Thesis (Ph. D.)--York University, 2002. Graduate Programme in Physics and Astronomy. / Typescript. Includes bibliographical references (leaves 111-116). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ76023.

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.

The asymptotic properties of discrete time stochastic systems operating under feedback is addressed. It is assumed that a Markov chain $ Phi$ evolving on Euclidean space exists, and that the input and output processes appear as functions of $ Phi$. The main objectives of the thesis are (i) to extend various asymptotic properties of Markov chains to hold for arbitrary initial distributions; and (ii) to develop a robustness theory for Markovian systems. / A condition called local stochastic controllability, a generalization of the concept of controllability from linear system theory, is introduced and is shown to be sufficient to ensure that the first objective is met. The second objective is explored by introducing a notion of convergence for stochastic systems and investigating the behavior of the invariant probabilities corresponding to a convergent sequence of stochastic systems. / These general results are applied to two previously unsolved problems: The asymptotic behavior of linear state space systems operating under nonlinear feedback, and the stability and asymptotic behavior of a class of random parameter AR (p) stochastic systems under optimal control.

Thesis (D. Phil.)(Systems Engineering)--University of Pretoria, 2006. / Includes summary. Includes bibliographical references. Available on the Internet via the World Wide Web.

Thesis (Ph. D.)--University of Washington, 1986. / Vita. Bibliography: leaves [81]-83.

Decoupling control is widely applied to multi-input multi-output industrial processes. The traditional decoupling control methods are based on accurate models, however it is difficult or impossible to obtain accurate models in practice. Moreover, the traditional decoupling control methods are not suitable for the analysis of the couplings among system outputs which are subjected to the random noises. To solve the problems mentioned above, we will look into the decoupling control problem in probability sense. To describe this control problem, probabilistic decoupling has been presented as a novel concept based on statistical independence. Using probability theory, a set of new control objectives has been extended by this presented concept. Conditions of probabilistic complete decoupling are given. Meanwhile, the relationship between the traditional decoupling and probabilistic decoupling has been analyzed in this thesis, theoretically. To achieve the control objectives of probabilistic decoupling, various control algorithms are developed for dynamic multi-variable stochastic systems, which are represented by linear stochastic models, bilinear stochastic models and stochastic nonlinear models, respectively. For linear stochastic models subjected to Gaussian noises, the covariance control theory has been used. The Output-feedback stabilization via block backstepping design has been considered for bilinear stochastic systems subjected to Gaussian noises. Furthermore, the minimum mutual information control has been proposed for stochastic nonlinear systems subjected to non-Gaussian noises. Some advanced topics are also considered in this thesis. The stochastic feedback linearization can be applied to a class of stochastic nonlinear systems and the reduced-order closed-form covariance control models are also presented, which can be applied in covariance control theory. Using kernel density estimation, data-based minimum mutual information control is given to extend the presented minimum mutual information control algorithm.

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Piecewise-smooth stochastic systems are widely used in engineering science. However, the theory of these systems is only in its infancy. In this thesis, we take as an example the Brownian motion with dry friction to illustrate dynamical properties of these systems with respect to three interesting topics: (i) weak-noise approximations, (ii) first-passage time (FPT) problems and (iii) functionals of stochastic processes. Firstly, we investigate the validity and accuracy of weak-noise approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example the Brownian motion with pure dry friction. For this model, we show that the weak-noise approximation of the path integral correctly reproduces the known propagator of the SDE at lowest order in the noise power, as well as the main features of the exact propagator with higher-order corrections, provided that the singularity of the path integral is treated with some heuristics. We also consider a smooth regularisation of this piecewise-constant SDE and study to what extent this regularisation can rectify some of the problems encountered in the non-smooth case. Secondly, we provide analytic solutions to the FPT problem of the Brownian motion with dry friction. For the pure dry friction case, we find a phase transition phenomenon in the spectrum which relates to the position of the exit point and affects the tail of the FPT distribution. For the model with dry and viscous friction, we evaluate quantitatively the impact of the corresponding stick-slip transition and of the transition to ballistic exit. We also derive analytically the distributions of the maximum velocity till the FPT for the dry friction model. Thirdly, we generalise the so-called backward Fokker-Planck technique and obtain a recursive ordinary differential equation for the moments of functionals in the Laplace space. We then apply the developed results to analyse the local time, the occupation time and the displacement of the dry friction model. Finally, we conclude this thesis and state some related unsolved problems.

Thesis (Ph. D.)--University of Chicago, Dept. of Statistics, August 2003. / Includes bibliographical references. Also available on the Internet.

In the operation of stochastic systems, and of queues in particular, it is important to recognize quickly the development in time of situations not compatible with their design criteria. Once such an anomalous condition is detected, it has to be decided, if the occurrence of that event can be attributed to chance or is due to a change in the parameters governing the system. This procedure of tracking the system is defined as monitoring. The design of a monitor and the selection of suitable threshold regions for monitoring a single server queue are the subjects of this thesis. The notion of profile curves, useful in formalizing monitoring schemes for queues, is also discussed. Finally, some numerical examples are presented to illustrate the performance of the monitor designed.