×

Minimal indices cancellation and rank revealing factorizations for rational matrix functions. (English) Zbl 1179.15013

The authors investigate general rational matrix functions, in particular, they consider their finite and infinite poles and zeros, their partial multiplicities and the left (right) indices of a minimal polynomial basis of the null spaces to the left (right). They concentrate on the left minimal displacement problem and the left rank compression problem.
The authors obtain the class of minimal solutions to the minimal indices displacement problem formulated for a completely general rational matrix function. They show that minimal indices can be canceled by an invertible factor that may have arbitrary poles and zeros. They study minimal solutions that feature some symmetry with respect to the imaginary axis or the unit circle and have given necessary and sufficient existence conditions and the class of solutions. They also extend the well-known rank revealing factorization from constant to rational matrices.

MSC:

15A23 Factorization of matrices
15A54 Matrices over function rings in one or more variables

Software:

SLICOT
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] V. Belevitch, Classical Network Theory, Holden Day, San Francisco, 1968.; V. Belevitch, Classical Network Theory, Holden Day, San Francisco, 1968. · Zbl 0172.20404
[2] P. Benner, V. Mehrmann, V. Sima, S. Van Huffel, A. Varga, SLICOT - a subroutine library in systems and control theory, in: Biswa N. Datta (Ed.), Applied and Computational Control, Signal and Circuits, Birkauser, vol. 1, 1999, pp. 499-539 (Chapter 10).; P. Benner, V. Mehrmann, V. Sima, S. Van Huffel, A. Varga, SLICOT - a subroutine library in systems and control theory, in: Biswa N. Datta (Ed.), Applied and Computational Control, Signal and Circuits, Birkauser, vol. 1, 1999, pp. 499-539 (Chapter 10). · Zbl 1051.93500
[3] P. Dewilde, J. Vandewalle, On the factorization of a nonsingular rational matrix, IEEE Trans. Circuits and Systems, CAS 22, 1975, pp. 637-645.; P. Dewilde, J. Vandewalle, On the factorization of a nonsingular rational matrix, IEEE Trans. Circuits and Systems, CAS 22, 1975, pp. 637-645. · Zbl 0319.93062
[4] Dym, H.; Nevo, S., Pole cancellation, Linear Algbra Appl., 404, 27-57 (2005) · Zbl 1086.30007
[5] Dym, H.; Nevo, S., Zero cancellation, Linear Algebra Appl., 404, 1-26 (2005) · Zbl 1086.30006
[6] Forney, G. D., Minimal bases of rational vector spaces, with applications to multivariable linear systems, SIAM J. Control, 13, 493-520 (1975) · Zbl 0269.93011
[7] F.R. Gantmacher, The Theory of Matrices, Chelsea, New York, 1960.; F.R. Gantmacher, The Theory of Matrices, Chelsea, New York, 1960. · Zbl 0088.25103
[8] Gohberg, I.; Kaashoek, M. A.; Ran, A. C.M., Partial pole and zero displacement by cascade connection, SIAM J. Matrix Anal. Appl., 10, 3, 316-325 (1989) · Zbl 0677.93032
[9] Gohberg, I.; Kaashoek, M. A.; Ran, A. C.M., Factorizations of and extensions to \(J\)-unitary rational matrix functions on the unit circle, Int. Equat. Oper. Theory, 15, 262-300 (1992) · Zbl 0792.47012
[10] G.H. Golub, Ch.F. van Loan, Matrix Computations, John Hopkins University Press, Baltimore, 1989.; G.H. Golub, Ch.F. van Loan, Matrix Computations, John Hopkins University Press, Baltimore, 1989. · Zbl 0733.65016
[11] G.J. Groenewald, Wiener-Hopf Factorization of Rational Matrix Functions in Terms of Realizations, Ph.D. Thesis, Faculty of Sciences, Vrije Universiteit Amsterdam, 1993.; G.J. Groenewald, Wiener-Hopf Factorization of Rational Matrix Functions in Terms of Realizations, Ph.D. Thesis, Faculty of Sciences, Vrije Universiteit Amsterdam, 1993.
[12] Groenewald, G. J., Inversion of Toeplitz operators with rational symbols via factorization, Quest. Math., 19, 357-378 (1996) · Zbl 0855.47019
[13] D. Henrion, P. Hippe, Hyperbolic QR factorization for \(J\)-spectral factorization of polynomial matrices, in: Proc. 42nd IEEE Conf. Decision Contr., Maui, Hawaii, USA, 2003, pp. 3479-3484.; D. Henrion, P. Hippe, Hyperbolic QR factorization for \(J\)-spectral factorization of polynomial matrices, in: Proc. 42nd IEEE Conf. Decision Contr., Maui, Hawaii, USA, 2003, pp. 3479-3484.
[14] Ionescu, V.; Oară, C.; Weiss, M., Generalized Riccati Theory and Robust Control: A Popov Function Approach (1999), John Wiley & Sons: John Wiley & Sons New York
[15] Kimura, H., Conjugation interpolation and model matching in \(H^\infty \), Int. J. Control, 49, 269-307 (1989) · Zbl 0666.93015
[16] Kimura, H., \((J, J \prime)\)-lossless factorization based on conjugation, Syst. Contr. Lett., 19, 95-109 (1992) · Zbl 0763.93017
[17] Kimura, H., Chain-scattering representation, \(J\)-lossless factorization and \(H^\infty\) control, J. Math. Syst. Estim. Contr., 203-255 (1995) · Zbl 0863.93015
[18] Kimura, H., Chain-Scattering Approach to \(H^\infty \)-Control (1997), Birkhauser: Birkhauser Boston · Zbl 0862.93001
[19] Oară, C.; Van Dooren, P., An improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects, Syst. Cont. Lett., 30, 39-48 (1997) · Zbl 0901.93007
[20] Oară, C.; Varga, A., Minimal degree coprime factorization of rational matrices, SIAM J. Matrix Anal. Appl., 21, 1, 245-278 (1999) · Zbl 0971.65057
[21] Oară, C.; Varga, A., Computation of general inner-outer and spectral factorizations, IEEE Trans. Autom. Contr. AC, 45, 12, 2307-2325 (2000) · Zbl 0990.93019
[22] Oară, C., Constructive solutions to spectral and inner-outer factorizations with respect to the disk, Automatica, 41, 11, 1855-1866 (2005) · Zbl 1125.93326
[23] C. Oară, P. Van Dooren, A. Varga, Generalized Eigenvalue Problems, draft.; C. Oară, P. Van Dooren, A. Varga, Generalized Eigenvalue Problems, draft.
[24] Rosenbrock, H. H., State-Space and Multivariable Theory (1970), Wiley: Wiley New York · Zbl 0246.93010
[25] Vandewalle, J.; Dewilde, P., On the irreducible cascade synthesis of a system with a real rational transfer function, IEEE Trans. Circuits Syst. CAS, 24, 481-494 (1977) · Zbl 0365.94059
[26] Van Dooren, P., The computation of Kronecker‘s canonical form of a singular pencil, Linear Algebra Appl., 27, 103-141 (1979) · Zbl 0416.65026
[27] Verghese, G.; Van Dooren, P.; Kailath, T., Properties of the system matrix of a generalized state-space system, Int. J. Control, 30, 235-243 (1979) · Zbl 0418.93016
[28] Van Dooren, P., The generalized eigenstructure problem in linear systems theory, IEEE Trans. Auto. Contr., 26, 111-129 (1981) · Zbl 0462.93013
[29] Van Dooren, P., Rational and polynomial matrix factorizations via recursive pole-zero cancelation, Linear Algebra Appl., 137/138, 663-697 (1990) · Zbl 0709.15011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.