Fixed Poles in Transfer Function Equations

Abstract

In this paper we study the pole structure of the solutions $H(z)$, if any, of equations of the form $T(z) = H(z)G(z)$ or, dually, $T(z) = F(z)H(z)$, where $T(z)$ and $G(z)$, or $F(z)$, are given matrices of rational functions (multivariable transfer functions) over a field K. This study is motivated by various design problems in linear dynamical system theory (such as model matching or factorization problems), whose solutions are systems with an internal dynamics determined by the pole structure of $H(z)$. The methods we use are algebraic and module theoretic methods and the main tools are represented by the modules of the poles and of the zeros associated with a transfer function. The main result is a complete description of the “essential” pole structure which is common to all the solutions $H(z)$. This is given by means of a module whose invariant factors are explicitly computed in terms of fractional representations of the data $T(z)$ and $G(z)$ or $F(z)$. The essential pole structure is shown to consist, in a suitable algebraic sense, exactly of the poles of $T(z)$ which do not appear as poles of $G(z)$ (resp. $F(z)$) together with the zeros of $G(z)$ (resp. $F(z)$) which do not appear as poles of $[T'(z)G'(z)]'$ (resp. $[T(z)G(z)]$. The possibility of stating this fact, which agrees with the basic intuition, in a precise meaningful way, is due to the chosen algebraic framework. Furthermore, it is shown that no design limitations, except a technical one, apply to the remaining “inessential” part of the pole structure of any solutions.