Abstract
We consider the problem of constructing a regular rational n × n matrix function W(z) = C(zI - A) -1B + D + zE(I - zG)-1F such that WR +n = W1R +n and WR -n = W2R -n . Here R +n (respectively R -n ) is the space of rational ℂn-valued functions analytic inside (respectively outside) a smooth closed con our in the complex plane, and realizations Wj(z) = Cj (zI - Aj)-1Bj + Dj + zEj (I - zGj)-1Fj are given for j = 1, 2. The special case where the contour Γ is the unit circle and W1 (∞) = W2 (∞) = I was studied recently in [BR3]. As applications we consider the model reduction problem from linear systems theory for both the discrete time and continuous time settings and the special case of matrix polynomials.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ball, J.A., Gohberg, I and Rodman, L.: Minimal factorization in terms of spectral data, Integral Equations and Operator Theory, 10 (1987), 309–348.
Ball, J.A. and Helton, J.W.: A Beurling-Lax theorem for the Lie group U(m,n) which contains most classical interpolation theory, J. Operator Theory 9 (1983), 107–142.
Ball, J.A. and Helton, J.W.: Beurling-Lax representations using classical Lie Groups with many applications II: GL(n,(math)) and Wiener-Hopf factorization, Integral Equations and Operator Theory 7(1984), 291–309.
Ball, J.A. and Ran, A.C.M.: Hankel norm approximation of a rational matrix function in terms of its realization, in Modelling, Identification and Robust Control, C.I. Byrnes and A. Lindquist (eds.), North Holland, 1986, 285–296.
Ball, J.A. and Ran, A.C.M.: Optimal Hankel norm model reductions and Wiener-Hopf factorization I: The canonical case, SIAM J. Control and Opt. 25 (1987), 362–382.
Ball, J.A. and Ran, A.C.M.: Global inverse spectral problems for rational matrix functions, Linear Algebra and Applications 86 (1987), 237–282.
Ball, J.A. and Ran, A.C.M.: Local inverse spectral problems for rational matrix funct ions, Integral Equations and Operator Theory, 10 (1987), 349–415.
Clancey, K, and Gohberg, I.: Factorization of Matrix Functions and Singular Integral Operators, OT3, Birkhäuser, Basel, 1981.
Cohen, N.: On minimal factorizations of rational matrix functions, Integral Equations and Operator Theory 6 (1983), 647–671.
Cohen, N.: Spectral analysis of regular matrix polynomials, Integral Equations and Operator Theory 6 (1983), 161–183.
Cohen, N.: On spectral analysis and factorization of rational matrix functions, Ph. D Thesis, Weizmann Institute of Science, Rehovot, Isreal, 1984.
Cohen, N.: Realization of nonproper rational matrices, to appear: Int. J. Control.
Glover, K.: All optimal Hankel-norm approximations of linear multivariable systems and their L-error bounds, Int. J. Control 39 (1984), 1115–1193.
Gohberg, I., Lancaster, P. and Rodman, L.: Matrix polynomials, Academic Press, New York etc., 1982.
Gohberg, I., Kaashoek, M.A., Lerer, L. and Rodman, L.: Minimal divisors of rational matrix functions with prescribed zero and pole structure, in: Topics in Operator Theory Systems and Networks, H. Dym and I. Gohberg (eds.) OT12, Birkhäuser, Basel, 1984.
Gohberg, I., Kaashoek, M.A. and Van Schagen, F.: Rational matrix and operator functions with prescribed singularities, Integral Equations and Operator Theory 5 (1982), 673–717.
Gohberg, I. and Rodman, L.: Analytic matrix functions with prescribed local data, J. de Analyse Mathematique 40 (1981), 90–128.
Gohberg, I. and Rodman, L.: Interpolation and local data for meromorphic matrix and operator functions, Integral Equations and Operator Theory 9 (1986), 60–94.
Gohberg, I. and Rodman, L.: On spectral analysis of nonmonic matrix and operator polynomials II. Dependence on the finite spectral data, Isreal J. Math. 30 (1978), 321–334.
Kailath, T.: Linear systems, Prentice Hall Inc., Englewood Cliffs N.J., 1980.
Halmos, P.: Shifts on Hilbert space, J. Reine Angew. Math. 208 (1961), 102–112.
McMillan, B.: Introduction to formal realizability theory I.II. Bell Sys. Tech. J. 31 (1952), 541.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Springer Basel AG
About this chapter
Cite this chapter
Ball, J.A., Cohen, N., Ran, A.C.M. (1988). Inverse Spectral Problems for Regular Improper Rational Matrix Functions. In: Gohberg, I. (eds) Topics in Interpolation Theory of Rational Matrix-valued Functions. Operator Theory: Advances and Applications, vol 33. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5469-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5469-6_4
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5471-9
Online ISBN: 978-3-0348-5469-6
eBook Packages: Springer Book Archive