State space realization of even generalized positive and odd rational function. Applications to static output feedback

TL;DR

This paper extends the Positive Real Lemma to complex matrix-valued rational functions, characterizes system minimality, and explores static output feedback applications for generalized positive even and odd functions.

Contribution

It specializes the Positive Real Lemma for specific classes of rational functions and analyzes system minimality and pole placement via static output feedback.

Findings

Characterization of minimality through state matrix.

Application of static output feedback to pole placement.

Extension of the Positive Real Lemma to new function classes.

Abstract

We here specialize the well known Positive Real Lemma (also known as the Kalman-Yakubovich-Popov Lemma) to complex matrix-valued rational functions, (i) generalized positive even and (ii) odd. On the way we characterize the (non) minimality of realization of arbitrary systems through (i) the corresponding state matrix and (ii) moving the poles by applying static output feedback. We then explore the application of static output feedback to both generalized positive even and to odd functions.