Abstract
This paper concerns two topics: (1) minimal factorizations in the class ofJ-unitary rational matrix functions on the unit circle and (2) completions of contractive rational matrix functions on the unit circle to two by two block unitary rational matrix functions which do not increase the McMillan degree. The results are given in terms of a special realization which does not require any additional properties at zero and at infinity. The unitary completion result may be viewed as a generalization of Darlington synthesis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[AG] Alpay, D., Gohberg, I.: Unitary rational matrix functions, in: Topics in Interpolation Theory of Rational Matrix-valued Functions, OT 33, ed. I. Gohberg, Birkhäuser, Basel, 1988, 175–222.
[AD] Alpay, D., Dym, H.: On applications of reproducing kernel spaces to the Schur algorithm and rational J unitary factorization, in: I. Schur methods in Operator Theory and Signal Processing, OT 18, ed. I. Gohberg, Birkhäuser, Basel, 1986, 89–159.
[AV] Anderson, B.D.O., Vongpanitlerd, S.: Network Analysis and Synthesis, Prentice Hall, Englewood Cliffs, N. J., 1973.
[BGK] Bart, H., Gohberg, I., Kaashoek, M.A.: Minimal factorizations of matrix and operator functions, OT 1, Birkhäuser, Basel, 1979.
[BGR] Ball, J.A., Gohberg, I., Rodman, L.: Interpolation of rational matrix functions, OT 45, Birkhäuser, Basel, 1990.
[C] Churilov, A.N.: Nonlinear vibrations in control theory, Siberian Mathematical Journal 20 (1979), 600.
[D] DeWilde, P.: Input-output descriptions of roomy systems, SIAM J. Control and Optimization 14(1976), 712–736.
[Dy] Dym, H.: On reproducing kernel spaces, J unitary matrix functions, interpolation and displacement rank, in: The Gohberg anniversary collection, Vol II: Topics in Analysis and Operator Theory, eds. H. Dym, S. Goldberg, M.A. Kaashoek, P. Lancaster, OT41, Birkhäuser, Basel, 1989, 173–239.
[GK] Gohberg, I., Kaashoek, M.A.: Regular rational matrix functions with prescribed pole and zero structure, in: Topics in Interpolation Theory of Rational Matrixvalued Functions, OT 33, ed. I. Gohberg, Birkhäuser, Basel, 1988, 109–122.
[GLR] Gohberg, I., Lancaster, P., Rodman, L.: Matrices and Indefinite Scalar Products, OT 8, Birkhäuser, Basel, 1983.
[GR] Gohberg, I., Rubinstein, S.: Proper contractions and their unitary minimal completions, in: Topics in Interpolation Theory of Rational Matrix-valued Functions, OT 33, ed. I. Gohberg, Birkhäuser, Basel, 1988, 109–122.
[IKL] Iohvidov, I.S., Krein, M.G., Langer, H.: Introduction to spectral theory of operators in spaces with an indefinite metric, Reihe “Math. Forschung”, Akademie Verlag, Berlin, 1981.
[LRR] Lancaster, P., Ran, A.C.M., Rodman, L.: Hermitian solutions of the discrete algebraic Ricati equation, International J. of Control 44 (1986), 777–802.
[LR] Lancaster, P., Rodman, L. Existence and uniqueness theorem for the algebraic Riccati equation. Int. J. Control 32 (1980), 285–309.
[R] Rellich, F.: Perturbation theory of eigenvalue problems, Lecture Notes (New York Univ. 1953).
[S] Shayman, M.A.: Geometry of the algebraic Riccati equation. Parts I, II. SIAM J. Control and Optimization 21 (1983), 374–394, 395–409.
[W] Willems, J.C.: Least squares stationary optimal control and the algebraic Riccati equation, IEEE Transactions on Automatic Control, Vol AC-16 (1971), 621–634.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gohberg, I., Kaashoek, M.A. & Ran, A.C.M. Factorizations of and extensions to J-unitary rational matrix functions on the unit circle. Integr equ oper theory 15, 262–300 (1992). https://doi.org/10.1007/BF01204239
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01204239