Abstract
We study the asymptotic stability of a singularly perturbed nonlinear time-invariant systemS ɛv , which has three vastly different time scales. The systemS ɛv is approximated by three simpler systems over different time intervals. We give a straightforward proof of the fact that the asymptotic stability ofS ɛv is guaranteed when the equilibrium points of the three simpler systems are exponentially stable and when the parametersɛ andν are sufficiently small.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. N. Tihonov, A. B. Vasiléva, and V. M. Volosov, Ordinary differential equations, inMathematics Applied to Physics (E. Roubine, ed.), Springer-Verlag, New York, 1970.
R. E. O'Malley, Jr.,Introduction to Singular Perturbations, Academic Press, New York, 1974.
P. V. Kokotovic, R. E. O'Malley, Jr., and P. Sannuti, Singular perturbations and order reduction in control theory—an overview,Automatica,12, 123–132, 1976.
V. R. Saksena, J. O. O'Reilly, and P. V. Kokotovic, Singular perturbations and timescale methods in control theory: survey 1976–1983,Automatica,20, 273–294.
F. Hoppensteadt, Singular perturbations on the infinite interval,Trans. Amer. Math. Soc.,123, 521–535, 1966.
F. Hoppensteadt, On systems of ordinary differential equations with several parameters multiplying the derivatives,J. Differential Equations,5, 106–116, 1969.
P. Habets, Stabilite asymptotique pour des problems de perturbations singulieres, inCentro Internazionale Matematico Estivo, Cremonese, Ed., Firenze, 1974.
J. H. Chow, Asymptotic stability of a class of nonlinear singularity perturbed systems,J. Franklin Inst.,305, 275–281, 1978.
L. T. Grujic, Uniform asymptotic stability of nonlinear singularly perturbed and large scale systems,Internat. J. Control,33, 481–504, 1981.
A. Saberi and H. Khalil, An initial value theorem for nonlinear singularly perturbed systems,Systems Control Lett.,4, 301–305, 1984.
A. Saberi and H. Khalil, Quadratic-type Lyapunov functions for singularly perturbed systems,IEEE Trans. Automat. Control,29, 542–550, 1984.
H. K. Khalil, Asymptotic stability of nonlinear multiparameter singularly perturbed systems,Automatica,17, 797–804, 1981.
M. Coderch, A. S. Willsky, S. S. Sastry, and D. A. Castanon, Hierarchical aggregation of linear systems with multiple time scales,IEEE Trans. Automat. Control,28, 1017–1030, 1983.
R. Silva-Madriz and S. S. Sastry, Multiple time scales for nonlinear systems,Circuit, Systems Signal Process.,5, 153–169, 1986.
R. Silva-Madriz and S. S. Sastry, Input-output description of linear systems with multiple time scales,Internat. J. Control,40, 699–721, 1984.
C. A. Desoer and M. J. Shensa, Networks with very small and very large parasitics: natural frequencies and stability,Proc. IEEE,58, 1933–1938, 1970.
J. Dieudonné,Foundations of Modern Analysis, Academic Press, New York, 1969.
W. M. Boothby,An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, New York, 1975.
W. Hahn,Stability of Motion, Springer-Verlag, New York, 1967.
Author information
Authors and Affiliations
Additional information
Research sponsored by the Joint Services Electronics Program, Contract Number F4962084-C-0057, and NASA, Grant NAG2-243.
Rights and permissions
About this article
Cite this article
Desoer, C.A., Shahruz, S.M. Stability of nonlinear systems with three time scales. Circuits Systems and Signal Process 5, 449–464 (1986). https://doi.org/10.1007/BF01599620
Issue Date:
DOI: https://doi.org/10.1007/BF01599620