Implicit Runge-Kutta Methods for Stiff and Constrained Optimal Control Problems

Abstract

The purpose of the research presented in this thesis is to better understand and improve direct transcription methods for stiff and state constrained optimal control problems. When some implicit Runge-Kutta methods are implemented as approximations to the dynamics of an optimal control problem, a loss of accuracy occurs when the dynamics are stiff or constrained. A new grid refinement strategy which exploits the variation of accuracy is discussed. In addition, the use of a residual function in place of classical error estimation techniques is proven to work well for stiff systems. Computational experience reveals the improvement in efficiency and reliability when the new strategies are incorporated as part of a direct transcription algorithm. For index three differential-algebraic equations, the solutions of some implicit Runge-Kutta methods may not converge. However, computational experience reveals apparent convergence for the same methods used when index three state inequality constraints become active. It is shown that the solution chatters along the constraint boundary allowing for better approximations. Moreover, the consistency of the nonlinear programming problem formed by a direct transcription algorithm using an implicit Runge-Kutta approximation is proven for state constraints of arbitrary index.