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We study a graph-theoretic approach to the $\mathcal{H}_{\infty}$ performanceof leader following consensus dynamics on directed and undirected graphs. Wefirst provide graph-theoretic bounds on the system $\mathcal{H}_{\infty}$ normof the leader following dynamics and show the tightness of the proposed bounds.Then, we discuss the relation between the system $\mathcal{H}_{\infty}$ normfor directed and undirected networks for specific classes of graphs, i.e.,balanced digraphs and directed trees. Moreover, we investigate the effects ofadding directed edges to a directed tree on the resulting system$\mathcal{H}_{\infty}$ norm. In the end, we apply these theoretical results toa reference velocity tracking problem in a platoon of connected vehicles anddiscuss the effect of the location of the leading vehicle on the overall$\mathcal{H}_{\infty}$ performance of the system.

A Graph-Theoretic Approach to the $\mathcal{H}_{\infty}$ Performance of Dynamical Systems on Directed and Undirected Networks