# What are fuzzifier and Defuzzifier?

## What is the Fuzzification?

Fuzzification is the process of converting a clear input to a fuzzy value. It converts a clear point value of the process state variable to be compatible with the representation of the fuzzy set of the system state variable in the precedent of the rule.

Fuzzification is done based on the type of the inference engine or the strategy of inference like disjunction rule-based or composition based.

## What is the defuzzification?

Defuzzification is the process of convert the set of controller output values into a single pointwise value and performs output renormalization that maps the pointwise value of the controller output into its physical domain.

## Fuzzifier:

Fuzzification is the process of converting a crisp input to a fuzzy value. The manipulation of data in an FLC is based on the theory of fuzzy sets, fuzzification is necessary and desirable at an early stage. Therefore, the fuzzifier can be defined as a mapping from an observed input space to fuzzy set labels in a universe of specified input universe of discourse.

The mapping function takes care of the associated measurement uncertainty for each input variable. The purpose of the mapping function is to interpret measurements of input variables, each expressed by a real number as more realistic fuzzy approximations of the respective real numbers. If f is the mapping function applied to a variable x it can be written as:

f (xi) : [– k, + k] → R+

where R+ is the set of fuzzy numbers and f(x0) is a real number chosen by f as a fuzzy approximation of the measurement xi = x0. A possible definition for this fuzzy number for any xi ∈ [– a, + a] is shown in the figure below, where ε is a parameter to be determined in the context of each application. If it is desirable, other shapes of mapping functions may be used for the fuzzy number f (x0). For each measurement xi = x0 the fuzzy set f (x0) enters into the inference mechanism.

A natural and simple fuzzification approach, however, is to convert a crisp value x0 into a fuzzy singleton A within the specified universe of discourse, since in fuzzy control applications, the observed data are usually crisp. The membership function of fuzzy singleton A, μA(x0), is equal to 1 at the point x0 and zero elsewhere.

When a singleton fuzzifier is used for a measured state xi(t) at time t, it is mapped to the linguistic term set Txi.

## Defuzzifier:

Conceptually, the task of the defuzzifier is to specify a point in W that best represents the fuzzy set C’, obtained as the result of the inference engine. Then, defuzzification can be defined as a mapping of the set C 'from a space of fuzzy control actions in the universe of speech output, W ⊂ R to a space of clear control actions z * ∈ W.

There are a number of choices in determining this crisp controller output z*, however, the following three criteria should be considered in choosing a defuzzification scheme:

1. Plausibility: The point z* should represent C′ from an intuitive point of view; for example, it may lie approximately in the middle of the support of C′ or has a high degree of membership in C′.

2. Computational simplicity: This criterion is particularly important for fuzzy control because fuzzy controllers operate in real-time.

3. Continuity: A small change in C′ should not result in a large change in z*.

4. Disambiguity: It means that the defuzzification method should always produce an unique value for z*.