Abstract
Decomposition is an important tool that is used in many differential systems for solving real engineering problems and improving the stability of a system. It involves breaking down of high-order linear systems into lower-order commutative pairs. Commutativity plays an essential role in mathematics, and its applications are extended in physical science and engineering. This paper explicitly expresses all form of necessary and sufficient conditions for decomposition of any kind of fourth-order linear time-varying system as commutative pairs of two second-order systems. Regarding the nonzero initial conditions, additional requirements are derived in order to satisfy the decomposition process. In this paper, explicit method for reducing fourth-order linear time-varying systems (LTVS) into two second-order commutative pairs is derived and solved. The method points out the effect of disturbance and sensitivity on the systems and also highlights the necessary and sufficient conditions for commutativity of the decomposed systems. The results are illustrated by solving some examples.
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References
C.A. Deoser, Notes for a Second Course on Linear Systems (Van Nostrand Reinhold Company, New York, 1970).
E. García, L. Littlejohn, J.S. López, E.P. Sinusía, Factorization of second-order linear differential equations and Liouville–Neumann expansions. Math. Comput. Model. 57, 514–1530 (2013)
A.G.J. Holt, K.M. Reineck, Transfer function synthesis for a cascade connection network. IEEE Trans. Circuit Theory 15(2), 162–163 (1968)
S. Ibrahim, M.E. Koksal, Decomposition of fourth-order linear time-varying differential system into its twin second-order commutative pairs (2+2), in 3rd International Symposium on Multidisciplinary Studies and Innovative Technologies, Ankara, Oct 11–13, pp. 224–226 (2019)
E.L. Ince, Ordinary Differential Equations (Dover Books on Mathematics, Charleston, 1956).
M. Koksal, Commutativity of second-order time-varying systems. Int. J. Control 36(3), 541–544 (1982)
M. Koksal, General conditions for the commutativity of time-varying systems’ of second-order time-varying systems, in International Conference on Telecommunication and Control, Halkidiki, Greece, August, pp. 223–225 (1984)
M. Koksal, Commutativity of 4th order systems and Euler systems, in Presented in National Congress of Electrical Engineers, Adana, Turkey, Paper no: BI-6 (1985)
M. Koksal, Exhaustive study on the commutativity of time-varying systems. Int. J. Control 47(5), 1521–1537 (1988)
M. Koksal, M.E. Koksal, Commutativity of linear time-varying differential systems with non-zero initial conditions: a review and some new extensions. Math. Probl. Eng. 2011, 1–25 (2011)
M.E. Koksal, Decomposition of a second-order linear time-varying differencial system as a series connection of two first-order commutative pairs. Open Math. 14, 693–704 (2016)
M.E. Koksal, A. Yakar, Decomposition of a third-order linear time-system into its second and first-order commutative pairs. Circuits Syst. Signal Process. 38(10), 4446–4464 (2019)
E. Marshall, Commutativity of time-varying systems. Electron. Lett. 18, 539–540 (1977)
W. Robin, Operator factorization and the solution of second-order linear ordinary differential equations. Int. J. Math. Educ. Sci. Technol. 38(2), 189–211 (2007)
J. Walczak, A. Piwowar, Cascade connection of parametric sections and its properties. Przeglad Elektrotechniczny 86(1), 56–58 (2010)
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Ibrahim, S., Koksal, M.E. Realization of a Fourth-Order Linear Time-Varying Differential System with Nonzero Initial Conditions by Cascaded two Second-Order Commutative Pairs. Circuits Syst Signal Process 40, 3107–3123 (2021). https://doi.org/10.1007/s00034-020-01617-1
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DOI: https://doi.org/10.1007/s00034-020-01617-1