Journal of Applied Mathematics

Robust Synchronization of Incommensurate Fractional-Order Chaotic Systems via Second-Order Sliding Mode Technique

Hua Chen, Wen Chen, Binwu Zhang, and Haitao Cao

Full-text: Open access

Abstract

A second-order sliding mode (SOSM) controller is proposed to synchronize a class of incommensurate fractional-order chaotic systems with model uncertainties and external disturbances. Based on the chattering free SOSM control scheme, it can be rigorously proved that the dynamics of the synchronization error is globally asymptotically stable by using the Lyapunov stability theorem. Finally, numerical examples are provided to illustrate the effectiveness of the proposed controller design approach.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 321253, 11 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808060

Digital Object Identifier
doi:10.1155/2013/321253

Mathematical Reviews number (MathSciNet)
MR3082040

Zentralblatt MATH identifier
1271.93130

Citation

Chen, Hua; Chen, Wen; Zhang, Binwu; Cao, Haitao. Robust Synchronization of Incommensurate Fractional-Order Chaotic Systems via Second-Order Sliding Mode Technique. J. Appl. Math. 2013 (2013), Article ID 321253, 11 pages. doi:10.1155/2013/321253. https://projecteuclid.org/euclid.jam/1394808060


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References

  • I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
  • R. Hilfer, Application of Fractional Calculus in Physics, World Science Publishing, Singapore, 2000.
  • M. D. Ortigueira and J. A. Tenreiro Machado, “Fractional signal processing and applications,” Signal Processing, vol. 83, no. 11, pp. 2285–2286, 2003.
  • P. Lanusse, H. Benlaoukli, D. Nelson-Gruel, and A. Oustaloup, “Fractional-order control and interval analysis of SISO systems with time-delayed state,” IET Control Theory & Applications, vol. 2, no. 1, pp. 16–23, 2008.
  • http://mechatronics.ece.usu.edu/foc/cdc02tw/cdrom/Lectures/ AppendixB/appendixB.pdf.
  • I. Petráš, “Tuning and implementation methods for fractional-order controllers,” Fractional Calculus and Applied Analysis, vol. 15, no. 2, pp. 282–303, 2012.
  • L. Dorcak, “Numerical models for the simulation of the fractional-order control systems,” 2002, http://arxiv.org/abs/ math/0204108v1.
  • K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
  • B. B. Mandelbrot and J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Review, vol. 10, pp. 422–437, 1968.
  • N. Engheta, “On fractional calculus and fractional multipoles in electromagnetism,” IEEE Transactions on Antennas and Propagation, vol. 44, no. 4, pp. 554–566, 1996.
  • D. Baleanu, A. K. Golmankhaneh, R. Nigmatullin, and A. K. Golmankhaneh, “Fractional Newtonian mechanics,” Central European Journal of Physics, vol. 8, no. 1, pp. 120–125, 2010.
  • H. G. Sun, W. Chen, H. Wei, and Y. Q. Chen, “A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems,” European Physical Journal, vol. 193, no. 1, pp. 185–192, 2011.
  • H. Sun, Y. Chen, and W. Chen, “Random-order fractional differential equation models,” Signal Processing, vol. 91, no. 3, pp. 525–530, 2011.
  • W. Chen, J. Lin, and F. Wang, “Regularized meshless method for nonhomogeneous problems,” Engineering Analysis with Boundary Elements, vol. 35, no. 2, pp. 253–257, 2011.
  • A. Oustaloup, La Derivation Non Entiere: Theorie. Synthese et Application S, Editions Hermes, Paris, France, 1995.
  • R. L. Magin, “Fractional calculus in bioengineering, part 3,” Critical Reviews in Biomedical Engineering, vol. 32, no. 3-4, pp. 195–377, 2004.
  • I. Podlubny, “Fractional-order systems and $P{I}^{\lambda }{D}^{\mu }$-controllers,” IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208–214, 1999.
  • M. S. Tavazoei, M. Haeri, S. Jafari, S. Bolouki, and M. Siami, “Some applications of fractional calculus in suppression of chaotic oscillations,” IEEE Transactions on Industrial Electronics, vol. 55, no. 11, pp. 4094–4101, 2008.
  • H. Linares, C. Baillot, A. Oustaloup, and C. Ceyral, “Generation of a fractional ground: application in robotics,” in Proceedings of the International Congress IEE-Smc (CESA '96), Lille, France, July 1996.
  • F. B. M. Duarte and J. A. Tenreiro Machado, “Chaotic phenomena and fractional-order dynamics in the trajectory control of redundant manipulators,” Nonlinear Dynamics, vol. 29, no. 1-4, pp. 315–342, 2002.
  • C. Li and G. Chen, “Chaos in the fractional order Chen system and its control,” Chaos, Solitons & Fractals, vol. 22, no. 3, pp. 549–554, 2004.
  • L.-J. Sheu, H.-K. Chen, J.-H. Chen et al., “Chaos in the Newton-Leipnik system with fractional order,” Chaos, Solitons & Fractals, vol. 36, no. 1, pp. 98–103, 2008.
  • W. M. Ahmad and J. C. Sprott, “Chaos in fractional-order autonomous nonlinear systems,” Chaos, Solitons & Fractals, vol. 16, no. 2, pp. 339–351, 2003.
  • I. Grigorenko and E. Grigorenko, “Chaotic dynamics of the fractional Lorenz system,” Physical Review Letters, vol. 91, no. 3, Article ID 034101, 4 pages, 2003.
  • M.-F. Danca, “Chaotic behavior of a class of discontinuous dynamical systems of fractional-order,” Nonlinear Dynamics, vol. 60, no. 4, pp. 525–534, 2010.
  • M. M. Asheghan, M. T. H. Beheshti, and M. S. Tavazoei, “Robust synchronization of perturbed Chen's fractional-order chaotic system,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, pp. 1044–1051, 2011.
  • M. P. Aghababa, S. Khanmohammadi, and G. Alizadeh, “Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique,” Applied Mathematical Modelling, vol. 35, no. 6, pp. 3080–3091, 2011.
  • R. Zhang and S. Yang, “Stabilization of fractional-order chaotic system via a single state adaptive-feedback controller,” Nonlinear Dynamics, vol. 68, no. 1-2, pp. 45–51, 2012.
  • S. Wang and Y. G. Yu, “Generalized projective synchronization of fractional order chaotic systems with different dimensions,” Chinese Physics Letters, vol. 29, no. 2, Article ID 020505, 3 pages, 2012.
  • C. Yin, S. Dadras, S. M. Zhang, and Y. Q. Chen, “Control of a novel class of fractional-order chaotic systems via adaptive sliding mode control approach,” Applied Mathematical Modelling, vol. 37, no. 4, pp. 1607–2600, 2013.
  • S. H. Hosseinnia, R. Ghaderi, A. Ranjbar N., M. Mahmoudian, and S. Momani, “Sliding mode synchronization of an uncertain fractional order chaotic system,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1637–1643, 2010.
  • X. J. Wu, H. T. Lu, and S. L. Shen, “Synchronization of a new fractional-order hyperchaotic system,” Physics Letters A, vol. 373, no. 27-28, pp. 2329–2337, 2009.
  • D. M. Senejohnny and H. Delavari, “Active sliding observer scheme based fractional chaos synchronization,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4373–4383, 2012.
  • C. Yin, S. Dadras, and S.-M. Zhong, “Design an adaptive sliding mode controller for drive-response synchronization of two different uncertain fractional-order chaotic systems with fully unknown parameters,” Journal of the Franklin Institute, vol. 349, no. 10, pp. 3078–3101, 2012.
  • 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.
  • X. J. Wu, D. R. Lai, and H. T. Lu, “Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes,” Nonlinear Dynamics, vol. 69, no. 1-2, pp. 667–683, 2012.
  • G. Bartolini, L. Fridman, A. Pisano, and E. Usai, Eds., Modern Sliding Mode Control Theory. New Perspectives and Applications, vol. 375 of Lecture Note in Control and Information Sciences, Springer, Berlin, Germany, 2008.
  • A. Levant, “Quasi-continuous high-order sliding-mode controllers,” IEEE Transactions on Automatic Control, vol. 50, no. 11, pp. 1812–1816, 2005.
  • A. Levant, “Principles of 2-sliding mode design,” Automatica, vol. 43, no. 4, pp. 576–586, 2007.
  • J. A. Moreno and M. Osorio, “A Lyapunov approach to second-order sliding mode controllers and observers,” in Proceedings of the 47th IEEE Conference on Decision and Control (CDC '08), pp. 2856–2861, December 2008.
  • M. P. 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, no. 1-2, pp. 247–261, 2012.
  • S. P. Bhat and D. S. Bernstein, “Finite-time stability of continuous autonomous systems,” SIAM Journal on Control and Optimization, vol. 38, no. 3, pp. 751–766, 2000.
  • A. Pisano, M. R. Rapaić, Z. D. Jeličić, and E. Usai, “Sliding mode control approaches to the robust regulation of linear multivariable fractional-order dynamics,” International Journal of Robust and Nonlinear Control, vol. 20, no. 18, pp. 2045–2056, 2010.
  • S. Dadras and H. R. Momeni, “Control of a fractional-order economical system via sliding mode,” Physica A, vol. 389, no. 12, pp. 2434–2442, 2010.
  • 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.
  • http://www.mathworks.com/matlabcentral/fileexchange/ 27336-fractional-order-chaotic-systems.
  • 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.
  • A. S. Hegazi and A. E. Matouk, “Dynamical behaviors and synchronization in the fractional order hyperchaotic Chen system,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1938–1944, 2011.