Advertisement

On the Characterization and Formation of Local Convex Directions for Hurwitz Stability

  • Ezra Zeheb
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 121)

Abstract

Let P o ( s) be a real given Hurwitz polynomial. A local convex direction with respect to P o ( s) is a polynomial P 1( s), such that all polynomials which belong to the convex combination of P o ( s) and P 1( s) are Hurwitz. The main result of this paper is the characterization of local convex directions, which enables the derivation of a continuum of pertinent real polynomials P 1( s), by a very simple algorithm. In arriving at this result, first a sufficient condition and then a necessary and sufficient condition are derived for a polynomial P 1( s) to be a local convex direction of a given Hurwitz polynomial P o ( s). These conditions are related to the property of strict positive realness of a rational function, and can be tested by tractable methods.

Keywords

Convex Combination Positive Realness Real Polynomial Stable Polynomial Single Input Single Output 
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]
    S. Bialas, A necessary and sufficient condition for the stability of convex combinations of stable polynomials and matrices, Bulletin Polish Acad, of Sci., Tech. Sci., Vol. 33, 1985, pp. 474–480.MathSciNetGoogle Scholar
  2. [2]
    S. Bialas and J. Garloff, Convex combinations of stable polynomials, J. Franklin Inst., Vol. 319, 1985, pp. 375–377.MathSciNetCrossRefGoogle Scholar
  3. [3]
    B.S. Bollepalli and L.K. Pujara, On the stability of a segment of polynomials, IEEE Trans. Circ. and Syst. I, Vol. 41, 1994, pp. 898–901.zbMATHCrossRefGoogle Scholar
  4. [4]
    N.K. Bose, A system theoretic approach to stability of sets of polynomials, Contemporary Mathematics, Vol. 47, AMS, 1985, pp. 25–34.Google Scholar
  5. [5]
    M. Fu, A class of weak Kharitonov regions for robust stability of linear uncertain systems, IEEE Trans. Automat. Contr., Vol. 36, 1991, pp. 975–978.zbMATHCrossRefGoogle Scholar
  6. [6]
    M. Fu, Test of convex directions for robust stability, Proc. of the 32nd Conf. on Decision and Control (CDC), San Antonio, Tx. 1993, pp. 502–507.Google Scholar
  7. [7]
    B.K. Ghosh, Some new results on the simultaneous stabilization of a family of single input single output systems, Syst. Contr. Lett., Vol. 6, 1985, pp. 39–45.zbMATHCrossRefGoogle Scholar
  8. [8]
    E.A. Guillemin, Synthesis of Passive Networks, ch. 1, p. 34, Wiley, N.Y., 1957.Google Scholar
  9. [9]
    D. Hinrichsen and V.L. Kharitonov, On convex directions for stable polynomials, Report 309, University of Bremen, Institute of Dynamic Systems, June 1994, pp. 1–22.Google Scholar
  10. [10]
    C.V. Hollot and F. Yang, Robust stabilization of interval plants using lead or lag compensators, Systems and Control Lett., Vol. 14, 1990, pp. 9–12.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    I.R. Petersen, A class of stability regions for which a Kharitonov like theorem holds, IEEE Trans. Automat. Contr., Vol. 34, 1989, pp. 1111–1115.zbMATHCrossRefGoogle Scholar
  12. [12]
    A. Rantzer, Stability conditions for polytopes of polynomials, IEEE Trans. Automat. Contr., Vol. 37, 1992, pp. 79–89.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    A. Rantzer, Hurwitz testing sets for parallel polytopes of polynomials, Systems and Control Lett., Vol. 15, 1990, pp. 99–104.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    E. Zeheb, Necessary and sufficient conditions for root clustering of a polytope of polynomials in a simply connected domain, IEEE Trans. Automat. Contr., Vol. 34, 1989, pp. 986–990.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel 1996

Authors and Affiliations

  • Ezra Zeheb
    • 1
  1. 1.Department of Electrical EngineeringTechnionHaifaIsrael

Personalised recommendations