This paper concerns the problem of stabilization of uncertain fractional-order chaotic systems in finite time. On the basis of fractional Lyapunov stability theory, a robust finite-time fractional controller is introduced to control chaos of fractional-order chaotic systems in the presence of system uncertainties. The finite-time stability of the closed-loop system is analytically proved. An estimation of the convergence time is also given. Some numerical simulations are provided to illustrate the usefulness and applicability of the proposed robust finite-time control approach. It is worth noting that the proposed fractional control method is applicable for stabilizing a broad range of uncertain fractional-order nonlinear systems in a given finite time.
Issue Section:
Research Papers
References
1.
Hilfer
, R.
, 2001, Applications of Fractional Calculus in Physics
, World Scientific
, Singapore
.2.
Müller
, S.
, Kästner
, M.
, Brummund
, J.
, and Ulbricht
, V.
, 2011, “A Nonlinear Fractional Viscoelastic Material Model for Polymers
,” Comput. Mater. Sci.
, 50
, pp. 2938
–2949
.3.
Luo
, Y.
, Chen
, Y. Q.
, and Pi
, Y.
, 2011, “Experimental Study of Fractional Order Proportional Derivative Controller Synthesis for Fractional Order Systems
,” Mechatronics
, 21
, pp. 204
–214
.4.
5.
Rivero
, M.
, Trujillo
, J. J.
, Vázquez
, L.
, and Velasco
, M. P.
, 2011, “Fractional Dynamics of Populations
,” Appl. Math. Comput.
, 218
, pp. 1089
–1095
.6.
Özalp
, N.
, and Demirci
, E.
, 2011, “A Fractional Order SEIR Model With Vertical Transmission
,” Math. Comput. Model.
, 54
, pp. 1
–6
.7.
Cao
, J.
, Ma
, C.
, Jiang
, Z.
, and Liu
, S.
, 2011, “Nonlinear Dynamic Analysis of Fractional Order Rub-Impact Rotor System
,” Commun. Nonlinear Sci. Numer. Simul.
, 16
, pp. 1443
–1463
.8.
Ionescu
, C.
, Machado
, J. T.
, and De Keyser
, R.
, 2011, “Fractional-Order Impulse Response of the Respiratory System
,” Comput. Math. Appl.
, 62
, pp. 845
–854
.9.
Pourmahmood
, M.
, Khanmohammadi
, S.
, and Alizadeh
, G.
, 2011, “Synchronization of Two Different Uncertain Chaotic Systems With Unknown Parameters using a Robust Adaptive Sliding Mode Controller
,” Commun. Nonlinear Sci. Numer. Simul.
, 16
, pp. 2853
–2868
.10.
Aghababa
, M. P.
, Khanmohammadi
, S.
, and Alizadeh
, G.
, 2011, “Finite-Time Synchronization of Two Different Chaotic Systems With Unknown Parameters Via Sliding Mode Technique
,” Appl. Math. Model
, 35
, pp. 3080
–3091
.11.
Aghababa
, M. P.
, and Aghababa
, H. P.
, 2011, “Adaptive Finite-Time Stabilization of Uncertain Non-Autonomous Chaotic Electromechanical Gyrostat Systems with Unknown Parameters
,” Mech. Res. Commun.
, 38
, pp. 500
–505
.12.
Aghababa
, M. P.
, 2011, “A Novel Adaptive Finite-Time Controller for Synchronizing Chaotic Gyros with Nonlinear Inputs
,” Chin. Phys. B
, 20
, p. 090505
.13.
Lu
, J. G.
, and Chen
, G.
, 2006, “A Note on the Fractional-Order Chen System
,” Chaos Soliton Fract.
, 27
, pp. 685
–688
.14.
Lu
, J. G.
, 2006, “Chaotic Dynamics of the Fractional-Order Lu System and its Synchronization
,” Phys. Lett. A
, 354
, pp. 305
–311
.15.
Lu
, J. G.
, 2005, “Chaotic Dynamics and Synchronization of Fractional-Order Arneodo’s Systems
,” Chaos Soliton Fract.
, 26
, pp. 1125
–1133
.16.
Yang
, Q.
, and Zeng
, C.
, 2010, “Chaos in Fractional Conjugate Lorenz System and its Scaling Attractors
,” Commun. Nonlinear Sci. Numer. Simulat.
, 15
, pp. 4041
–4051
.17.
Lu
, J. G.
, 2005, “Chaotic Dynamics and Synchronization of Fractional-Order Genesio-Tesi Systems
,” Chin. Phys. B
, 14
, pp. 1517
–1521
.18.
Zhu
, H.
, Zhou
, S.
, and Zhang
, J.
, 2009, “Chaos and Synchronization of the Fractional-Order Chua’s System
,” Chaos Soliton Fract.
, 14
, pp. 1595
–1603
.19.
Ge
, Z. M.
, and Qu
, C. Y.
, 2007, “Chaos in a Fractional Order Modified Duffing System
,” Chaos Soliton Fract.
, 34
, pp. 262
–291
.20.
Wang
, X. Y.
, and Song
, J. M.
, 2009, “Synchronization of the Fractional Order Hyperchaos Lorenz Systems With Activation Feedback Control
,” Commun. Nonlinear Sci. Numer. Simul.
, 14
, pp. 3351
–335
.21.
Hegazi
, A. S.
, and Matouk
, A. E.
, 2011, “Dynamical Behaviors and Synchronization in the Fractional Order Hyperchaotic Chen System
,” Appl. Math. Lett.
, 24
, pp. 1938
–1944
.22.
Zhong
, Q. S.
, Bao
, J. F.
, Yu
, Y. B.
, and Liao
, X. F.
, 2008, “Impulsive Control for Fractional-Order Chaotic Systems
,” Chin. Phys. Lett.
, 25
, pp. 2812
.23.
Lin
, T. C.
, and Kua
, C. H.
, 2011, “H∞ Synchronization of Uncertain Fractional Order Chaotic Systems: Adaptive Fuzzy Approach
,” ISA Trans.
, 24
, pp. 548
–556
.24.
Aghababa
, M. P.
, 2011, “Comments On H∞ Synchronization of Uncertain Fractional Order Chaotic Systems: Adaptive Fuzzy Approach
,” ISA Trans.
, 50
, pp. 548
–556
.25.
Bhalekar
, S.
, and Daftardar-Gejji
, V.
, 2010, “Synchronization of Different Fractional Order Chaotic Systems using Active Control
,” Commun. Nonlinear Sci. Numer. Simulat.
, 15
, pp. 3536
–3546
.26.
Song
, L.
, Yang
, J.
, and Xu
, S.
, 2010, “Chaos Synchronization for a Class of Nonlinear Oscillators with Fractional Order
,” Nonlinear Anal.
, 72
, pp. 2326
–2336
.27.
Pan
, L.
, Zhou
, W.
, Zhou
, L. G.
, and Sun
, K.
, 2011, “Chaos Synchronization between Two Different Fractional-Order Hyperchaotic Systems
,” Commun. Nonlinear Sci. Numer. Simul.
, 16
, pp. 2628
–2640
.28.
Yin
, C.
, Zhong
, S. M.
, and Chen
, W. F.
, 2012, “Design of Sliding Mode Controller for a Class of Fractional-Order Chaotic Systems
,” Commun. Nonlinear Sci. Numer. Simul.
, 17
, pp. 356
–366
.29.
Chen
, D. Y.
, Liu
, Y. X.
, Ma
, X. Y.
, and Zhang
, R. F.
, 2011, “Control of a Class of Fractional-Order Chaotic Systems Via Sliding Mode
,” Nonlinear Dyn.
, to be published.30.
Aghababa
, M. P.
, 2011, “Comments on Control of a Class of Fractional-Order Chaotic Systems Via Sliding Mode
,” Nonlinear Dyn.
, to be published.31.
Aghababa
, M. P.
, 2012, “Comments on Design of Sliding Mode Controller for a class of Fractional-Order Chaotic Systems
,” Commun. Nonlinear Sci. Numer. Simul.
, 17
, pp. 1485
–1488
.32.
Li
, Y.
, Chen
, Y. Q.
, and Podlubny
, I.
, 2009, “Mittag-Leffler Stability of Fractional Order Nonlinear Dynamic Systems
,” Automatica
, 45
, pp. 1965
–1969
.33.
Podlubny
, I.
, 1999, Fractional Differential Equations
, Academic Press
, New York
.34.
Hardy
, G. H.
, Littlewood
, J. E.
, and Polya
, G.
, 1952, Inequalities
, Cambridge University Press
, Cambridge
.35.
Utkin
, V. I.
, 1992, Sliding Modes in Control Optimization
, Springer Verlag
, Berlin, Germany
.36.
Diethelm
, K.
, Ford
, N. J.
, and Freed
, A. D.
, 2002, “A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
,” Nonlinear Dyn.
, 29
, pp. 3
–22
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