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Decomposition of Second-Order Discrete-Time Linear Time-Varying Systems into First-Order Commutative Pairs

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Abstract

In this study, necessary and sufficient conditions for the decomposition of any second-order discrete-time linear time-varying system as a commutative pair of two first-order systems are presented. Commutativity conditions with zero initial conditions are first expressed by Theorem 1. Then, the conditions under nonzero initial conditions are studied and presented by Theorem 2. The results including decomposition formulas are well verified by examples worked by using MATLAB Simulink tool. The importance of the paper subject is emphasized in view of some engineering applications for getting better performance characteristics such as sensitivity, robustness and stability. In fact, the results of this paper can be used when the synthesis of a second-order system is made by cascaded and commutative pairs of simple first-order subsystems; and by using the explicit decomposition formulas derived in the paper this can be achieved very easily. Since different decompositions lead to different performance characteristics, one should choose the appropriate ones among them.

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References

  1. M.R. Ainbund, I.P. Maslenkov, Improving the characteristics of microchannel plates in cascade connection. Instrum. Exp. Tech. 26(3), 650–652 (1983)

    Google Scholar 

  2. D.V. Balandin, R.S. Biryukov, M.M. Kogan, Minimax control of deviations for the outputs of a linear discrete time-varying system. Autom. Remote Control 80(12), 2091–2107 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. I. Gohberg, M.A. Kaashoek, A.C.M. Ran, Partial role and zero displacements by cascade connection. SIAM J. Matrix Anal. Appl. 10(3), 316–325 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. A.G.J. Holt, K.M. Reineck, Transfer function synthesis for a cascade connection network. IEEE Trans. Circuit Theory 15(2), 162–163 (1968)

    Article  Google Scholar 

  5. D.N. Ibragimov, N.M. Novozhilin, E. Yu, Portseva1, on sufficient optimality conditions for a guaranteed control in the speed problem for a linear time-varying discrete-time system with bounded control. Autom. Remote Control 82(12), 2076–2096 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Ibrahim, M.E. Koksal, Realization of a fourth-order linear time-varying differential system with non-zero initial conditions by cascaded two second-order commutative Pairs. Circuits Syst. Signal Process. 40(6), 3107–3123 (2021)

    Article  Google Scholar 

  7. S. Ibrahim, M.E. Koksal, Commutativity of Sixth-order time-varying linear systems. Circuits Syst. Signal Process. 40(10), 4799–4832 (2021)

    Article  Google Scholar 

  8. M. Koksal, Commutativity of 2ND order time-varying systems. Int. J. Control 36, 541–544 (1982)

    Article  MathSciNet  Google Scholar 

  9. M. Koksal, General conditions for the commutativity of time-varying systems, in International Conference on Telecommunication and Control, Halkidiki, Greece (1984), pp. 223–225

  10. M. Koksal, A Survey on the Commutativity of Time-Varying Systems. METU, Technical Report no: GEEE CAS-85/1 (1985)

  11. M. Koksal, Effects of nonzero initial conditions on commutativity and those of commutativity on system sensitivity, in 2nd National Congress of Electrical Engineers, Ankara, Turkey, vol. 2/2 (1987), pp. 570–573

  12. M. Koksal, Effects of nonzero initial conditions on the commutativity of linear time-varying systems, in International Conference on Modeling and Simulation, Istanbul, Turkey, vol. 1A (1988), pp. 49–55

  13. M. Koksal, Effects of commutativity on system's sensitivity, in 6th International Symposium on Networks, Systems and Signal Processing, Zagreb, Yugoslavia (1989), pp. 61–62

  14. 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(2011), 1–25 (2011)

    Article  MATH  Google Scholar 

  15. M. Koksal, M.E. Koksal, Commutativity of cascade connected discrete-time linear time-varying systems. Trans. Inst. Meas. Control. 37(5), 615–622 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. M.E. Koksal, Commutativity of first-order discrete-time linear time-varying systems. Math. Methods Appl. Sci. 42(16), 5274–5292 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. M.E. Koksal, A. Yakar, Decomposition of a third-order linear time-varying differential system into its second and first-order commutative pairs. Circuits Syst. Signal Process. 38(10), 4446–4464 (2019)

    Article  Google Scholar 

  19. M.E. Koksal, An alternative method for cryptology in secret communication. J. Inf. Sci. Eng. 32(5), 1–23 (2020)

    Google Scholar 

  20. X. Lu, Q.Y. Zhang, X. Liang, H.X. Wang, C.Y. Sheng, Z.G. Zhang, W. Cui, Optimal linear quadratic Gaussian control for discrete time-varying system with simultaneous input delay and state/control-dependent noises. Optimal Control Appl. Methods 41(3), 882–897 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. R.G. Lyons, Understanding Digital Signal Processing, 3rd edn. (Prentice Hall, New York, 2011)

    Google Scholar 

  22. E. Marshal, Commutativity of time varying systems. Electron. Lett. 13, 539–540 (1977)

    Article  Google Scholar 

  23. B.T. Polyak, A.N. Vishnyakov, Multiplying disks: robust stability of a cascade connection. Eur. J. Control 2(2), 101–111 (1986)

    Article  MATH  Google Scholar 

  24. J. Walczak, A. Piwowar, Cascade connection of a parametric sections and its properties. Przeglad Elektrotechniczny 86(1), 56–58 (2010)

    Google Scholar 

  25. Z. Zhang, W. Lin, L. Zheng, P. Zhang, X. Qu, Y. Feng, A power-type varying gain discrete-time recurrent neural network for solving time-varying linear system. Neurocomputing 388, 24–33 (2020)

    Article  Google Scholar 

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  1. Department of Mathematics, Ondokuz Mayis University, 55139, Atakum, Samsun, Turkey

    Sümeyye Ar Güneş & Mehmet Emir Köksal

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  1. Sümeyye Ar Güneş
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  2. Mehmet Emir Köksal
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Correspondence to Mehmet Emir Köksal.

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Güneş, S.A., Köksal, M.E. Decomposition of Second-Order Discrete-Time Linear Time-Varying Systems into First-Order Commutative Pairs. Circuits Syst Signal Process 42, 2723–2739 (2023). https://doi.org/10.1007/s00034-022-02259-1

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