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Research Papers

A Quadratic Numerical Scheme for Fractional Optimal Control Problems

[+] Author and Article Information
Om P. Agrawal

Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901om@engr.siu.edu

J. Dyn. Sys., Meas., Control 130(1), 011010 (Dec 27, 2007) (6 pages) doi:10.1115/1.2814055 History: Received July 02, 2006; Revised April 06, 2007; Published December 27, 2007

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.

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Copyright © 2008 by American Society of Mechanical Engineers
Topics: Optimal control
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References

Figures

Grahic Jump Location
Figure 1

Convergence of the state variable x(t) for α=0.8 (˟, N=10; ○, N=20; ▵, N=40; +, N=80; ◻, N=160; ◇, N=320)

Grahic Jump Location
Figure 3

State variable x(t) as a function of time for different values of α and N=10 [˟, α=0.6; ○, α=0.7; ▵, α=0.8; +, α=0.9; ◻, α=1.0; ◇, α=1.0 (analytical)]

Grahic Jump Location
Figure 6

Convergence of the control variable u(t) for α=0.8 (˟, N=10; ○, N=20; ▵, N=40; +, N=80; ◻, N=160; ◇, N=320)

Grahic Jump Location
Figure 7

State variable x(t) as a function of time for different values of α and N=10 (˟, α=0.6; ○, α=0.7; ▵, α=0.8; +, α=0.9; ◻, α=1.0)

Grahic Jump Location
Figure 8

Control variable u(t) as a function of time for different values of α and N=10 (˟, α=0.6, ○, α=0.7; ▵, α=0.8; +, α=0.9; ◻ α=1.0)

Grahic Jump Location
Figure 2

Convergence of the control variable u(t) for α=0.8 (˟, N=10; ○, N=20; ▵, N=40; +, N=80; ◻, N=160; ◇, N=320)

Grahic Jump Location
Figure 4

Control variable u(t) as a function of time for different values of α and N=10 (˟, α=0.6; ○, α=0.7; ▵, α=0.8; +, α=0.9; ◻, α=1.0; ◇, α=1.0 (analytical))

Grahic Jump Location
Figure 5

Convergence of the state variable x(t) for α=0.8 (˟, N=10; ○, N=20; ▵, N=40; +, N=80; ◻, N=160; ◇, N=320)

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