Abstract and Applied Analysis

Synchronization of Fractional-Order Chaotic Systems with Gaussian Fluctuation by Sliding Mode Control

Yong Xu and Hua Wang

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Abstract

Chaotic systems are always influenced by some uncertainties and external disturbances. This paper investigates the problem of practical synchronization of fractional-order chaotic systems with Gaussian fluctuation. A fractional integral (FI) sliding surface is proposed for synchronizing the uncertain fractional-order system, and then the sliding mode control technique is carried out to realize the synchronization of the given systems. One theorem about sliding mode controller is presented to prove that the proposed controller can make the system achieve synchronization. As a case study, the presented method is applied to the fractional-order Chen-Lü system, and simulation results show that the proposed control approach is capable to go against Gaussian noise well.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 948782, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393512163

Digital Object Identifier
doi:10.1155/2013/948782

Mathematical Reviews number (MathSciNet)
MR3132559

Zentralblatt MATH identifier
07095528

Citation

Xu, Yong; Wang, Hua. Synchronization of Fractional-Order Chaotic Systems with Gaussian Fluctuation by Sliding Mode Control. Abstr. Appl. Anal. 2013 (2013), Article ID 948782, 7 pages. doi:10.1155/2013/948782. https://projecteuclid.org/euclid.aaa/1393512163


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References

  • R. Zhang and S. Yang, “Robust synchronization of two different fractional-order chaotic systems with unknown parameters using adaptive sliding mode approach,” Nonlinear Dynamics, vol. 71, pp. 269–278, 2013.
  • X. Wang, X. Zhang, and C. Ma, “Modified projective synchronization of fractional-order chaotic systems via active sliding mode control,” Nonlinear Dynamics, vol. 69, pp. 511–517, 2012.
  • P. Zhou, R. Ding, and Y. Cao, “Multi drive-one response synchronization for fractional-order chaotic systems,” Nonlinear Dynamics, vol. 70, pp. 1263–1271, 2012.
  • C. P. Li, W. H. Deng, and D. Xu, “Chaos synchronization of the Chua system with a fractional order,” Physica A, vol. 360, no. 2, pp. 171–185, 2006.
  • K. P. Wilkie, C. S. Drapaca, and S. Sivaloganathan, “A nonlinear viscoelastic fractional derivative model of infant hydrocephalus,” Applied Mathematics and Computation, vol. 217, no. 21, pp. 8693–8704, 2011.
  • Y. Luo, Y. Chen, and Y. Pi, “Experimental study of fractional order proportional derivative controller synthesis for fractional order systems,” Mechatronics, vol. 21, no. 1, pp. 204–214, 2011.
  • M. Rivero, J. J. Trujillo, L. Vázquez, and M. Pilar Velasco, “Fractional dynamics of populations,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 1089–1095, 2011.
  • N. Laskin, “Fractional market dynamics,” Physica A, vol. 287, no. 3-4, pp. 482–492, 2000.
  • W. M. Ahmad and J. C. Sprott, “Chaos in fractional-order autonomous nonlinear systems,” Chaos, Solitons and Fractals, vol. 16, no. 2, pp. 339–351, 2003.
  • T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, “Chaos in a fractional order Chua's system,” IEEE Transactions on Circuits and Systems I, vol. 42, no. 8, pp. 485–490, 1995.
  • C. Li and G. Chen, “Chaos and hyperchaos in the fractional-order Rössler equations,” Physica A, vol. 341, no. 1–4, pp. 55–61, 2004.
  • C. Li and G. Peng, “Chaos in Chen's system with a fractional order,” Chaos, Solitons and Fractals, vol. 22, no. 2, pp. 443–450, 2004.
  • L.-J. Sheu, H.-K. Chen, J.-H. Chen et al., “Chaos in the Newton-Leipnik system with fractional order,” Chaos, Solitons and Fractals, vol. 36, no. 1, pp. 98–103, 2008.
  • P. Zhou and R. Ding, “Modified function projective synchronization between different dimension fractional-order chaotic systems,” Abstract and Applied Analysis, vol. 2012, Article ID 862989, 12 pages, 2012.
  • A. Kiani-B, K. Fallahi, N. Pariz, and H. Leung, “A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 863–879, 2009.
  • L. J. Sheu, “A speech encryption using fractional chaotic systems,” Nonlinear Dynamics, vol. 65, no. 1-2, pp. 103–108, 2011.
  • N. Singh and A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Optics and Lasers in Engineering, vol. 46, no. 2, pp. 117–123, 2008.
  • N. Zhou, Y. Wang, L. Gong, H. He, and J. Wu, “Novel single-channel color image encryption algorithm based on chaos and fractional Fourier transform,” Optics Communications, vol. 284, no. 12, pp. 2789–2796, 2011.
  • J. Lu, X. Wu, and J. Lü, “Synchronization of a unified chaotic system and the application in secure communication,” Physics Letters A, vol. 305, no. 6, pp. 365–370, 2002.
  • J. Lü and G. Chen, “A time-varying complex dynamical network model and its controlled synchronization criteria,” IEEE Transactions on Automatic Control, vol. 50, no. 6, pp. 841–846, 2005.
  • C. Li and W. Deng, “Chaos synchronization of fractional-order differential systems,” International Journal of Modern Physics B, vol. 20, no. 7, pp. 791–803, 2006.
  • S. Bhalekar and V. Daftardar-Gejji, “Synchronization of different fractional order chaotic systems using active control,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, pp. 3536–3546, 2010.
  • R. Zhang and S. Yang, “Adaptive synchronization of fractional-order chaotic systems via a single driving variable,” Nonlinear Dynamics, vol. 66, no. 4, pp. 831–837, 2011.
  • Z. M. Odibat, “Adaptive feedback control and synchronization of non-identical chaotic fractional order systems,” Nonlinear Dynamics, vol. 60, no. 4, pp. 479–487, 2010.
  • D.-Y. Chen, Y.-X. Liu, X.-Y. Ma, and R.-F. Zhang, “Control of a class of fractional-order chaotic systems via sliding mode,” Nonlinear Dynamics, vol. 67, no. 1, pp. 893–901, 2012.
  • J. G. Lu, “Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal,” Chaos, Solitons and Fractals, vol. 27, no. 2, pp. 519–525, 2006.
  • I. Boiko, L. Fridman, R. Iriarte, A. Pisano, and E. Usai, “Parameter tuning of second-order sliding mode controllers for linear plants with dynamic actuators,” Automatica, vol. 42, no. 5, pp. 833–839, 2006.
  • J. J. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, Upper Saddle River, NJ, USA, 1991.
  • M. S. Tavazoei and M. Haeri, “Synchronization of chaotic fractional-order systems via active sliding mode controller,” Physica A, vol. 387, no. 1, pp. 57–70, 2008.
  • M. Aghababa, “Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique,” Nonlinear Dynamics, vol. 69, pp. 247–261, 2012.
  • A. S. Pikovsky, “Comment on `chaos, noise, and synchronization',” Physical Review Letters, vol. 73, no. 21, p. 2931, 1994.
  • K. H. Nagai and H. Kori, “Noise-induced synchronization of a large population of globally coupled nonidentical oscillators,” Physical Review E, vol. 81, no. 6, Article ID 065202, 2010.
  • H. Herzel and J. Freund, “Chaos, noise, and synchronization reconsidered,” Physical Review E, vol. 52, no. 3, pp. 3238–3241, 1995.
  • C.-H. Lai and C. Zhou, “Synchronization of chaotic maps by symmetric common noise,” Europhysics Letters, vol. 43, no. 4, pp. 376–380, 1998.
  • J.-S. Lin, J.-J. Yan, and T.-L. Liao, “Chaotic synchronization via adaptive sliding mode observers subject to input nonlinearity,” Chaos, Solitons and Fractals, vol. 24, no. 1, pp. 371–381, 2005.
  • H.-T. Yau, “Design of adaptive sliding mode controller for chaos synchronization with uncertainties,” Chaos, Solitons and Fractals, vol. 22, no. 2, pp. 341–347, 2004.
  • H. Salarieh and A. Alasty, “Chaos synchronization of nonlinear gyros in presence of stochastic excitation via sliding mode control,” Journal of Sound and Vibration, vol. 313, no. 3–5, pp. 760–771, 2008.
  • G. Tao, “A simple alternative to the Barbǎlat lemma,” IEEE Transactions on Automatic Control, vol. 42, no. 5, p. 698, 1997.
  • J. Lü, G. Chen, D. Cheng, and S. Celikovsky, “Bridge the gap between the Lorenz system and the Chen system,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 12, pp. 2917–2926, 2002.
  • J. Lü and G. Chen, “A new chaotic attractor coined,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 3, pp. 659–661, 2002.
  • I. Petráš, Fractional-Order Nonlinear Systems Modeling, Analysis and Simulation, Springer, Berlin, Germany, 2011.
  • T. Škovránek, I. Podlubny, and I. Petráš, “Modeling of thenational economies in state-space: a fractional calculus approach,” Economic Modelling, vol. 29, pp. 1322–1327, 2012.