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Research Papers

Fractional Optimal Control of Continuum Systems

[+] Author and Article Information
X. W. Tangpong1

Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58108annie.tangpong@ndsu.edu

Om P. Agrawal

Department of Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901

1

Corresponding author.

J. Vib. Acoust 131(2), 021012 (Feb 19, 2009) (6 pages) doi:10.1115/1.3025833 History: Received November 10, 2007; Revised August 27, 2008; Published February 19, 2009

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.

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Copyright © 2009 by American Society of Mechanical Engineers
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References

Figures

Grahic Jump Location
Figure 1

Convergence of the state variable x; α=0.6 and y=0.5

Grahic Jump Location
Figure 2

Convergence of the control variable u; α=0.6 and y=0.5

Grahic Jump Location
Figure 3

State variable x for different values of α; h=0.025 and y=0.5

Grahic Jump Location
Figure 4

Control variable u for different values of α; h=0.025 and y=0.5

Grahic Jump Location
Figure 5

State variable vector x(y,t); α=0.6 and h=0.025

Grahic Jump Location
Figure 6

Control variable vector u(y,t); α=0.6 and h=0.025

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