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, M₁(s) and M₂(s), with state space realizations of order n₁ and n₂ respectively, with M₁(s), M₂(-s) bounded and invertible over the closed right half complex plane, such that M(s)=M ₂(s)M₁(s), and n=n₁+n₂. 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