Existence of minimal nonsquare J -symmetric factorizations for self-adjoint rational matrix functions

Linear Algebra and Its Applications

Volumn 379, Issue 1-3 SPEC. ISS, 2004, Pages 159-178

  Lerer, L ^a   Petersen, M A ^b   Ran, A C M ^c  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

^b NORTH WEST UNIVERSITY   (South Africa)

^c VU UNIVERSITY AMSTERDAM   (Netherlands)

Author keywords

Bezoutians; Common zeros; Minimal factorizations; Rational matrix functions; Riccati equations; Symmetric factorization

Indexed keywords

FUNCTION EVALUATION; POLES AND ZEROS; RICCATI EQUATIONS; THEOREM PROVING;

BEZOUTIANS; HERMITIAN MATRICES; MATRIX ADJOINTS; MINIMAL FACTORIZATIONS; RATIONAL MATRIX FUNCTIONS; SYMMETRIC FACTORIZATION;

MATRIX ALGEBRA;

EID: 0742319244     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(03)00457-9     Document Type: Conference Paper

References (61)

1. Ball J.A., Ran A.C.M. Left versus right canonical Wiener-Hopf factorization. in: Constructive Methods of Wiener-Hopf Factorizations. Operator Theory Advances and Applications. vol. OT 21:1986;9-38 Birkhäuser Verlag, Basel.
2. Ball J.A., Ran A.C.M. Optimal Hankel-norm model reduction and Wiener-Hopf factorization I: the canonical case. SIAM Journal on Control and Optimization. 25:1987;362-382.
3. Bart H., Gohberg I., Kaashoek M.A. Minimal Factorization of Matrix and Operator Functions, vol. OT 1. 1979;Birkhäuser, Basel.
4. Bart H., Gohberg I., Kaashoek M.A., van Dooren P. Factorizations of transfer functions. SIAM Journal on Control and Optimization. 18:1980;675-696.
5. Clancey K., Gohberg I. Factorization of matrix functions and singular integral operators. in: Operator Theory: Advances and Applications. vol. 3:1981;Birkhäuser Verlag, Basel, Boston, MA.
6. Clements D.J. Rational spectral factorization using state space methods. Systems and Control Letters. 20:1993;335-343.
7. Clements D.J. A state-space approach to indefinite spectral factorization. SIAM Journal Matrix Analysis and Applications. 21:2000;743-767.
8. Clements D.J., Glover K. Spectral factorization by hermitian pencils. Linear Algebra and its Applications. 122-124:1989;797-846.
9. Cohen N. On minimal factorizations of rational matrix functions. Integral Equations and Operator Theory. 6:1983;647-671.
10. Ferrante A., Michaletzky G., Pavon M. Parametrization of all minimal square spectral factors. Systems and Control Letters. 21:1993;249-254.
11. Finesso L., Picci G. A characterization of minimal spectral factors. IEEE Transactions of Automatic Control. AC-27:1982;122-127.
12. Fuhrmann P. On symmetric rational transfer functions. Linear Algebra and its Applications. 50:1983;167-250.
13. Fuhrmann P. On the characterization and parametrization of minimal spectral factors. Journal of Mathematical Systems, Estimation and Control. 5:1995;383-444.
14. Fuhrmann P., Gombani A. On a Hardy space approach to the analysis of spectral factors. International Journal of Control. 71:1998;277-357.
15. Fuhrmann P., Gombani A. On the Lyapunov equation, coinvariant subspaces and some problems related to spectral factorizations. International Journal of Control. 73:2000;1129-1159.
16. Gohberg I., Feldman D. Convolution equations and projective methods for their solution. Translations of Mathematical Monographs. 41:1979.
17. Gohberg I., Krein M.G. Systems of integral equations on a half-line with kernel depending upon the difference of the arguments. Uspekhi Matematicheskikh Nauk. 13:1958;3-72. English Translation: American Mathematical Society Translations 14 (1960) 217-287.
18. Gohberg I., Krein M.G. Systems of integral equations on a half-line with kernel depending upon the difference of the arguments. Uspekhi Matematicheskikh Nauk. 13:1958;3-72. English Translation: American Mathematical Society Translations 14 (1960) 217-287.
19. Gohberg I., Krupnik N. One-dimensional Linear Singular Integral Equations, Volume I, Introduction, vol. 53. 1992;Birkhäuser Verlag, Basel, Boston.
20. Gohberg I., Lancaster P., Rodman L. in: Matrices and Indefinite Scalar Products. Operator Theory: Advances and Applications. vol. OT 8:1983;Birkhäuser Verlag, Basel.
21. Gohberg I., Lancaster P., Rodman L. Matrix Polynomials. 1982;Academic Press.
22. Gohberg I., Lancaster P., Rodman L. Factorization of selfadjoint matrix polynomials with constant signature. Linear and Multilinear Algebra. 11:1982;209-224.
23. ∞ -control SIAM Journal of Control and Optimization. 28:1990;1350-1371.
24. Greenberg W., van der Mee C.V.M., Protopopescu V. in: Boundary Value Problems in Abstract Kinetic Theory. Operator Theory: Advances and Applications. vol. OT 23:1987;Birkhäuser Verlag.
25. Ionescu V., Weiss M., Oara C. Generalized Riccati theory and robust control, A Popov function approach. 1999;John Wiley, Chichester.
26. Jacubovič V.A. Factorization of symmetric matrix polynomials. Soviet Mathematics Doklady. 11:1970;1261-1264.
27. Kaashoek M.A., Ran A.C.M. Symmetric Wiener-Hopf factorization of self-adjoint rational matrix functions and realization. in: Constructive Methods of Wiener-Hopf factorizations. Operator Theory: Advances and Applications. vol. OT 21:1986;373-409 Birkhäuser Verlag, Basel.
28. Karelin I., Lerer L. Generalized Bezoutian, factorization of rational matrix functions and matrix quadratic equations. Operator Theory and Analysis, The M.A. Kaashoek Anniversary Volume. Bart H., Gohberg I., Ran A.C.M. Operator Theory: Advances and Applications. vol. OT 122:1997;303-321 Birkhäuser Verlag, Basel.
29. Karelin I., Lerer L., Ran A.C.M. J -symmetric factorizations and algebraic Riccati equations Recent Advances in Operator Theory, The Israel Gohberg Anniversary Volume. Dijksma A., Kaashoek M.A., Ran A.C.M. Operator Theory: Advances and Applications. vol. OT 124:1997;319-360 Birkhäuser Verlag, Basel.
30. Lancaster P., Rodman L. Algebraic Riccati Equations. 1995;Oxford University Press.
31. Lerer L. The matrix quadratic equation and factorization of matrix polynomials. in: Operator Theory: Advances and Applications. vol. 40:1989;279-324 Birkhäuser Verlag, Basel.
32. Lerer L., Rodman L. Bezoutians of rational matrix functions. Journal of Functional Analysis. 141:1996;1-36.
33. Lerer L., Rodman L. Bezoutians of rational matrix functions, matrix equations and factorizations. Linear Algebra and its Applications. 302/303:1999;105-135.
34. Lerer L., Rodman L. Common zero structure of rational matrix functions. Journal of Functional Analysis. 136:1996;1-38.
35. Lerer L., Rodman L. Symmetric factorizations and localization of zeroes of rational matrix functions. Linear and Multilinear Algebra. 40:1996;259-281.
36. Lindquist A., Michaletzky G., Picci G. Zeros of spectral factors, the geometry of splitting subspaces and the algebraic Riccati inequality. SIAM Journal on Control and Optimization. 33:1995;365-401.
37. Lindquist A., Picci G. Forward and backward semimartingale representations for stationary increment processes. Stochastics. 15:1985;1-50.
38. Lindquist A., Picci G. A geometric approach to modelling and estimation of linear stochastic systems. Journal of Mathematical Systems, Estimation and Control. 1:1991;241-333.
39. van der Mee C.V.M. Semigroup and Factorization Methods in Transport Theory. 1981;Mathematisch Centrum, Amsterdam.
40. Molinari B.P. The stabilizing solution of the algebraic Riccati equation. SIAM Journal of Control and Optimization. 11:1973;262-271.
41. Nikolaïčuk A.M., Spitkovskiï I.M. On the Riemann boundary-value problem with hermitian matrix. Soviet Mathematics Doklady. 16:1975;533-536.
42. C. Oara, A. Varga, Solutions to the general inner-outer and spectral factorization problems, in: Proceedings of the 37th IEEE Conference on Decision and Control, Tampa Florida USA, December 1998, pp. 2774-2779.
43. Pavon M. On the parametrization of nonsquare spectral factors. Helmke U., Mennicken R., Sauer J. Systems and Networks: Mathematical Theory and Applications, vol. II, Mathematical Research. vol. 79:1994;413-416 Akademie Verlag, Berlin.
44. M.A. Petersen, A.C.M. Ran, Minimal nonsquare J -spectral factorization, generalized bezoutians and common zeros of rational matrix functions, Integral Equations and Operator Theory, in press.
45. Petersen M.A., Ran A.C.M. Minimal nonsquare spectral factors. Linear Algebra and its Applications. 351-352:2002;553-565.
46. Petersen M.A., Ran A.C.M. Minimal nonsquare spectral factors via factorizations of unitary functions. Linear Algebra and its Applications. 351-352:2002;567-583.
47. Petersen M.A., Ran A.C.M. Minimal square spectral factors via triples. SIAM Journal of Matrix Analysis and Applications. 22(4):2001;1222-1244.
48. Popov V.M. Hyperstability of control systems. Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen. vol. 204:1973;Springer Verlag, Berlin. Translated and expanded version of the 1966 original in Romanian.
49. Ran A.C.M. Minimal factorization of selfadjoint rational matrix functions. Integral Equations and Operator Theory. 5:1982;850-869.
50. Ran A.C.M. Minimal square spectral factors. Systems and Control Letters. 24:1995;307-316.
51. Ran A.C.M., Rodman L. Stability of invariant lagrangian subspaces. I. in: Topics in Operator Theory. Operator Theory: Advances and Applications. vol. OT 32:1988;181-218 Birkhäuser, Basel.
52. Ran A.C.M., Rodman L. Stable invariant lagrangian subspaces: factorization of symmetric rational matrix functions and applications. Linear Algebra and its Applications. 137/138:1990;576-620.
53. Ran A.C.M., Rodman L. On symmetric factorizations of rational matrix functions. Linear and Multilinear Algebra. 29:1991;243-261.
54. Ran A.C.M., Zizler P. On selfadjoint matrix polynomials with constant signature. Linear Algebra and its Applications. 259:1997;133-153.
55. Roozemond L. Canonical pseudo-spectral factorization and Wiener-Hopf integral equations. Gohberg I., Kaashoek M.A. Constructive Methods of Wiener-Hopf Factorization. vol. OT 21:1986;127-156 Birkhäuser Verlag, Basel.
56. Rosenbrock H. State-Space Multivariable Theory. 1970;Nelson.
57. Sahnovic L.A. On the factorization of an operator valued transfer function. Soviet Mathematics Doklady. 17:1976;203-207.
58. Tan S.H., Vandewalle J. On factorizations of rational matrices. IEEE Transactions on Circuits and Systems. 35:1988;1179-1182.
59. Tan S.H., Vandewalle J. An alternative proof for the minimal factorization theorem for rational matrices. Systems and Control Letters. 14:1990;37-40.
60. Vandewalle J., Tan S.H. Properties of minimal first degree factorizations for arbitrary rational matrices. Kaashoek M.A., van Schuppen J.H., Ran A.C.M. Proceedings of MTNS 89, Progress in Systems and Control Theory. vol. 5:1990;309-314 Birkhäuser Verlag, Boston.
61. Willems J.C. Least squares stationary optimal control and the algebraic Riccati equation. IEEE Transactions on Automatic Control. AC-16:1971;621-634.