### International Journal of Modern Education and Computer Science (IJMECS)

*ISSN: *2075-0161 (Print), *ISSN: *2075-017X (Online)

*Published By: *MECS Press

*IJMECS Vol.3, No.1, Feb. 2011*

#### A Criterion for Hurwitz Polynomials and its Applications

Full Text (PDF, 110KB), PP.38-44

Views:144 Downloads:5

#### Author(s)

#### Index Terms

#### Abstract

We present a new criterion to determine the stability of polynomial with real coefficients. Combing with the existing results of the real and negative roots discrimination, we deduced the explicit conditions of stability for any real polynomial with a degree no more than four. Meanwhile, we discussed the problem of controls system stability and inertia of Bezout matrix as the applications of the criterion. A necessary and sufficient condition to determine the stability of the characteristic polynomial of the continuous time control systems was proposed. And also, we discussed a pathological case of the bilinear transformation, which can convert the stability analysis of a given discrete time system to the corresponding continuous time system, and brought forward an alternative one.

#### Cite This Paper

Liejun Xie ,"A Criterion for Hurwitz Polynomials and its Applications", IJMECS, vol.3, no.1, pp.38-44, 2011.

#### Reference

[1]Gantmacher F R, “The theory of matrices”, New York: Chelsea, 1959.

[2]Li nard-Chipart, “Sur la signe de partie reelle des raciness d’une equation algebrique”, J. Math. Pures Appl., 1914, 10(6), pp. 291-346.

[3]Olga M. Katkova, Anna M. Vishnyakova, “A sufficient condition for a polynomial to be stable”, J. Math. Anal. Appl. 2008, (347), pp. 81-89.

[4]Xiaojing Yang, “Some necessary conditions for Hurwitz stability”, Automatica, 2004, (40), pp. 527-529.

[5]Xiaojing Yang, “Necessary conditions of Hurwitz polynomials”, Linear Algebra and its Applications, 2003, (359), pp. 21-27.

[6]Xiaojing Yang, “Generalized form of Hurwitz-Routh Criterion and hopf bifurcation of high order”, Applied Mathematics Letters, 2002 (15), pp. 615-621.

[7]Alberto Borobia, Sebastian Dormido, “Three coefficients of a polynomial can determine its instability”, Linear Algebra and its Applications, 2001, (338), pp. 67-76.

[8]G. Heining and K. Rost, “Using Algebraic Geometry”, Springer-Verlag, New York, 1989.

[9]Waerden Vander B L, “Algebra”, Springer-Verlag, New York, 1956.

[10]Yang Lu, “Recent Advances on Determining the Number of Real Roots of Parametric Polynomial”, Journal of Symbolic Computation, 1999, (28), pp. 225- 242.

[11]Yang Lu, Hou Xiaorong, Zeng Zhenbing, “A Complete Discrimination System for Polynomials”, Science in China (series E), 1996, 39(6), pp. 628-646.

[12]Yang Lu, Xia Bican, “An Explicit Criterion to Determine the Number of Roots in An Interval of A Polynomial”, Progress in Natural Science, 2000, 10(12), pp. 897-910.

[13]K.Ogata, “Discrete-time Control System” 2nd ed., Englewood Cliffs, NJ: Prentice-Hall, 1995.

[14]E. I. Jury, “Theory and application of the z-transform method”, New York NY: Wiley, 1964.

[15]Stephen Barnet, Polynomials and Linear Control Systems, New York, Marcel Decker, Inc., 1983.

[16]Xie Liejun, “A Note on Sturm Theorem”, Journal of Mathematics in Practice and Theory, 2007, 37(1), pp. 121-125(in Chinese).

[17]Luca Gemignani, “A fast iterative method for determining the stability of a polynomial”, Journal of Computational and Applied Mathematics, 1996, (76), pp. 1-11.

[18]Luca Gemignani, “Computationally efficient applications of the Euclidean algorithm to zero location”, Linear Algebra and its Applications, 1996, (249), pp. 79-91.

[19]G. Heining and K. Rost, “Algebraic Methods for Toeplitz-like Matrices and Operators”, Birkhauser, Boston, 1984.

[20]M. G. Krein and M. A. Naimark, “The method of symmetric and Hermitian forms in the theory of separation of the roots of algebraic equations”, Linear and Multilinear Algebra, 1981, (10), pp. 265-308.

[21]Luca Gemignani, “Compute the inertia of Bezout and Hankel matrix”, CALCOLO, 1991, (28), pp. 267-274.

[22]I S IOHVIDOV. “Hankel and Toeplitz Matrices and Forms”, Birkh$ddot{a}$user, Boston, 1984.

[23]Feng Qin Rong, “A fast method to compute the inertia of Bezout matrix and its application”, Chinese Quarterly Journal of Mathematics, 2001, (16)1, pp. 52-58.

[24]Luca Gemignani, “A hybrid approach to computation of the inertia of a parametric family of Bezoutians with application to some stability problems for bivariate polynomials”, Linear Algebra and its Applications, 1998, (283), pp. 221-238.