State Estimation—Beyond Gaussian Filtering

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