Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal. (English) Zbl 1086.94007

Summary: A drive-response synchronization method with linear output error feedback is presented for synchronizing a class of fractional-order chaotic systems via a scalar transmitted signal. Based on stability theory of fractional-order systems and linear system theory, a necessary and sufficient condition for the existence of the feedback gain vector such that global synchronization between the fractional-order drive system and response system can be achieved and its design method are given. This synchronization approach that is simple, global and theoretically rigorous enables synchronization of fractional-order chaotic systems be achieved in a systematic way and does not require the computation of the conditional Lyapunov exponents. An example is used to illustrate the effectiveness of the proposed synchronization method.


94A12 Signal theory (characterization, reconstruction, filtering, etc.)
26A33 Fractional derivatives and integrals
93E12 Identification in stochastic control theory
37N35 Dynamical systems in control
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[1] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[2] Bagley, R.L.; Calico, R.A., Fractional order state equations for the control of viscoelastically damped structures, J guid, contr dyn, 14, 304-311, (1991)
[3] Sun, H.H.; Abdelwahad, A.A.; Onaral, B., Linear approximation of transfer function with a pole of fractional order, IEEE trans auto contr, 29, 441-444, (1984) · Zbl 0532.93025
[4] Ichise, M.; Nagayanagi, Y.; Kojima, T., An analog simulation of noninteger order transfer functions for analysis of electrode process, J electroanal chem, 33, 253-265, (1971)
[5] Heaviside, O., Electromagnetic theory, (1971), Chelsea New York · JFM 30.0801.03
[6] Laskin, N., Fractional market dynamics, Physica A, 287, 482-492, (2000)
[7] Kusnezov, D.; Bulgac, A.; Dang, G.D., Quantum levy processes and fractional kinetics, Phys rev lett, 82, 1136-1139, (1999)
[8] Oustaloup, A.; Levron, F.; Nanot, F.; Mathieu, B., Frequency band complex non integer differentiator: characterization and synthesis, IEEE trans CAS-I, 47, 25-40, (2000)
[9] Chen, Y.Q.; Moore, K., Discretization schemes for fractional-order differentiators and integrators, IEEE trans CAS-I, 49, 363-367, (2002) · Zbl 1368.65035
[10] Hartley, T.T.; Lorenzo, C.F., Dynamics and control of initialized fractional-order systems, Nonlinear dyn, 29, 201-233, (2002) · Zbl 1021.93019
[11] Hwang, C.; Leu, J.-F.; Tsay, S.-Y., A note on time-domain simulation of feedback fractional-order systems, IEEE trans auto contr, 47, 625-631, (2002) · Zbl 1364.93772
[12] Podlubny, I.; Petras, I.; Vinagre, B.M.; O’Leary, P.; Dorcak, L., Analogue realizations of fractional-order controllers, Nonlinear dyn, 29, 281-296, (2002) · Zbl 1041.93022
[13] Hartley, T.T.; Lorenzo, C.F.; Qammer, H.K., Chaos in a fractional order chua’s system, IEEE trans CAS-I, 42, 485-490, (1995)
[14] Arena P, Caponetto R, Fortuna L, Porto D. Chaos in a fractional order Duffing system. In: Proc ECCTD, Budapest; 1997. p. 1259-62.
[15] Ahmad, W.M.; Sprott, J.C., Chaos in fractional-order autonomous nonlinear systems, Chaos, solitons & fractals, 16, 339-351, (2003) · Zbl 1033.37019
[16] Ahmad, W.M.; Harb, W.M., On nonlinear control design for autonomous chaotic systems of integer and fractional orders, Chaos, solitons & fractals, 18, 693-701, (2003) · Zbl 1073.93027
[17] Ahmad, W.; El-Khazali, R.; El-Wakil, A., Fractional-order wien-bridge oscillator, Electr lett, 37, 1110-1112, (2001)
[18] Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys rev lett, 91, 034101, (2003)
[19] Arena, P.; Caponetto, R.; Fortuna, L.; Porto, D., Bifurcation and chaos in noninteger order cellular neural networks, Int J bifur chaos, 7, 1527-1539, (1998) · Zbl 0936.92006
[20] Arena, P.; Fortuna, L.; Porto, D., Chaotic behavior in noninteger-order cellular neural networks, Phys rev E, 61, 776-781, (2000)
[21] Li, C.G.; Chen, G., Chaos and hyperchaos in fractional order Rössler equations, Phycica A, 341, 55-61, (2004)
[22] Li, C.G.; Chen, G., Chaos in the fractional order Chen system and its control, Chaos, solitons & fractals, 22, 549-554, (2004) · Zbl 1069.37025
[23] Li, C.P.; Peng, G.J., Chaos in chen’s system with a fractional order, Chaos, solitons & fractals, 22, 443-450, (2004) · Zbl 1060.37026
[24] Zaslavsky, G.M., Chaos, fractional kinetics, and anomalous transport, Phys rep, 371, 461-580, (2002) · Zbl 0999.82053
[25] Lu JG. Chaotic dynamics and synchronization of fractional-order Arneodo’s systems. Chaos, Solitons & Fractals 2005, in press. · Zbl 1074.65146
[26] Chen G, Fradkov AL. Chaos control and synchronization bibliographies (1987-2001). Available from: http://www.ee.cityu.edu.hk/ gchen/chaos-papers.html.
[27] Pecora, L.M.; Carroll, T.L., Synchronization in chaotic systems, Phys rev lett, 64, 821-824, (1990) · Zbl 0938.37019
[28] Li, C.; Liao, X.; Yu, J., Synchronization of fractional order chaotic systems, Phys rev E, 68, 067203, (2003)
[29] Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fract calculus appl anal, 5, 4, 367-386, (2002) · Zbl 1042.26003
[30] Charef, A.; Sun, H.H.; Tsao, Y.Y.; Onaral, B., Fractal system as represented by singularity function, IEEE trans auto contr, 37, 1465-1470, (1992) · Zbl 0825.58027
[31] Matignon D. Stability results of fractional differential equations with applications to control processing. In: IMACS, IEEE-SMC, Lille, France; 1996. p. 963-8.
[32] Chen, C.T., Linear system theory and design, (1984), Holt, Rinehart & Winston New York
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