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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2019, Vol. 36 ›› Issue (1): 99-105.doi: 10.3969/j.issn.1005-3085.2019.01.008

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分数阶单摆系统的终端滑模控制混沌同步

程春蕊,   朱军辉,   毛北行   

  1. 郑州航空工业管理学院理学院，郑州  450015
  • 收稿日期:2017-01-18 接受日期:2017-09-12 出版日期:2019-02-15 发布日期:2019-04-15
  • 基金资助:
    国家自然科学基金(11501525)；河南省高等学校重点科研项目(16B110014; 17A110034)；郑州航院青年科研基金(2014113002; 2015113001).

Chaos Synchronization of Fractional-order Simple Pendulum Systems Based on Terminal Sliding Mode Control

CHENG Chun-rui,   ZHU Jun-hui,   MAO Bei-xing   

  1. College of Science, Zhengzhou Institute of Aeronautical Industry Management, Zhengzhou 450015
  • Received:2017-01-18 Accepted:2017-09-12 Online:2019-02-15 Published:2019-04-15
  • Supported by:
    The National Natural Science Foundation of China (11501525); the Natural Vital Project of Higher Education of Henan Province (16B110014; 17A110034); the Youth Natural Science Foundation of Zhengzhou Institute of Aeronautical Industry Management (2014113002; 2015113001).

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摘要： 针对一类具有不确定性和外部扰动的分数阶单摆系统，本文提出了一种新的分数阶滑模同步控制方法．首先，在分数阶微积分的基础上，引入了一种新的非奇异分数阶终端滑模面，并利用分数阶 Lyapunov 稳定性定理，证明了在滑模面上误差系统能够在有限时间内收敛到平衡点．在此基础上，针对事先未知系统中不确定性和外部扰动的界的情况，设计了适当的自适应律．基于自适应律和有限时间控制思想，提出了一种自适应滑模控制器，以保证系统在给定的时间内发生滑模运动，并证明了所提出的滑模控制方法在到达和滑模阶段都具有有限时间收敛性和稳定性．最后，通过数值算例验证了该方案的适用性和有效性．

关键词: 分数阶系统, 终端滑模, 混沌同步, 自适应控制, 单摆系统

Abstract: In this paper, a novel adaptive sliding mode approach for synchronization combining the fractional calculus with terminal sliding mode control is proposed. The scheme is applied to synchronize a class of fractional-order simple pendulum chaotic systems in the presence of model uncertainties and external disturbances. First, based on the fractional calculus a new nonsingular fractional-order terminal sliding surface is introduced and its finite-time convergence to the zero equilibrium is analytically proved using the fractional Lyapunov stability theorem. Then, for the case that the bounds of the uncertainties and external disturbances are assumed to be unknown in advance, appropriate adaptive laws are proposed. Afterwards, based on ideas of the adaptive laws and finite-time control, an adaptive sliding mode controller is proposed to ensure the occurrence of the sliding motion in a given finite peroid. It is mathematically proved that the introduced sliding mode technique has finite-time convergence and stability in both reaching and sliding mode phases. Finally, a numerical simulation is presented to demonstrate the applicability and effectiveness of the proposed scheme.

Key words: fractional-order systems, terminal sliding mode, chaos synchronization, adaptive control, simple pendulum systems

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