Quickly Finding Recursively Feasible Solutions for MPC with Discrete Variables

In this paper, we consider designing real-time Model Predictive Control (MPC) for embedded control applications where both continuous-valued and discrete-valued control inputs are present. The online optimization in MPC is formulated as a Mixed Integer Quadratic Program (MIQP), and can be achieved using a branch and bound algorithm where multiple relaxed Quadratic Programs (QPs) are solved. Due to the computational constraints for embedded applications, we impose a limit on the number of relaxed problems to be solved. Tailored heuristics in the branch and bound algorithm are developed taking into account the problem structure in MPC framework to generate early feasible solutions. To provide recursive feasibility guarantee to the MPC solved with such limited branch and bound algorithm, we propose to supervise the MPC with a correct-by-construction switching protocol. The paper describes these concepts, provides chronometric estimates for some problems, and gives a numerical example to explain the behavior. Moreover, a fuel cell control problem from the literature with five continuous states, three continuous inputs and three discrete inputs is used to show that the proposed approach reaches a feasible solution much faster than standard branch and bound solutions.