Some properties of a positive real matrix (Q1066983)

Let \(Z=P^{-1}R\) be a real rational square matrix, where P and R are left coprime polynomial matrices. The authors show that if det Z(S) is not identically zero then Z is positive real if and only if \(Z^{-1}\) is positive real. They also show that if \((P+R)^{-1} P\) is proper then Z is positive real if and only if (i) \(P+R\) is a stable polynomial matrix and (ii) for all \(\omega\) such that \(i\omega\) is not a pole of Z, \(Z(i\omega)+Z(-i\omega)^ T\geq 0\).