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Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive Control

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  • J. Humberto Pérez-Cruz

Abstract

Standard adaptive control is the preferred approach for stabilization and synchronization of chaotic systems when the structure of such systems is a priori known but the parameters are unknown. However, in the presence of unmodeled dynamics and/or disturbance, this approach is not effective anymore due to the drift of the parameter estimations, which eventually causes the instability of the closed-loop system. In this paper, a robustifying term, which consists of a saturation function, is used to avoid this problem. The robustifying term is added to the adaptive control law. Consequently, the learning law is also modified. The boundedness of the states and the parameter estimations is rigorously and thoroughly proven by means of Lyapunov like analysis based on Barbalat’s lemma. On these new conditions, the convergence to zero cannot be achieved due to the presence of unmodeled dynamics and/or disturbance. However, it is still possible to guarantee the asymptotic convergence to a bounded zone around zero. The width of this zone can be adjusted by the designer. The performance of this robust approach is verified by numerical simulations. Although, for simplicity, this strategy is only applied to the stabilization and synchronization of Zhang system, the procedure can easily be generalized to a broad class of chaotic systems.

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  • J. Humberto Pérez-Cruz, 2018. "Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive Control," Complexity, Hindawi, vol. 2018, pages 1-19, December.
    • Handle: RePEc:hin:complx:4989520
    DOI: 10.1155/2018/4989520
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    1. Gao, Wei & Yan, Li & Saeedi, Mohammadhossein & Saberi Nik, Hassan, 2018. "Ultimate bound estimation set and chaos synchronization for a financial risk system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 154(C), pages 19-33.
    2. Zribi, Mohamed & Oteafy, Ahmed & Smaoui, Nejib, 2009. "Controlling chaos in the permanent magnet synchronous motor," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1266-1276.
    3. Yan, Zhenya & Yu, Pei, 2007. "Linear feedback control, adaptive feedback control and their combination for chaos (lag) synchronization of LC chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 419-435.
    4. Chen, Hsien-Keng, 2005. "Global chaos synchronization of new chaotic systems via nonlinear control," Chaos, Solitons & Fractals, Elsevier, vol. 23(4), pages 1245-1251.
    5. Qian Ye & Zhengxian Jiang & Tiane Chen, 2018. "Adaptive Feedback Control for Synchronization of Chaotic Neural Systems with Parameter Mismatches," Complexity, Hindawi, vol. 2018, pages 1-8, September.
    6. Messadi, M. & Mellit, A., 2017. "Control of chaos in an induction motor system with LMI predictive control and experimental circuit validation," Chaos, Solitons & Fractals, Elsevier, vol. 97(C), pages 51-58.
    7. Lei, Youming & Xu, Wei & Xie, Wenxian, 2007. "Synchronization of two chaotic four-dimensional systems using active control," Chaos, Solitons & Fractals, Elsevier, vol. 32(5), pages 1823-1829.
    8. Mahmoud, Emad E. & Abo-Dahab, S.M., 2018. "Dynamical properties and complex anti synchronization with applications to secure communications for a novel chaotic complex nonlinear model," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 273-284.
    9. Jajarmi, Amin & Hajipour, Mojtaba & Baleanu, Dumitru, 2017. "New aspects of the adaptive synchronization and hyperchaos suppression of a financial model," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 285-296.
    10. Wang, Cong & Zhang, Hongli & Fan, Wenhui & Ma, Ping, 2018. "Adaptive control method for chaotic power systems based on finite-time stability theory and passivity-based control approach," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 159-167.
    11. Singh, Piyush Pratap & Singh, Jay Prakash & Roy, B.K., 2014. "Synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems using nonlinear active control," Chaos, Solitons & Fractals, Elsevier, vol. 69(C), pages 31-39.
    12. Vafamand, Navid & Khorshidi, Shapour & Khayatian, Alireza, 2018. "Secure communication for non-ideal channel via robust TS fuzzy observer-based hyperchaotic synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 116-124.
    13. Volos, Ch. K. & Kyprianidis, I.M. & Stouboulos, I.N. & Vaidyanathan, S. & Pham, V.-T., 2016. "Analysis, adaptive control and circuit simulation of a novel nonlinear finance systemAuthor-Name: Tacha, O.I," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 200-217.
    14. Uçar, Ahmet & Lonngren, Karl E. & Bai, Er-Wei, 2006. "Synchronization of the unified chaotic systems via active control," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1292-1297.
    15. Zhang, Qunjiao & Lu, Jun-an, 2008. "Chaos synchronization of a new chaotic system via nonlinear control," Chaos, Solitons & Fractals, Elsevier, vol. 37(1), pages 175-179.
    16. Vincent, U.E., 2008. "Synchronization of identical and non-identical 4-D chaotic systems using active control," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 1065-1075.
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