Minimal-degree coprime factorizations of rational matrix functions
Let \(G(z)\) be a given regular \(n \times n\) rational matrix function which is described in the form of a left stable coprime factorization of \(G(z)\). It means that \(G(z)=D(z)^{-1}N(z)\), where \(D(z)\) and \(N(z)\) are rational \(n \times n\) matrix functions satisfying: 1) \(D(z)\), \(N(z)\) are stable; 2) \(D(z)\), \(N(z)\) are left coprime, i.e. there exist stable rational matrix functions \(X(z)\), \(Y(z)\) such that \(N(z)X(z)+D(z)Y(z)=I\). Denote by \(\delta(G)\) the McMillan degree of \(G(z)\). \(\delta(G)\) equals the dimension of any minimal state-space realization of \(G(z)\), i.e. \(G(z)=I+C(zI-B)^{-1}B\) with appropriately dimensioned matrices \(A,B,C\). Considering a left coprime factorization of \(G=D^{-1}N\) in the matrix case, a sufficient condition for the minimal possible sum \(\delta(N)+\delta(D)\) is proved using the concept of null-pole triples for rational matrix functions. This concept is completely explained in \textit{J. A. Ball}, \textit{I. Gohberg} and \textit{L. Rodman} [Interpolation of Rational Matrix Functions. OT 45, Birkhäuser, Basel, 1990]. Illustrative examples are supplied.
scientific article; zbMATH DE number 205207
scientific article