Fractional Optimal Control of Continuum Systems

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
X. W. Tangpong ◽  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Numerical Scheme ◽  
Numerical Technique ◽  
Fractional Integration ◽  
Scalar Case ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Time Step ◽  
Fractional Optimal Control ◽  
Continuum Systems

This paper presents a formulation and a numerical scheme for fractional optimal control (FOC) of a class of continuum systems. The fractional derivative is defined in the Caputo sense. The performance index of a fractional optimal control problem is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a partial fractional differential equation. The scheme presented relies on reducing the equations of a continuum system into a set of equations that have no space parameter. Several strategies are pointed out for this task, and one of them is discussed in detail. The numerical scheme involves discretizing the space domain into several segments, and expressing the spatial derivatives in terms of variables at spatial node points. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain the Euler–Lagrange equations for the problem. The numerical technique presented in the work of Agrawal (2006, “A Formulation and a Numerical Scheme for Fractional Optimal Control Problems,” Proceedings of the Second IFAC Conference on Fractional Differentiations and Its Applications, FDA ‘06, Porto, Portugal) for the scalar case is extended for the vector case. In this method, the FOC equations are reduced to the Volterra type integral equations. The time domain is also discretized into a number of subintervals. For the linear case, the numerical technique results in a set of algebraic equations that can be solved using a direct or an iterative scheme. An example problem is solved for various orders of fractional derivatives and different spatial and temporal discretizations. For the problem considered, only a few space grid points are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced. The formulation presented is simple and can be extended to FOC of other continuum systems.

Fractional Optimal Control of Distributed Systems

2007 ◽  
Author(s):  
X. W. Tangpong ◽  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Distributed Systems ◽  
Numerical Technique ◽  
Fractional Integration ◽  
Scalar Case ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Time Step ◽  
Fractional Optimal Control ◽  
The Time Domain

This paper presents a formulation and a numerical scheme for Fractional Optimal Control (FOC) for a class of distributed systems. The fractional derivative is defined in the Caputo sense. The performance index of a Fractional Optimal Control Problem (FOCP) is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). The scheme presented rely on reducing the equations for distributed system into a set of equations that have no space parameter. Several strategies are pointed out for this task, and one of them is discussed in detail. This involves discretizing the space domain into several segments, and writing the spatial derivatives in terms of variables at space node points. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the problem. The numerical technique presented in [1] for scalar case is extended for the vector case. In this technique, the FOC equations are reduced to Volterra type integral equations. The time domain is also descretized into several segments. For the linear case, the numerical technique results into a set of algebraic equations which can be solved using a direct or an iterative scheme. The problem is solved for various order of fractional derivatives and various order of space and time discretizations. Numerical results show that for the problem considered, only a few space grid points are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced. The formulation presented is simple and can be extended to FOC of other distributed systems.

Fractional Optimal Control of a Distributed System Using Eigenfunctions

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Distributed Systems ◽  
Dimensional Space ◽  
Numerical Technique ◽  
Finite Dimensional Space ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Time Step ◽  
Control Variables ◽  
Fractional Optimal Control

This paper presents a formulation and a numerical scheme for fractional optimal control (FOC) for a class of distributed systems. The fractional derivative is defined in the Caputo sense. The performance index of an FOC problem (FOCP) is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by partial fractional differential equations. Eigenfunctions are used to eliminate the space parameter and to define the problem in terms of a set of state and control variables. This leads to a multi-FOCP in which each FOCP could be solved independently. Several other strategies are pointed out to reduce the problem to a finite dimensional space, some of which may not provide a decoupled set of 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 problem. In the proposed technique, the FOC equations are reduced to Volterra-type integral equations. The time domain is discretized into several segments and a time marching scheme is used to obtain the response at discrete time points. For a linear case, the numerical technique results into a set of algebraic equations, which can be solved using a direct or an iterative scheme. The problem is solved for different number of eigenfunctions and time discretizations. Numerical results show that only a few eigenfunctions are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced. The formulation presented is simple and can be extended to FOC of other distributed systems.

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.

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.

Fractional Optimal Control of a Distributed System Using Eigenfunctions

2007 ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Distributed Systems ◽  
Dimensional Space ◽  
Numerical Technique ◽  
The State ◽  
Finite Dimensional Space ◽  
Lagrange Equations ◽  
Time Step ◽  
Control Variables ◽  
Fractional Optimal Control

This paper presents a formulation and a numerical scheme for Fractional Optimal Control (FOC) for a class of distributed systems. The fractional derivative is defined 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 Partial Fractional Differential Equations (PFDEs). Eigenfunctions are used to eliminate the space parameter, and to define the problem in terms of a set of state and control variables. This leads to a multi FOCP in which each FOCP could be solved independently. Several other strategies are pointed out to reduce the problem to a finite dimensional space, some of which may not provide a decoupled set of 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 problem. The numerical technique presented in [1] is used to obtain the state and the control variables. In this technique, the FOC equations are reduced to Volterra type integral equations. The time domain is descretized into several segments and a time marching scheme is used to obtain the response at discrete time points. For a linear case, the numerical technique results into a set of algebraic equations which can be solved using a direct or an iterative scheme. The problem is solved for different number of eigenfunctions and time discretizations. Numerical results show that only a few eigenfunctions are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced. The formulation presented is simple and can be extended to FOC of other distributed systems.

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

2021 ◽  
pp. 107754632110169
Author(s):  
Hossein Jafari ◽  
Roghayeh M Ganji ◽  
Khosro Sayevand ◽  
Dumitru Baleanu
Keyword(s):  
Optimal Control ◽  
Approximate Solution ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Numerical Approach ◽  
Control Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

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.

Numerical solution for fractional optimal control problems by Hermite polynomials

2020 ◽  
pp. 107754632093312
Author(s):  
Ayatollah Yari
Keyword(s):  
Optimal Control ◽  
Numerical Method ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Control Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

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.

scholarly journals Application of Bernoulli Polynomials for Solving Variable-Order Fractional Optimal Control-Affine Problems

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 114 ◽  
Author(s):  
Somayeh Nemati ◽  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Initial Conditions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Bernoulli Polynomials ◽  
Test Problems ◽  
Variable Order ◽  
State Function ◽  
Algebraic Equations ◽  
Fractional Optimal Control

We propose two efficient numerical approaches for solving variable-order fractional optimal control-affine problems. The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann–Liouville integral operator is used in our new techniques. An accurate operational matrix of variable-order fractional integration for Bernoulli polynomials is introduced. Our methods proceed as follows. First, a specific approximation of the differentiation order of the state function is considered, in terms of Bernoulli polynomials. Such approximation, together with the initial conditions, help us to obtain some approximations for the other existing functions in the dynamical control-affine system. Using these approximations, and the Gauss—Legendre integration formula, the problem is reduced to a system of nonlinear algebraic equations. Some error bounds are then given for the approximate optimal state and control functions, which allow us to obtain an error bound for the approximate value of the performance index. We end by solving some test problems, which demonstrate the high accuracy of our results.

An efficient approximate method for solving two-dimensional fractional optimal control problems using generalized fractional order of Bernstein functions

2020 ◽  
Author(s):  
Ali Ketabdari ◽  
Mohammad Hadi Farahi ◽  
Sohrab Effati
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Operational Matrix ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Control Variables ◽  
Fractional Optimal Control ◽  
Bernstein Functions ◽  
Fractional Optimal Control Problems ◽  
And Control

Abstract We define a new operational matrix of fractional derivative in the Caputo type and apply a spectral method to solve a two-dimensional fractional optimal control problem (2D-FOCP). To acquire this aim, first we expand the state and control variables based on the fractional order of Bernstein functions. Then we reduce the constraints of 2D-FOCP to a system of algebraic equations through the operational matrix. Now, one can solve straightforward the problem and drive the approximate solution of state and control variables. The convergence of the method in approximating the 2D-FOCP is proved. We demonstrate the efficiency and superiority of the method by comparing the results obtained by the presented method with the results of previous methods in some examples.

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