[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Nonlinear estimation and filtering have been intensively studied for decades since it has been widely used in engineering and science such as navigation, radar signal processing and target tracking systems. Because the posterior density function is not a Gaussian distribution, then the optimal solution is intractable. The nonlinear/non-Gaussian estimation problem is more challenging than the linear/Gaussian case, which has an optimal closed form solution, i.e. the celebrated Kalman filter. Many nonlinear filters including the extended Kalman filter, the unscented Kalman filter and the Gaussian-approximation filters, have been proposed to address nonlinear/non-Gaussian estimation problems in the past decades. Although the estimate yield by Gaussian-approximation filters such as cubature Kalman filters and Gaussian-Hermite quadrature filters is satisfied in many applications, there are two obvious drawbacks embedded in the use of Gaussian filters. On the one hand, with the increase of the quadrature points, much computational effort is devoted to approximate Gaussian integrals, which is not worthy sometimes. On the other hand, by the use of the update rule, the estimate constrains to be a linear function of the observation. In this dissertation, we aim to address this two shortcoming associated with the conventional nonlinear filters. We propose two nonlinear filters in the dissertation. Based on an adaptive strategy, the first one tries to reduce the computation cost during filtering without sacrificing much accuracy, because when the system is close to be linear, the lower level Gaussian quadrature filter is sufficient to provide accurate estimate. The adaptive strategy is used to evaluate the nonlinearity of the system at current time first and then utilize different quadrature rule for filtering. Another filter aims to modify the conventional update rule, i.e. the linear minimum mean square error (LMMSE) rule, to involve a nonlinear transformation of the observation, which is proven to be an efficient way to exploit more information from the original observation. According to the orthogonal property, we propose a novel approach to construct the nonlinear transformation systematically. The augmented nonlinear filter outperforms Gaussian filters and other conventional augmented filters in terms of the root mean square error and onsistency. Furthermore, we also extend the work to the more general case. The higher order moments can be utilized to construct the nonlinear transformation and in turn, the measurement space can be expand efficiently. Without the Gaussian assumption, the construction of the nonlinear transformation only demand the existence of a finite number of moments. Finally, the simulation results validate and demonstrate the superiority of the adaptive and augmented nonlinear filters.
Gaussian approximation of general non-parametric posterior distributions