Consensus of first-order multi-agent systems based on cooperation and competition

This paper studies the consensus of multiple single integrators with a leader, where the interaction graph is not necessarily nonnegative and is not necessarily structurally balanced. By matrix analysis, some sufficient conditions for consensus are established. It is shown that consensus can be reached if the interaction graph among followers and the weights for the edges connecting the leader and followers satisfy some inequalities. Some other results are also revealed. For example, if the interaction graph has some negative weights, then having a spanning tree is just a necessary condition but not a sufficient condition for consensus. Simulation examples are further provided to explain the effectiveness of the theoretical results.