Foundation of Fuzzy Systems

Fuzzy logic theory was developed developed by Lofti A. Zadeh in the 60‘s and is based on the theory of fuzzy sets.

The membership of an element is not strictly false or

true({0,1}) like in Boolean logic, but rather gradual.

The degree of membership of an element in fuzzy logic

can be any real number in the interval [0,1].

To model systems that have to process some kind of vague terms like old, young, tall, high, very, extremely, not so much, etc.. Crisp sets as a particular case of fuzzy sets, where [0,1] is restricted to {0,1}.

A fuzzy set A in U is defined as a set of ordered pairs ∈ defined as a set of ordered pairs A ≡ {|x ∈ U} where μ A(x) is called the membership membership function.

function for the set of all objects x in U, where to each x is related a real number(membership grade) in the closed interval [0,1].

Basic Terms and Operations

Union: The membership function of the union of two fuzzy sets A and B with membership functions µA and µB respectively is defined as the maximum of the two individual membership functions: µA∪B(x) := max{µA(x), µB(x)}

Intersection: The membership function of the intersection of two fuzzy sets A and B with membership functions µA and µB respectively is defined as the minimum of the two individual membership functions: µA∩B(x) := min{µA(x), µB(x)}

Complement: The membership function of the complement of a Fuzzy set A with membership function is defined as µ¬A(x) := 1 − µA(x).

The fuzzy set operations of union, intersection and

complement corresponds to the logical operations

disjunction, conjunction and negation, respectively.

1) Disjunction (OR):

µA∨B(x) := max{µA(x), µB(x)}

2)Conjunction (AND):

µA∧B(x) := min{µA(x), µB(x)}

3)Complement(NOT): µ¬A(x) := 1 − µA(x)

Ex. Conjunction (AND):

µA∧B(x) := min{µA(x), µB(x)}

μyoung(John) = 0.5 AND μtall(John) = 0.8

then

μyoung AND tall(John) = min(μyoung(John),μtall(John))

young AND tall (0.5, 0.8 )= min(0.5,0.8)=0.5

Linguistic Description

By a linguistic variable we mean a variable whose values are words or sentences in a natural or artifical language. For example, Age is a linguistic variable if its values are linguistic rather than numerical, i.e., than numerical, i.e., young, not young not young, very young, quite young, old, not very old and not very young, etc.

The name of a linguistic linguistic variable variable is its label

Example:

Let x be a linguistic variable labelled ’Age’. Its term set T could be defined as

T (age)= {very young, young, not very young, more or less old, old}

Fuzzification

Fuzzification may be defined as the process of transforming a crisp set to a fuzzy set or a fuzzy set to fuzzier set. Basically, this operation translates accurate crisp input values into linguistic variables.

Defuzzification

Defuzzification It may be defined as the process of reducing a fuzzy set into a crisp set or to convert a fuzzy member into a crisp member.

Fuzzification	Defuzzification
Fuzzification is the process of transforming a crisp set to a fuzzy set or a fuzzy set to fuzzier set.	Defuzzification is the process of reducing a fuzzy set into a crisp set or converting a fuzzy member into a crisp member.
Fuzzification converts a precise data into imprecise data.	Defuzzification converts an imprecise data into precise data.
Voltmeter.	Stepper motor, D/A converter.
Inference, Rank ordering, Angular fuzzy sets, Neural network.	Maximum membership principle, Centroid method, Weighted average method, Center of sums.
Fuzzification is easy.	Defuzzification is quite complex to implement.
Fuzzification uses if-then rules to fuzzify the crisp value.	Defuzzification uses center of gravity methods to get centroid of sets.

Advantages of Fuzzy System

Robust approach to solve many real-world problems.

• Employable in very complex systems, when there is no

simple mathematical model for highly nonlinear processes.

• Hence, low computational costs and ease at using it in

embedded systems.

• Expert knowledge in complex systems can be formulated in ordinary language

Milan Tomic

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