Numerical solution of time-delay optimal control problems by the operational matrix based on Hartley series

Numerical Technique ◽

Operational Matrix ◽

Main Idea ◽

Control Problems ◽

Operational Matrices ◽

Algebraic Equations ◽

The Cost ◽

And Control

In this paper, a numerical technique based on the Hartley series for solving a class of time-delayed optimal control problems (TDOCPs) is introduced. The main idea is converting such TDOCPs into a system of algebraic equations. Thus, we first expand the state and control variables in terms of the Hartley series with undetermined coefficients. The delay terms in the problem under consideration are expanded in terms of the Hartley series. Applying the operational matrices of the Hartley series including integration, differentiation, dual, product, delay, and substituting the estimated functions into the cost function, the given TDOCP is reduced to a system of algebraic equations to be solved. The convergence of the proposed method is extensively investigated. At last, the precision and applicability of the proposed method is studied through different types of numerical examples.

Numerical solution for fractional optimal control problems by Hermite polynomials

10.1177/1077546320933129 ◽

2020 ◽

pp. 107754632093312

Author(s):

Ayatollah Yari

Keyword(s):

Optimal Control ◽

Numerical Method ◽

Fractional Integration ◽

Operational Matrix ◽

Control Problems ◽

Operational Matrices ◽

Algebraic Equations ◽

In this study, a numerical method based on Hermite polynomial approximation for solving a class of fractional optimal control problems is presented. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Operational matrices of integration by using such known formulas as Caputo and Riemann–Liouville operators for computing fractional derivatives and integration of polynomials is introduced and used to reduce the problem of a system of algebraic equations. The convergence of the proposed method is analyzed, and the error upper bound for the operational matrix of the fractional integration is obtained. To confirm the validity and accuracy of the proposed numerical method, three numerical examples are presented along with a comparison between our numerical results and those obtained using Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

A numerical approach for solving fractional optimal control problems with mittag-leffler kernel

10.1177/10775463211016967 ◽

2021 ◽

pp. 107754632110169

Author(s):

Hossein Jafari ◽

Roghayeh M Ganji ◽

Khosro Sayevand ◽

Dumitru Baleanu

Keyword(s):

Optimal Control ◽

Approximate Solution ◽

Fractional Integration ◽

Numerical Approach ◽

Control Problems ◽

Operational Matrices ◽

Algebraic Equations ◽

In this work, we present a numerical approach based on the shifted Legendre polynomials for solving a class of fractional optimal control problems. The derivative is described in the Atangana–Baleanu derivative sense. To solve the problem, operational matrices of AB-fractional integration and multiplication, together with the Lagrange multiplier method for the constrained extremum, are considered. The method reduces the main problem to a system of nonlinear algebraic equations. In this framework by solving the obtained system, the approximate solution is calculated. An error estimate of the numerical solution is also proved for the approximate solution obtained by the proposed method. Finally, some illustrative examples are presented to demonstrate the accuracy and validity of the proposed scheme.

Solving Two-Dimensional Variable-Order Fractional Optimal Control Problems With Transcendental Bernstein Series

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4042997 ◽

2019 ◽

Vol 14 (6) ◽

Cited By ~ 12

Author(s):

Hossein Hassani ◽

Zakieh Avazzadeh ◽

José António Tenreiro Machado

Keyword(s):

Optimal Control ◽

Control Problems ◽

Operational Matrices ◽

Variable Order ◽

Two Dimensional ◽

Control Functions ◽

Fractional Optimal Control Problems ◽

Dimensional Variable ◽

And Control

This paper studies two-dimensional variable-order fractional optimal control problems (2D-VFOCPs) having dynamic constraints contain partial differential equations such as the convection–diffusion, diffusion-wave, and Burgers' equations. The variable-order time fractional derivative is described in the Caputo sense. To overcome computational difficulties, a novel numerical method based on transcendental Bernstein series (TBS) is proposed. In fact, we generalize the Bernstein polynomials to the larger class of functions which can provide more accurate approximate solutions. In this paper, we introduce the TBS and their properties, and subsequently, the privileges and effectiveness of these functions are demonstrated. Furthermore, we describe the approximation procedure which shows for solving 2D-VFOCPs how the needed basis functions can be determined. To do this, first we derive a number of new operational matrices of TBS. Second, the state and control functions are expanded in terms of the TBS with unknown free coefficients and control parameters. Then, based on these operational matrices and the Lagrange multipliers method, an optimization method is presented to an approximate solution of the state and control functions. Additionally, the convergence of the proposed method is analyzed. The results for several illustrative examples show that the proposed method is efficient and accurate.

A new operational matrix based on Boubaker wavelet for solving optimal control problems of arbitrary order

Transactions of the Institute of Measurement and Control ◽

10.1177/0142331219898343 ◽

2020 ◽

Vol 42 (10) ◽

pp. 1858-1870 ◽

Cited By ~ 1

Author(s):

Kobra Rabiei ◽

Yadollah Ordokhani

Keyword(s):

Optimal Control ◽

Arbitrary Order ◽

Ritz Method ◽

Operational Matrix ◽

Similar Process ◽

Control Problems ◽

Two Dimensional ◽

Algebraic Equations ◽

The One

This paper presents numerical solution for solving the nonlinear one and two-dimensional optimal control problems of arbitrary order. First, we have constructed Boubaker wavelet for the first time and defined a general formulation for its fractional derivative operational matrix. To solve the one-dimensional problem, we have transformed the problems into an optimization one. The similar process together with the Ritz method are applied to find a solution for two-dimensional problems as well. Then, the necessary conditions of optimality result in a system of algebraic equations with unknown coefficients and then control parameters can be simply solved. The error vector is considered to show the convergence of the used approximation in this method. Finally, some illustrative examples are given to demonstrate accuracy and efficiency of the proposed method.

Numerical solution for a class of fractional optimal control problems using the fractional-order Bernoulli functions

Transactions of the Institute of Measurement and Control ◽

10.1177/01423312211047033 ◽

2021 ◽

pp. 014233122110470

Author(s):

Forugh Valian ◽

Yadollah Ordokhani ◽

Mohammad Ali Vali

Keyword(s):

Optimal Control ◽

Fractional Order ◽

Operational Matrix ◽

Control Problems ◽

Algebraic Equations ◽

Newton’S Iterative Method ◽

Bernoulli Functions ◽

The main purpose of this paper is to provide an efficient method for solving some types of fractional optimal control problems governed by integro-differential and differential equations, and because finding the analytical solutions to these problems is usually difficult, a numerical method is proposed. In this study, the fractional-order Bernoulli functions (F-BFs) are applied as basis functions and a new operational matrix of fractional integration is constructed for these functions. In the first step, the problem is transformed into an equivalent variational problem. Then the F-BFs, the constructed operational matrix, the Gauss quadrature formula, and necessary conditions for optimization are used to convert the problem into a system of algebraic equations. Finally, with the aid of Newton’s iterative method, the system of algebraic equations is solved and the approximate solution of the problem is obtained. Several numerical examples have been analysed for illustrating the efficiency and accuracy of the proposed method, and the results have been compared with the exact solutions and the results of other methods. The results show that the method provides accurate solutions.

Solving Optimal Control Linear Systems by Using New Third kind Chebyshev Wavelets Operational Matrix of Derivative

Baghdad Science Journal ◽

10.21123/bsj.11.2.229-234 ◽

2014 ◽

Vol 11 (2) ◽

pp. 229-234

Author(s):

Baghdad Science Journal

Keyword(s):

Optimal Control ◽

Linear System ◽

Optimal Control Problem ◽

Control Problem ◽

Operational Matrix ◽

Control Problems ◽

State Variables ◽

Algebraic Equations ◽

Chebyshev Wavelets

In this paper, a new third kind Chebyshev wavelets operational matrix of derivative is presented, then the operational matrix of derivative is applied for solving optimal control problems using, third kind Chebyshev wavelets expansions. The proposed method consists of reducing the linear system of optimal control problem into a system of algebraic equations, by expanding the state variables, as a series in terms of third kind Chebyshev wavelets with unknown coefficients. Example to illustrate the effectiveness of the method has been presented.

A new efficient numerical scheme for solving fractional optimal control problems via a Genocchi operational matrix of integration

10.1177/1077546317698909 ◽

2017 ◽

Vol 24 (14) ◽

pp. 3036-3048 ◽

Cited By ~ 6

Author(s):

Chang Phang ◽

Noratiqah Farhana Ismail ◽

Abdulnasir Isah ◽

Jian Rong Loh

Keyword(s):

Optimal Control ◽

Operational Matrix ◽

Control Problems ◽

Algebraic Equations ◽

Dynamic Constraints ◽

Genocchi Polynomials ◽

Fractional Optimal Control Problems ◽

Operational Matrix Of Integration

In this paper, a new operational matrix of integration is derived using Genocchi polynomials, which is one of the Appell polynomials. By using the matrix, we develop an efficient, direct and new numerical method for solving a class of fractional optimal control problems. The fractional derivative in the dynamic constraints was replaced with the Genocchi polynomials with unknown coefficients and a Genocchi operational matrix of fractional integration. Then, the equation derived from the dynamic constraints was put into the performance index. Hence, the fractional optimal control problems will be reduced to fractional variational problems. By finding a necessary condition for the optimality for the performance index, we will obtain a system of algebraic equations that can be easily solved by using any numerical method. Hence, we obtain the value of unknown coefficients of Genocchi polynomials. Lastly, the solution of the fractional optimal control problems will be obtained. In short, the properties of Genocchi polynomials are utilized to reduce the given problems to a system of algebraic equations. The approximation approach is simple to use and computer oriented. Illustrative examples are given to show the simplicity, accuracy and applicability of the method.

Fibonacci wavelets and Galerkin method to investigate fractional optimal control problems with bibliometric analysis

10.1177/1077546320948346 ◽

2020 ◽

pp. 107754632094834 ◽

Cited By ~ 1

Author(s):

Sedigheh Sabermahani ◽

Yadollah Ordokhani

Keyword(s):

Optimal Control ◽

Galerkin Method ◽

Operational Matrix ◽

Inequality Constraints ◽

Computational Method ◽

Control Problems ◽

Algebraic Equations ◽

This study presents a computational method for the solution of the fractional optimal control problems subject to fractional systems with equality and inequality constraints. The proposed procedure is based upon Fibonacci wavelets. The fractional derivative is described in the Caputo sense. The Riemann–Liouville operational matrix for Fibonacci wavelets is obtained. Then, we use this operational matrix and the Galerkin method to reduce the given problem into a system of algebraic equations. We discuss the convergence of the algorithm. Several numerical examples are included to observe the validity, effectiveness, and accuracy of the suggested scheme. Moreover, fractional optimal control problems are studied through a bibliometric viewpoint.

Feedback Control Method Using Haar Wavelet Operational Matrices for Solving Optimal Control Problems

Abstract and Applied Analysis ◽

10.1155/2013/240352 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 3

Author(s):

Waleeda Swaidan ◽

Amran Hussin

Keyword(s):

Optimal Control ◽

Feedback Control ◽

Control Method ◽

Numerical Technique ◽

Direct Methods ◽

Haar Wavelet ◽

Control Problems ◽

Operational Matrices ◽

Feedback Control Method

Most of the direct methods solve optimal control problems with nonlinear programming solver. In this paper we propose a novel feedback control method for solving for solving affine control system, with quadratic cost functional, which makes use of only linear systems. This method is a numerical technique, which is based on the combination of Haar wavelet collocation method and successive Generalized Hamilton-Jacobi-Bellman equation. We formulate some new Haar wavelet operational matrices in order to manipulate Haar wavelet series. The proposed method has been applied to solve linear and nonlinear optimal control problems with infinite time horizon. The simulation results indicate that the accuracy of the control and cost can be improved by increasing the wavelet resolution.

Proceedings of the Institution of Mechanical Engineers Part G Journal of Aerospace Engineering ◽

A modified pseudospectral method for indirect solving a class of switching optimal control problems

10.1177/0954410020916303 ◽

2020 ◽

Vol 234 (9) ◽

pp. 1531-1542

Author(s):

Mohammad A Mehrpouya

Keyword(s):

Optimal Control ◽

Minimum Principle ◽

Pseudospectral Method ◽

Piecewise Constant Function ◽

Control Problems ◽

Time Interval ◽

Algebraic Equations ◽

Control Functions ◽

And Control

In the present paper, an efficient pseudospectral method for solving the Hamiltonian boundary value problems arising from a class of switching optimal control problems is presented. For this purpose, based on the Pontryagin's minimum principle, the first-order necessary conditions of optimality are derived. Then, by partitioning the time interval related to the problem under study into some subintervals, the states (and costates) and control functions are approximated on each subintervals with piecewise interpolating polynomials based on Legendre–Gauss–Radau points and a piecewise constant function, respectively. As a result, solution of the problem is turned into solution of a number of algebraic equations, in which the values of the states (and costates) and control functions at Legendre–Gauss–Radau points as well as switching and terminal points are allowed to be unknown. Numerical examples are presented at the end to show the efficiency of the proposed method.