From Characteristic Values of PR‐Inverse Systems to Model Order Reduction

In this note, a new result is proved: the inverse of a positive real (PR) system has the same characteristic values as the original system. Specifically, given a proper impedance transfer function matrix Z(s), its corresponding admittance transfer function matrix Y(s) = Z(s)^-1 shares the same characteristic values. This observation enables the derivation of approximated models of the same order for both Z(s) and Y(s). Moreover, by considering a system of order n, if the truncated models Zr(s) and its inverse Yr(s)=Zr (s)^-1 are constructed based on the characteristic values mu_i and a balancing procedure involving the PR Riccati equation, then the error bounds for both models are governed by the same quantity, which depends on the discarded characteristic values mu_r+1, mu_r+2, … , mu_n. A significant numerical example referring to an extended RLC cascade system is presented, along with the procedure to obtain the PR Riccati-based balanced representation, which is provided at the end of the paper.