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Decomposition of a second-order linear time-varying differential system as the series connection of two first order commutative pairs. (English) Zbl 1350.93032
Summary: Necessary and sufficiently conditions are derived for the decomposition of a second order linear time-varying system into two cascade connected commutative first order linear time-varying subsystems. Explicit formulas describing these subsystems are presented. It is shown that a very small class of systems satisfies the stated conditions. The results are well verified by simulations. It is also shown that its cascade synthesis is less sensitive to numerical errors than the direct simulation of the system itself.

MSC:
93B17 Transformations
93B11 System structure simplification
93A30 Mathematical modelling of systems (MSC2010)
93B35 Sensitivity (robustness)
93C05 Linear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
93C99 Model systems in control theory
34A30 Linear ordinary differential equations and systems
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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[1] Holt A.G.J., Reineck K.M., Transfer function synthesis for a cascade connection network, IEEE Transactions on Circuit Theory, 1968, 15 (2), 162-163
[2] Ainbund M.R., Maslenkov I.P., Improving the characteristics of microchannel plates in cascade connection, Instruments and Experimental Techniques, 1983, 26 (3), 650-652
[3] Gohberg I., Kaashoek M.A., Ran A.C.M., Partial role and zero displacement by cascade connection, SIAM Journal on Matrix Analysis and Applications, 1989, 10 (3), 316-325 · Zbl 0677.93032
[4] Polyak B.T., Vishnyakov A.N., Multiplying disks: robust stability of a cascade connection, European Journal of Control, 1996, 2 (2), 101-111 · Zbl 0868.93051
[5] Walczak J., Piwowar A., Cascade connection of a parametric section and its properties, Przeglad Elektrotechniczny, 2010, 86 (1), 56-58
[6] Marshall E., Commutativity of time-varying systems, Electro Letters, 1977, 18, 539-540
[7] Koksal M., Commutativity of second order time-varying systems, International Journal of Control, 1982, 36 (3), 541-544 · Zbl 0488.93022
[8] Saleh S.V., Comments on ‘Commutativity of second-order time-varying systems’, International Journal of Control, 1983, 37, 1195-1195 · Zbl 0508.93031
[9] Koksal M., General conditions for commutativity of time-varying systems, Proceedings of the International Conference on Telecommunication and Control, (1984, Halkidiki, Greece), 1984, 223-225
[10] Koksal M., A survey on the commutativity of time-varying systems. (Circuit and Systems, Middle East Technical University, Department of Electrics & Electronics Engineering, Technical Report no: GEEE CAS-85/1, 1985), p. 5
[11] Koksal M., An exhaustive study on the commutativity of time-varying systems, International Journal of Control, 1988, 47 (5), 1521-1537 · Zbl 0643.93015
[12] Koksal M., Koksal M.E., Commutativity of linear time-varying differential systems with non-zero initial conditions: A review and some new extensions, Mathematical Problems in Engineering, 2011, 2011 (2011), 1-25 · Zbl 1241.34015
[13] Koksal M., Effects of nonzero initial conditions on the commutativity of linear time-varying systems, Proceedings of the International AMSE Conference on Modeling and Simulation, (June 1988, Istanbul, Turkey), 1988, 1A, 49-55
[14] Koksal, M., Effects of commutativity on system sensitivity, Proceedings of the 6th International Symposium on Networks, Systems and Signal Processing, (June 1989, Zagreb, Yugoslavia), 1989, 61-62
[15] Koksal M., Koksal M.E., Commutativity of cascade connected discrete-time linear time-varying systems, Transaction of the Institute of Measurement and Control, 2015, 37 (5), 615-622.
[16] C.A. Desoer, Slowly varying system \(\dot x = a\left( t \right)x\), IEEE Transactions on Automatic Control, 14 (1969), pp. 780-781
[17] Okano R., Kida T., Nagashio T., Asymptotic stability of second-order linear time-varying systems, Journal of Guidance, Control, and Dynamics, 2006, 29 (6), 1472-1476
[18] Ratchagit K., Stability of linear time-varying systems, International Journal of Pure and Applied Mathematics, 2010, 63 (4), 411-417 · Zbl 1219.93102
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