Bi-level trajectory optimization by stochastic collocation for uncertainty interval calculation

This study presents a bi-level framework designed for the optimal control (OC) of aircraft. Overall, the goal is to calculate trajectories for the aircraft under external and internal uncertainties that give a worst-case approximation of the uncertainty interval. The bi-level OC framework is set up as follows: Within the lower level, standard deterministic trajectory optimization problems by gradient-based optimization are solved. The solved problems only differ in their numerical values set for the uncertain parameters. It should be noted that this makes it easy to parallelize them, as they are independent of each other. The calculation of the uncertain response of the system is based on the generalized polynomial chaos (gPC) method. The upper level problem provides the numerical values of the uncertain parameters and is optimized using a differential evolution (DE) strategy. Thus, the connection from the upper to the lower level are the numerical values of the uncertain parameters. Conversely, the lower level provides the upper level with the optimized trajectory at each of the uncertain parameter's positions yielding the statistical moments. We use case studies from a vertical take-off and landing vehicle (VTOL) transition maneuvers to show the viability of the approach.