Group Consensus of Linear Multi-agent Systems under Nonnegative Directed Graphs

TL;DR

This paper establishes a new necessary and sufficient condition for group consensus in linear multi-agent systems with nonnegative directed graphs, linking graph structure to consensus achievement and characterizing the resulting cluster states.

Contribution

It introduces a novel graph-theoretic condition for group consensus, connecting spanning forests and cluster spanning trees, and provides bounds on coupling strength for multi-cluster formation.

Findings

Derived a new necessary and sufficient condition for group consensus.

Established the equivalence of the condition to cluster spanning trees.

Characterized the pattern of multiple consensus states for large coupling strengths.

Abstract

Group consensus implies reaching multiple groups where agents belonging to the same cluster reach state consensus. This paper focuses on linear multi-agent systems under nonnegative directed graphs. A new necessary and sufficient condition for ensuring group consensus is derived, which requires the spanning forest of the underlying directed graph and that of its quotient graph induced with respect to a clustering partition to contain equal minimum number of directed trees. This condition is further shown to be equivalent to containing cluster spanning trees, a commonly used topology for the underlying graph in the literature. Under a designed controller gain, lower bound of the overall coupling strength for achieving group consensus is specified. Moreover, the pattern of the multiple consensus states formed by all clusters is characterized when the overall coupling strength is large enough.