UK Nonlinear News, October 1996

Chaos, Nonlinearity and Control

Mario di Bernardo's new regular column

UK Nonlinear News is proud to welcome Mario di Bernardo as its first regular columnist. Mario has published a number of papers on chaotic control, working with Dr. Alan Champneys and Prof. David Stoten at the Department of Engineering Mathematics of and the Department of Mechanical Engineering at the University of Bristol. He will be filing quarterly dispatches for us from the frontiers of nonlinearity in Control Engineering and Electronics.

We hope he will be the first of many columnists, reporting back from the different corners of the nonlinear world. If you would like to become another of them, do contact the editors at

Starting from this issue of UK Nonlinear News, I shall be reporting on new developments and results in the fascinating task of exploiting chaos and other nonlinear phenomena in Control Engineering and Electronics.

Since this is a very rich and timely area of research, it is not easy to choose where to start from and what topics to cover. The format of this column will therefore be quite flexible as I try to outline the most exciting achievements in my field of interest, and also report on current research and conferences in the area.

I hope to cover any interesting new developments and applications which should arise in the future. If you have any news, conferences, or reviews of papers of interest to this column, then I would be happy to include them. Please e-mail them to me directly at

This issue I should like to start with a bit of history and an overview of how the task of ``taming'' chaos has been tackled, and in many cases successfully performed. I shall also try to highlight interesting links to other sources of related information on the net. Needless to say, I would greatly appreciate any sort of feedback, suggestions and advice.

So, let's get started then !

ENGINEERING CHAOS : the control problem

The traditionally strong temptation to ``linearise everything'' and to treat nonlinearities as disturbances to the linear interpretation of nature is now often discarded in favour of a truly nonlinear description of the world. Nonlinear models are now used by engineers all around the world, and areas of engineering such as the Control of Nonlinear Systems have been revived by a new-born wave of interest. At the same time, new exciting results have been achieved in the understanding and detection of nonlinear phenomena, so that many synthesis and analysis tools are now available to engineers working in the field.

It was against this background that at the end of the last decade it was increasingly realised that bifurcations, and in particular chaotic dynamics, could be exploited ``ad hoc'' in Engineering applications. At the beginning of the nineties, the traditional antipathy of engineers toward chaos became subject to a radical change. From a phenomenon to avoid, Chaos became something to use ... to consider eventually in the design stage.

One key result responsible for this change of mind has been the well known OGY method to control chaos, taking its name from the initials of its three inventors (E. Ott, C. Grebogi and J. Yorke, "Controlling Chaos", Phys. Review Letters, 64, pp.1196-1199, 1990). They proposed that small parameter perturbations can be exploited to stabilise a chaotic system to one of the many unstable periodic orbits embedded in the chaotic attractor itself. In so doing, several properties of chaos are explicitly used to solve the control problem (for more details see

Firstly, once a target periodic orbit has been chosen among the many embedded in the attractor, the ergodicity of chaotic evolutions guarantees that they will come close enough to it at a certain time. Then, thanks to the trajectories' extreme sensitivity to small perturbations, an appropriate control strategy can use quite small changes in the parameters to stabilise the desired periodic solution. Moreover, the fact that the system to control is chaotic also implies that the attractors ``often have embedded within them an infinite, dense set of unstable periodic orbits'' among which to choose the control goal. Hence the OGY method can often be used to take a given system from one desired behaviour to another with a minimal control effort.

During the last few years, the method has been applied to several situations ( such as control of lasers, models of the heart and the brain, buckling beams and so on.

This was enough to arouse wide interest in many areas of Engineering, and of course in the traditional branch of Control Engineering. This discipline has one of the longest tradition among applied science, and though usually focused on the control of Linear Systems, has seen a rapid development in the area of the control of Nonlinear Systems since the sixties. The now well understood and widely used results by Lyapunov, for instance, have set the basis for the solid Theory of Nonlinear Control which is still quickly expanding.

With respect to these results, the control goal proposed by Ott, Grebogi and Yorke was easily classified as ``non-standard'' and that has led to much skepticism among the control engineers. Problems commonly addressed in the control area such as the robustness against external noise were not sufficiently investigated in the case of the OGY method. Moreover, no performance indexes were set to allow a comparison with other existing ``standard'' methods and so on.

Nevertheless, since the early nineties, several engineers around the world have started looking at the problem of controlling and synchronising chaos, discovering that ``standard'' control engineering approaches such as Linear and Nonlinear State Feedback, Adaptive Control, Stochastic Control etc. can be successfully used to achieve the control goal (see G. Chen, X. Dong, "From chaos to order--Perspectives and Methodologies in controlling nonlinear dynamical systems", Intl. Journal of Bifurcation and Chaos, 3, pp. 93-108, 1993). It is relevant to point out that in many of the published results, the main goal of the control scheme is still achieved without using explicitly any of the properties exhibited by chaotic systems.

On the one hand, one challenge of the next few years may be to analyse standard methods using the techniques and language of nonlinear dynamics and chaos. On the other, a control engineering analysis of the OGY method, addressing issues such as robustness, is pressing. Only then perhaps can a proper comparison be made between standard and non-standard methods and the reticence of control engineering community to embrace nonlinear dynamics and chaos overcome.

Either way, the future looks interesting; and it is does not seem unrealistic to state that the "dream" of using chaos in many everyday applications will soon become true.

Mario di Bernardo

 Applied Nonlinear Mathematics Group
 Department of Engineering Mathematics
 University of Bristol
 Bristol, BS8 1TR
 Tel. 0117-9466802
 Fax: 0117-9251154

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Last Updated: 4th October 1996.