On the Orthogonal Collocation and Mathematical Programming Approach for State Constrained Optimal Control Problems

Abstract

The orthogonal collocation approach is now well known to solve, effectively, the state constrained optimal control problems. Mathematical programming technique was also used as an effective tool to construct the optimal trajectories. In this paper, a study is done on the efficiency and accuracy requirements of the combined orthogonal collocation and mathematical programming approach, as regarding the employed optimization algorithm, and the number of orthogonal collocation points. It is shown, by experimentation with numerical examples that Fletcher-Powell optimization algorithm is much more faster to produce convergence than Fletcher-Reeves algorithm. The efficiency can be a ratio of six-to-one. The results are compared with an alternative approach to solve the same problem. It is shown that the present algorithm is less costly than the alternative approach, although requiring more computation time. The choice is then a compromise one. As the number of orthogonal points increases, the resulting solutions are more accurate, but the convergence speed decreases. Experimentation with N, shows a save of five-to-one in computing time can be achieved with almost the same cost function. Finally, it is shown, by a numerical example, that uniformly distributed collocation points result in non-optimal solutions, which also violate the problem constraints. It is a numerical proof of the superiority of the orthogonal collocation approach.