Edge-based Synchronization over Signed Digraphs with Multiple Leaders

Abstract: We address 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 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 uses the 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 compute the equilibrium to which all edge states converge. Numerical simulations validate our theoretical results.