The Mean Square Linear Bidirectional Convergence Of A Second-order Integral Multi-self-host System

In multi-agent systems, not only cooperation but also antagonism exist between a-gents. For this kind of multi-agent systems, this paper studies the bipartite consensus problem of multi-agent systems in the presence of measurement noise. A distributed bipartite consensus protocol with time-varying consensus gain is employed to solve the bipartite consensus problem for the first time. By using tools of state transition matrix,and algebraic graph theory, the state of the closed-loop system is analysed and then neces-sary and sufficient conditions for the continuous-time and discrete-time double-integrator multi-agent systems to achieve bipartite consensus are given, respectively. It is shown that conditions for the signed digraph to be structurally balanced and having a spanning tree are the weakest topology assumptions on connectivity. In particular, if the signed digraph is also weight balanced, then the protocol is proved to be a mean square bipartite average consensus protocol. In this case, the positions of all agents converge in mean square to random variables which agree in modulus but differ in sign. Moreover, the mathematical expectation of the modulus is the weighted average of all agents' initial positions and velocities.