On Structured Realizability and Stabilizability of Linear Systems

We study the notion of structured realizability for linear systems defined over graphs. A stabilizable and detectable realization is structured if the state-space matrices inherit the sparsity pattern of the adjacency matrix of the associated graph. In this paper, we demonstrate that not every structured transfer matrix has a structured realization and we reveal the practical meaning of this fact. We also uncover a close connection between the structured realizability of a plant and whether the plant can be stabilized by a structured controller. In particular, we show that a structured stabilizing controller can only exist when the plant admits a structured realization. Finally, we give a parameterization of all structured stabilizing controllers and show that they always have structured realizations

Zero forcing number, constrained matchings and strong structural controllability

The zero forcing number is a graph invariant introduced to study the minimum rank of the graph. In 2008, Aazami proved the NP-hardness of computing the zero forcing number of a simple undirected graph. We complete this NP-hardness result by showing that the non-equivalent problem of computing the zero forcing number of a directed graph allowing loops is also NP-hard. The rest of the paper is devoted to the strong controllability of a networked system. This kind of controllability takes into account only the structure of the interconnection graph, but not the interconnection strengths along the edges. We provide a necessary and sufficient condition in terms of zero forcing sets for the strong controllability of a system whose underlying graph is a directed graph allowing loops. Moreover, we explain how our result differs from a recent related result discovered by Monshizadeh et al. Finally, we show how to solve the problem of finding efficiently a minimum-size input set for the strong controllability of a self-damped system with a tree-structure.Comment: Submitted as a journal paper in May 201