This paper deals with the problem of master-slave synchronization of fractional-order chaotic systems with input saturation. Sufficient stability conditions for achieving the synchronization are derived from the basis of a fractional-order extension of the Lyapunov direct method, a new lemma of the Caputo fractional derivative, and a local sector condition. The stability conditions are formulated in linear matrix inequality (LMI) forms and therefore are readily solved. The fractional-order chaotic Lorenz and hyperchaotic Lü systems with input saturation are utilized as illustrative examples. The feasibility of the proposed synchronization scheme is demonstrated through numerical simulations.
Issue Section:
Special Issue Papers
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.
Baleanu
, D.
, Caponetto
, R.
, and Machado
, J. A. T.
, 2016
, “Challenges in Fractional Dynamics and Control Theory
,” J. Vib. Control
, 22
(9
), pp. 2151
–2152
.5.
Atıcı
, F. M.
, and Şengül
, S.
, 2010
, “Modeling With Fractional Difference Equations
,” J. Math. Anal. Appl.
, 369
(1
), pp. 1
–9
.6.
Wu
, F.
, and Liu
, J.-F.
, 2016
, “Discrete Fractional Creep Model of Salt Rock
,” J. Comput. Complex. Appl.
, 2
(1
), pp. 1
–6
.http://www.computcomplex.com/upLoad/file/20151120/14480005132124280.pdf7.
Hartly
, T. T.
, Lorenzo
, C. F.
, and Qammer
, H. K.
, 1995
, “Chaos in a Fractional Order Chua's System
,” IEEE Trans. Circuits Syst. I
, 42
(8
), pp. 485
–490
.8.
Grigorenko
, L.
, and Grigorenko
, E.
, 2003
, “Chaotic Dynamics of the Fractional Lorenz System
,” Phys. Rev. Lett.
, 91
(3
), p. 034101
.9.
Pan
, L.
, Zhou
, W.
, Zhou
, L.
, and Sun
, K.
, 2011
, “Chaos Synchronization Between Two Different Fractional-Order Hyperchaotic Systems
,” Commun. Nonlinear Sci. Numer. Simul.
, 16
(6), pp. 2628
–2640
.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.
Baleanu
, D.
, Magin
, R. L.
, Bhalekar
, S.
, and Daftardar-Gejji
, V.
, 2015
, “Chaos in the Fractional Order Nonlinear Bloch Equation With Delay
,” Commun. Nonlinear Sci. Numer. Simul.
, 25
(1–3), pp. 41
–49
.12.
Wu
, G.-C.
, and Baleanu
, D.
, 2014
, “Discrete Fractional Logistic Map and Its Chaos
,” Nonlinear Dyn.
, 75
(1–2
), pp. 283
–287
.13.
Wu
, G.-C.
, and Baleanu
, D.
, 2015
, “Discrete Chaos in Fractional Delayed Logistic Maps
,” Nonlinear Dyn.
, 80
(4
), pp. 1697
–1703
.14.
Ott
, E.
, Grebogi
, C.
, and Yorke
, J. A.
, 1990
, “Controlling Chaos
,” Phys. Rev. Lett.
, 64
(11
), pp. 1196
–1199
.15.
Kuntanapreeda
, S.
, 2012
, “Robust Synchronization of Fractional-Order Unified Chaotic Systems Via Linear Control
,” Comput. Math. Appl.
, 63
(1
), pp. 183
–190
.16.
Razminia
, A.
, and Baleanu
, D.
, 2013
, “Complete Synchronization of Commensurate Fractional Order Chaotic System Using Sliding Mode Control
,” Mechatronics
, 23
(7
), pp. 873
–879
.17.
Wu
, G.-C.
, and Baleanu
, D.
, 2014
, “Chaos Synchronization of Discrete Fractional Logistic Map
,” Signal Process.
, 102
, pp. 96
–99
.18.
Golmankhaneh
, A. K.
, Arefi
, R.
, and Baleanu
, D.
, 2015
, “Synchronization in a Nonidentical Fractional Order of a Proposed Modified System
,” J. Vib. Control
, 21
(6
), pp. 1154
–1161
.19.
Lopes
, A. M.
, and Machado
, J. A. T.
, 2015
, “Visualizing Control Systems Performance: A Fractional Perspective
,” Adv. Mech. Energy
, 7
(12
), pp. 1
–8
.20.
Khamsuwan
, P.
, and Kuntanapreeda
, S.
, 2016
, “A Linear Matrix Inequality Approach to Output Feedback Control of Fractional-Order Unified Chaotic Systems With One Control Input
,” ASME J. Comput. Nonlinear Dyn.
, 11
(5
), p. 051021
.21.
Shahri
, E.
, Alfi
, A.
, and Machado
, J. A. T.
, 2016
, “Stabilization of Fractional-Order Systems Subject to Saturation Element Using Fractional Dynamic Output Feedback Sliding Mode Control
,” ASME J. Comput. Nonlinear Dyn.
, 12
(3
), p. 031014
.22.
David
, S. A.
, Machado
, J. A. T.
, Quintino
, D. D.
, and Balthazar
, J. M.
, 2016
, “Partial Chaos Suppression in a Fractional Order Macroeconomic Model
,” Math. Comput. Simul.
, 122
, pp. 55
–68
.23.
Gao
, Y.-F.
, Sun
, X.-M.
, Wen
, C.
, and Wang
, W.
, 2017
, “Adaptive Tracking Control for a Class of Stochastic Uncertain Nonlinear Systems With Input Saturation
,” IEEE Trans. Autom. Control
, 62
(5
), pp. 2498
–2504
.24.
Hu
, Q.
, Zhang
, J.
, and Friswell
, M. I.
, 2015
, “Finite-Time Coordinated Attitude Control for Spacecraft Formation Flying Under Input Saturation
,” ASME J. Dyn. Sys., Meas., Control
, 137
(6
), p. 061012
.25.
Rehan
, M.
, Tufail
, M.
, Ahn
, C. K.
, and Chadli
, M.
, 2017
, “Stabilisation of Locally Lipschitz Non-Linear Systems Under Input Saturation and Quantisation
,” IET Control Theory Appl.
, 11
(9
), pp. 1459
–1466
.26.
Du
, J.
, Hu
, X.
, Krstić
, M.
, and Sun
, Y.
, 2016
, “Robust Dynamic Positioning of Ships With Disturbances Under Input Saturation
,” Automatica
, 73
, pp. 207
–214
.27.
Zelei
, A.
, Bencsik
, L.
, and Stépán
, G.
, 2016
, “Handling Actuator Saturation as Underactuation: Case Study With Acroboter Service Robot
,” ASME J. Comput. Nonlinear Dyn.
, 12
(3
), p. 031011
.28.
Castelan
, E. B.
, Tarbouriech
, S.
, and Queinnec
, I.
, 2005
, “Stability and Stabilization of a Class of Nonlinear Systems With Saturating Actuators
,” IFAC World Congr.
, 38
(1), pp. 729
–734
.29.
Tarbouriech
, S.
, Prieur
, C.
, and da Silva
, J. M. G.
, 2006
, “Stability Analysis and Stabilization of Systems Presenting Nested Saturations
,” IEEE Trans. Autom. Control
, 51
(8
), pp. 1364
–1371
.30.
Rehan
, M.
, 2013
, “Synchronization and Anti-Synchronization of Chaotic Oscillators Under Input Saturation
,” Appl. Math. Model.
, 37
(10–11
), pp. 6829
–6837
.31.
Ma
, Y.
, and Jing
, Y.
, 2014
, “Robust H∞ Synchronization of Chaotic System With Input Saturation and Time-Varying Delay
,” Adv. Differ. Equations
, 2014
(1
), p. 124
.32.
Iqbal
, M.
, Rehan
, M.
, Hong
, K.-S.
, Khaliq
, A.
, and Rehman
, S.-U.
, 2015
, “Sector-Condition-Based Result for Adaptive Control and Synchronization of Chaotic Systems Under Input Saturation
,” Chaos, Solitons Fractals
, 77
, pp. 158
–169
.33.
Li
, Y.
, Chen
, Y.
, and Podlubny
, I.
, 2009
, “Mittag- Leffler Stability of Fractional Order Nonlinear Dynamic Systems
,” Automatica
, 45
(8
), pp. 1965
–1969
.34.
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
.35.
Alikhanov
, A. A.
, 2010
, “A Priori Estimates for Solutions of Boundary Value Problems for Fractional-Order Equations
,” Differ. Equations
, 46
(5
), pp. 660
–666
.36.
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
.37.
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
.38.
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
.http://www.computcomplex.com/upLoad/file/20151120/14480007099843825.pdf39.
Trigeassou
, J.
, Maamri
, N.
, and Oustaloup
, A.
, 2015
, “Lyapunov Stability of Noncommensurate Fractional Order Systems: An Energy Balance Approach
,” ASME J. Comput. Nonlinear Dyn.
, 11
(4
), p. 041007
.40.
Trigeassou
, J.
, Maamri
, N.
, and Oustaloup
, A.
, 2016
, “Lyapunov Stability of Commensurate Fractional Order Systems: A Physical Interpretation
,” ASME J. Comput. Nonlinear Dyn.
, 11
(5
), p. 051007
.41.
Shahri
, E. S. A.
, Alfi
, A.
, and Machado
, J. A. T.
, 2015
, “An Extension of Estimation of Domain of Attraction for Fractional Order Linear System Subject to Saturation Control
,” App. Math. Lett.
, 47
, pp. 26
–34
.42.
Chen
, F.
, and Liu
, Z.
, 2012
, “Asymptotic Stability Results for Nonlinear Fractional Difference Equations
,” J. Appl. Math.
, 2012
, p. 879657
.43.
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
.44.
Chen
, F.-L.
, 2015
, “A Review of Existence and Stability Results for Discrete Fractional Equations
,” J. Comput. Complex. Appl.
, 1
(1
), pp. 22
–53
.http://www.computcomplex.com/aspcms/news/2015-8-16/100.html45.
Aguila-Camacho
, N.
, Duarte-Mermoud
, M. A.
, and Delgado-Aguilera
, E.
, 2016
, “Adaptive Synchronization of Fractional Lorenz Systems Using a Reduced Number of Control Signals and Parameters
,” Chaos, Solitons Fractals
, 87
, pp. 1
–11
.46.
Lenka
, B. K.
, and Banerjee
, S.
, 2018
, “Sufficient Conditions for Asymptotic Stability and Stabilization of Autonomous Fractional Order Systems
,” Commun. Nonlinear Sci. Numer. Simul.
, 56
, pp. 365
–379
.47.
Diethelm
, K.
, Ford
, N. J.
, and Freed
, A. D.
, 2002
, “Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
,” Nonlinear Dyn..
, 29
, pp. 3
–22
.48.
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
.Copyright © 2018 by ASME
You do not currently have access to this content.