Computational Optimal Control Of Nonlinear Systems With Parameter Uncertainty
- Author(s): Phelps, Chris
- Advisor(s): Gong, Qi
- et al.
A number of emerging applications in the field of optimal control theory require the computation of an open-loop control for a dynamical system with uncertain parameters. In this dissertation we examine a class of uncertain optimal control problems, in which the goal is to minimize the expectation of a predetermined cost functional subject to such an uncertain system. We provide a computational framework for this class of problems based on a discretization of the parameter space. In this approach, a set of nodes from the parameter space and corresponding weights are selected, and the expectation of the cost functional is approximated by a finite sum. This process results in a sequence of standard optimal control problems which can be solved using existing techniques. However, it is well-known that an inappropriately designed discretization scheme for an optimal control problem may fail to converge to the optimal solution, therefore further analysis must be performed to examine the convergence properties of the scheme. We provide this analysis for a scheme based on quadrature methods for the approximation of the expectation in the cost functional. This analysis demonstrates that an accumulation point of a sequence of optimal solutions to the approximate problem is an optimal solution of the original problem. Furthermore, we examine the convergence of the adjoint states for the approximation based on the quadrature scheme, which leads to a Pontryagin-like necessary condition which must be satisfied by these accumulation points. To address the exponential growth of computational cost with respect to the dimension of the parameters, we introduce a numerical algorithm based on sample average approximations, in which an independently drawn random sample is taken from the parameter space, and the expectation in the objective functional is approximated by the sample mean. Using a generalization of the strong law of large numbers, we analyze the convergence properties of this approximation. In addition, we develop an optimality function for the class of uncertain optimal control problems based on the L&ndash Frechet derivative of the objective functional, which provides a necessary condition for an optimal solution. By demonstrating that an accumulation point of a sequence of stationary points for the approximate problem is a stationary point of the original problem, we demonstrate the approximation scheme based on sample averages is consistent in the sense of Polak. These numerical algorithms for the uncertain optimal control problem are applied to real-world scenarios from the fields of optimal search theory and ensemble control.