Fractional Optimal Control of Continuum Systems

  • Tangpong X
  • Agrawal O
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Abstract

Thispaper presents a formulation and a numerical scheme for fractionaloptimal control (FOC) of a class of continuum systems. Thefractional derivative is defined in the Caputo sense. The performanceindex of a fractional optimal control problem is considered asa function of both the state and the control variables,and the dynamic constraints are expressed by a partial fractionaldifferential equation. The scheme presented relies on reducing the equationsof a continuum system into a set of equations thathave no space parameter. Several strategies are pointed out forthis task, and one of them is discussed in detail.The numerical scheme involves discretizing the space domain into severalsegments, and expressing the spatial derivatives in terms of variablesat spatial node points. The calculus of variations, the Lagrangemultiplier, and the formula for fractional integration by parts areused to obtain the EulerLagrange equations for the problem. Thenumerical technique presented in the work of Agrawal (2006, AFormulation and a Numerical Scheme for Fractional Optimal Control Problems,Proceedings of the Second IFAC Conference on Fractional Differentiations andIts Applications, FDA `06, Porto, Portugal) for the scalar caseis extended for the vector case. In this method, theFOC equations are reduced to the Volterra type integral equations.The time domain is also discretized into a number ofsubintervals. For the linear case, the numerical technique results ina set of algebraic equations that can be solved usinga direct or an iterative scheme. An example problem issolved for various orders of fractional derivatives and different spatialand temporal discretizations. For the problem considered, only a fewspace grid points are sufficient to obtain good results, andthe solutions converge as the size of the time stepis reduced. The formulation presented is simple and can beextended to FOC of other continuum systems. 2009 American Society of Mechanical Engineers

Author-supplied keywords

  • algebra
  • optimal control
  • partial differential equations
  • time domain analysis
  • volterra equations

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Authors

  • X W Tangpong

  • Om P Agrawal

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