EXPLICIT EXPRESSIONS FOR CASCADE FACTORIZATIONS OF GENERAL NONSTRICTLY PROPER SYSTEMS

This paper presents explicit expressions for two different cascade fac torizations of any detectable system which is not necessarily left inv ertible and which is not necessarily strictly proper. The first one is a well known minimum phase/all-pass factorization by which G(s) is wr itten 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 functi on matrix which has all unstable invariant zeros of G(s) as its invari ant zeros. The second one is a generalized cascade factorization by wh ich G(s) is written as G(M)(s)U(s), where G(M)(s) is left invertible a nd of minimum-phase with its invariant zeros at desired locations in t he open left-half s-plane while U(s) is a stable right invertible syst em which has all ''awkward'' invariant zeros, including the unstable i nvariant zeros of G(s), as its invariant zeros, and is ''asymptoticall y'' 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.