Bipartite consensus on networks of agents with antagonistic interactions and measurement noises

This study considers the effects of measurement noises on bipartite consensus over undirected signed graphs. Each agent has to design a protocol based on imprecise information caused by noises. To reduce the detrimental effects of measurement noises, a time-varying consensus gain a(t) is introduced and then a time-varying stochastic-type protocol is presented to solve the bipartite consensus problem for the first time. By means of stochastic Lyapunov analysis and algebraic graph theory, the protocol is proved to be a mean-square bipartite consensus protocol. Particularly, in the noise-free case, not only sufficient, but also necessary conditions for ensuring a bipartite consensus are given. Conditions for the undirected signed graph to be structurally balanced and connected are shown to be the weakest assumptions on connectivity. Moreover, the structural unbalance case is studied in the presence of measurement noises. In this case, bipartite consensus value is proved to converge to zero in mean square for arbitrary initial conditions.