Conjugate Unscented Transform Based Methods for Uncertainty Quantification, Nonlinear Filtering, Optimal Feedback Control and Dynamic Sensing

The fusion of observational data with numerical simulation promises to provide greater understanding of physical phenomenon than either approach alone can achieve. While traditional methods use Kalman filtering paradigms to fuse model outputs with the observation data, the presence of nonlinearities in observation functions, boundary conditions and the dynamics of the flow render the statistical linearization and Gaussian statistical assumptions insufficient for accurate forecasting. Furthermore, sensor planning to make statistically significant observations using mobile sensors to carry out effective model data fusion is a significant challenge. The model data fusion and accurate forecasting necessitates an uncertainty propagation paradigm. The ensuing dynamic data acquisition problem naturally presents itself as a dynamic optimization problem, that has the mobility, power and sensor modality constraints of the sensor platforms embedded in the mathematical formalism.;This dissertation integrates fundamental studies in uncertainty quantification, information theory, model data fusion and optimal control theory to realize a scalable framework for dynamic data driven sensor management framework to achieve accurate forecasting of stochastic dynamic systems. The Conjugate Unscented Transformation (CUT) technique is used as an enabling tool to drive the model data fusion, information theoretic optimization and optimal feedback control realization processes.;The CUT approach can be considered an extension of the conventional unscented transformation method that satisfies additional higher order moment constraints. Rather than using tensor products as in Gauss quadrature, the CUT approach judiciously selects special structures to extract symmetric quadrature points constrained to lie on specially defined axes. These new sets of so-called sigma points are guaranteed to exactly evaluate expectation integrals involving polynomial functions with significantly fewer points. Hence, the CUT points are used to efficiently propagate the statistical moments of system state vector through the nonlinear flow of the dynamical system. The Principle of Maximum Entropy (PME) is then used to construct the probability density function of system state vector from propagated statistical moments of system state vector. Furthermore, the minimum variance framework is used in conjunction with the CUT approach to derive nonlinear filtering algorithms for state and parameter estimation.;The optimal sensing problem is posed as an optimal control framework, which formalizes the lifecycle of the information flow as a feedback control system. The model data fusion module in the feedback loop drives the feedback control system modeling the sensor management process. The plant in this case is the collection of mobile sensor platforms along with their sensor modalities. However, the objective of the control system is not to affect the sensing environment but to glean the information content by actuating appropriate sensing modalities. This abstract effect of the sensing is denoted by the mutual information of measurement and system state vector. By optimally routing the mobile sensors, better estimates of system states can be made while utilizing as little energy as possible. Given the fact that the mutual information is not additive over time, i.e., total cost associated with the mutual information metric over all time is not equal to the sum of information costs incurred at each time step, one needs to compute the mutual information metric for all combinations of possible locations in space and time for each mobile sensor and perform an exhaustive search. Several simplifications and engineering approximations motivated by physical scenarios are proposed to alleviate the computational complexity associated with the combinatorial exhaustive search of optimal sensing actions. In this dissertation, the applicability and utility of each approximation vis a vis the number of mobile sensors, number of time steps involved in the finite horizon, and the number of targets are evaluated systematically.;Finally, the development of a computationally efficient approach is discussed to generate optimal feedback control laws for infinite time problems by solving the corresponding Hamilton Jacobi Bellman (HJB) equation. The solution process consists of iteratively solving the linear Generalized Hamilton Jacobi Bellman (GHJB) equation starting with an admissible stable controller. The collocation methods are exploited to solve the GHJB equation in the specified domain of interest. The CUT approach is used to manage the curse of dimensionality associated with the growth of collocation points with increase in state dimension. Recent advances in sparse approximation are leveraged to automatically generate the appropriate polynomial basis function set for the collocation based approximation from an over-complete dictionary of basis functions. The solution process uses the basis function selection process to automatically identify a form for the feedback control law, which is frequently unknown.;The applicability and feasibility of these new ideas is demonstrated on many benchmark problems throughout the dissertation and some real world problems such as computing probability of collision between two resident space objects, tracking ground targets and resident space objects and optimal attitude control of rigid body. (Abstract shortened by UMI.).