UK Nonlinear News , February 1997.

# Chaos, Nonlinearity and Control

## Mario di Bernardo's regular column

Mario di Bernardo has published a number of papers on the analysis and control of chaotic systems, working with Dr. Alan Champneys of the Department of Engineering Mathematics and Professor David Stoten of the Department of Mechanical Engineering at the University of Bristol. He writes quarterly columns for us from the frontiers of nonlinearity in Control Engineering and Electronics.

We are looking for additional columnists to report back from the different corners of the nonlinear world. If you would like to become another of them, do contact the editors at uk-nonl@ucl.ac.uk.

### Chaos in Power Electronics

Power Electronics is the branch of Electronics concerned with the processing of electrical energy. Power converters are used to convert electrical energy from one form to the other. Since electrical sources can be either DC or AC there are four basic types of converters: AC-DC, DC-AC, AC-AC, and DC-DC.

Although nonlinear effects in electronic circuits were first observed by Van der Pol in 1927[1], it is only recently that the existence of bifurcations and chaos in power converters has been proposed. (Actually, engineers were aware of its occurrence even earlier[2] due to the sound made by vibrating magnetics, often referred to as "raucuous whine" or "frying bacon" !)

Much research effort has been spent analysing nonlinear phenomena in the DC/DC "buck" converter. This circuit is used when a DC voltage needs to be converted to a lower value. This is generally achieved by ``chopping'' and filtering the input voltage through an appropriate ``switching'' action.

In a noise and perturbation free environment, given the desired output voltage, the switching frequency can be selected and the switches turned on and off according to a fixed pattern. This functioning mode is called "open loop" behaviour. However, in real applications noise is always present; moreover, the load usually is not fixed but varies and the input voltage source can be affected by external disturbances; hence the need for some kind of control action or "closed loop". The control usually implemented is PWM (Pulse Width Modulation), which is a form of state feedback control; the switching element is turned on or off whenever a linear combination of the circuit states (current and voltage) crosses a given reference signal.

From the mathematical view point, PWM controlled DC/DC buck converters can be modelled by piece-wise linear models, whose topologies change according to the given PWM law. Both experiments and simulations have shown that this type of circuit exhibits several types of bifurcation and chaos as the input voltage is varied. Their bifurcation diagram, shown in Figure 1, contains several interesting features[3]. In particular, a first period doubling cascade is interrupted by a sudden enlargement of the resulting chaotic attractor which has a well-defined structure and exists for a wide parameter range. Moreover, many secondary bifurcations have been detected[4].

The analysis of this system has traditionally used stroboscopic maps[5]. This left many of these phenomena unexplained, due to the discontinuous nature of the system. Recently a deeper insight into the system dynamics has been gained by using tools typical of piece-wise linear systems. For instance, appropriately defined "impact maps" have been used to locate bifurcation points analytically and to explain the nature of the chaotic attractor. Unexpected links with other piece-wise linear system, such as impact oscillators and suspension bridges, were also found[6].

The wide spread use of these circuits motivates the need for a deeper understanding of the appearance and nature of their bifurcations and their chaotic attractors. Such knowledge could be used to improve their performance and in finding new fields of application. For example, the aim of the control scheme of a DC/DC converter is to stabilise its output onto a desired periodic solution. Here chaos control techniques could be exploited to solve the problem. Moreover, given the density of periodic orbits embedded in a chaotic attractor, flexibility could be gained. It is possible to think of a circuit whose output quickly switches from one periodic orbit to the other by slight parameter perturbations (in the OGY control method sense). Current research in this direction seems to confirm that this is indeed possible and recently a chaos control scheme for this class of dynamical systems has been proposed and experimentally implemented[7].

In a noise sensitive environment the sudden broadening of the sub-harmonic content of a converter output, due to chaos, is undesired. In this case knowledge of the chaotic regime of the converter could be exploited to delay or avoid its appearance.

Finally, Power Electronics is incredibly rich with nonlinear phenomena. Many electronic circuits have been investigated, both numerically and experimentally, and in almost all bifurcations and chaos were detected[8]. The study of these phenomena have immediate consequences in applications and hence there is the opportunity for nonlinear dynamics to make a significant contribution.

Some links to further information on the net

### References

1. B. Van der Pol & J. van der Mark, "Frequency demultiplication", Nature, 120:3019, pp. 363-364, Sept. 1927.
2. Cecil W. Deisch, "Simple Switching Control Method Changes Power Converter Into a Current Source", PESC'78, IEEE Power Electronics Specialists Conference, 1978, pp. 300-306.
3. J.B. Deane & D.C. Hamill, "Analysis, simulation and experimental study of chaos in the DC/DC buck converter", Power Electronics Specialists Conf., pp. 491-498, San Antonio TX, 1990.
4. M. di Bernardo, E. Fossas, G. Olivar & F. Vasca, "Secondary Bifurcations and High Periodic Orbits in Voltage Controlled Buck Converter", submitted to International Journal of Bifurcations and Chaos, 1996.
5. E. Fossas & G. Olivar, "Study of Chaos in the Buck Converter", IEEE Trans. on Circuits and Systems I, pp. 13-25 , 43, January 1996.
6. M. di Bernardo, F. Garofalo, L. Glielmo, & F. Vasca, "Impacts, Bifurcations and Chaos in the DC/DC buck converter", to appear in IEEE Trans. on Circuits and Systems.
7. G. Poddar, K. Chakrabarty & S. Banerjee, "Experimental Control of Chaotic Behaviour of Buck Converter", IEEE Trans. on Circuits and Systems, pp. 502-504, 42, August 1995.
8. J.B. Deane & D.C. Hamill, "Instability, subharmonics and chaos in power electronics systems", IEEE Trans. on Power Electronics, 5:3, pp. 260-268, July 1990.

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