Distributed containment control of Euler–Lagrange systems over directed graphs via distributed continuous controllers

This study considers the distributed containment problem of Euler–Lagrange systems over directed graphs in the presence of multiple dynamic leaders having varying vectors of generalised coordinate derivatives and accelerations. A distributed continuous algorithm is proposed, which can guarantee the containment error is uniformly ultimately bounded and converges to a small adjustable residual set. The authors also design adaptive protocols for the case where the physical parameters are unknown. Then, as a special case, they design a continuous distributed protocol for Euler–Lagrange systems to solve the coordinated tracking consensus problem over directed graphs containing a directed spanning tree with the leader as the root node. Compared to the previous related works, the authors' main contribution is that the proposed distributed continuous algorithms, using the sliding mode control and the boundary layer technique, can avoid the undesirable chattering effect and meanwhile are applicable to general directed graphs.