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Projective synchronization of two different fractional-order chaotic systems via adaptive fuzzy control

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Abstract

In this paper, the projective synchronization problem of two fractional-order different chaotic (or hyperchaotic) systems with both uncertain dynamics and external disturbances is considered. More particularly, a fuzzy adaptive control system is investigated for achieving an appropriate projective synchronization of unknown fractional-order chaotic systems. The adaptive fuzzy logic systems are used to approximate some uncertain nonlinear functions appearing in the system model. These latter are augmented by a robust control term to compensate for the unavoidable fuzzy approximation errors and external disturbances as well as residual error due to the use of the so-called e-modification in the adaptive laws. A Lyapunov approach is adopted for the design of the parameter adaptation laws and the proof of the corresponding stability as well as the asymptotic convergence of the underlying synchronization errors towards zero. The effectiveness of the proposed synchronization system is illustrated through numerical experiment results.

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References

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

    Article  Google Scholar 

  2. Sun H, Abdelwahad A, Onaral B (1984) Linear approximation of transfer function with a pole of fractional power. IEEE Trans Autom Control 29(5):441–444

    Article  MATH  Google Scholar 

  3. 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(2):253–265

    Article  Google Scholar 

  4. Heaviside O (1971) Electromagnetic theory. Chelsea, New York

    MATH  Google Scholar 

  5. Yin-He W, Yong-Qing F, Qing-Yun W, Yun Z (2012) Adaptive feedback stabilization with quantized state measurements for a class of chaotic systems. Commun Theor Phys 57(5):808–816

    Article  MATH  Google Scholar 

  6. Ginarsa IM, Soeprijanto A, Purnomo MH (2013) Controlling chaos and voltage collapse using an ANFIS-based composite controller-static var compensator in power systems. Int J Electr Power Energy Syst 46:79–88

    Article  Google Scholar 

  7. 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 

  8. Hartley TT, Lorenzo CF, Qammer HK (1995) Chaos in a fractional order Chua’s system. IEEE Trans Circuits Syst I 42(8):485–490

    Article  Google Scholar 

  9. Yu Y, Li H, Wang S, Yu J (2009) Dynamic analysis of a fractional-order Lorenz chaotic system. Chaos Solitons Fractals 42(2):1181–1189

    Article  MathSciNet  MATH  Google Scholar 

  10. Li C, Peng G (2004) Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22(2):443–450

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Daftardar-Gejji V, Bhalekar S (2010) Chaos in fractional ordered Liu system. Comput Math Appl 59:1117–1127

    Article  MathSciNet  MATH  Google Scholar 

  13. Lu JG (2005) Chaotic dynamics and synchronization of fractional order Arneodo’s systems. Chaos Solitons Fractal 26(4):1125–1133

    Article  MATH  Google Scholar 

  14. Carroll TL, Heagy JF, Pecora LM (1996) Transforming signals with chaotic synchronization. Phys Rev E 54(5):4676

    Article  Google Scholar 

  15. Sun J, Zhang Y (2004) Impulsive control and synchronization of Chua’s oscillators. Math Comput Simul 66(6):499–508

    Article  MATH  Google Scholar 

  16. Bowonga S, Kakmenib M, Koinac R (2006) Chaos synchronization and duration time of a class of uncertain systems. Math Comput Simul 71(3):212–228

    Article  MathSciNet  Google Scholar 

  17. Rosenblum MG, Pikovsky AS, Kurths J (1996) Phase synchronization of chaotic oscillators. Phys Rev Lett 76(11):1804

    Article  MATH  Google Scholar 

  18. Pikovsky AS, Rosenblum MG, Osipov GV, Kurths J (1997) Phase synchronization of chaotic oscillators by external driving. Phys D 104(3):219–238

    Article  MathSciNet  MATH  Google Scholar 

  19. 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

  20. Morgul O, Solak E (1996) Observer based synchronization of chaotic systems. Phys Rev E 54(5):4803

    Article  Google Scholar 

  21. Morgul O, Solak E (1997) On the synchronization of chaotic systems by using state observers. Int J Bifurcat Chaos 7(6):1307–1322

    Article  MathSciNet  MATH  Google Scholar 

  22. Li Z, Xu D (2004) A secure communication scheme using projective chaos synchronization. Chaos Solitons Fractals 22(2):477–481

    Article  MATH  Google Scholar 

  23. Yan J, Li C (2005) Generalized projective synchronization of a unified chaotic system. Chaos Solitons Fractals 26(4):1119–1124

    Article  MathSciNet  MATH  Google Scholar 

  24. Li GH (2006) Projective synchronization of chaotic system using backstepping control. Chaos Solitons Fractals 29(2):490–494

    Article  MATH  Google Scholar 

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

    Google Scholar 

  26. 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(5):1332–1339

    Article  MATH  Google Scholar 

  27. Roopaei M, Jahromi MZ (2008) Synchronization of two different chaotic systems using novel adaptive fuzzy sliding mode control. Chaos 18(3):033133

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  29. 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 

  30. 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 

  31. 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 

  32. Liu YJ, Zheng YQ (2009) Adaptive robust fuzzy control for a class of uncertain chaotic systems. Nonlinear Dyn 57(3):431–439

    Article  MathSciNet  MATH  Google Scholar 

  33. Lin TC, Lee TY, Balas VE (2011) Adaptive fuzzy sliding mode control for synchronization of uncertain fractional order chaotic systems. Chaos Solitons Fractals 44(10):791–801

    Article  MATH  Google Scholar 

  34. Precup RE, Tomescu ML, Dragos CA (2014) Stabilization of Rössler chaotic dynamical system using fuzzy logic control algorithm. Int J Gen Syst 43(5):413–433

    Article  MathSciNet  MATH  Google Scholar 

  35. Jing CG, He P, Fan T, Li Y, Chen C, Song X (2015) Single state feedback stabilization of unified chaotic systems and circuit implementation. Open Phys 13:111–122

    Google Scholar 

  36. Precup RE, Tomescu ML (2015) Stable fuzzy logic control of a general class of chaotic systems. Neural Comput Appl 26(3):541–550

    Article  Google Scholar 

  37. Peng G (2007) Synchronization of fractional order chaotic systems. Phys Lett A 363(5):426–432

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang JW, Zhang YB (2009) Synchronization in coupled nonidentical incommensurate fractional-order systems. Phys Lett A 374(2):202–207

    Article  MATH  Google Scholar 

  39. 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 Simul 15(12):3754–3762

    Article  MathSciNet  MATH  Google Scholar 

  40. Hosseinnia SH, Ghaderi R, Ranjbar NA, Mahmoudian M, Momani S (2010) Sliding mode synchronization of an uncertain fractional order chaotic system. Comput Math Appl 59(5):1637–1643

    Article  MathSciNet  MATH  Google Scholar 

  41. 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(4):623–635

    Article  Google Scholar 

  42. Lin TC, Kuo CH (2011) H∞ synchronization of uncertain fractional order chaotic systems: adaptive fuzzy approach. ISA Trans 50(4):548–556

    Article  Google Scholar 

  43. Chen LP, Qu JF, Chai Y, Wu RC, Qi GY (2013) Synchronization of a class of fractional-order chaotic neural networks. Entropy 15(8):3265–3276

    Article  MathSciNet  MATH  Google Scholar 

  44. 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(5):993–995

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    MATH  Google Scholar 

  47. Diethelm K, Ford NJ (2002) Analysis of fractional differential equations. J Math Anal Appl 265(2):229–248

    Article  MathSciNet  MATH  Google Scholar 

  48. Matignon D (1996) Stability result on fractional differential equations with applications to control processing. In: Proceedings of the IMACS-SMC proceedings, 1996, Lille, France, pp 963–968

  49. Utkin VI (1992) Sliding modes in control optimization. Springer, Berlin

    Book  MATH  Google Scholar 

  50. Tong SC, Li YM, Feng G, Li TS (2011) Observer-based adaptive fuzzy backstepping dynamic surface control for a class of MIMO nonlinear systems. IEEE Trans Syst Man Cybern Part B 41(4):1124–1135

    Article  Google Scholar 

  51. Tong S, Li Y, Shi P (2012) Observer-based adaptive fuzzy backstepping output feedback control of uncertain MIMO pure-feedback nonlinear systems. IEEE Trans Fuzzy Syst 20(4):771–785

    Article  Google Scholar 

  52. Tong S, Sui S, Li Y (2014) Fuzzy adaptive output feedback control of MIMO nonlinear systems with partial tracking errors constrained. IEEE Trans Fuzzy Syst. doi:10.1109/TFUZZ.2014.2327987

    MATH  Google Scholar 

  53. Li Y, Tong S, Li T (2014) Observer-based adaptive fuzzy tracking control of MIMO stochastic nonlinear systems with unknown control direction and unknown dead-zones. IEEE Trans Fuzzy Syst. doi:10.1109/TFUZZ.2014.2348017

    MATH  Google Scholar 

  54. Tong S, Li Y (2013) Adaptive fuzzy output feedback control of MIMO nonlinear systems with unknown dead-zone input. IEEE Trans Fuzzy Syst 21(1):134–146

    Article  Google Scholar 

  55. Li Y, Tong S, Li T (2013) Direct adaptive fuzzy backstepping control of uncertain nonlinear systems in the presence of input saturation. Neural Comput Appl 23(5):1207–1216

    Article  MathSciNet  Google Scholar 

  56. 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 

  57. Boulkroune A, Bouzeriba A, Hamel S, Bouden T (2014) Adaptive fuzzy control-based projective synchronization of uncertain nonaffine chaotic systems. Complexity. doi:10.1002/cplx.21596

    MathSciNet  MATH  Google Scholar 

  58. 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(8):926–948

    Article  MathSciNet  MATH  Google Scholar 

  59. 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 

  60. Boulkroune A, M’saad M, Farza M (2012) Fuzzy approximation-based indirect adaptive controller for multi-input multi-output Non-affine systems with unknown control direction. IET Control Theory Appl 6(17):2619–2629

    Article  MathSciNet  Google Scholar 

  61. Boulkroune A, M’saad M, Farza M (2012) Adaptive fuzzy tracking control for a class of MIMO nonaffine uncertain systems. Neurocomputing 93:48–55

    Article  Google Scholar 

  62. Boulkroune A, Bounar N, Msaad M, Farza M (2014) Indirect adaptive fuzzy control scheme based on observer for nonlinear systems: a novel SPR-filter approach. Neurocomputing 135:378–387

    Article  Google Scholar 

  63. Boulkroune A, Tadjine M, Msaad M, Farza M (2014) Design of a unified adaptive fuzzy observer for uncertain nonlinear systems. Inf Sci 265:139–153

    Article  MathSciNet  MATH  Google Scholar 

  64. Tavazoei MS, Haeri M (2008) Synchronization of chaotic fractional-order systems via active sliding mode controller. Phys A 387(1):57–70

    Article  Google Scholar 

  65. Agrawal SK, Das S (2013) A modified adaptive control method for synchronization of some fractional chaotic systems with unknown parameters. Nonlinear Dyn 73(1–2):907–919

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    A. Bouzeriba & A. Boulkroune

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

    T. Bouden

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  1. A. Bouzeriba
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Correspondence to A. Boulkroune.

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Bouzeriba, A., Boulkroune, A. & Bouden, T. Projective synchronization of two different fractional-order chaotic systems via adaptive fuzzy control. Neural Comput & Applic 27, 1349–1360 (2016). https://doi.org/10.1007/s00521-015-1938-4

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