Directional Input Adaptation in Parametric Optimal Control Problems

This paper deals with input adaptation in dynamic processes in order to guarantee feasible and optimal operation despite the presence of uncertainty. For those optimal control problems having terminal and mixed control-state path constraints, two orthogonal sets of adaptation directions can be distinguished in the input space: the sensitivity-seeking directions, along which a small variation from an optimal nominal solution will not affect the respective active constraints, and the complementary constraint-seeking directions, along which a variation will affect the respective constraints. It follows that selective input adaptation strategies can be defined, namely, adaptation in the sensitivity- and constraint-seeking directions. This paper proves the important result that, for small parametric perturbations, the cost variation resulting from adaptation in the sensitivity-seeking directions (over no input adaptation) is typically smaller than the cost variation due to adaptation in the constraint-seeking directions. It is also established that no selective input adaptation along a sensitivity-seeking direction can reduce the dominant, first-order term in the optimality gap; adaptation along a constraint-seeking direction is necessary to cancel it out. These results are illustrated with two numerical case studies.