Advertisement

Robust Stability Conditions of Quasipolynomials by Frequency Sweeping

  • Jie Chen
  • Silviu-Iulian Niculescu
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 38)

Abstract

In this chapter we study the robust stability independent of delay of some class of uncertain quasipolynomials, whose coefficients may vary in a certain prescribed range. OUr main contributions include frequency-sweeping conditions for interval, diamond and spherical quasipolynomial families. The correspoding results provide necessary and sufficient conditions, and are easy to check, requiring only the computation of two simple frequency-dependent functions. Various extensions (polytopic uncertainty, multivariate polynomials) are also presented.

Keywords

Robust Stability Frequency Domain Approach Polynomial Family Spherical Polynomial Robust Stability Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B.R. Barmish, New Toolsfor Robllstlless of Lillear Systems, New York, NY Macmillan, 1994.Google Scholar
  2. 2.
    K.D. Kim and N. K. Bose, “Vertex implications of stability for a class of delay-differential interval systems.” IEEE Trans. Circ. Syst., vol. 37.no. 7. pp. 969–972, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    S. P. Bhattacharyya, H. Chapellat. and L. H. Keel. Robust control. The parametric approach, Prentice Hall, 1995.Google Scholar
  4. 4.
    J. Chen, “On computing the maximal delay intervals for stability of linear delay systems.” IEEE Trans. Automat. COntr., vol. 40, pp. 1087–1093, 1995.zbMATHCrossRefGoogle Scholar
  5. 5.
    J. J. Chen, G. Gu, and C. N. Nett, “A new method for computing delay margins for stability of linear delay systems.’” Syst. & COntr. Lett., vol. 26, pp. 101–117, 1995.MathSciNetGoogle Scholar
  6. 6.
    J. Chen, and H. A. Latchman, “Frequency sweeping tests for stability independent of delay”, IEEE TranS. Automat. COntr., vol. 40, pp. 1640–1645, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    J. Chen, M.K.H. Fan, and C.N. Nett. “Structured singular values and stability analysis of uncertain polynomials, part 1: the generalized µ,” Syst. & COntr. Lett., vol. 23, pp. 53–65, 1994MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    J. Chen, M.K.H. Fan, and C.N. Nett, “Structured singular values and stability analysis of uncertain polynomials. part 2: a missing link.” Syst. & Contr. Lett., vol. 23, pp. 97–109, 1994.MathSciNetCrossRefGoogle Scholar
  9. 9.
    J. Chen and S.-I. Niculescu. “Robust stability of quasipolynomials: Frequency-sweeping conditions and vertex tests,” Internal Note HeuDillSyC’03, March 2003.Google Scholar
  10. 10.
    J. Chen and A.L. Tits. “Robust control analysis,” in Ellcyclopedia of Electrical and Electronics Engineering. vol. 18, pp. 602–616, J.G.Webster, Ed., Wiley, 1998Google Scholar
  11. 11.
    M. Fu, A. W. Olbrot, and M. P. Polis, “Robust stability for time-delay systems: The edge theorem and graphical tests,” IEEE Trans. Automat. Contr., vol. 34, pp. 813–820, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    K. Gu, V. L. Khantonov, and J. Chen, Stability and robust stability of time-delay systems, Birkhauser Boston. 2003CrossRefGoogle Scholar
  13. 13.
    J.K. Hale, E.F. Infante, and F.S.P. Tsen. “Stability in linear delay equations,” J. Math. Anal. Appl., vo1.105, pp. 533–555, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    V. L. Khantonov, and L. Atanassova. “On stability of wheighted diamond of real quasipolynomials,” IEEE Trans. Automat. Contr., vol. 42, pp. 831–835, 1997.CrossRefGoogle Scholar
  15. 15.
    V. L. Kharitonov, and A. Zhabko, “Robust stability of time delay systems,” in IEEE Trans. Automat. Contr., vol. 39, pp. 2388–2397, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    J. Kogan, Robust Stability and Convexity, LNCIS., vol. 201. Springer New York, 1995.Google Scholar
  17. 17.
    Y. Li, K. M. Nagpal, and E. Bruce Lee, “Stabil ity analysis of polynomials with coefficients in disks,” IEEE Trans. Automat. COntr., vol. 37, pp. 509–513, 1992MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    M.S. Mahmoud, Robust Control alld Filterillg for Time-Delay Systems, Mercel Dekker, 2000.Google Scholar
  19. 19.
    S.-I. Niculescu, Delay effects on stability. A robust control approach, Heidelberg. Germany Springer-Verlag, LNCIS, vol. 269, 2001.Google Scholar
  20. 20.
    A.W. OlbrOl and C.U.T. Igwe, “Necessary and sufficient conditions for robust stability independent of delay and coefficient perturbations.’” Proc. 34th IEEE Conf. Decision COntr., New Orleans, LA, Dec. 1995, pp. 392–394.Google Scholar
  21. 21.
    E. R. Panier, M. K. H. Fan. and A.L. Tits, “On the robust stability of polynomials with no cross-coupling between the perturbations in the coefficients of even and odd parts,” Syst. & COntr. Lett., vol. 12, pp. 291–299, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    O. Toker, and H. Ozbay, “Complexity issues in robust stability of linear delay-differential systems.” Math., Contr., Signals, Syst., vol. 9, pp. 386–400, 1996.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jie Chen
    • 1
  • Silviu-Iulian Niculescu
    • 2
  1. 1.Department of Electrical EngineeringUniversity of CaliforniaRiversideUSA
  2. 2.HeuDiaSyC (UMR CNRS 6599), UT CompiègneCompiègneFrance

Personalised recommendations