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# Crisp and fuzzy logic

Fuzzy logic forms a bridge between the two areas of qualitative and quantitative modelling. Although the input-output mapping of such a model is integrated into a system as a quantitative map, internally it can be considered as a set of qualitative linguistic rules. Since the pioneering work of Zadeh in 1965 and Mamdani in 1975, the models formed by fuzzy logic have been applied to many varied types of information processing including control systems.

The term Fuzzy Logic is a misnomer. It implies that in some way the methodology is vague or ill-defined. This is in fact far from the case. Fuzzy logic just evolved from the need to model the type of of vague or ill-defined systems that are difficult to handle using conventional binary valued logic, but the methodology itself is based on mathematical theory.

We are all familiar with binary valued logic and set theory. An element belongs to a set of all possible elements and given any specific subset, it can be said accurately, whether that element is or is not a member of it. For example, a person belongs to the set of all human beings, and given a specific subset, such as all males, one can say whether or not each particular person (element) belongs to this set. This is appealing since it seems to describe the way human reason. Collecting many elements into sets allows to describe many occurrences with few rules. For example, the simple statement

IF person is male AND a parent THEN person is a father

applies to many people across the world with complete precision. The rules are formed using operators. Here, the intersection operator AND is used, which manipulates the sets.

Unfortunately, not everything can be described using binary valued sets. The classifications of persons into males and females is easy, but it is problematic to classify them as being tall or not tall. The set of tall people is far more difficult to define, because there is no distinct cut-off point at which tall begins. This is not a measurement problem, and measuring the height of all elements more precisely is not helpful. Such a problem is often distorted so that it can be described using the well-known existing methodology. Here, one could simply select a height, e.g. 1.80m, at which the set tall begins, see Figure 14.1a. The output of a reasoning system using this definition would not be smooth with respect to the height of a person. A person of height 1.79m would produce a different output than a person of 1.81m. In human reasoning this property is not observed and it is also undesirable for reasoning systems that are part of a control system.

Fuzzy logic was suggested by Zadeh as a method for mimicking the ability of human reasoning using a small number of rules and still producing a smooth output via a process of interpolation. It forms rules that are based upon multi-valued logic and so introduced the concept of set membership. With fuzzy logic an element could partially belong to a set and this is represented by the set membership. For example, a person of height 1.79m would belong to both tall and not tall sets with a particular degree of membership. As the height of a person increases the membership grade within the tall set would increase whilst the membership grade within the not tall set would decrease, see Figure 14.1b. The output of a fuzzy reasoning system would produce similar results for similar inputs. Fuzzy logic is simply the extension of conventional logic to the case where partial set membership can exist, rule conditions can be satisfied partially and system outputs are calculated by interpolation and, therefore, have output smoothness over the equivalent binary-valued rule base. This property is particularly relevant to control systems.

A fuzzy logic control system is one that has at least one system component that uses fuzzy logic for its internal knowledge representation. Although it is possible for fuzzy systems to communicate information using fuzzy sets, most applications have a single fuzzy system component communicating with conventional system components via deterministic values. In this case, and also in this chapter, fuzzy logic is used purely for internal knowledge representation and, externally, can be considered as any other system component.

Demonstration Example 14.1   Tumbler full or empty ?

Demonstration Example 14.2   The fuzzy set of full and empty tumblers

Next: Why use fuzzy logic Up: Introduction to fuzzy techniques Previous: Introduction to fuzzy techniques   Contents
Christian Schmid 2005-05-09