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Synchronization of the Chaotic Fractional-Order Genesio–Tesi Systems Using the Adaptive Sliding Mode Fractional-Order Controller

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Abstract

This paper introduces an adaptive fractional-order sliding mode controller for synchronization of two chaotic Genesio–Tesi systems with fractional dynamics. For this purpose, first the error dynamics is defined; then the adaptive sliding mode synchronization controller is designed through defining suitable sliding surface and estimating uncertainty parameters and stability of the control system is verified using related theorems. Robustness against input uncertainty and disturbance is considered as a primary design objective. Simulation results using MATLAB show that the proposed adaptive fractional-order sliding mode controller is able to synchronize the mentioned chaotic system with acceptable performance.

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References

  • Arena, P., Caponetto, R., Fortuna, L., Porto, D. (1997). Chaos in a fractional order Duffing system. In Proceedings ECCTD, Budapest (pp. 1259–1262).

  • Belkhatir, Z., Meriem, T., & Kirati, L. (2017). High-order sliding mode observer for fractional commensurate linear systems with unknown input. Automatica, 82(8), 209–217.

    Article  MathSciNet  MATH  Google Scholar 

  • Caponetto, R., Dongola, G., & Fortuna, L. (2010). Fractional order systems modeling and control applications, World Scientific Series on Nonlinear Science, Series A 72 (pp. 53–77).

  • Das, S. (2008). Functional fractional calculus for system identification and controls. Berlin: Springer.

    MATH  Google Scholar 

  • Deng, W., & Li, C. P. (2005). Chaos synchronization of the fractional Lu system. Physica A, 353, 61–72.

    Article  Google Scholar 

  • Deng, W., & Li, C. (2005). Synchronization of chaotic fractional Chen system. Journal of the Physical Society of Japan, 74(6), 1645–1648.

    Article  MATH  Google Scholar 

  • Faieghi, M. R., & Delavari, H. (2012). Chaos in fractional-order Genesio–Tesi system and its synchronization. Communications in Nonlinear Science and Numerical Simulation, 17(2), 731–741.

    Article  MathSciNet  MATH  Google Scholar 

  • Genesio, R., & Tesi, A. (1992). Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica, 28(3), 531–548.

    Article  MATH  Google Scholar 

  • Genesio, R., & Tesi, A. (1996). Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics. Automatica, 32(9), 1255–1271.

    Article  MathSciNet  MATH  Google Scholar 

  • Hegazi, A. S., & Matouk, A. E. (2011). Dynamical behaviors and synchronization in the fractional order hyperchaotic Chen system. Applied Mathematics Letters, 24(11), 1938–1944.

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, H., & Yang, J. (2015). Sliding-mode synchronization control for uncertain fractional-order chaotic systems with time delay. Entropy, 17, 4202–4214.

    Article  Google Scholar 

  • Lu, J. (2005). Chaotic dynamics and synchronization of fractional-order Genesio. Chinese Physics, 14(8), 1517–1521.

    Article  Google Scholar 

  • Luo, R., & Zeng, Y. (2017). The control and synchronization of fractional-order Genesio–Tesi system. Nonlinear Dynamics, 88(3), 2111–2121.

  • Manabe, S. (1961). The non integer and its application to control system. Jpn Inst Electr Eng, 6(3/4), 83–87.

    Google Scholar 

  • Manabe, S. (1963). The system design by the use of a model consisting of a saturation and noninteger integrals. Jpn Inst Electr Eng, 8(3/4), 147–150.

    Google Scholar 

  • Matignon, D. (1996). Stability results for fractional differential equations with applications to control processing. In Computational engineering in systems and application multi-conference, IMACS, IEEE-SMC Proceedings, Lille, France (Vol. 2, pp. 963–968).

  • Matouk, A. E. (2011). Chaos, feedback control and synchronization of a fractional-order modified autonomous Van der Pol-Duffing circuit. Communications in Nonlinear Science and Numerical Simulation, 16(2), 975–986.

    Article  MathSciNet  MATH  Google Scholar 

  • Miller, K., & Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations. San Fransisco: Wiley.

    MATH  Google Scholar 

  • Oustaloup, A. (2006). The CRONE approach: Theoretical developments and major applications. In proceedings of the second IFAC workshop on fractional differentiation and its applications (pp. 39–69), Porto, Portugal.

  • Podlubny, I. (1999). Fractional order system and \(PI^{\lambda }D^{\mu }\)—controllers. IEEE Transactions on Automatic Control, 44(1), 208–214.

    Article  MathSciNet  Google Scholar 

  • Pourmahmood, M., Khanmohammadi, S., & Alizadeh, G. (2011). Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller. Communications in Nonlinear Science and Numerical Simulation, 16(7), 2853–2868.

    Article  MathSciNet  MATH  Google Scholar 

  • Sheu, L. J., Chen, H. K., Chen, J. H., Tam, L. M., Chen, W. C., Lin, K. T., et al. (2008). Chaos in the Newton–Leipnik system with fractional order. Chaos Solitons Fractals, 36(1), 98–103.

    Article  MathSciNet  MATH  Google Scholar 

  • Tang, Y., Zhang, X., Hua, C., Li, L., & Yang, Y. (2012). Parameter identification of commensurate fractional-order chaotic system via differential evolution. Physics Letters A, 376(4), 457–464.

    Article  MATH  Google Scholar 

  • Tavazoei, M. S., & Haeri, M. (2007). Determination of active sliding mode controller parameters in synchronizing different chaotic systems. Chaos, Solitons & Fractals, 32(2), 583–591.

    Article  Google Scholar 

  • Tavazoei, M. S., & Haeri, M. (2008). Synchronization of chaotic fractional-order systems via active sliding mode controller. Physica A: Statistical Mechanics and its Applications, 387(1), 57–70.

    Article  Google Scholar 

  • Tustin, A., Allason, J.M., Jakeways, R.J. (1958). The design of systems for automatic control of the position of massive object. In Proceedings of the institution of electrical engineers, Part C, 105 (pp. 1–57)

  • Vafaeih, A., Kheiri, H., & Javadi, M. (2015). Chaotic dynamics and synchronization of fractional order PMSM system. Sahand Communications in Mathematical Analysis (SCMA), 2(2), 83–90.

    Google Scholar 

  • Xu, Y., & Wang, H. (2013). Synchronization of fractional-order chaotic systems with Gaussian fluctuation by sliding mode control. Abstract and Applied Analysis, 2013, 948782. https://doi.org/10.1155/2013/948782.

  • Yassen, M. T. (2005). Controlling Chaos and synchronization for new chaotic system using linear feedback control. Chaos, Solitons & Fractals, 26(3), 913–920.

    Article  MathSciNet  MATH  Google Scholar 

  • Yau, H. T. (2004). Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos, Solitons & Fractals, 22(2), 341–347.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, R., & Gong, J. (2014). Synchronization of the fractional-order chaotic system via adaptive observer. Systems Science & Control Engineering, 2(1), 751–754.

    Article  Google Scholar 

  • Zhang, H., Xi-Kui, M., & Wei-Zeng, L. (2004). Synchronization of chaotic systems with parametric uncertainty using active sliding mode control. Chaos, Solitons & Fractals, 21(3), 1249–1257.

    Article  MATH  Google Scholar 

  • Zhao, L. D., & Hu, J. B. (2012). Synchronizing fractional chaotic Genesio–Tesi system via backstepping approach. Applied Mechanics and Materials, 220–223, 1244–1248.

    Article  Google Scholar 

  • Zhu, H., Zhou, S., & Zhang, J. (2009). Chaos and synchronization of the fractional order Chua’s system. Chaos, Solitons & Fractals, 39(4), 1595–1603.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, Gonabad Branch, Islamic Azad University, Gonabad, Khorasan-e-Razavi, Iran

    Mehran Tabasi & Saeed Balochian

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  1. Mehran Tabasi
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  2. Saeed Balochian
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Correspondence to Saeed Balochian.

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Tabasi, M., Balochian, S. Synchronization of the Chaotic Fractional-Order Genesio–Tesi Systems Using the Adaptive Sliding Mode Fractional-Order Controller. J Control Autom Electr Syst 29, 15–21 (2018). https://doi.org/10.1007/s40313-017-0350-y

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  • DOI: https://doi.org/10.1007/s40313-017-0350-y

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