Control of chaos: Methods and applications in engineering,☆☆

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Abstract

A survey of the emerging field termed “control of chaos” is given. Several major branches of research are discussed in detail: feedforward or “nonfeedback control” (based on periodic excitation of the system); “OGY method” (based on linearization of the Poincaré map), “Pyragas method” (based on a time-delay feedback), traditional control engineering methods including linear, nonlinear and adaptive control, neural networks and fuzzy control. Some unsolved problems concerning the justification of chaos control methods are presented. Other directions of active research such as chaotic mixing, chaotization, etc. are outlined. Applications in various fields of engineering are discussed.

Introduction

For almost three decades, the unusual behavior of nonlinear dynamics and chaos has attracted the attention of several different scientific communities. Chaotic phenomena and chaotic behavior have been observed in numerous natural and model systems in physics, chemistry, biology, ecology, etc. Engineering applications are rapidly developing in areas such as lasers and plasma technologies, mechanical and chemical engineering and telecommunications. It is then not surprising that the matter of controlling chaotic systems has come under detailed investigation. Publication activity in this field has grown rapidly during the past decade. Starting with a few papers in 1990, the number of publications in peer reviewed journals1 exceeded 2700 in 2000, with more than half published in 1997–2000. Although different interpretations of the term “control” are in use2 the intensity of publications is unusually high. The control of chaos has been addressed in a number of monographs and surveys (see Section A of the Reference list). At the same time, no survey has yet appeared in a control related journal. This survey is aimed at filling this gap.

Surprisingly, the development of the field was triggered by essentially one paper, by E. Ott, C. Grebogi and J. Yorke from the University of Maryland, published in Physical Review Letters in 1990 (Ott, Grebogi, & Yorke, 1990C2), where the term “controlling chaos” was coined. Perhaps, the key achievement of the paper (Ott et al., 1990C2) was a demonstration of the fact that a significant change in the behavior of a chaotic system can be made by a “tiny” adjustment of its parameters. This observation opened possibilities for changing behavior of natural systems without interfering with their inherent properties. The idea was quickly appreciated in physics and other natural sciences. Such a situation is likely to attract additional attention from the control community because it opens up new markets for control theory.

It is worth noting that, in spite of the enormous number of published papers, very few rigorous results are so far available. Most results are justified by computer simulations rather than by analytical tools. Therefore, many problems remain unsolved.

This survey aims to outline the field and describe some of the open problems. Most attention will be paid to control of continuous-time chaotic systems focusing on three approaches: feedforward or “nonfeedback” control, the OGY method and the Pyragas method. These approaches were historically the first in the field and produced the largest number of publications. Also, application of conventional nonlinear and adaptive control methods will be addressed.

In Section 2, some preliminaries are given concerning system models, control goals and properties of chaotic systems. Because of space limitations we will not discuss definitions and properties of chaotic systems in detail. Section 3 is devoted to surveying the above mentioned approaches. In Section 4, the discrete-time case will be discussed, while in Sections 5 Neural networks, 6 Fuzzy systems, a brief account of other approaches based on neural network and fuzzy system frameworks will be given. Section 7 is devoted to generation of chaos (chaotification). Control of mixing is considered in Section 8. A number of other problems are briefly listed in Section 9. Section 10 is devoted to applications of chaos control in engineering.

The aim of the applications part of the survey is to demonstrate a variety of potential applications in different fields of science and engineering. Since the number of reported applications is very high, only a few typical examples from each field are described. More information can be found in the references.

In order to facilitate reading, the references are structured in a few sections in accordance with the structure of the survey as follows:

  • A. Books and surveys.

  • B. Background.

  • C. Control of chaos in continuous-time systems.

  • C1. Feedforward (open loop) control.

  • C2. Linearization of Poincaré map (OGY method).

  • C3. Delayed feedback.

  • C4. Linear and nonlinear control.

  • C5. Adaptive control.

  • D. Discrete-time systems.

  • E. Neural networks.

  • F. Fuzzy control.

  • G. Generation of chaos (chaotization).

  • I. Chaotic mixing.

  • J. Other problems.

  • L. Applications in engineering (mechanical engineering and motion control, electrical engineering, telecommunications, information systems, chemical and material engineering, miscellaneous applications).

When quoting the reference in the text the section number is added to the publication year. For example, citation (Fradkov & Markov, 2000C5) points to the reference from Section C5 of the Reference list. References in each section are listed alphabetically. Besides the references actually cited in this survey, the bibliography contains some other important papers. However, the bibliography is by no means complete and contains only what we consider to be the most relevant and useful references. Further references can be found in the bibliography on control of chaos (papers of 1997–2000) at http://www.rusycon.ru/chaos-control.html.

Section snippets

Models of controlled systems and control goals

Several classes of model are considered in the literature related to control of chaos. The most common class consists of continuous systems with lumped parameters described in state space by differential equationsx˙=F(x,u),where x is a n-dimensional vector of state variables x˙=d/dt and u is an m-dimensional vector of inputs (control variables). The vector-function F(x,u) is usually assumed to be continuously differentiable. Models (1) encompass two physically different cases: (A) input

Feedforward control

The idea of feedforward control (also called nonfeedback or open loop control) is to change the behavior of a nonlinear system by applying a properly chosen input function or external excitation u(t). The excitation can reflect influence of some physical action, e.g. external force/field, or it can be some parameter perturbation (modulation). Such an approach is attractive because of its simplicity: no measurements or extra sensors are needed. It is especially advantageous for ultrafast

Discrete-time control

Discrete-time algorithms were mentioned in Section 3.2 (when discussing methods based on the Poincaré map) and in Section 3.3. They can be considered as special forms of sampled-data control. There are many results on the stability of sampled-data feedback control systems. Stability analysis in the context of chaotic systems was undertaken by Yang and Chua (1998D).

Although many authors use the term “optimal control”, in most cases only locally optimal solutions are proposed, based on

Neural networks

There are several methods of employing neural networks in the field of chaos control. Firstly, many authors exploit the universal ability of neural networks to predict and control nonlinear systems. Since systems possessing chaotic behavior are necessarily nonlinear, the potential for neural networks to control chaotic systems is not surprising. Several universal neural-like learning networks for control of nonlinear systems have been proposed (Hirasawa, Wang, Murata, Hu, & Jin, 2000E; Poznyak,

Fuzzy systems

A number of control algorithms have been reported based on fuzzy modeling of the controlled system. Perhaps, the most convenient fuzzy models for control design are the so-called Takagi–Sugeno fuzzy models (T–S systems), described by sets of fuzzy rulesIFz1(t)F1iANDANDzp(t)F1pTHENx˙=Aix+Biu,y=Cix+Diu,i=1,2,,r,where x(t)Rn,u(t)Rm,y(t)Rm are vectors of state, input and output, respectively; zj(t) are premise variables that are functions of the state, inputs and, possibly, time; Fji are

Generation of chaos (chaotization)

The problem of chaotization of a given system by feedback (called also chaos synthesis, chaos generation, anticontrol of chaos or chaotification) appears when it is necessary to design (generate) chaotic signals, e.g. for information encryption, broadband communications, computation with pseudorandom numbers (Monte Carlo method), etc. (Andrecut, 1998G; Chan & Tse, 1998G). Studying ways of creating chaotic signals with prescribed properties may shed light upon mechanisms of biological systems,

Chaotic mixing

An important stream of research is related to chaotic mixing, particularly, mixing (stirring) of fluids and mixing of granular flows. Mixing properties of flows are important in a variety of applications such as chemical production in continuously stirred chemical reactors (CSCR), production of powders, polymers, design of combustion processes, heat exchangers, etc. (see Ottino, 1989I; Sharma & Gupte, 1997A). A typical control goal is to enhance mixing, i.e. to increase its rate and quality.

Other problems

We give a brief account of other directions of research related to control of chaos.

Applications in engineering

The number of papers in peer reviewed journals in 1997–2000 devoted to the control of chaos in various application fields exceeds 200. Among the fields of science and engineering where control of chaotic behavior is actively developing are the physics of turbulence, laser physics and optics, physics of plasma, molecular and quantum physics, mechanics, chemistry and electrochemistry, biology and ecology, economics and finance, medicine, mechanical, electrical and chemical engineering, motion

Conclusions

In the conclusion to the first version of the survey (see also Fradkov & Evans, 2002A), we wrote that “It seems that the publication rate in the field has achieved saturation”. However, the statistics of the years 2001–2002 shows that this is not the case. Control of chaos is still an emerging field of research. Its three major and historically the first branches: feedforward or “nonfeedback” control, the OGY method and the Pyragas method are currently flourishing. The potential for the methods

Other important papers

Schuster, 1999A, Chen, 1999A, Kapitaniak, 2000A, Boccaletti et al., 2002A, Fradkov and Evans, 2001A, Guckenheimer and Holmes, 1983B, Wiggins, 1988B, Bezruchko et al., 1999C2, Grebogi and Lai, 1997a, Grebogi and Lai, 1997b, Lim and Mareels, 2000C2, Ritz et al., 1997C2, Hikihara and Ueda, 1999C3, Ushio and Yamamoto, 1998C3, Ciofini et al., 1999C5, Ding et al., 1997C5, Dong et al., 1997C5, Ge et al., 2000C5, Gluckman et al., 1997C5, Tian and Yu, 2000C5a, Tian and Yu, 2000C5b, Yu, 1999C5, Zhang et

Acknowledgements

The work was supported in part by the Cooperative research Centre for Systems, Signals and Information Processing, University of Melbourne and by the Russian Foundation for Basic Research, project 02-01-00765.

Alexander Fradkov received his Diploma degree in mathematics from St. Petersburg State University (Department of Theoretical Cybernetics) in 1971; Candidate of Sciences (PhD) degree in engineering cybernetics from Leningrad Mechanical Institute (now Baltic State Technical University, BSTU) in 1975; Doctor of Sciences degree in control engineering in 1986 from St. Petersburg Electrotechnical Institute. From 1971 to 1987, he occupied different research positions and in 1987 became professor of

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Alexander Fradkov received his Diploma degree in mathematics from St. Petersburg State University (Department of Theoretical Cybernetics) in 1971; Candidate of Sciences (PhD) degree in engineering cybernetics from Leningrad Mechanical Institute (now Baltic State Technical University, BSTU) in 1975; Doctor of Sciences degree in control engineering in 1986 from St. Petersburg Electrotechnical Institute. From 1971 to 1987, he occupied different research positions and in 1987 became professor of computer science with BSTU. Since 1990, he has been the head of the “Control of Complex Systems” Lab of the Institute for Problems of Mechanical Engineering of Russian Academy of Sciences. He is also part-time professor with St. Petersburg State University (Department of Theoretical Cybernetics). His research interests are in fields of nonlinear and adaptive control, control of oscillations and chaos, cybernetical physics (borderland field between physics and control). Dr. Fradkov co-authored 350 journal and conference papers, 10 patents, 15 books and textbooks, including: “Introduction to control of oscillations and chaos” (World Scientific, 1998); “Nonlinear and adaptive control of complex systems” (Kluwer, 1999); “Selected chapters of automatic control theory with MATLAB examples” (Nauka, 1999, in Russian); “Cybernetical physics” (Nauka, 2003, in Russian).

Dr. Fradkov is the vice president of the St. Petersburg Informatics and Control Society since 1991, member of the Russian National Committee of Automatic Control. He organized and co-chaired 1st–10th International (Baltic) Student Olympiads on Automatic Control in 1991–2004; 1st and 2nd International IEEE–IUTAM Conference “Control of oscillations and chaos” in 1997 and 2000; 5th IFAC Symposium on Nonlinear Control Systems (NOLCOS’01); 1st IEEE–IUTAM–EPS Conference “Physics and control 2003”. He is a member of IEEE CSS Conference Editorial Board (1998–2004). Dr. Fradkov was awarded JSPS fellowship for research in Japan in 1998–1999. In 1991–2004, he visited and gave invited lectures in 70 universities and research centers of 22 countries.

Rob Evans was born in Melbourne, Australia, in 1947. After completing a BE degree in electrical engineering at the University of Melbourne in 1969, he worked as a radar systems engineering officer with the Royal Australian Airforce. He completed a PhD in 1975 at the University of Newcastle followed by postdoctoral studies at the Laboratory for Information and Decision Systems, MIT and the Control and Management Department, Cambridge University.

In 1977, he took up an academic position at the University of Newcastle, where he was head of the Department of Electrical and Computer Engineering from 1986 to 1991, and co-director of the Centre on Industrial Control Systems between 1988 and 1991.

In 1992, he moved to the University of Melbourne, where he was head of the Department of Electrical and Electronic Engineering until 1996. He was research leader for the Cooperative Centre for Sensor Signal and Information Processing until 2000 and director of the Centre for Networked Decision Systems until 2004. He is currently director of the Victoria Laboratory of National ICT Australia.

His research has ranged across many areas including theory and applications in industrial control, radar systems, signal processing and telecommunications. He is a fellow of the Australian Academy of Science, a fellow of the Australian Academy of Technological Sciences and Engineering, a fellow of the Institution of Electrical and Electronic Engineers (USA).

The preliminary version of the paper was presented at the 15th IFAC World Congress, Barcelona, 2002.

☆☆

An earlier version of this paper was presented at the IFAC World Congress, Barcelona, Spain, July 21–26, 2002.

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