Robustness of Leader–Follower Networked Dynamical Systems

We present a graph–theoretic approach to analyze the robustness of leader–follower consensus dynamics to disturbances and time delays. Robustness to disturbances is captured via the system <inline-formula><tex-math notation="LaTeX">$\mathcal{H}_2$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathcal{H}_{\infty }$</tex-math></inline-formula> norms, and robustness to time delay is defined as the maximum-allowable delay for the system to remain asymptotically stable. Our analysis is built on understanding certain spectral properties of the grounded Laplacian matrix that play a key role in such dynamics. Specifically, we give graph–theoretic bounds on the extreme eigenvalues of the grounded Laplacian matrix that quantify the impact of disturbances and time delays on the leader–follower dynamics. We then provide tight characterizations of these robustness metrics in Erdős–R <inline-formula><tex-math notation="LaTeX">$\acute{e}$</tex-math></inline-formula> nyi random graphs and random d-regular graphs. Finally, we view robustness to disturbances and time delay as network centrality metrics, and provide conditions under which a leader in a network optimizes each robustness objective. Furthermore, we propose a sufficient condition under which a single leader optimizes both robustness objectives simultaneously.