Collective behavior of second-order multi-agent system in directed network

In this paper the collective behavior of multi-agent system in directed network, where the agents are with dynamical order two, is studied. The control protocol depends on two parameters, i.e. position-cooperative parameter w_x > 0 and velocity-cooperation parameter w_v > 0, as well as the Laplacian associated with communication network. We use the notion called inertia to describe the collective behavior of the system and study the conditions when system asymptotically remain in inertia state. We also use the notion consensus in this paper, but it has different meaning from that used in some existent literature. Based on a matrix decomposition of the Laplacian, we decompose a system into some basic independent systems and basic non-independent systems of the multi-agent system under directed networks. Furthermore, using matrix analysis approach the necessary and sufficient conditions for inertial state and position and/or velocity inertia-consensus, are given for the second-order systems. The collective behavior of the system is discussed in both cases that the system achieves consensus and not. We also provide some simulation results to show the validation of our results.