linear algebra and linear programming.

Numerical examples validate the theoretical conditions.

Abstract

In this paper, we describe sufficient conditions when block-diagonal solutions to Lyapunov and $\mathcal{H}_{\infty}$ Riccati inequalities exist. In order to derive our results, we define a new type of comparison systems, which are positive and are computed using the state-space matrices of the original (possibly nonpositive) systems. Computing the comparison system involves only the calculation of $\mathcal{H}_{\infty}$ norms of its subsystems. We show that the stability of this comparison system implies the existence of block-diagonal solutions to Lyapunov and Riccati inequalities. Furthermore, our proof is constructive and the overall framework allows the computation of block-diagonal solutions to these matrix inequalities with linear algebra and linear programming. Numerical examples illustrate our theoretical results.