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

TL;DR

This paper develops graph-theoretic bounds for the $$ performance measure of leader-following consensus dynamics on networks, analyzing the impact of network structure and leader placement on system robustness.

Contribution

It introduces tight graph-theoretic bounds for the $$ norm in leader-following systems and explores the effects of network modifications and leader positioning.

Findings

Bounds are tight and applicable to various network types.

Adding directed edges can improve or degrade performance.

Leader placement significantly influences system robustness.

Abstract

We study a graph-theoretic approach to the $\mathcal{H}_{\infty}$ performance of leader following consensus dynamics on directed and undirected graphs. We first provide graph-theoretic bounds on the system $\mathcal{H}_{\infty}$ norm of the leader following dynamics and show the tightness of the proposed bounds. Then, we discuss the relation between the system $\mathcal{H}_{\infty}$ 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 $\mathcal{H}_{\infty}$ norm. In the end, we apply these theoretical results to a reference velocity tracking problem in a platoon of connected vehicles and discuss the effect of the location of the leading vehicle on the overall $\mathcal{H}_{\infty}$ performance of the system.