LAUSR

We study the performance of leader–follower noisy consensus networks and, in particular, the relationship between this performance and the locations of the leader nodes. Two types of dynamics are considered: 1) noise-free leaders, in which leaders dictate the trajectory exactly and followers are subject to external disturbances and 2) noise-corrupted leaders, in which both leaders and followers are subject to external perturbations. We measure the performance of a network by its coherence, an $H_2$ norm that quantifies how closely the followers track the leaders’ trajectory. For both dynamics, there is a relationship between the coherence and resistance distances in an electrical network. Using this relationship, we derive closed-form expressions for coherence as a function of the locations of the leaders, and we give analytical solutions to the optimal leader selection problem for several classes of graphs.