Transitivity of Commutativity for Second-Order Linear Time-Varying Analog Systems

TL;DR

This paper investigates the transitivity of commutativity in second-order linear time-varying analog systems, establishing conditions under which transitivity holds, especially considering initial states, and validating findings through MATLAB simulations.

Contribution

It derives inverse commutativity conditions for these systems and proves transitivity is always valid with zero or non-zero initial states, under specific assumptions.

Findings

Transitivity holds for systems with zero initial states.

Transitivity remains valid with non-zero initial states without additional conditions.

Results are validated through MATLAB simulations.

Abstract

After reviewing commutativity of second-order linear time-varying analog systems, the inverse commutativity conditions are derived for these systems by considering non-zero initial conditions. On the base of these conditions, the transitivity property is studied for second order linear time-varying unrelaxed analog systems. It is proven that this property is always valid for such systems when their initial states are zero; when non-zero initial states are present, it is shown that the validity of transitivity does not require any more conditions and it is still valid. Throughout the study it is assumed that the subsystems considered can not be obtained from each other by any feed-forward and feed-back structure. The results are well validated by MATLAB simulations.