Robustness of Leader-Follower Networked Dynamical Systems

TL;DR

This paper develops a graph-theoretic framework to analyze the robustness of leader-follower networked systems against disturbances and delays, providing bounds, characterizations, and optimization conditions.

Contribution

It introduces spectral bounds on the grounded Laplacian for robustness analysis and characterizes these metrics in random graph models, also exploring optimal leader placement.

Findings

Spectral bounds on grounded Laplacian eigenvalues quantify robustness.

Tight robustness characterizations in Erdős-Rényi and regular graphs.

Conditions for a leader to optimize both disturbance and delay robustness.

Abstract

We present a graph-theoretic approach to analyzing the robustness of leader-follower consensus dynamics to disturbances and time delays. Robustness to disturbances is captured via the system $\mathcal{H}_2$ and $\mathcal{H}_{\infty}$ 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 which quantify the impact of disturbances and time-delays on the leader-follower dynamics. We then provide tight characterizations of these robustness metrics in Erdos-Renyi random graphs and random 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.