Matching zeros: a fixed constraint in multivariable synthesis

The model-matching equation (T_z)=P( z)M(z) induces constraints on the multivariate zero structures of P(z) and M( z), and the nature of the constraint is best explained by extending the usual notion of zero. In particular, the extended Γ-zero module of P(z) must contain as a submodule the module Z_Γ of matching Γ-zeros, which depends only on T(z) and M (z), and the extended Ω-zero module of M( z) must contain as a factor module the module Z_Ω of matching Ω-zeros, which depends only on T(z) and P(z). Essential solutions, in which the constraint is by module isomorphism, are possible if and only if the nullity of P(z) does not exceed the nullity of T(z), on the one hand, or the conullity of M(z) does not exceed the conullity of T(z), on the other