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.

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.

No description available.

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.

This thesis studies three models: Multi-type TASEP in discrete time, long-range last- passage percolation on the line and convoy formation in a travelling servers model. All three models are relatively easy to state but they show a very rich and interesting behaviour. The TASEP is a basic model for a one-dimensional interacting particle system with non-reversible dynamics. We study some aspects of the TASEP in discrete time and compare the results to recently obtained results for the TASEP in continuous time. In particular we focus on stationary distributions for multi-type models, speeds of second- class particles, collision probabilities and the speed process. We consider various natural update rules.

The theory of large deviations can help to shed light on systems in non-equilibrium statistical mechanics and, more generically, on non-reversible stochastic processes. For this purpose, we target trajectories in space time rather than static configurations and study time-extensive observables. This suggests that the details of the evolution law such as the presence of time correlations take on a major role. In this thesis, we investigate selected models with stochastic dynamics that incorporate memory by means of different mechanisms, devise a numerical approach for such models, and quantify to what extent the memory affects the large deviation functionals. The results are relevant for real-world situations, where simplified memoryless (Markovian) models may not always be appropriate. After an original introduction to the mathematics of stochastic processes, we explore, analytically and numerically, an open-boundary zero-range process which incorporates memory by means of hidden variables that affect particle congestion. We derive the exact solution for the steady state of the one-site system, as well as a mean-field approximation for larger one-dimensional lattices. Then, we focus on the large deviation properties of the particle current in such a system. This reveals that the time correlations can be apparently absorbed in a memoryless description for the steady state and the small fluctuation regime. However, they can dramatically alter the probability of rare currents. Different regimes are separated by dynamical phase transitions. Subsequently, we address systems in which the memory cannot be encoded in hidden variables or the waiting-time distributions depend on the whole trajectory. Here, the difficulty in obtaining exact analytical results is exacerbated. To tackle these systems, we have proposed a version of the so-called 'cloning' algorithm for the evaluation of large deviations that can be applied consistently for both Markovian and non-Markovian dynamics. The efficacy of this approach is confirmed by numerical results for some of the rare non-Markovian models whose large deviation functions can be obtained exactly. We finally adapt this machinery to a technological problem, specifically the performance evaluation of communication systems, where temporal correlations and large deviations are important.

Piecewise-smooth stochastic systems have attracted a lot of interest in the last decades in engineering science and mathematics. Many investigations have focused only on one-dimensional problems. This thesis deals with simple two-dimensional piecewise-smooth stochastic systems in the absence of detailed balance. We investigate the simplest example of such a system, which is a pure dry friction model subjected to coloured Gaussian noise. The nite correlation time of the noise establishes an additional dimension in the phase space and gives rise to a non-vanishing probability current. Our investigation focuses on stick-slip transitions, which can be related to a critical value of the noise correlation time. Analytical insight is provided by applying the uni ed coloured noise approximation. Afterwards, we extend our previous model by adding viscous friction and a constant force. Then we perform a similar analysis as for the pure dry friction case. With parameter values close to the deterministic stick-slip transition, we observe a non-monotonic behaviour of the probability of sticking by increasing the correlation time of the noise. As the eigenvalue spectrum is not accessible for the systems with coloured noise, we consider the eigenvalue problem of a dry friction model with displacement, velocity and Gaussian white noise. By imposing periodic boundary conditions on the displacement and using a Fourier ansatz, we can derive an eigenvalue equation, which has a similar form in comparison to the known one-dimensional problem for the velocity only. The eigenvalue analysis is done for the case without a constant force and with a constant force separately. Finally, we conclude our ndings and provide an outlook on related open problems.