Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Comparison Between Davidson-Cole and Frequency-Band Limited Fractional Differentiator I/O Type Transfer Function with Speed and Acceleration Inputs in Path Tracking Design

Journal of Applied Nonlinear Dynamics 3(1) (2014) 1--16 | DOI:10.5890/JAND.2014.03.001

N. Yousfi$^{1}$, P. Melchior$^{2}$, C. Rekik$^{1}$, N. Derbel$^{1}$, A. Oustaloup$^{2}$

$^{1}$ Control & Energy Management laboratory (CEM) University of Sfax, Sfax Engineering School, Tunisia

$^{2}$ IMS (UMR 5218 CNRS, Universit Bordeaux 1 - ENSEIRB - ENSCPB), Department LAPS, TAL ENCE cedex, France

Download Full Text PDF



A new approach to path tracking design based on fractional prefilter was developed in this paper. In path tracking design, the dynamic of actuators must be taken into account in order to reduce over- shoots appearing for small displacements. Taking into consideration the maximum velocity, acceleration, jerk, and the bandwidth of the closed-loop on which the input is applied, it permits the generation of an optimal movement reference-input giving a minimum path comple- tion time. An approach to path tracking based on fractional prefilter has been developed. This approach based on a Davidson-Cole (DC) and Frequency Band Limited Fractional Differentiator (FBLFD) pre- filters, with position input. This work describes an extension of this method. It consists of a path tracking using fractional differentiation and comparison between different types of prefilters by direct opti- mization of an Input/Output (I/O) transfer function with speed and acceleration inputs. Fractional differentiation has been used through a Davidson-Cole and frequency band-limited fractional differentiator (FBLFD) prefilters. A simulation on a motor model validates the developed methodology.


  1. [1]  Orsoni, B. (2002), Drive gnralise en planification de trajectoire et gnration de mouvement, PhD thesis, University of Bordeaux 1, France.
  2. [2]  Melchior, P., Poty, A., Orsoni, B., and Oustaloup, A. (2003), Preshaping command inputs for 2nd generation crone control: application on an instrumented DC Motor Bench, In: Proceedings of ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Chicago, USA, 695-702,
  3. [3]  Oustaloup, A. (1995), La Drivation Non Entire: Thorie, Synthse et Applications, Herms Ed., Paris.
  4. [4]  Tenreiro Machado, J.A. (2012), Entropy analysis of fractional derivatives and their approximation, Journal of Applied Nonlinear Dynamics, 1 (1), 109-112.
  5. [5]  Melchior, P., Robin, G., L'Hostis, S., and Levron, F. (1998), Non integer order movement generation in path planning, In; Proceedings of IEEE-SMC CESA 98 IMACS, Nabeul-Hammamet, Tunisia, 4, 371-375.
  6. [6]  Poty, A., Melchior, P., and Oustaloup, A.(2006), Frequency band-limited fractional differentiator in path tracking design, In: Proceedings of Second IFACWorkshop on Fractional Differentiation and its Applications, Portugal.
  7. [7]  Melchior, P., Orsoni, B., Badie, Th., Robin, G., and Oustaloup, A. (2000), Non-Integer Motion Control: Application to an XY Cutting Table, In: Proceedings of first IFAC Conference on Mechatronic Systems, Darmstadt, Germany.
  8. [8]  Orsoni, B., Melchior, P., and Oustaloup, A. (2001), Davidson-Cole transfer function in path tracking design, In Proceedings of the 6th IEEE-ECC'2001, Porto, Portugal, 4( 7), 1174-1179.
  9. [9]  Orsoni, B., Melchior, P. Badie, Th., Robin, G., and Oustaloup, A. (2002), Fractional motion control: Application to an XY Cutting Table, International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, Special Issue on Fractional Order Calculus and its Applications, 29 (1-4),297-314.
  10. [10]  Poty, A. (2006), Planification de trajectoire dans un environnement dynamique et gnration de mouvement d'ordre non entier, PhD thesis, University of Bordeaux 1, France.
  11. [11]  Lin, C.S., Chang, P.R., and Luh., J.Y.S. (1983), Formulation and optimization of cubic polynomial joint trajectories for industrial robots, IEEE Transactions on Automatic Control , 28(12), 1066-1073.
  12. [12]  Melchior, P., Poty, A., and Oustaloup, A. (2007), Frequency band-limited fractional differentiator prefilter in path tracking design, chapter in the book of Advances in Fractional Calculus, 7, 477-492.
  13. [13]  Davidson, D.W. and Cole, R.H. (1951), Dielectric relaxation in glycerol, propylene glycol and n-propanol, Journal of Chemical Physics, 19(12), 1484-1490.
  14. [14]  Yousfi, N., Melchior, P., Rekik, C., Derbel, N., and Oustaloup, A. (2011), Path tracking design by input/ output Davidson-Cole transfer function, In Proceedings of the IEEE SSD'2011, Sousse, Tunisie, March.
  15. [15]  Melchior, P., Yousfi, N., Rekik, C., Derbel, N., and Oustaloup, A. (2012), Path tracking design by input/ output Davidson-Cole transfer function, In Proceedings of the 5th IFAC Workshop on Fractional Differentiation and its Applications, Hohai university, Nanjing, China, 14-17 May.
  16. [16]  Oustaloup, A., Melchior, P., Lanusse, P., Cois, O., and Dancla, F. (2000), The CRONE toolbox for Matlab, In Proceedings of IEEE International Symposium on Compute-Aided Control-System Design, Anchorage, USA, 190-195.
  17. [17]  Melchior, P., Lanusse, P., Dancla, F., and Cois, O. (1999), Valorisation de l'approche non entire par le logiciel CRONE, In: Proceedings of CETSIS-EEA'99, Montpellier, France.
  18. [18]  Melchior, P., Poty, A., and Oustaloup, A. (2005), Path tracking design by frequency band-limited fractional differentiator prefilter, In Proceedings of ENOC-2005, Eindhoven, Netherlands.