Data driven modeling and state estimation of nonlinear dynamical systems
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Deriving a mathematical model which properly describes a given system is a fundamental step in solving many science and engineering problems. Such mathematical model can be derived using first principles such as Newton's laws based on understanding of physics and using data driven modeling methods based on the measured input-output data. While the physics based modeling approaches are limited to the modeling of relatively simple systems due to the requirement of knowing explicit knowledge of physical phenomenon of given systems, the data driven modeling methods can be extended to the modeling of real world systems. Due to the complexity involved in real world systems, empirical learning process based on the observed time series input-output data which form a basis of data driven modeling framework is often more suitable to describe such systems. To develop an effective data driven modeling method, the sparse approximation and Global-Local Orthogonal Mapping (GLOMAP) algorithm are used in this dissertation. The sparse approximation based GLOMAP provides a means of constructing independent local models from the input-output data and integrating the local models seamlessly as forcing a global continuity. The utility of the sparse approximation based GLOMAP algorithm is extended to real world applications of the unmanned aerial vehicle (UAV) based aeromagnetic sensing. In addition to the importance of finding proper system models, the proper characterization of model uncertainty involved in the system models is also an essential step. The system models derived from any chosen modeling process involve model uncertainty and nearly every real world sensor designed to measure state of systems has measurement uncertainty. In fact, there exists such model uncertainty and measurement uncertainty for virtually all real world systems being considered in science and engineering problems. In the presence of model uncertainty and measurement uncertainty, the state of system must be properly addressed at any particular time to make our estimates reflect closely reality. To accomplish this, a computationally-efficient method for uncertainty propagation and a high order filter are developed in this dissertation. These frameworks seek to provide a means of achieving the propagation and update of high order statistical moments for accurate and reliable state estimation in nonlinear dynamical systems. Benchmark problems and orbit estimation problem are considered to demonstrate the efficiency and utility of the developed methods.