Optimization of local control of chaos by an evolutionary algorithm

An evolutionary algorithm for optimizing local control of chaos is presented. Based on a Lyapunov approach, a linear control law and the state-space region in which this control law is activated are determined. In addition, we study a relation between certain adjustable design parameters and a particular measure of the uncontrolled chaotic attractor in the state-space region of control (SSRC). From this relation the objective function to be optimized is derived. In that context, we assume a linear control law to be given and optimize size and shape of the SSRC using an evolutionary algorithm. It is shown by examples how the algorithm can also be applied to higher-dimensional systems with possibly more than one positive Lyapunov exponent. © 2000 Elsevier Science B.V. All rights reserved.

[1]  W. Martienssen,et al.  Local control of chaotic motion , 1993 .

[2]  Wang Jiao,et al.  General method of controlling chaos. , 1996 .

[3]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .

[4]  E. Ott.  Chaos in Dynamical Systems: Contents , 1993 .

[5]  Thomas Bäck,et al.  Genetic Algorithms and Evolution Strategies - Similarities and Differences , 1990, PPSN.

[6]  Visarath In,et al.  Control and synchronization of chaos in high dimensional systems: Review of some recent results. , 1997, Chaos.

[7]  E. Ott,et al.  Controlling Chaotic Dynamical Systems , 1991, 1991 American Control Conference.

[8]  Jürgen Ackermann,et al.  Sampled-Data Control Systems , 1985 .

[9]  Hendrik Richter,et al.  Local Control of Chaotic Systems — A Lyapunov Approach , 1998 .

[10]  A. Hammad,et al.  Stabilization of chaotic dynamics: a modern control approach , 1996 .

[11]  G. Baier,et al.  Maximum hyperchaos in generalized Hénon maps , 1990 .

[12]  Ying-Cheng Lai,et al.  Controlling chaos , 1994 .

[13]  Zbigniew Michalewicz,et al.  Genetic Algorithms + Data Structures = Evolution Programs , 1996, Springer Berlin Heidelberg.

[14]  J. Ackermann.  Sampled-Data Control Systems: Analysis and Synthesis, Robust System Design , 1985 .

[15]  Ute Dressler,et al.  Controlling chaotic dynamical systems using time delay coordinates , 1992 .

[16]  Edward Ott,et al.  Controlling chaos , 2006, Scholarpedia.

[17]  Lawrence. Davis,et al.  Handbook Of Genetic Algorithms , 1990 .

[18]  Earl H. Dowell,et al.  On the optimality of the Ott-Grebogi-Yorke control scheme , 1998 .

[19]  Kok Lay Teo,et al.  Directing Orbits of Chaotic Dynamical Systems , 1995 .

[20]  Zbigniew Michalewicz,et al.  Genetic algorithms + data structures = evolution programs (3rd ed.) , 1996 .

[21]  M. Paskota.  On Local Control of Chaos: the Neighbourhood Size , 1996 .

[22]  Grebogi,et al.  Controlling chaos in high dimensional systems. , 1992, Physical review letters.

[23]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[24]  Thomas Bäck,et al.  Evolutionary algorithms in theory and practice - evolution strategies, evolutionary programming, genetic algorithms , 1996 .

[25]  J. Greene,et al.  The calculation of Lyapunov spectra , 1987 .

[26]  Grebogi,et al.  Using chaos to direct trajectories to targets. , 1990, Physical review letters.

[27]  Y. Lai,et al.  Controlling chaotic dynamical systems , 1997 .

[28]  Kok Lay Teo,et al.  Directing Orbits of Chaotic Systems in the Presence of Noise: Feedback Correction , 1997 .

[29]  M. Hénon.  A two-dimensional mapping with a strange attractor , 1976 .

[30]  de Sousa Vieira M,et al.  Controlling chaos using nonlinear feedback with delay. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[31]  M. J. D. Powell,et al.  Variable Metric Methods for Constrained Optimization , 1982, ISMP.

[32]  Nostrand Reinhold,et al.  the utility of using the genetic algorithm approach on the problem of Davis, L. (1991), Handbook of Genetic Algorithms. Van Nostrand Reinhold, New York. , 1991 .

[33]  Arjan van der Schaft,et al.  Non-linear dynamical control systems , 1990 .

[34]  C. Desoer,et al.  Linear System Theory , 1963 .

[35]  Grebogi,et al.  Higher-dimensional targeting. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[36]  D. E. Goldberg,et al.  Genetic Algorithms in Search , 1989 .

[37]  Thomas Kailath,et al.  Linear Systems , 1980 .