Mixed–Integer Optimal Control

Abstract

In this chapter we extend the problem class of continuous optimal control problems discussed in chapter 1 to include control functions that may at each point in time attain only a finite number of values from a discrete set. We briefly survey different approaches for the solution of the discretized Mixed–Integer Optimal Control Problem (MIOCP), such as Dynamic Programming, branching techniques, and Mixed–Integer Nonlinear Programming methods. These approaches however turn out to be computationally very demanding already for off–line optimal control, and this fact becomes even more apparent in a real–time on–line context. The introduction of an outer convexification and relaxation approach for the MIOCP class allows to obtain an approximation of the MIOCP’s solution by solving only one single but potentially much larger continuous optimal control problem using the techniques of chapter 1. We describe theoretical properties of this problem reformulation that provide bounds on the loss of optimality for the infinite dimensional MIOCP. Rounding schemes are presented for the discretized case that maintain these guarantees. We develop techniques for the introduction of switch costs into the class of MIOCPs and give reformulations that allow for a combination with direct multiple shooting and outer convexification.