Applications of Algebraic Geometry to Systems Theory: The M<scp>c</scp>Millan Degree and Kronecker Indices of Transfer Functions as Topological and Holomorphic System Invariants

Abstract

It is shown that every rational transfer function determines a mapping from the sphere $S^2 $ into the Grassman manifold $G^m (\mathcal{C}^{m + p} )$. Based on this embedding, it is proved that the McMillan degree of a multivariable rational transfer function can be defined using mixed algebro- geometric and algebro- topological methods. The pullback of the map from $S^2 $ into $G^m (\mathcal{C}^{m + p} )$ associates a vector bundle on $S^2 $ with each such transfer function. The Grothendieck invariants of this bundle are shown to be feedback invariants of the transfer function. A complete systems theoretic interpretation of these invariants is obtained by relating the pullback bundle to the kernel bundle of a pencil of matrices associated with a minimal realization of the rational transfer function.