Phase synchronization control of complex networks of Lagrangian systems on adaptive digraphs

This paper presents a formation control and synchronization method that utilizes adaptive network topologies for a class of complex dynamical networks comprised of a large number of highly-nonlinear Euler–Lagrange (EL) systems. A time-varying and switching network topology, constructed by the adaptive graph Laplacian matrix, relaxes the standard requirement of consensus stability, even permitting exponential synchronization on an unbalanced digraph or a weakly connected digraph that can sporadically lose connectivity. The time-varying graph Laplacian matrix is adapted by an adaptive control scheme based on relative positions and errors of synchronization and tracking. The adaptive graph Laplacian is integrated with a phase synchronization controller that synchronizes the relative motions of EL systems moving in elliptical orbits, thereby yielding a smaller synchronization error than an uncoupled tracking control law in the presence of bounded disturbances and modeling errors. An example of reconfiguring hundreds of spacecraft in Low Earth Orbit shows the effectiveness of the proposed phase synchronization controller for a large number of complex EL systems moving in periodic elliptical orbits.