Table of Contents Author Guidelines Submit a Manuscript
Journal of Control Science and Engineering
Volume 2013 (2013), Article ID 947428, 10 pages
Research Article

Fractional-Order Control of a Micrometric Linear Axis

DIME, Department of Mechanical, Energetic, Management and Transport Engineering, University of Genova, via Opera Pia 15A, 16145 Genova, Italy

Received 13 December 2012; Accepted 18 February 2013

Academic Editor: Mohamed Zribi

Copyright © 2013 Luca Bruzzone and Pietro Fanghella. 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. K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order, Academic Press, New York, NY, USA, 1974.
  2. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, 1993.
  3. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
  4. D. Baleanu, “Fractional variational principles in action,” Physica Scripta, vol. T136, 2009. View at Google Scholar
  5. A. Golmankhaneh, A. Golmankhaneh, D. Baleanu, and M. C. Baleanu, “Hamiltonian structure of fractional first order lagrangian,” International Journal of Theoretical Physics, vol. 49, no. 2, pp. 365–375, 2010. View at Google Scholar
  6. S. I. Muslih, O. P. Agrawal, and D. Baleanu, “A fractional Dirac equation and its solution,” Journal of Physics A, vol. 43, no. 5, Article ID 055203, 2010. View at Publisher · View at Google Scholar · View at Scopus
  7. S. Grimnes and O. G. Martinsen, Bioimpedance and Bioelectricity Basics, Academic Press, San Diego, Calif, USA, 2000.
  8. R. S. Lakes, Viscoelastic Solids, CRC Press, Boca Raton, Fla, USA, 1999.
  9. M. Sasso, G. Palmieri, and D. Amodio, “Application of fractional derivative models in linear viscoelastic problems,” Mechanics of Time-Dependent Materials, vol. 15, no. 4, pp. 367–387, 2011. View at Google Scholar
  10. R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Redding, Conn, USA, 2006.
  11. A. J. Bard and L. R. Faulkner, Electrochemical Methods: Fundamentals and Applications, John Wiley & Sons, New York, NY, USA, 2nd edition, 2001.
  12. D. Baleanu, A. K. Golmankhaneh, and A. K. Golmankhaneh, “Fractional nambu mechanics,” International Journal of Theoretical Physics, vol. 48, no. 4, pp. 1044–1052, 2009. View at Publisher · View at Google Scholar · View at Scopus
  13. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000.
  14. N. Heymans and I. Podlubny, “Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives,” Rheologica Acta, vol. 45, no. 5, pp. 765–771, 2006. View at Publisher · View at Google Scholar · View at Scopus
  15. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  16. R. E. Gutiérrez, J. M. Rosário, and J. T. Machado, “Fractional order calculus: basic concepts and engineering applications,” Mathematical Problems in Engineering, vol. 2010, Article ID 375858, 19 pages, 2010. View at Publisher · View at Google Scholar
  17. M. Axtell and M. E. Bise, “Fractional calculus applications in control systems,” in Proceedings of the IEEE National Aerospace and Electronics Conference (NAECON '90), pp. 563–566, Dayton, Ohio, USA, May 1990. View at Scopus
  18. S. E. Hamamci and M. Koksal, “Calculation of all stabilizing fractional-order PD controllers for integrating time delay systems,” Computers and Mathematics with Applications, vol. 59, no. 5, pp. 1621–1629, 2010. View at Publisher · View at Google Scholar · View at Scopus
  19. S. E. Hamamci, “Stabilization using fractional-order PI and PID controllers,” Nonlinear Dynamics, vol. 51, no. 1-2, pp. 329–343, 2008. View at Publisher · View at Google Scholar · View at Scopus
  20. R. Matušů, “Application of fractional order calculus to control theory,” International Journal of Mathematical Models and Methods in Applied Sciences, vol. 5, no. 7, pp. 1162–1169, 2011. View at Google Scholar
  21. I. Podlubny, “Fractional-order systems and PIλDμ controllers,” IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208–213, 1999. View at Google Scholar
  22. C. Yeroglu and N. Tan, “Note on fractional-order proportional-integral-differential controller design,” IET Control Theory & Applications, vol. 5, no. 17, pp. 1978–1989, 2012. View at Google Scholar
  23. G. Ruikun, L. Aiwu, F. Min, G. Lihui, and G. Huanyao, “Study of fractional order PIλDμ controller designing method,” in Proceedings of the IEEE Symposium on Robotics and Applications (ISRA '12), pp. 277–281, Kuala Lumpur, Malaysia, June 2012.
  24. P. Ostalczyk and P. Duch, “Closed-Loop system synthesis with the variable-, fractional-Order PID controller,” in Proceedings of the 17th International Conference on Methods and Models in Automation and Robotics (MMAR '12), pp. 589–594, Miedzyzdroje, Poland, August 2012.
  25. A. Tepljakov, E. Petlenkov, and J. Belikov, “A flexible MATLAB tool for optimal fractional-order PID controller design subject to specifications,” in Proceedings of the 31st Chinese Control Conference (CCC '12), pp. 4698–4703, Hefei, China, July 2012.
  26. W. Guo, Y. Song, L. Zhou, and L. Deng, “A novel model algorithmic controller with fractional order PID structure,” in Proceedings of the 10th World Congress on Intelligent Control and Automation (WCICA '12), pp. 2517–2522, Beijing, China, July 2012.
  27. K. Erenturk, “Fractional order PIλDμ and active disturbance rejection control of nonlinear two mass drive system,” IEEE Transactions on Industrial Electronics, vol. PP, no. 99, 2012. View at Google Scholar
  28. M. O. Efe, “Neural network assisted computationally simple PIλDμ control of a quadrotor UAV,” IEEE Transactions on Industrial Informatics, vol. 7, no. 2, pp. 354–361, 2011. View at Publisher · View at Google Scholar · View at Scopus
  29. R. Duma, P. Dobra, and M. Trusca, “Embedded application of fractional order control,” Electronics Letters, vol. 48, no. 24, pp. 1526–1528, 2012. View at Google Scholar
  30. L. Bruzzone and G. Bozzini, “Fractional-order derivatives and their application to the position control of robots,” International Journal of Mechanics and Control, vol. 10, no. 1, pp. 39–44, 2009. View at Google Scholar
  31. L. Bruzzone and G. Bozzini, “Nondimensional analysis of fractional-order PDD1/2 control of purely inertial systems,” Journal of Mechatronics and Applications, vol. 2010, Article ID 903420, 10 pages, 2010. View at Publisher · View at Google Scholar
  32. L. Bruzzone and P. Fanghella, “Influence of the half-derivative term on fractional-order control of mechatronic systems,” in Proceedings of the 21st International Workshop on Robotics in Alpe-Adria-Danube Region (RAAD '12), pp. 292–298, Naples, Italy, September 2012.
  33. J. A. T. Machado, “Fractional-order derivative approximations in discrete-time control systems,” Systems Analysis Modelling Simulation, vol. 34, no. 4, pp. 419–434, 1999. View at Google Scholar · View at Scopus
  34. Y. Q. Chen, I. Petráš, and D. Xue, “Fractional order control—a tutorial,” in Proceedings of the American Control Conference, pp. 1397–1411, St. Louis, Mo, USA, June 2009.
  35. D. Matignon, “Generalized fractional differential and difference equations: stability properties and modelling issues,” in Proceedings of the Mathematical Theory of Networks and Systems Symposium, Padova, Italy, 1998.
  36. S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, 2008.