Abstract
This paper studies commutativity of sixth-order differential systems with zero and nonzero initial conditions (ICs). Given a sixth-order differential system \(A\), we find its commutative pair, that is a new differential system \(B\) of order \(m \le 6\); the obtained result will be used to show the input–output equivalency between the cascaded-connected systems \(AB\) and \(BA\). Necessary and sufficient conditions for commutativity of sixth-order continuous-time linear time-varying systems (CTLTVSs) are derived. Based on our findings, we discovered that only a small group of sixth-order CTLTVSs has commutative pairs other than their feedback conjugates obtained by constant feedback and forward path gains. The explicit results obtained in this paper will help to fill the gap on commutativity theory, system behaviors which has significant application to science and play an important role in engineering. This research will also address the problems of sensitivity due to changes in ICs or parameters, and also disturbance as a result of external noise. The results are well verified by examples as well as validated by Matlab Simulink tool and Wolfram Mathematica 11.
Similar content being viewed by others
Data availability
My manuscript has no associated data.
References
M.R. Ainbund, I.P. Maslenkov, Improving the characteristics of micro channel plates in cascade connection. Instrum. Exp. Tech. 26(3), 650–652 (1983)
C. Chicone, Ordinary Differential Equations with Applications, 2nd edn. (Springer, Berlin, 2006).
I. Gohberg, M.A. Kaashoekve, A.C.M. Ran, Partial role and zero displacement by cascade connection. SIAM J. Matrix Anal. Appl. 10(3), 316–325 (1989)
A.G.J. Holt, K.M. Reineck, Transfer function synthesis for a cascade connection network. IEEE Trans. Circuit Theory 15(2), 162–163 (1968)
M. Koksal, Commutativity of second order time-varying systems. Int. J. Control 36(3), 541–544 (1982)
M. Koksal, General commutativity conditions for time-varying systems, in 2nd National Congress of Electrical Engineers, vol. 2, no. 2 (1987a), pp. 566–569.
M. Koksal, Effects of nonzero initial conditions on commutativity and those of commutativity on system sensitivity, in 2nd National Congress of Electrical Engineers vol. 2, no, 2 (1987b), pp. 570–573.
M. Koksal, Effects of nonzero initial conditions on commutativity and those of commutativity on system sensitivity, in 2nd National Congress of Electrical Engineers vol. 2, no. 2 (1987c) pp. 570–573.
M. Koksal, Effects of nonzero initial conditions on the commutativity of linear time-varying systems. Int. Conf. Model. Simul. 1A, 49–55 (1988)
M. Koksal, Commutativity of 4th order systems and Euler systems, in Symposium of Yildiz Technical University on the Role of Engineering on the Development of our Country (1988a), pp. 398–408.
M. Koksal, An exhaustive study on the commutativity of time-varying systems. Int. J. Control 47(5), 1521–1537 (1988)
M. Koksal, Effects of commutativity on system's sensitivity, in 6th International Symposium on Networks, Systems and Signal Processing, (1989), pp. 61–62.
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. Koksal, M.E. Koksal, Ardışık Bağlı Zamanla Değişen Doğrusal Ayrık-zaman Sistemlerinin Sıra-değişim Özelliği, Turkish National Meeting on Automatics Control, (2013), pp. 1128–1131
M. Koksal, M.E. Koksal, Commutativity of cascade connected discrete time linear varying systems. Trans. Inst. Meas. Control. 37, 615–622 (2015)
M.E. Koksal, Inverse Commutativity Conditions for Second-order Linear Time-Varying Systems. J. Math. (2017), pp. 1–6.
M.E. Koksal, Commutativity of systems with their feedback conjugates. Trans. Inst. Meas. Control. 41(3), 696–700 (2019)
M.E. Koksal, Explicit commutativity conditions for second order linear time varying systems with non-zero initial conditions. Archiv. Control Sci. 29(3), 413–432 (2019)
M.E. Koksal, Transitivity of commutativity for second-order linear time-varying analogue systems. Circuits Syst. Signal Process. 38(3), 1385–1395 (2019)
M.E. Koksal, Commutativity of first-order discrete-time linear time-varying systems. Math. Methods Appl. Sci. 42(16), 5274–5292 (2019)
M.E.A. KoksalYakar, 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)
B.T. Polyak, A.N. Vishnyakov, Multiplying disks, robust stability of a cascade connection. Eur. J. Control. 2(2), 101–111 (1976)
S.V. Saleh, Comments on commutativity of second-order time-varying systems. Int. J. Control 37, 1195–1195 (1983)
J. Walczak, A. Piwowar, Cascade connection of a parametric sections and its properties. Przeglad Elektrotechniczny 86(1), 56–58 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ibrahim, S., Koksal, M.E. Commutativity of Sixth-Order Time-Varying Linear Systems. Circuits Syst Signal Process 40, 4799–4832 (2021). https://doi.org/10.1007/s00034-021-01709-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-021-01709-6