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In principle, general approaches to optimal nonlinear filtering can be described in a unified way from the recursive Bayesian approach. The central idea to this recur- sive Bayesian estimation is to determine the probability density function of the state vector of the nonlinear systems conditioned on the available measurements. However, the optimal exact solution to this Bayesian filtering problem is intractable since it requires an infinite dimensional process. For practical nonlinear filtering applications approximate solutions are required. Recently efficient and accurate approximate non- linear filters as alternatives to the extended Kalman filter are proposed for recursive nonlinear estimation of the states and parameters of dynamical systems. First, as sampling-based nonlinear filters, the sigma point filters, the unscented Kalman fil- ter and the divided difference filter are investigated. Secondly, a direct numerical nonlinear filter is introduced where the state conditional probability density is calcu- lated by applying fast numerical solvers to the Fokker-Planck equation in continuous- discrete system models. As simulation-based nonlinear filters, a universally effective algorithm, called the sequential Monte Carlo filter, that recursively utilizes a set of weighted samples to approximate the distributions of the state variables or param- eters, is investigated for dealing with nonlinear and non-Gaussian systems. Recentparticle filtering algorithms, which are developed independently in various engineer- ing fields, are investigated in a unified way. Furthermore, a new type of particle filter is proposed by integrating the divided difference filter with a particle filtering framework, leading to the divided difference particle filter. Sub-optimality of the ap- proximate nonlinear filters due to unknown system uncertainties can be compensated by using an adaptive filtering method that estimates both the state and system error statistics. For accurate identification of the time-varying parameters of dynamic sys- tems, new adaptive nonlinear filters that integrate the presented nonlinear filtering algorithms with noise estimation algorithms are derived. For qualitative and quantitative performance analysis among the proposed non- linear filters, systematic methods for measuring the nonlinearities, biasness, and op- timality of the proposed nonlinear filters are introduced. The proposed nonlinear optimal and sub-optimal filtering algorithms with applications to spacecraft orbit es- timation and autonomous navigation are investigated. Simulation results indicate that the advantages of the proposed nonlinear filters make these attractive alterna- tives to the extended Kalman filter.

<p>The object of this thesis is to provide a MATLAB framework for nonlinear filtering in general, and particle filtering in particular. This is done by using the object-oriented programming paradigm, resulting in truly expandable code. Three types of discrete and nonlinear state-space models are supported by default, as well as three filter algorithms: the Extended Kalman Filter and the SIS and SIR particle filters. Symbolic expressions are differentiated automatically, which allows for comfortable EKF filtering. A graphical user interface is also provided to make the process of filtering even more convenient. By implementing a specified interface, programming new classes for use within the framework is easy and guidelines for this are presented.</p>

The object of this thesis is to provide a MATLAB framework for nonlinear filtering in general, and particle filtering in particular. This is done by using the object-oriented programming paradigm, resulting in truly expandable code. Three types of discrete and nonlinear state-space models are supported by default, as well as three filter algorithms: the Extended Kalman Filter and the SIS and SIR particle filters. Symbolic expressions are differentiated automatically, which allows for comfortable EKF filtering. A graphical user interface is also provided to make the process of filtering even more convenient. By implementing a specified interface, programming new classes for use within the framework is easy and guidelines for this are presented.

Master of Science / Department of Electrical and Computer Engineering / Alexei Nikitin / Balasubramaniam Natarajan / It has been shown that the performance of communications systems can be severely limited by non-Gaussian and impulsive interference from a variety of sources. The non-Gaussian nature of this interference provides an opportunity for its effective mitigation by nonlinear filtering. In this thesis, we describe blind adaptive analog nonlinear filters, referred to as Adaptive Nonlinear Differential Limiters (ANDLs), that are characterized by several methodological distinctions from the existing digital solutions. When ANDLs are incorporated into a communications receiver, these methodological differences can translate into significant practical advantages, improving the receiver performance in the presence of non-Gaussian interference. A Nonlinear Differential Limiter (NDL) is obtained from a linear analog filter by introducing an appropriately chosen feedback-based nonlinearity into the response of the filter, and the degree of nonlinearity is controlled by a single parameter. ANDLs are similarly controlled by a single parameter, and are suitable for improving quality of non-stationary signals under time-varying noise conditions. ANDLs are designed to be fully compatible with existing linear devices and systems (i.e., ANDLs’ behavior is linear in the absence of impulsive interference), and to be used as an enhancement, or as a simple low-cost alternative, to state-of-the-art interference mitigation methods. We provide an introduction to the NDLs and illustrate their potential use for noise mitigation in communications systems. We also develop a digital implementation of an ANDL. This allows for rapid prototyping and performance analysis of various ANDL configurations and use cases.

Filtering or estimation is of great importance to virtually all disciplines of engineering and science that need inference, learning, information fusion, and knowledge discovery of dynamical systems. The filtering problem is to recursively determine the states and/or parameters of a dynamical system from a sequence of noisy measurements made on the system. The theory and practice of optimal estimation of linear Gaussian dynamical systems have been well established and successful, but optimal estimation of nonlinear and non-Gaussian dynamical systems is much more challenging and in general requires solving partial differential equations and intractable high-dimensional integrations. Hence, Gaussian approximation filters are widely used. In this dissertation, three innovative point-based Gaussian approximation filters including sparse Gauss-Hermite quadrature filter, sparse-grid quadrature filter, and the anisotropic sparse-grid quadrature filter are proposed. The relationship between the proposed filters and conventional Gaussian approximation filters is analyzed. In particular, it is proven that the popular unscented Kalman filter and the cubature Kalman filter are subset of the proposed sparse-grid filters. The sparse-grid filters are employed in three aerospace applications including spacecraft attitude estimation, orbit determination, and relative navigation. The results show that the proposed filters can achieve better estimation accuracy than the conventional Gaussian approximation filters, such as the extended Kalman filter, the cubature Kalman filter, the unscented Kalman filter, and is computationally more efficient than the Gauss-Hermite quadrature filter.

During strong earthquakes structural systems exhibit nonlinear behavior due to low-cycle fatigue, cracking, yielding and/or fracture of constituent elements. After a seismic event it is essential to assess the state of damage of structures and determine if they can safely resist aftershocks or future strong motions. The current practice in post-earthquake damage assessment relies mainly on visual inspections and local testing. These approaches are limited to the ability of inspectors to reach all potentially damaged locations, and are typically intended to detect damage near the outer surfaces of the structure leaving the possibility of hidden undetected damage. Some structures in seismic prone-regions are instrumented with an array of sensors that measure their acceleration at different locations. We operate under the premise that acceleration response measurements contain information about the state of damage of structures, and it is of interest to extract this information and use it in post-earthquake damage assessment and decision making strategies. The objective of this dissertation is to show that Bayesian filters can be successfully employed to estimate the nonlinear dynamic response of instrumented structural systems. The estimated response is subsequently used for structural damage diagnosis. Bayesian filters combine dynamic response measurements at limited spatial locations with a nonlinear dynamic model to estimate the response of stochastic dynamical systems at the model degrees-of-freedom. The application of five filters is investigated: the extended, unscented and ensemble Kalman filters, the particle filter and the model-based observer. The main contributions of this dissertation are summarized as follows: i) Development of a filtering-based mechanistic damage assessment framework; ii) Experimental validation of Bayesian filters in small and large-scale structures; iii) Uncertainty quantification and propagation of response and damage estimates computed using Bayesian filters.

In battery management systems, the main figure of merit is the battery's SOC, typically obtained from voltage and current measurements. Present estimation methods use simplified battery models that do not fully capture the electrical characteristics of the battery, which are useful for system design. This thesis studied SOC estimation for a lithium-ion battery using a nonlinear, electrical-circuit battery model that better describes the electrical characteristics of the battery. The extended Kalman filter, unscented Kalman filter, third-order and fifth-order cubature Kalman filter, and the statistically linearized filter were tested on their ability to estimate the SOC through numerical simulation. Their performances were compared based on their root-mean-square error over one hundred Monte Carlo runs as well as the time they took to complete those runs. The results show that the extended Kalman filter is a good choice for estimating the SOC of a lithium-ion battery.

We are interested in a nonlinear filtering problem motivated by an information-based approach for modelling the dynamic evolution of a portfolio of credit risky securities. We solve this problem by `change of measure method\\\' and show the existence of the density of the unnormalized conditional distribution which is a solution to the Zakai equation. Zakai equation is a linear SPDE which, in general, cannot be solved analytically. We apply Galerkin method to solve it numerically and show the convergence of Galerkin approximation in mean square. Lastly, we design an adaptive Galerkin filter with a basis of Hermite polynomials and we present numerical examples to illustrate the effectiveness of the proposed method. The work is closely related to the paper Frey and Schmidt (2010).