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.
Issue Section:
Research Papers
References
1.
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
.10.1109/TBCAS.2010.20776362.
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
.10.1109/LAWP.2012.21900293.
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
.10.1016/j.econmod.2012.03.0194.
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
.10.1049/iet-cta.2010.07185.
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
.10.1049/iet-cta.2010.02226.
Victor
, S.
, Malti
, R.
, Garnier
, H.
, and Oustaloup
, A.
, 2013
, “Parameter and Differentiation Order Estimation in Fractional Models
,” Automatica
, 49
, pp. 926
–935
.10.1016/j.automatica.2013.01.0267.
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
.10.1016/j.jprocont.2010.04.0118.
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
.10.1109/TCST.2009.20191209.
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
.10.1016/j.conengprac.2007.08.00610.
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
.10.1016/j.isatra.2011.02.00211.
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
.10.3166/ejc.16.680-69412.
Astrom
, K.
, and Hagglund
, T.
, 1995
, PID Controllers: Theory, Design, and Tuning
, Instrument Society of America
, Research Triangle Park, NC
.13.
Ogunnaike
, B. A.
, and Ray
, W. H.
, 1994
, Process Dynamics, Modeling, and Control
, Oxford University Press
, New York
.14.
Ahmed
, S.
, Huang
, B.
, and Shah
, S. L.
, 2007
, “Novel Identification Method From Step Response
,” Control Eng. Pract.
, 15
, pp. 545
–556
.10.1016/j.conengprac.2006.10.00515.
Wang
, Q.
, and Zhang
, Y.
, 2001
, “Robust Identification of Continuous Systems With Dead-Time From Step Responses
,” Automatica
, 37
, pp. 377
–390
.10.1016/S0005-1098(00)00177-116.
Ahmed
, S
., 2010
, “Recent Developments in Identification From Step Response
,” Vol. 2, Proceedings of the 2nd Annual Gas Processing Symposium
, Jan. 10–14, Qatar.17.
Ahmed
, S.
, Huang
, B.
, and Shah
, S. L.
, 2008
, “Identification From Step Responses With Transient Initial Conditions
,” J. Process Control
, 18
, pp. 121
–130
.10.1016/j.jprocont.2007.07.00918.
Podlubny
, I.
, 1999
, Fractional Differential Equations
, Academic Press
, San Diego, CA
.19.
Tavazoei
, M. S.
, 2012
, “Overshoot in the Step Response of Fractional-Order Control Systems
,” J. Process Control
, 22
, pp. 90
–94
.10.1016/j.jprocont.2011.10.00520.
Tavazoei
, M. S.
, 2011
, “On Monotonic and Non-Monotonic Step Responses in Fractional Order Systems
,” IEEE Trans. Circuits Syst. II
, 58
, pp. 447
–451
.10.1109/TCSII.2011.215825821.
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
.10.1177/014233120935615722.
Tavazoei
, M. S.
, 2010
, “Notes on Integral Performance Indices in Fractional-Order Control Systems
,” J. Process Control
, 20
, pp. 285
–291
.10.1016/j.jprocont.2009.09.00523.
Mainardi
, F.
, and Gorenflo
, R.
, 2000
, “On Mittag-Leffler-Type Functions in Fractional Evolution Processes
,” J. Comput. Appl. Math.
, 118
, pp. 283
–299
.10.1016/S0377-0427(00)00294-624.
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
.10.1109/MCS.2007.36500825.
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
.10.1016/j.cma.2004.06.00626.
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
.10.1098/rsta.2012.015227.
Li
, J. R.
, 2010
, “A Fast Time Stepping Method for Evaluating Fractional Integrals
,” SIAM J. Sci. Comput.
, 31
, pp. 4696
–4714
.10.1137/08073653328.
Ferdi
, Y.
, 2006
, “Computation of Fractional Order Derivative and Integral Via Power Series Expansion and Signal Modelling
,” Nonlin. Dyn.
, 46
, pp. 1
–15
.10.1007/s11071-005-9000-129.
Zhu
, Z.
, Li
, G.
, and Cheng
, C.
, 2003
, “A Numerical Method for Fractional Integral With Applications
,” Appl. Math. Mech.
, 24
, pp. 373
–384
.10.1007/BF0243961630.
Marinov
, T. M.
, Ramirez
, N.
, and Santamaria
, F.
, 2013
, “Fractional Integration Toolbox
,” Fractional Calc. Appl. Anal.
, 16
, pp. 670
–681
.10.2478/s13540-013-0042-731.
Forssell
, U.
, and Ljung
, L.
, 1999
, “Closed-Loop Identification Revisited
,” Automatica
, 35
, pp. 1215
–1241
.10.1016/S0005-1098(99)00022-932.
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
.10.1016/S0167-6911(97)00137-033.
Van den Hof
, P. M. J.
, and Schrama
, R. J. P.
, 1995
, “Identification and Control-Closed Loop Issues
,” Automatica
, 31
, pp. 1751
–1770
.10.1016/0005-1098(95)00094-X34.
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
.10.1109/JETCAS.2013.227385535.
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
.10.1016/j.jfranklin.2008.05.004Copyright © 2014 by ASME
You do not currently have access to this content.