Edge-based Synchronization over Signed Digraphs with Multiple Leaders

TL;DR

This paper develops a new edge-based synchronization protocol for multi-agent systems with signed interactions and multiple leaders, analyzing stability and convergence to various synchronization states.

Contribution

It introduces a signed edge-Laplacian framework with novel spectral properties and characterizes the equilibrium behavior in complex signed networks.

Findings

Proves global exponential stability of synchronization errors.

Characterizes the equilibrium states for multi-leader signed networks.

Validates theoretical results with numerical simulations.

Abstract

This work addresses the edge-based synchronization problem in first-order multi-agent systems containing both cooperative and antagonistic interactions with one or multiple leader groups. The presence of multiple leaders and antagonistic interactions means that the multi-agent system typically does not achieve consensus, unless specific conditions (on the number of leaders and on the signed graph) are met, in which case the agents reach a trivial form of consensus. In general, we show that the multi-agent system exhibits a more general form of synchronization, including bipartite consensus and containment. Our approach proposes a signed edge-based agreement protocol for signed networks described by signed edge-Laplacian matrices. In particular, in this work, we present new spectral properties of signed edge-Laplacian matrices containing multiple zero eigenvalues and establish global exponential stability of the synchronization errors. Moreover, we explicitly compute the equilibrium to which all edge states converge, thereby characterizing the resulting synchronization behavior. Numerical simulations validate our theoretical results.