Stability analysis and control of stochastic dynamic systems using polynomial chaos

Abstract

Recently, there has been a growing interest in analyzing stability and developing\ncontrols for stochastic dynamic systems. This interest arises out of a need to develop\nrobust control strategies for systems with uncertain dynamics. While traditional\nrobust control techniques ensure robustness, these techniques can be conservative as\nthey do not utilize the risk associated with the uncertainty variation. To improve\ncontroller performance, it is possible to include the probability of each parameter\nvalue in the control design. In this manner, risk can be taken for parameter values\nwith low probability and performance can be improved for those of higher probability.\nTo accomplish this, one must solve the resulting stability and control problems\nfor the associated stochastic system. In general, this is accomplished using sampling\nbased methods by creating a grid of parameter values and solving the problem for\neach associated parameter. This can lead to problems that are difficult to solve and\nmay possess no analytical solution.\nThe novelty of this dissertation is the utilization of non-sampling based methods\nto solve stochastic stability and optimal control problems. The polynomial chaos expansion\nis able to approximate the evolution of the uncertainty in state trajectories\ninduced by stochastic system uncertainty with arbitrary accuracy. This approximation\nis used to transform the stochastic dynamic system into a deterministic system\nthat can be analyzed in an analytical framework. In this dissertation, we describe the generalized polynomial chaos expansion and\npresent a framework for transforming stochastic systems into deterministic systems.\nWe present conditions for analyzing the stability of the resulting systems. In addition,\na framework for solving L2 optimal control problems is presented. For linear systems,\nfeedback laws for the infinite-horizon L2 optimal control problem are presented. A\nframework for solving finite-horizon optimal control problems with time-correlated\nstochastic forcing is also presented. The stochastic receding horizon control problem\nis also solved using the new deterministic framework. Results are presented that\ndemonstrate the links between stability of the original stochastic system and the\napproximate system determined from the polynomial chaos approximation. The solutions\nof these stochastic stability and control problems are illustrated throughout\nwith examples.

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