Commutativity of Systems with their Feedback Conjugates

TL;DR

This paper investigates the conditions under which a linear time-varying system commutes with its feedback conjugates, proving that only constant feedback conjugates are commutative, unlike their time-varying counterparts.

Contribution

It establishes that among linear time-varying feedback conjugates, only constant feedback conjugates are commutative with the original system.

Findings

Constant feedback conjugates are the only commutative pairs with the original system.

Time-varying feedback conjugates do not form commutative pairs with the original system.

The work extends the understanding of feedback conjugate commutativity in linear time-varying systems.

Abstract

After introducing commutativity concept and summarizing the relevant literature, this work is focused on the commutativity of feedback conjugates. It is already known that a linear time-varying differential system describing a single input-single output dynamical system is always commutative with its constant gain feedback pairs. In this article, it is proven that among the time-varying feedback conjugates of a linear time-varying system, constant feedback conjugates are the only commutative feedback pairs and any of the time-varying feedback conjugates cannot constitutes a commutative pair of a linear time-varying system.