# Fuzzy Inference and Reasoning. Proposition 2 Logic variable 3

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30-Jan-2016Category

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Fuzzy Inference and Reasoning

Fuzzy Inference and ReasoningProposition2

Logic variable3

3Basic connectives for logic variables4(1)Negation

(2)Conjunction

45(3) Disjunction

(4)Implication

Basic connectives for logic variables5Logical function6

Logic Formula 7

7

Tautology9

9Tautology10

10

Predicate logic11

11Fuzzy PropositionsAssuming that truthand falsity are expressed by values 1 and 0, respectively, the degree of truth of each fuzzy proposition is expressed by a number in the unit interval [0, 1].Fuzzy Propositions

p : temperature (V) is high (F).

p : V is F is S

V is a variable that takes values v from some universal set VF is a fuzzy set onV that represents a fuzzy predicate S is a fuzzy truth qualifierIn general, the degree of truth, T(p), of any truth-qualified proposition p is given for each v e V by the equation

T(p) = S(F(v)).Fuzzy Propositionsp : Age (V) is very(S) young (F).

Representation of Fuzzy Rule17

17Representation of Fuzzy Rule18

18Fuzzy rule as a relation19

Fuzzy implications20

Example of Fuzzy implications21

Example of Fuzzy implications22

Example of Fuzzy implications23

Representation of Fuzzy Rule24

Single input and single output

Multiple inputs and single output

Multiple inputs and Multiple outputs24Representation of Fuzzy Rule25Multiple rules

25Compositional rule of inference26

The inference procedure is called as the compositional rule of inference. The inference is determined by two factors : implication operator and composition operator.

For the implication, the two operators are often used:For the composition, the two operators are often used:

Representation of Fuzzy Rule27Max-min composition operator

Mamdani: min operator for the implicationLarsen: product operator for the implication27One singleton input and one fuzzy output28

Mamdani

One singleton input and one fuzzy output29

Mamdani

One singleton input and one fuzzy output30

Larsen

One singleton input and one fuzzy output31Larsen

One fuzzy input and one fuzzy output32

Mamdani

One fuzzy input and one fuzzy output33

Mamdani

Ri consists of R1 and R234

Example35

Two singleton inputs and one fuzzy output36Mamdani

Two singleton inputs and one fuzzy output37

MamdaniExample38

Two fuzzy inputs and one fuzzy output39Mamdani

Two fuzzy inputs and one fuzzy output40

MamdaniTwo fuzzy inputs and one fuzzy output41

MamdaniExample42

Multiple rules43

Multiple rules44

Multiple rules45

Example46

Mamdani method47

Mamdani method48

Mamdani method49

Mamdani method50

Larsen method51

Larsen method52

Larsen method53

Larsen method54

54Fuzzy Logic Controller55

Inference56

Inference57

Inference58

Inference59

DefuzzificationMean of Maximum Method (MOM)60

DefuzzificationCenter of Area Method (COA)61

DefuzzificationBisector of Area (BOA)62