双状态切换下BVP振子的复杂行为分析

陈章耀; 王亚茗; 张春; 毕勤胜

doi:10.6052/0459-1879-16-044

双状态切换下BVP振子的复杂行为分析

江苏大学土木工程与力学学院, 镇江 212013

基金项目: 国家自然科学基金(11472115, 11572141, 11502091) 和镇江市科技攻关基金(GY2013032, GY2013052) 资助项目.

详细信息

通讯作者:
毕勤胜,教授,主要研究方向:非线性动力学.E-mail:qbi@ujs.edu.cn

中图分类号: O322
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- 文章访问数: 833
- HTML全文浏览量: 99
- PDF下载量: 635
出版历程
- 收稿日期: 2016-02-01
- 修回日期: 2016-04-20
- 刊出日期: 2016-07-17

COMPLICATED BEHAVIORS AS WELL AS THE MECHANISM IN BVP OSCILLATOR WITH SWITCHES RELATED TO TWO STATES

Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China

摘要

摘要: 非线性切换系统具有广泛的工程背景，而传统的非线性理论不能直接用来解决其中的问题，因而成为当前国内外热点和前沿课题之一. 目前相关工作大都是围绕固定时间或单状态切换开展的，而实际工程系统大都属于多状态切换问题，同时多状态切换涉及到更为丰富的动力学行为. 本文基于两广义BVP 振子，通过引入双向切换开关，构建了双状态切换下的非线性动力学模型，进而研究状态切换导致的各种运动模式及其相应的产生机制. 应用非光滑系统的Poincaré映射理论，推导了双状态切换下的Lyapunov 指数的计算公式，结合子系统的分岔分析，得到了切换系统随分岔参数变化的动力学演化过程及其相应的最大Lyapunov 指数的变化情况. 得到了双状态切换条件下系统存在着各种形式的振荡行为，分析了诸如周期突变等现象及通往混沌的倍周期分岔道路，揭示了不同运动模式的产生机制及倍周期序列的本质. 与固定时间切换和单状态切换系统不同，双临界状态切换系统存在着更为丰富的非线性现象，其主要原因在于双状态切换会产生更多的切换点，且切换点的位置更加多变. 同时切换系统的倍周期分岔序列与光滑系统中的倍周期分岔序列不同，切换系统的倍周期分岔序列只对应于切换点数目的成倍增加，而其相应的周期一般不对应于严格的周期倍化过程.
- 切换系统 /
- 广义BVP 振子 /
- 双状态切换 /
- Lyapunov 指数
Abstract: The dynamics of nonlinear switched systems which possess wide engineering background and cannot be explored directly by traditional nonlinear theory, become one of hot and frontier tasks at home and abroad for the time being. The complicated behaviors as well as the mechanism of the vector field alternated between two subsystems by two different critical states are investigated in this paper. Upon employing the typical generalized BVP oscillator as an example, by introducing bilateral switch, the nonlinear dynamical model alternated between two subsystems related two states is established, the different movement forms as well as the dynamical evolution of which caused by switches are explored in details. Based on the Poincaré theory of nonlinear system, the computational equation of Lyapunov exponents of switched system is derived. Combined with the bifurcation analysis of subsystems, different oscillations of the system are discussed, upon which the nonlinear behaviors such as sudden changes of period in periodic oscillations and the route to chaos with period-doubling bifurcations as well as the related essence are presented. Different from the systems with fixed time or single state switch, much more nonlinear phenomena may be observed in the dynamic systems with two state switches in which there may exist more switch points with changeable positions. Furthermore, different from the cascading of period-doubling bifurcations in smooth systems, the period-doubling bifurcations in switched systems correspond to the doubling of the number of switch points, which usually does not correspond to the doubling of the real periodic length of the movements.
- switched system /
- generalized BVP oscillator /
- switch related to two states /
- Lyapunov exponents

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参考文献(31)

1 Pérez C, Bensítez F. Switched convergence of second-order switch nonlinear systems. Journal of Control, Automation, and Systems, 2012, 10(5): 920-930

2 王仁明, 关治洪, 刘新芝. 一类非线性切换系统的稳定性分析. 系统工程与电子技术. 2004, 26(1):68-71 (Wang Renming, Guan Zhihong, Liu Xinzhi. Stability analysis for a class of nonlinear switched systems. Systems Engineering and Electronics, 2004, 26(1): 68-71(in Chinese))

3 Davila J, Pisano A, Usai E. Continuous and discrete state reconstruction for nonlinear switched system via high-order sliding-mode observers. International Journal of System Science, 2011, 42(5): 725-735

4 刘正凡, 蔡晨晓, 段文勇等. 不确定切换时滞非线性系统状态切换的指数稳定性. 控制与决策, 2014, 29(12): 2247-2252(Liu Zhengfan, Cai Chenxiao, Duan Wenyong, et al. Exponential stability of uncertain switched nonlinear time-delay systems with statedependent switching. Control and Decision, 2014, 29(12): 2247-2252(in Chinese))

5 Oishi M, Mitchell I, Bayen AM, et al. Hybrid verification of an interface for an automatic landing. Proceeding of the IEEE Conference on Decision and Control, 2002, 2: 1607-1613

6 Bishop BE, Spong MW. Control of redundant manipulators using logic-based switching. Proceedings of the 36th IEEE Conference on Decision and Control, 1998, 2:16-18

7 Horowitz R, Varaiya P. Control design of an automated highway system. Proceedings of IEEE, 2000, 88(7): 913-925

8 方志明. 切换系统稳定性分析与优化控制若干问题研究. [博士论文]. 南京: 南京理工大学, 2012(Fang Zhiming. Study on several problems of stability and optimal control of switched systems. [PhD Thesis]. Nanjing: Nanjing University of Science & Technology, 2012 (in Chinese))

9 Goncalves JM. Constructive global analysis of hybrid systems. [PhD thesis]. MA, USA:Massachusetts Institute of Technology, 2000

10 Zhang JF, Han ZZ, Zhu FB, et al. Absolute exponential stability and stabilization of switched nonlinear systems, Systems & Control Letters, 2014, 66(1): 51-57

11 Xiang WM, Xiao J, Iqba MN. Robust fault detection for a class of uncertain switched nonlinear systems via the state updating approach. Nonlinear Analysis: Hybrid Systems, 2014, 12(2): 132-146

12 Jin Y, Fu J, Zhang YM, et al. Reliable control of a class of switched cascade nonlinear systems with its application to flight control. Nonlinear Analysis: Hybrid Systems, 2014, 11(1): 11-21

13 Yu Aleksandrov A, Chen Y, Platonov AV, et al. Stability analysis for a class of switched nonlinear systems. Automatica, 2011, 47(10): 2286-2291

14 Moulay E, Bourdais R, Perruquetti W. Stabilization of nonlinear switched systems using control Lyapunov functions. Nonlinear Analysis: Hybrid Systems, 2007, 1(4): 482-490

15 Liu YY, Zhao J. Stabilization of switched nonlinear systems with passive and non-passive subsystems. Nonlinear Dyn, 2012, 67(3): 1709-1716

16 Colaneria P, Geromelb JC, Astolfic A. Stabilization of continuoustime switched nonlinear systems. Systems & Control, 2008, 57(1): 95-103

17 Alessandria A, Sanguineti M. Connections between Lp stability and asymptotic stability of nonlinear switched systems. Nonlinear Analysis: Hybrid Systems, 2007, 1(1): 501-509

18 Hajiahmadi M, De Schutter B, Hellendoorn H. Stabilization and robust H1 control for sector-bounded switched nonlinear systems. Automatica, 2014, 50(10): 2726-2731

19 Han TT, Ge SS, Lee TH. Adaptive neural control for a class of switched nonlinear systems. Systems & Control Letters, 2009, 58(2): 109-118

20 Elfarra NH, Mhaskar P, Christofider PD. Output feedback control of switched nonlinear systems using multiple Lyapunov functions. Systems & Control Letters, 2005, 54(12): 1163-1182

21 Chiang ML, Fu LC. Rubust output feedback stabilization of switched nonlinear systems with average dwell time. Asian Journal of Control, 2014, 16(1): 264-276

22 Jouili K, Braiek NB. Stabilization of non-minimum phase switched nonlinear systems with the concept of multi-diffeomorphism. Commun Nonlinear Sci Numer Simulat, 2015, 23(1-3): 282-293

23 Mojica-Nava E, Quijano N, Rakoto-Ravalontsalama N. A polynomial approach for optimal control of switched nonlinear systems. Robust Nonlinear Control, 2014, 24(12): 1797-1808

24 Lin Q, Loxton R, Teo KL. Optimal control of nonlinear switched systems: computational methods and applications. Journal of the Operations Research Society of China, 2013, 1(3): 275–311

25 陈章耀, 雪增红, 张春等. 周期切换下Rayleigh 振子的振荡行为及机理. 物理学报, 2014, 63(1): 1-8 (Chen Zhangyao, Xue Zenghong, Zhang Chun, et al. Oscillation behaviors and mechanism of Rayleigh oscillator with periodic switches. Acta Phys. Sin, 2014, 63(1): 1-8 (in Chinese))

26 Jerome JW. Time dependent closed quantum systems: nonlinear Kohn–Sham potential operators and weak solutions. Journal of Mathematical Analysis and Applications, 2015, 429(2): 995-1006

27 Yang JQ, Chen YT, Zhu FL, et al. Synchronous switching observer for nonlinear switched systems with minimum dwell time constraint. Journal of the Franklin Institute, 2015, 352(11): 4665-4681

28 Ma RC, Zhao SZ, Wang M. Global robust stabilisation of a class of uncertain switched nonlinear systems with dwell time specifications. International Journal of Control, 2014, 87(3): 589-599

29 Kousaka T, Ueta T, Kawakami H. Bifurcation of switched nonlinear dynamical systems. IEEE Trans. Circuits Syst, 1999, 46(7): 878-885

30 Zhang C, Bi QS, Han XJ, et al. On two-parameter bifurcation analysis of switched system composed of Duffing and van der Pol oscillators. Communications in Nonlinear Science and Numerical Simulation, 2014, 19(3): 750-757

31 Nishiuchi Y, Ueta T, Kawakami H. Stable torus and its bifurcation phenomena in a simple three-dimensional autonomous circuit. Chaos, Solitons and Fractals, 2006, 27(4): 941-951

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双状态切换下BVP振子的复杂行为分析

通讯作者:
毕勤胜,教授,主要研究方向:非线性动力学.E-mail:qbi@ujs.edu.cn

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COMPLICATED BEHAVIORS AS WELL AS THE MECHANISM IN BVP OSCILLATOR WITH SWITCHES RELATED TO TWO STATES

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双状态切换下BVP振子的复杂行为分析

通讯作者: 毕勤胜,教授,主要研究方向:非线性动力学.E-mail:qbi@ujs.edu.cn

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COMPLICATED BEHAVIORS AS WELL AS THE MECHANISM IN BVP OSCILLATOR WITH SWITCHES RELATED TO TWO STATES

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通讯作者:
毕勤胜,教授,主要研究方向:非线性动力学.E-mail:qbi@ujs.edu.cn