On the consensus and bipartite consensus in high-order multi-agent dynamical systems with antagonistic interactions

The aim of this paper is to address consensus and bipartite consensus for a group of homogeneous agents, under the assumption that their mutual interactions can be described by a weighted, signed, connected and structurally balanced communication graph. This amounts to assuming that the agents can be split into two antagonistic groups such that interactions between agents belonging to the same group are cooperative, and hence represented by nonnegative weights, while interactions between agents belonging to opposite groups are antagonistic, and hence represented by nonpositive weights. In this framework, bipartite consensus can always be reached under the stabilizability assumption on the state-space model describing the dynamics of each agent. On the other hand, (nontrivial) standard consensus may be achieved only under very demanding requirements, both on the Laplacian associated with the communication graph and on the agents' description. In particular, consensus may be achieved only if there is a sort of “equilibrium” between the two groups, both in terms of cardinality and in terms of the weights of the “conflicting interactions” amongst agents.