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.

References

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.2010.2077636
2.
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.2012.2190029
3.
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.2012.03.019
4.
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.2010.0718
5.
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.2010.0222
6.
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.2013.01.026
7.
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.2010.04.011
8.
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.2009.2019120
9.
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.2007.08.006
10.
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.2011.02.002
11.
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.16.680-694
12.
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.2006.10.005
15.
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)00177-1
16.
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.2007.07.009
18.
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.2011.10.005
20.
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.2011.2158258
21.
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
.10.1177/0142331209356157
22.
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.2009.09.005
23.
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)00294-6
24.
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.2007.365008
25.
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.2004.06.006
26.
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.2012.0152
27.
Li
,
J. R.
,
2010
, “
A Fast Time Stepping Method for Evaluating Fractional Integrals
,”
SIAM J. Sci. Comput.
,
31
, pp.
4696
4714
.10.1137/080736533
28.
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/s11071-005-9000-1
29.
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/BF02439616
30.
Marinov
,
T. M.
,
Ramirez
,
N.
, and
Santamaria
,
F.
,
2013
, “
Fractional Integration Toolbox
,”
Fractional Calc. Appl. Anal.
,
16
, pp.
670
681
.10.2478/s13540-013-0042-7
31.
Forssell
,
U.
, and
Ljung
,
L.
,
1999
, “
Closed-Loop Identification Revisited
,”
Automatica
,
35
, pp.
1215
1241
.10.1016/S0005-1098(99)00022-9
32.
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)00137-0
33.
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-X
34.
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.2013.2273855
35.
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.2008.05.004
You do not currently have access to this content.