Controllability-Gramian Submatrices for a Network Consensus Model

TL;DR

This paper analyzes principal submatrices of the controllability Gramian in network consensus models, revealing their properties and how they relate to network structure, with implications for control strategies.

Contribution

It characterizes properties of Gramian submatrices and their inverses, linking them to network graph cutsets and long-term controllability features.

Findings

Eigenvalues and eigenvectors of Gramian submatrices are characterized.

Majorizations relate Gramian properties to network cutsets.

Asymptotic structure of the Gramian is analyzed for long-term control.

Abstract

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.