Weighted consensus for multiple Lagrangian systems under a directed graph

In this paper, we study the leaderless consensus problem for multiple Lagrange systems in the presence of parametric uncertainties under a directed graph. By introducing an integrate term in the auxiliary variable design, the final consensus equilibrium can be explicitly derived. We show that this equilibrium is dependent on three factors, namely, the interactive topology, the initial positions of the agents, and the control gains of the proposed control algorithm. For the case where the graph associated with the interactive topology is strongly connected, a Lyapunov based method is presented to show the consensus convergence, where the input-to-state stability is repeatedly used. We also give discussions on the consensus convergence when the graph contains a directed spanning tree. The leader-follower tracking problem and average consensus problem are also presented under special conditions.