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本文研究两个非线性电路系统通过开关组成的时间切换系统的复杂振荡行为及其产生机理. 利用开环运算放大器放大倍数为极大值的特性,即运算放大器总是处于正的或负的饱和状态, 当输入电压从负过零变正时,输出电压从正饱和状态跃变为负饱和状态,本文选择子电路系统中的非线性部分为跃变函数. 首先对两个子系统进行了稳定性分析,给出了不同参数条件下的振荡行为,然后在子系统单个参数在一定范围内变化, 而其他参数保持不变的情况下,研究了切换系统的复杂振荡特征,并分析了其产生机理. 由于子系统方程的非光滑性和切换带来的整个系统的非光滑性,使得整个系统的周期振荡轨迹有四个切换点, 随着参数的变化,周期振荡轨线与非光滑分界面发生擦边分岔,导致周期振荡分裂成两个对称的周期振荡. 并且研究了切换点位置改变对整个系统周期振荡行为的影响以及切换点处的分岔机理.The complex dynamical evolution of a circuit system composed of two nonlinear circuit subsystems, which is switched by a periodic switching, is investigated. According to the fact that the magnification of an open-loop operational amplifier is maximum magnification, namely, the operational amplifier is always in a positive or negative saturated state, when an input voltage becomes positive from negative through zero, the output voltage jumps from the positive saturation into negative saturation. In this paper the jump function is selected as a nonlinear part in subsystems. Firstly through the stability analysis of the subsystems, their oscillation behaviors in the parameter space are given correspondingly. Secondly the complex oscillation behavior and mechanism of the switched system are discussed in the parameter space of one subsystem. The periodic orbit of the switched system is divided into four parts, influenced by non-smooth characteristics of the subsystems and switching. With the variation of the parameters, grazing bifurcation appears, and then the whole periodic orbit is separated into two symmetrical periodic oscillations. Finally the convesion of switching points into the periodic oscillation is given,and the mechanism at switching point is discussed.
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Keywords:
- jump circuit /
- switch /
- non-smooth /
- periodic oscillation
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[2] Varaiya P P 1993 IEEE Transactions on Automatic Control 38 195
[3] Dalvi A, Guay M 2009 Control Engineering Practice 17 924
[4] Yildirim H, Frank G, Bernard B 2004 Automatica 40 1647
[5] Yoshiyasu S, Masaru N, Hirokazu N, Yoshiyuki Y, Masatoshi Y, Sigeru M 2006 Computer Aided Chemical Engineering 21 1515
[6] Sun Z D, Zheng D Z 2001 IEEE Transactions on Automatic Control 46 291
[7] Leine R I 2006 Physica D 223 121
[8] Guo S Q, Yang S P, Guo J B 2005 Journal of Vibration Engineering 18 276 (in Chinese) [郭树起, 杨绍普, 郭京波 2005 振动工程学报 18 276]
[9] Li S L, Zhang Z D, Wu T Y, Bi Q S 2012 Acta Phys. Sin. 61 060504 (in Chinese) [李绍龙, 张正娣, 吴天一, 毕勤胜 2012 物理学报 61 060504]
[10] Xu X P, Antsaklis P J 2000 International Journal of Control 73 1261
[11] Hu B, Xu X, Antsaklis P J 1999 Systems & Control Letters 38 197
[12] Zhang L G, Chen Y Z, Cui P Y 2005 Nonlinear Analysis 62 1527
[13] Cheng D Z 2003 Systems & Control Letters 51 79
[14] Wu T Y, Zhang Z D, Bi Q S 2012 Acta Phys. Sin. 61 070502 (in Chinese) [吴天一, 张正娣, 毕勤胜 2012 物理学报 61 070502]
[15] Lu T X 2002 Overview of Nonlinear Physics (Hefei: China University of Science and Technology Press) p173 (in Chinese) [陆同兴 2002 非线性物理概论(合肥:中国科技大学出版社) 第173页]
[16] Contou-Carrere M N, Daoutidis P 2005 IEEE Transactions on Automatic Control 50 1831
[17] Sprott J C 2000 Physics Letters A 266 19
[18] Galvenetto U 2001 Journal of Sound and Vibration 248 653
[19] Zhang X G, Ma Y D, Li S L 2011 Nonlinear Circuit: Basic Analysis and Design (Beijing: Higher Education Press) p18 (in Chinese) [张新国, 马义德, 李守亮 2011 非线性电路:基础分析与设计(北京:高等教育出版社) 第18页]
[20] Gao W H, Guo X, Jiang J 2011 Journal of Dynamics and Control 9 363 (in Chinese) [高文辉, 郭旭, 江俊 2011 动力学与控制学报 9 363]
[21] Qin Z Y, Lu Q S 2009 Journal of Vibration and Shock 28 80 (in Chinese) [秦志英, 陆启韶 2009 振动与冲击 28 80]
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[1] Wyczalek F A 2001 IEEE Aerospace and Electronics Systems Magazine 16 15
[2] Varaiya P P 1993 IEEE Transactions on Automatic Control 38 195
[3] Dalvi A, Guay M 2009 Control Engineering Practice 17 924
[4] Yildirim H, Frank G, Bernard B 2004 Automatica 40 1647
[5] Yoshiyasu S, Masaru N, Hirokazu N, Yoshiyuki Y, Masatoshi Y, Sigeru M 2006 Computer Aided Chemical Engineering 21 1515
[6] Sun Z D, Zheng D Z 2001 IEEE Transactions on Automatic Control 46 291
[7] Leine R I 2006 Physica D 223 121
[8] Guo S Q, Yang S P, Guo J B 2005 Journal of Vibration Engineering 18 276 (in Chinese) [郭树起, 杨绍普, 郭京波 2005 振动工程学报 18 276]
[9] Li S L, Zhang Z D, Wu T Y, Bi Q S 2012 Acta Phys. Sin. 61 060504 (in Chinese) [李绍龙, 张正娣, 吴天一, 毕勤胜 2012 物理学报 61 060504]
[10] Xu X P, Antsaklis P J 2000 International Journal of Control 73 1261
[11] Hu B, Xu X, Antsaklis P J 1999 Systems & Control Letters 38 197
[12] Zhang L G, Chen Y Z, Cui P Y 2005 Nonlinear Analysis 62 1527
[13] Cheng D Z 2003 Systems & Control Letters 51 79
[14] Wu T Y, Zhang Z D, Bi Q S 2012 Acta Phys. Sin. 61 070502 (in Chinese) [吴天一, 张正娣, 毕勤胜 2012 物理学报 61 070502]
[15] Lu T X 2002 Overview of Nonlinear Physics (Hefei: China University of Science and Technology Press) p173 (in Chinese) [陆同兴 2002 非线性物理概论(合肥:中国科技大学出版社) 第173页]
[16] Contou-Carrere M N, Daoutidis P 2005 IEEE Transactions on Automatic Control 50 1831
[17] Sprott J C 2000 Physics Letters A 266 19
[18] Galvenetto U 2001 Journal of Sound and Vibration 248 653
[19] Zhang X G, Ma Y D, Li S L 2011 Nonlinear Circuit: Basic Analysis and Design (Beijing: Higher Education Press) p18 (in Chinese) [张新国, 马义德, 李守亮 2011 非线性电路:基础分析与设计(北京:高等教育出版社) 第18页]
[20] Gao W H, Guo X, Jiang J 2011 Journal of Dynamics and Control 9 363 (in Chinese) [高文辉, 郭旭, 江俊 2011 动力学与控制学报 9 363]
[21] Qin Z Y, Lu Q S 2009 Journal of Vibration and Shock 28 80 (in Chinese) [秦志英, 陆启韶 2009 振动与冲击 28 80]
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