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In this note, we describe sufficient conditions when block-diagonal solutions to Lyapunov and H<sub>&#x221E;</sub> 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 H<sub>&#x221E;</sub> 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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On the Existence of Block-Diagonal Solutions to Lyapunov and <inline-formula><tex-math notation="LaTeX">${\mathcal {H}_\infty }$</tex-math></inline-formula> Riccati Inequalities

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