Nonlinear Estimation with Perron-Frobenius Operator and Karhunen-Loeve Expansion

Abstract : In this paper, a novel methodology for state estimation of stochastic dynamical systems is proposed. In this formulation, finite-term Karhunen-Loeve (KL) expansion is used to approximate the process noise, thus resulting in a non-autonomous deterministic approximation (with parametric uncertainty) of the original stochastic nonlinear system. It is proved that the solutions of the approximate dynamical system, asymptotically converge to the true solutions, in mean square sense. The evolution of uncertainty for the KL approximated system is predicted via Perron-Frobenius (PF) operator. Furthermore, a nonlinear estimation algorithm, using the proposed uncertainty propagation scheme, is developed. It is found that for finite dimensional linear and nonlinear filters, the evolving posterior densities obtained from the KLPF based estimator is closer, than particle filter, to the true posterior densities. The methodology is then applied to estimate states of a hypersonic reentry vehicle. It is observed that, the KLPF based estimator outperformed particle filter in terms of capturing localization of uncertainty through posterior densities, and reduction of uncertainty.
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IEEE Transactions on Aerospace and Electronic Systems, Institute of Electrical and Electronics Engineers, 2015, 51 (4), pp.3210 - 3225 <10.1109/TAES.2015.140591>
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Parikshit Dutta, Abhishek Halder, Raktim Bhattacharya. Nonlinear Estimation with Perron-Frobenius Operator and Karhunen-Loeve Expansion. IEEE Transactions on Aerospace and Electronic Systems, Institute of Electrical and Electronics Engineers, 2015, 51 (4), pp.3210 - 3225 <10.1109/TAES.2015.140591>. <hal-00773533>

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