State Estimation—Beyond Gaussian Filtering

5-2022

Dissertation

Ph.D.

Degree Program

Engineering and Applied Science - Electrical

Department

Electrical Engineering

Li, Xiao-Rong

Jilkov, Vesselin

Chen, Huimin

Li, Linxiong

Yu, Xiaochuan

Abstract

This dissertation considers the state estimation problems with symmetric Gaussian/asymmetric skew-Gaussian assumption under linear/nonlinear systems. It consists of three parts. The first part proposes a new recursive finite-dimensional exact density filter based on the linear skew-Gaussian system. The second part adopts a skew-symmetric representation (SSR) of distribution for nonlinear skew-Gaussian estimation. The third part gives an optimized Gauss-Hermite quadrature (GHQ) rule for numerical integration with respect to Gaussian integrals and applies it to nonlinear Gaussian filters.

We first develop a linear system model driven by skew-Gaussian processes and present the exact filter for the posterior density with fixed dimensional recursive representation, i.e., the skew-Gaussian filter (SGF). The SGF not only has an analytical recursion of a small dimension akin to the Kalman filter, but also possesses an efficiency comparable to the Kalman filter. The minimum mean-square error (MMSE) estimator based on our proposed skew-Gaussian filter is demonstrated via a simulation study.

Next, we propose a skew-symmetric presentation of the posterior density to handle the discrete-time filtering problem for a nonlinear system driven by non-Gaussian processes. The skew-symmetric representation of distributions, which has a product form of a symmetric pdf (known as the base pdf) times a perturbation function (known as the skewing function), is employed in this dissertation. Based on a first-order skew-symmetric representation of Gaussian distribution, we propose the first-order skew-Gaussian filter (FOSGF) and demonstrate it by applications to the radar tracking problem.

For the filtering problem where Gaussian integrals are adopted in the state update, we propose a new set of Gauss-Hermite quadrature rules using an optimized proposal density. The optimized GHQ rule, proposed in this dissertation, finds an optimized way to improve GHQ-based Gaussian integration when the integrand is not close to a polynomial by transforming it to one approximated by a polynomial. The solution is formulated as a nonlinear least-squares problem with linear constraints. Several numerical examples based on the optimized GHQ rule are studied and compared with the traditional methods.

Rights

The University of New Orleans and its agents retain the non-exclusive license to archive and make accessible this dissertation or thesis in whole or in part in all forms of media, now or hereafter known. The author retains all other ownership rights to the copyright of the thesis or dissertation.

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