Bibliography: leaves 127-128.

Includes summary. Bibliography: leaves 108-112.

Bibliography: leaves 146-152 / vi, 152 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1987

Response behavior of a nonlinear structural system subject to environmental loadings is investigated in this study. The system contains a nonlinear restoring force due to large geometric displacement. The external excitation is modeled as a narrowband stochastic process possessing dynamic characteristics of typical environmental loadings. A semi-analytical method is developed to predict the stochastic nonlinear response behavior under narrowband excitations in both the primary and the subharmonic resonance regions. Preservation of deterministic response characteristics under the narrowband random field is assumed. The stochastic system response induced by variations in the narrowband excitations is considered as a sequence of successive transient states. Due to the system nonlinearity, under a combination of excitation conditions, several response attraction domains may co-exist. Presence of co-existence of attraction domains and variations in the excitation amplitude often induce complex response inter-domain transitions. The response characteristics are found to be attraction domain dependent. Among different response attraction domains, their corresponding response amplitude domains overlap. In addition, within an individual attraction domain, response amplitude domains corresponding to different excitation amplitudes also overlap. Overlapping of response amplitude domains and the time-dependent variations in the excitation parameters induce response intra-domain transitions. Stationary Markovian assumption is employed to characterize the stochastic behavior of the response amplitude process and the excitation parameter processes. Based on the stochastic excitation properties and the deterministic response characteristics, governing equations of the response amplitude probability inter- and intra-domain transitions are formulated. Numerical techniques and an iteration procedure are employed to evaluate the stationary response amplitude probability distribution. The proposed semi-analytical method is validated by extensive numerical simulations. The capability of the method is demonstrated by good agreements among the predicted response amplitude distributions and the simulation results in both the primary and the subharmonic resonance regions. Variations in the stochastic response behavior under varying excitation bandwidth and variance are also predicted accurately. Repeated occurrences of various subharmonic responses observed in the numerical simulations are taken into account in the proposed analysis. Comparisons of prediction results with those obtained by existing analytical methods and simulation histograms show that a significant improvement in the prediction accuracy is achieved. / Graduation date: 1999

Stochastic analysis procedures have been recently applied to analyze nonlinear dynamical systems. In this study, nonlinear responses, stochastic and/or chaotic, are examined and interpreted from a probabilistic perspective. A multi-point-moored ocean structural system under regular and irregular wave excitations is analytically examined via a generalized stochastic Melnikov function and Markov process approach. Time domain simulations and associated experimental observations are employed to assist in the interpretation of the analytical predictions. Taking into account the presence of random noise, a generalized stochastic Melnikov function associated with the corresponding averaged system, where a homoclinic connection exists near the primary resonance, is derived. The effects of random noise on the boundary of regions of possible existence of chaotic response is demonstrated via a mean-squared Melnikov criterion. The random wave field is approximated as random perturbations on regular and nearly regular (with very narrow-band spectrum) waves by adding a white noise component, or using a filtered white noise process to fit the JONSWAP spectrum. A Markov process approach is then applied explicitly to analyze the response. The evolution of the probability density function (PDF) of nonlinear stochastic response under the Markov process approach is characterized by a deterministic partial differential equation called the Fokker-Planck equation, which in this study is solved by a path integral solution procedure. Numerical evaluation of the path integral solution is based on path sum, and the short-time propagator is discretized accordingly. Short-time propagation is performed by using a fourth order Runge-Kutta scheme to calculate the most probable (i.e. mean) position in the phase space and to establish the fact that discrete contributions to the random response are locally Gaussian. Transient and steady-state PDF's can be obtained by repeat application of the short-time propagation. Based on depictions of the joint probability density functions and time domain simulations, it is observed that the presence of random noise may expedite the occurrence of "noisy" chaotic response. The noise intensity governs the transition among various types of stochastic nonlinear responses and the relative strengths of coexisting response attractors. Experimental observations confirm the general behavior depicted by the analytical predictions. / Graduation date: 1995

Thesis (Ph. D.)--Ohio State University, 1983. / Includes vita. Includes bibliographical references (leaves 80-81). Available online via OhioLINK's ETD Center.

In this thesis we consider estimation of the tail index for heavy tailed stochastic volatility models with long memory. We prove a central limit theorem for a Hill estimator. In particular, it is shown that neither the rate of convergence nor the asymptotic variance is affected by long memory. The theoretical findings are verified by simulation studies.

The design of a stochastic optimal controller using state feedback and output feedback is developed for unified, decentralized, singularly perturbed systems with Gaussian noise. To filter out the external noises contained in the system signals, a unified optimal observer (Kalman filter) is used for the decentralized, singularly perturbed system with a reduced-order model. The reduced-order stabilizing observer is also derived by the unified Riccati equation approach. Rationalization of the decentralized, singularly perturbed system with time delays is presented in the frequency domain by using the delta operator approach. It is shown that the discrete-time system is realized into the discrete-time state-space model. The stability robustness of a unified decentralized singularly perturbed stochastic system is investigated by exploring stability bounds under system uncertainties. A new unified stochastic bound is derived for both "unstructured" and "structured" time-varying independent perturbations. / Thesis (Ph.D.)--Wichita State University, College of Engineering. / Includes bibliographic references (leaves 128-132). / "May 2006." / Includes bibliographic references (leaves 128-132)

In this thesis we consider estimation of the tail index for heavy tailed stochastic volatility models with long memory. We prove a central limit theorem for a Hill estimator. In particular, it is shown that neither the rate of convergence nor the asymptotic variance is affected by long memory. The theoretical findings are verified by simulation studies.

Recently, there has been a growing interest in analyzing stability and developing controls for stochastic dynamic systems. This interest arises out of a need to develop robust control strategies for systems with uncertain dynamics. While traditional robust control techniques ensure robustness, these techniques can be conservative as they do not utilize the risk associated with the uncertainty variation. To improve controller performance, it is possible to include the probability of each parameter value in the control design. In this manner, risk can be taken for parameter values with low probability and performance can be improved for those of higher probability. To accomplish this, one must solve the resulting stability and control problems for the associated stochastic system. In general, this is accomplished using sampling based methods by creating a grid of parameter values and solving the problem for each associated parameter. This can lead to problems that are difficult to solve and may possess no analytical solution. The novelty of this dissertation is the utilization of non-sampling based methods to solve stochastic stability and optimal control problems. The polynomial chaos expansion is able to approximate the evolution of the uncertainty in state trajectories induced by stochastic system uncertainty with arbitrary accuracy. This approximation is used to transform the stochastic dynamic system into a deterministic system that can be analyzed in an analytical framework. In this dissertation, we describe the generalized polynomial chaos expansion and present a framework for transforming stochastic systems into deterministic systems. We present conditions for analyzing the stability of the resulting systems. In addition, a framework for solving L2 optimal control problems is presented. For linear systems, feedback laws for the infinite-horizon L2 optimal control problem are presented. A framework for solving finite-horizon optimal control problems with time-correlated stochastic forcing is also presented. The stochastic receding horizon control problem is also solved using the new deterministic framework. Results are presented that demonstrate the links between stability of the original stochastic system and the approximate system determined from the polynomial chaos approximation. The solutions of these stochastic stability and control problems are illustrated throughout with examples.