Explicit expressions for cascade factorizations of general non-strictly proper systems

The authors present explicit expressions for two different cascade factorizations of any detectable system which is not necessarily left invertible and which is not necessarily strictly proper. The first is a well-known minimum phase/all-pass factorization by which G(s) is written as G_m(s) V(s), where G_m(s) is left invertible and of minimum phase, while V(s) is a stable right invertible all-pass transfer function matrix which has all unstable invariant zeros of G(s) as its invariant zeros. The second is a generalized cascade factorization by which G (s) is written as G_M(s)U (s), where G_M(s) is left invertible and of minimum-phase with its invariant zeros at desired locations in the open left-half s-plane, while U(s ) is a stable right invertible system which has all awkward invariant zeros, including the unstable invariant zeros of G( s), as its invariant zeros, and is asymptotically all-pass