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Time-optimal control of systems with fractional dynamics. (English) Zbl 1203.49031
Summary: We introduce a formulation for the time-optimal control problems of systems displaying fractional dynamics in the sense of the Riemann-Liouville fractional derivatives operator. To propose a solution to the general time-optimal problem, a rational approximation based on the Hankel data matrix of the impulse response is considered to emulate the behavior of the fractional differentiation operator. The original problem is then reformulated according to the new model which can be solved by traditional optimal control problem solvers. The time-optimal problem is extensively investigated for a double fractional integrator and its solution is obtained using numerical optimization time-domain analysis.

MSC:
49K21 Optimality conditions for problems involving relations other than differential equations
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
Software:
RIOTS_95
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References:
[1] A. Oustaloup, F. Levron, B. Mathieu, and F. M. Nanot, “Frequency-band complex noninteger differentiator: characterization and synthesis,” IEEE Transactions on Circuits and Systems I, vol. 47, no. 1, pp. 25-39, 2000. · doi:10.1109/81.817385
[2] M. Zamani, M. Karimi-Ghartemani, and N. Sadati, “FOPID controller design for robust performance using particle swarm optimization,” Fractional Calculus & Applied Analysis, vol. 10, no. 2, pp. 169-187, 2007. · Zbl 1141.93351 · eudml:11326
[3] L. Sommacal, P. Melchior, A. Oustaloup, J.-M. Cabelguen, and A. J. Ijspeert, “Fractional multi-models of the frog gastrocnemius muscle,” Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1415-1430, 2008. · Zbl 1229.92019 · doi:10.1177/1077546307087440
[4] D. E. Kirk, Optimal Control Theory: An Introduction, Dover, New York, NY, USA, 2004. · Zbl 1083.53061
[5] O. P. Agrawal, “A general formulation and solution scheme for fractional optimal control problems,” Nonlinear Dynamics, vol. 38, no. 1-4, pp. 323-337, 2004. · Zbl 1121.70019 · doi:10.1007/s11071-004-3764-6
[6] C. Tricaud and Y. Q. Chen, “Solving fractional order optimal control problems in riots 95-a general-purpose optimal control problem solver,” in Proceedings of the 3rd IFAC Workshop on Fractional Differentiation and Its Applications, Ankara, Turkey, November 2008.
[7] O. P. Agrawal and D. Baleanu, “A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems,” Journal of Vibration and Control, vol. 13, no. 9-10, pp. 1269-1281, 2007. · Zbl 1182.70047 · doi:10.1177/1077546307077467
[8] O. P. Agrawal, “A quadratic numerical scheme for fractional optimal control problems,” ASME Journal of Dynamic Systems, Measurement and Control, vol. 130, no. 1, Article ID 011010, 6 pages, 2008. · doi:10.1115/1.2814055
[9] O. P. Agrawal, “Fractional optimal control of a distributed system using eigenfunctions,” Journal of Computational and Nonlinear Dynamics, vol. 3, no. 2, Article ID 021204, 6 pages, 2008. · doi:10.1115/1.2833873
[10] N. Özdemir, D. Karadeniz, and B. B. \DIskender, “Fractional optimal control problem of a distributed system in cylindrical coordinates,” Physics Letters A, vol. 373, no. 2, pp. 221-226, 2009. · Zbl 1227.49007 · doi:10.1016/j.physleta.2008.11.019
[11] N. Özdemir, O. P. Agrawal, B. B. \DIskender, and D. Karadeniz, “Fractional optimal control of a 2-dimensional distributed system using eigenfunctions,” Nonlinear Dynamics, vol. 55, no. 3, pp. 251-260, 2009. · Zbl 1170.70397 · doi:10.1007/s11071-008-9360-4
[12] D. Baleanu, O. Defterli, and O. P. Agrawal, “A central difference numerical scheme for fractional optimal control problems,” Journal of Vibration and Control, vol. 15, no. 4, pp. 583-597, 2009. · Zbl 1272.49068 · doi:10.1177/1077546308088565
[13] G. S. F. Frederico and D. F. M. Torres, “Noether/s theorem for fractional optimal control problems,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, Porto, Portugal, July 2006.
[14] G. S. F. Frederico and D. F. M. Torres, “Fractional conservation laws in optimal control theory,” Nonlinear Dynamics, vol. 53, no. 3, pp. 215-222, 2008. · Zbl 1170.49017 · doi:10.1007/s11071-007-9309-z
[15] G. S. F. Frederico and D. F. M. Torres, “Fractional optimal control in the sense of Caputo and the fractional Noether/s theorem,” International Mathematical Forum, vol. 3, no. 10, pp. 479-493, 2008. · Zbl 1154.49016
[16] Z. D. Jelicic and N. Petrovacki, “Optimality conditions and a solution scheme for fractional optimal control problems,” Structural and Multidisciplinary Optimization, vol. 38, no. 6, pp. 571-581, 2009. · Zbl 1274.49035 · doi:10.1007/s00158-008-0307-7
[17] P. L. Butzer and U. Westphal, “An introduction to fractional calculus,” in Applications of Fractional Calculus in Physics, pp. 1-85, World Scientific, River Edge, NJ, USA, 2000. · Zbl 0987.26005
[18] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0789.26002
[19] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[20] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 11 of Mathematics in Science and Engineering, Academic Press, London, UK, 1974. · Zbl 0292.26011
[21] D. S. Naidu, Optimal Control Systems, vol. 2 of Electrical Engineering Series, CRC Press, Boca Raton, Fla, USA, 2002. · Zbl 1041.93002
[22] C. Tricaud and Y. Q. Chen, “Solution of fractional order optimal control problems using SVD-based rational approximations,” in Proceedings of the American Control Conference (ACC /09), pp. 1430-1435, St. Louis, Mo, USA, June 2009. · doi:10.1109/ACC.2009.5160677
[23] A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere, New York, NY, USA, 1975.
[24] A. L. Schwartz, Theory and implementation of numerical methods based on runge-kutta integration for solving optimal control problems, Ph.D. thesis, University of California at Berkeley, Berkeley, Calif, USA, 1989.
[25] A. L. Schwartz, E. Polak, and Y. Q. Chen, “RIOTS-A MATLAB Toolbox for Solving Optimal Control Problems,” 1997, http://www.accesscom.com/ adam/RIOTS/.
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