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.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2853 |
| Date | 15 May 2009 |
| Creators | Fisher, James Robert |
| Contributors | Bhattacharya, Raktim |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | electronic, application/pdf, born digital |
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