Stability analysis and control of stochastic dynamic systems using polynomial chaos

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Title: Stability analysis and control of stochastic dynamic systems using polynomial chaos
Author: Fisher, James Robert
Abstract: 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 .
URI: http : / /hdl .handle .net /1969 .1 /ETD -TAMU -2853
Date: 2009-05-15

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Stability analysis and control of stochastic dynamic systems using polynomial chaos. Available electronically from http : / /hdl .handle .net /1969 .1 /ETD -TAMU -2853 .

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