Model-based Reconstruction of Distributed Phenomena using Discretized Representations of Partial Differential Equations

This article addresses the model-based reconstruction and prediction of distributed phenomena characterized by partial differential equations, which are monitored by sensor networks. The novelty of the proposed reconstruction method is the systematic approach and the integrated treatment of uncertainties, which occur in the physical model and arise naturally from noisy measurements. By this means it is possible not only to reconstruct the entire phenomenon, even at non-measurement points, but also to reconstruct the complete density function of the state characterizing the distributed phenomenon. In the first step, the partial differential equation, i.e.,

distributed

-parameter system, is spatially and temporally decomposed leading to a

finite

-dimensional state space form. In the next step, the state of the resulting

lumped

-parameter system, which provides an approximation of the solution of the underlying partial differential equation, is dynamically estimated under consideration of uncertainties. By using the estimation results, several additional tasks can be achieved by the sensor network, e.g. optimal sensor placement, optimal scheduling, model improvement, and system identification. The performance of the proposed model-based reconstruction method is demonstrated by means of simulations.