Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2010, Article ID 375858, 19 pages
http://dx.doi.org/10.1155/2010/375858
Review Article

Fractional Order Calculus: Basic Concepts and Engineering Applications

1Department of Mechanical Engineering, UNICAMP, 13083-970 Campinas, Brazil
2Department of Electrical Engineering, ISEP, 4200-072 Oporto, Portugal

Received 7 October 2009; Revised 19 February 2010; Accepted 2 March 2010

Academic Editor: Katica R. (Stevanovic) Hedrih

Copyright © 2010 Ricardo Enrique Gutiérrez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. A. T. Machado, “A probabilistic interpretation of the fractional-order differentiation,” Fractional Calculus and applied Analysis, vol. 6, no. 1, pp. 73–80, 2003. View at Google Scholar · View at Zentralblatt MATH
  2. Q.-S. Zeng, G.-Y. Cao, and X.-J. Zhu, “The effect of the fractional-order controller's orders variation on the fractional-order control systems,” in Proceedings of the 1st International Conference on Machine Learning and Cybernetics, vol. 1, pp. 367–372, 2002. View at Scopus
  3. R. L. Magin and M. Ovadia, “Modeling the cardiac tissue electrode interface using fractional calculus,” Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1431–1442, 2008. View at Publisher · View at Google Scholar · View at Scopus
  4. 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. View at Publisher · View at Google Scholar · View at Scopus
  5. N. Heymans, “Dynamic measurements in long-memory materials: fractional calculus evaluation of approach to steady state,” Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1587–1596, 2008. View at Publisher · View at Google Scholar · View at Scopus
  6. J. De Espíndola, C. Bavastri, and E. De Oliveira Lopes, “Design of optimum systems of viscoelastic vibration absorbers for a given material based on the fractional calculus model,” Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1607–1630, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  7. B. T. Krishna and K. V. V. S. Reddy, “Active and passive realization of fractance device of order 1/2,” Active and Passive Electronic Components, vol. 2008, Article ID 369421, 5 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus
  8. Y. Pu, X. Yuan, K. Liao et al., “A recursive two-circuits series analog fractance circuit for any order fractional calculus,” in ICO20: Optical Information Processing, vol. 6027 of Proceedings of SPIE, pp. 509–519, August 2006. View at Publisher · View at Google Scholar · View at Scopus
  9. M. F. M. Lima, J. A. T. Machado, and M. Crisóstomo, “Experimental signal analysis of robot impacts in a fractional calculus perspective,” Journal of Advanced Computational Intelligence and Intelligent Informatics, vol. 11, no. 9, pp. 1079–1085, 2007. View at Google Scholar
  10. J. Rosario, D. Dumur, and J. T. Machado, “Analysis of fractional-order robot axis dynamics,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  11. L. Debnath, “Recent applications of fractional calculus to science and engineering,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 54, pp. 3413–3442, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. G. W. Bohannan, “Analog fractional order controller in temperature and motor control applications,” Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1487–1498, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  13. J. Cervera and A. Baños, “Automatic loop shaping in QFT using CRONE structures,” Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1513–1529, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  14. R. Panda and M. Dash, “Fractional generalized splines and signal processing,” Signal Processing, vol. 86, no. 9, pp. 2340–2350, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. Z.-Z. Yang and J.-L. Zhou, “An improved design for the IIR-type digital fractional order differential filter,” in Proceedings of the International Seminar on Future BioMedical Information Engineering (FBIE '08), pp. 473–476, December 2008. View at Publisher · View at Google Scholar · View at Scopus
  16. I. Petráš, “A note on the fractional-order cellular neural networks,” in Proceedings of the International Joint Conference on Neural Networks, pp. 1021–1024, July 2006. View at Scopus
  17. L. Dorcak, I. Petras, I. Kostial, and J. Terpak, “Fractional-order state space models,” in Proceedings of the International Carpathian Control Conference, pp. 193–198, 2002.
  18. D. Cafagna, “Past and present—fractional calculus: a mathematical tool from the past for present engineers,” IEEE Industrial Electronics Magazine, vol. 1, no. 2, pp. 35–40, 2007. View at Google Scholar · View at Scopus
  19. A. Benchellal, T. Poinot, and J.-C. Trigeassou, “Fractional modelling and identification of a thermal process,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  20. I. S. Jesus, J. A. T. Machado, and J. B. Cunha, “Fractional electrical dynamics in fruits and vegetables,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  21. W. M. Ahmad and R. El-Khazali, “Fractional-order dynamical models of love,” Chaos, Solitons and Fractals, vol. 33, no. 4, pp. 1367–1375, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. A. Oustaloup, J. Sabatier, and X. Moreau, “From fractal robustness to the CRONE approach,” in Systèmes Différentiels Fractionnaires (Paris, 1998), vol. 5, pp. 177–192, SIAM, Paris, France, 1998. View at Google Scholar
  23. I. Podlubny, “The laplace transform method for linear differential equations of the fractional order,” Tech. Rep., Slovak Academy of Sciences, Institute of Experimental Physics, 1994. View at Google Scholar
  24. D. Xue, C. Zhao, and Y. Chen, “A modified approximation method of fractional order system,” in Proceedings of the IEEE International Conference on Mechatronics and Automation (ICMA '06), pp. 1043–1048, 2006. View at Publisher · View at Google Scholar · View at Scopus
  25. J. L. Adams, T. T. Hartley, and C. F. Lorenzo, “Fractional-order system identification using complex order-distributions,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  26. M. D. Ortigueira, J. A. T. Machado, and J. S. Da Costa, “Which differintegration?” IEE Proceedings: Vision, Image and Signal Processing, vol. 152, no. 6, pp. 846–850, 2005. View at Publisher · View at Google Scholar · View at Scopus
  27. D. Xue and Y. Q. Chen, “A comparative introduction of four fractional order controllers,” in Proceedings of the 4th World Congress on Intelligent Control and Automation, vol. 4, pp. 3228–3235, 2002. View at Scopus
  28. R. L. Magin and M. Ovadia, “Modeling the cardiac tissue electrode interface using fractional calculus,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  29. K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 1, Dover, New York, NY, USA, 2006.
  30. C. Ma and Y. Hori, “Fractional order control and its application of PIαD controller for robust two-inertia speed control,” in Proceedings of the 4th International Power Electronics and Motion Control Conference (IPEMC '04), vol. 3, pp. 1477–1482, August 2004. View at Scopus
  31. I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. M. Moshrefi-Torbati and J. K. Hammond, “Physical and geometrical interpretation of fractional operators,” Journal of the Franklin Institute, vol. 335, no. 6, pp. 1077–1086, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. S. Miyazima, Y. Oota, and Y. Hasegawa, “Fractality of a modified Cantor set and modified Koch curve,” Physica A, vol. 233, no. 3-4, pp. 879–883, 1996. View at Publisher · View at Google Scholar · View at Scopus
  34. F. B. M. Duarte and J. A. T. Machado, “Fractional dynamics in the describing function analysis of nonlinear friction,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  35. P. J. Torvik and R. L. Bagley, “On the appearance of the fractional derivative in the behaviour of real materials,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 294–298, 1984. View at Google Scholar · View at Scopus
  36. D. Baleanu and S. I. Muslih, “Nonconservative systems within fractional generalized derivatives,” Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1301–1311, 2008. View at Publisher · View at Google Scholar · View at Scopus
  37. T. Poinot and J.-C. Trigeassou, “A method for modelling and simulation of fractional systems,” Signal Processing, vol. 83, no. 11, pp. 2319–2333, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  38. S. Oh and Y. Hori, “Realization of fractional order impedance by feedback control,” in Proceedings of the 33rd Annual Conference of the IEEE Industrial Electronics Society (IECON '07), pp. 299–304, 2007. View at Publisher · View at Google Scholar · View at Scopus
  39. R. Caponetto and D. Porto, “Analog implementation of non integer order integrator via field programmable analog array,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  40. T. C. Haba, G. Ablart, T. Camps, and F. Olivie, “Influence of the electrical parameters on the input impedance of a fractal structure realised on silicon,” Chaos, Solitons and Fractals, vol. 24, no. 2, pp. 479–490, 2005. View at Publisher · View at Google Scholar · View at Scopus
  41. I. Podlubny, I. Petráš, B. M. Vinagre, P. O'Leary, and L. Dorčák, “Analogue realizations of fractional-order controllers,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 281–296, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  42. J. J. De Espiíndola, J. M. Da Silva Neto, and E. M. O. Lopes, “A generalised fractional derivative approach to viscoelastic material properties measurement,” Applied Mathematics and Computation, vol. 164, no. 2, pp. 493–506, 2005. View at Publisher · View at Google Scholar · View at Scopus
  43. B. M. Vinagre and V. Feliu, “Optimal fractional controllers for rational order systems: a special case of the Wiener-Hopf spectral factorization method,” IEEE Transactions on Automatic Control, vol. 52, no. 12, pp. 2385–2389, 2007. View at Publisher · View at Google Scholar · View at Scopus
  44. T. T. Hartley and C. F. Lorenzo, “Fractional-order system identification based on continuous order-distributions,” Signal Processing, vol. 83, no. 11, pp. 2287–2300, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  45. A. Djouambi, A. Charef, and A. V. Besan, “Approximation and synthesis of non integer order systems,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  46. B. M. Vinagre, V. Feliu, and J. J. Feliu, “Frequency domain identification of a flexible structure with piezoelectric actuators using irrational transfer function models,” in Proceedings of the 37th IEEE Conference on Decision and Control, vol. 17, pp. 1278–1280, December 1998.
  47. F. Benoit-Marand, L. Signac, T. Poinot, and J.-C. Trigeassou, “Identification of non linear fractional systems using continuous time neural networks,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  48. D. Xue, C. Zhao, and Y. Chen, “Fractional order PID control of A DC-motor with elastic shaft: a case study,” in Proceedings of the American Control Conference, pp. 3182–3187, 2006. View at Scopus
  49. Y. Chen, “Ubiquitous fractional order controls?” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  50. A. Oustaloup, J. Sabatier, and X. Moreau, “From fractal robustness to the CRONE approach,” in Fractional Differential Systemas: Models, Methods and Applications, vol. 5, pp. 177–192, SIAM, Paris, France, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  51. N. Sadati, M. Zamani, and P. Mohajerin, “Optimum design of fractional order PID for MIMO and SISO systems using particle swarm optimization techniques,” in Proceedings of the 4th IEEE International Conference on Mechatronics (ICM '07), pp. 1–5, 2007. View at Publisher · View at Google Scholar · View at Scopus
  52. J. Y. Cao, J. Liang, and B.-G. Cao, “Optimization of fractional order PID controllers based on genetic algorithms,” in Proceedings of the 4th International Conference on Machine Learning and Cybernetics, pp. 5686–5689, 2005.
  53. O. P. Agrawal, “A formulation and a numerical scheme for fractional optimal control problems,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  54. F. B. M. Duarte, M. da Graça Marcos, and J. A. T. Machado, “Fractional-order harmonics in the trajectory control of redundant manipulators,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  55. F. B. M. Duarte and J. A. Machado, “Pseudoinverse trajectory control of redundant manipulators: a fractional calculus perspective,” in Proceedings of the IEEE International Conference on Robotics and Automation, vol. 3, pp. 2406–2411, 2002. View at Scopus
  56. F. B. M. Duarte and J. A. T. Machado, “Chaotic phenomena and fractional-order dynamics in the trajectory control of redundant manipulators,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 315–342, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  57. N. M. F. Ferreira and J. A. T. Machado, “Fractional-order position/force control of two cooperating manipulators,” in Proceedings of the IEEE International Conference on Computational Cybertnetics, August 2003.
  58. M. F. M. Lima, J. T. Machado, and M. Cris, “Fractional order fourier spectra in robotic manipulators with vibrations,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  59. C. A. Monje, F. Ramos, V. Feliu, and B. M. Vinagre, “Tip position control of a lightweight flexible manipulator using a fractional order controller,” IET Control Theory and Applications, vol. 1, no. 5, pp. 1451–1460, 2007. View at Publisher · View at Google Scholar · View at Scopus
  60. N. M. F. Ferreira and J. A. T. Machado, “Fractional-order hybrid control of robotic manipulators,” in Proceedings of the 11th International Conference on Advanced Robotics, pp. 393–398, 2003. View at Publisher · View at Google Scholar
  61. N. M. F. Ferreira, J. A. T. Machado, A. M. S. F. Galhano, and J. B. Cunha, “Fractional control of two arms working in cooperation,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.
  62. B. Orsoni, P. Melchior, A. Oustaloup, Th. Badie, and G. Robin, “Fractional motion control: application to an XY cutting table,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 297–314, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  63. M. F. Silva and J. A. T. MacHado, “Fractional order PDα joint control of legged robots,” Journal of Vibration and Control, vol. 12, no. 12, pp. 1483–1501, 2006. View at Publisher · View at Google Scholar · View at Scopus
  64. M. F. Silva, J. A. T. Machado, and R. S. Barbosa, “Comparison of different orders pad fractional order pd0.5 control algorithm implementations,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006.