681.513.5 Stabilization of an unstable limit cycle of relay chaotic system

Krasnoschechenko V. I. (Bauman Moscow State Technical University)

CHAOS, POINCARE MAP, LIMIT CYCLE, STABILIZATION, RELAY SYSTEM, REGULATOR SYNTHESIS, OGY METHOD


doi: 10.18698/2309-3684-2015-2-87104


The article presents an algorithm of synthesis for stabilization of an unstable limit cycle of relay chaotic system. One-dimensional discrete Poincare map is used in algorithm for finding fixed points of the period one (limit cycles of initial continuous system). It is shown, that classical OGY method of dead beat regulator synthesis does not solve the problem as it takes into account only speed of the target coordinate what is not sufficient for stabilizing. The proposed algorithm is based on search of the necessary regulator factor by solving an inverse problem: at first some factor is assigned and then two-step procedure of system transition to the following switching point (with correction) is carried out. The task of correction is performed in a complete neighborhood of target coordinate position and speed, and it provides stabilization of a limit cycle by adjusting small amplitude pulses in the chosen area of entry conditions (area of stabilization) as evidenced by the simulation results.


[1] Lorenz E.N. Deterministic Nonperiodic Flow. J. Atmosferic Sci., 1963, vol. 20 (2), pp. 130–148.
[2] Andrievskiy B.R., Fradkov A.L. Avtomatika i telemekhanika – Automatics and Remote Control, 2003, no. 5, pp. 3–45; 2004, no. 4, pp. 3–34.
[3] Cook P.A. Simple feedback systems with chaotic behavior. Syst. & Cont. Letters, 1985, no. 6, pp. 223–227.
[4] Ott E., Grebogi C., Yorke J. Controlling Chaos. Phys. Rev. Letters, 1990,
vol. 64, no. 11, pp. 1196–1199.
[5] Holzhuter T., Klinker T. Ein einfaches Relais-System mit chaotischem Verhalten. Nichtlineare Dynimik, Chaos und Strukturbildung. Meyer-Spasche R., Rast M. und Zenger C., eds. Munchen, Akademicher Verlag, 1997, pp. 83–96.
[6] Holzhuter T., Klinker T. Control of a chaotic relay system using the OGY method. Z. Naturforsch, 1998, vol. 53a, pp. 1029–1036.
[7] PyragasV.K. Continuos Control of Chaos by Self-Controlling Feedback. Phys. Letters. A, 1992, vol. 170, pp. 421–428.
[8] Morgil O. On the Stability of Delayed Feedback Controllers for Discrete Time Systems. Phys. Letters. A, 2005, vol. 335, pp. 31–42.
[9] Ushio T. Limitation of Delayed Feedback Control in Nonlinear Discrete Time Systems. IEEE Trans. Circ. Syst. I, 1996, vol. 43, pp. 815–816.
[10] Bhajekar S., Joncheere E.A., Hammad A. Hinf Control of Chaos. Proceedings of 33d Conf. Decision Control, Lake Buena Vista, FL, 1994, pp. 3285–3286.
[11] Boccletti S., Arecchi F.T. Adaptive Control of Chaos. Europhys. Letters, 1995, vol. 31, pp. 127–132.
[12] Yau H.T., Chen C.K., Chen C.L. Sliding Mode Control of Chaotic Systems with Uncertainties. Int. J. Bifurcat. Chaos, 2000, vol. 10, no. 5, pp. 1139–1147.


Krasnoschechenko V. Stabilization of an unstable limit cycle of relay chaotic system. Маthematical Modeling and Coтputational Methods, 2015, №2 (6), pp. 87-104



Download article

Колличество скачиваний: 322