Optimal and suboptimal minimum time-to-climb trajectories

Minimum time-to-climb trajectories are generated for a high performance atmospheric flight vehicle. Variational solutions obtained via Pontryagin's minimum principle are used as a benchmark result. For comparison, near-optimal solutions are generated using a recently introduced direct optimization technique. The new approach, which relies on a representation of the dynamical system in terms of differential inclusions, has excellent convergence properties and does not require initial guesses for the controls. The results show excellent agreement. I n t r o d u c t i o n The precise solution of an optimal control problem via Pontryagin's Minimum Principle involves the numerical treatment of nonlinear multipoint boundary value problems (BVPs). The structure of these BVPs depends on the sequence in which the optimal control switches between singular/nonsingular and constrained/unconstrained arcs, and is not known to the analyst in advance. Additionally, these BVPs involve costate variables whose physical meaning offers little help in determining reasonable initial guesses from which gradient search methods can converge. For these reasons, rapid trajectory generation is usually attempted by applying direct optimization techniques to some type of discretized problem formulation. This approach leads to the numerical task of solving nonlinear programming problems. The performance of the optimization algorithm involved and the precision of the obtained solutions depends strongly on the chosen problem discretization and on the dimension of the associated parameter search space. Recently, a new algorithm was introduced [I] based upon a representation of the dynamical system in differential inclusion format with the right-hand side of the equations of motion being set-valued. As a result of this representation the new approach is completely void of controls. Besides the lower dimensionality of the discretized problem and the reduced number of initial guesses that need to be provided by the user, the absence of control variables seems to have a positive effect on the convergence behavior, especially for singular *Supervising Engineer, Analytical klechanics Associates, Inc., 17 Research Drive, Hampton, VA 23666, working under contract a t the Spacecraft Controls Branch, NASA-LARC, Member AIAA. Copyright 01994 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. control problems. For the example of a minimum-time-to-climb problem for a high performance atmospheric flight vehicle, solutions obtained with the new algorithm are compared to solutions obtained using a variational approach. The agreement is excellent. The results are particularly interesting in view of the numerically challenging task of NASP ascent guidance For the present vehicle model the minimum-time-to-climb solution is identical to the minimum fuel ascent solution. P r o b l e m Formula t ion The problem under consideration is that of finding control functions r](t) and n(t) that steer an atmospheric flight vehicle from prescribed initial states t o the farmost point to the right on the level flight envelope in minimum time. Explicitly the problem can be stated in Mayer form as follows: