Transitivity of Commutativity for Linear Time-Varying Analog Systems
This is an incremental theoretical result for control theory researchers working on linear time-varying systems.
The paper proves that commutativity is transitive for first-order linear time-varying systems, with and without initial conditions, and extends the proof to certain combinations of first- and second-order relaxed systems.
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.