A general formulation and solution scheme for fractional optimal control problems. (English) Zbl 1121.70019

Agrawal, Om Prakash

Nonlinear Dyn. 38, No. 1-4, 323-337 (2004).

The concept of fractional derivative (derivative of arbitrary order) leads to fractional differential equations and fractional dynamic systems (FDS). There are several definitions of fractional derivatives. Here the author formulates a fractional optimal control problem (FOCP) for fractional differential equations, written with the use of Riemann-Liouville definitions. This problem consists in finding the optimal control \(u(t)\) that minimizes the integral performance index subject to system dynamic constraints and initial conditions. With the use of the calculus of variations, Lagrange multipliers and the formula for fractional integration by parts, necessary conditions of optimality of the FOCP are obtained in the form of Euler-Lagrange equations, including left and right Riemann-Liouville fractional derivatives. As a special case, the FOCP is studied for linear FDS with the performance index in the form of integral of quadratic form in the state and in the control. For this case, the author proposes numerical scheme based on the approximation of FOCP solutions by a set of basic functions (shifted Legendre polynomials). The numerical scheme results in a system of linear algebraic equations for unknown coefficients.

Reviewer: Boris Ivanovich Konosevich (Donetsk)

MSC:

70Q05	Control of mechanical systems
26A33	Fractional derivatives and integrals

Keywords:

fractional derivative; necessary conditions for optimality; Euler-Lagrange equations

References:

[1]	Hestenes, M. R.,Calculus of Variations and Optimal Control Theory, Wiley, New York, 1966. · Zbl 0173.35703
[2]	Bryson, Jr. A. E. and Ho, Y. C.,Applied Optimal Control: Optimization, Estimation, and Control, Blaisdell, Waltham, Massachusetts, 1975.
[3]	Sage, A. P. and White, III, C. C.,Optimum Systems Control, Prentice-Hall, Englewood Cliffs, New Jersey, 1977. · Zbl 0388.49002
[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.
[5]	Carpinteri, A. and Mainardi, F.,Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Vienna, 1997. · Zbl 0917.73004
[6]	Podlubny, I.,Fractional Differential Equations, Academic Press, New York, 1999. · Zbl 0924.34008
[7]	Hilfer, R.,Applications of Fractional Calculus in Physics, World Scientific, River Edge, New Jersey, 2000. · Zbl 0998.26002
[8]	Machado, J. A. T. (guest editor),?Special issue on fractional calculus and applications?, Nonlinear Dynamics29, 2002, 1-386.
[9]	Miller, K. S. and Ross, B.,An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. · Zbl 0789.26002
[10]	Samko, S. G., Kilbas, A. A., and Marichev, O. I.,Fractional Integrals and Derivatives ? Theory and Applications, Gordon and Breach, Longhorne, Pennsylvania, 1993. · Zbl 0818.26003
[11]	Oldham, K. B. and Spanier, J.,The Fractional Calculus, Academic Press, New York, 1974. · Zbl 0292.26011
[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. · Zbl 0917.73004
[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. · Zbl 0917.73004
[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.
[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.
[16]	Bode, H. W.,Network Analysis and Feedback Amplifier Design, Van Nostrand, New York, 1945.
[17]	Manabe, S.,?The non-integer integral and its application to control?, Japanese Institute of Electrical Engineers80, 1960, 589-597.
[18]	Skaar, S. B., Michel, A. N., and Miller, R. K.,?Stability of viscoelastic control systems?, IEEE Transactions on Automatic Control33, 1988, 348-357. · Zbl 0641.93051
[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.
[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.
[21]	Mbodje, B. and Montseny, G.,?Boundary fractional derivative control of the wave equation?, IEEE Transactions on Automatic Control40, 1995, 378-382. · Zbl 0820.93034
[22]	Machado, J. A. T.,?Analysis and design of fractional-order digital control systems?, Systems Analysis Modelling Simulation27, 1997, 107-122. · Zbl 0875.93154
[23]	Machado, J. A. T.,?Fractional-order derivative approximations in discrete-time control systems?, Systems Analysis Modelling Simulation34, 1999, 419-434. · Zbl 0979.93519
[24]	Podlubny, I., Dorcak, L., and Kostial, I.,?On fractional derivatives, fractional-order dynamic systems and PI?D?-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.
[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.
[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. · Zbl 1021.93014
[27]	Hotzel, R.,?Some stability conditions for fractional delay systems?, Journal of Mathematical Systems, Estimation, and Control8, 1998, 499-502. · Zbl 0913.93068
[28]	Hartley, T. and Lorenzo, C. F.,?Dynamics and control of initialized fractional-order systems?, Nonlinear Dynamics29, 2002, 201-233. · Zbl 1021.93019
[29]	Riewe, F.,?Nonconservative {Lagrangian and Hamiltonian mechanics}?, Physical Review E53, 1996, 1890-1899.
[30]	Riewe, F.,?Mechanics with fractional derivatives?, Physical Review E55, 1997, 3582-3592.
[31]	Agrawal, O. P.,?Formulation of Euler-Lagrange equations for fractional variational problems?, Mathematical Analysis and Applications272, 2002, 368-379. · Zbl 1070.49013
[32]	Agrawal, O. P.,?General formulation for the numerical solution of optimal control problems?, International Journal of Control50, 1989, 627-638. · Zbl 0679.49031
[33]	Lorenzo, C. F. and Hartley, T. T.,?Initialized fractional calculus?, International Journal of Applied Mathematics3, 2000, 249-265. · Zbl 1172.26301
[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. · Zbl 1009.65049
[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. · Zbl 0987.26005
[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.