Application of fractional order calculus to control theory

International Journal of Mathematical Models and Methods in Applied Sciences

Volumn 5, Issue 7, 2011, Pages 1162-1169

  Matušů, Radek ^a  

^a TOMAS BATA UNIVERSITY IN ZLIN   (Czech Republic)

Author keywords

Control systems; Control theory; Differintegration; Fractional order calculus; Fractional order controllers; Matlab toolboxes

Indexed keywords

EID: 80055034584     PISSN: 19980140     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (36)

1. K. B. Oldham, and J. Spanier, Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. New York-London: Academic Press, 1974.
2. K. S. Miller, and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. New York, USA: John Wiley and Sons, 1993.
3. I. Podlubný, Fractional Differential Equations. San Diego, CA, USA: Academic Press, 1999.
4. Y. Chen, I. Petráš, and D. Xue, "Fractional Order Control-A Tutorial" presented at the 2009 American Control Conference, St. Louis, MO, USA, 2009.
5. I. Petráš, "Stability of fractional-order systems with rational orders: A survey" Fractional Calculus & Applied Analysis, vol. 12, no. 3, pp. 269-298, 2009.
6. I. Petráš, "Stability of fractional-order systems with rational orders" Mathematics-dynamical systems, pp. 1-25, 2008, arXiv:0811.4102v2.
7. R. E. Gutiérrez, J. M. Rosário, and J. T. Machado, "Fractional Order Calculus: Basic Concepts and Engineering Applications" Mathematical Problems in Engineering, vol. 2010, 19 p., 2010, DOI: 10.1155/2010/375858.
8. M. Axtell, and M. E. Bise, "Fractional calculus applications in control systems" presented at the 1990 National Aerospace and Electronics Conference, Dayton, OH, USA, 1990.
9. S. E. Hamamci, "Stabilization using fractional order PI and PID controllers" Nonlinear Dynamics, vol. 51, no. 1-2, pp. 329-343, 2008, DOI: 10.1007/s11071-007-9214-5.
10. S. E. Hamamci, and M. Koksal, "Calculation of all stabilizing fractionalorder PD controllers for integrating time delay systems" Computers and Mathematics with Applications, vol. 59, no. 5, pp. 1621-1629, 2010, DOI: 10.1016/j.camwa.2009.08.049.
11. μ-Controllers" IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208-214, 1999.
12. L.-Y. Chang, and H.-Ch. Chen, "Tuning of Fractional PID Controllers Using Adaptive Genetic Algorithm for Active Magnetic Bearing System" WSEAS Transactions on Systems, vol. 8, no. 1, pp. 158-167, 2009.
13. T. Emami, and J. M. Watkins, "A Unified Approach for Sensitivity Design of PID Controllers in the Frequency Domain" WSEAS Transactions on Systems and Control, vol. 4, no. 5, pp. 221-231, 2009.
14. T. Emami, and J. M. Watkins, "Robust Performance Characterization of PID Controllers in the Frequency Domain" WSEAS Transactions on Systems and Control, vol. 4, no. 5, pp. 232-242, 2009.
15. Md. S. Islam, N. Amin, M. Zaman, and M. S. Bhuyan, "Fuzzy based PID Controller using VHDL for Transportation Application" International Journal of Mathematical Models and Methods in Applied Sciences, vol. 2, no. 2, pp. 143-147, 2008.
16. T. N. L. Vu, and M. Lee, "Multi-loop PI Controller Design for Enhanced Disturbance Rejection in Multi-delay Processes" International Journal of Mathematics and Computers in Simulation, vol. 2, no. 1, pp. 89-94, 2008.
17. R. Matušů, and R. Prokop, Experimental Verification of Design Methods for Conventional PI/PID Controllers WSEAS Transactions on Systems and Control, vol. 5, no. 5, pp. 269-280, 2010.
18. R. Matušů, "Fractional Order Calculus in Control Theory" presented at the 13th WSEAS International Conference on Automatic Control, Modelling and Simulation, Lanzarote, Spain, 2011.
19. F. Merrikh-Bayat, and M. Afshar, "Extending the Root-Locus Method to Fractional-Order Systems" Journal of Applied Mathematics, vol. 2008, 13 p., 2008, DOI: 10.1155/2008/528934.
20. D. Matignon, "Stability properties for generalized fractional differential systems" ESAIM: Proc., vol. 5, pp. 145-158, 1998, DOI: 10.1051/proc:1998004.
21. Y. Chen, "Ubiquitous fractional order controls?" presented at the 2nd IFAC Workshop on Fractional Differentiation and its Applications, Porto, Portugal, 2006.
22. D. Xue, and Y. Chen, "A Comparative Introduction of Four Fractional Order Controllers" presented at the 4th World Congress on Intelligent Control and Automation, Shanghai, China, 2002.
23. B. J. Lurie, "Three-parameter tunable tilt-integral-derivative (TID) controller" US Patent US5371670, 1994.
24. A. Oustaloup, B. Mathieu, and P. Lanusse, "The CRONE control of resonant plants: application to a flexible transmission" European Journal of Control, vol. 1, no. 2, pp. 113-121, 1995.
25. A. Oustaloup, X. Moreau, and M. Nouillant, "The CRONE suspension" Control Engineering Practice, vol. 4, no. 8, pp. 1101-1108, 1996.
26. A. Oustaloup, J. Sabatier, and X. Moreau, "From fractal robustness to the CRONE approach" ESAIM: Proc., vol. 5, pp. 177-192, 1998, DOI: 10.1051/proc:1998006.
27. P. Lanusse, J. Sabatier, X. Moreau, and A. Oustaloup, "Robust Control" presented as a Lecture 5 at the Tutorial Workshop on Fractional Calculus Applications in Automatic Control and Robotics at the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, USA, 2002. Available from: http://mechatronics.ece.usu.edu/foc/cdc02tw/ cdrom/Slides/Lecture5/Tutorial_Workshop_CC.pdf.
28. CRONE Toolbox-Official Website, Available from: www.ims-bordeaux.fr/CRONE/toolbox/.
29. A. Oustaloup, P. Melchior, P. Lanusse, O. Cois, and F. Dancla, "The CRONE toolbox for Matlab" presented at the 2000 IEEE International Symposium on Computer-Aided Control System Design, Anchorage, Alaska, USA, 2000.
30. P. Melchior, B. Orsoni, O. Lavialle, and A. Oustaloup, "The CRONE Toolbox for Matlab: Fractional Path Planning Design in Robotics" presented at the 10th IEEE International Workshop on Robot and Human Interactive Communication, Bordeaux and Paris, France, 2001.
31. I. Podlubný, "Derivácie neceločíselného rádu: história, teória, aplikácie (Fractional-order derivatives: History, theory, and applications)" plenary lecture presented at the Conference of the Slovak Mathematical Society, Jasná, Slovakia, 2002 (In Slovak), Available from: http://people.tuke.sk/igor.podlubny/pspdf/jasna.pdf.
32. H.-F. Raynaud, and A. Zergaïnoh, "State-space representation for fractional order controllers" Automatica, vol. 36, no. 7, pp. 1017-1021, 2000, DOI: 10.1016/S0005-1098(00)00011-X.
33. C. A. Monje, B. M. Vinagre, A. J. Calderón, V. Feliu, and Y. Chen, "Self-tuning of Fractional Lead-Lag Compensators" presented at the 16th IFAC World Congress, Prague, Czech Republic, 2005.
34. D. Valério, Toolbox ninteger for MatLab, Available from: http://web.ist.utl.pt/duarte.valerio/ninteger/ninteger.htm.
35. D. Valério, and J. S. da Costa, "Ninteger: A non-integer control toolbox for Matlab" presented at the 1st IFAC Workshop on Fractional Differentiation and its Applications, Bordeaux, France, 2004.
36. I. Petráš, "Software pages of Dr. Ivo Petráš", Available from: http://people.tuke.sk/ivo.petras/SW.html.