Stochastic state estimation via incremental iterative sparse polynomial chaos based Bayesian-Gauss-Newton-Markov-Kalman filter

Abstract

In this paper is proposed a novel incremental iterative Gauss-Newton-Markov-Kalman filter method for state estimation of dynamic models given noisy measurements. The mathematical formulation of the proposed filter is based on the construction of an optimal nonlinear map between the observable and parameter (state) spaces via a convergent sequence of linear maps obtained by successive linearisation of the observation operator in a Gauss-Newton-like form. To allow automatic linearisation of the dynamical system in a sparse form, the smoother is designed in a hierarchical setting such that the forward map and its linearised counterpart are estimated in a Bayesian manner given a forecasted data set. To improve the algorithm convergence, the smoother is further reformulated in its incremental form in which the current and intermediate states are assimilated before the initial one, and the corresponding posterior estimates are taken as pseudo-measurements. As the latter ones are random variables, and not deterministic any more, the novel stochastic iterative filter is designed to take this into account. To correct the bias in the posterior outcome, the procedure is built in a predictor-corrector form in which the predictor phase is used to assimilate noisy measurement data, whereas the corrector phase is constructed to correct the mean bias. The resulting filter is further discretised via time-adapting sparse polynomial chaos expansions obtained either via modified Gram-Schmidt orthogonalisation or by a carefully chosen nonlinear mapping, both of which are estimated in a Bayesian manner by promoting the sparsity of the outcomes. The time adaptive basis with non-Gaussian arguments is further mapped to the polynomial chaos one by a suitably chosen isoprobabilistic transformation. Finally, the proposed method is tested on a chaotic nonlinear Lorenz 1984 system.