Self-Organizing Control Systems for Unstable and Deterministic Chaotic Processes

Authors: M. A. Beisenbi, N. M. Kissikova, S. E. Beisembina, S. T. Suleimenova, S. A. Kaliyeva

Abstract:

The paper proposes a method for constructing a self-organizing control system for unstable and deterministic chaotic processes in the class of catastrophe “hyperbolic umbilic” for objects with m-inputs and n-outputs. The self-organizing control system is investigated by the universal gradient-velocity method of Lyapunov vector-functions. The conditions for self-organization of the control system in the class of catastrophes “hyperbolic umbilic” are shown in the form of a system of algebraic inequalities that characterize the aperiodic robust stability in the stationary states of the system.

Keywords: Gradient-velocity method of Lyapunov vector-functions, hyperbolic umbilic, self-organizing control system, stability.

References:

[1] Loskutov A.Yu., Mikhajlov A.S. Osnovy` teorii slozhny`kh sistem – M. Izhevsk: Institut komp`yuterny`kh issledovanij, 2007, p. 620.
[2] Brok U. Teoriya khaosa. – M.: Nauka, 2011, p. 424.
[3] Andrievskij B.R., Fradkov A.L. Izbranny`e glavy` teorii avtomaticheskogo upravleniya. Gl. 13 Upravlenie nelinejny`mi kolebatel`ny`mi i khaoticheskimi sistemami – SPb.: Nauka, 1999, p. 475.
[4] Polyak B.T., Shherbakov P.S. Robastnaya ustojchivost` i upravlenie. – M.: Nauka, 2002, p.303.
[5] Dorato Peter, Yedavalli Rama K. Recent Advances in Robust Control. – New York: IEE Press – 1990.
[6] Beisenbi M.A. Modeli i metody` sistemnogo analiza i upravlenie determinirovanny`m khaosam v e`konomike. Astana, 2011, p. 201.
[7] Beisenbi M.A. Upravlyaemy`j khaos v razvitii e`konomicheskoj sistemy`. Nur-Sultan: TOO «Master Po», 2019, p. 168.
[8] Beisenbi M.A., Erzhanov B.A. Sistemy` upravleniya s povy`shenny`m potenczialom robastnoj ustojchivosti. – Astana, 2002, p. 164.
[9] Gilmor R. Prikladnaya teoriya katastrof. V 2-kh tomakh. T. 1. – M.: Mir, 1984, p. 287.
[10] Beisenbi M.A. Metody` povy`sheniya potencziala robastnoj ustojchivosti sistem upravleniya. – Astana, 2011, p.292.
[11] Beisenbi M.A. Issledovanie robastnoj ustojchivosti sistem avtomaticheskogo upravleniya metodom funkczii A.M. Lyapunova. – Astana, 2015, p.204.
[12] M. Beisenbi, S. Beisembina, A. Satpayeva. Synthesis of a control system of a deterministic chaotic process in the class of two-parameter structurally stable mappings: Stambul, Vol.3. – pp. 171-175, May 2021.
[13] M.A. Beisenbi, S.A. Kaliyeva, Zh. Abdugulova, A. Ostayeva. A new approach for synthesis of the control system by gradient-velocity method of Lyapunov vector functions. Journal of Theoretical and Applied Information Technology. E-ISSN: 1817-3195. №02(99), 2021. pp. 381-389.