# Uncertainty in Fuzzy Logic Systems

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## Rule-Based Fuzzy Logic Systems

A rule-based fuzzy logic system (FLS) contains four components—rules, fuzzifier, inference engine, and output processor—that are interconnected, as shown in Figure 1. Once the rules have been established, an FLS can be viewed as a mapping from inputs to outputs (the solid path in Figure 1, from crisp inputs to crisp outputs), and this mapping can be expressed quantitatively as y = f(x). This kind of FLS is very widely used in many engineering applications of FL, such as in FL controllers and signal processors, and is also known as a fuzzy controller, fuzzy system, fuzzy expert system, or fuzzy model.

Figure 1 Fuzzy logic system.

Rules are the heart of an FLS and may be provided by experts or extracted from numerical data. In either case, the rules that we are interested in can be expressed as a collection of IF-THEN statements, such as this example:

IF the total average input rate of real-time voice and video traffic is a moderate amount, and the total average input rate of the non–real-time data traffic is some, THEN the confidence of accepting the telephone call is a large amount.

The IF part of a rule is its antecedent, and the THEN part of a rule is its consequent.

Fuzzy sets are associated with terms that appear in the antecedents or consequents of rules and with the inputs to and output of the FLS. Membership functions are used to describe these fuzzy sets. Two kinds of fuzzy sets can be used in an FLS, type-1 and type-2. Type-1 fuzzy sets are described by membership functions that are totally certain, whereas type-2 fuzzy sets are described by membership functions that are themselves fuzzy. The latter type lets us quantify different kinds of uncertainties that can occur in an FLS.

An FLS that is described completely in terms of type-1 fuzzy sets is called a type-1 FLS, whereas an FLS that is described using at least one type-2 fuzzy set is called a type-2 FLS.

Returning to the Figure 1 FLS, the fuzzifier maps crisp numbers into fuzzy sets. It is needed to activate rules in terms of linguistic variables, which have fuzzy sets associated with them. The inputs to the FLS prior to fuzzification may be certain (for example, perfect measurements) or uncertain (for example, noisy measurements). Type-1 or type-2 fuzzy sets can be used to model such measurements.

The inference engine of the Figure 1 FLS maps fuzzy sets into fuzzy sets. It handles the way in which rules are activated and combined. Just as we humans use many different types of inferential procedures to help us understand things or to make decisions, there are many different FL inferential procedures.

In many applications of an FLS, crisp numbers must be obtained at its output. This is accomplished by the output processor and is known as defuzzification. In a control-system application, for example, such a number corresponds to a control action. In a signal-processing application, such a number could correspond to the prediction of next year's sun-spot activity, a financial forecast, or the location of a target. The output processor for a type-1 FLS is just a defuzzifier; however, the output processor of a type-2 FLS contains two components: The first maps a type-2 fuzzy set into a type-1 fuzzy set and is called type-reduction, and the second performs defuzzification on the latter set.

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