On the conditions for cycles existence in a second-order discrete-time system with sector-nonlinearity
DOI:
https://doi.org/10.21638/spbu01.2021.106Abstract
In this paper, a second-order discrete-time automatic control system is studied. This work is a continuation of the research presented in the author’s papers “On the Aizerman problem: coefficient conditions for the existence of a four-period cycle in a second-order discrete-time system” and “On the Aizerman problem: coefficient conditions for the existence of threeand six-period cycles in a second-order discrete-time system”, where systems with two- and three-periodic nonlinearities lying in the Hurwitz angle were considered. The systems with nonlinearities subjected to stronger constraints are discussed in this paper. It is assumed that the nonlinearity not only lies in the Hurwitz angle, but also satisfies the additional sector-condition. This formulation of the problem is found in many works devoted to theoretical and applied questions of the automatic control theory. In this paper, a system with such nonlinearity is explored for all possible values of the parameters. It is shown that in this case there are parameter values for which a system with a two-periodic nonlinearity has a family of four-period cycles, and a system with a three-periodic nonlinearity has a family of three- or six-period cycles. The conditions on the parameters under which the system can have a family of periodic solutions are written out explicitly. The proofs of the theorems provide a method for constructing nonlinearity in such a way that any solution of the system with initial data lying on some definite ray is periodic.Keywords:
second-order discrete-time system, Aizerman conjecture, sector nonlinearity, absolute stability, periodic solution
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Литература
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6. Heath W. P., Carrasco J., de la Sen M. Second-order counterexample to the discrete-time Kalman conjecture. Proceedings of the European Control Conference (ECC15), 2015, Linz, Austria, 981–985 (2015). https://doi.org/10.1109/ECC.2015.7330669
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References
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5. Zvyagintseva T.E. On the Aizerman problem: coefficient conditions for the existence of threeand six-period cycles in a second-order discrete-time system. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 7 (65), iss. 2, 309–318 (2020). https://doi.org/10.21638/11701/spbu01.2020.213 (In Russian) [Engl. transl.: Vestnik St. Petersb. Univ. Math. 53, iss. 2, 206–213 (2020). https://doi.org/10.1134/S106345412002017X].
6. Heath W. P., Carrasco J., de la Sen M. Second-order counterexample to the discrete-time Kalman conjecture. Proceedings of the European Control Conference (ECC15). 2015. Linz, Austria, 981–985 (2015). https://doi.org/10.1109/ECC.2015.7330669
7. HeathW. P., Carrasco J., de la Sen M. Second-order counterexamples to the discrete-time Kalman conjecture. Automatica 60, 140–144 (2015). https://doi.org/10.1016/j.automatica.2015.07.005
8. Heath W. P., Carrasco J. Global asymptotic stability for a class of discrete-time systems. Proceedings of the European Control Conference (ECC15), 2015, Linz, Austria, 969–974 (2015). https://doi.org/10.1109/ECC.2015.7330667
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Published
2021-05-29
How to Cite
Zvyagintseva, T. E. (2021). On the conditions for cycles existence in a second-order discrete-time system with sector-nonlinearity. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(1), 63–72. https://doi.org/10.21638/spbu01.2021.106
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
