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This paper considers the problem of finite dimensional output feedback H∞ control for a class of nonlinear spatially distributed processes described by highly dissipative partial differential equations (PDEs), whose state is observed by a sensor network (SN) with a given topology. This class of systems typically involves a spatial differential operator whose eigenspectrum can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Motivated by this fact, the modal decomposition and singular perturbation techniques are initially applied to the PDE system to derive a finite-dimensional ordinary differential equation model, which accurately captures the dominant dynamics of the PDE system. Subsequently, based on the slow system and the SN topology, a set of finite-dimensional distributed consensus observers is constructed to estimate the state of the slow system. Then, a centralized control scheme, which only uses the available estimates from a specified group of SN nodes, is proposed for the PDE system. An H∞ control design is developed in terms of a bilinear matrix inequality (BMI), such that the closed-loop PDE system is exponentially stable and a prescribed level of disturbance attenuation is satisfied for the slow system. Furthermore, a suboptimal H∞ controller is also provided to make the attenuation level as small as possible, which can be obtained via a local optimization algorithm that treats the BMI as a double linear matrix inequality. Finally, the proposed method is applied to the control of the 1-D Kuramoto-Sivashinsky equation system.