Multiparametric/explicit nonlinear model predictive control for quadratically constrained problems

Abstract

Explicit model predictive control is an established methodology for the offline determination of the optimal control policy for linear discrete time-invariant systems with linear constraints. Nevertheless, nonlinearities in the form of quadratic constraints naturally appear in process models or are imposed for stability purposes in model predictive control formulations. In this manuscript, we present the theoretical developments and propose an algorithm for the exact solution of explicit nonlinear model predictive control problems with convex quadratic constraints. Our approach is based on a second-order Taylor approximation of Fiacco’s Basic Sensitivity Theorem, which allows for the existence and the analytic derivation of the optimal control actions. The complete exploration of the parameter space is founded on an active set strategy, which employs a pruning criterion to eliminate infeasible active sets. Based on that, the optimal map of solutions is constructed along with the corresponding control actions. The proposed strategy is applied to an explicit nonlinear model predictive control problem with an ellipsoidal terminal set, and comparisons with approximate solutions are drawn to demonstrate the benefits of the presented approach. Furthermore, as a practical application, the optimal operation of a chemostat in the presence of disturbances is exhibited.