Geodesic
Stabilization of an unstable limit cycle of relay chaotic system
Matematičeskoe modelirovanie i čislennye metody, no. 6 (2015), pp. 87-104 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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 fac-tor 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.
Mots-clés : chaos
Keywords: Poincare map, limit cycle, stabilization, relay system, regulator syn-thesis, OGY method. 
@article{MMCM_2015_6_a5,
     author = {V. I. Krasnoschechenko},
     title = {Stabilization of an unstable limit cycle of relay chaotic system},
     journal = {Matemati\v{c}eskoe modelirovanie i \v{c}islennye metody},
     pages = {87--104},
     year = {2015},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MMCM_2015_6_a5/}
}
TY  - JOUR
AU  - V. I. Krasnoschechenko
TI  - Stabilization of an unstable limit cycle of relay chaotic system
JO  - Matematičeskoe modelirovanie i čislennye metody
PY  - 2015
SP  - 87
EP  - 104
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/MMCM_2015_6_a5/
LA  - ru
ID  - MMCM_2015_6_a5
ER  - 
%0 Journal Article
%A V. I. Krasnoschechenko
%T Stabilization of an unstable limit cycle of relay chaotic system
%J Matematičeskoe modelirovanie i čislennye metody
%D 2015
%P 87-104
%N 6
%U http://geodesic.mathdoc.fr/item/MMCM_2015_6_a5/
%G ru
%F MMCM_2015_6_a5
V. I. Krasnoschechenko. Stabilization of an unstable limit cycle of relay chaotic system. Matematičeskoe modelirovanie i čislennye metody, no. 6 (2015), pp. 87-104. http://geodesic.mathdoc.fr/item/MMCM_2015_6_a5/

[1] Lorenz E.N., “Deterministic Nonperiodic Flow”, J. Atmosferic Sci., 20:2 (1963), 130–141 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[2] Andrievskiy B.R., Fradkov A.L., Automatics and Remote Control, 2003, no. 5, 3–45 | MR | Zbl

[3] Cook P.A., “Simple feedback systems with chaotic behavior”, Syst. Cont. Letters, 1985, no. 6, 223–227 | DOI | MR | Zbl

[4] Ott E., Grebogi C., Yorke J., “Controlling Chaos”, Phys. Rev. Letters, 64:11 (1990), 1196–1199 | DOI | MR | Zbl

[5] Holzhuter T., Klinker T., “Ein einfaches Relais-System mit chaotischem Verhalten”, Nichtlineare Dynimik, 1997, 83–96

[6] Holzhuter T., Klinker T., “Control of a chaotic relay system using the OGY method”, Z. Naturforsch, 53a (1998), 1029–1036

[7] PyragasV.K., “Continuos Control of Chaos by Self-Controlling Feedback”, Phys. Letters. A, 170 (1992), 421–428 | DOI

[8] Morgil O., “On the Stability of Delayed Feedback Controllers for Discrete Time Systems”, Phys. Letters. A, 335 (2005), 31–42 | DOI

[9] Ushio T., “Limitation of Delayed Feedback Control in Nonlinear Discrete Time Systems”, IEEE Trans. Circ. Syst. I, 43 (1996), 815–816 | DOI

[10] Bhajekar S., Joncheere E.A., Hammad A., “Hinf Control of Chaos”, Proceedings of 33d Conf. Decision Control (Lake Buena Vista, FL), 1994, 3285–3286

[11] Boccletti S., Arecchi F.T., “Adaptive Control of Chaos”, Europhys. Letters, 31 (1995), 127–132 | DOI | MR

[12] Yau H.T., Chen C.K., Chen C.L., “Sliding Mode Control of Chaotic Systems with Uncertainties”, Int. J. Bifurcat. Chaos, 10:5 (2000), 1139–1147 | DOI