Consensus Problem Of Multi-agent Systems On Signed Networks

Consensus plays a central role in multi-agent coordination,where state deviations between neighboring agents are employed to construct the consensus protocol,leading to the agreement on the states of all the agents in the network.Consensus problem has been extensively examined in corporative control of multi-agent systems,modeling,and analysis of swarming behaviors,and opinion dynamics in social systems etc.In the last two decades,a common assumption on the consensus problem is that each agent in the network intends to cooperate with its neighboring agents,that is,only positively weighted interactions are allowed.However,competitive or antag-onistic interactions modeled by negatively weighted edges can coexist in the underlying network.Recently,several works have delved into the consensus problem on signed networks.This the-sis examines the consensus problem on networks with generalized weighted interactions and the following main contributions have been obtained.1.The negatively weighted edges in a signed network can often lead to an unstable Laplacian matrix,essentially due to the loss of matrix diagonal dominance.In fact,the difference between a Laplacian matrix and its corresponding signed Laplacian matrix only lies in their diagonal entries,referred to as the stability gap in this thesis.The main results in this part indicate that the stability gap is closely related to the structural balance of the underlying signed network.Necessary and sufficient conditions on the stability gap are obtained for both undirected and directed signed networks.2.The leader-following bipartite consensus problem over structurally balanced signed net-works is examined,and it is shown that the absolute value of the steady-state of bipartite consensus converges to the absolute value of the external input,and the clustering is determined by the sign of weight of external input.An event-based distributed control protocol is proposed for the multi-agent system under signed networks.Furthermore,the Zeno phenomenon of the closed-loop system is shown to be non-existed when employing the proposed event-based control protocol.3.The verification of structural balance for multilayer signed networks is examined.The sufficient and/or necessary conditions for the structural balance of multiplex signed net-works and Cartesian product graph are provided from a graph-theoretic perspective.A data-driven approach is further proposed for the distributed verification of the structural balance of the large-scale multiplex signed network and Cartesian product graph even if without the knowledge of network topology.4.The bipartite consensus problem for matrix-valued weighted networks is studied.The necessary/sufficient conditions for matrix-valued weighted networks to achieve bipartite consensus are provided from both the algebraic and graph-theoretic perspectives.In addition,we show that in the case where edges are weighted by either positive definite or negative definite matrices,the bipartite consensus is achieved if and only if the network is structurally balanced;on the other hand,asymptotic convergence to the origin is achieved if and only if the network is structurally imbalanced.