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Fuzzy adaptive synchronization of uncertain fractional-order chaotic systems

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Abstract

This paper deals with the issue of projective synchronization of two distinct fractional-order chaotic systems with the presence of both uncertain dynamics and external disturbances. More precisely, this study is an attempt to investigate a novel fuzzy adaptive controller for achieving an appropriate projective synchronization of uncertain fractional-order chaotic systems. The adaptive fuzzy systems are utilized to online estimate unknown system nonlinearities. The proposed controller, which is derived based on a Lyapunov approach, is continuous and ensures the stability of the closed-loop system and the exponential convergence of the underlying synchronization errors to a small residual set. Finally, three simulation examples are provided to verify the effectiveness of the proposed synchronization method.

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References

  1. Aghababa MP (2012) Comments on “H∞ synchronization of uncertain fractional order chaotic systems: adaptive fuzzy approach”. ISA Trans 51:11–12

    Article  Google Scholar 

  2. Bagley RL, Calico RA (1991) Fractional order state equations for the control of viscoelastically damped structures. J Guid Control Dyn 14:304–311

    Article  Google Scholar 

  3. Boulkroune A, Bouzeriba A, Hamel S, Bouden T (2014) A projective synchronization scheme based on fuzzy adaptive control for unknown multivariable chaotic systems. Nonlinear Dyn 78:433–447

    Article  MathSciNet  MATH  Google Scholar 

  4. Boulkroune A, M’saad M M, Farza M (2011) Adaptive fuzzy controller for multivariable nonlinear state time-varying delay systems subject to input nonlinearities. Fuzzy Sets Syst 164:45–65

    Article  MathSciNet  MATH  Google Scholar 

  5. Boulkroune A, M’saad M (2012) On the design of observer-based fuzzy adaptive controller for nonlinear systems with unknown control gain sign. Fuzzy Sets Syst 201:71–85

    Article  MathSciNet  MATH  Google Scholar 

  6. Boulkroune A, M’saad M (2012) Fuzzy adaptive observer-based projective synchronization for nonlinear systems with input nonlinearity. J Vib Control 18:437–450

    Article  MathSciNet  MATH  Google Scholar 

  7. Boulkroune A, Tadjine M, M’saad M, Farza M (2008) How to design a fuzzy adaptive control based on observers for uncertain affine nonlinear systems. Fuzzy Sets Syst 159:926–948

    Article  MathSciNet  MATH  Google Scholar 

  8. Bowonga S, Kakmenib M, Koinac R (2006) Chaos synchronization and duration time of a class of uncertain systems. Math Comput Simul 71:212–228

    Article  MathSciNet  Google Scholar 

  9. Cailian C, Gang F, Xinping G (2005) An adaptive lag-synchronization method for time-delay chaotic systems. In: Proceedings of the American control conference, pp 4277–4282

  10. Carpinteri A, Mainardi F (1997) Fractals and fractional calculus. Springer, New York

    Book  MATH  Google Scholar 

  11. Carroll TL, Heagy JF, Pecora LM (1996) Transforming signals with chaotic synchronization. Phys Rev E 54:4676–4680

    Article  Google Scholar 

  12. Chen CS, Chen HH (2009) Robust adaptive neural-fuzzy-network control for the synchronization of uncertain chaotic systems. Nonlinear Anal: Real World Appl 10(3):1466–1479

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen LP, Qu JF, Chai Y, Wu RC, Qi GY (2013) Synchronization of a class of fractional-order chaotic neural networks. Entropy 15:3265–3276

    Article  MathSciNet  MATH  Google Scholar 

  14. Diethelm K, Ford NJ, Freed AD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29:3–22

    Article  MathSciNet  MATH  Google Scholar 

  15. Diethelm K, Ford NJ (2002) Analysis of fractional differential equations. J Math Anal Appl 265:229–248

    Article  MathSciNet  MATH  Google Scholar 

  16. Dong-Feng W, Jin-Ying Z, Xiao-Yan W (2011) Synchronization of uncertain fractional-order chaotic systems with disturbance based on a fractional terminal sliding mode controller. Chin Phys B 20:110506

    Article  Google Scholar 

  17. Gao X, Yu J (2005) Chaos in the fractional order periodically forced complex Duffing’s oscillators. Chaos Solitons Fractals 24(4):1097–1104

    Article  MATH  Google Scholar 

  18. Hartley TT, Lorenzo CF, Qammer HK (1995) Chaos in a fractional order Chua’s system. IEEE Trans Circuits Syst I 42:485–490

    Article  Google Scholar 

  19. Heaviside O (1971) Electromagnetic Theory. Chelsea, New York

    MATH  Google Scholar 

  20. Hosseinnia SH, Ghaderi R, Ranjbar A, Mahmoudian M, Momani S (2010) Sliding mode synchronization of an uncertain fractional order chaotic system. Comput Math Appl 59:1637–1643

    Article  MathSciNet  MATH  Google Scholar 

  21. Hwang EJ, Hyun CH, Kim E, Park M (2009) Fuzzy model based adaptive synchronization of uncertain chaotic systems: robust tracking control approach. Phys Lett A 373(22):1935–1939

    Article  MATH  Google Scholar 

  22. Ichise M, Nagayanagi Y, Kojima T (1971) An analog simulation of non-integer order transfer functions for analysis of electrode process. J Electroanal Chem Interfacial Electrochem 33:253–265

    Article  Google Scholar 

  23. Jianping Y, Changpin L (2005) Generalized projective synchronization of a unified chaotic system. Chaos Solitons Fractals 26:1119–1124

    Article  MATH  Google Scholar 

  24. Li C, Chen G (2004) Chaos and hyperchaos in the fractional-order Rössler equations. Physica A 341:55–61

    Article  MathSciNet  Google Scholar 

  25. Li C, Liao X, Wong KW (2005) Lag synchronization of hyperchaos with application to secure communications. Chaos Solitons Fractals 23(1):183–193

    Article  MathSciNet  MATH  Google Scholar 

  26. Li C, Peng G (2004) Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22:443–450

    Article  MathSciNet  MATH  Google Scholar 

  27. Li G (2006) Projective synchronization of chaotic system using backstepping control. Chaos Solitons Fractals 29:490–598

    Article  MATH  Google Scholar 

  28. Li Y, Chen YQ, Podlubny I (2010) Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput Math Appl 59:1810–1821

    Article  MathSciNet  MATH  Google Scholar 

  29. Li Z, Xu D (2004) A secure communication scheme using projective chaos synchronization. Chaos Solitons Fractals 22:477–481

    Article  MATH  Google Scholar 

  30. Lin TC, Kuo CH (2011) H∞ synchronization of uncertain fractional order chaotic systems: adaptive fuzzy approach. ISA Trans 50:548–556

    Article  Google Scholar 

  31. Lin TC, Lee TY (2011) Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy sliding mode control. IEEE Trans Fuzzy Syst 19:623–635

    Article  Google Scholar 

  32. Liu Y, Zheng Y (2009) Adaptive fuzzy approach to control unified chaotic systems. Nonlinear Dyn 57:431–439

    Article  MATH  Google Scholar 

  33. Lu CH, Tsai CC (2007) Generalized predictive control using recurrent fuzzy neural networks for industrial processes. J Process Control 17(1):83–92

    Article  MathSciNet  Google Scholar 

  34. Lu JG (2005) Chaotic dynamics and synchronization of fractional order Arneodo’s systems. Chaos Solitons Fractal 26:1125–1133

    Article  MATH  Google Scholar 

  35. Lu JG (2005) Chaotic dynamics and synchronization of fractional-order Chua’s circuits with a piecewise-linear nonlinearity. Int J Mod Phys B 19(20):3249–3259

    Article  MATH  Google Scholar 

  36. Pan L, Zhou W, Fang J, Li D (2010) Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems via novel active pinning control. Commun Nonlinear Sci Numer Simulat 15:3754–3762

    Article  MathSciNet  MATH  Google Scholar 

  37. Peng G (2007) Synchronization of fractional order chaotic systems. Phys Lett A 363:426–432

    Article  MathSciNet  MATH  Google Scholar 

  38. Petras I (2011) Fractional-order nonlinear systems: modeling, analysis and simulation. Springer, Berlin

    Book  MATH  Google Scholar 

  39. Pikovsky AS, Rosenblum MG, Osipov GV, Kurths J (1997) Phase synchronization of chaotic oscillators by external driving. Physica D 104:219–238

    Article  MathSciNet  MATH  Google Scholar 

  40. Podlubny I (1999) Fractional differential equations. Academic Press, New York

    MATH  Google Scholar 

  41. Polycarpou MM, Ioannou PA (1996) A robust adaptive nonlinear control design. Automatica 32:423–427

    Article  MathSciNet  MATH  Google Scholar 

  42. Poursamad A, Davaie-Markazi AH (2009) Robust adaptive fuzzy control of unknown chaotic systems. Appl Soft Comput 9(3):970–976

    Article  Google Scholar 

  43. Roopaei M, Jahromi MZ (2008) Synchronization of two different chaotic systems using novel adaptive fuzzy sliding mode control. Chaos 18:033133

    Article  MathSciNet  MATH  Google Scholar 

  44. Rosenblum MG, Pikovsky AS, Kurths J (1996) Phase synchronization of chaotic oscillators. Phys Rev Lett 76:1804–1807

    Article  MATH  Google Scholar 

  45. Ruo-Xun Z, Shi-Ping Y (2011) Adaptive stabilization of an incommensurate fractional order chaotic system via a single state controller. Chin Phys B 20:110506

    Article  Google Scholar 

  46. Sheu LJ, Chen HK, Chen JH, Tam LM, Chen WC, Lin KT, Kang Y (2008) Chaos in the Newton–Leipnik system with fractional order. Chaos Solitons Fractals 36(1):98–103

    Article  MathSciNet  MATH  Google Scholar 

  47. Sun H, Abdelwahad A, Onaral B (1984) Linear approximation of transfer function with a pole of fractional power. IEEE Trans Autom Control 29:441–444

    Article  MATH  Google Scholar 

  48. Sun J, Zhan Y (2004) Impulsive control and synchronization of Chua’s oscillators. Math Comput Simul 66:499–508

    Article  MATH  Google Scholar 

  49. Tavazoei MS (2012) Comments on “Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy sliding mode control”. IEEE Trans Fuzzy Syst 20:993–995

    Article  Google Scholar 

  50. Varsha D, Sachin B (2010) Chaos in fractional ordered Liu system. Comput Math Appl 59:1117–1127

    Article  MathSciNet  MATH  Google Scholar 

  51. Wang J, Chen L, Deng B (2009) Synchronization of Ghostburster neuron in external electrical stimulation via H∞ variable universe fuzzy adaptive control. Chaos Solitons Fractals 39(5):2076–2085

    Article  Google Scholar 

  52. Wang J, Zhang Z, Li H (2008) Synchronization of FitzHugh–Nagumo systems in EES via H∞ variable universe adaptive fuzzy control. Chaos Solitons Fractals 36:1332–1339

    Article  MATH  Google Scholar 

  53. Wang JW, Zhang YB (2009) Synchronization in coupled nonidentical incommensurate fractional-order systems. Phys Lett A 374:202–207

    Article  MATH  Google Scholar 

  54. Wang LX (1994) Adaptive fuzzy systems and control: design and stability analysis. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  55. Wang XZ, Xing HJ, Li Y, Hua Q, Dong CR, Pedrycz W (2014) A study on relationship between generalization abilities and fuzziness of base classifiers in ensemble learning. IEEE Trans Fuzzy Syst. doi:10.1109/TFUZZ.2014.2371479

    Google Scholar 

  56. Wang XZ, Dong CR (2009) Improving generalization of fuzzy if-then rules by maximizing fuzzy entropy. IEEE Trans Fuzzy Syst 17(3):556–567

    Article  Google Scholar 

  57. Wang XZ, Dong LC, Yan JH (2012) Maximum ambiguity based sample selection in fuzzy decision tree induction. IEEE Trans Knowl Data Eng 24(8):1491–1505

    Article  Google Scholar 

  58. Wang YW, Guan ZH (2006) Generalized synchronization of continuous chaotic systems. Chaos Solitons Fractals 27:97–101

    Article  MATH  Google Scholar 

  59. Yu Y, Li H, Wang S, Yu J (2009) Dynamic analysis of a fractional-order Lorenz chaotic system. Chaos Solitons Fractals 42:1181–1189

    Article  MathSciNet  MATH  Google Scholar 

  60. Zhang R, Yang S (2012) Robust chaos synchronization of fractional-order chaotic systems with unknown parameters and uncertain perturbations. Nonlinear Dyn 69:983–992

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. LAJ, University of Jijel, BP. 98 Ouled-Aissa, 18000, Jijel, Algeria

    A. Bouzeriba & A. Boulkroune

  2. NDT, University of Jijel, BP. 98, Ouled-Aissa, 18000, Jijel, Algeria

    A. Bouzeriba & T. Bouden

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  1. A. Bouzeriba
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  2. A. Boulkroune
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  3. T. Bouden
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Correspondence to A. Boulkroune.

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Bouzeriba, A., Boulkroune, A. & Bouden, T. Fuzzy adaptive synchronization of uncertain fractional-order chaotic systems. Int. J. Mach. Learn. & Cyber. 7, 893–908 (2016). https://doi.org/10.1007/s13042-015-0425-7

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