  This paper considers the problem of finite dimensional output feedback
H-infinity control for a class of nonlinear spatially distributed processes
(SDPs) described by highly dissipative partial differential equations (PDEs),
whose state is observed by a sensor network (SN) with a given topology. A
highly dissipative PDE system typically involves a spatial differential
operator with eigenspectrum that 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 dynamics of the slow
modes of the PDE system. Subsequently, based on the slow system and the
topology of the SN, a set of finite dimensional distributed consensus observers
are 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-infinity control design
method is developed in terms of bilinear matrix inequality (BMI), such that the
original closed-loop PDE system is exponentially stable and a prescribed level
of disturbance attenuation is satisfied for the slow system. Furthermore, a
suboptimal H-infinity 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 double linear matrix inequality. Finally, the proposed
method is applied to the control of one dimensional Kuramoto-Sivashinsky
equation (KSE) system.
