Research output: Contribution to journal › Article › peer-review
On the Aizerman Problem : Coefficient Conditions for the Existence of a Four-Period Cycle in a Second-Order Discrete-Time System. / Zvyagintseva, T. E. .
In: Vestnik St. Petersburg University: Mathematics, Vol. 53, No. 1, 26.03.2020, p. 37-44.Research output: Contribution to journal › Article › peer-review
}
TY - JOUR
T1 - On the Aizerman Problem
T2 - Coefficient Conditions for the Existence of a Four-Period Cycle in a Second-Order Discrete-Time System
AU - Zvyagintseva, T. E.
N1 - Publisher Copyright: © 2020, Pleiades Publishing, Ltd.
PY - 2020/3/26
Y1 - 2020/3/26
N2 - We consider in this paper an automatic control second-order discrete-time system whose nonlinearity satisfies the generalized Routh–Hurwitz conditions. Systems of this type are widely used in solving modern application problems that arise in engineering, theory of motion control, mechanics, physics, and robotics. Two constructed examples of discrete-time systems with nonlinearities that lie in a Hurwitz angle were presented in recent papers by W. Heath, J. Carrasco, and M. de la Sen. These examples demonstrate that in the discrete case, the Aizerman and Kalman conjectures are untrue even for second-order systems. One such system in these examples has a three-period cycle and the other system, a four-period cycle. We assume in the present paper that the nonlinearity is two-periodic and lies in a Hurwitz angle; here, we study a system for all possible parameter values. We explicitly present the conditions (for the parameters) under which it is possible to construct a two-periodic nonlinearity in such a way that a system with it is not globally asymptotically stable. Such a nonlinearity can be constructed in more than one way. We propose a method for constructing the nonlinearity in such a way that a family of four-period cycles is found in the system. The cycles are nonisolated; any solution of the system with the initial data, which lies on a certain specified ray, is a periodic solution.
AB - We consider in this paper an automatic control second-order discrete-time system whose nonlinearity satisfies the generalized Routh–Hurwitz conditions. Systems of this type are widely used in solving modern application problems that arise in engineering, theory of motion control, mechanics, physics, and robotics. Two constructed examples of discrete-time systems with nonlinearities that lie in a Hurwitz angle were presented in recent papers by W. Heath, J. Carrasco, and M. de la Sen. These examples demonstrate that in the discrete case, the Aizerman and Kalman conjectures are untrue even for second-order systems. One such system in these examples has a three-period cycle and the other system, a four-period cycle. We assume in the present paper that the nonlinearity is two-periodic and lies in a Hurwitz angle; here, we study a system for all possible parameter values. We explicitly present the conditions (for the parameters) under which it is possible to construct a two-periodic nonlinearity in such a way that a system with it is not globally asymptotically stable. Such a nonlinearity can be constructed in more than one way. We propose a method for constructing the nonlinearity in such a way that a family of four-period cycles is found in the system. The cycles are nonisolated; any solution of the system with the initial data, which lies on a certain specified ray, is a periodic solution.
KW - period cycle
KW - discrete systems
KW - Aizerman problem
KW - second-order discrete-time system
KW - Aizerman problem
KW - absolute stability
KW - Periodic solution
KW - periodic solution
UR - http://www.scopus.com/inward/record.url?scp=85082618903&partnerID=8YFLogxK
U2 - 10.1134/S1063454120010161
DO - 10.1134/S1063454120010161
M3 - Article
VL - 53
SP - 37
EP - 44
JO - Vestnik St. Petersburg University: Mathematics
JF - Vestnik St. Petersburg University: Mathematics
SN - 1063-4541
IS - 1
ER -
ID: 52526556