An equivalent matrix pencil for bivariate polynomial matrices

Boudellioua, M. S.

International Journal of Applied Mathematics and Computer Science, Tome 16 (2006) no. 2, pp. 175-181

Résumé

In this paper, we present a simple algorithm for the reduction of a given bivariate polynomial matrix to a pencil form which is encountered in Fornasini-Marchesini’s type of singular systems. It is shown that the resulting matrix pencil is related to the original polynomial matrix by the transformation of zero coprime equivalence. The exact form of both the matrix pencil and the transformation connecting it to the original matrix are established.

Détail

Keywords: matrix pencils, 2-D singular systems, zero-coprime-equivalence, invariant polynomials, invariant zeros
Mots-clés : macierz, układ singularny, teoria niezmienniczości, zera niezmienne

Bibliographie

[1] Blomberg H. and Ylinen R. (1983): Algebraic Theory for Multivariable Linear Systems. - London: Academic Press.

[2] Fuhrmann P.A. (1977): On strict system equivalence and similarity.- Int. J. Contr., Vol. 25, No. 1, pp. 5-10.

[3] Hayton G.E., Walker A.B. and Pugh A.C. (1990): Infinite frequency structure-preserving transformations for general polynomial system matrices. - Int. J. Contr., Vol. 33, No. 52, pp. 1-14.

[4] Johnson D.S. (1993): Coprimeness in multidimensional system theory and symbolic computation. - Ph.D. thesis, Loughborough University of Technology, UK.

[5] Kaczorek T. (1988): The singular general model of 2-D systems and its solution. - IEEE Trans. Automat. Contr., Vol. 33, No. 11, pp. 1060-1061.

[6] Karampetakis N.K., Vardulakis A.I. and Pugh A.C. (1995): A classification of generalized state-space reduction methods for linear multivariable systems. - Kybernetica, Vol. 31, No. 6, pp. 547-557.

[7] Levy B.C. (1981): 2-D polynomial and rational matrices and their applications for the modelling of 2-D dynamical systems.- Ph.D. thesis, Stanford University, USA.

[8] Oberst U. (1990): Multidimensional constant linear systems. - Acta Applicande Mathematicae, Vol. 20, pp. 1-175.

[9] Polderman J.W. and Willems J.C. (1998): Introduction to Mathematical System Theory: A Behavioral Approach. - New York: Springer.

[10] Pugh A.C., McInerney S.J., Hou M. and Hayton G.E. (1996): A transformation for 2-D systems and its invariants. - Proc. 35-th IEEE Conf. Decision and Control, Kobe, Japan, pp. 2157-2158.

[11] Pugh A.C., McInerney S.J., Boudellioua M.S. and Hayton G.E. (1998a): Matrix pencil equivalents of a general 2-D polynomial matrix. - Int. J. Contr., Vol. 71, No. 6, pp. 1027- 1050.

[12] Pugh A.C., McInerney S.J., Boudellioua M.S., Johnson D.S. and Hayton G.E. (1998): A transformation for 2-D linear systems and a generalization of a theorem of Rosenbrock. - Int. J. Contr., Vol. 71, No. 3, pp. 491-503.

[13] Pugh A.C., McInerney S.J. and El-Nabrawy E.M.O. (2005a): Equivalence and reduction of 2-D systems.-IEEE Trans. Circ. Syst., Vol. 52, No. 5, pp. 371-375.

[14] Pugh A.C., McInerney S.J. and El-Nabrawy E.M.O. (2005b): Zero structures of n-D systems. - Int. J. Contr., Vol. 78, No. 4, pp. 277-285.

[15] Rosenbrock H.H. (1970): State Space and Multivariable Theory. - London: Nelson-Wiley.

[16] Sontag E.D. (1980): On generalized inverses of polynomial and other matrices. - IEEE Trans. Automat. Contr., Vol. AC- 25, No. 3, pp. 514-517.

[17] Verghese G.C. (1978): Infinite-frequency behaviour in generalized dynamical systems. - Ph.D. thesis, Stanford University, USA.

[18] Wolovich W.A. (1974): Linear Multivariable Systems. - New York: Springer.

[19] Youla D.C. and Gnavi G. (1979): Notes on n-dimensional system theory. - IEEE Trans. Circ. Syst., Vol. CAS-26, No. 2, pp. 105-111.

[20] Zerz E. (1996): Primeness of multivariate polynomial matrices. Syst. Contr. Lett., Vol. 29, pp. 139-145.