Containment control for networked unknown Lagrangian systems with multiple dynamic leaders under a directed graph

In this paper, we address the containment control problem for multiple Lagrangian systems with multiple dynamic leaders in the presence of unknown nonlinearities and external disturbances under a directed graph. A distributed adaptive control algorithm with an adaptive gain design using both relative position and velocity feedback is proposed based on the approximation capability of neural networks. We present a necessary and sufficient condition on the directed graph such that the containment error can be reduced as small as desired. As a byproduct, we show a necessary and sufficient condition on leaderless consensus for networked Lagrangian systems under a directed graph with unknown nonlinearities and external disturbances, in which the systems achieve consensus asymptotically. We then propose a distributed containment control algorithm without using neighbors' velocity information.