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arXiv 1910.024692019-10-06math.OC

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

Aivar Sootla, Yang Zheng, Antonis Papachristodoulou

备注：: 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
DOI：: 10.1109/TAC.2019.2948194

英文摘要：

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.

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