State space sets with common optimal feedback laws for nonlinear MPC

We extend an approach to nonlinear model predictive control that was recently proposed for linear model predictive control. A solution to an optimal control problem for a point in state space provides an optimal feedback law and a region in state space where this optimal feedback law is valid. Also, an optimal active set results from the solution. We state a criterion for a subset of an active set to already define the optimal feedback law. Any optimal active set containing a subset such that the criterion holds defines the same optimal feedback law as the subset. This enables to identify regions with identical optimal feedback laws if the optimal active sets of the regions are known. We use this result to propose a model predictive control approach that reduces the number of optimal control problems that are solved online. The approach initially unites regions with identical optimal feedback laws offline. This results in large regions with known optimal feedback laws. Online, the calculation of the optimal control problem is avoided whenever the current state is detected to be part of a region with known optimal feedback law. We illustrate the achieved savings with an example.