Robust Conversion of Chaos into an Arbitrary Periodic Motion
One of the most attractive and important field of chaos theory is control of chaos. In this paper, we try to present a simple framework for chaotic motion control using the feedback linearization method. Using this approach, we derive a strategy, which can be easily applied to the other chaotic systems. This task presents two novel results: the desired periodic orbit need not be a solution of the original dynamics and the other is the robustness of response against parameter variations. The illustrated simulations show the ability of these. In addition, by a comparison between a conventional state feedback and our proposed method it is demonstrated that the introduced technique is more efficient.
[1] R.M.May, "Simple Mathematical Models With Very Complicated
Dynamics", Nature 261, 459. 1976.
[2] R.M.Crownover, "Introduction to Fractals and chaos", Jones and Bartlett
Publishers, 1995.
[3] A.J.Lichtenber, and M.A.Liebermann,"Regular and Stochastic Motion.
Springer, Heidelberg-New York.1982.
[4] B.R. Andrievskii, and A.L.Fradkov, " Control of chaos: Methods and
applications. I. Methods:, Avtom.Telemkh, no. 5, pp.3-45. 2003.
[5] Li, T. and Yorke, J.A., Period Three Implies Chaos, Am. Math.
Monthly.,1975, vol. 82, pp. 985-992..
[6] A.Razminia, and M.A.Sadrnia, "Chua-s circuit regulation using a
nonlinear adaptive feedback technique", International journal of
electronics, circuits and systems, Vol 2. No1. 2008.
[7] Yongai Zheng," Controlling chaos based on an adaptive adjustment
mechanism", Chaos, Solitons and Fractals,2005.
[8] Battle, E.Fossas, " Stabilization of periodic orbits of the buck converter
by time delayed feedback", Int. J. Circ. Theory Appl., vol. 27, pp. 617-
631. 1999.
[9] H.Khalil, " Nonlinear Systems", Prentice hall, 2006..
[10] J Chaohong Cai , Zhenyuan Xu, Wenbo Xu, Converting chaos into
periodic motion by state feedback control, Automatica 38 (2002),pp.
1927 - 1933.
[11] R. Ghidossi, D. Veyret, P. Moulin, "Computational fluid dynamics
applied to membranes: State of the art and opportunities", Chemical
Engineering and Processing, Volume 45, Issue 6, June 2006, Pages 437-
454.
[12] P. Magni, M. Simeoni, I. Poggesi, M. Rocchetti, G. De Nicolao, "A
mathematical model to study the effects of drugs administration on
tumor growth dynamics", Mathematical Biosciences, Volume 200, Issue
2, April 2006, Pages 127-151.
[13] Shu Chen, Guo-Hua Hu, Chen Guo, Hui-Zhou Liu, "Experimental study
and dissipative particle dynamics simulation of the formation and
stabilization of gold nanoparticles in PEO-PPO-PEO block copolymer
micelles" Chemical Engineering Science, Volume 62, Issues 18-20,
September-October 2007, Pages 5251-5256.
[14] Franz S. Hover, Michael S. Triantafyllou, "Application of polynomial
chaos in stability and control" Automatica, Volume 42, Issue 5, May
2006, Pages 789-795.
[15] R.L.Devaney, "A First Course in Chaotic Dynamical Systems: Theory
and Experiment" Addison-Wesley, 1997.
[16] J.Guckenheimer and Holmes, "Nonlinear oscillation, Dynamical
Systems, and Bifurcation of Vector Fields", SpringerVerlag, New York,
1990.
[17] David A. Towers, Vicente R. Varea, "Elementary Lie algebras and Lie
A-algebras" Journal of Algebra, Volume 312, Issue 2, 15 June 2007,
Pages 891-901.
[18] Jorge Lauret, Cynthia E. Will, "On Anosov automorphisms of
manifolds" Journal of Pure and Applied Algebra, Volume 212, Issue 7,
July 2008, Pages 1747-1755.
[19] Chen, G., 1996, "Controlling Chaotic Trajectories to Unstable Limit
Cycles", Proceedings of the American Control Conference, San Diego,
California, pp. 2413-2414.
[20] C.D.Meyer, "Matrix analysis and applied linear algebra", SIAM 2000J.
[21] Wang, and J.Zhang, "Chaotic secure communication based on nonlinear
autoregressive filter with changeable parameters", Phy.Lett.A 357,
pp.323-329, 2006.
[22] Sinha, S.C., and Joseph, P., "Control of General Dynamical Systems
with Periodically Varying Parameters Via Lyapunov-Floquet
Transformation", ASME Journal of Dynamical Systems, Measurement,
and Control, Vol. 116, pp. 650-658, 1994.
[23] R.L.Devaney, " Topological Bifurcation" Topology Proceedings 28
pp.99-112, 2004.
[24] Alexander L. Fradkov, Robin J. Evans, "Control of chaos: Methods and
applications in engineering,", Annual Reviews in Control, Volume 29,
Issue 1, 2005, Pages 33-56.
[1] R.M.May, "Simple Mathematical Models With Very Complicated
Dynamics", Nature 261, 459. 1976.
[2] R.M.Crownover, "Introduction to Fractals and chaos", Jones and Bartlett
Publishers, 1995.
[3] A.J.Lichtenber, and M.A.Liebermann,"Regular and Stochastic Motion.
Springer, Heidelberg-New York.1982.
[4] B.R. Andrievskii, and A.L.Fradkov, " Control of chaos: Methods and
applications. I. Methods:, Avtom.Telemkh, no. 5, pp.3-45. 2003.
[5] Li, T. and Yorke, J.A., Period Three Implies Chaos, Am. Math.
Monthly.,1975, vol. 82, pp. 985-992..
[6] A.Razminia, and M.A.Sadrnia, "Chua-s circuit regulation using a
nonlinear adaptive feedback technique", International journal of
electronics, circuits and systems, Vol 2. No1. 2008.
[7] Yongai Zheng," Controlling chaos based on an adaptive adjustment
mechanism", Chaos, Solitons and Fractals,2005.
[8] Battle, E.Fossas, " Stabilization of periodic orbits of the buck converter
by time delayed feedback", Int. J. Circ. Theory Appl., vol. 27, pp. 617-
631. 1999.
[9] H.Khalil, " Nonlinear Systems", Prentice hall, 2006..
[10] J Chaohong Cai , Zhenyuan Xu, Wenbo Xu, Converting chaos into
periodic motion by state feedback control, Automatica 38 (2002),pp.
1927 - 1933.
[11] R. Ghidossi, D. Veyret, P. Moulin, "Computational fluid dynamics
applied to membranes: State of the art and opportunities", Chemical
Engineering and Processing, Volume 45, Issue 6, June 2006, Pages 437-
454.
[12] P. Magni, M. Simeoni, I. Poggesi, M. Rocchetti, G. De Nicolao, "A
mathematical model to study the effects of drugs administration on
tumor growth dynamics", Mathematical Biosciences, Volume 200, Issue
2, April 2006, Pages 127-151.
[13] Shu Chen, Guo-Hua Hu, Chen Guo, Hui-Zhou Liu, "Experimental study
and dissipative particle dynamics simulation of the formation and
stabilization of gold nanoparticles in PEO-PPO-PEO block copolymer
micelles" Chemical Engineering Science, Volume 62, Issues 18-20,
September-October 2007, Pages 5251-5256.
[14] Franz S. Hover, Michael S. Triantafyllou, "Application of polynomial
chaos in stability and control" Automatica, Volume 42, Issue 5, May
2006, Pages 789-795.
[15] R.L.Devaney, "A First Course in Chaotic Dynamical Systems: Theory
and Experiment" Addison-Wesley, 1997.
[16] J.Guckenheimer and Holmes, "Nonlinear oscillation, Dynamical
Systems, and Bifurcation of Vector Fields", SpringerVerlag, New York,
1990.
[17] David A. Towers, Vicente R. Varea, "Elementary Lie algebras and Lie
A-algebras" Journal of Algebra, Volume 312, Issue 2, 15 June 2007,
Pages 891-901.
[18] Jorge Lauret, Cynthia E. Will, "On Anosov automorphisms of
manifolds" Journal of Pure and Applied Algebra, Volume 212, Issue 7,
July 2008, Pages 1747-1755.
[19] Chen, G., 1996, "Controlling Chaotic Trajectories to Unstable Limit
Cycles", Proceedings of the American Control Conference, San Diego,
California, pp. 2413-2414.
[20] C.D.Meyer, "Matrix analysis and applied linear algebra", SIAM 2000J.
[21] Wang, and J.Zhang, "Chaotic secure communication based on nonlinear
autoregressive filter with changeable parameters", Phy.Lett.A 357,
pp.323-329, 2006.
[22] Sinha, S.C., and Joseph, P., "Control of General Dynamical Systems
with Periodically Varying Parameters Via Lyapunov-Floquet
Transformation", ASME Journal of Dynamical Systems, Measurement,
and Control, Vol. 116, pp. 650-658, 1994.
[23] R.L.Devaney, " Topological Bifurcation" Topology Proceedings 28
pp.99-112, 2004.
[24] Alexander L. Fradkov, Robin J. Evans, "Control of chaos: Methods and
applications in engineering,", Annual Reviews in Control, Volume 29,
Issue 1, 2005, Pages 33-56.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:59828", author = "Abolhassan Razminia and Mohammad-Ali Sadrnia", title = "Robust Conversion of Chaos into an Arbitrary Periodic Motion", abstract = "One of the most attractive and important field of chaos theory is control of chaos. In this paper, we try to present a simple framework for chaotic motion control using the feedback linearization method. Using this approach, we derive a strategy, which can be easily applied to the other chaotic systems. This task presents two novel results: the desired periodic orbit need not be a solution of the original dynamics and the other is the robustness of response against parameter variations. The illustrated simulations show the ability of these. In addition, by a comparison between a conventional state feedback and our proposed method it is demonstrated that the introduced technique is more efficient.
", keywords = "chaos, feedback linearization, robust control,periodic motion.", volume = "4", number = "8", pages = "1189-4", }
