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

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.

This is an extended technical report. The main results have been accepted for publication as a technical note in the IEEE Transactions on Automatic Control

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