A Graph-Theoretic Approach to the ${\mathcal{H}}_{\infty}$ Performance of Leader–Follower Consensus on Directed Networks

We study a graph-theoretic approach to the H
_∞
performance of leader following consensus dynamics in the presence of external disturbances when the underlying graph is a directed network. We first provide graph-theoretic necessary and sufficient conditions for the consensus dynamics to have the system H
_∞
norm from external disturbances to the state of each agent to be lower than a certain value. Moreover, we discuss the tightness of the proposed conditions via examples. Then, we study the relation between the system H
_∞
norm for directed and undirected networks for specific classes of graphs, i.e., balanced digraphs and directed trees. Moreover, we investigate the effects of adding directed edges to a directed tree on the resulting system H
_∞
norm.