zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Double delayed feedback control for the stabilization of unstable steady states in chaotic systems. (English) Zbl 1221.37217
Summary: Double delayed feedback control (DDFC) method with two mutually prime delays is analytically analyzed for the stabilization of unstable steady states. Some stabilization criteria are proposed by utilizing Lyapunov theory and matrix inequality technique. The relation between the feedback gain matrices and the controller delays can be implicitly represented by the given criterion. Numerical results are also presented.
MSC:
37N35Dynamical systems in control
34D20Stability of ODE
34H10Chaos control (ODE)
37D45Strange attractors, chaotic dynamics
93B52Feedback control
References:
[1]Chen, G. R.; Dong, X.: From chaos to order: methodologies, perspectives, and applications, (1998)
[2]Boccaletti, S.; Grebogi, C.; Lai, Y. C.; Mancini, H.; Maza, D.: The control of chaos: theory and application, Phys rep 329, 103-197 (2000)
[3]Boccaletti, S.; Kurths, J.; Osipov, G.; Valladares, D. L.; Zhou, C. S.: The synchronization of chaotic systems, Phys rep 366, 1-101 (2002) · Zbl 0995.37022 · doi:10.1016/S0370-1573(02)00137-0
[4]Lu, J. F.: Generalized (complete, lag, anticipated) synchronization of discrete-time chaotic systems, Commun nonlinear sci numer simulat 13, 1851-1859 (2008) · Zbl 1221.37216 · doi:10.1016/j.cnsns.2007.04.022
[5]Ott, E.; Grebogi, C.; Yorke, J. A.: Controlling chaos, Phys rev lett 64, 1196-1199 (1990) · Zbl 0964.37501 · doi:10.1103/PhysRevLett.64.1196
[6]Pecora, L. M.; Carroll, T. L.: Synchronization in chaotic systems, Phys rev lett 64, No. 8, 821-824 (1990)
[7]Tanaka, K.; Ikeda, T.; Wang, H. O.: A unified approach to controlling chaos via an LMI-based fuzzy control system design, IEEE trans circuits syst I 45, 1021-1040 (1998) · Zbl 0951.93046 · doi:10.1109/81.728857
[8]Yang, T.; Yang, L. B.; Yang, C. M.: Impulsive control of Lorenz system, Physica D 110, 18-24 (1997) · Zbl 0925.93414 · doi:10.1016/S0167-2789(97)00116-4
[9]Sun, J. T.; Zhang, Y. P.; Wu, Q. D.: Less conservative conditions for asymptotic stability of impulsive control systems, IEEE trans auto contr 48, No. 5, 829-831 (2003)
[10]Rafikov, M.; Balthazar, J. M.: On control and synchronization in chaotic and hyperchaotic systems via linear feedback control, Commun nonlinear sci numer simulat 13, 1246-1255 (2008) · Zbl 1221.93230 · doi:10.1016/j.cnsns.2006.12.011
[11]Lu, J. Q.; Cao, J. D.: Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters, Chaos 15, No. 4, 043901 (2005) · Zbl 1144.37378 · doi:10.1063/1.2089207
[12]Cao, J. D.; Lu, J. Q.: Adaptive synchronization of neural networks with or without time-varying delay, Chaos 16, No. 1, 013133 (2006) · Zbl 1144.37331 · doi:10.1063/1.2178448
[13]Huang, D. B.: Stabilizing near-nonhyperbolic chaotic systems with applications, Phys rev lett 93, 214101 (2004)
[14]Cao, J. D.; Li, H. X.; Ho, D. W. C.: Synchronization criteria of Lur’e systems with time-delay feedback control, Chaos soliton fract 23, No. 4, 1285-1298 (2005) · Zbl 1086.93050 · doi:10.1016/j.chaos.2004.06.025
[15]Pyragas, K.: Continuous control of chaos by self-controlling feedback, Phys lett A 170, No. 6, 421-428 (1992)
[16]Pyragas, V.; Pyragas, K.: Delayed feedback control of the Lorenz system: an analytical treatment at a subcritical Hopf bifurcation, Phys rev E 73, No. 3, 036215 (2006)
[17]Xu, S. Y.; Lam, J.; Zou, Y.: Delay-dependent approach to stabilization of time-delay chaotic systems via standard and delayed feedback controllers, Int J bifur chaos 15, No. 4, 1455-1465 (2005) · Zbl 1089.93017 · doi:10.1142/S0218127405012570
[18]Guan, X. P.; Feng, G.; Chen, C. L.: A stabilization method of chaotic systems based on full delayed feedback controller design, Phys lett A 348, No. 3-6, 210-221 (2006) · Zbl 1195.37022 · doi:10.1016/j.physleta.2005.08.061
[19]Park, J. H.; Kwon, O. M.: A novel criterion for delayed feedback control of time-delay chaotic systems, Chaos solitons fract 23, No. 2, 495-501 (2005) · Zbl 1061.93507 · doi:10.1016/j.chaos.2004.05.023
[20]Ahlborn, A.; Parlitz, U.: Stabilizing unstable steady states using multiple delay feedback control, Phys rev lett 93, No. 26, 264101 (2004)
[21]Ahlborn, A.; Parlitz, U.: Controlling dynamical systems using multiple delay feedback control, Phys rev E 72, No. 1, 016206 (2005)
[22]Yang, L.; Liu, Z. R.; Mao, J. M.: Controlling hyperchaos, Phys rev lett 84, No. 1, 67-70 (2000)
[23]Baer, T.: Large-amplitude fluctuations due to longitudinal mode-coupling in diode-pumped intracavity-doubled ND-YAG lasers, J opt soc am B 3, No. 9, 1175-1180 (1986)
[24]Myneni, K.; Barr, T. A.; Corron, N. J.; Pethel, S. D.: New method for the control of fast chaotic oscillations, Phys rev lett 83, No. 11, 2175-2178 (1999)
[25]Pyragas, K.: Control of chaos via extended delay feedback, Phys lett A 206, No. 5-6, 323-330 (1995) · Zbl 0963.93523 · doi:10.1016/0375-9601(95)00654-L
[26]Socolar, J. E. S.; Gauthier, D. J.: Analysis and comparison of multiple-delay schemes for controlling unstable fixed points of discrete maps, Phys rev E 57, No. 6, 6589-6595 (1998)
[27]Tian, Y. P.; Chen, G. R.: A separation principle for dynamical delayed output feedback control of chaos, Phys lett A 284, No. 1, 31-42 (2001) · Zbl 0985.37049 · doi:10.1016/S0375-9601(01)00275-4
[28]Nakajima, H.: Delayed feedback control with state predictor for continuous-time chaotic systems, Int J bifur chaos 12, No. 5, 1067-1077 (2002) · Zbl 1051.93527 · doi:10.1142/S0218127402004917
[29]Bielawski, S.; Derozier, D.; Glorieux, P.: Controlling unstable periodic-orbit by a delayed continuous feedback, Phys rev E 49, No. 2, R971-R974 (1994)
[30]Namajunas, A.: Stabilization of an unstable steady-state in a MacKey-Glass system, Phys lett A 204, 255-262 (1995)
[31]Hikihara, T.: An experimental study on stabilization of unstable periodic motion in magneto-elastic chaos, Phys lett A 211, 29-36 (1996)
[32]Ushio, T.: Limitation of delayed feedback control in nonlinear discrete-time systems, IEEE trans circuits syst I 43, 815-816 (1996)
[33]Nakajima, H.; Ueda, Y.: Limitation of generalized delayed feedback control, Physica D 111, No. 1-4, 143-150 (1998) · Zbl 0927.93030 · doi:10.1016/S0167-2789(97)80009-7
[34]Yamamoto, S.: Dynamic delayed feedback controllers for chaotic discrete-time systems, IEEE trans circuits syst I 48, 785-789 (2001) · Zbl 1159.93329 · doi:10.1109/81.928162
[35]Hövel, P.; Schöll, E.: Control of unstable steady states by time-delayed feedback methods, Phys rev E 72, No. 4, 046203 (2005)
[36]Xu, S. Y.; Lam, J.: Improved delay-dependent stability criteria for time-delay systems, IEEE trans auto contr 50, No. 3, 384-387 (2005)
[37]Kolmanovskii, V. B.; Myshkis, A. D.: Introduction to the theory and applications of functional differential equations, (1999)
[38]Hale, J. K.; Lunel, S. M. V.: Introduction to functional differential equations, (1993)
[39]Boyd, S.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V.: Linear matrix inequalities in system and control theory, (1994)
[40]Elghaoui, L.; Oustry, F.; Aitrami, M.: A cone complementarity linearization algorithm for static output-feedback and related problems, IEEE trans auto contr 42, No. 8, 1171-1176 (1997) · Zbl 0887.93017 · doi:10.1109/9.618250
[41]Lorenz, E. N.: Deterministic non-periodic flows, J atmos sci 20, 130-141 (1963)
[42]Nesterov, Y.; Nemirovskii, A.: Interior-point polynomial methods in convex programming, (1994)
[43]Gahinet, P.; Mathworks, I.: LMI control toolbox for use with Matlab, Mathworks (1995)