A Deterministic Filter for non-Gaussian State Estimation

The usual mathematical method to represent uncertain quantities, for example the state of a dynamical system with uncertain initial conditions, are random variables (RVs). In many problems the space of elementary events Ω, on which the RVs are defined as functions of these events, is not concretely accessible, so that the usual idea of a function (e.g. given as a formula) loses much of its meaning. The representation of RVs is therefore often strikingly different from what is used for “normal” functions. With the help of RVs one can formulate Bayesian estimators for the uncertain quantity when additional information (usually noisy, incomplete measurements) becomes available. A common way to derive such an estimator is to use an instance of the projection theorem for Hilbert spaces. In this work we present a linear Bayesian estimation method which results from using a recently popular representation of an RV, the polynomial chaos expansion (PCE), also known as “white noise analysis”. The resulting method is completely deterministic, as well as computationally efficient. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)