Technical Stability of Nonlinear Time-Varying Systems With Small Parameters

CHU Tian-guang; WANG Zhao-lin

Volume 21 Issue 11

Turn off MathJax

Article Contents

Abstract

References

Article Navigation > Applied Mathematics and Mechanics > 2000 > 21(11): 1140-1146

CHU Tian-guang, WANG Zhao-lin. Technical Stability of Nonlinear Time-Varying Systems With Small Parameters[J]. Applied Mathematics and Mechanics, 2000, 21(11): 1140-1146.

Citation:

CHU Tian-guang, WANG Zhao-lin. Technical Stability of Nonlinear Time-Varying Systems With Small Parameters[J]. Applied Mathematics and Mechanics, 2000, 21(11): 1140-1146.

Citation:

CHU Tian-guang, WANG Zhao-lin. Technical Stability of Nonlinear Time-Varying Systems With Small Parameters[J]. Applied Mathematics and Mechanics, 2000, 21(11): 1140-1146.

PDF( 291 KB)

Technical Stability of Nonlinear Time-Varying Systems With Small Parameters

CHU Tian-guang¹,
WANG Zhao-lin²

1.
Department of Mechanics and Engineering Science, Peking University, Beijing 100871, P R China;
2.
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P R China

Received Date: 1999-08-03
Rev Recd Date: 2000-05-25
Publish Date: 2000-11-15

Abstract

Abstract

Technical stability allowing quantitative estimation of trajectory behavior of a dynamical system over a given time interval was considered.Based on a differential comparison principle and a basic monotonicity condition,technical stability relative to certain prescribed state constraint sets of a class of nonlinear time-varying systems with small parameters was analyzed by means of vector Liapunov function method.Explicit criteria of technical stability are established in terms of coefficients of the system under consideration.Conditions under which the technical stability of the system can be derived from its reduced linear time-varying(LTV) system were further examined,as well as a condition for linearization approach to technical stability of general nonlinear systems.Also,a simple algebraic condition of exponential asymptotic stability of LTV systems is presented.Two illustrative examples are given to demonstrate the availability of the presently proposed method.
- nonlinear time-varying system,
- small parameter,
- technical stability,
- vector comparison principle,
- reduced system,
- linearization technique,
- exponential asymptotic stability

FullText(HTML)

References(11)

References

[1]	LaSalle J P,Lefschetz S.Stability by Lia punov'sDir ect Method with Applications[M].New York:Academic Press,1961.
[2]	Michel A N.Quantitative analysis of systems:stability,boundedness and trajectory behavior[J].Arch Rat Mech Anal,1970,38(2):107-122.
[3]	Martynyuk A A.Technical Stability in Dynamics[M].Kiev:Technika,1973.(in Russian)
[4]	王照林.运动稳定性及其应用[M].北京:高等教育出版社,1992.
[5]	Lakshmikantham V,Leela S,Martynyuk A A.Practical Stability of Nonlinear Systems[M].Singapore:World Scientific,1990.
[6]	Vukobratovic M,Stokic D.Applied Control of Manipulation Robots[M].Berlin:Springer-Verlag,1987.
[7]	Chen Y H.On the robustness of mismatched uncertain dynamical systems[J].ASME Tr an s J Dynam ic Systems,Measurement,Con tr ol,1987,109(3):29-35.
[8]	Skowronski J M.Parameter and state identification in non-linearizable uncertain systems[J].Int J Non-Linear Mech,1984,19(5):421-429.
[9]	王照林,楚天广.向量V函数方法与时变非线性大系统的轨线性态[J].非线性动力学学报,1994,1(3):209-213.
[10]	Siljak D D.Large Scale Dynamic Systems,Stability and Structure[M].New York:Elsevier North-Holland,Inc,1978.
[11]	Mori T,Fukuma N,Kuwahara M.A stability criterion for linear time-varying systems[J].Int J Control,1981,34(3):585-591.

Relative Articles

[1]	YAN Rui, LIU Guirong, LI Xiaocui. Nonlinear Stability of Traveling Wavefronts for a Discrete Cooperative Lotka-Volterra System With Delays[J]. Applied Mathematics and Mechanics, 2023, 44(4): 461-470. doi: 10.21656/1000-0887.430172
[2]	ZHAI Guoqing, CHEN Qiaoyu, TONG Dongbing, ZHOU Wuneng. Fixed-Time Asymptotic Stability and Energy Consumption Estimation of Nonlinear Systems[J]. Applied Mathematics and Mechanics, 2023, 44(10): 1180-1186. doi: 10.21656/1000-0887.440041
[3]	TIAN Hongqiao, ZHANG Zhixin, JIANG Wei. Finite-Time Stability of Fractional-Oder Linear Differential Systems With Delays[J]. Applied Mathematics and Mechanics, 2020, 41(8): 921-930. doi: 10.21656/1000-0887.400365
[4]	V. Filipovic. Global Exponential Stability of Switched Systems[J]. Applied Mathematics and Mechanics, 2011, 32(9): 1118-1126. doi: 10.3879/j.issn.1000-0887.2011.09.011
[5]	V. Čović, M. Vesković, D. Djuri, A. Obradović. On the Stability of Equilibria of Nonholonomic Systems With Nonlinear Constraints[J]. Applied Mathematics and Mechanics, 2010, 31(6): 722-723. doi: 10.3879/j.issn.1000-0887.2010.06.009
[6]	SU Chun-hua, LIU Si-feng. p-Moment Exponential Robust Stability for Stochastic Systems With Distributed Delays and Interval Parameters[J]. Applied Mathematics and Mechanics, 2009, 30(7): 856-864. doi: 10.3879/j.issn.1000-0887.2009.07.010
[7]	NI Ming-kang, LIN Wu-zhong. Minimizing Sequence of Variational Problems With Small Parameters[J]. Applied Mathematics and Mechanics, 2009, 30(6): 648-654. doi: 10.3879/j.issn.1000-0887.2009.06.003
[8]	RONG Hai-wu, WANG Xiang-dong, MENG Guang, XU Wei, FANG Tong. On Double Peak Probability Density Functions of a Duffing Oscillator to Combined Deterministic and Random Excitations[J]. Applied Mathematics and Mechanics, 2006, 27(11): 1373-1379.
[9]	ZHENG Hui-ping, XUE Yu-sheng, CHEN Yu-shu. Quantitative Methodology for the Stability Analysis of Nonlinear Rotor Systems[J]. Applied Mathematics and Mechanics, 2005, 26(9): 1038-1044.
[10]	SUN JI-tao, ZHANG Yin-ping, LIU Yong-qing, DENG Fei-qi. Exponential Stability of Interval Dynamical System With Multidelay[J]. Applied Mathematics and Mechanics, 2002, 23(1): 87-91.
[11]	LIU Jun. Research of the Periodic Motion and Stability of Two-Degree-of-Freedom Nonlinear Oscillating Systems[J]. Applied Mathematics and Mechanics, 2002, 23(10): 1093-1100.
[12]	CONG Yu-hao. NGP_G-Stability of Linear Multistep Methods for Systems of Generalized Neutral Delay Differential Equations[J]. Applied Mathematics and Mechanics, 2001, 22(7): 735-742.
[13]	Li Junfeng, Wang Zhaolin. Stability of Non-Conservative Linear Gyroscopic Systems[J]. Applied Mathematics and Mechanics, 1996, 17(12): 1107-1111.
[14]	Chen Daduan, Liu Xiaoming, Shi Weihui. On the Stability of Forced Dissipative Nonlinear System[J]. Applied Mathematics and Mechanics, 1996, 17(6): 515-522.
[15]	Hu Chao-yuang. The Asymptotic Stability of the Linear Discrete Large Scale Systems[J]. Applied Mathematics and Mechanics, 1993, 14(1): 89-94.
[16]	Shu Huang. Stability of Nonlinear Comparison Equations for Discrete Large-Scale Systems[J]. Applied Mathematics and Mechanics, 1990, 11(8): 731-737.
[17]	Zhao Jun-san. A Criterion for the Stability of Motion of Nonlinear Systems[J]. Applied Mathematics and Mechanics, 1988, 9(11): 1033-1036.
[18]	Liu Lian-sheng, Huo Quan-zhong, Huang Ke-lei. A Method of Finding the Principal Modes of Nonlinear Vibration Systems and Their Stabilities[J]. Applied Mathematics and Mechanics, 1987, 8(6): 505-512.
[19]	Chen Xiao-lin, Huang Ling. The Relationship between the Stability and the Optimality of Linear Systems (Ⅱ)[J]. Applied Mathematics and Mechanics, 1987, 8(3): 253-257.
[20]	Zhang Shu-shun. The Theorem of the Stability of Linear Nonautonomous ystems under the Frequently-Acting Perturbation[J]. Applied Mathematics and Mechanics, 1986, 7(2): 169-171.