Delayed state feedback and chaos control for time-periodic systems via a symbolic approach

Communications in Nonlinear Science and Numerical Simulation

Volumn 10, Issue 5, 2005, Pages 479-497

  Ma, Haitao ^a   Deshmukh, Venkatesh ^a   Butcher, Eric ^a   Averina, Victoria ^b,c  

^a University of Alaska   (United States)

^b UNIVERSITY OF ALASKA FAIRBANKS   (United States)

^c UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Chaos control; Symbolic computation; Time delay

Indexed keywords

CHAOS THEORY; COMPUTER SIMULATION; ERROR ANALYSIS; MATRIX ALGEBRA; PRODUCT DESIGN;

LINEAR TIME-PERIODIC DELAY (LTPD) SYSTEMS; NONLINEAR TIME-PERIODIC DELAY (NPTD) SYSTEMS;

FEEDBACK CONTROL;

MATHEMATICAL ANALYSIS;

EID: 10644293960     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2003.12.007     Document Type: Article

References (31)

1. Averina Victoria. Symbolic stability of delay differential equations, M.S. thesis, Department of Mathematical Science, University of Alaska Fairbanks, AK, August 2002
2. Barnett S. Polynomials and linear control systems. 1983 Madison Dekker Inc. New York
3. Butcher E.A. Ma H. Bueler E. Averina V. Szabo Z. Stability of linear time-periodic delay-differential equations via Chebyshev polynomials. Int. J. Num. Meth. Engr. 59 2004 895-922
4. Curtain R.F. Zwart H.J. An introduction to infinite-dimensional linear systems theory 1995 Springer-Verlag New York
5. Fox L. Parker I.B. Chebyshev polynomials in numerical analysis. 1968 Oxford Univ. Press London
6. Ma H-.T. Butcher E.A. Bueler E. Chebyshev expansion of linear dynamic systems with time delay and periodic coefficients under control excitations. J. Dyn. Syst. Meas. and Control 125 2003 236-243
7. Hale J. Verduyn Lunel S.M. Introduction to functional differential equations. 1993 Spring
8. Sinha S.C. Butcher E.A. Symbolic computation of fundamental solution matrices for linear time-periodic dynamical systems. J. Sound Vib. 206 1 1997 61-85
9. Sinha S.C. Wu D.H. An efficient computational scheme for the analysis of periodic systems. J. Sound Vib. 151 1 1991 91-117
10. Yakubovich V.A. Starzhinski V.M. Linear differential equations with periodic coefficients, Part. 1975 Wiley New York
11. Sinha S.C. Henrichs J.T. Ravindra B. A general approach in the design of active controllers for nonlinear systems exhibiting chaos Int. J. Bifurcat. Chaos 10 1 2000 165-178
12. Sinha S.C. Joseph P. Control of general dynamic systems with periodically varying parameters via Liapunov-Floquet transformation. ASME J. Dyn. Syst. Meas. Control 116 1994 650-658
13. Shampine LF, Thompson S. Solving DDEs in Matlab, Available from: http://www.radford.edu/%7Ethompson/webddes/
14. Malkus M.V.R. Non-periodic convection at high and low Prandtl number Mem. Soc. Roy. Sci. Liege, Ser. 6 4 1972 125-128
15. Yorke E.D. Yorke J.A. Metastable chaos: Transition to sustained chaotic behavior in the Lorenz model. J. Stat. Phys. 21 1979 263-277
16. Knobloch E. Chaos in a segmented disc dynamo. Phys. Lett. 82A 1981 439-440
17. Pedlosky J. Limit cycles and unstable baroclinic waves. J. Atmos. Sci. 29 1972 53
18. Pedlosky J. Frenzen C. Chaotic and periodic behavior of finite amplitude baroclinic waves J. Atmos. Sci. 37 1980 1177-1196
19. Chen G. Dong X. From chaos to order, perspectives and methodologies in controlling chaotic nonlinear dynamical systems. Int. J. Bifurcat. Chaos 3 1993 1363-1409
20. Jackson E.A. Controls of dynamic flows with attractors. Phys. Rev. A 44 1991 4839-4853
21. Ott E. Grebogi C. Yorke J.A. Controlling chaos. Phys. Lett. A 64 1990 1169-1199
22. Kapitaniak T. Controlling chaos. Theoretical and practical methods in nonlinear dynamics 1996 Academic Press
23. David A, Sinha SC. Control of chaos in nonlinear systems with time-periodic coefficients. In: Proceedings of the 2000 American Control Conference, Chicago, Illinois, 28-30 June 2000
24. Ma H, Deshmukh V, Butcher E, Averina V. Controller design for linear time-periodic delay systems via a symbolic approach. In: Proceedings of 2003 American Control Conference, Denver, CO, 4-6 June 2003
25. Guan X.-P. Chen C.-L. Peng H.-P. Fan Z.-P. Time-delayed feedback control of time-delay chaotic systems. Int. J. Bifurcat. Chaos 13 1 2003 193-205
26. Liao X.-X. Chen G.-R. Chaos synchronization of general Lur'e systems via time-delay feedback control. Int. J. Bifurcat. Chaos 13 1 2003 207-213
27. Butcher E.A. Sinha S.C. A hybrid formulation for the analysis of time-periodic linear systems via Chebyshev polynomials. J. Sound Vib. 195 3 1996 518-527
28. Phat Vu Ngoc Stabilization of linear continuous time-varying systems with state delays in Hilbert spaces. Electron. J. Differ. Equat. 67 2001 1-13
29. Gowrdon G, Sinha SC. Design of controllers with symbolic computation for nonlinear system exhibiting chaos, Technical Report, Department of Mechanical Engineering, Auburn University, 2003
30. Bleich ME, Socolar JES. Stability of periodic orbits controlled by time-delay feedback, Available from: http://arxiv.org/abs/chao-dyn/9510019, 28 October 1995
31. Maccari A. The response of a parametrically excited van der Pol oscillator to a time delay state feedback Nonlinear Dyn. 26 2001 105-119