Commutativity and Commutative Pairs of Some Differential Equations

Mehmet Emir Koksal

Abstract


In this study, explicit differential equations representing commutative pairs of some well-known second-order linear time-varying systems have been derived. The commutativity of these systems are investigated by considering 30 second-order linear differential equations with variable coefficients. It is shown that the system modeled by each one of these equations has a commutative pair with(or without) some conditions or not. There appear special cases such that both, only one or neither of the original system and its commutative pair has explicit analytic solution. Some benefits of commutativity have already been mentioned in the literature but a new application in cryptology for obscuring transmitted signals in telecommunication is illustrated in this paper.

Keywords


Commutativity, differential equations, analytic solutions, analogue control, robust control, cryptology

Full Text:

PDF

References


M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th edition, National bureau of Standards (1972).

K.J. Astrom, R.M. Murray, Feedback Systems - An Introduction for Scientists and Engineers, Version v 2.10c, Princeton University Press, Aniston and Oxford (2010).

R.L. Boylestad and L. Nashelsky, Electronic Devices and Circuit Theory, Prentice Hall (2002).

R.C. Dorf and J.A. Svadova, Introduction to Electric Circuits, Wiley International Edition (2004).

R. Dorf and R. Bishop, Modern Control Systems, Pearson New International Edition, 12th Edition (2013).

C. Freytag, Design of Feedback Control Systems, 4th Edition (2016).

I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 7th Edition, Academic Press (2007).

M. Koksal, Commutativity of second order time-varying systems, International Journal of Control, 3 (1982), 541 – 544.

M. Koksal, Corrections on ‘Commutativity of second-order time-varying systems’, International Journal of Control 1 (1983), 273 – 274.

M.E. Koksal, Decomposition of a second-order linear time-varying differential system as the series connection of two first-order commutative pairs, Open Mathematics 14 (2016), 693 – 704.

M. Koksal, Effects of commutativity on system sensitivity, In: Proceeding of the 6th Int. Symposium on Networks, Systems and Signal Processing, Zargeb, Yugoslavia, pp. 61 – 62 (1989).

M. Koksal and M.E. Koksal, Commutativity of linear time-varying differential systems with nonzero initial conditions: a review and some new extensions, Mathematical Problems in Engineering 2011 (2011), 1 – 25.

M. Koksal and M.E. Koksal, Commutativity of cascade connected discrete time linear time-varying systems, Transactions of the Institute of Measurement and Control 37 (2015), 615 – 622.

M.E. Koksal, The second order commutative pairs of a first-order linear time-varying system, Applied Mathematics and Information Sciences 9 (2015), 1 – 6.

M. Koksal, An exhaustive study on the commutativity of time varying systems, International Journal of Control 5 (1988), 1521 – 1537.

R.G. Lyons, Understanding Digital Signal Processing, 3rd Ed., Prentice Hall (2011).

E. Marshall, Commutativity of time varying systems, Electro Letters 18 (1977), 539 – 540.

G.M. Miller and J.S. Beasley, Modern Electronic Communication, Prentice Hall (2002).

A.V. Oppenheim and R.W. Schafer, Discrete-Time Signal Processing, Prentice-Hall, Engle-wood Cliffs, New Jersey, p. 406 (1989).

A.D. Polyanin and V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall, CRC Press (2003).

S.V. Saleh, Comments on ‘Commutativity of second-order time-varying systems’, International Journal of Control 37 (1983), 1195.

R.T. Stefani, B. Shahian, C.J. Savant and G.H. Hostetter, Design of Feedback Control Systems, Oxford Series in Electrical and Computer Engineering, 4th Edition (2001).

P.H. Young, Electronic Communication Techniques, Engle Wood Cliffs (1994).

B. Zhou, On asymptotic stability of linear time-varying systems, Automatica 68 (2016), 266 – 276.

D. Zwillinger, Handbook of Differential Equations, 3rd Edition, Academic Press (1997).


Refbacks

  • There are currently no refbacks.


eISSN 0975-8607; pISSN 0976-5905