Abstract
This paper presents decomposition of the fourth-order Euler-type linear time-varying system (LTVS) as a commutative pair of two second-order Euler-type systems. All necessary and sufficient conditions for the decomposition are deployed to investigate the commutativity, sensitivity, and the effect of disturbance on the fourth-order LTVS. Some systems are commutative, and some are not commutative, while some are commutative under certain conditions. Based on this fact, the commutativity of fourth-order Euler-type LTVS is investigated by introducing the commutative requirements, theories, and conditions. The fourth-order Euler-type LTVSs are investigated into commutative pairs of twice Euler-type second-order linear time-varying systems (LTVSs). The decomposition theories and conditions are derived, proved, and solved to simplify the use of commutativity for practical and industrial uses. Some fourth-order systems are sensitive toward change in initial conditions or parameters while others are not, and the effect due to disturbance also varies within systems. Furthermore, the stability and robustness of systems have so many issues. But we consider fourth-order Euler-type LTVS to observe, investigate, and tackle these issues. Lastly, the realization of fourth-order LTVS from cascaded two second-order systems can be laboratory experimented which is an open problem for future engineers to investigate. However, the theoretical results show a good agreement with the simulation results is considered in this work. Perhaps it might have unlimited physical applications in science and engineering as well as theoretical contribution. But beyond any reasonable doubt, the novelty is guaranteed because this study is the first of its kind that introduces the decomposition of the fourth-order Euler-type linear time-varying system (LTVS) as a commutative pair of two second-order Euler-type systems. Illustrative examples are presented to support the results.
Highlights
The commutativity of fourth-order Euler-type LTVS is investigated by introducing the commutative requirements, theories, and conditions. e fourth-order Eulertype LTVSs are investigated into commutative pairs of twice Euler-type second-order linear time-varying systems (LTVSs). e decomposition theories and conditions are derived, proved, and solved to simplify the use of commutativity for practical and industrial uses
Beyond any reasonable doubt, the novelty is guaranteed because this study is the first of its kind that introduces the decomposition of the fourth-order Euler-type linear time-varying system (LTVS) as a commutative pair of two second-order Euler-type systems
Decomposition formulas for 2nd order continuous-time LTVS were proved in [5] in 2016 by Koksal. e theoretical results and application for the realization of the 4th order LTVS were studied in [6] by Ibrahim and Koksal
Summary
C: C4(t)t4y(4)(t) + C3(t)t3y(3)(t) + C2(t)t2y′′(t) + C1(t)ty′(t) + C0(t)y(t) x(t),. Where the input and output are x(t) and y(t) and Ci(t) represent the coefficients of the time-varying system, which are piecewise continuous functions on [t0, ∞). E decomposition of C as the cascade connection of second-order systems A and B is given as. A: a2(t)yA′′(t) + a1(t)yA′(t) + a0(t)yA(t) xA(t), (2) B: b2(t)y′′B (t) + b1(t)y′B(t) + b0(t)yB(t) xB(t), with ICs yA t0, yA′ t0. YB t0, y′B t0, where a2(t) ≠ 0 and b2(t) ≠ 0. E systems A and B are called commutative, while (A, B) represents the commutative pair provided that the input-output relations of AB and BA are equivalent. For the cascade connection AB, the authors in [6] obtained a 4th order LTVS for the connection AB as a2b2y(4) + a2b1 + a1b2 + 2a2b2′y(3) + a1b1 + a0b2 + a2b0 + 2a2b1′ + a1b2′ + a2b2′′y′′ (4).
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