# The behaviour of swingers

Variable Structure Systems

This control-engineering monograph, which has 31 contributors from around the globe, came to my attention after I had already used the term "variable structure system" to describe an adaptive and possibly evolutionary electronic system that reconfigures itself depending on its own outputs.

Although the term as used in the narrow sense of this book seems rather arcane and technical, we are all used to the behaviour of variable structure control systems, as every few years in the West we elect a new administration with limited powers to alter the rules under which we all operate. Therefore, what can be learnt from the mathematically erudite control engineers about the behaviour of such classes of system is interesting.

In the domain of real-world control engineering problems, many (if not most) systems that one seeks to control have the properties of being non-linear; imperfectly understood or specified; possibly containing time delays between causes and effects; and potentially dangerous if the control fails. These factors provide a strong motivation for the control engineer to develop a general method that may be algorithmically applied in diverse cases. From reading this book, I am convinced that such a pot of gold at the end of the rainbow may be possible.

Some idea of the power of a variable structure control system may be obtained by considering the escapement mechanism in an 18th-century long case (grandfather) clock. The escapement senses the displacement of the pendulum from vertical, and at a prescribed position allows a cogwheel to administer a small kick to the pendulum. The energy for this comes from the lead weights, moving down over eight days under gravity. The swing of the damped pendulum is maintained on what is termed a "limit cycle", at constant amplitude and period. The periodic movements of the cogwheel are transmitted through the drive train to move the hands around the clock face.

We can immediately grasp why this is a variable structure system, with configuration that is changed by a measure of the output variable of the system (in this case the displacement of the pendulum from vertical). What is not so obvious is the advantage of this system over other possible methods of maintaining the swing of the pendulum. In the case of my family grandfather clock, which was built about 1770, there have only been three recorded times when the mechanism needed to be maintained. The pallets of the cogwheel wear, the bearings of the drive train wear, dust and airborne tar accumulate and alter the friction, the temperature and humidity cycle between day and night and summer and winter, and yet no allowance needs to be made by the control system for these unquantifiable effects. The clock just goes on, century after century, keeping accurate time, give or take a few seconds a week.

Occasionally, the cumulative effects of wear and ageing cause the clock to stop. We may say with some confidence that if we wait long enough, the clock will certainly eventually stop due to causes other than that the weights have run down. So the control engineer's best efforts to build a system that is robust against variations in the system being controlled may eventually fail, and it is probably important to know when this may happen.

I have for some years been constructing and studying chaotic electronic circuits, some of which are variable structure systems in the wider sense. It is possible to construct a system that chatters along chaotically for an unpredictable and indefinite time before falling into a trap, or stable fixed point, allowing the system behaviour to "die".

Chattering is a common problem in variable structure control systems, and many pages of this book are devoted to its minimisation. Chattering is a concomitant side-effect of the advantages of the method; it is suggestive of the behaviour of the economy under successive changes in regulatory instruments, such as interest rates and taxes.

The ability to alter itself as a consequence of its own behaviour is a property of what is termed a "complex adaptive system", including biological and ecological models. The advent of powerful computing resources lets us study such models. The question then arises as to how well the model translates into a prediction of the behaviour in the real world. Also suggestive is the idea of man-made evolutionary systems, such as those studied in electronic hardware by a group at Sussex University.

The book contains introductory material and a selection of well worked-out practical applications, including power electronics, neural networks, motion control, automobile applications and underwater objects. For anyone seeking a way in to the regulation of non-linear and uncertain systems, this unique book may provide valuable reading.

David Jefferies is senior lecturer in engineering, Surrey University.

## Variable Structure Systems: From Principles to Implementation

Editor - Asif Sabanovic, Leonid M. Fridman and Sarah Spurgeon
Publisher - Institution of Electrical Engineers
Pages - 409
Price - £65.00
ISBN - 0 86341 350 1

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