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

International audience ; We study the structure at infinity of transfer matrices with time-varyingcoefficients. 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 latterforming a ring $\mathbf{F}_{pr}$ (a subring of $\mathbf{F}$) on which a\textquotedblleft valuation at infinity\textquotedblright\ is defined. Atransfer matrix $G$ has both a \textquotedblleft generalizeddegree\textquotedblright\ and a valuation at infinity, the sum of which isthe opposite of the \textquotedblleft defect\textquotedblright\ of $G$. Thelatter was first defined by Forney in the time-invariant case to be thedifference between the total number of poles and the total number of zerosof $G$ (poles and zeros at infinity included and multiplicities accountedfor). In our framework, which covers both continuous- and discrete-timesystems, the classic relation between the defect and Forney's left- andright-minimal indices is extended to the time-varying case. The exactmodel-matching problem is also completely solved. These results areillustrated through an example belonging to the area of power systems.%