Abstract
This letter investigates the synchronization of a class of three-dimensional fractional-order chaotic systems. Based on sliding mode variable structure control theory and adaptive control technique, a single-state adaptive-feedback controller containing a novel fractional integral sliding surface is developed to synchronize a class of fractional-order chaotic systems. The present controller, which only contains a single driving variable, is simple both in design and implementation. Simulation results for three fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.
Similar content being viewed by others
References
Bagley, R.L., Calico, R.A.: Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control Dyn. 14, 304–311 (1991)
Sun, H.H., Abdelwahad, A.A., Onaral, B.: Linear approximation of transfer function with a pole of fractional power. IEEE Trans. Autom. Control 29, 441–444 (1984)
Ichise, M., Nagayanagi, Y., Kojima, T.: An analog simulation of non-integer order transfer functions for analysis of electrode processes. J. Electroanal. Chem. Interfacial Electrochem. 33, 253–265 (1971)
Heaviside, O.: Electromagnetic Theory. Chelsea, New York (1971)
Mandelbrot, B., Van Ness, J.W.: Fractional. Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)
Oustaloup, A.: La dérivation non entière: theorie, synthèse et applications. Editions Hermes, Paris (1995)
Podlubny, I.: Fractional-order systems and PI λ D μ-controllers. IEEE Trans. Autom. Control 44, 208–214 (1999)
Linares, H., Baillot, Ch., Oustaloup, A., Ceyral, Ch.: Generation of a fractal ground: application in robotics. In: International Congress in IEEE-SMC CESA’96 IMACS Multiconf., Lille, July 1996
Duarte, F.B.M., Macado, J.A.T.: Chaotic phenomena and fractional-order dynamics in the trajectory control of redundant manipulators. Nonlinear Dyn. 29, 315–342 (2002)
Deng, W.H., Li, C.P.: Chaos synchronization of the fractional Lü system. Physica A 353, 61–72 (2005)
Li, C.P., Deng, W.H.: Chaos synchronization of fractional-order differential systems. Int. J. Mod. Phys. B 20, 791–803 (2006)
Lu, J.G.: Nonlinear observer design to synchronize fractional-order chaotic system via a scaler transmitted signal. Physica A 359, 107–118 (2006)
Wu, X.J., Lu, H.T., Shen, S.L.: Synchronization of a new fractional-order hyperchaotic system. Phys. Lett. A 373, 2329–2337 (2009)
Zhang, R.X., Yang, S.P.: Adaptive synchronization of fractional-order chaotic systems. Chin. Phys. B 19, 020510 (2010)
Peng, G.J.: Synchronization of fractional order chaotic systems. Phys. Lett. A 363, 426 (2007)
Tavazoei, M.S., Haeri, M.: Synchronization of chaotic fractional-order systems via active sliding mode controller. Physica A 387, 57–70 (2008)
Lu, J.G.: Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal. Chaos Solitons Fractals 27, 519–525 (2006)
Lu, J.G.: Chaotic dynamics and synchronization of fractional-order Arneodo’s systems. Chaos Solitons Fractals 26, 1125–1133 (2005)
Wang, X.J., Li, J., Chen, G.R.: Chaos in the fractional order unified system and its synchronization. J. Franklin Inst. 345, 392 (2008)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, R., Yang, S. Adaptive synchronization of fractional-order chaotic systems via a single driving variable. Nonlinear Dyn 66, 831–837 (2011). https://doi.org/10.1007/s11071-011-9944-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-011-9944-2