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arXiv · 1910.02469

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

Aivar Sootla·Yang Zheng·Antonis Papachristodoulou
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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.

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Aivar Sootla, Yang Zheng, Antonis Papachristodoulou. 2019-10-06. On the Existence of Block-Diagonal Solutions to Lyapunov and $\mathcal{H}_{\infty}$ Riccati Inequalities. https://doi.org/10.1109/tac.2019.2948194

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