This paper focuses on stabilization of fractional-order unified chaotic systems. In contrast to existing methods in literature, the proposed method requires only the system output for feedback and uses only one control input. The controller consists of a state feedback control law and a dynamic estimator. Sufficient stability conditions are derived using a fractional-order extension of the Lyapunov direct method and a new lemma of the Caputo fractional derivative. The conditions are expressed in the form of linear matrix inequalities (LMIs). All the parameters of the controller can be simultaneously obtained by solving the LMIs. Numerical simulations are provided to illustrate the feasibility and effectiveness of the proposed method.

References

References
1.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
San Diego, CA.
2.
Monje
,
C. A.
,
Chen
,
Y. Q.
,
Vinagre
,
B. M.
,
Xue
,
D.
, and
Feliu
,
V.
,
2010
,
Fractional-Order Systems and Controls: Fundamentals and Applications
,
Springer-Verlag
,
London
.
3.
Petras
,
I.
,
2011
,
Fractional-Order Nonlinear Systems
,
Springer-Verlag
,
Berlin
.
4.
Atıcı
,
F. M.
, and
Şengül
,
S.
,
2010
, “
Modeling With Fractional Difference Equations
,”
J. Math. Anal. Appl.
,
369
(
1
), pp.
1
9
.
5.
Wu
,
F.
, and
Liu
,
J.-F.
,
2016
, “
Discrete Fractional Creep Model of Salt Rock
,”
J. Comput. Complex. Appl.
,
2
(
1
), pp.
1
6
.
6.
Wu
,
G.-C.
, and
Baleanu
,
D.
,
2014
, “
Discrete Fractional Logistic Map and Its Chaos
,”
Nonlinear Dyn.
,
75
(
1
), pp.
283
287
.
7.
Ge
,
Z.-M.
, and
Ou
,
C.-Y.
,
2007
, “
Chaos in a Fractional Order Modified Duffing System
,”
Chaos, Solitons Fractals
,
34
(
2
), pp.
262
291
.
8.
Wu
,
X.
,
Li
,
J.
, and
Chen
,
G.
,
2008
, “
Chaos in the Fractional Order Unified System and Its Synchronization
,”
J. Franklin Inst.
,
345
(
4
), pp.
392
401
.
9.
Deng
,
W.
, and
Li
,
C.
,
2008
, “
The Evolution of Chaotic Dynamics for Fractional Unified System
,”
Phys. Lett. A
,
372
(
4
), pp.
401
407
.
10.
Golmankhaneh
,
A. K.
,
Arefi
,
R.
, and
Baleanu
,
D.
,
2013
, “
The Proposed Modified Liu System With Fractional Order
,”
Adv. Math. Phys.
,
2013
, p.
186037
.
11.
Ding
,
Y.
,
Wang
,
Z.
, and
Ye
,
H.
,
2012
, “
Optimal Control of a Fractional-Order HIV-Immune System With Memory
,”
IEEE Trans. Control Syst. Technol.
,
20
(
3
), pp.
763
769
.
12.
Rhouma
,
A.
, and
Bouani
,
F.
,
2014
, “
Robust Model Predictive Control of Uncertain Fractional Systems: A Thermal Application
,”
IET Control Theory Appl.
,
8
(
17
), pp.
1986
1994
.
13.
Wei
,
Y.
,
Chen
,
Y.
,
Liang
,
S.
, and
Wang
,
Y.
,
2015
, “
A Novel Algorithm on Adaptive Backstepping Control of Fractional Order Systems
,”
Neurocomputing
,
165
, pp.
395
402
.
14.
Peng
,
C.
, and
Chen
,
C.
,
2008
, “
Robust Chaotic Control of Lorenz System by Backstepping Design
,”
Chaos, Solitons Fractals
,
37
(
2
), pp.
598
608
.
15.
Sangpet
,
T.
, and
Kuntanapreeda
,
S.
,
2010
, “
Output Feedback Control of Unified Chaotic Systems Based on Feedback Passivity
,”
Int. J. Bifurcation Chaos
,
20
(
5
), pp.
1519
1525
.
16.
Chen
,
G.
,
2011
, “
A Simple Adaptive Feedback Control Method for Chaos and Hyper-Chaos Control
,”
Appl. Math. Comput.
,
217
(
17
), pp.
7258
7264
.
17.
Kuntanapreeda
,
S.
, and
Sangpet
,
T.
,
2012
, “
Synchronization of Chaotic Systems With Unknown Parameters Using Adaptive Passivity-Based Control
,”
J. Franklin Inst.
,
349
(
8
), pp.
2547
2569
.
18.
Chadli
,
M.
, and
Zelinka
,
I.
,
2014
, “
Chaos Synchronization of Unknown Inputs Takagi–Sugeno Fuzzy: Application to Secure Communications
,”
Comput. Math. Appl.
,
68
(
12
), pp.
2142
2147
.
19.
Ott
,
E.
,
Grebogi
,
C.
, and
Yorke
,
J. A.
,
1990
, “
Controlling Chaos
,”
Phys. Rev. Lett.
64
(
11
), pp.
1196
1199
.
20.
Hartly
,
T. T.
,
Lorenzo
,
C. F.
, and
Qammer
,
H. K.
,
1995
, “
Chaos in a Fractional Order Chua's System
,”
IEEE Trans. Circuit. Syst. I
,
42
(
8
), pp.
485
490
.
21.
Hegazi
,
A. S.
,
Ahmed
,
E.
, and
Matouk
,
A. E.
,
2013
, “
On Chaos Control and Synchronization of the Commensurate Fractional Order Liu System
,”
Commun. Nonlinear Sci. Numer. Simul.
,
18
(
5
), pp.
1193
1202
.
22.
Faieghi
,
M. R.
,
Delavari
,
H.
, and
Baleanu
,
D.
,
2013
, “
A Note on Stability of Sliding Mode Dynamic in Suppression of Fractional-Order Chaotic Systems
,”
Comput. Math. Appl.
,
66
(
5
), pp.
832
837
.
23.
Aghababa
,
M. P.
,
2014
, “
Control of Fractional-Order Using Chatter-Free Sliding Mode Approach
,”
ASME J. Comput. Nonlinear Dyn.
,
9
(
3
), p.
031003
.
24.
Faieghi
,
M. R.
,
Kuntanapreeda
,
S.
,
Delavari
,
H.
, and
Baleanu
,
D.
,
2014
, “
Robust Stabilization of Fractional-Order Chaotic Systems With Linear Controllers: LMI-Based Sufficient Conditions
,”
J. Vib. Control
,
20
(
7
), pp.
1042
1051
.
25.
Wang
,
B.
,
Xue
,
J.
, and
Chen
,
D.
,
2014
, “
Takagi–Sugeno Fuzzy Control for a Wide Class of Fractional-Order Chaotic Systems With Uncertain Parameters Via Linear Matrix Inequality
,”
J. Vib. Control
,
2014
, pp.
1
14
.
26.
Kuntanapreeda
,
S.
,
2015
, “
Tensor Product Model Transformation Based Control and Synchronization of a Class of Fractional-Order Chaotic Systems
,”
Asian J. Control
,
17
(
2
), pp.
371
380
.
27.
Li
,
R.
, and
Li
,
W.
,
2015
, “
Suppressing Chaos for a Class of Fractional-Order Chaotic Systems by Adaptive Integer-Order and Fractional-Order Feedback Control
,”
Optik
,
126
(
21
), pp.
2965
2973
.
28.
Danca
,
M.-F.
, and
Garrappa
,
R.
,
2015
, “
Suppressing Chaos in Discontinuous Systems of Fractional Order by Active Control
,”
Appl. Math. Comput.
,
257
, pp.
89
102
.
29.
Golmankhaneh
,
A. K.
,
Arefi
,
R.
, and
Baleanu
,
D.
,
2015
, “
Synchronization in a Nonidentical Fractional Order of a Proposed Modified System
,”
J. Vib. Control
,
216
(
6
), pp.
1154
1161
.
30.
Wu
,
G.-C.
, and
Baleanu
,
D.
,
2014
, “
Chaos Synchronization of Discrete Fractional Logistic Map
,”
Signal Process.
,
102
, pp.
96
99
.
31.
Li
,
Y.
,
Chen
,
Y.
, and
Podlubny
,
I.
,
2009
, “
Mittag–Leffler Stability of Fractional Order Nonlinear Dynamic Systems
,”
Automatica
,
45
(
2
), pp.
1965
1969
.
32.
Li
,
Y.
,
Chen
,
Y.
, and
Podlubny
,
I.
,
2010
, “
Stability of Fractional-Order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag Leffler Stability
,”
Comput. Math. Appl.
,
59
(
5
), pp.
1810
1821
.
33.
Tarasov
,
V. E.
,
2013
, “
No Violation of the Leibniz Rule. No Fractional Derivative
,”
Commun. Nonlinear Sci. Numer. Simul.
,
18
(
11
), pp.
2945
2948
.
34.
Aguila-Camacho
,
N.
,
Duarte-Mermoud
,
M. A.
, and
Gallegos
,
J. A.
,
2014
, “
Lyapunov Functions for Fractional Order Systems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
19
(
9
), pp.
2951
2957
.
35.
Duarte-Mermoud
,
M. A.
,
Aguila-Camacho
,
N.
,
Gallegos
,
J. A.
, and
Castro-Linares
,
R.
,
2015
Using General Quadratic Lyapunov Function to Prove Lyapunov Uniform Stability for Fractional Order Systems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
22
(
1–3
), pp.
650
659
.
36.
Keshtkar
,
F.
,
Erjaee
,
G. H.
, and
Kheiri
,
H.
,
2016
, “
On Global Stability of Nonlinear Fractional Dynamical Systems
,”
J. Comput. Complex. Appl.
,
2
(
1
), pp.
16
23
.
37.
Chen
,
F.
, and
Liu
,
Z.
,
2012
, “
Asymptotic Stability Results for Nonlinear Fractional Difference Equations
,”
J. Appl. Math.
,
2012
, p.
879657
.
38.
Abu-Saris
,
R.
, and
Al-Mdallal
,
Q.
,
2013
, “
On the Asymptotic Stability of Linear System of Fractional-Order Difference Equations
,”
Frac. Calc. Appl. Anal.
16
(
3
), pp.
613
629
.
39.
Chen
,
F.-L.
,
2015
, “
A Review of Existence and Stability Results for Discrete Fractional Equations
,”
J. Comput. Complex. Appl.
,
1
(
1
), pp.
22
53
.
40.
Zhang
,
X.
,
Khadra
,
A.
,
Yang
,
D.
, and
Li
,
D.
,
2010
, “
Unified Impulsive Fuzzy-Model-Based Controllers for Chaotic Systems With Parameter Uncertainties Via LMI
,”
Commun. Nonlinear Sci. Numer. Simul.
,
15
(
1
), pp.
105
114
.
41.
Mobayen
,
S.
,
2015
, “
An LMI-Based Robust Controller Design Using Global Nonlinear Sliding Surfaces and Application to Chaotic Systems
,”
Nonlinear Dyn.
,
79
(
2
), pp.
1075
1084
.
42.
Diethelm
,
K.
,
Ford
,
N. J.
, and
Freed
,
A. D.
,
2002
, “
Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
,”
Nonlinear Dyn.
,
29
(
1
), pp.
3
22
.
43.
Diethelm
,
K.
,
Ford
,
N. J.
, and
Freed
,
A. D.
,
2004
, “
Detailed Error Analysis for a Fractional Adams Method
,”
Numer. Algorithms
,
36
(
1
), pp.
31
52
.
You do not currently have access to this content.