Explicit predictive control with non-convex polyhedral constraints

This paper proposes an explicit solution to the model predictive control of linear systems subject to non-convex polyhedral constraints. These constraints are modeled as the union of a finite number of convex polyhedra. The algorithm is based on calculating the explicit solution to a modified problem with linear constraints defined as the convex hull of the original ones and classifying its regions by their relation with the regions of the explicit solution to the original problem. Some of the regions are divided, and a procedure based on sum-of-squares programming is designed to determine which of the possible solutions are in fact optimal. Finally, the online algorithm is shown to be better in terms of computational cost and memory requirements than an algorithm based on obtaining and comparing the solutions of the problem using as constraints the polyhedra whose union forms the non-convex regions, both theoretically and by the results of an example.