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

In this note, 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.