Research Papers

Estimation of the Order and Parameters of a Fractional Order Model From a Noisy Step Response Data1

[+] Author and Article Information
Mahsan Tavakoli-Kakhki

Faculty of Electrical Engineering,
K. N. Toosi University of Technology,
P.O. Box 16315-1355,
Tehran 19697, Iran
e-mail: matavakoli@kntu.ac.ir

Mohammad Saleh Tavazoei

Electrical Engineering Department,
Sharif University of Technology,
P.O. Box 11155-4363,
Tehran, Iran
e-mail: tavazoei@sharif.edu

A short version of this paper has been presented in the 6th IFAC Workshop on Fractional Differentiation and its Applications, Grenoble, France, Feb. 4–6, 2013.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 24, 2013; final manuscript received November 28, 2013; published online February 24, 2014. Assoc. Editor: YangQuan Chen.

J. Dyn. Sys., Meas., Control 136(3), 031020 (Feb 24, 2014) (7 pages) Paper No: DS-13-1214; doi: 10.1115/1.4026345 History: Received May 24, 2013; Revised November 28, 2013

This paper deals with integral based methods to estimate the order and parameters of simple fractional order models from the extracted noisy step response data of a process. This data can be obtained from both open-loop and closed-loop tests. Numerical simulation results are presented to verify the robustness of these proposed methods in the presence of the measurement noise.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Ionescu, C. M., Machado, J. A. T., and De Keyser, R., 2011, “Modeling of the Lung Impedance Using a Fractional-Order Ladder Network With Constant Phase Elements,” IEEE Trans. Biomed. Circuits Syst., 5, pp. 83–89. [CrossRef] [PubMed]
Abdullah, H. H., Elsadek, H. A., ElDeeb, H. E., and Bagherzadeh, N., 2012, “Fractional Derivatives Based Scheme for FDTD Modeling of th-Order Cole–Cole Dispersive Media,” IEEE Antennas Wireless Propag. Lett., 11, pp. 281–284. [CrossRef]
Škovránek, T., Podlubny, I., and Petráš, I., 2012, “Modeling of the National Economies in State-Space: A Fractional Calculus Approach,” Econ. Modell., 29, pp. 1322–1327. [CrossRef]
Narang, A., Shah, S. L., and Chen, T., 2011, “Continuous-Time Model Identification of Fractional-Order Models With Time Delays,” IET Control Theory Appl., 5, pp. 900–912. [CrossRef]
Gabano, J. D., Poinot, T., and Kanoun, H., 2011, “Identification of a Thermal System Using Continuous Linear Parameter-Varying Fractional Modeling,” IET Control Theory Appl., 5, pp. 889–899. [CrossRef]
Victor, S., Malti, R., Garnier, H., and Oustaloup, A., 2013, “Parameter and Differentiation Order Estimation in Fractional Models,” Automatica, 49, pp. 926–935. [CrossRef]
Luo, Y., Chen, Y. Q., Wang, C. Y., and Pi, Y. G., 2010, “Tuning Fractional Order Proportional Integral Controllers for Fractional Order Systems,” J. Process Control, 20, pp. 823–831. [CrossRef]
Li, H. S., Luo, Y., and Chen, Y. Q., 2010, “A Fractional Order Proportional and Derivative (FOPD) Motion Controller: Tuning Rule and Experiments,” IEEE Trans. Control Syst. Technol., 18, pp. 516–520. [CrossRef]
Monje, A., Vinagre, B. M., Feliu, V., and Chen, Y. Q., 2008, “Tuning and Auto-Tuning of Fractional Order Controllers for Industry Applications,” Control Eng. Pract., 16, pp. 798–812. [CrossRef]
Tavakoli-Kakhki, M., and Haeri, M., 2011, “Fractional Order Model Reduction Approach Based on Retention of the Dominant Dynamics: Application in IMC Based Tuning of FOPI and FOPID Controllers,” ISA Trans., 50, pp. 432–442. [CrossRef] [PubMed]
Tavakoli-Kakhki, M., Haeri, M., and Tavazoei, M. S., 2010, “Simple Fractional Order Model Structures and Their Applications in Control System Design,” Eur. J. Control, 6, pp. 680–694. [CrossRef]
Astrom, K., and Hagglund, T., 1995, PID Controllers: Theory, Design, and Tuning, Instrument Society of America, Research Triangle Park, NC.
Ogunnaike, B. A., and Ray, W. H., 1994, Process Dynamics, Modeling, and Control, Oxford University Press, New York.
Ahmed, S., Huang, B., and Shah, S. L., 2007, “Novel Identification Method From Step Response,” Control Eng. Pract., 15, pp. 545–556. [CrossRef]
Wang, Q., and Zhang, Y., 2001, “Robust Identification of Continuous Systems With Dead-Time From Step Responses,” Automatica, 37, pp. 377–390. [CrossRef]
Ahmed, S., 2010, “Recent Developments in Identification From Step Response,” Vol. 2, Proceedings of the 2nd Annual Gas Processing Symposium, Jan. 10–14, Qatar.
Ahmed, S., Huang, B., and Shah, S. L., 2008, “Identification From Step Responses With Transient Initial Conditions,” J. Process Control, 18, pp. 121–130. [CrossRef]
Podlubny, I., 1999, Fractional Differential Equations, Academic Press, San Diego, CA.
Tavazoei, M. S., 2012, “Overshoot in the Step Response of Fractional-Order Control Systems,” J. Process Control, 22, pp. 90–94. [CrossRef]
Tavazoei, M. S., 2011, “On Monotonic and Non-Monotonic Step Responses in Fractional Order Systems,” IEEE Trans. Circuits Syst. II, 58, pp. 447–451. [CrossRef]
Tavakoli-Kakhki, M., Haeri, M., and Tavazoei, M. S., 2010, “Over and Under Convergent Step Responses in Fractional Order Transfer Functions,” Trans. Inst. Meas. Control, 32, pp. 376–394. [CrossRef]
Tavazoei, M. S., 2010, “Notes on Integral Performance Indices in Fractional-Order Control Systems,” J. Process Control, 20, pp. 285–291. [CrossRef]
Mainardi, F., and Gorenflo, R., 2000, “On Mittag-Leffler-Type Functions in Fractional Evolution Processes,” J. Comput. Appl. Math., 118, pp. 283–299. [CrossRef]
Chen, J., Lundberg, K. H., Davison, D. E., and Bernstein, D. S., 2007, “The Final Value Theorem Revisited Infinite Limits and Irrational Functions,” IEEE Control Syst. Mag., 27, pp. 97–99. [CrossRef]
Diethelm, K., Ford, N. J., Freed, A. D., and Luchko, Y., 2005, “Algorithms for the Fractional Calculus: A Selection of Numerical Methods,” Comput. Methods Appl. Mech. Eng., 194, pp. 743–773. [CrossRef]
Fukunaga, M., and Nobuyuki, S., 2013, “A High-Speed Algorithm for Computation of Fractional Differentiation and Fractional Integration,” Philos. Trans. R. Soc. A, 371, p. 20120152. [CrossRef]
Li, J. R., 2010, “A Fast Time Stepping Method for Evaluating Fractional Integrals,” SIAM J. Sci. Comput., 31, pp. 4696–4714. [CrossRef]
Ferdi, Y., 2006, “Computation of Fractional Order Derivative and Integral Via Power Series Expansion and Signal Modelling,” Nonlin. Dyn., 46, pp. 1–15. [CrossRef]
Zhu, Z., Li, G., and Cheng, C., 2003, “A Numerical Method for Fractional Integral With Applications,” Appl. Math. Mech., 24, pp. 373–384. [CrossRef]
Marinov, T. M., Ramirez, N., and Santamaria, F., 2013, “Fractional Integration Toolbox,” Fractional Calc. Appl. Anal., 16, pp. 670–681. [CrossRef]
Forssell, U., and Ljung, L., 1999, “Closed-Loop Identification Revisited,” Automatica, 35, pp. 1215–1241. [CrossRef]
Karimi, A., and Landau, I. D., 1998, “Comparison of the Closed-Loop Identification Methods in Terms of the Bias Distribution,” Syst. Control Lett., 34, pp. 159–167. [CrossRef]
Van den Hof, P. M. J., and Schrama, R. J. P., 1995, “Identification and Control-Closed Loop Issues,” Automatica, 31, pp. 1751–1770. [CrossRef]
Tavakoli-Kakhki, M., Haeri, M., and Tavazoei, M. S., 2013, “Study on Control Input Energy Efficiency of Fractional Order Control Systems,” IEEE J. Emerg. Sel. Top. Circuits Syst., 3, pp. 475–482. [CrossRef]
Fedele, G., 2009, “A New Method to Estimate a First-Order Plus Time Delay Model From Step Response,” J. Franklin Inst., 346, pp. 1–9. [CrossRef]


Grahic Jump Location
Fig. 1

Estimation of α and T by fractional order integration

Grahic Jump Location
Fig. 5

The control system considered in Sec. 5

Grahic Jump Location
Fig. 6

Estimation of α and T by fractional order integration from closed-loop data

Grahic Jump Location
Fig. 7

Noisy unit step response of G(s) in Eq. (36) with SNR=20

Grahic Jump Location
Fig. 8

fγ(t) for different values of integral order γ in Example 1

Grahic Jump Location
Fig. 9

Noisy unit step response of G(s) in Eq. (41) with SNR=20

Grahic Jump Location
Fig. 10

fγ(t) for different values of integral order γ in Example 3

Grahic Jump Location
Fig. 11

Values of integral 1k∫0tg(τ)dτ in Example 3

Grahic Jump Location
Fig. 12

Noisy control signal in Example 4

Grahic Jump Location
Fig. 13

hγ(t) for different values of integral order γ in Example 4




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In