Finite-Time Stabilization of Uncertain Non-Autonomous Chaotic Gyroscopes With Nonlinear Inputs

Mohammad Pourmahmood Aghababa; Hasan Pourmahmood Aghababa

doi:10.3879/j.issn.1000-0887.2012.02.002

Volume 33 Issue 2

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Mohammad Pourmahmood Aghababa, Hasan Pourmahmood Aghababa. Finite-Time Stabilization of Uncertain Non-Autonomous Chaotic Gyroscopes With Nonlinear Inputs[J]. Applied Mathematics and Mechanics, 2012, 33(2): 153-163. doi: 10.3879/j.issn.1000-0887.2012.02.002

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Finite-Time Stabilization of Uncertain Non-Autonomous Chaotic Gyroscopes With Nonlinear Inputs

doi: 10.3879/j.issn.1000-0887.2012.02.002

1.
¹Electrical Engineering Department, Urmia University of Technology, Urmia, Iran;²Department of Mathematics, University of Tabriz, Tabriz, Iran;³Research Center for Industrial Mathematics of University of Tabriz, Tabriz, Iran

Received Date: 2011-01-04
Rev Recd Date: 2011-10-31
Publish Date: 2012-02-15

Abstract

Abstract

Gyroscopes were one of the most interesting and everlasting onlinear non-autonomous dynamical systems that exhibited very complex dynamical behavior such as chaos. The problem of robust stabilization of the nonlinear non-autonomous gyroscopes in a given finite time was studied. It was assumed that the gyroscope system was perturbed by model uncertainties, external disturbances and unknown parameters. Besides, the effects of input nonlinearities were taken into account. Appropriate adaptive laws were proposed to tackle the unknown parameters. Based on the adaptive laws and the finite-time control theory, discontinuous finite-time control laws were proposed to ensure the finite-time stability of the system. The finite-time stability and convergence of the closed-loop system are analytically proved. Some numerical simulations are presented to show the efficiency of the proposed finite-time control scheme and to validate the theoretical results.
- non-autonomous chaotic gyroscope,
- finite-time control,
- uncertainty,
- unknown parameter,
- nonlinear input

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References(23)

References

[1]	Ott E, Grebogi C, Yorke J A. Using chaos to direct trajectories to targets[J]. Phys Rev Lett, 1990, 65(26): 3215-2158.
[2]	胡满峰，徐振源. 两个模态耦合的Ginzburg-Landau方程的时空混沌同步化[J]. 应用数学和力学, 27(8): 1001-1008. (HU Man-feng, XU Zhen-yuan. Patio-temporal chaotic synchronization for modes coupled two Ginzburg-Landau equations[J]. Applied Mathematics and Mechanics(English Edition), 2006, 27(8): 1149-1156.)
[3]	阿衡 C K. 基于混沌同步化的广义无源性[J]. 应用数学和力学, 2010, 31(8): 961-970. (Ahn C K. Generalized passivity-based chaos synchronization[J]. Applied Mathematics and Mechanics(English Edition), 2010, 31(8): 1009-1018.)
[4]	刘艳, 吕翎. N个异结构混沌系统的环链耦合同步[J]. 应用数学和力学, 2008, 29(10): 1181-1190.(LIU Yan, L Ling. Synchronization of N different coupled chaotic systems with ring and chain connections[J]. Applied Mathematics and Mechanics(English Edition), 2008, 29(10): 1299-1308.)
[5]	Pourmahmood M, Khanmohammadi S, Alizadeh G. Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller[J]. Commun Nonlinear Sci Numer Simulat, 2011, 16(7): 2853-2868.
[6]	Aghababa M P, Khanmohammadi S, Alizadeh G. Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique[J]. Appl Math Model, 2001, 30(6): 3080-3091.
[7]	唐林俊, 李东, 王汉兴. 基于模糊观测器的模糊混沌系统的延迟同步[J]. 应用数学和力学, 2009, 30(6): 750-756. (TANG Lin-jun, LI Dong, WANG Han-xing. Lag synchronization for fuzzy chaotic system based on fuzzy observer[J]. Applied Mathematics and Mechanics(English Edition), 2009, 30(6): 803-810.)
[8]	勃布特索夫 A，尼克拉伊夫 N，斯利塔O. 卫星系统中的混沌控制[J].应用数学和力学, 2007, 28(7): 798-804.(Bobtsov A, Nikolaev N, Slita O. Control of chaotic oscillations of a satellite[J]. Applied Mathematics and Mechanics(English Edition), 2007, 28(7): 893-900.)
[9]	Chen H K. Chaos and chaos synchronization of a symmetric gyro with linear-plus-cubic damping[J]. J Sound Vibr, 2002, 255(4): 719-740.
[10]	van Dooren R, CHEN Hsien-keng. Comments on chaos and chaos synchronization of a symmetric gyro with linear-plus-cubic damping[J]. J Sound Vibr, 2008, 268(3): 632-634.
[11]	Ge Z M, Chen H K. Bifurcations and chaos in a rate gyro with harmonic excitation[J]. J Sound Vibr, 1996, 194(1):107-117.
[12]	Tong X, Mrad N. Chaotic motion of a symmetric gyro subjected to a harmonic base excitation[J]. J Applied Mech Trans Amer Soc Mech Eng, 2001, 68(4): 681-684.
[13]	Leipnik R B, Newton T A. Double strange attractors in rigid body motion with linear feedback control[J]. Phys Lett A, 1981, 86(2): 63-67.
[14]	Ge Z M, Chen H K, Chen H H. The regular and chaotic motion of a symmetric heavy gyroscope with harmonic excitation[J]. J Sound Vibr, 1996, 198(2): 131-147.
[15]	Lei Y, Xu W, Zheng H. Synchronization of two chaotic nonlinear gyros using active control[J]. Phys Lett A, 2005, 343(1/3): 153-158.
[16]	Hung M, Yan J, Liao T. Generalized projective synchronization of chaotic nonlinear gyros coupled with dead-zone input[J]. Chaos Soliton Fract, 35(1): 181-187.
[17]	Yau H. Nonlinear rule-based controller for chaos synchronization of two gyros with linear-plus-cubic damping[J]. Chaos Soliton Fract, 2007, 34(4): 1357-1365.
[18]	Yau H. Chaos synchronization of two uncertain chaotic nonlinear gyros using fuzzy sliding mode control[J]. Mech Sys Signal Proc, 2008, 22(2): 408-418.
[19]	YAN Jun-jun, HUANG Meei-ling, LIAO Teh-lu. Adaptive sliding mode control for synchronization of chaotic gyros with fully unknown parameters[J]. J Sound Vibr, 2006, 298(1/2): 298-306.
[20]	YAN Jun-jun, HUANG Meei-ling, LINB Jui-Sheng , LIAO Teh-lu . Controlling chaos of a chaotic nonlinear gyro using variable structure control[J]. Mech Sys Signal Proc, 2007, 21(6): 2515-2522.
[21]	Bwidehat S P, Bernstein D S. Finite-time stability of continuous autonomous systems[J]. SIAM J Control Optim, 2008, 38(3): 751-766.
[22]	Curran P F, Chua L O. Absolute stability theory and the synchronization problem[J]. Int J Bifurcat Chaos, 1997, 7(6): 1357-1382.
[23]	WANG Hua, HAN Zheng-zhi, XIE Qi-yue, ZHANG Wei. Finite-time chaos control via nonsingular terminal sliding mode control[J]. Commun Nonlinear Sci Numer Simulat, 2009, 14(6): 2728-2733.