On the zeros of a minimal realization

In an earlier work, the authors have introduced a coordinate-free, module-theoretic definition of zeros for the transfer function G(s) of a linear multivariable system (A,B,C). The first contribution of this paper is the construction of an explicit k[z]-module isomorphism from that zero module, Z(G), to V∗/R∗, where V∗ is the supremal (A,B)-invariant subspace contained in kerC and R∗ is the supremal (A,B)-controllable subspace contained in kerC, and where (A,B,C) constitutes a minimal realization of G(s). The isomorphism is developed from an exact commutative diagram of k-vector spaces. The second contribution is the introduction of a zero-signal generator and the establishment of a relation between this generator and the classic notion of blocked signal transmissions.