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Zeros, poles and modules in linear system theory. (English) Zbl 0701.93016
Three decades of mathematical system theory, Collect. Surv. Occas. 50th Birthday of Jan C. Willems, Lect. Notes Control Inf. Sci. 135, 79-100 (1989).
[For the entire collection see Zbl 0676.00025.]
A module theoretic approach is given to the notions of pole/zero of the linear transfer function of a multivariable system. Algebraic tools are developed for the consideration of the transfer function equation \(T(z)=H(z)G(z)\), where T(z) and G(z) are two given transfer functions with the same input space and H(z) must be determined. The main result says that there exists a module P, described in terms of T(z) and G(z), which is naturally contained in the pole module of any solution H(z). The invariant factors of P characterize the essential part of the pole structure of any solution H(z). An extension of this approach is given to the case of poles and zeros at infinity and to the case of poles and zeros in a specific region of interest.
Reviewer: K.Marti

93B25 Algebraic methods
93C35 Multivariable systems, multidimensional control systems