Uncertainty Characterization Methods for Dynamic System Identification and State Estimation
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Numerous fields of science and engineering present the problem of uncertainty propagation through nonlinear dynamic systems with stochastic excitation and uncertain initial conditions. One may be interested in predicting the probability of collision of asteroid with Earth, diffusion of chemical plumes through air, control of movement and planning of actions of autonomous systems, the optimization of financial policies, active control of structural vibrations, the motion of particles under the influence of stochastic force fields, or simply the computation of the prediction step in the design of a Bayes filter. All these applications require the study of the relevant stochastic dynamic system. This dissertation work focus recent development of mathematical and algorithmic fundamentals of long-term propagation of state probability density function (pdf) through the nonlinear stochastic dynamical system, estimation of model parameter and characterizing the uncertainty associated with identified models from sensor data. The central idea is to replace evolution of initial conditions for a dynamical system by evolution of pdf for state variables. The use of Fokker-Planck-Kolmogorov equation (FPKE) to determine evolution of state pdf due to probabilistic uncertainty in initial or boundary conditions, model parameters and forcing function is discussed. In order to solve FPKE, a recently developed adaptive Gaussian mixture models (AGMM) is used in conjunction with an adaptive scheme to introduce and prune Gaussian kernels to accurately solve the FPKE. In contrast to the classical Gaussian mixture models, AGMM updates weights even during pure propagation by minimizing FPKE residual error. When properly formulated, the mixture problem can be solved efficiently and accurately using convex optimization solvers, even if the mixture model includes many terms. This methodology effectively decouples a large uncertainty propagation problem into many small problems. As a consequence, the solution algorithm can be parallelized on most High Performance Computing (HPC) systems. The FPKE error provides the metric to evaluate the accuracy of the AGMM and identify the Gaussian kernel which contribute most to the FPKE error. Two different approaches are developed to split one Gaussian kernel into multiple Gaussian kernels based upon FPKE error feedback. The first approach is based upon the conventional methods for splitting a single Gaussian kernel into multiple Gaussian kernels while the other approach identify the most nonlinear direction to split the Gaussian kernel into two Gaussian kernels. Furthermore, information theoretic measures are used to identify eligible Gaussian components for merging. Furthermore, this dissertation work provides semi-analytical tools to carry out a detailed probabilistic error analysis for the system model identified by the Eigenvalue Realization Algorithm (ERA). The System Identification is the term associated with the estimation and validation of mathematical models of physical phenomena from measured input-output data. The ERA and the Observer/Kalman filter IDentification (OKID) algorithms are among the most popular for dynamic system modeling, and have been successfully used in various system identification problems for structural analysis. In this dissertation work, the Random Matrix Theory (RMT) is exploited to derive the pdf of singular values of the system Hankel matrix. It is assumed that the columns of the Hankel matrix are corrupted by independent zero mean Gaussian random noise. The probabilistic analysis of singular values is further exploited to compute the bounds on system model error identified by the ERA algorithm. Finally, the pdf of singular values is used to compute the appropriate model order when input-output data is corrupted by Gaussian noise. Finally, this dissertation work exploits the recent advances in RMT to compute the analytical expressions for the covariance matrix associated with the maximum likelihood estimates corresponding to the Total Least Square (TLS) problem. This work leads to the formulation of a statistical version of the Optimal Linear Attitude Estimation (OLAE) algorithm while highlighting its similarities with the total least square algorithm. The OLAE algorithm reformulates the inherently nonlinear constrained attitude estimation problem as a rigorously linear, unconstrained problem to solve the single point optimal attitude estimation problem. Assuming the observations to be corrupted by zero mean Gaussian noise, the distribution of the linear model for the OLAE algorithm is represented by the nonsymmetric Wishart distribution. From the Wishart distribution, the joint density function of the eigenvalue and eigenvector of the linear model are constructed which are further used to derive the covariance matrix for the OLAE estimate. Throughout the dissertation, selected academic and real-world problems such as space domain awareness, system identification of flexible structures, attitude estimation are considered is to assess the reliability and limitations of the newly established methods and quantifying performance relative to state of art existing methods.