Uncontrollability of controlled consensus networks characterized by Faria vectors

In this paper we investigate the controllability of a controlled agreement problem where the interaction dynamics is given by a nearest-neighbor averaging. A single agent or a group of agents is selected to be the leader(s) and act(s) as control input to all other nodes. This opens the question, where the leader(s) should be placed such that arbitrary configurations of the nodes can be achieved. Based on the observation that a zero-entry in the Laplacian eigenvector at the position of a leader affects an uncontrollable subspace we study the characterization of the uncontrollable subspace by means of a generalized version of Faria vectors. Faria vectors are eigenvectors of a Laplacian which have two entries unequal to zero, +1, -1. This leads to a novel topological characterization of the uncontrollable subspace. The results are valid not only for the single leader but also the multi-leader case. Numerical investigations show the advantages of the proposed approach using Faria vectors to characterize the uncontrollable subspace under certain conditions.