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.

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.

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 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.

A Numerical Scheme and an Error Analysis for a Class of Fractional Optimal Control Problems

2009 ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Error Analysis ◽  
Fractional Derivative ◽  
Performance Index ◽  
Numerical Solutions ◽  
Fractional Power ◽  
The State ◽  
Control Variables ◽  
Fractional Optimal Control ◽  
Linear Algebraic Equations

There has been a growing interest in recent years in the area of Fractional Optimal Control (FOC). In this paper, we present a formulation for a class of FOC problems, in which a performance index is defined as an integral of a quadratic function of the state and the control variables, and a dynamic constraint is defined as a Fractional Differential Equation (FDE) linear in both the state and the control variables. The fractional derivative is defined in the Caputo sense. In this formulation, the FOC problem is reduced to a Fractional Variational Problem (FVP), and the necessary differential equations for the problems are obtained using the recently developed theories for FVPs. For the numerical solutions of the problems, a direct approach is taken in which the solutions are approximated using a truncated Fractional Power Series (FPS). An error analysis is also performed. It is demonstrated that the solution converges from above in the sense that the value of the approximate performance index is always higher than the optimum performance index. An expression for the error in the performance index is also given. The application of a FPS and an optimality criterion reduces the FOC to a set of linear algebraic equations which are solved using a linear solver. It is demonstrated numerically that the solution converges as the number of terms in the series increases, and the approximate solution approaches to the analytical solution as the order of the fractional derivative approaches to an integer order derivative. Numerical results are presented to demonstrate the performance of the Formulation.

Fractional optimal control of distributed systems in spherical and cylindrical coordinates

2011 ◽  
Vol 18 (10) ◽  
pp. 1506-1525 ◽  
Author(s):  
M Mehedi Hasan ◽  
Xiangqing W Tangpong ◽  
Om Prakash Agrawal
Keyword(s):  
Optimal Control ◽  
Distributed Systems ◽  
Fractional Derivatives ◽  
Separation Of Variables ◽  
Cylindrical Coordinates ◽  
Control Variables ◽  
Fractional Optimal Control ◽  
Dynamic Constraints ◽  
The Time Domain ◽  
And Control

This paper presents a general formulation and numerical scheme for the fractional optimal control problem (FOCP) of distributed systems in spherical and cylindrical coordinates. The fractional derivatives are expressed in the Caputo-Sense. The performance index of 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. A method of separation of variables is employed to separate the time and space terms, and the eigenfunction approach is used to eliminate the terms containing space parameter and define the formulation in terms of countable number of infinite state and control variables. The fractional optimal control equations are reduced to the Volterra-type integral equations. For the problems considered, only a few eigenfunctions in each direction are sufficient for the calculations to converge. The time domain is discretized into several subintervals and the result is more stable for a larger number of time segments. Various orders of fractional derivatives are analyzed and the results converge toward those of integer optimal control problems as the order approaches the integer value of 1.

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.

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.

scholarly journals Numerical Solution of Some Types of Fractional Optimal Control Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Nasser Hassan Sweilam ◽  
Tamer Mostafa Al-Ajami ◽  
Ronald H. W. Hoppe
Keyword(s):  
Optimal Control ◽  
Numerical Solution ◽  
Optimal Control Problems ◽  
Ritz Method ◽  
Necessary Optimality Conditions ◽  
State Equation ◽  
The State ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

We present two different approaches for the numerical solution of fractional optimal control problems (FOCPs) based on a spectral method using Chebyshev polynomials. The fractional derivative is described in the Caputo sense. The first approach follows the paradigm “optimize first, then discretize” and relies on the approximation of the necessary optimality conditions in terms of the associated Hamiltonian. In the second approach, the state equation is discretized first using the Clenshaw and Curtis scheme for the numerical integration of nonsingular functions followed by the Rayleigh-Ritz method to evaluate both the state and control variables. Two illustrative examples are included to demonstrate the validity and applicability of the suggested approaches.

Fractional Optimal Control Within Caputo’s Derivative

2011 ◽  
Author(s):  
Raj Kumar Biswas ◽  
Siddhartha Sen
Keyword(s):  
Optimal Control ◽  
Arbitrary Order ◽  
Numerical Technique ◽  
Space Representation ◽  
Caputo Fractional Derivatives ◽  
Fractional Optimal Control ◽  
State Space Representation ◽  
Dynamic Constraint ◽  
Caputo’S Derivative ◽  
And Control

A general formulation and solution of fractional optimal control problems (FOCPs) in terms of Caputo fractional derivatives (CFDs) of arbitrary order have been considered in this paper. The performance index (PI) of a FOCP is considered as a function of both the state and control. The dynamic constraint is expressed by a fractional differential equation (FDE) of arbitrary order. A general pseudo-state-space representation of the FDE is presented and based on that, FOCP has been developed. A numerical technique based on Gru¨nwald-Letnikov (G-L) approximation of the FDs is used for solving the resulting equations. Numerical example is presented to show the effectiveness of the formulation and solution scheme.

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