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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 732503, 6 pages
Synchronization of an Uncertain Fractional-Order Chaotic System via Backstepping Sliding Mode Control
1College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China
2State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing 210093, China
Received 19 March 2013; Accepted 7 June 2013
Academic Editor: Sridhar Seshagiri
Copyright © 2013 Zhen Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, New York, NY, USA, 1999.
- J. Wang and Y. Zhou, “Complete controllability of fractional evolution systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4346–4355, 2012.
- Z. Wang, X. Huang, Y. Li, and X. Song, “A new image encryption algorithm based on the fractional-order hyperchaotic Lorenz system,” Chinese Physics B, vol. 22, no. 1, Article ID 010504, 2013.
- R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Hackensack, NJ, USA, 2001.
- G. R. Chen, Controlling Chaos and Bifurcations in Engineering Systems, CRC Press, 1999.
- 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,” Transactions on Circuits and Systems, vol. 42, no. 8, pp. 485–490, 1995.
- C. Li and G. Peng, “Chaos in Chen's system with a fractional order,” Chaos, Solitons & Fractals, vol. 22, no. 2, pp. 443–450, 2004.
- Y. Yu, H.-X. Li, S. Wang, and J. Yu, “Dynamic analysis of a fractional-order Lorenz chaotic system,” Chaos, Solitons and Fractals, vol. 42, no. 2, pp. 1181–1189, 2009.
- 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.
- V. Daftardar-Gejji and S. Bhalekar, “Chaos in fractional ordered Liu system,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1117–1127, 2010.
- J. Lu and J. Cao, “Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters,” Chaos, vol. 15, no. 4, 2005.
- J. Cao and J. Lu, “Adaptive synchronization of neural networks with or without time-varying delay,” Chaos, vol. 16, no. 1, 2006.
- J. Lu, J. Cao, and D. W. C. Ho, “Adaptive stabilization and synchronization for chaotic Lur'e systems with time-varying delay,” IEEE Transactions on Circuits and Systems, vol. 55, no. 5, pp. 1347–1356, 2008.
- W. Deng and C. Li, “Chaos synchronization of the fractional order Lü system,” Physica A, vol. 353, pp. 61–72, 2005.
- M. R. Faieghi and H. Delavari, “Chaos in fractional-order Genesio-Tesi system and its synchronization,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 731–741, 2012.
- 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.
- T. C. Lin and T. Y. Lee, “Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy sliding mode control,” IEEE Transactions on Fuzzy Systems, vol. 4, pp. 623–635, 2011.
- T. C. Lin and C. H. Kuo, “ synchronization of uncertain fractional order chaotic systems: adaptive fuzzy approach,” ISA Transactions, vol. 50, no. 4, pp. 548–556, 2011.
- L. Song, J. Yang, and S. Xu, “Chaos synchronization for a class of nonlinear oscillators with fractional order,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 5, pp. 2326–2336, 2010.
- X. Y. Wang and J. M. Song, “Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 8, pp. 3351–3357, 2009.
- J. W. Wang and Y. B. Zhang, “Synchronization in coupled nonidentical incommensurate fractional order systems,” Physics Letters A, vol. 374, no. 25, pp. 202–207, 2009.
- X. Wu, H. Lu, and S. Shen, “Synchronization of a new fractional-order hyperchaotic system,” Physics Letters A, vol. 373, no. 27-28, pp. 2329–2337, 2009.
- K. Zhang, H. Wang, and H. Fang, “Feedback control and hybrid projective synchronization of a fractional-order Newton-Leipnik system,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 317–328, 2012.
- P. Zhou and W. Zhu, “Function projective synchronization for fractional-order chaotic systems,” Nonlinear Analysis, vol. 12, no. 2, pp. 811–816, 2011.
- S. Wang, Y. G. Yu, and M. Diao, “Hybrid projective synchronization of chaotic fractional order systems with different dimensions,” Physica A, vol. 389, no. 21, pp. 4981–4988, 2010.
- M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, New York, NY, USA, 1995.
- Z. Zhang, S. Xu, and H. Shen, “Reduced-order observer-based output-feedback tracking control of nonlinear systems with state delay and disturbance,” International Journal of Robust and Nonlinear Control, vol. 20, no. 15, pp. 1723–1738, 2010.
- Z. Zhang, S. Xu, and B. Wang, “Adaptive actuator failure compensation with unknown control gain signs,” IET Control Theory & Applications, vol. 5, no. 16, pp. 1859–1867, 2011.
- H. N. Pishkenari, N. Jalili, S. H. Mahboobi, A. Alasty, and A. Meghdari, “Robust adaptive backstepping control of uncertain Lorenz system,” Chaos, vol. 20, no. 2, Article ID 023105, 5 pages, 2010.
- H. Shi, “A novel scheme for the design of backstepping control for a class of nonlinear systems,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 35, no. 4, pp. 1893–1903, 2011.
- Y. Li, Y. Chen, and I. Podlubny, “Mittag-Leffler stability of fractional order nonlinear dynamic systems,” Automatica, vol. 45, no. 8, pp. 1965–1969, 2009.
- K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 3–22, 2002, Fractional order calculus and its applications.