Efficient online solution of multi-parametric mixed-integer quadratic problems

Abstract

This paper considers model predictive control of a class of piecewise affine systems and introduces a method to reduce the complexity of the optimisation problem which is solved online. The optimisation problem is a mixed-integer quadratic program which is parameterised by the initial state of the system dynamics. The complexity of the problem solved online is reduced by moving part of the computational burden offline. In the offline computations, the structure of the mixed-integer problem is explored to determine a polytopic partition of the set of initial states. Each polytope of the partition is then considered and it is determined which integer combinations can/cannot be optimal, given that the initial state is contained in the current polytope. The result of the offline computations is that each polytope is associated with the set of potentially optimal integer combinations. The online solution of the mixed-integer problem consists of a two-step procedure: first, it is determined which polytope contains the initial state. Then, the mixed-integer problem is solved, taking into account only those integer combinations associated with the current polytope. The solution approach is shown to give a significant complexity reduction when applied to hybrid control of switched-mode power converters.