An efficient discretization scheme for solving nonlinear optimal control problems with multiple time delays

Abstract

Summary This paper presents a composite Chebyshev finite difference method to numerically solve nonlinear optimal control problems with multiple time delays. The proposed discretization scheme is based on a hybrid of block‐pulse functions and Chebyshev polynomials using the well‐known Chebyshev Gauss–Lobatto points. Our approach is an extension and also a modification of the Chebyshev finite difference scheme. A direct approach is used to transform the delayed optimal control problem into a nonlinear programming problem whose solution is much more easier than the original one. Some useful error bounds are established. In addition, the convergence of the method is discussed. A wide variety of numerical experiments are investigated to show the usefulness and effectiveness of the proposed discretization procedure. The method has a simple structure and can be implemented without too much effort. Copyright © 2015 John Wiley & Sons, Ltd.

A novel approach for the numerical investigation of optimal control problems containing multiple delays

“…is minimized. This problem has been studied in the works of Marzban and Hoseini 25 and Göllmann et al, 36 Koshkouei et al, 37 Hashemi Borzabadi and Asadi, 38 Maleki and Hashim, 39 and Dadkhah et al 40 In the work of Göllmann et al, 36 a Euler discretization scheme has been implemented. In the proposed approach, the control problem under study has been converted into a nonlinear programming one.…”

“…An iterative-adaptive shifted Legendre-Gauss PS method has been devised in the work of Maleki and Hashim 39 for solving constrained optimal control problems with 2 delays. In the work of Marzban and Hoseini, 25 a composite Chebyshev finite-difference method has been proposed to numerically solve nonlinear optimal control problems involving multiple delays. The method developed in the work of Marzban and Hoseini 25 is based on a hybrid of block-pulse functions and Chebyshev polynomials using the Gauss-Lobatto points.…”

“…They have been extensively implemented to numerically solve various types of problems arising in diverse areas of science and engineering. [19][20][21][22][23][24][25][26][27] More recently, an efficient and effective approximation scheme has been successfully applied to numerically solve linear optimal control problems involving multiple delays. 28 It has been shown that the developed method actually works well and provides very accurate results.…”

Hybrid Functions Direct Approach and State Feedback Optimal Solutions for a Class of Nonlinear Polynomial Time Delay Systems

“…Particularly, many attempts have been made in literature to solve optimal control problems for many classes of linear [8][9][10][11] and nonlinear [6,12,13] time delay systems. Among them, we recall the application of Pontryagins maximum principle to the optimization of control systems with time delays which was firstly proposed by [14].…”

Spatio-temporal decomposition: a knowledge-based initialization strategy for parallel parking motion optimization

“…Given the complexity and unification of this formulated optimal control problem, analytical solutions are generally unobtainable [18]. An alternative method to solve optimal control problems is by converting them into parameter optimization problems, that is, nonlinear programming (NLP) problems [19][20][21]. However, the corresponding NLP problems are NP-hard and have hindered a vast of research efforts in robotics [18].…”