Flocking of networked Euler-Lagrange systems with uncertain parameters and time-delays under directed graphs

In this paper, the flocking problem for multiple nonlinear Euler-Lagrange systems is investigated with both parametric uncertainties and time-delays under directed communication graphs. With the help of a new coordinate transformation method, an adaptive controller, which is dependent on position and velocity measurements, is proposed to guarantee flocking. By frequency domain analysis, input-output stability analysis and the final value theorem, the synchronization of positions/velocities of all the agents is guaranteed, and in the meantime all agents ultimately track a designated velocity. It is shown that to ensure flocking the sufficient condition is closely relevant to a constant gain in the proposed controller and the nonzero eigenvalues of the Laplacian matrix whose moduli are less than one. Finally, one numerical simulation is presented to demonstrate the efficiency of the theoretical results.