Bipartite consensus of integrator multi-agent systems with measurement noise

The bipartite consensus problem for integrator multi-agent systems over signed fixed digraphs is investigated in the presence of measurement noise. A time-varying consensus gain is introduced and then a stochastic type protocol is proposed, whose performance is analysed using the state transition matrix of the closed-loop system. Necessary and sufficient conditions for ensuring a mean square bipartite consensus protocol are obtained in the presence of noise. Furthermore, in the absence of noise it is shown that these conditions are also necessary and sufficient for ensuring the bipartite consensus except for the quadratic integrability of the consensus gain. It is found that the signed digraph being structurally balanced and having a spanning tree are the weakest assumptions on connectivity for achieving bipartite consensus regardless of the measurement noise. In particular, if the signed digraph is structurally unbalanced, then under some mild conditions, the states of the closed-loop system converge to zero in mean square, regardless of the initial states.