Explicit Commutativity Conditions for Second-order Linear Time-Varying Systems with Non-Zero Initial Conditions

TL;DR

This paper derives explicit necessary and sufficient conditions for the commutativity of second-order linear time-varying systems with non-zero initial conditions, including switched systems, supported by simulations.

Contribution

It provides the first explicit conditions for commutativity with non-zero initial conditions and analyzes the impact of switching on commutativity.

Findings

Derived explicit conditions for non-zero initial conditions

Analyzed commutativity in switched systems

Supported results with simulations

Abstract

Although the explicit commutativitiy conditions for second-order linear time-varying systems have been appeared in some literature, these are all for initially relaxed systems. This paper presents explicit necessary and sufficient commutativity conditions for commutativity of second-order linear time-varying systems with non-zero initial conditions. It has appeared interesting that the second requirement for the commutativity of non-relaxed systems plays an important role on the commutativity conditions when non-zero initial conditions exist. Another highlight is that the commutativity of switched systems is considered and spoiling of commutativity at the switching instants is illustrated for the first time. The simulation results support the theory developed in the paper.