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

On Structured Realizability and Stabilizability of Linear Systems