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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
LaSalle J P, Lefschetz S.Stability by Liapunov's Direct Method With Applications[M]. New York: Academic Press, 1961.
Michel A N. Quantitative analysis of systems: stability, boundedness and trajectory behavior[J].Arch Rat Mech Anal 1970,38(2):107–122.
Martynyuk A A.Technical Stability in Dynamics [M]. Kiev: Technika, 1973 (in Russian)
WANG Zhao-lin.Stability of Motion and Its Applications [M]. Beijing: Higher Education Publishing House, 1992. (in Chinese)
Lakshmikantham V, Leela S, Martynyuk A A.Practical Stability of Nonlinear Systems [M]. Singapore: World Scientific, 1990.
Vukobratovic M, Stokic D.Applied Control of Manipulation Robots [M]. Berlin: Springer-Verlag, 1987.
Chen Y H. On the robustness of mismatched uncertain dynamical systems[J]ASME Trans J Dynamic Systems, Measurement, Control, 1987,109(3):29–35.
Skowronski J M. Parameter and state identification in non-linearizable uncertain systems[J].Int J Non-Linear Mech, 1984,19(5):421–429.
WANG Zhao-lin, CHU Tian-guang. The method of vectorV-function and the trajectory behavior of time-varying nonlinear large scale system[J].J Nonlinear Dynamics in Science and Technology, 1994,1(3):209–213. (in Chinese)
Siljak D D.Large Scale Dynamic Systems, Stability and Structure [M]. New York: Elsevier North-Holland, Inc, 1978.
Mori T, Fukuma N, Kuwahara M. A stability criterion for linear time-varying systems[J].Int J Control, 1981,34(3):585–591.
Author information
Authors and Affiliations
Additional information
Communicated by YE Qing-kai
Foundation item: the National Natural Science Foundation of China (19872005); the Research Fund for the Doctoral Program of Higher Education (97000130); Foundation of Chinese Academy of Space Technology
Biographies: CHU Tian-guang(1964-) WANG Zhao-lin(1928-)
Rights and permissions
About this article
Cite this article
Tian-guang, C., Zhao-lin, W. Technical stability of nonlinear time-varying systems with small parameters. Appl Math Mech 21, 1264–1271 (2000). https://doi.org/10.1007/BF02459247
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02459247
Key words
- nonlinear time-varying system
- small parameter
- technical stability
- vector comparison principle
- reduced system
- linearization technique
- exponential asymptotic stability