Design of nonlinear coupling for efficient synchronization in networks of nonlinear systems

This paper proposes a design methodology of nonlinear coupling functions for guaranteed network synchronization. Compared to commonly used linear coupling, the proposed nonlinear coupling allows for a significant reduction of coupling energy cost and output noise sensitivity. This is achieved by activating the coupling only where necessary. Using the novel concept of strict incremental feedback passivity with a nonlinear gain, we estimate the magnitude and state-space location of potential incremental instabilities present in the systems ' intrinsic dynamics, which could drive systems apart in the absence of coupling. Then we introduce a nonlinear coupling design that provides a gain only in the part of the coupled systems ' state-space where the estimated incremental instabilities need to be suppressed. We provide constructive methods to design the nonlinear couplings for guaranteed synchronization over any connected, undirected, weighted network. By means of a numerical example, we demonstrate that our nonlinear coupling design, compared to linear couplings, results in significant performance improvements in terms of noise sensitivity and required coupling energy.