Nonlinear Dynamics

, Volume 38, Issue 1–4, pp 323–337 | Cite as

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

  • Om Prakash AgrawalEmail author


Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville 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 FDEs. 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 system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) 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. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.


Lagrange Multiplier Fractional Derivative Fractional Differential Equation Virtual Work Solution Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Hestenes, M. R.,Calculus of Variations and Optimal Control Theory, Wiley, New York, 1966.Google Scholar
  2. 2.
    Bryson, Jr. A. E. and Ho, Y. C.,Applied Optimal Control: Optimization, Estimation, and Control, Blaisdell, Waltham, Massachusetts, 1975.Google Scholar
  3. 3.
    Sage, A. P. and White, III, C. C.,Optimum Systems Control, Prentice-Hall, Englewood Cliffs, New Jersey, 1977.Google Scholar
  4. 4.
    Bagley, R. L. and Calico, R. A.,‘Fractional order state equations for the control of viscoelastically damped structures’, Journal of Guidance, Control, and Dynamics14, 1991, 304–311.Google Scholar
  5. 5.
    Carpinteri, A. and Mainardi, F.,Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Vienna, 1997.Google Scholar
  6. 6.
    Podlubny, I.,Fractional Differential Equations, Academic Press, New York, 1999.Google Scholar
  7. 7.
    Hilfer, R.,Applications of Fractional Calculus in Physics, World Scientific, River Edge, New Jersey, 2000.Google Scholar
  8. 8.
    Machado, J. A. T. (guest editor),‘Special issue on fractional calculus and applications’, Nonlinear Dynamics29, 2002, 1–386.Google Scholar
  9. 9.
    Miller, K. S. and Ross, B.,An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.Google Scholar
  10. 10.
    Samko, S. G., Kilbas, A. A., and Marichev, O. I.,Fractional Integrals and Derivatives – Theory and Applications, Gordon and Breach, Longhorne, Pennsylvania, 1993.Google Scholar
  11. 11.
    Oldham, K. B. and Spanier, J.,The Fractional Calculus, Academic Press, New York, 1974.Google Scholar
  12. 12.
    Gorenflo, R. and Mainardi, F.,‘Fractional calculus: Integral and differential equations of fractional order’, in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi (eds), Springer-Verlag, Vienna, 1997, pp. 291–348.Google Scholar
  13. 13.
    Mainardi, F.,‘Fractional calculus: Some basic problems in continuum and statistical mechanics’, in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi (eds), Springer-Verlag, Vienna, 1997, pp. 291–348.Google Scholar
  14. 14.
    Rossikhin, Y. A. and Shitikova, M. V.,‘Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids’, Applied Mechanics Reviews50, 1997, 15–67.Google Scholar
  15. 15.
    Manabe, S.,‘Early development of fractional order control’, DETC2003/VIB-48370, in Proceedings of DETC–03, ASME 2003 Design Engineering Technical Conference, Chicago, Illinois, September 2–6, 2003.Google Scholar
  16. 16.
    Bode, H. W.,Network Analysis and Feedback Amplifier Design, Van Nostrand, New York, 1945.Google Scholar
  17. 17.
    Manabe, S.,‘The non-integer integral and its application to control’, Japanese Institute of Electrical Engineers80, 1960, 589–597.Google Scholar
  18. 18.
    Skaar, S. B., Michel, A. N., and Miller, R. K.,‘Stability of viscoelastic control systems’, IEEE Transactions on Automatic Control33, 1988, 348–357.Google Scholar
  19. 19.
    Axtell, M. and Bise, M. E.,‘Fractional calculus applications in control systems’, IEEE Proceedings of the National Aerospace and Electronics Conference, Dayton, OH, USA, May 21–25, 1990, pp. 563–566.Google Scholar
  20. 20.
    Makroglou, A., Miller, R. K., and Skaar, S.,Computational results for a feedback control for a rotating viscoelastic beam’, Journal of Guidance, Control, and Dynamics17, 1994, 84–90.Google Scholar
  21. 21.
    Mbodje, B. and Montseny, G.,‘Boundary fractional derivative control of the wave equation’, IEEE Transactions on Automatic Control40, 1995, 378–382.Google Scholar
  22. 22.
    Machado, J. A. T.,‘Analysis and design of fractional-order digital control systems’, Systems Analysis Modelling Simulation27, 1997, 107–122.Google Scholar
  23. 23.
    Machado, J. A. T.,‘Fractional-order derivative approximations in discrete-time control systems’, Systems Analysis Modelling Simulation34, 1999, 419–434.Google Scholar
  24. 24.
    Podlubny, I., Dorcak, L., and Kostial, I.,‘On fractional derivatives, fractional-order dynamic systems and PIλDμ-controllers’, in Proceedings of the 1997 36th IEEE Conference on Decision and Control, Part 5, San Diego, California, December 10–12, 1997, pp. 4985–4990.Google Scholar
  25. 25.
    Oustaloup, A., Levron, F., Mathieu, B., and Nanot, F. M.,‘Frequency-band complex noninteger differentiator: characterization and synthesis’, IEEE Transactions on Circuits and Systems – Fundamental Theory and Applications40, 2000, 25–39.Google Scholar
  26. 26.
    Sabatier, J., Oustaloup, A., Iturricha, A. G., and Lanusse, P.,‘CRONE control: principles and extension to time-variant plants with asymptotically constant coefficients’, Nonlinear Dynamics29, 2002, 363–385.Google Scholar
  27. 27.
    Hotzel, R.,‘Some stability conditions for fractional delay systems’, Journal of Mathematical Systems, Estimation, and Control8, 1998, 499–502.Google Scholar
  28. 28.
    Hartley, T. and Lorenzo, C. F.,‘Dynamics and control of initialized fractional-order systems’, Nonlinear Dynamics29, 2002, 201–233.Google Scholar
  29. 29.
    Riewe, F.,‘Nonconservative {Lagrangian and Hamiltonian mechanics}’, Physical Review E53, 1996, 1890–1899.Google Scholar
  30. 30.
    Riewe, F.,‘Mechanics with fractional derivatives’, Physical Review E55, 1997, 3582–3592.Google Scholar
  31. 31.
    Agrawal, O. P.,‘Formulation of Euler-Lagrange equations for fractional variational problems’, Mathematical Analysis and Applications272, 2002, 368–379.Google Scholar
  32. 32.
    Agrawal, O. P.,‘General formulation for the numerical solution of optimal control problems’, International Journal of Control50, 1989, 627–638.Google Scholar
  33. 33.
    Lorenzo, C. F. and Hartley, T. T.,‘Initialized fractional calculus’, International Journal of Applied Mathematics3, 2000, 249–265.Google Scholar
  34. 34.
    Deithelm, K., Ford, N. J., and Freed, A. D.,‘A predictor-corrector approach for the numerical solution of fractional differential equations’, Nonlinear Dynamics29, 2002, 3–22.Google Scholar
  35. 35.
    Butzer, P. L. and Westphal, U,‘An introduction to fractional calculus’, in Applications of Fractional Calculus in Physics, R. Hilfer (ed), World Scientific, New Jersey, 2000, pp. 1–85.Google Scholar
  36. 36.
    Agrawal, O. P. and Saigal, S.,‘A novel, computationally efficient approach for hamilton–s law of varying action’, International Journal of Mechanical Sciences29, 1987, 285–292.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringSouthern Illinois UniversityCarbondaleU.S.A.

Personalised recommendations