Reentry Trajectory Optimization under Atmospheric Uncertainty as a Differential Game

Abstract

If solutions of optimal control problems are to be realized in practical applications, one has to take into account the influence of unpredictable disturbances. If lower and upper bounds for the disturbances are known, one can investigate the so-called worst case. This worst case can be formulated as a two-person differential game and its solution in closed form provides the optimal feedback controller against all possible disturbances. Following this approach, realistically modelled optimal control problems lead in general to nonseparable, non zero-sum differential games where inequality constraints have to be taken into account. A maximum cross-range reentry of a space-shuttle in the presence of uncertain air density fluctuations serves as an example. For the treatment of terminal conditions and path constraints in which controls of both players are involved, the responsibility of the players for obeying these constraints is investigated, and a new transformation technique is used. The open-loop representation of the optimal feedback controller is computed along various saddle-point trajectories representing the worst case. The computational method is based on the numerical solution of the multipoint boundary-value problem which arises from the necessary conditions for a saddle-point solution of the game. Extensive numerical results are presented for a space-shuttle reentry under a dynamic pressure and an aerodynamic heating constraint. It is outlined, that the proposed method allows not only the analysis of the worst case, but is also the first step towards the construction of the optimal feedback controller via a successive computation of the open-loop representation or via a neural network training.