Journal of Shandong University(Engineering Science) ›› 2020, Vol. 50 ›› Issue (4): 46-51.doi: 10.6040/j.issn.1672-3961.0.2019.475

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Three control schemes of chaos synchronization for fractional-order Brussel system

CHENG Chunrui   

  1. College of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou 450015, Henan, China
  • Published:2020-08-13

Abstract: Based on the fractional calculus theory, three synchronous control schemes were proposed to make the error system of the fractional Brussel system converge the error system states to the equilibrium point. An appropriate controller was designed in the first control scheme and the convergence of the error system was obtained by using Mittag-Leffler function. In the second control scheme, the fractional sliding mode surface was introduced, and the chaos synchronization of the fractional order Brussel master-slave systems was achieved based on the fractional version of the Lyapunov stability and the sliding mode control method. The effects of model uncertainties and external disturbances were fully taken into account in the third control scheme. A new sliding mode reaching law was designed and the fast convergence of the error system to the equilibrium point was obtained based on fractional order terminal sliding mode control. It was proved that master-slave systems were chaos synchronization under proper controllers. Numerical simulations were presented to illustrate the effectiveness and applicability of the proposed schemes and to validate the theoretical results of the paper.

Key words: Brussel system, fractional-order, chaos synchronization, Mittag-Leffler function, terminal slding mode

CLC Number: 

  • O482.4
[1] 马莉. Brussel系统的稳定性分析与动力学行为研究[J]. 自动化与仪器仪表,2016,21(7):73-75. MA Li. The stability analysis and dynamic behavior research of Brussel systems[J].Automation and Instrumentation, 2016, 21(7):73-75.
[2] 马莉. Brussel模型的混沌控制[J]. 电气自动化,2015,37(5):17-19. MA Li. Chaos control of Brussel model [J]. Electric Automatization, 2015, 37(5):17-19.
[3] MANDELBROT B B, VAN NESS J W. Fractional Brownian motions, fractional noises and applications[J]. Siam Review, 1968, 10(4):422-437.
[4] HAMAMCI, KOLKSAL M. Calculation of all stabilizing fractional-order PD controllers for integrating time delay systems[J]. Comput Math Appl, 2010, 52(2):1267-1278.
[5] MATOUK A. Chaos feedback and synchronization of fractional-order modified autonomous Van der pol-Duffling circuit[J]. Commun Nonlinear Sci Numer Simul, 2011, 46(5):975-986.
[6] MOHAMMAD P A. Robust finite-time stabilization of fractional-order chaotic systems based on fractional Lyapunov stability theory[J]. Journal of Computation and Nonlinear Dynamic, 2012, 32(7):1011-1015.
[7] MILAD M, HADI D. Synchronization of fractional order hyper-chaotic systems based on a new adaptive sliding mode control[J].International Journal of Dynamics and Control, 2017, 5(1):124-134.
[8] DELAVARI H, GHADERI R, RANJBAR A, et al. Fuzzy fractional order sliding mode controller for nonlinear systems[J]. Communications in Nonlinear Science and Numerical Simulation, 2010, 15(4):963-978.
[9] 毛北行,李巧利.一类分数阶 Duffling-Van der pol 系统的混沌同步[J]. 吉林大学学报(理学版),2016,54(2):369-373. MAO Beixing, LI Qiaoli. Chaos synchronization of a class of fractional-order Duffling-Van der pol systems[J]. Journal of Jilin University(Science Edition), 2016, 54(2):369-373.
[10] 王悍枭,刘凌,吴华伟.改进型滑模观测器的永磁同步电机无传感器控制策略[J]. 西安交通大学报,2016, 50(6):104-109. WANG Hanniao, LIU Ling, WU Huawei. A senseless permanent magnet synchronous motor control strategy for improved sliding mode observers with stator parameters identification[J]. Journal of Xi'an Jiaotong University, 2016, 50(6):104-109.
[11] 毛北行,王东晓.分数阶多涡卷系统滑模控制混沌同步[J]. 山东大学学报(工学版),2017,47(3):79-83. MAO Beixing, WANG Dongxiao. Sliding mode chaos synchronization control of a class of fractional-order multi-scroll systems[J]. Journal of Shandong University(Engineering Science), 2017, 47(3):79-83.
[12] DADRAS S, MOMENI H R. Fractional terminal sliding mode control design for a class of dynamical systems with uncertainty[J]. Communications in Nonlinear Science and Numerical Simulation, 2012, 17(1):367-377.
[13] 徐瑞萍, 高明美. 自适应终端滑模控制不确定混沌系统的同步[J]. 控制工程, 2016, 23(5):715-719. XU Ruiping, GAO Mingmei. Synchronization of chaotic systems with uncertainty using an adaptive terminal sliding mode controller[J]. Control Engineering of China, 2016, 23(5):715-719.
[14] MOHAMMAD P A, SOHRAB K, GHASSEM A. Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique[J]. Applied Mathematical Modelling, 2011, 35(6):3080-3091.
[15] 王战伟, 张伟, 毛北行. 分数阶Brussel混沌系统终端滑模控制[J]. 重庆师范大学学报(自然版), 2018,35(2): 100-103. WANG Zhanwei, ZHANG Wei, MAO Beixing. Terminal sliding mode synchronization control of fractional-order Brussel systems[J]. Journal of Chongqing Normal university(Science Edition), 2018, 35(2):100-103.
[16] PODLUBNY. Fractional differential equation[M]. San Diego, USA: Academic Press, 1999:715-719.
[17] 梅生伟, 申铁龙, 刘康志. 现代鲁棒控制理论与应用[M]. 北京:清华大学出版社, 2003.
[18] BHAT S P, BERNSTEIN D S. Geometric homogeneity with applications to finite-time stability[J]. Mathematics of Control Signals and Systems, 2005, 17(2):101-127.
[19] BAGLEY R L, TORYIK P J. On the appearance of the fractional derivative in the behavior of real materials[J]. Journal of Applied Mechanics, 1984, 51(4): 294-298.
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