Learning Robustly Stabilizing Explicit Model Predictive Controllers: A Non-Regular Sampling Approach

Off-line supervised learning from data of robustly-stabilizing nonlinear explicit model predictive controllers (EMPC) is dealt with in this letter. The learning procedure relies on the construction of a suitably large set of specifically chosen sampling points of the state space in which the values of the optimal EMPC control function have to be computed. When bounding the magnitude of approximation errors is important for stability or performance specifications, regular gridding techniques are not feasible due to the curse of dimensionality arising from the structural exponential growth of the number of points with the state dimension. In this note, we consider non-regular sampling techniques - namely, i.i.d. sampling with uniform distribution, low-discrepancy sequences and lattice point sets - that offer a good covering of the state space without suffering from an unfeasible growth of the number of points, while preserving at the same time the method guarantees in terms of robustness and stability. Some theoretical properties of the proposed sampling schemes are briefly discussed, and their successful application is showcased in a practically-relevant optimal heating problem involving a 21-dimensional state space that rules out the use of regular gridding techniques.