Canonical factorization for generalized positive real transfer functions

We prove that given any square multi-input multi-output generalized positive real transfer function matrix, M(s), with minimal state space realization of order n, there always exist two square transfer function matrices, M/sub 1/(s) and M/sub 2/(s), with state space realizations of order n/sub 1/ and n/sub 2/ respectively, with M/sub 1/(s), M/sub 2/(-s) bounded and invertible over the closed right half complex plane, such that M(s)=M/sub 2/(s)M/sub 1/(s), and n=n/sub 1/+n/sub 2/. The existence of such a factorization, commonly termed a canonical factorization, is important in absolute and robust stability results for diagonal LTI parametric uncertainty, which require multi-input multi-output non-causal positive real multipliers. Explicit state space formulae are presented for the canonical factors in terms of a stabilizing solution to a generalized Riccati equation, which is shown to always exist.