Controllability-Gramian Submatrices for a Network Consensus Model
For researchers in network control and consensus, this provides theoretical insights into controllability Gramian submatrices, but the results are incremental and specific to the consensus model.
The paper characterizes properties of principal submatrices of the controllability Gramian for a network-consensus model, including eigenvalues, eigenvectors, diagonal entries, and sign patterns, using the doubly-nonnegative structure. It provides majorizations in terms of graph cutsets and analyzes the asymptotic structure, applying results to target control metrics.
Principal submatrices of the controllability Gramian and their inverses are examined, for a network-consensus model with inputs at a subset of network nodes. Specifically, several properties of the Gramian submatrices and their inverses -- including dominant eigenvalues and eigenvectors, diagonal entries, and sign patterns -- are characterized by exploiting the special doubly-nonnegative structure of the matrices. In addition, majorizations for these properties are obtained in terms of cutsets in the network's graph, based on the diffusive form of the model. The asymptotic (long time horizon) structure of the controllability Gramian is also analyzed. The results on the Gramian are used to study metrics for target control of the network-consensus model.