Abstract
This article studies dynamical systems under perturbations. We prove an Equilibrium Equivalence Theorem that guarantees that the dynamics of the system remains unchanged under perturbations with certain fairly general assumptions. We also prove that a power system is robust with respect to parameter changes in generic situations. Concepts of equilibrium equivalence and equilibrium equivalence structural stability are developed and are applied to studies of bifurcations of vector fields on noncompact manifolds. A constructive approach to equilibrium equivalence structural stability verification is emphasized. General results on structural stability of vector fields on differential manifolds are established and important applications of this theory to stability analysis are considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Luxemburg and G. Huang, On the number of unstable equilibrium points for a class of nonlinear systems, inIEEE 26th CDC (Los Angeles, CA) 1987.
G. Huang and L. Luxemburg, On the lower bound of the number of (n−1)st order equilibrium points on the stability boundary, inMTNS conference (Phoenix, AZ) June, 1987.
L. Luxemburg and G. Huang, “Characterization of equilibrium points on the stability boundary via algebraic topology approach,” inAnalysis and Control of Nonlinear Systems (C. Byrnes, C. Martin, and R. Saeks, eds.), pp. 71–76, Amsterdam: North-Holland, 1988.
L. Luxemburg and G. Huang, Generalized Morse Theory and its applications to power systems,CSSP Journal, Vol. 10, No. 2, pp. 175–209, 1991.
J. Zaborszky, G. Huang, B. Zheng, and T. Leung, New Results for stability monitoring of the large electric power systems: The phase portrait of the power system, inIFAC Proceedings (Peking), August 1986.
H. Chiang, F. Wu, and P. Varaiya, Foundations of direct methods for power system transient stability analysis,IEEE Trans. on Circuits and Systems, Vol. CAS-34, No. 2, pp. 160–173, Feb. 1987.
H. Yee and B. Spalding, Transient stability analysis of multimachine power systems by the method of hyperplanes,IEEE Trans. on PAS, Vol. 96, pp. 276–284, Jan. Feb. 1977.
A. Arapostathis, Sastry, and P. Varaiya, Global analysis of swing dynamics,IEEE Trans. on Circuits and Systems, Vol. 29, No. 10, pp. 673–678, 1982.
S. Ushiki, Analytical expression of unstable manifold,Proc. Japan Acad., Vol. 56, Ser. A, p. 259, 1980.
L. Luxemburg, Structural stability analysis and it applications to power systems, Ph.D. Thesis, Electrical Engineering Dept., Texas A & M University, Dec. 1987.
T. Palis, Jr., and W. de Melo,Geometric Theory of Dynamical Systems, New York: Springer-Verlag, 1982.
M. Peixoto, On approximation theorem of Kupka-Smale,Journal of Differential Equations, Vol. 3, pp. 214–227, 1967.
S. Smale,The Mathematics of Time, New York: Springer-Verlag, 1980.
H.D. Chiang, F. Wu, and P. Varaiya, Foundations of the potential energy boundary surface method for power systems transient stability analysis,IEEE Trans. on Circuits and Systems, Vol. 35, No. 6, pp. 712–728, June 1988.
M.A. Pai,Energy Function Analysis for Power Systems Stability, New York: Kluwer, 1989.
P.A. Cook and A.M. Eskicioglu, Transient stability analysis of electric power systems by the method of tangent hypersurfaces,Proc. of IEEE, Part C, pp. 183–193, July 1983.
H.D. Chiang and J.S. Thorp, Stability region of nonlinear dynamical systems: A constructive methodology,IEEE Trans. on Automatic Control, Vol. 34, No. 12, Dec. 1989, pp. 1229–1241.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Luxemburg, L.A., Huang, G.M. Equilibrium Equivalence Theorem and its applications to control and stability analysis. Circuits Systems and Signal Process 14, 111–134 (1995). https://doi.org/10.1007/BF01183751
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01183751