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Spatial proportional-integral-derivative penalization of distributed consensus filters for spatially distributed processes

The main thrust of this work is on the penalization of the pairwise state estimates used to enforce consensus in spatially distributed filters. It is assumed that a spatially distributed process has a network of in-domain sensors with spatially distributed filters corresponding to each sensor in the network. To better improve the agreement of the distributed filters, the spatial gradient of the pairwise difference of state estimates is used as a means to penalize their disagreement. Additionally, a proportional penalization and an integral penalization for the pairwise differences are also examined in order to lay down the foundation for a spatial proportional-integral-derivative penalization of the spatially distributed filters. Addressing the partial connectivity issue, a condition that resembles the Lagrangian potential for infinite dimensional systems is given in terms of the inner product of the state errors and their pairwise differences. In a forward looking approach, the extension to a more general class of partial differential equations, written as evolution equations in an appropriate Hilbert space, are examined and the conditions regarding the network connectivity are expressed as conditions on the inner product of the consensus operator and the pairwise difference of the state estimation errors.