Transitivity of Commutativity for Linear Time-Varying Analog Systems

TL;DR

This paper investigates the transitivity of commutativity in first-order linear time-varying systems, establishing conditions under which this property holds with and without initial conditions, and extending it to certain higher-order systems.

Contribution

It provides a formal proof of transitivity of commutativity for linear time-varying systems, including special cases involving higher-order systems and initial conditions.

Findings

Transitivity holds for first-order systems with and without initial conditions.

Formulation of transitivity based on impulse response functions.

Extension of transitivity results to specific second-order systems.

Abstract

In this contribution, the transitivity property of commutative first-order linear time-varying systems is investigated with and without initial conditions. It is proven that transitivity property of first-order systems holds with and without initial conditions. On the base of impulse response function, transitivity of commutation property is formulated for any triplet of commutative linear time-varying relaxed systems. Transitivity proves are given for some special combinations of first and second-order linear time-varying systems which are initially relaxed.