吉林大学学报(工学版) ›› 2017, Vol. 47 ›› Issue (4): 1225-1230.doi: 10.13229/j.cnki.jdxbgxb201704030
邵克勇, 陈丰, 王婷婷, 王季驰, 周立朋
SHAO Ke-yong, CHEN Feng, WANG Ting-ting, WANG Ji-chi, ZHOU Li-peng
摘要: 针对一类无平衡点的分数阶混沌系统,首先通过分数阶微分变换方法(FDTM)得到它的解序列。然后,研究了系统的Kaplan-Yorke维数和耗散性,基于系统的离散映射通过QR分解得到最大Lyapunov特征指数,通过该特征指数可以判断系统是否保持混沌。最后,给出一种全状态自适应控制方法,使系统的状态变量追踪期望轨迹,并通过数值模拟验证了本文算法的可行性。
中图分类号:
[1] Meral F, Royston T, Magin R. Fractional calculus in viscoelasticity: an experimental study[J]. Communications in Nonlinear Science and Numerical Simulation, 2010,15(4):939-945. [2] Magin R L. Fractional calculus models of complex dynamics in biological tissues[J]. Computers & Mathematics with Applications, 2010,59(5):1586-1593. [3] Drapaca C, Sivaloganathan S. A fractional model of continuum mechanics[J]. Journal of Elasticity, 2012,107(2) :105-123. [4] Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos[M]//Marsden J E, Sirovich L, Antman S S. Texts in Applied Mathematics. Berlin: Springer, 2003. [5] Hoppensteadt F C. Analysis and simulation of chaotic systems[J]. Journal of Applied Mathematics and Mechanics,2002,82(7):472. [6] Li C, Peng G. Chaos in Chen's system with a fractional order[J]. Chaos, Solitons & Fractals, 2004,22(2):443-450. [7] Xu Y, Gu R, Zhang H, et al. Chaos in diffusionless lorenz system with a fractional order and its control[J]. International Journal of Bifurcation and Chaos, 2012,22(4):1250088. [8] Danca M F, Garrappa R. Suppressing chaos in discontinuous systems of fractional order by active control[J]. Applied Mathematics and Computation, 2015,257: 89-102. [9] Tacha O I, Volos C K, Stouboulos I N, et al. Analysis, adaptive control and circuit simulation of a novel finance system with dissaving[J]. Archives of Control Sciences, 2016, 26(1): 95-115. [10] Kuntanapreeda S. Adaptive control of fractional-order unified chaotic systems using a passivity-based control approach[J]. Nonlinear Dynamics, 2016:84(4):2505-2515. [11] Wang Q, Zhang J, Ding D, et al. Adaptive mittag-leffler stabilization of a class of fractional order uncertain nonlinear systems[J]. Asian Journal of Control, 2016, 18(6):2343-2351. [12] Arikoglu A, Ozkol I. Solution of fractional differential equations by using differential transform method[J]. Chaos, Solitons & Fractals, 2007,34(5):1473-1481. [13] Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[J]. Academic Press, 1999, 91(3):427-436. [14] Odibat Z, Momani S. A generalized differential transform method for linear partial differential equations of fractional order[J]. Applied Mathematics Letters, 2008,21(2):194-199. [15] Eckmann J P, Kamphorst S O, Ruelle D, et al. Liapunov exponents from time series[J]. Physical Review A, 1986,34(6): 4971-4979. [16] Takens F. Detecting Strange Attractors in Turbulence[M]//Smale S. Dynamical Systems and Turbulence. Berlin: Springer, 1981: 366-381. [17] Duarte-Mermoud M A, Aguila-Camacho N, Gallegos J A, et al. Using general quadratic Lyapunov functions to prove lyapunov uniform stability for fractional order systems[J]. Communications in Nonlinear Science and Numerical Simulation, 2015,22 (1): 650-659. [18] Li Y, Chen Y, Podlubny I. Mittag-leffler stability of fractional order nonlinear dynamic systems[J]. Automatica, 2009,45(8):1965-1969. [19] Li Y, Chen Y, Podlubny I. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized mittag-leffler stability[J]. Computers & Mathematics with Applications, 2010,59(5):1810-1821. [20] Caponetto R, Fazzino S. A semi-analytical method for the computation of the lyapunov exponents of fractional-order systems[J]. Communications in Nonlinear Science and Numerical Simulation, 2013,18(1): 22-27. [21] Frederickson P, Kaplan J L, Yorke E D, et al. The Liapunov dimension of strange attractors[J]. Journal of Differential Equations,1983, 49(2):185-207. |
[1] | 顾万里,王萍,胡云峰,蔡硕,陈虹. 具有H∞性能的轮式移动机器人非线性控制器设计[J]. 吉林大学学报(工学版), 2018, 48(6): 1811-1819. |
[2] | 李战东,陶建国,罗阳,孙浩,丁亮,邓宗全. 核电水池推力附着机器人系统设计[J]. 吉林大学学报(工学版), 2018, 48(6): 1820-1826. |
[3] | 赵爽,沈继红,张刘,赵晗,陈柯帆. 微细电火花加工表面粗糙度快速高斯评定[J]. 吉林大学学报(工学版), 2018, 48(6): 1838-1843. |
[4] | 王德军, 魏薇郦, 鲍亚新. 考虑侧风干扰的电子稳定控制系统执行器故障诊断[J]. 吉林大学学报(工学版), 2018, 48(5): 1548-1555. |
[5] | 闫冬梅, 钟辉, 任丽莉, 王若琳, 李红梅. 具有区间时变时滞的线性系统稳定性分析[J]. 吉林大学学报(工学版), 2018, 48(5): 1556-1562. |
[6] | 董飒, 刘大有, 欧阳若川, 朱允刚, 李丽娜. 引入二阶马尔可夫假设的逻辑回归异质性网络分类方法[J]. 吉林大学学报(工学版), 2018, 48(5): 1571-1577. |
[7] | 顾海军, 田雅倩, 崔莹. 基于行为语言的智能交互代理[J]. 吉林大学学报(工学版), 2018, 48(5): 1578-1585. |
[8] | 张茹斌, 占礼葵, 彭伟, 孙少明, 刘骏富, 任雷. 心肺功能评估训练系统的恒功率控制[J]. 吉林大学学报(工学版), 2018, 48(4): 1184-1190. |
[9] | 董惠娟, 于震, 樊继壮. 基于激光测振仪的非轴对称超声驻波声场的识别[J]. 吉林大学学报(工学版), 2018, 48(4): 1191-1198. |
[10] | 王旭, 欧阳继红, 陈桂芬. 基于垂直维序列动态时间规整方法的图相似度度量[J]. 吉林大学学报(工学版), 2018, 48(4): 1199-1205. |
[11] | 张浩, 占萌苹, 郭刘香, 李誌, 刘元宁, 张春鹤, 常浩武, 王志强. 基于高通量数据的人体外源性植物miRNA跨界调控建模[J]. 吉林大学学报(工学版), 2018, 48(4): 1206-1213. |
[12] | 田彦涛, 张宇, 王晓玉, 陈华. 基于平方根无迹卡尔曼滤波算法的电动汽车质心侧偏角估计[J]. 吉林大学学报(工学版), 2018, 48(3): 845-852. |
[13] | 张士涛, 张葆, 李贤涛, 王正玺, 田大鹏. 基于零相差轨迹控制方法提升快速反射镜性能[J]. 吉林大学学报(工学版), 2018, 48(3): 853-858. |
[14] | 黄岚, 纪林影, 姚刚, 翟睿峰, 白天. 面向误诊提示的疾病-症状语义网构建[J]. 吉林大学学报(工学版), 2018, 48(3): 859-865. |
[15] | 李雄飞, 冯婷婷, 骆实, 张小利. 基于递归神经网络的自动作曲算法[J]. 吉林大学学报(工学版), 2018, 48(3): 866-873. |
|