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In this paper, the H∞ theory is introduced to investigate the robustness and design of synchronization nonlinear coupled network. The H∞ synchronization performance is defined as the disturbance attenuation ability for a synchronized coupled network. To measure the H∞ synchronization performance of a nonlinear coupled network, we need to solve a Hamilton-Jacobi inequality (HJI), which is hard to treat directly. Hence, a Takagi-Sugeno fuzzy system is employed to approximate the nonlinear coupled network, so that the HJI can be replaced by a set of linear matrix inequalities. Furthermore, based on this H∞ synchronization performance, a robust nonlinear coupled network with a prescribed H∞ synchronization performance can be designed for a given network topology. In the robust H∞ synchronization network, our design task is to specify the minimum coupling strengths of the corresponding links in the network topology such that the coupled network cannot only synchronize but also attenuate the external disturbance below a prescribed level. Since the design of robust H∞ synchronization network leads to a set of bilinear matrix inequalities (BMIs), a two-step algorithm is proposed to solve the BMI-constrained optimization problem. The time-delay effect on the synchronization of coupled network is also discussed.