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.
REFERENCES
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.
J. G.
Lu
, Phys. Lett. A
354
(4
), 305
(2006
).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.
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.
40.
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.
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
).© 2012 American Institute of Physics.
2012
American Institute of Physics
You do not currently have access to this content.