Block-Diagonal Solutions to Lyapunov Inequalities and Generalisations of Diagonal Dominance

TL;DR

This paper extends the concept of diagonal dominance to block matrices, enabling scalable stability analysis of complex systems through block-diagonal Lyapunov solutions and decoupling techniques.

Contribution

It introduces a new definition of scaled block diagonal dominance, allowing for block-diagonal Lyapunov solutions and improved scalability in stability analysis.

Findings

Block-diagonal Lyapunov solutions can be computed using linear algebraic tools.

Decoupling Lyapunov inequalities into lower-dimensional LMIs enhances scalability.

Numerical examples demonstrate advantages and limitations of the proposed methods.

Abstract

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.