EXPLICIT EXPRESSIONS FOR CASCADE FACTORIZATIONS OF GENERAL NONSTRICTLY PROPER SYSTEMS

This paper presents 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 one 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 one 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. These factorizations are useful in several applications including loop transfer recovery, H-2 and H(infinity) optimal control. This paper is an extension of the results of Chen, Saberi and Sannuti (1992) who consider only strictly proper left invertible systems.