A Hybrid Optimization Method for Constrained Optimal Control Problem

Abstract

A new hybrid algorithm by integrating a nested partitions (NP) method with successive quadratic programming (SQP) is presented for global optimization of general optimal control problems involving lumped parameter system. The control parameterization technique first employed to reduce the control problem into a parameter selection problem. Then, in the global phase a vicinity of global optimizer is approximated by an appropriate NP method. Subsequently, the SQP algorithm in the local phase promotes the accuracy of final solution. The effectiveness of the hybrid NP-SQP algorithm is also illustrated by means of numerical simulations. I. INTRODUCTION In the early years of optimal control theory traditional methods based on calculus of variations could furnish some relatively simple problems with an analytical solution. As the analytical solution might not always be available, researchers have been interested in computational method for optimal control problems, yet when reliable and powerful computers became everywhere available, it was the end of an era. The software environments have become a platform for implementation of many practical methods to solve arising control problems. The iterative traditional methods both directly (3) or indirectly (using the Pontryagin's minimum principle) (8) rely on gradient information, and perform efficiently for convex problems, otherwise the performance of these methods highly depend on selected initial solution. As an alternative to approximate a solution for complex nonconvex problems, many metaheuristic methods have been proposed (4). In general, these methods iteratively sample amongst potential solutions, and the search is directed using the value of the objective function only, hence reducing the need for gradient information. Moreover, as metaheuristic methods globally search for optima, they are usually characterized as global optimizer due to their capability to efficiently arrive at the vicinity of a global optimum. Nested partitions (NP) (10) is an exemplary method in this category, where concisely described in the third section. We use this method to globally search for optimal control for the class of the control problems described in the next section. Then, in the subsequent sections we propose the hybrid NP-SQP algorithm. II. THE CONTROL PROBLEMS In this study, we consider a general optimal control problem where the system of dynamics is described by a first-order ordinary differential equation (ODE),