An improved Toeplitz algorithm for polynomial matrix null-space computation

In this paper, we present an improved algorithm to compute the minimal null-space basis of polynomial matrices, a problem which has many applications in control and systems theory. This algorithm takes advantage of the block Toeplitz structure of the Sylvester matrix associated with the polynomial matrix. The analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well-known pencil (state-space) algorithms found in the literature.

[1]  Panos J. Antsaklis,et al.  Polynomial and Rational Matrix Interpolation: Systems and Control Applications , 1993, 1993 American Control Conference.

[2]  J. H. Wilkinson,et al.  Reliable Numerical Computation. , 1992 .

[3]  Jean Jacques Loiseau,et al.  On the modification of a structure at infinity by a return of static state , 1988 .

[4]  Michael Sebek,et al.  Reliable numerical methods for polynomial matrix triangularization , 1999, IEEE Trans. Autom. Control..

[5]  J. C. Basilio,et al.  A robust solution of the generalized polynomial Bezout identity , 2004 .

[6]  Hong Zhang,et al.  NUMERICAL CONDITION OF POLYNOMIALS IN DIFFERENT FORMS , 2001 .

[7]  Paul Van Dooren,et al.  Normwise Scaling of Second Order Polynomial Matrices , 2004, SIAM J. Matrix Anal. Appl..

[8]  P. Voutier,et al.  Primitive divisors of Lucas and Lehmer sequences , 1995 .

[9]  Zhonggang Zeng.  Computing multiple roots of inexact polynomials , 2005, Math. Comput..

[10]  C. Praagman,et al.  Column reduction of polynomial matrices , 1993 .

[11]  P. Dooren,et al.  The eigenstructure of an arbitrary polynomial matrix : Computational aspects , 1983 .

[12]  A. Morse,et al.  Decoupling and Pole Assignment in Linear Multivariable Systems: A Geometric Approach , 1970 .

[13]  E. N. Antoniou,et al.  Numerical computation of minimal polynomial bases: A generalized resultant approach , 2005 .

[14]  Didier Henrion,et al.  A Toeplitz algorithm for polynomial J-spectral factorization , 2006, Autom..

[15]  Vladimír Kucera,et al.  Discrete linear control: The polynomial equation approach , 1985, IEEE Transactions on Systems, Man, and Cybernetics.

[16]  D. Henrion,et al.  Block Toeplitz Methods in Polynomial Matrix Computations , 2004 .

[17]  H. Zha.  A componentwise perturbation analysis of the QR decomposition , 1993 .

[18]  Erik Frisk,et al.  RESIDUAL GENERATION FOR FAULT DIAGNOSIS , 2001 .

[19]  P. Dooren,et al.  An improved algorithm for the computation of Kronecker's canonical form of a singular pencil , 1988 .

[20]  Frank M. Callier.  Book review: J. W. Polderman and J.C. Willems, "Introduction to Mathematical Systems Theory: a Behavioral Approach" (Springer Verlag 1998) , 2002 .

[21]  Hans J. Stetter,et al.  Numerical polynomial algebra , 2004 .

[22]  Nicholas J. Higham,et al.  Structured Backward Error and Condition of Generalized Eigenvalue Problems , 1999, SIAM J. Matrix Anal. Appl..

[23]  Jr. G. Forney,et al.  Minimal Bases of Rational Vector Spaces, with Applications to Multivariable Linear Systems , 1975 .

[24]  A. Varga.  Computation of least order solutions of linear rational equations , 2004 .

[25]  A. Morse.  Structural Invariants of Linear Multivariable Systems , 1973 .

[26]  Jeffrey S. Rosenthal,et al.  Introduction to mathematical systems theory. a behavioral approach [Book Review] , 2002, IEEE Transactions on Automatic Control.

[27]  A. Edelman,et al.  Polynomial roots from companion matrix eigenvalues , 1995 .

[28]  N. Higham.  Analysis of the Cholesky Decomposition of a Semi-definite Matrix , 1990 .

[29]  D. Henrion,et al.  Reliable Algorithms for Polynomial Matrices , 1998 .

[30]  T. Beelen,et al.  Numerical computation of a coprime factorization of a transfer function matrix , 1987 .

[31]  Thomas Kailath,et al.  Fast reliable algorithms for matrices with structure , 1999 .

[32]  B. Anderson,et al.  Greatest common divisor via generalized Sylvester and Bezout matrices , 1978 .

[33]  Thomas Kailath,et al.  Linear Systems , 1980 .

[34]  G. W. Stewart,et al.  Matrix algorithms , 1998 .

[35]  Jan C. Willems,et al.  Introduction to mathematical systems theory: a behavioral approach, Texts in Applied Mathematics 26 , 1999 .

[36]  Vladimír Kučera,et al.  Diophantine equations in control - A survey , 1993, Autom..

[37]  Joos Vandewalle,et al.  On the determination of the Smith-Macmillan form of a rational matrix from its Laurent expansion , 1970 .

[38]  A. Pugh,et al.  A novel method to determine the finite and infinite frequency structure of a rational matrix , 2001 .

[39]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[40]  B. Molinari.  Structural invariants of linear multivariable systems , 1978 .

[41]  Didier Henrion,et al.  On the application of displacement structure methods to obtain null-spaces of polynomial matrices , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[42]  Gene H. Golub,et al.  Matrix computations , 1983 .

[43]  D. Henrion,et al.  On the application of different numerical methods to obtain null-spaces of polynomial matrices . Part 2 : block displacement structure algorithms , 2004 .

[44]  Panos J. Antsaklis,et al.  Polynomial and rational matrix interpolation: theory and control applications , 1993 .

[45]  P. Dooren.  The Computation of Kronecker's Canonical Form of a Singular Pencil , 1979 .

[46]  Andrew P. Paplinski,et al.  Numerical Operations with Polynomial Matrices: Application To Multi-Variable Dynamic Compensator Design , 1992 .

[47]  D. Henrion,et al.  On the application of different numerical methods , 2004 .

[48]  A. Vardulakis.  Linear Multivariable Control: Algebraic Analysis and Synthesis Methods , 1991 .

[49]  W. M. Wonham,et al.  Linear Multivariable Control , 1979 .

[50]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[51]  A. Varga.  A Descriptor Systems Toolbox for MATLAB , 2000, CACSD. Conference Proceedings. IEEE International Symposium on Computer-Aided Control System Design (Cat. No.00TH8537).

[52]  Didier Henrion,et al.  NUMERICAL STABILITY OF BLOCK TOEPLITZ ALGORITHMS IN POLYNOMIAL MATRIX COMPUTATIONS , 2005 .

[53]  Didier Henrion,et al.  Detecting infinite zeros in polynomial matrices , 2005, IEEE Transactions on Circuits and Systems II: Express Briefs.