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Multi-band modal consensus filters for parabolic partial differential equations

This paper presents a new formulation of consensus filters for parabolic PDEs. Using modal decompositions, the information a given distributed filter transmits to and receives from the remaining networked filters depends on the modal information needed. If a given distributed filter can completely reconstruct a specific mode or modes of the PDE, then it does not need any information from any of the networked filters. Similarly, if a distributed filter cannot adequately reconstruct a given mode, then it receives information from the filter that can completely reconstruct that specific mode. This then presents a connectivity which is based on the information needed. This consensus protocol which is dictated by the information a filter does not have but needs, is essentially a projection of information needed onto the unobservable space. This is demonstrated for a diffusion PDE in 1D and subsequently its abstraction is formulated for Riesz-spectral systems. Numerical studies demonstrate the proposed modal consensus filters.