Adaptive practical synchronisation of Lagrangian networks with a directed graph via pinning control

This study investigates the practical synchronisation of networked Lagrangian systems with a directed graph by pinning adaptive control. Some simple yet general algebraic criteria are proposed based on practical stability theory on dynamical systems, and the proposed criteria ensure that all the robotic agents described by Lagrangian dynamics can achieve desired practical synchronisation. The main features of the present investigation include, an efficient pinning adaptive strategy is implemented by applying local linear feedback injections to a small fraction of agents, which means that the presented control strategy does not require the prerequisite knowledge of system models. Moreover, an allowable attraction region is explicitly given by Lagrangian dynamics parameter and the controlled network structure, and so it is very convenient to application in practice. Besides, this study addresses the fundamental problems in the pinning control of networked Lagrangian systems: what kind of agents and how many agents should be pinned? As a direct application of the theoretical results, practical synchronisation of eight two-link revolute manipulators is discussed in detail. Numerical experiments verify the effectiveness of the proposed control technique.