Convex Combinations of Hurwitz Functions and Its Applications to Robustness Analysis

  • Y. K. Foo
  • Y. C. Soh
Part of the Progress in Systems and Control Theory book series (PSCT, volume 4)


In this paper, we shall show that a family of analytic functions, constructed from the convex hull of a finite number of vertex functions, will have no zero within a simply-connected region in the complex plane if and only if all its edge functions have no zero within the simply-connected region. The result is in fact a generalization of the Edge Theorem which has been derived for polynomial functions [1,2,3]. We then proceed to show how the result can be used to analyse the stability of uncertain systems.


Complex Plane Convex Combination Uncertain System Vertex Function Robustness Analysis 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A.C. Bartlett, C.V. Hollot and H. Lin, “Root Locations of an Entire Polytope of Polynomials: It Suffices to Check the Edges,” in Proc. Amer. Contr. Conf., Minneapolis, MN, 1987. Also in Mathematics of Control, Signals, and Systems, 1 (1988), 61–71.Google Scholar
  2. [2]
    M. Fu and B.R. Barmish, “Polytopes of Polynomials with Zeros in a Prescribed Region,” submitted to IEEE Trans. Auto. Contr. for publication, (1988).Google Scholar
  3. [3]
    A.C. Bartlett, C.V. Hollot, “A Necessary and Sufficient Condition for Schur Invariance and Generalised Stability of Polytope of Polynomials,” IEEE Trans. Auto. Contr., 33 (1988), 575–578.CrossRefGoogle Scholar
  4. [4]
    V.L. Kharitonov, “Asymptotic Stability of an Equilibrium Position of a Family of Systems of Linear Differential Equations,” Differential’nye Urauneniya, 14, No. 11 (1978), 1483–1485.Google Scholar
  5. [5]
    N.K. Bose, “A System-Theoretic Approach to Stability of Sets of Polynomials,” Contemp. Math., 47 (1985), 25–34.Google Scholar
  6. [6]
    C.V. Hollot and A.C. Bartlett, “Some Discrete-Time Counterparts to Kharitonov’s Stability Criterion for Uncertain Systems,” IEEE Trans. Auto. Contr., 31 (1986), 355–356.CrossRefGoogle Scholar
  7. [7]
    J. Cieslik, “On Possibilities of the Extension of Kharitonov’s Stability Test for Interval Polynomials to the Discrete-Time Case,” IEEE Trans. Auto. Contr., 32 (1987), 237–238.CrossRefGoogle Scholar
  8. [8]
    F. Kraus, B.D.O. Anderson and M. Mansour, “Robust Schur Polynomial Stability and Kharitonov’s Theorem,” Int. J. Contr., 47 (1988), 1213–1225.CrossRefGoogle Scholar
  9. [9]
    C.B. Soh and Y.C. Soh, “On the Existence of Kharitonov-Like Theorems,” Technical Report, School of Electrical and Electronic Engineering, NTI, Singapore 2263, (1988).Google Scholar
  10. [10]
    I.R. Petersen, “A Class of Stability Regions for which a Kharitonov-Like Theorem Holds,” Proc. of IEEE Conf. Decision and Control, Los Angeles, (1987).Google Scholar
  11. [11]
    C. B. Soh and C.S. Berger, “Damping Margin of Polynomials with Perturbed Coefficients;” IEEE Trans. Auto. Contr., 32 (1988), 509–511.CrossRefGoogle Scholar
  12. [12]
    N.K. Bose and E. Zeheb, “Kharitonov’s Theorem and a Stability Test of Multidimensional Digital Filters,” IEE Proc., 133, Pt. 6 (1986), 187–190.Google Scholar
  13. [13]
    R. DeCarlo and R. Saek, “The Encirclement Condition: An Approach Using Algebraic Topology,” Int. J. Contr., 26 (1977), 279–287.CrossRefGoogle Scholar
  14. [14]
    J. C. Doyle and G. Stein, “Multivariable Feedback Design Concepts for a Classical/Modern Synthesis,” IEEE Trans. Auto. Contr., 26 (1981), 14–16.CrossRefGoogle Scholar
  15. [15]
    N. A. Lehtomaki, N. R. Sandell, Jr. and M. Athens, “Robustness Results in Linear-Quadratic Gaussian Based Multivariable Control Design,” IEEE Trans. Auto. Contr.. 26 (1981), 75–93.CrossRefGoogle Scholar
  16. [16]
    I. Postlethwaite and Y. K. Foo, “Robustness with Simultaneous Pole and Zero Movement Across the jw-axis,” Automatica, 21 (1985), 433–443.CrossRefGoogle Scholar
  17. [17]
    M. Saeki, “A Method of Robust Stability Analysis with Highly Structured Uncertainties,” IEEE Trans. Auto. Contr., 31 (1986), 935–940.CrossRefGoogle Scholar
  18. [18]
    J. C. Doyle, “Analysis of Feed-back Systems with Structured Uncertainties,” IEE Proc., Pt. D, 129 (1982), 242–250.Google Scholar

Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • Y. K. Foo
  • Y. C. Soh

There are no affiliations available

Personalised recommendations