On canonical Wiener-Hopf factorizations. (English) Zbl 0933.93043
Picci, Giorgio (ed.) et al., Dynamical systems, control, coding, computing vision. New trends, interfaces, and interplay. Basel: Birkhäuser. Prog. Syst. Control Theory. 25, 69-91 (1999).
The author has considered some problems arising in Wiener-Hopf factorizations. The results obtained are slight generalizations of previous results of I. Gohberg and Y. Zucker [Integral Equations Oper. Theory 19, No. 2, 216-239 (1994; Zbl 0806.47021); ibid. 25, No. 1, 73-93 (1996; Zbl 0856.47012)] and Y. Zucker [Constructive factorization and partial indices of rational matrix functions, Ph.D. thesis, Tel-Aviv Univ. (1998)]. The different, functional oriented, technique used in this paper allows the establishment of a clearer connection between factorization theory and geometry.
Assuming that a rational matrix function is positive on the imaginary axis, including the point at infinity, its spectral factorization coincides with a Wiener-Hopf factorization. In this case, both factors are invertible at infinity. However, the situation changes in case of canonical factorizations with respect to the unit circle. Throughout the paper, the method of polynomial models and in particular the shift realization, introduced by the author [IEEE Trans. Autom. Control AC-26, 284-295 (1981; Zbl 0459.93032)] has been used.
In Section 2 (Rational function factorizations), the author considered the inverse problem, that is, given a rational function, which can be considered without loss of generality to be the transfer function of a finite-dimensional linear system, one likes to obtain a characterization of its factorization into the product of two transfer functions. A transfer function which is a normalized biproper function has been considered to start with.
In Section 3, left and right canonical factorizations have been discussed, since in the case of the unit circle, the existence of canonical factorizations for a rational matrix \(W\) does not imply that it is necessarily biproper. In Section 4 (State-space analysis) it is shown how one can be more specific on the existence of canonical factorizations and the realization of the canonical factors if one is more specific on a minimal realization of \(W\).
For the entire collection see [Zbl 0911.00015].
Assuming that a rational matrix function is positive on the imaginary axis, including the point at infinity, its spectral factorization coincides with a Wiener-Hopf factorization. In this case, both factors are invertible at infinity. However, the situation changes in case of canonical factorizations with respect to the unit circle. Throughout the paper, the method of polynomial models and in particular the shift realization, introduced by the author [IEEE Trans. Autom. Control AC-26, 284-295 (1981; Zbl 0459.93032)] has been used.
In Section 2 (Rational function factorizations), the author considered the inverse problem, that is, given a rational function, which can be considered without loss of generality to be the transfer function of a finite-dimensional linear system, one likes to obtain a characterization of its factorization into the product of two transfer functions. A transfer function which is a normalized biproper function has been considered to start with.
In Section 3, left and right canonical factorizations have been discussed, since in the case of the unit circle, the existence of canonical factorizations for a rational matrix \(W\) does not imply that it is necessarily biproper. In Section 4 (State-space analysis) it is shown how one can be more specific on the existence of canonical factorizations and the realization of the canonical factors if one is more specific on a minimal realization of \(W\).
For the entire collection see [Zbl 0911.00015].
Reviewer: R.K.Bose (New Delhi)
MSC:
| 93C05 | Linear systems in control theory |
| 93B15 | Realizations from input-output data |
| 15A23 | Factorization of matrices |
