Gradient method using pseudospectral collocation scheme for two-stage optimal control with an unspecified switching time

Abstract

A two-stage gradient method (TGM) based on traditional backward integration is firstly proposed for solving two-stage nonlinear optimal control problem with arbitrary performance index and unspecified switching time. To reduce the computational cost, pseudospectral collocation scheme is embedded so as to transfer a family of time-varying influence function dynamics into a set of linear algebraic equations. Therefore, a series of symbolic formulation can be analytically derived to efficiently calculate the influence functions. Also, some special integrals using to update the improvements can be efficiently determined by Gaussian Quadrature. Finally, the performance of both TGMs is investigated with reduced-order reentry trajectory planning problem. The results show that both methods only take a few steps to find the solution satisfying the necessary condition for optimality and terminal constraints. Furthermore, the TGM with pseudospectral collocation scheme considerably improves the computational efficiency and has the potential to be applicable in the framework of real-time guidance.