A Mixed-Integer Nonlinear Problem Algorithm to Control Finite State Machines using Branch and Bound

The dynamics of a large variety of systems such as micro-grids and parallel hybrid cars can be described using finite state machines. The optimal control of these systems leads to mixed-integer non-linear programming (MINLP) problems. This class of problems typically belongs to the non-deterministic polynomial-time hard problems since they combine both the computational intensity of solving problems with discrete variables and the complexity of solving nonlinear functions. In the last decade, different techniques are developed to reach a global optimal solution for convex MINLP problems, such as branch & bound, outer approximation or the generalized benders decomposition algorithm. These techniques often have difficulties in efficiently handling the constraints of state machines. These constraints include the minimum time to stay within a state or the minimum cycle time before the same state can be reached again. In order to tackle these issues, the paper proposes a new technique that exploits the properties of a state machine within the branch & bound tree to reduce execution time. This reduction is achieved by separately solving the state machine variables and relaxing the switches between states using sigmoid functions. As a result, the technique reduces the number of explored nodes in the branch & bound tree significantly. The algorithm can be used in an online optimization strategy when employing a model predictive control framework that preserves added constraints from a previous iteration in the next iteration. A numerical simulation demonstrates the computational efficiency of this algorithm.