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Description:
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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 . |