An Improved Algorithm for Solving Constrained Optimal Control Problems

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## Abstract

Motivated by the need to have an algorithm which (1) can solve generally constrained optimal control problems, (2) is globally convergent, (3) has a fast local convergence rate, a new algorithm, which solves fixed end-time optimal control problems with hard control constraints, end-point inequality constraints, and a variable initial state, is developed. This algorithm is based on a second-order approximation to the change of the cost functional due to a change in the control and a change in the initial state. Further approximation produces a simple convex functional. An exact penalty function is employed to penalize any violated end-point inequality constraints. We then show that the solution of the minimization of the convex functional, subject to linearized system dynamics, the original hard control constraints, the original hard initial state constraints, and linearized end-point constraints, generates a descent direction for that exact penalty function.

We then show that the algorithm developed in this dissertation can also solve the following types of optimal control problems: (1) problems with a free end-time; (2) problems with path constraints; (3) problems with some design parameters that are also to be optimized.

Global convergence properties of a version of the algorithm are analyzed. In particular, it is shown that the algorithm is globally convergent under some conditions. The local convergence rate of the algorithm can be better than that of the first-order algorithms when some matrices are properly updated.

A version of the algorithm is implemented in a package which is easy to use. A variety of benchmark problems are solved. Finally, the algorithm is employed in solving two challenging biomechanics problems: (1) a human moving his arm from an initial resting position so as to touch an stop at a target with the tip of the index finger while the muscular stress, the joint constraint forces, and the neural excitations are minimized; (2) a human pedaling s stationary bicycle as fast as possible from rest. Those results demonstrate that the algorithm developed in this dissertation is effective in dealing with generally constrained optimal control problems.