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Stability analysis of complex dynamical systems

Some computational methods

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Abstract

We present two new algorithms to estimate the domain of attraction of the equilibriumx=0 of a nonlinear systemx=f(x). One of these algorithms utilizes quadratic Lyapunov functions while the second algorithm makes use of norm Lyapunov functions. Both of these procedures yield estimates for the domain of attraction which are comparable to those obtained by existing methods; however, the present algorithms appear to be significantly more efficient than existing algorithms.

We also show how sometimes the applicability of the above results can be extended to high order systems by invoking the comparison principle. In doing so, we establish some results for the comparison principle which are of interest in their own right. Specifically, we relate the domain of attraction of a low order comparison system to the domain of attraction of a higher order system and we give an interpretation of the comparison principle in terms of stability preserving maps.

In order to demonstrate the applicability of the present results, and in order to compare the present results with existing results, several specific examples are presented.

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References

  1. E. J. Davison and E. M. Kurak, “A Computational Method for Determining Quadratic Lyapunov Functions for Nonlinear Systems,”Automatica, Vol. 17, 1971, pp. 627–636.

    Google Scholar 

  2. R. K. Brayton and C. H. Tong, “Stability of Dynamical Systems: A Constructive Approach,”IEEE Transactions on Circuits and Systems, Vol. 26, April 1979, pp. 224–234.

    Google Scholar 

  3. R. K. Brayton and C. H. Tong, “Constructive Stability and Asymptotic Stability of Dynamical Systems,”IEEE Transactions on Circuits and Systems, Vol. 27, November 1980, pp. 1121–1130.

    Google Scholar 

  4. A. N. Michel and R. K. Miller,Qualitative Analysis of Large-Scale Dynamical Systems, Academic Press, New York, 1977.

    Google Scholar 

  5. D. N. Shields and C. Storey, “The Behavior of Optimal Lyapunov Functions,”International Journal of Control, Vol. 21, No. 4, 1975, pp. 561–573.

    Google Scholar 

  6. H. R. Schwarz, H. Rutishauser, and E. Stiefel,Numerical Analysis of Symmetric Matrices (Transl. by P. Hertelendy), Prentice-Hall, Englewood Cliffs, N.J., 1973.

    Google Scholar 

  7. H. H. Rosenbrock and C. Storey,Computational Techniques for Chemical Engineers, Pergamon Press, London, 1966.

    Google Scholar 

  8. J. J. Rodden, “Numerical Application of Lyapunov Stability Theory,”Proceedings of the 1964 JACC, pp. 261–268.

  9. H. H. Rosenbrock, “An Automatic Method for Finding the Greatest or Least Value of a Function,”Computer Journal, Vol. 3, 1960, pp. 175–184.

    Google Scholar 

  10. R. Subramanian and R. K. Bansal, “Estimation of Power System Stability Domains Using Quadratic Lyapunov Functions,”Proceedings of the IEE, Vol. 124, No. 7, July 1977, pp. 597–601.

    Google Scholar 

  11. W. Hahn,Stability of Motion, Springer-Verlag, New York, 1967.

    Google Scholar 

  12. J. LaSalle and S. Lefschetz,Stability by Lyapunov's Direct Method with Applications, Academic Press, New York, 1961.

    Google Scholar 

  13. M. A. Pai, “Stability of Large Scale Interconnected Power Systems,”Proceedings of the 1980 IEEE International Conference on Circuits and Computers, Port Chester, New York, October 1980, pp. 413–416.

    Google Scholar 

  14. Y. K. Chen and R. Schinzinger, “Lyapunov Stability of Multi-Machine Power Systems Using Decomposition — Aggregation Method,” Paper A-80–036–4, IEEE PES Winter Meeting, New York, 1980.

  15. J. Thomas, “Über die Invarianz der Stabilität be einem PhasenraumHomöomorphismus,”J. für reine und angewandte Mathematik, Vol. 213, 1964, pp. 147–150.

    Google Scholar 

  16. W. Hahn, “Über Stabilitätserhaltende Abbildungen und Ljapunovsche Funktionen,”J. für die reine und angewandte Mathematik, Vol. 228, 1967, pp.189–192.

    Google Scholar 

  17. A. N. Michel, R. K. Miller and W. Tang, “Lyapunov Stability of Interconnected Systems: Decomposition into Strongly Connected Subsystems,”IEEE Transactions on Circuits and Systems, Vol. 28, September 1978, pp. 799–809.

    Google Scholar 

  18. R. K. Bansal, “Estimation of Stability Domains for the Transient Stability Investigation of Power Systems,” Ph.D. Dissertation, I.I.T., Kanpur, India, 1975.

    Google Scholar 

  19. J. K. Kuester and J. H. Mize,Optimization Techniques with FORTRAN, McGraw-Hill Company, New York, 1973.

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, Iowa State University, 50011, Ames, Iowa, USA

    A. N. Michel & N. R. Sarabudla

  2. Department of Mathematics, Iowa State University, 50011, Ames, Iowa, USA

    R. K. Miller

Authors
  1. A. N. Michel
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  2. N. R. Sarabudla
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  3. R. K. Miller
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Additional information

Supported in part by the National Science Foundation under grants ENG77-28446 and ECS-81-00690.

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Michel, A.N., Sarabudla, N.R. & Miller, R.K. Stability analysis of complex dynamical systems. Circuits Systems and Signal Process 1, 171–202 (1982). https://doi.org/10.1007/BF01600051

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  • DOI: https://doi.org/10.1007/BF01600051

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