Bipartite Linear χ ‐Consensus of Double‐Integrator Multi‐Agent Systems With Measurement Noise

The bipartite consensus problem is investigated for double‐integrator multi‐agent systems in the presence of measurement noise. A distributed protocol with time‐varying consensus gain is proposed. By using tools of state transition matrix and algebraic graph theory, necessary and sufficient conditions for the designed protocol to be a mean square bipartite linear χ ‐consensus protocol are given. It is shown that the signed digraph being structurally balanced and having a spanning tree are not only sufficient, but also necessary for bipartite consensus. Furthermore, the protocol is proved to be a mean square bipartite average consensus protocol if the signed graph is weight balanced.