Comparison of Derivative Estimation Methods in Optimal Control Using Direct Collocation

Abstract

A study is conducted to evaluate four derivative estimation methods when solving a large sparse nonlinear programming problem that arises from the approximation of an optimal control problem using a direct collocation method. In particular, the Taylor series-based finite difference, bicomplex-step, and hyper-dual derivative estimation methods are evaluated and compared alongside a well-known automatic differentiation method. The performance of each derivative estimation method is assessed based on the number of iterations, the computation time per iteration, and the total computation time required to solve the nonlinear programming problem. The efficiency of each of the four derivative estimation methods is compared by solving three benchmark optimal control problems. It is found that, although central finite differencing is typically more efficient per iteration than either the hyper-dual or the bicomplex-step method, the latter two methods have significantly lower overall computation times due to the fact that fewer iterations are required by the nonlinear programming problem when compared with central finite differencing. Furthermore, although the bicomplex-step and hyper-dual methods are similar in performance, the hyper-dual method is significantly easier to implement. Moreover, the automatic differentiation method is found to be substantially less computationally efficient than any of the three Taylor series-based methods. The results of this study show that the hyper-dual method offers several benefits over the other three methods: both in terms of computational efficiency and ease of implementation.