Optimal control problem and its solution by grey wolf optimizer algorithm

Abstract

The paper is devoted to a numerical method for solving the optimal control problem. The main approach to the numerical solution of the optimal control problem is the reduction of the optimal control problem to the problem of nonlinear programming and its following solution by classical gradient optimization methods. For this purpose, optimal control problem, which is a problem of searching time-dependent function, is replaced by the problem of searching of control values at discrete instants of time. An increase in the number of sampling points increases the accuracy of function approximation, but at the same time increases the dimensionality of the search space in the non-linear programming problem. In complex problems of non-linear programming with an unknown topology of the objective function, the statement that using classical gradient methods ensures finding a solution is not justified. The optimal control problem after the discretization and other modifications is often transformed to a non-linear programming problem with a non-unimodal objective function for which gradient methods are not applicable. In this paper we propose to solve the optimal control problem by evolutionary algorithms that do not use gradients and are able to find solutions of problems with nonunimodal objective function. The paper presents the modern evolutionary algorithm Grey wolf optimizer. The problem of the optimal combat turn of the aircraft is considered. In this problem the mathematical model of the control object is described by a system of seven ordinary differential equations. Also constraints on the value and rate of change of control are given. It is experimentally shown that the evolutionary algorithm Grey wolf optimizer successfully solves this optimal control problem.