Containment control for multiple euler-lagrange systems with parametric uncertainties in directed networks

In this paper, we study the distributed containment control problem for networked Lagrangian systems with multiple stationary or dynamic leaders in the presence of parametric uncertainties under a directed graph that characterizes the interaction among the leaders and the followers. When the leaders are stationary, a distributed adaptive control algorithm is proposed. We present a necessary and sufficient condition on the directed graph such that all followers converge to the stationary convex hull spanned by the stationary leaders asymptotically. As a byproduct, we show a necessary and sufficient condition on leaderless consensus for networked Lagrangian systems under a directed graph. When the leaders are dynamic, two cases are considered: i) The leaders have constant vectors of generalized coordinate derivatives; ii) The leaders have varying vectors of generalized coordinate derivatives. In the first case, we propose a distributed continuous estimator and a distributed adaptive control algorithm. In the second case, we propose a distributed adaptive control algorithm combined with distributed sliding-mode estimators. In both cases, a necessary and sufficient condition on the directed graph is presented such that all followers converge to the dynamic convex hull spanned by the dynamic leaders asymptotically.