Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T11:29:31.605Z Has data issue: false hasContentIssue false
  • English
  • Français

A new formulation of the Liénard–Chipart stability criterion

Published online by Cambridge University Press:  24 October 2008

S. Barnett
S. Barnett
Affiliation:
School of Mathematics, University of Bradford, Yorkshire
Get access Rights & Permissions [Opens in a new window]

Abstract

The Liénard–Chipart criterion for determining whether all the zeros of a real nth degree polynomial a(λ) have negative real parts involves calculation of only about half the Hurwitz determinants, the order of the largest being n − 1. It is shown, by using the companion matrix of a polynomial formed from a(λ), that the required determinants are equal to minors of a matrix whose order is ½ n or ½(n−1) according as n is even or odd. In either case this matrix is very easy to obtain A simple application of the result provides a criterion for a(λ) to be stable and aperiodic.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 70 , Issue 2 , September 1971 , pp. 269 - 274
Copyright
Copyright © Cambridge Philosophical Society 1971

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Barnett, S.Greatest common divisor of two polynomials. Linear Algebra and Appl. 3 (1970) 7–9.Google Scholar
(2)Barnett, S.Number of zeros of a complex polynomial inside the unit circle. Electronics Letters 6 (1970), 164–165.CrossRefGoogle Scholar
(3)Barnett, S.Qualitative analysis of polynomials using matrices, IEEE Trans. Automatic Control 15 (1970), 380–382.CrossRefGoogle Scholar
(4)Barnett, S.Location of zeros of a complex polynomial. Linear Algebra and Appl. 4 (1971), 71–76.Google Scholar
(5)Barnett, S.Greatest common divisor of several polynomials. Proc. Cambridge Philos. Soc. 70 (1971), 263–268.Google Scholar
(6)Fuller, A. T.Stability criteria for linear systems and realizability criteria for RC networks. Proc. Cambridge Phios. Soc. 53 (1957), 878–896.CrossRefGoogle Scholar
(7)Gantmacher, F. R.The theory of matrices, vol. 2 (New York, 1959).Google Scholar
(8)MacDuffee, C. C.Some applications of matrices in the theory of equations. Amer. Math. Monthly 57 (1950), 154–161.Google Scholar