On operators on positive real functions and related issues

In this paper, we proved that the sets of strict positive real functions with zero relative degree (SPR0) and positive real (PR) single-input--single-output functions are closed under linear operators preserving stability and for rational functions by differentiation of the numerator and denominator. Also, we show that any PR system can be approximated by an SPR0 system. These results are extended to strict bounded real and bounded real functions when the numerator and denominator polynomials are transformed by a class of linear operators. Additionally, for functions in RH∞, the H∞-norm is preserved under linear operators applied on numerator and denominator polynomials. Three possible applications are given.