Stability analysis and control of stochastic dynamic systems using polynomial chaos

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.