Optimization Problems in Model Predictive Control

This paper provides a review of the types of optimization problems that arise when implementing model predictive control, a feedback control method based on the on-line solution of open-loop optimal control problems. For linear, unconstrained processes, the issues are fully resolved; the controller is linear and the optimization reduces to a least squares problem that can be solved off line. For linear, constrained processes, the controller is nonlinear and the optimization is a convex quadratic program that must be solved on line. Maintaining feasibility of the optimization problem in the presence of the constraints is the remaining open issue. For nonlinear processes, the controller is nonlinear and the optimization is a nonlinear and non-convex problem. Because on-line solution of non-convex problems is difficult, alternatives to this optimization problem are presented. The available theory to establish robustness of the nonlinear controller to perturbations is summarized. Throughout the paper, we highlight the interplay between optimization theory and model predictive control theory.