On the Controllability of Matrix-Weighted Networks

This letter examines the controllability of matrix-weighed networks from a graph-theoretic perspective. As distinct from the scalar-weighted networks, the rank of weight matrices introduce additional intricacies into characterizing the dimension of the controllable subspace for such networks. Specifically, we investigate how the definiteness of weight matrices, encoding a generalized characterization of inter-agent connectivity on matrix-weighted networks, influences the lower and upper bounds of the associated controllable subspaces. We show that such a lower bound is determined by the existence of a certain positive path in the distance partition of the network. By introducing the notion of matrix-valued almost equitable partitions, we show that the corresponding upper bound is determined by the product of the dimension of the weight matrices and the cardinality of the associated matrix-valued almost equitable partition. Furthermore, the structure of an uncontrollable input for such networks is examined.