Transitivity of Commutativity for Linear Time-Varying Physical Systems
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ORIGINAL ARTICLE
Transitivity of Commutativity for Linear Time‑Varying Physical Systems Mehmet Emir Koksal1 Received: 10 March 2020 / Revised: 14 July 2020 / Accepted: 4 November 2020 © The Korean Institute of Electrical Engineers 2020
Abstract In this contribution, the transitivity property of commutative first-order linear time-varying systems with and without initial conditions is investigated. 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. Keywords Commutativity · Time-varying systems · Linear systems · Differential equations · Impulse response Mathematics Subject Classification 34A30 · 34HXX · 37N35 · 45A05
1 Introduction A∶ As the main branch of applied mathematics, linear-time varying systems described by differential equations arise in many different physical problems including acoustics, electromagnetic, electrodynamics, fluid dynamics, etc. There is a great deal of papers on the theory, technique and applications of linear time-varying systems. Especially, they are used as a major tool in order to achieve many developments in real engineering problems (see [1–4]) by modelling, analyzing and solving natural problems. For example, as an interdisciplinary branch of applied mathematics and electric-electronics engineering, they play a pioneering role in system and control theory that deal with the behavior of dynamical systems with inputs and how their behavior is modified by different combinations such as cascade and feedback connections. When the cascade connection in system design is considered, the commutativity concept places a prominent role to improve different system performances. Consider a system A described by a linear time-varying differential equation of the form
* Mehmet Emir Koksal [email protected] 1
Department of Mathematics, Ondokuz Mayis University, Atakum, 55139 Samsun, Turkey
nA ∑ i=0
ai (t)
di yA (t) = uA (t); dti
(1)
(uA − yA ) represents input–output pair of the system at any time t ∈ R , ai (t) are time-varying coefficients ( )with an (t) ≠ 0 , nA ≥ 0 is the order of system; and y(i) t ∈ R, i = 0, 1, ⋯ , nA − 1 are the initial conditions at A 0 the initial time t0 ∈ R. When two systems of this type are interconnected sequentially so that the output of the former feeds the input of the later, it is said that they are connected in cascade [5]. If the order of connection does not affect the input–output relation of the combined system AB or BA , it is said that the pair of systems (A, B) is commutative. If the combined system has an overall input–output relation invariant with the sequence of connection, it is said that these systems are commutative [6]. In [6], Marshall has proven that “for commutativity, either
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