Nonlinear estimation with Perron-Frobenius operator and Karhunen-Loève expansion

In this paper, a novel methodology for state estimation of stochastic dynamical systems is proposed. In this formulation, finite-term Karhunen-Loève (KL) expansion is used to approximate the process noise, resulting in a nonautonomous 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 a mean-square sense. The evolution of uncertainty for the KL-approximated system is predicted via the 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 are closer than those obtained from the 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 the particle filter in terms of capturing localization of uncertainty through posterior densities and reduction of uncertainty.