This paper concerns the control of a time fractional diffusion system defined in the Riemann–Liouville sense. It is assumed that the system is subject to hysteresis nonlinearity at its input, where the hysteresis is mathematically modeled with the Duhem operator. To compensate the effects of hysteresis nonlinearity, a fractional order Proportional+Integral+Derivative (PID) controller is designed by minimizing integral square error. For numerical computation, the Riemann–Liouville fractional derivative is approximated by the Grünwald–Letnikov approach. A set of algebraic equations arises from this approximation, which can be solved numerically. Performance of the fractional order PID controllers are analyzed in comparison with integer order PID controllers by simulation results, and it is shown that the fractional order controllers are more advantageous than the integer ones.

1.
Mainardi
,
F.
, 1996, “
The Fundamental Solutions for the Fractional Diffusion-Wave Equation
,”
Appl. Math. Lett.
0893-9659,
9
, pp.
23
28
.
2.
Wyss
,
W.
, 1986, “
The Fractional Diffusion Equation
,”
J. Math. Phys.
0022-2488,
27
, pp.
2782
2785
.
3.
Agrawal
,
O. P.
, 2002, “
Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain
,”
Nonlinear Dyn.
0924-090X,
29
, pp.
145
155
.
4.
Heymans
,
N.
, and
Podlubny
,
I.
, 2006, “
Physical Interpretation of Initial Conditions for Fractional Differential Equations With Riemann-Liouville Fractional Derivatives
,”
Rheol Acta
,
45
, pp.
765
771
.
5.
Hilfer
,
R.
, 2000, “
Fractional Diffusion Based on Riemann-Liouville Fractional Derivatives
,”
J. Phys. Chem. B
1089-5647,
104
, pp.
3914
3917
.
6.
Kilbas
,
A. A.
,
Trujillo
,
J. J.
, and
Voroshilov
,
A. A.
, 2005, “
Cauchy-Type Problem for Diffusion-Wave Equation With the Riemann-Liouville Partial Derivative
,”
Fractional Calculus Appl. Anal.
1311-0454,
8
, pp.
403
430
.
7.
Özdemir
,
N.
,
Agrawal
,
O. P.
,
Karadeniz
,
D.
, and
İskender
,
B. B.
, 2008, “
Axis-Symmetric Fractional Diffusion-Wave Problem Part I—Analysis
,”
ENOC-2008
,
Saint Petersburg, Russia
.
8.
Özdemir
,
N.
, and
Karadeniz
,
D.
, 2008, “
Fractional Diffusion-Wave Problem in Cylindrical Coordinates
,”
Phys. Lett. A
0375-9601,
372
, pp.
5968
5972
.
9.
Povstenko
,
Y. Z.
, 2008, “
Time Fractional Radial Diffusion in a Sphere
,”
Nonlinear Dyn.
0924-090X,
53
, pp.
55
65
.
10.
Oustaloup
,
A.
, 1995,
La Derivation Non Enteire
,
Hermes
,
Paris
.
11.
Podlubny
,
I.
, 1994, “
Fractional-Order Systems and Fractional Order Controllers
,” Institute of Experimental Physics,
Slovak Academy of Sciences
,
Kosice
.
12.
Podlubny
,
I.
,
Dorcak
,
L.
, and
Kostial
,
I.
, 1997, “
On Fractional Derivatives, Fractional-Order Dynamic Systems and -Controllers
,”
Proceedings of the 36th Conference on Decision & Control
,
San Diego, CA
.
13.
Zhao
,
C.
,
Xue
,
D.
, and
Chen
,
Y. Q.
, 2005, “
A Fractional Order PID
Tuning Algorithm for a Class of Fractional Order Plants,”
Proceedings of the IEEE International Conference on Mechatronics & Automation
,
Niagara Falls, Canada
, pp.
216
221
.
14.
Maione
,
G.
, and
Lino
,
P.
, 2007, “
New Tuning Rules for Fractional PIα Controllers
,”
Nonlinear Dyn.
0924-090X,
49
, pp.
251
257
.
15.
Barbosa
,
R. S.
,
Silva
,
M. F.
, and
Machado
,
J. A. T.
, 2008, “
Tuning and Application of Integer and Fractional Order PID
Controllers,”
Intelligent Engineering Systems and Computational Cybernetics
,
Springer
,
The Netherlands
, pp.
245
255
.
16.
Jesus
,
I.
, and
Machado
,
J. T.
, 2007, “
Application of Fractional Calculus in the Control of Heat Systems
,”
Journal of Advanced Computational Intelligence and Intelligent Informatics, Fuji Technology Press
,
11
, pp.
1086
1091
.
17.
Jesus
,
I.
,
Machado
,
J. T.
, and
Barbosa
,
R. S.
, 2008, “
On the Fractional Order Control of Heat Systems
,”
Intelligent Engineering Systems and Computational Cybernetics-IESCC
,
Springer
,
Netherlands
, pp.
375
385
.
18.
Agrawal
,
O. P.
, 2004, “
A General Formulation and Solution Scheme for Fractional Optimal Control Problems
,”
Nonlinear Dyn.
0924-090X,
38
, pp.
323
337
.
19.
Agrawal
,
O. P.
, 2007, “
Fractional Optimal Control of a Distributed System Using Eigenfunctions
,”
Proceedings of the ASME 2007 International Design Engineering Technical Conferences
,
Las Vegas, NV
.
20.
Özdemir
,
N.
,
Agrawal
,
O. P.
,
İskender
,
B. B.
, and
Karadeniz
,
D.
, 2009, “
Fractional Optimal Control of a 2-Dimensional Distributed System Using Eigenfunctions
,”
Nonlinear Dyn.
0924-090X,
55
, pp.
251
260
.
21.
Özdemir
,
N.
,
Karadeniz
,
D.
, and
İskender
,
B. B.
, 2009, “
Fractional Optimal Control Problem of a Distributed System in Cylindrical Coordinates
,”
Phys. Lett. A
0375-9601,
373
, pp.
221
226
.
22.
Barbosa
,
R. S.
,
Machado
,
J. A. T.
, and
Galhano
,
A. M.
, 2007, “
Performance of Fractional PID Algorithms Controlling Nonlinear Systems With Saturation and Backlash Phenomena
,”
J. Vibr. Control
,
13
(
9–10
), pp.
1407
1418
.
23.
Özdemir
,
N.
, and
İskender
,
B. B.
, 2008, “
Fractional PIλ
Controller for Fractional Order Linear System With Input Hysteresis,”
ENOC-2008
,
Saint Petersburg, Russia
.
24.
Tao
,
G.
, and
Kokotovic
,
P. V.
, 1994, “
Discrete-Time Adaptive Control of Systems With Unknown Output Hysteresis
,”
Proceedings of the American Control Conference
.
Baltimore, MD
.
25.
Sain
,
P. M.
,
Sain
,
M. K.
, and
Spencer
,
B. F.
, 1997, “
Models for Hysteresis and Applications to Structural Control
,”
Proceedings of the American Control Conference
, Vol.
1
, pp.
16
20
.
26.
Logemann
,
H.
, and
Mawby
,
A. D.
, 1998, “
Integral Control of Distributed Parameter Systems With Input Relay Hysteresis
,”
UKACC International Conference on Control ‘98
,
University of Wales, Swansea, United Kingdom
.
27.
Bagley
,
R. L.
, and
Torvik
,
P. J.
, 1986, “
On the Fractional Calculus Model of Viscoelastic Behavior
,”
J. Rheol.
0148-6055,
30
, pp.
133
155
.
28.
Padovan
,
J.
, and
Sawicki
,
J. T.
, 1997, “
Diophantine Type Fractional Derivative Representation of Structural Hysteresis
,”
Comput. Mech.
0178-7675,
19
, pp.
335
340
.
29.
Darwish
,
M. A.
, and
El-Bary
,
A. A.
, 2006, “
Existence of Fractional Integral Equation With Hysteresis
,”
Appl. Math. Comput.
0096-3003,
176
, pp.
684
687
.
30.
Schafer
,
I.
, and
Kruger
,
K.
, 2006, “
Modeling of Coils Using Fractional Derivatives
,”
J. Magn. Magn. Mater.
0304-8853,
307
, pp.
91
98
.
31.
Deng
,
W.
, and
,
J.
, 2007, “
Generating Multi-Directional Multi-Scroll Chaotic Attractors Via a Fractional Differential Hysteresis System
,”
Phys. Lett. A
0375-9601,
369
, pp.
438
443
.
32.
Oldham
,
K. B.
, and
Spanier
,
J.
, 1974,
The Fractional Calculus
,
Academic
,
New York
.
33.
Miller
,
K. S.
, and
Ross
,
B.
, 1993,
An Introduction to the Fractional Calculus and Fractional Differential Equations
,
Wiley
,
New York
.
34.
Podlubny
,
I.
, 1999,
Fractional Differential Equations
,
Academic
,
San Diego
.
35.
Krasnosel’skii
,
M. A.
, and
Pokrovskii
,
A. V.
, 1989,
Systems With Hysteresis
,
Springer
,
New York
.
36.
Mayergoyz
,
I. D.
, 1991,
Mathematical Models of Hysteresis
,
Springer-Verlag
,
Berlin
.
37.
Macki
,
J. W.
,
Nistri
,
P.
, and
Zecca
,
P.
, 1993, “
Mathematical Models of Hysteresis
,”
SIAM Rev.
0036-1445,
35
, pp.
94
123
.
38.
Visintin
,
A.
, 1994,
Differential Models of Hysteresis
,
Springer
,
Berlin
.
39.
Coleman
,
B. D.
, and
Hodgdon
,
M. L.
, 1986, “
A Constitutive Relation for Rate-Independent Hysteresis in Ferromagnetically Soft Materials
,”
Int. J. Eng. Sci.
0020-7225,
24
, pp.
897
919
.
40.
Coleman
,
B. D.
, and
Hodgdon
,
M. L.
, 1987, “
On a Class of Constitutive Relations for Ferromagnetic Hysteresis
,”
Arch. Ration. Mech. Anal.
0003-9527,
99
, pp.
375
396
.
41.
Moradi
,
M. H.
, and
Johnson
,
M. A.
, 2005,
PID Control
,
Springer-Verlag
,
London
.
You do not currently have access to this content.