Suboptimal control of hybrid systems using approximate multi-parametric MILP

As a way to reduce the on-line computational burden, explicit solution to the problem of optimal control for some classes of hybrid systems can be found by reformulating the problem as multi-parametric MILP problems. The main contribution of this paper is the introduction of an approximation algorithm for solving a general class of mp-MILP problems. The algorithm wisely selects those binary sequences which make important improvement in the objective function if considered. It is shown that considerable reduction in computational complexity could be achieved by introduction of adjustable level of suboptimality. So a family of suboptimal controllers would be obtained for which the level of error and complexity can be adjusted by a tuning parameter. Several important theoretical results about approximate solutions to the mp-MILP problem are presented. It is shown that no part of the parameter space is lost during the approximation. Also it is proved that the error in the achieved approximate solutions is monotonically increasing function of the tuning parameter. The reduced complexity achieved by the proposed approach is clarified through an illustrative example.