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