Characterizing bipartite consensus on signed matrix-weighted networks via balancing set. (July 2022)

• Record Type: Journal Article
• Authors: Wang, Chongzhi; Pan, Lulu; Shao, Haibin; Li, Dewei; Xi, Yugeng
• Is Part Of: Automatica. Volume 141(2022)
• Publication Date: 2022-07
• Subjects: Matrix-weighted networks -- Bipartite consensus -- Balancing set -- Structural balance -- Null space; Automatic control -- Periodicals; Automation -- Periodicals; 629.805
• DOI: 10.1016/j.automatica.2022.110237
• Languages: English
• ISSNs: 0005-1098

Abstract

Abstract: In contrast with the scalar-weighted networks, where bipartite consensus can be achieved if and only if the underlying signed network is structurally balanced, the structural balance property is no longer a graph-theoretic equivalence to the bipartite consensus in the case of signed matrix-weighted networks. To re-establish the relationship between the network structure and the bipartite consensus solution, the non-trivial balancing set is introduced which is a set of edges whose sign negation can transform a structurally imbalanced network into a structurally balanced one and the weight matrices associated with edges in this set have a non-trivial intersection of null spaces. We show that necessary and/or sufficient conditions for bipartite consensus on matrix-weighted networks can be characterized by the uniqueness of the non-trivial balancing set, while the contribution of the associated non-trivial intersection of null spaces to the steady-state of the matrix-weighted network is examined. Moreover, for matrix-weighted networks with a positive–negative spanning tree, necessary and sufficient condition for bipartite consensus using the non-trivial balancing set is obtained. Simulation examples are provided to demonstrate the theoretical results.