Block-diagonal solutions to Lyapunov inequalities and generalisations of diagonal dominance

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.

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

“…The main question is how to choose F , G, H, J so that analysis of the comparison system yields meaningful properties of system (2). We first present a new comparison matrix inspired by [9,19]. Definition 1.…”

“…Definition 1 is in the spirit of the generalisations of scaled diagonally dominant matrices discussed in [20,21,19] and is a direct generalisation of the definition in [9]. In order to streamline the presentation we discuss the connection to [9] in the Appendix.…”

“…Recently, the diagonal dominant approach was applied to block-partitioned matrices, which lead to conditions for the existence of block-diagonal solutions to Lyapunov inequalities [9]. In this paper, we generalise the existence results from [9] and derive conditions on the existence of blockdiagonal solutions to H ∞ Riccati inequalities. The main idea of the approach is to partition the state-space and compute a comparison system, which is positive and its dimension is equal to the number of clusters in the state partition.…”

“…In this paper, we focus on a generalisation of positive systems based on diagonally dominant matrices since it is known that for this class of systems separable Lyapunov functions exist [7] and can be computed using linear programming [8]. Recently, the diagonal dominant approach was applied to block-partitioned matrices, which lead to conditions for the existence of block-diagonal solutions to Lyapunov inequalities [9]. In this paper, we generalise the existence results from [9] and derive conditions on the existence of blockdiagonal solutions to H ∞ Riccati inequalities.…”

“…Even though we took inspiration from the linear algebra literature, in our previous work [9] we reconstructed and generalised some existing control theory results, in particular, the stability criteria in [10]. Therefore, our comparison system approach is tightly related to previous work on comparison systems reported in [11,12] and more recently in [13].…”