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Lag-synchronization of two different fractional-order time-delayed chaotic systems using fractional adaptive sliding mode controller

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Abstract

This paper investigates the lag-synchronization of two non-idantical fractional-order time-delayed chaotic systems in the presence of uncertainties and external disturbances. A fractional-order adaptive sliding mode control for lag-synchronizing two non-identical fractional-order time-delayed chaotic systems with unknown uncertainties, external disturbances and uncertain parameters is proposed. The suggested technique can be used for a large range of chaotic systems. Appropriate adaptive laws are established to overcome the uncertain parameters and bounds of parameters. Furthermore, to eliminate the undesirable phenomenon of chattering, instead of using the discontinuous sign function, the continuous tanh function with adaptive amplitude and slope is used. Using the Lyapunov theorem, the stability of the suggested strategy control is proved. Finally, the simulation results demonstrate the feasibility and robustness of our suggested scheme.

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References

  1. Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64(8):821

    MathSciNet  MATH  Google Scholar 

  2. Rong CG, Xiaoning D (1998) From chaos to order: methodologies, perspectives and applications, vol 24. World Scientific, Singapore

    Google Scholar 

  3. Noroozi N, Roopaei M, Karimaghaee P, Safavi AA (2010) Simple adaptive variable structure control for unknown chaotic systems. Commun Nonlinear Sci Numer Simul 15(3):707–727

    MathSciNet  MATH  Google Scholar 

  4. Xu B, Shi Z, Yang C (2015) Composite fuzzy control of a class of uncertain nonlinear systems with disturbance observer. Nonlinear Dyn 80(1–2):341–351

    MathSciNet  MATH  Google Scholar 

  5. Luo S (2014) Adaptive fuzzy dynamic surface control for the chaotic permanent magnet synchronous motor using Nussbaum gain. Chaos Interdiscip J Nonlinear Sci 24(3):033135

    MathSciNet  MATH  Google Scholar 

  6. Wang Y, Yu H (2018) Fuzzy synchronization of chaotic systems via intermittent control. Chaos, Solitons Fractals 106:154–160

    MathSciNet  MATH  Google Scholar 

  7. Heydari ZR, Karimaghaee P (2019) Projective synchronization of different uncertain fractional-order multiple chaotic systems with input nonlinearity via adaptive sliding mode control. Adv Differ Equ 2019(1):1–23

    MathSciNet  Google Scholar 

  8. Deepika D, Kaur S, Narayan S (2018) Uncertainty and disturbance estimator based robust synchronization for a class of uncertain fractional chaotic system via fractional order sliding mode control. Chaos, Solitons Fractals 115:196–203

    MathSciNet  MATH  Google Scholar 

  9. Kocamaz UE, Cevher B, Uyaroğlu Y (2017) Control and synchronization of chaos with sliding mode control based on cubic reaching rule. Chaos, Solitons Fractals 105:92–98

    MathSciNet  MATH  Google Scholar 

  10. Hallaji M, Dideban A, Khanesar MA (2018) Optimal synchronization of non-smooth fractional order chaotic systems with uncertainty based on extension of a numerical approach in fractional optimal control problems. Chaos, Solitons Fractals 115:325–340

    MathSciNet  MATH  Google Scholar 

  11. Behinfaraz R, Badamchizadeh M (2016) Optimal synchronization of two different in-commensurate fractional-order chaotic systems with fractional cost function. Complexity 21(S1):401–416

    MathSciNet  Google Scholar 

  12. Kountchou M, Louodop P, Bowong S, Fotsin H (2016) Analog circuit design and optimal synchronization of a modified Rayleigh system. Nonlinear Dyn 85(1):399–414

    MathSciNet  MATH  Google Scholar 

  13. Adloo H, Noroozi N, Karimaghaee P (2012) Observer-based model reference adaptive control for unknown time-delay chaotic systems with input nonlinearity. Nonlinear Dyn 67(2):1337–1356

    MathSciNet  MATH  Google Scholar 

  14. Kebriaei H, Yazdanpanah MJ (2010) Robust adaptive synchronization of different uncertain chaotic systems subject to input nonlinearity. Commun Nonlinear Sci Numer Simul 15(2):430–441

    MathSciNet  MATH  Google Scholar 

  15. Lin W, Chen X, Zhou S (2017) Achieving control and synchronization merely through a stochastically adaptive feedback coupling. Chaos Interdiscip J Nonlinear Sci 27(7):073110

    MathSciNet  MATH  Google Scholar 

  16. Singh PP, Singh JP, Roy BK (2014) Synchronization and anti-synchronization of Lu and Bhalekar-Gejji chaotic systems using nonlinear active control. Chaos, Solitons Fractals 69:31–39

    MathSciNet  MATH  Google Scholar 

  17. Prakash SJ, Pratap SP, Roy BK (2018) Anti-synchronization of Bhalekar-Gejji chaotic system via nonlinear active control. Res Rev J Phys 4(1):1–6

    Google Scholar 

  18. Huang C, Cao J (2017) Active control strategy for synchronization and anti-synchronization of a fractional chaotic financial system. Phys A 473:262–275

    MathSciNet  MATH  Google Scholar 

  19. Pishkenari HN, Jalili N, Mahboobi SH, Alasty A, Meghdari A (2010) Robust adaptive backstepping control of uncertain Lorenz system. Chaos Interdiscip J Nonlinear Sci 20(2):023105

    MATH  Google Scholar 

  20. Vaidyanathan S, Kingni ST, Sambas A, Mohamed MA, Mamat M (2018) A new chaotic jerk system with three nonlinearities and synchronization via adaptive backstepping control. Int J Eng Technol 7(3):1936–1943

    Google Scholar 

  21. Kocamaz UE, Göksu A, Taşkın H, Uyaroğlu Y (2015) Synchronization of chaos in nonlinear finance system by means of sliding mode and passive control methods: a comparative study. Inf Technol Control 44(2):172–181

    Google Scholar 

  22. Lai BC, He JJ (2018) Dynamic analysis, circuit implementation and passive control of a novel four-dimensional chaotic system with multiscroll attractor and multiple coexisting attractors. Pramana 90(3):33

    Google Scholar 

  23. Pourdehi S, Karimaghaee P (2012) Simple adaptive output-feedback lag-synchronization of multiple time-delayed chaotic systems. Chaos Interdiscip J Nonlinear Sci 22(2):023145

    MathSciNet  MATH  Google Scholar 

  24. Khalil HK (2003) Nonlinear systems, 2nd edn. Englewood Cliffs, NJ Prentice Hall

    Google Scholar 

  25. Vaseghi B, Pourmina MA, Mobayen S (2017) Secure communication in wireless sensor networks based on chaos synchronization using adaptive sliding mode control. Nonlinear Dyn 89(3):1689–1704

    MathSciNet  MATH  Google Scholar 

  26. Muthukumar P, Balasubramaniam P, Ratnavelu K (2017) Sliding mode control design for synchronization of fractional order chaotic systems and its application to a new cryptosystem. Int J Dyn Control 5(1):115–123

    MathSciNet  Google Scholar 

  27. Vaidyanathan S (2015) Global chaos synchronization of chemical chaotic reactors via novel sliding mode control method. Parameters 1:4

    Google Scholar 

  28. Yang XJ, Gao F, Srivastava HM (2018) A new computational approach for solving nonlinear local fractional PDEs. J Comput Appl Math 339:285–296

    MathSciNet  MATH  Google Scholar 

  29. Xiao-Jun XJ, Srivastava HM, Machado JT (2016) A new fractional derivative without singular kernel. Therm Sci 20(2):753–756

    Google Scholar 

  30. Yang AM, Han Y, Li J, Liu WX (2016) On steady heat flow problem involving Yang-Srivastava-Machado fractional derivative without singular kernel. Therm Sci 20(suppl 3):S719–S723

    Google Scholar 

  31. Yang XJ, Gao F, Machado JT, Baleanu D (2017) A new fractional derivative involving the normalized sinc function without singular kernel. Eur Phys J Spec Top 226(16–18):3567–3575

    Google Scholar 

  32. Yang XJ (2019) New general calculi with respect to another functions applied to describe the newton-like dashpot models in anomalous viscoelasticity. Therm Sci (00):260

  33. Yang XJ, Feng YY, Cattani C, Inc M (2019) Fundamental solutions of anomalous diffusion equations with the decay exponential kernel. Math Methods Appl Sci 42(11):4054–4060

    MathSciNet  MATH  Google Scholar 

  34. Yang XJ, Tenreiro Machado JA (2019) A new fractal nonlinear Burgers’ equation arising in the acoustic signals propagation. Math Methods Appl Sci 42(18):7539–7544

    MathSciNet  MATH  Google Scholar 

  35. Yang XJ (2019) General fractional derivatives: theory, methods and applications. Chapman and Hall/CRC, Boca Raton

    MATH  Google Scholar 

  36. Yang XJ, Abdel-Aty M, Cattani C (2019) A new general fractional-order derivataive with Rabotnov fractional-exponential kernel applied to model the anomalous heat transfer. Therm Sci 23(3 Part A):1677–1681

    Google Scholar 

  37. Yang XJ, Gao F, Ju Y, Zhou HW (2018) Fundamental solutions of the general fractional-order diffusion equations. Math Methods Appl Sci 41(18):9312–9320

    MathSciNet  MATH  Google Scholar 

  38. Yang XJ, Tenreiro Machado JA (2017) Baleanu D (2017) Anomalous diffusion models with general fractional derivatives within the kernels of the extended Mittag-Leffler type functions. Rom Reports Phys 69:115

    Google Scholar 

  39. Yang XJ (2017) New rheological problems involving general fractional derivatives with nonsingular power-law kernels. Proc Rom Acad Ser A-Math Phys Tech Sci Inf Sci б/н:1–8

    Google Scholar 

  40. Alain KST, Azar AT, Kengne R, Bertrand FH (2020) Stability analysis and robust synchronisation of fractional-order modified Colpitts oscillators. Int J Autom Control 14(1):52–79

    Google Scholar 

  41. Khan A (2017) Hybrid function projective synchronization of chaotic systems via adaptive control. Int J Dyn Control 5(4):1114–1121

    MathSciNet  Google Scholar 

  42. Xi X, Mobayen S, Ren H, Jafari S (2018) Robust finite-time synchronization of a class of chaotic systems via adaptive global sliding mode control. J Vib Control 24(17):3842–3854

    MathSciNet  Google Scholar 

  43. Yin C, Zhong SM, Chen WF (2012) Design of sliding mode controller for a class of fractional-order chaotic systems. Commun Nonlinear Sci Numer Simul 17(1):356–366

    MathSciNet  MATH  Google Scholar 

  44. Chen D, Zhang R, Sprott JC, Chen H, Ma X (2012) Synchronization between integer-order chaotic systems and a class of fractional-order chaotic systems via sliding mode control. Chaos Interdiscip J Nonlinear Sci 22(2):023130

    MathSciNet  MATH  Google Scholar 

  45. Yin C, Chen Y, Zhong SM (2014) Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems. Automatica 50(12):3173–3181

    MathSciNet  MATH  Google Scholar 

  46. Zhang H, Huang W, Wang Z, Chai T (2006) Adaptive synchronization between two different chaotic systems with unknown parameters. Phys Lett A 350(5–6):363–366

    MATH  Google Scholar 

  47. Aghababa MP, Hashtarkhani B (2015) Synchronization of unknown uncertain chaotic systems via adaptive control method. J Comput Nonlinear Dyn 10(5):051004

    Google Scholar 

  48. Chen X, Park JH, Cao J, Qiu J (2018) Adaptive synchronization of multiple uncertain coupled chaotic systems via sliding mode control. Neurocomputing 273:9–21

    Google Scholar 

  49. Li XF, Chu YD, Leung AY, Zhang H (2017) Synchronization of uncertain chaotic systems via complete-adaptive-impulsive controls. Chaos, Solitons Fractals 100:24–30

    MathSciNet  MATH  Google Scholar 

  50. Liu JG, Yang XJ, Feng YY (2019) On integrability of the time fractional nonlinear heat conduction equation. J Geometry Phys 144:190–198

    MathSciNet  MATH  Google Scholar 

  51. Yang XJ (2017) New general fractional-order rheological models with kernels of Mittag-Leffler functions. Rom Rep Phys 69(4):118

    Google Scholar 

  52. Khan A, Budhraja M, Ibraheem A (2018) Combination–combination synchronisation of time-delay chaotic systems for unknown parameters with uncertainties and external disturbances. Pramana 91(2):20

    Google Scholar 

  53. Yang XJ (2019) New non-conventional methods for quantitative concepts of anomalous rheology. Therm Sci:427

  54. Mackey M, Glass L (1997) Oscillation and chaos in physiological control system. Science 197:287

    MATH  Google Scholar 

  55. Mohammadzadeh A, Ghaemi S, Kaynak O (2019) Robust predictive synchronization of uncertain fractional-order time-delayed chaotic systems. Soft Comput 23(16):6883–6898

    MATH  Google Scholar 

  56. Huang C, Cai L, Cao J (2018) Linear control for synchronization of a fractional-order time-delayed chaotic financial system. Chaos, Solitons Fractals 113:326–332

    MathSciNet  MATH  Google Scholar 

  57. He S, Sun K, Wang H (2016) Synchronisation of fractional-order time delayed chaotic systems with ring connection. Eur Phys J Spec Top 225(1):97–106

    Google Scholar 

  58. Podlubny I (1994) Fractional-order systems and fractional-order controllers. Inst Exp Phys Slovak Acad Sci Kosice 12(3):1–18

    Google Scholar 

  59. Shi L, Chen G, Zhong S, Li X, Wang W (2018) Lag synchronisation of master–slave dynamical systems via intermittent control. Int J Syst Sci 49(16):3346–3353

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran

    Paknosh Karimaghaee & Zahra Rashidnejad Heydari

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  1. Paknosh Karimaghaee
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  2. Zahra Rashidnejad Heydari
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Correspondence to Paknosh Karimaghaee.

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Karimaghaee, P., Rashidnejad Heydari, Z. Lag-synchronization of two different fractional-order time-delayed chaotic systems using fractional adaptive sliding mode controller. Int. J. Dynam. Control 9, 211–224 (2021). https://doi.org/10.1007/s40435-020-00628-9

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  • DOI: https://doi.org/10.1007/s40435-020-00628-9

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