Block-Diagonal Solutions to Lyapunov Inequalities and Generalisations of Diagonal Dominance
For control theorists and systems engineers, this provides a scalable stability analysis tool for linear dynamical systems with block-structured drift matrices.
This paper extends scaled diagonal dominance to block-partitioned matrices, showing that such matrices admit block-diagonal solutions to Lyapunov inequalities, which can be computed with linear algebra and sometimes decoupled for improved scalability.
Diagonally dominant matrices have many applications in systems and control theory. Linear dynamical systems with scaled diagonally dominant drift matrices, which include stable positive systems, allow for scalable stability analysis. For example, it is known that Lyapunov inequalities for this class of systems admit diagonal solutions. In this paper, we present an extension of scaled diagonally dominance to block partitioned matrices. We show that our definition describes matrices admitting block-diagonal solutions to Lyapunov inequalities and that these solutions can be computed using linear algebraic tools. We also show how in some cases the Lyapunov inequalities can be decoupled into a set of lower dimensional linear matrix inequalities, thus leading to improved scalability. We conclude by illustrating some advantages and limitations of our results with numerical examples.