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 syn-chronization 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.