L2L2 norm performance index of synchronization and LQR control synthesis of complex networks

In this paper, the synchronizability problem of dynamical networks is addressed, where better synchronizability means faster network synchronization with lower vibration. The L2L2 norm of the error vector ee is taken as a performance index to measure this kind of synchronizability. For the equilibrium synchronization case, it is shown that there is a close relationship between the L2L2 norm of the error vector ee and the H2H2 norm of the transfer function GG of the linearized network about the equilibrium point. Consequently, the effect of the network coupling topology on the H2H2 norm of the transfer function GG is analyzed. Finally, an LQR controller is designed, according to the LQR   problem in modern control theory, which can drive the whole network to its equilibrium point and meanwhile minimize the L2L2 norm of the output of the linearized network.