Positive Realness of a Transfer Function Neither Implies Nor is Implied by the External Positivity of their Associate Realizations

TL;DR

This paper clarifies that positive realness of transfer functions and external positivity of their realizations are independent properties, with neither implying the other, highlighting fundamental distinctions in system theory.

Contribution

It provides a formal proof that positive realness and external positivity are mutually non-implicative properties in linear systems.

Findings

Positive realness does not imply external positivity.

External positivity does not imply positive realness.

The properties are fundamentally independent.

Abstract

This letter discusses the differences between the properties of positive realness of transfer functions and external positivity in linear time-invariant dynamic systems. It is proved that each one of both properties does not imply to each other.