In this paper, sliding mode control is utilized for stabilization of a particular class of nonlinear polytopic differential inclusion systems with fractional-order-0 < q < 1. This class of fractional order differential inclusion systems is used to model physical chaotic fractional order Chen and Lu systems. By defining a sliding surface with fractional integral formula, exploiting the concept of the state space norm, and obtaining sufficient conditions for stability of the sliding surface, a special feedback law is presented which enables the system states to reach the sliding surface and consequently creates a sliding mode control. Finally, simulation results are used to illustrate the effectiveness of the proposed method.

1.
M. A.
Aizerman
and
F. R.
Gantmacher
,
Absolute Stability of Regulator Systems
(
Holden-Day
,
San Francisco, CA
,
1964
).
2.
K. S.
Narendra
and
J.
Taylor
,
Frequency Domain Methods for Absolute Stability
(
Academic
,
New York
,
1973
).
3.
S.
Boyd
,
L.
El Ghaoui
,
E.
Feron
, and
V.
Balakrishnan
,
Linear Matrix Inequalities in System and Control Theory
(
Society for Industrial and Applied Mathematics (SIAM)
,
Philadelphia
,
1994
).
4.
P.
Arena
,
R.
Caponetto
,
L.
Fortuna
, and
D.
Porto
,
Nonlinear Noninteger Order Circuits and Systems—An Introduction
, World Scientific Series on Nonlinear Science Vol.
38
(
World Scientific
,
Singapore
,
2000
).
5.
D.
Baleanu
,
Z. B.
Guvenc
, and
J. A.
Tenreiro Machado
,
New Trends in Nanotechnology and Fractional Calculus Applications
(
Springer
,
New York
,
2010
).
6.
M. S.
Tavazoei
and
M.
Haeri
,
Nonlinear Anal.
69
,
1299
(
2008
).
7.
J.
Sabatier
,
O. P.
Agrawal
, and
J. A. T.
Machado
,
Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering
(
Springer
,
London
,
2007
).
8.
M. S.
Tavazoei
and
M.
Haeri
,
Phys. Lett. A
367
,
102
(
2007
).
9.
M. S.
Tavazoei
and
M.
Haeri
,
Physica D
237
,
2628
(
2008
).
10.
M. D.
Ortigueira
,
IEEE Circuits Syst. Mag.
8
(
3
),
19
(
2008
).
11.
Y. K.
Chang
,
A.
Anguraj
, and
M.
Mallika Arjunan
,
Chaos, Solitons Fractals
39
,
1864
(
2009
).
12.
A.
Ouahab
,
Nonlinear Anal. Theory, Methods Appl.
69
,
3877
(
2008
).
13.
M.
Benchohra
,
J.
Henderson
,
S. K.
Ntouyas
, and
A.
Ouahab
,
Fractional Calculus Appl. Anal.
11
,
35
(
2008
).
14.
15.
A.
Si-Ammour
,
S.
Djennoune
, and
M.
Bettayeb
,
Commun. Nonlinear Sci. Numer. Simul.
14
,
2310
(
2009
).
16.
R.
El-Khazali
,
W.
Ahmad
, and
Y.
Al-Assaf
, “
Sliding mode control of fractional chaotic systems
,” in
First IFAC Workshop on Fractional Differentiation and Its Applications
, 19–21 July (
Bordeaux
,
France
,
2004
), pp.
495
500
.
17.
R.
El-Khazali
,
W.
Ahmad
, and
Y.
Al-Assaf
,
Int. J. Bifurcation Chaos
16
(
10
),
1
(
2006
).
18.
A. A.
Kilbas
,
H. M.
Srivastava
, and
J. J.
Trujillo
,
Theory and Applications of Fractional Differential Equations
(
Elsevier B.V.
,
The Netherlands
,
2006
).
19.
D.
Xue
and
Y. Q.
Chen
, “
A comparative introduction of four fractional order controllers
,” in
Proceedings of the 4th IEEE World Congress on Intelligent Control and Automation (WCICA02)
(
Shanghai, China
,
2002
), pp.
3228
3235
.
20.
X.
Cai
,
L.
Liu
, and
W.
Zhang
,
Nonlinear Dyn.
58
(
3
),
487
(
2009
).
21.
C.
Li
and
G.
Chen
,
Phys. A: Stat. Mech. Appl.
341
,
55
(
2004
).
22.
J. L.
Wu
,
IEEE Trans. Autom. Control
51
,
1492
(
2006
).
23.
T. H.
Hu
,
A. R.
Teel
, and
L.
Zaccarian
,
Automatica
43
,
685
(
2007
).
24.
Y. J.
Sun
,
Chaos, Solitons Fractals
39
(
5
),
2386
(
2009
).
25.
J. G.
Lu
,
Chaos, Solitons Fractals
26
(
4
),
1125
(
2005
).
26.
J. W.
Chen
,
J. Math. Anal. Appl.
261
,
369
(
2001
).
27.
J. G.
Lu
and
G.
Chen
,
Chaos, Solitons Fractals
27
(
3
),
685
(
2006
).
28.
I.
Podlubny
,
Fractional Differential Equations
(
Academic
,
San Diego
,
1999
).
29.
S.
Balochian
,
A. K.
Sedigh
, and
A.
Zare
,
ISA Trans.
50
,
21
(
2011
).
30.
D.
Matignon
, “
Stability properties for generalized fractional differential systems
,” in
ESAIM: Proceedings
,
Paris, France
, December
1998
, Vol.
5
, pp.
145
158
.
31.
G. V.
Smirnov
,
Introduction to the Theory of Differential Inclusions
(
American Mathematical Society
,
USA
,
2002
).
32.
H.
Salarieh
and
A.
Alasty
,
Chaos, Solitons Fractals
41
(
1
),
67
(
2009
).
33.
A.
Pisano
,
M. R.
Rapaic
,
Z. D.
Jelecic
, and
E.
Usai
,
Int. J. Robust Nonlinear Control
20
(
18
),
2045
(
2010
).
34.
R. I.
Leine
and
N.
Wouw
,
Stability and Convergence of Mechanical Systems With Unilateral Constraints
(
Springer
,
Berlin
,
2008
).
35.
S.
Balochian
,
A. K.
Sedigh
, and
M.
Haeri
,
Nonlinear Dyn.
66
,
141
(
2011
).
36.
H. S.
Ahn
and
Y. Q.
Chen
,
Automatica
44
,
2985
(
2008
).
37.
J. W.
Chen
,
J. F.
Huang
, and
L. Y.
Lo
,
J. Math. Anal. Appl.
315
,
41
(
2006
).
38.
M. S.
Tavazoei
and
M.
Haeri
,
Physica A
387
(
1
),
57
(
2008
).
39.
J. A. T.
Machado
, ,
Syst. Anal. Model. Simul.
,
27
(
2–3
),
107
(
1997
).
40.
A.
Oustaloup
,
La commande CRONE
(
Hermes
,
Paris
,
1999
).
41.
H.
Linares
,
Ch.
Baillot
,
A.
Oustaloup
, and
Ch.
Ceyral
, “
Generation of a fractal ground: Application in robotics
,” in
International Congress in IEEE-SMC CESA’96 IMACS Multiconference
(
Lille
,
1996
).
42.
F. B. M.
Duarte
and
J. A. T.
Macado
,
Nonlinear Dyn.
29
,
315
(
2002
).
43.
R.
de Levie
,
J. Electroanal. Chem.
281
,
1
(
1990
).
44.
S.
Westerlund
,
Phys. Scr.
43
(
2
),
174
(
1991
).
45.
A.
Le Mehaute
and
G.
Crepy
,
Solid State Ionics
9
,
17
(
1983
).
46.
M.
Nakagawa
and
K.
Sorimachi
,
IEICE Trans. Fundamentals
E75-A
(
12
),
1814
(
1992
).
47.
K. B.
Oldham
and
C. G.
Zoski
,
J. Electroanal. Chem.
157
,
27
(
1983
).
48.
G.
Chen
and
G.
Friedman
,
IEEE Trans. Comput.-Aided Des.
24
(
2
),
170
(
2005
).
49.
T. T.
Hartley
,
C. F.
Lorenzo
, and
H. K.
Qammer
,
IEEE Trans. Circuits Syst. I
42
,
485
(
1995
).
50.
P.
Arena
,
R.
Caponetto
,
L.
Fortuna
, and
D.
Porto
, “
Chaos in a fractional order Duffing system
,” in
Proceedings ECCTD
(
Budapest
,
1997
), pp.
1259
1262
.
51.
W. M.
Ahmad
and
J. C.
Sprott
,
Chaos, Solitons Fractals
16
,
339
(
2003
).
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