Consensus of cooperative–antagonistic multi-agent networks with asynchronous three-option decision mechanism

Instead of using the conventional discrete-time or continuous-time protocols with arbitrary values, this paper studies a cooperative–antagonistic multi-agent network with three-option decision mechanism. In particular, each agent over the network periodically makes decisions just from three available options (i.e., “1”, “0” and “−1”) at decision instants for the promotion of agreement with its cooperative neighbors or disagreement with its antagonistic neighbors. The asynchronous decisions of distinct agents are allowed herein. A digraph with signs is implemented to characterize the information delivery. Next, the consensus behaviors of the concerned cooperative–antagonistic multi-agent network are analyzed. In particular, on the basis of an intermediate modulus consensus result in terms of a strongly connected topology condition, it is first shown that structural balance is a sufficient and necessary condition for the bipartite consensus result. Besides this, a stabilizing result that all the agents reach an agreement at the origin is shown under the structurally unbalanced condition. On the other hand, for a relaxed signed digraph that contains a spanning tree, if only the root subgraph is strongly connected and structurally balanced, it is shown that only the root agents reach the bipartite consensus while the states of the non-root agents converge between the polarized values of the root agents. In addition, if the structural balance condition associated with the root subgraph cannot be guaranteed, the considered cooperative–antagonistic multi-agent network is shown stabilizing. Finally, simulations are carried out to verify the proposed theoretical results.