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Optimal synchronization of non-smooth fractional order chaotic systems with uncertainty based on extension of a numerical approach in fractional optimal control problems

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  • Hallaji, Majid
  • Dideban, Abbas
  • Khanesar, Mojtaba Ahmadieh
  • kamyad, Ali vahidyan

Abstract

In this paper, a control mechanism is presented for optimal synchronization of two non-smooth fractional order chaotic systems with parametric uncertainty based on nonlinear fractional order proportional derivative (NLFPD) controller combined with optimal periodic control signals. Unlike synchronization methods based on FPID controllers, in this approach, optimal tuning of the NLFPD controller and determination of optimal periodic control signals for synchronization process are presented in the form of non-smooth fractional order optimal control (FOC) problems. Optimal periodic control signals are demonstrated as a generalized expansion in the sense of Fourier expansion that is used to accelerate the synchronization process. Using the generalization of a numerical method in nonlinear optimal control problems and the Grunwald-Letnikov(GL) fractional derivative definition, the synchronization problem based on the suggested mechanism is transformed into the form of a smooth FOC problem. By defining a base-time soft switch, a supervisory approach is added to the proposed control mechanism for desired and robust performance against parametric uncertainty in the slave system. Finally, to illustrate the proposed control mechanism, the synchronization of two identical fractional order Chua systems with simulation results is presented. The Results show that using the suggested control mechanism, the synchronization is fast and robust against parametric uncertainty in Chua slave system.

Suggested Citation

  • Hallaji, Majid & Dideban, Abbas & Khanesar, Mojtaba Ahmadieh & kamyad, Ali vahidyan, 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, Elsevier, vol. 115(C), pages 325-340.
    • Handle: RePEc:eee:chsofr:v:115:y:2018:i:c:p:325-340
    DOI: 10.1016/j.chaos.2018.07.024
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    References listed on IDEAS

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    1. Hung, Meei-Ling & Lin, Jui-Sheng & Yan, Jun-Juh & Liao, Teh-Lu, 2008. "Optimal PID control design for synchronization of delayed discrete chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 781-785.
    2. Chang, Wei-Der, 2009. "PID control for chaotic synchronization using particle swarm optimization," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 910-917.
    3. Cang, Jie & Tan, Yue & Xu, Hang & Liao, Shi-Jun, 2009. "Series solutions of non-linear Riccati differential equations with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 1-9.
    4. Santos Coelho, Leandro dos & de Andrade Bernert, Diego Luis, 2009. "An improved harmony search algorithm for synchronization of discrete-time chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2526-2532.
    5. Odibat, Zaid & Momani, Shaher, 2008. "Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 167-174.
    6. Yau, Her-Terng & Chen, Chieh-Li, 2006. "Chattering-free fuzzy sliding-mode control strategy for uncertain chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 30(3), pages 709-718.
    7. Chunde Yang & Hao Cai & Ping Zhou, 2016. "Compound Generalized Function Projective Synchronization for Fractional-Order Chaotic Systems," Discrete Dynamics in Nature and Society, Hindawi, vol. 2016, pages 1-8, January.
    8. Zhu, Hao & Zhou, Shangbo & Zhang, Jun, 2009. "Chaos and synchronization of the fractional-order Chua’s system," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1595-1603.
    9. Li, C.P. & Deng, W.H. & Xu, D., 2006. "Chaos synchronization of the Chua system with a fractional order," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 360(2), pages 171-185.
    10. Joydeep Dutta, 2005. "Generalized derivatives and nonsmooth optimization, a finite dimensional tour," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(2), pages 185-279, December.
    11. Ge, Zheng-Ming & Ou, Chan-Yi, 2008. "Chaos synchronization of fractional order modified duffing systems with parameters excited by a chaotic signal," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 705-717.
    12. Tavazoei, Mohammad Saleh & Haeri, Mohammad, 2007. "Determination of active sliding mode controller parameters in synchronizing different chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 583-591.
    13. Ahmad, Wajdi M., 2005. "Hyperchaos in fractional order nonlinear systems," Chaos, Solitons & Fractals, Elsevier, vol. 26(5), pages 1459-1465.
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    2. Elgindy, Kareem T., 2024. "Fourier–Gegenbauer pseudospectral method for solving periodic fractional optimal control problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 225(C), pages 148-164.

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