High Order Optimal Feedback Control of Space Trajectories with Bounded Control

Optimal feedback control is classically based on linear approximations, whose accuracy drops off rapidly in highly nonlinear dynamics. Several nonlinear optimal feedback control strategies have appeared in recent years. Among them, differential algebraic techniques have been used to tackle nonlinearities by expanding the solution of the optimal control problem about a reference trajectory and reducing the computation of optimal feedback control laws to the evaluation of high order polynomials. However, the resulting high order method could not handle control saturation constraints, which remain a critical facet of nonlinear optimal feedback control. This work introduces the management of saturating actuators in the differential algebraic method. More specifically, the constraints are included in the optimal control problem formulation and differential algebra is used to expand the associated optimal bang-bang solution with respect to initial and terminal conditions. Optimal feedback control laws for thrust direction and switching times are again computed by evaluating the resulting polynomials. Illustrative applications are presented in the frame of the optimal low-thrust transfer to asteroid 1996 FG3.