L2 norm performance index of synchronization and optimal control synthesis of complex networks

TL;DR

This paper investigates the synchronizability of dynamical networks using the L2 norm of the error vector as a performance index, analyzing the impact of network topology and designing optimal controllers for improved synchronization.

Contribution

It establishes a relationship between the L2 norm of synchronization error and the H2 norm of the transfer function, and proposes an LQR-based optimal control method for network synchronization.

Findings

L2 norm effectively measures network synchronizability.

Network topology influences the H2 norm of the transfer function.

Optimal controllers can enhance synchronization performance.

Abstract

In this paper, the synchronizability problem of dynamical networks is addressed, where better synchronizability means that the network synchronizes faster with lower-overshoot. The L2 norm of the error vector e 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 L2 norm of the error vector e and the H2 norm of the transfer function G of the linearized network about the equilibrium point. Consequently, the effect of the network coupling topology on the H2 norm of the transfer function G is analyzed. Finally, an optimal controller is designed, according to the so-called LQR problem in modern control theory, which can drive the whole network to its equilibrium point and meanwhile minimize the L2 norm of the output of the linearized network.