搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

改进的保群算法及其在混沌系统中的应用

陆见光 唐卷 秦小林 冯勇

引用本文:
Citation:

改进的保群算法及其在混沌系统中的应用

陆见光, 唐卷, 秦小林, 冯勇

Modified group preserving methods and applications in chaotic systems

Lu Jian-Guang, Tang Juan, Qin Xiao-Lin, Feng Yong
PDF
导出引用
  • 混沌系统的跟踪控制是近年来非线性控制领域研究的热点之一. 本文提出了一种基于快速下降控制方法的保群算法, 此方法使受控混沌系统能够快速稳定到相空间的一个不动点; 另外提出一种基于滑模控制方法的保群算法, 此方法使受控混沌系统能够快速跟踪一个确定的运动轨迹. 应用这两种新方法分别对两个经典的混沌系统(Lorenz系统和Duffing系统)进行了相应的数值实验, 实验结果表明这两种方法都具用较高的精度和稳定性.
    The tracking control of chaotic system has been one of the research focus areas of nonlinear control in recent years, in which the vital problem is to enable chaotic system to stabilize to an equilibrium point or to track a deterministic trajectory quickly. The conventional chaos control methods make the control power unnecessarily large and generate the phenomenon of chattering easily, resulting in the instabilities of the systems.The problems above can be transformed into the solutions of differential algebraic equations effectively. Considering that the group preserving scheme not only approximates the original system, but also preserve as much as possible the geometric structure and invariants of the original system, this paper takes advantage of the group preserving method to study the control method in chaotic system from two different perspectives.A new group preserving scheme based on the fast descending control method is presented, which enables chaotic system to stabilize to an equilibrium point quickly. Firstly, we introduce a novel approach to replace the optimal control problem of nonlinear system by directly specifying a time-decaying Lagrangian function, which helps us to transform the optimal control problem into a system of differential algebraic equations. Then we derive a modified group preserving scheme for the system.Similarly, we propose a new group preserving scheme based on the sliding mode control method for chaotic system to track a deterministic trajectory quickly. Owing to numerical discretization errors, signal noises and structural uncertainties in dynamical systems, the conventional sliding mode control method cannot guarantee to maintain the trajectories on the sliding surface, unless the numerical integration method is designed to do so. On the other hand, the conventional sliding mode control method easily induces high frequency chattering of the control force. Therefore, we modify the conventional sliding mode control method and use the modified group preserving scheme to find the control force.The above two methods are the combination of traditional control method and the Lie-group method. An invariant manifold is properly designed, and the original system is transformed into the differential algebraic system, in which the modified group preserving scheme can be used to find the control force. The resulting controlled system is stable.Finally, the proposed methods are applied to the classic Lorenz system and Duffing system correspondingly. Numerical experimental results show that the new approaches are very accurate and stable. Since the two controlled methods are fast in convergence and chattering-free, each of them has a good application prospect in the tracking control of chaotic systems.
      通信作者: 唐卷, tangjuan0822@gmail.com
    • 基金项目: 国家重点基础研究发展计划(批准号: 2011CB302402)和国家自然科学基金(批准号: 61402537, 91118001)资助的课题.
      Corresponding author: Tang Juan, tangjuan0822@gmail.com
    • Funds: Project supported by the National Basic Research Program of China (Grant No. 2011CB302402) and the National Natural Science Foundation of China (Grant Nos. 61402537, 91118001).
    [1]

    Chen G R, Ueta T 1999 Int. J. Bifurcat. Chaos 9 1465

    [2]

    L J H, Chen G R 2002 Int. J. Bifurcat. Chaos 12 659

    [3]

    Zhang H G, Wang Z L,Huang W 2008 Control Theory of Chaotic Systems (Vol. 1) (Shenyang: Northeast Univesity Press) pp1-4 (in Chinese) [张化光, 王智良, 黄伟 2008 混沌系统的控制理论 (沈阳: 东北大学出版社) 第1-4页]

    [4]

    Zhang X H, Shen J, Mei L, Wang D M 2011 Syst. Eng. Elect. 33 603 (in Chinese) [张兴华, 沈捷, 梅磊, 王德明 2011 系统工程与电子技术 33 603]

    [5]

    Davies M J 1972 J. Inst. Math. Appl. 9 357

    [6]

    Van D R, Vlassenbroeck J 1982 J. Comput. Phys. 47 321

    [7]

    El-Kady M, Elbarbary E M E 2002 Appl. Math. Comput. 129 171

    [8]

    Razzaghi M, Elnagar G 1994 J. Comput. Appl. Math. 56 253

    [9]

    Lakestani M, Razzaghi M, Dehghan M 2006 Phys. Scr. 74 362

    [10]

    Song R Z, Xiao W D, Sun C Y, Wei Q L 2013 Chin. Phys. B 22 090502

    [11]

    Wei Q L, Liu D R, Xu Y C 2015 Chin. Phys. B 24 030502

    [12]

    Wei Q L, Song R Z, Sun Q Y, Xiao W D 2015 Chin. Phys. B 24 090504

    [13]

    Liu C S {2012 CMES: Comput. Model. Eng. Sci. 86 171

    [14]

    Liu C S 2014 Commun. Nonlinear Sci. Numer. Simul. 19 2012

    [15]

    Utkin V I 1977 IEEE Trans. Autom. Contr. 22 212

    [16]

    Utkin V I 1992 Sliding Modes in Control and Optimization (Vol. 1) (New York: Springer-Verlag) pp7-11

    [17]

    Slotine J J E, Sastry S S 1983 Int. J. Control 38 465

    [18]

    Levant A 1993 Int. J. Control 58 1247

    [19]

    Wong L K, Leung F H F, Tam P K S 1998 Mechatronics 8 765

    [20]

    Lee H, Utkin V I 2007 Ann. Rev. Control 31 179

    [21]

    Yang G L, Li H G 2009 Acta Phys. Sin. 58 7552 (in Chinese) [杨国良, 李惠光 2009 物理学报 58 7552]

    [22]

    Falahpoor M, Ataei M, Kiyoumarsi A 2009 Chaos Solitons Fract. 42 1755

    [23]

    Qi L 2013 Ph. D. Dissertation (Shanghai: East China University of Science and Technology) (in Chinese) [齐亮 2013 博士学位论文 (上海: 华东理工大学)]

    [24]

    Dong K W, Zhang X 2007 Electrotechnical Application 2 6 (in Chinese) [董克文, 张兴 2007 电气应用 2 6]

    [25]

    Liu C S 2001 Int. J. Non-Linear Mech. 36 1047

    [26]

    Liu C S {2006 CMES: Comput. Model. Eng. Sci. 12 83

    [27]

    Wu W G, Gu T X 2000 Acta Phys. Sin. 49 1922 (in Chinese) [伍维根, 古天祥 2000 物理学报 49 1922]

    [28]

    Cai G L, Tan Z M, Zhou W H, Tu W T 2007 Acta Phys. Sin. 56 6230 (in Chinese) [蔡国梁, 谭振梅, 周维怀, 涂文桃 2007 物理学报 56 6230]

    [29]

    Li G Y 2008 Optimal Control Theory and Application (Vol. 1) (Beijing: National Defence Industry Press) p4 (in Chinese) [李国勇 2008 最优化控制理论与应用 (北京: 国防工业出版社) 第4页]

    [30]

    Wang W 2009 M. S. Dissertation (Dalian: Dalian Jiaotong University) (in Chinese) [王文 2009 硕士学位论文 (大连: 大连交通大学)]

    [31]

    Roopaei M, Sahraei B R, Lin T C 2010 Commun. Nonlinear Sci. Numer. Simul. 15 4158

    [32]

    Shen C W, Yu S M, L J H, Chen G R 2014 IEEE Trans. Circuits Syst. I 61 854

    [33]

    Shen C W, Yu S M, L J H, Chen G R 2014 IEEE Trans. Circuits Syst. I 61 2380

  • [1]

    Chen G R, Ueta T 1999 Int. J. Bifurcat. Chaos 9 1465

    [2]

    L J H, Chen G R 2002 Int. J. Bifurcat. Chaos 12 659

    [3]

    Zhang H G, Wang Z L,Huang W 2008 Control Theory of Chaotic Systems (Vol. 1) (Shenyang: Northeast Univesity Press) pp1-4 (in Chinese) [张化光, 王智良, 黄伟 2008 混沌系统的控制理论 (沈阳: 东北大学出版社) 第1-4页]

    [4]

    Zhang X H, Shen J, Mei L, Wang D M 2011 Syst. Eng. Elect. 33 603 (in Chinese) [张兴华, 沈捷, 梅磊, 王德明 2011 系统工程与电子技术 33 603]

    [5]

    Davies M J 1972 J. Inst. Math. Appl. 9 357

    [6]

    Van D R, Vlassenbroeck J 1982 J. Comput. Phys. 47 321

    [7]

    El-Kady M, Elbarbary E M E 2002 Appl. Math. Comput. 129 171

    [8]

    Razzaghi M, Elnagar G 1994 J. Comput. Appl. Math. 56 253

    [9]

    Lakestani M, Razzaghi M, Dehghan M 2006 Phys. Scr. 74 362

    [10]

    Song R Z, Xiao W D, Sun C Y, Wei Q L 2013 Chin. Phys. B 22 090502

    [11]

    Wei Q L, Liu D R, Xu Y C 2015 Chin. Phys. B 24 030502

    [12]

    Wei Q L, Song R Z, Sun Q Y, Xiao W D 2015 Chin. Phys. B 24 090504

    [13]

    Liu C S {2012 CMES: Comput. Model. Eng. Sci. 86 171

    [14]

    Liu C S 2014 Commun. Nonlinear Sci. Numer. Simul. 19 2012

    [15]

    Utkin V I 1977 IEEE Trans. Autom. Contr. 22 212

    [16]

    Utkin V I 1992 Sliding Modes in Control and Optimization (Vol. 1) (New York: Springer-Verlag) pp7-11

    [17]

    Slotine J J E, Sastry S S 1983 Int. J. Control 38 465

    [18]

    Levant A 1993 Int. J. Control 58 1247

    [19]

    Wong L K, Leung F H F, Tam P K S 1998 Mechatronics 8 765

    [20]

    Lee H, Utkin V I 2007 Ann. Rev. Control 31 179

    [21]

    Yang G L, Li H G 2009 Acta Phys. Sin. 58 7552 (in Chinese) [杨国良, 李惠光 2009 物理学报 58 7552]

    [22]

    Falahpoor M, Ataei M, Kiyoumarsi A 2009 Chaos Solitons Fract. 42 1755

    [23]

    Qi L 2013 Ph. D. Dissertation (Shanghai: East China University of Science and Technology) (in Chinese) [齐亮 2013 博士学位论文 (上海: 华东理工大学)]

    [24]

    Dong K W, Zhang X 2007 Electrotechnical Application 2 6 (in Chinese) [董克文, 张兴 2007 电气应用 2 6]

    [25]

    Liu C S 2001 Int. J. Non-Linear Mech. 36 1047

    [26]

    Liu C S {2006 CMES: Comput. Model. Eng. Sci. 12 83

    [27]

    Wu W G, Gu T X 2000 Acta Phys. Sin. 49 1922 (in Chinese) [伍维根, 古天祥 2000 物理学报 49 1922]

    [28]

    Cai G L, Tan Z M, Zhou W H, Tu W T 2007 Acta Phys. Sin. 56 6230 (in Chinese) [蔡国梁, 谭振梅, 周维怀, 涂文桃 2007 物理学报 56 6230]

    [29]

    Li G Y 2008 Optimal Control Theory and Application (Vol. 1) (Beijing: National Defence Industry Press) p4 (in Chinese) [李国勇 2008 最优化控制理论与应用 (北京: 国防工业出版社) 第4页]

    [30]

    Wang W 2009 M. S. Dissertation (Dalian: Dalian Jiaotong University) (in Chinese) [王文 2009 硕士学位论文 (大连: 大连交通大学)]

    [31]

    Roopaei M, Sahraei B R, Lin T C 2010 Commun. Nonlinear Sci. Numer. Simul. 15 4158

    [32]

    Shen C W, Yu S M, L J H, Chen G R 2014 IEEE Trans. Circuits Syst. I 61 854

    [33]

    Shen C W, Yu S M, L J H, Chen G R 2014 IEEE Trans. Circuits Syst. I 61 2380

  • [1] 李航, 申永军, 杨绍普, 彭孟菲, 韩彦军. Duffing系统的主-超谐联合共振. 物理学报, 2021, 70(4): 040502. doi: 10.7498/aps.70.20201059
    [2] 黄泽徽, 李亚安, 陈哲, 刘恋. 基于多尺度熵的Duffing混沌系统阈值确定方法. 物理学报, 2020, 69(16): 160501. doi: 10.7498/aps.69.20191642
    [3] 李保生, 丁瑞强, 李建平, 钟权加. 强迫Lorenz系统的可预报性研究. 物理学报, 2017, 66(6): 060503. doi: 10.7498/aps.66.060503
    [4] 张志森, 龚志强, 支蓉. 利用传递熵对Lorenz系统和Walker环流信息传输方向的分析. 物理学报, 2013, 62(12): 129203. doi: 10.7498/aps.62.129203
    [5] 贾红艳, 陈增强, 薛薇. 分数阶Lorenz系统的分析及电路实现. 物理学报, 2013, 62(14): 140503. doi: 10.7498/aps.62.140503
    [6] 罗明伟, 罗小华, 李华青. 一类四维多翼混沌系统及其电路实现. 物理学报, 2013, 62(2): 020512. doi: 10.7498/aps.62.020512
    [7] 黎爱兵, 张立凤, 项杰. 外强迫对Lorenz系统初值可预报性的影响. 物理学报, 2012, 61(11): 119202. doi: 10.7498/aps.61.119202
    [8] 唐友福, 刘树林, 雷娜, 姜锐红, 刘颖慧. 基于广义局部频率的Duffing系统频域特征分析. 物理学报, 2012, 61(17): 170504. doi: 10.7498/aps.61.170504
    [9] 李清都, 谭宇玲, 杨芳艳. 连续时间系统二维不稳定流形的异构算法. 物理学报, 2011, 60(3): 030206. doi: 10.7498/aps.60.030206
    [10] 李小娟, 徐振源, 谢青春, 王兵. 单向耦合下两个不同Lorenz系统的广义同步. 物理学报, 2010, 59(3): 1532-1539. doi: 10.7498/aps.59.1532
    [11] 孙克辉, 杨静利, 丁家峰, 盛利元. 单参数Lorenz混沌系统的电路设计与实现. 物理学报, 2010, 59(12): 8385-8392. doi: 10.7498/aps.59.8385
    [12] 王启光, 支 蓉, 张增平. Lorenz系统长程相关性研究. 物理学报, 2008, 57(8): 5343-5350. doi: 10.7498/aps.57.5343
    [13] 仓诗建, 陈增强, 袁著祉. 一个新四维非自治超混沌系统的分析与电路实现. 物理学报, 2008, 57(3): 1493-1501. doi: 10.7498/aps.57.1493
    [14] 李文林, 宋运忠. 不确定非线性系统混沌反控制. 物理学报, 2008, 57(1): 51-55. doi: 10.7498/aps.57.51
    [15] 郝建红, 孙志华, 许海波. 干扰信号对两种混沌加密系统的影响及分析. 物理学报, 2007, 56(12): 6857-6864. doi: 10.7498/aps.56.6857
    [16] 王兴元, 王明军. 超混沌Lorenz系统. 物理学报, 2007, 56(9): 5136-5141. doi: 10.7498/aps.56.5136
    [17] 王杰智, 陈增强, 袁著祉. 一个新的混沌系统及其性质研究. 物理学报, 2006, 55(8): 3956-3963. doi: 10.7498/aps.55.3956
    [18] 李 爽, 徐 伟, 李瑞红, 李玉鹏. 异结构系统混沌同步的新方法. 物理学报, 2006, 55(11): 5681-5687. doi: 10.7498/aps.55.5681
    [19] 唐国宁, 罗晓曙. 混沌系统的预测反馈控制. 物理学报, 2004, 53(1): 15-20. doi: 10.7498/aps.53.15
    [20] 郭会军, 刘君华. 基于径向基函数神经网络的Lorenz混沌系统滑模控制. 物理学报, 2004, 53(12): 4080-4086. doi: 10.7498/aps.53.4080
计量
  • 文章访问数:  5473
  • PDF下载量:  375
  • 被引次数: 0
出版历程
  • 收稿日期:  2016-01-16
  • 修回日期:  2016-03-21
  • 刊出日期:  2016-06-05

/

返回文章
返回