Suboptimal control of fractional-order dynamic systems with delay argument

2017 ◽  
Vol 24 (12) ◽  
pp. 2430-2446 ◽  
Author(s):  
Amin Jajarmi ◽  
Dumitru Baleanu
Keyword(s):  
Optimal Control ◽  
Linear Programming ◽  
Calculus Of Variations ◽  
Fractional Order ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Suboptimal Control ◽  
Control Problems ◽  
Lagrange Equations ◽  
Delay Argument

In this paper, an efficient linear programming formulation is proposed for a class of fractional-order optimal control problems with delay argument. By means of the Lagrange multiplier in the calculus of variations and using the formula for fractional integration by parts, the Euler–Lagrange equations are derived in terms of a two-point fractional boundary value problem including an advance term as well as the delay argument. The derived equations are then reduced into a linear programming problem by using a Grünwald–Letnikov approximation for the fractional derivatives and introducing a new transformation in the calculus of variations. The new scheme is also effective for the delay fractional optimal control problems influenced by the external persistent disturbances. Numerical simulations and comparative results verify that the proposed approach is efficient and easy to implement.

A Quadratic Numerical Scheme for Fractional Optimal Control Problems

2007 ◽  
Vol 130 (1) ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Numerical Scheme ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Classical Solutions ◽  
Control Problems ◽  
Lagrange Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

This paper presents a quadratic numerical scheme for a class of fractional optimal control problems (FOCPs). The fractional derivative is described in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of fractional differential equations. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. For a linear system, this method results into a set of linear simultaneous equations, which can be solved using a direct or an iterative scheme. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs.

Formulation of Euler–Lagrange Equations for Multidelay Fractional Optimal Control Problems

2018 ◽  
Vol 13 (6) ◽  
Author(s):  
Sohrab Effati ◽  
Seyed Ali Rakhshan ◽  
Samane Saqi
Keyword(s):  
Optimal Control ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Control Problems ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

In this paper, a new numerical scheme is proposed for multidelay fractional order optimal control problems where its derivative is considered in the Grunwald–Letnikov sense. We develop generalized Euler–Lagrange equations that results from multidelay fractional optimal control problems (FOCP) with final terminal. These equations are created by using the calculus of variations and the formula for fractional integration by parts. The derived equations are then reduced into system of algebraic equations by using a Grunwald–Letnikov approximation for the fractional derivatives. Finally, for confirming the accuracy of the proposed approach, some illustrative numerical examples are solved.

Orthonormal piecewise Bernoulli functions: Application for optimal control problems generated using fractional integro-differential equations

2022 ◽  
pp. 107754632110593
Author(s):  
Mohammad Hossein Heydari ◽  
Mohsen Razzaghi ◽  
Zakieh Avazzadeh
Keyword(s):  
Optimal Control ◽  
Fractional Integral ◽  
Original Problem ◽  
Optimal Control Problems ◽  
Integration Method ◽  
Direct Method ◽  
Fractional Integration ◽  
Basis Functions ◽  
Control Problems ◽  
Bernoulli Functions

In this study, the orthonormal piecewise Bernoulli functions are generated as a new kind of basis functions. An explicit matrix related to fractional integration of these functions is obtained. An efficient direct method is developed to solve a novel set of optimal control problems defined using a fractional integro-differential equation. The presented technique is based on the expressed basis functions and their fractional integral matrix together with the Gauss–Legendre integration method and the Lagrange multipliers algorithm. This approach converts the original problem into a mathematical programming one. Three examples are investigated numerically to verify the capability and reliability of the approach.

Shifted Chebyshev schemes for solving fractional optimal control problems

2019 ◽  
Vol 25 (15) ◽  
pp. 2143-2150 ◽  
Author(s):  
M Abdelhakem ◽  
H Moussa ◽  
D Baleanu ◽  
M El-Kady
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Chebyshev Polynomials ◽  
Optimal Control Problems ◽  
Optimization Problems ◽  
Control Problems ◽  
Numerical Examples ◽  
Fractional Optimal Control ◽  
Functional Approximation ◽  
Fractional Optimal Control Problems

Two schemes to find approximated solutions of optimal control problems of fractional order (FOCPs) are investigated. Integration and differentiation matrices were used in these schemes. These schemes used Chebyshev polynomials in the shifted case as a functional approximation. The target of the presented schemes is to convert such problems to optimization problems (OPs). Numerical examples are included, showing the strength of the schemes.

scholarly journals SOLUTION OF INFINITE HORIZON NONLINEAR OPTIMAL CONTROL PROBLEMS BY PIECEWISE ADOMIAN DECOMPOSITION METHOD

2013 ◽  
Vol 18 (4) ◽  
pp. 543-560 ◽  
Author(s):  
Hassan Saberi Nik ◽  
Paulo Rebelo ◽  
Moosarreza Shamsyeh Zahedi
Keyword(s):  
Optimal Control ◽  
Decomposition Method ◽  
Optimal Control Problems ◽  
Infinite Horizon ◽  
Adomian Decomposition Method ◽  
Nonlinear Optimal Control ◽  
Suboptimal Control ◽  
Control Problems ◽  
Time Step ◽  
Adomian Decomposition

In this paper, a Piecewise Adomian Decomposition Method (PADM) is used to obtain the analytical approximate solution for a class of infinite horizon nonlinear optimal control problems (OCPs). The method is a new modification of the standard ADM, in which it is treated as an algorithm in a sequence of small intervals (i.e. with small time step) for finding accurate approximate solutions to the corresponding OCPs. Applying the PADM, the nonlinear two-point boundary value problem (TPBVP), derived from the application of Pontryagin's maximum principle (PMP), is transformed into a sequence of linear time-invariant TPBVP's. Through the finite iterations of algorithm, a suboptimal control law is obtained for the nonlinear optimal control problem. Comparing the methodology with some known techniques shows that the present approach is powerful and reliable. It is remarkable accuracy properties are finally demonstrated by two examples.

An iterative approach for solving fractional optimal control problems

2016 ◽  
Vol 24 (1) ◽  
pp. 18-36 ◽  
Author(s):  
Ali Alizadeh ◽  
Sohrab Effati
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Necessary Optimality Conditions ◽  
Variational Iteration Method ◽  
Successive Approximations ◽  
Classical Solutions ◽  
Control Problems ◽  
Lagrange Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

In this work, the variational iteration method (VIM) is used to solve a class of fractional optimal control problems (FOCPs). New Lagrange multipliers are determined and some new iterative formulas are presented. The fractional derivative (FD) in these problems is in the Caputo sense. The necessary optimality conditions are achieved for FOCPs in terms of associated Euler–Lagrange equations and then the VIM is used to solve the resulting fractional differential equations. This technique rapidly provides the convergent successive approximations of the exact solution and the solutions approach the classical solutions of the problem as the order of the FDs approaches 1. To achieve the solution of the FOCPs using VIM, four illustrative examples are included to demonstrate the validity and applicability of the proposed method.

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