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FUZZY MODELLING

Few of us would disagree that the world is difficult to understand clearly. In the mid-sixties Lofti Zadeh concluded that 'hardly anything is precise if we examine it with sufficient care'. Zadeh studied the mathematical techniques of the time and found that none of them allowed a satisfactory framework with which to deal with the imprecision's of reality. Fuzzy logic originated from the requirement for sound mathematical techniques to describe imprecision and uncertainty.

Fuzzy logic is constructed around the basic idea of the fuzzy set. Conventional set theory is based upon boolean sets - i.e. a value either belongs to a certain set or it doesn't. This works well for numerical theory - a number is either negative or positive, a certain value is either real or complex. However, for describing (or categorising) real world processes boolean sets are less helpful. For instance, consider the definition that cars going at 70mph or more belong to the set 'fast'. At first glance this looks like a reasonable proposition but looking in more detail shows a lack of flexibility in this definition. The first, and most obvious, question must be : "Is a car going at 69mph not going fast ?". This is the primary problem with boolean sets, their absolute nature does not reflect the way the world actually is. Almost every categorisation is made in shades of grey - not in black and white. For example, temperatures are quite hot or very cold whilst even supposedly precise measurements tend to have degrees of tolerance associated with them. In fuzzy set theory imprecision and vagueness is accounted for by permitting values to belong partially to any particular fuzzy set. This partial membership is generally termed the 'degree of belonging' and is usually expressed in the range 0 (no belonging) to 1 (complete belonging). This methodology allows for imprecision in measurements and also permits linguistic descriptions such as quite high or very big to be expressed within a solid mathematical framework. Typically a particular measurement range might be partitioned into a number of different fuzzy sets, for example a measurement of temperature might be represented by degrees of belonging to the fuzzy sets very cold, cold, tepid, warm, hot and very hot.

The control of chemical processes is often far from straightforward. Non-linearity, deadtime, variable interaction and measurement uncertainty all combine to complicate the task. The control field is awash with different techniques proclaiming to solve some or all of the problems but many processes are still controlled by a human operator. The human operator controls a process using his or her knowledge combined with experience and training. An operator tends to think in linguistic terms when controlling a process - a temperature may be 'quite high' or a valve may need opened 'a little'. Fuzzy logic allows an automatic controller to be programmed to 'think' like a human operator and the first implementations of fuzzy logic controllers used a set of fuzzy if-then rules to decide on control action.

A fairly recent technique in the process control field is 'model-based' control. This has become practical as computers have become ever cheaper and more powerful. In 'model-based' control a model of the process is used in determining controller action. This can again be related to the human operator's control of a process - the model in that case is a mental picture of the process and what will happen given any set of actions or conditions. The process model used in the controller can be built using a huge variety of methods including conventional mathematics such as differential equations. Unfortunately, it is usually very difficult to accurately describe a complicated process using mathematical equations and building even a reasonable model can be very time-consuming and require a great deal of detailed expert knowledge.

The traditional form of fuzzy logic controller where the rule-base was constructed by getting information from process operators was also heavily reliant on expert knowledge and this shortcoming led to the development of new fuzzy relational modelling techniques where a fuzzy model of a chemical process could be constructed simply by using input/output process data. These fuzzy models are easy to construct and require much less process knowledge than most other techniques. In addition they can handle non-linear processes and because they are based on fuzzy set theory are able to handle imprecision and uncertainty. By incorporating such models into a control scheme able to handle deadtime and variable interaction the automatic control of difficult chemical processes can be made possible.

The following are useful sources for further information on fuzzy logic and fuzzy control :-