Consensus problem of second-order multi-agent system in directed network: A matrix analysis approach

In this paper the consensus of multi-agent system in directed network, where the agent is described by a second-order dynamics, is studied. The control protocol depends on two parameters, i.e. position-cooperative parameter w_x >0 and velocity-cooperation parameter w_v >0, and the Laplacian associated with communication network. The definition of consensus in this paper is slightly different from that used in some existent literature. We define the notion called inertia-consensus. Based on a matrix decomposition of the Laplacian, we define the notions called basic independent system and basic non-independent system of a multi-agent system under directed networks. Furthermore, using matrix analysis approach the necessary and sufficient conditions for state inertia-consensus and/or velocity inertia-consensus, are given for the second-order systems. Also, the collective behavior of the system is discussed subject to the cases that system achieves consensus or does not. We also provide some simulation results to show the validation of our results.