Logic-Based Solution Methods for Optimal Control of Hybrid Systems

Abstract

Combinatorial optimization over continuous and integer variables is a useful tool for solving complex optimal control problems of dynamical systems formulated in discrete-time. Current approaches are based on mixed-integer linear (or quadratic) programming (MIP), which provides the solution after solving a sequence of relaxed linear (or quadratic) programs. MIP formulations require the translation of the discrete/logic part of the problem into mixed-integer inequalities. Although this operation can be done automatically, most of the original symbolic structure of the problem (e.g., transition functions of finite state machines, logic constraints, symbolic variables, etc.) is lost during the conversion, with a consequent loss of computational performance. In this paper, we attempt to overcome such a difficulty by combining numerical techniques for solving convex programming problems with symbolic techniques for solving constraint satisfaction problems (CSP). The resulting hybrid solver proposed here takes advantage of CSP solvers for dealing with satisfiability of logic constraints very efficiently. We propose a suitable model of the dynamics and a class of optimal control problems that embrace both symbolic and continuous variables/functions, and that are tailored to the use of the new solver. The superiority in terms of computational performance with respect to commercial MIP solvers is shown on a centralized supply chain management problem with uncertain forecast demand.