Structure at infinity and defect of transfer matrices with time-varying coefficients, with application to exact model-matching

We study the structure at infinity of transfer matrices with time-varying coefficients. Such transfer matrices have their entries in a skew field $% \mathbf{F}$ of rational fractions, i.e. of quotients of skew polynomials. Any skew rational fraction is the quotient of two proper ones, the latter forming a ring $\mathbf{F}_{pr}$ (a subring of $\mathbf{F}$) on which a \textquotedblleft valuation at infinity\textquotedblright\ is defined. A transfer matrix $G$ has both a \textquotedblleft generalized degree\textquotedblright\ and a valuation at infinity, the sum of which is the opposite of the \textquotedblleft defect\textquotedblright\ of $G$. The latter was first defined by Forney in the time-invariant case to be the difference between the total number of poles and the total number of zeros of $G$ (poles and zeros at infinity included and multiplicities accounted for). In our framework, which covers both continuous- and discrete-time systems, the classic relation between the defect and Forney's left- and right-minimal indices is extended to the time-varying case. The exact model-matching problem is also completely solved. These results are illustrated through an example belonging to the area of power systems.%