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Generalized Morse theory and its applications to control and stability analysis

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Abstract

This paper extends Morse theory to noncompact manifolds which is important since in many engineering applications the manifolds involved are usually noncompact. To demonstrate the application, generalized Morse theory is used to estimate the number of unstable equilibria on the stability boundary. The engineering significance of the estimation is explained in the paper.

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References

  1. J. Zaborszky, G. Huang, B. Zheng, and T. C. Leung, New Results for Stability Monitoring of the Large Electric Power System: the Phase Portrait of the Power System,IFAC Proceedings, Peking, pp. 311–313, August 1986.

  2. H. D. Chiang, F. F. Wu, and P. Varaiya, Foundations of Direct Methods for Power System Transient Stability Analysis,IEEE Trans. Circuits and Systems, Vol. 34, No. 2, pp. 160–173, February 1987.

    Google Scholar 

  3. L. A. Luxemburg, Structural Stability Analysis and Its Applications to Power Systems, Ph.D. dissertation, Department of Electrical Engineering, Texas A&M University, Dec., 1987.

  4. L. A. Luxemburg and G. Huang, On the Number of Unstable Equilibrium Points for a Class of Nonlinear Systems,Proc. IEEE 26th CDC, Los Angeles, CA, 1987.

  5. J. Baillieul and C. I. Byrnes, Geometric Critical Point Analysis of Lossless Power System Models,IEEE Trans. Circuits and Systems, Vol. 29, No. 11, pp. 723–727, November 1982.

    Google Scholar 

  6. J. Milnor,Morse Theory, Annals of Mathematical Studies, Princeton University Press, Princeton, 1963.

    Google Scholar 

  7. S. Smale,The Mathematics of Time, Springer-Verlag, New York, 1980.

    Google Scholar 

  8. H. Yee and B. D. Spalding, Transient Stability Analysis of Multimachine Power Systems by the Method of Hyperplanes,IEEE Trans. PAS, Vol. 96, pp. 276–284, January–February, 1977.

    Google Scholar 

  9. M. Ribben-Pavella and F. J. Evans, Direct Methods for Studying Dynamics of Large-Scale Electric Power Systems-A Survey,Automatica, Vol. 21, No. 1, pp. 1–21, January 1985.

    Google Scholar 

  10. M. A. Pai,Power System Stability, Control Series, North-Holland, Amsterdam, 1981.

    Google Scholar 

  11. A. N. Michel, R. K. Miller, and D. H. Nam, Stability Analysis of Interconnected Systems Using Computer Generated Lyapunov Functions,IEEE Trans. Circuits and Systems, Vol. 29, No. 7, pp. 431–440, July 1982.

    Google Scholar 

  12. A. Arapostathis, S. S. Sastry, and P. Varaiya, Global Analysis of Swing Dynamics,IEEE Trans. Circuits and Systems, Vol. 29, No. 10, pp. 673–678, 1982.

    Google Scholar 

  13. P. Varaiya, F. F. Wu, and R.-L. Chen, Direct Methods for Transient Stability Analysis of Power Systems: Recent Results,Proc. IEEE, Vol. 73, No. 12, pp. 1703–1715. 1985.

    Google Scholar 

  14. A. Vannelli and M. Vidyasagar, Maximal Lyapunov Functions and Domains of Attraction for Autonomous Nonlinear Systems,Automatica, Vol. 21, No. 1, pp. 69–75, 1985.

    Google Scholar 

  15. N. O. Kakimoto, Y. Ohaswa, and M. Hayashi, Transient Stability Analysis of Electric Power System via Lur'e Type Lyapunov Function with Effect of Transfer Conductances,Trans. IEEE Japan, Vol. 98, pp. 141–150, 1978.

    Google Scholar 

  16. R. Genesio, M. Tartaglia, and A. Vicino, On the Estimation of Asymptotic Stability Regions: State of Art and New Proposals,IEEE Trans. Automat. Control, Vol. 30, No. 8, pp. 747–755, 1985.

    Google Scholar 

  17. J. L. Willems and J. C. Willems, The Application of Lyapunov Methods to the Computation of Transient Stability Regions for Multimachine Power System,IEEE Trans. PAS, Vol. 89, pp. 795–801, 1970.

    Google Scholar 

  18. A. A. Fouad, V. Vittal, and T. K. Oh, Critical Energy for Transient Stability Assessment of a Multimachine Power System,IEEE Trans. PAS, Vol. 103, pp. 2199–2206, 1984.

    Google Scholar 

  19. L. Jocie, Transient Stability of Dissipative Power System,Proc. 9th World IFAC Congress, Vol. 1, pp. 26–30, 1984.

    Google Scholar 

  20. H. Miyagi and A. Bergen, Stability Studies of Multimachine Power Systems with the Effects of Automatic Voltage Regulators,IEEE Trans. Automat. Control, Vol. 31, No. 3, pp. 210–215, 1986.

    Google Scholar 

  21. H. G. Kwatny, L. Y. Bahar, and A. K. Parsija, Energy-Like Lyapunov Functions for Power System Stability Analysis,IEEE Trans. Circuits and Systems, Vol. 32, No. 11, pp. 1140–1149, 1985.

    Google Scholar 

  22. F. S. Prabhakara and A. H. El-Abiad, A Simplified Determination of Transient Stability Region for Lyapunov Method,IEEE Trans. PAS, Vol. 94, pp. 672–689, 1975.

    Google Scholar 

  23. T. Athay, R. Podmore, and S. Virmani, A Practical Method for the Direct Analysis of Transient Stability,IEEE Trans. PAS, Vol. 98, pp. 573–584, 1979.

    Google Scholar 

  24. J. Zaborsky, G. Huang, B. H. Zheng, and T. C. Leung, Phase Portrait of a Class of Nonlinear Systems such as Power Systems,IEEE Trans. Automat. Control, Vol. 33, No. 1, pp. 4–15, 1988.

    Google Scholar 

  25. . Zaborszky, G. Huang, B. H. Zheng, and T. C. Leung, Fast Stability Monitoring of Large Power System,Proc. IEEE Control and Decision Conference, pp. 500–02, 1987.

  26. J. Palis and W. De Melo,Geometric Theory of Dynamical Systems, Springer-Verlag, New York, 1980.

    Google Scholar 

  27. E. Spanier,Algebraic Topology, McGraw-Hill, New York, 1966.

    Google Scholar 

  28. K. Borsuk,Theory of Retracts, PWN, Warszawa, 1967.

    Google Scholar 

  29. K. Kuratowski,Topology, Academic Press, New York, 1966.

    Google Scholar 

  30. Walter Rudin,Principles of Mathematical Analysis, McGraw-Hill, New York, 1964.

    Google Scholar 

  31. M. Peixoto, On an Approximation Theorem of Kupka-Smale,J. Differential Equations, Vol. 3, pp. 214–227, 1967.

    Google Scholar 

  32. J. Zaborszky and G. Huang, On-Line Detection of System Instabilities, Final Report to Department of Energy, Nov., 1987.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Rice University, 77251-1892, Houston, Texas, USA

    L. A. Luxemburg

  2. Department of Electrical Engineering, Texas A&M University, 77843, College Station, Texas, USA

    G. Huang

Authors
  1. L. A. Luxemburg
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  2. G. Huang
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Additional information

This work was supported in part by NSF ECS-8307028, ECS-8421105, ECS-8900499 and Texas Advanced Research Program No 4659.

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Luxemburg, L.A., Huang, G. Generalized Morse theory and its applications to control and stability analysis. Circuits Systems and Signal Process 10, 175–209 (1991). https://doi.org/10.1007/BF01183770

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  • DOI: https://doi.org/10.1007/BF01183770

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