Fuzzy Inference and Reasoning

Proposition

2

Logic variable

3

Basic connectives for logic variables

4

(1)Negation

(2)Conjunction

5

(3) Disjunction

(4)Implication

Basic connectives for logic variables

Logical function

6

Logic Formula

7

Tautology

9

Tautology

10

Predicate logic

11

Fuzzy Propositions

• Assuming 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 V• F is a fuzzy set onV that represents a fuzzy predicate • S is a fuzzy truth qualifier• In 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 Propositions

p : Age (V) is very(S) young (F).

Representation of Fuzzy Rule

17

Representation of Fuzzy Rule

18

Fuzzy rule as a relation

19

BAin ),( of thoseinto

Bin andA in of degrees membership theing transformof

task theperforms ,function"n implicatiofuzzy " is f where

))(),((f),(

function membership dim-2set with fuzzy a considered becan ),R(

)B()A( :),R(

relationby drepresente becan

)B( then ),A( If

)B( ),A( predicatesfuzzy B is A, is

B is then A, is If

R

yx

yx

yxyx

yx

yxyx

yx

yxyx

yx

BA

Fuzzy implications

20

Example of Fuzzy implications

21

),/()()(BAh)R(t,

R(h)R(t):h)R(t,

B ish :R(h) A, ist :R(t)

B ish then A, is t If:h)R(t,

asrewritten becan rule then the

HB,high"fairly "B

TA ,high""A

H.h and T t variablesdefine and

humidity, and re temperatuof universe be H and TLet

htht BA

Example of Fuzzy implications

22

),/()()(BAh)R(t, htht BA

ht

20 50 70 90

20 0.1 0.1 0.1 0.130 0.2 0.5 0.5 0.540 0.2 0.6 0.7 0.9

Example of Fuzzy implications

23

TA ,A isor t high"fairly is etemperatur"When ''

) ,(R )R( )R(

R(h) find torelationsfuzzy ofn compositio usecan We

C' htth

ht

20 50 70 90

20 0.1 0.1 0.1 0.130 0.2 0.5 0.5 0.540 0.2 0.6 0.7 0.9

Representation of Fuzzy Rule

24

Fact: is ' : ( )

Rule: If is then is : ( , )

Result: is ' : ( ) ( ) ( , )

u A R u

u A w C R u w

w C R w R u R u w

Single input and single output

' ' '1 1 2 2

1 1 2 2

Fact: is ' and is ' and ... and is '

Rule: If is and is and ... and is then is

Result: is '

n n

n n

u A u A u A

u A u A u A w C

w C

Multiple inputs and single output

' ' '1 1 2 2

1 1 2 2 1 1 2 2

Fact: is and is and ... and is

Rule: If is and is and ... and is then is , is ,..., is

Res

n n

n n m m

u A u A u A

u A u A u A w C w C w C' ' '

1 1 2 2ult: is , is ,..., is m mw C w C w C

Multiple inputs and Multiple outputs

Representation of Fuzzy Rule

25

Multiple rules

m'

m2'

21'

1

mj'

mj2j'

2j1j'

1j2211

2211

C is w..., ,C is w,C is w:Result

C is w..., ,C is w,C is then w, is and ... and is and is If :j Rule

is and ... and is and is :Fact

nj'

njj'

n'

n'

AuAuAu

AuAuAu'

'

Compositional rule of inference

26

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 Rule

27

Max-min composition operator

Fact: is ' : ( )

Rule: If is then is : ( , )

Result: is ' : ( ) ( ) ( , )

u A R u

u A w C R u w

w C R w R u R u w

( , ) :R u w A C

Mamdani: min operator for the implicationLarsen: product operator for the implication

One singleton input and one fuzzy output

28

Fact: is ' : ( )

Rule: If is then is : ( , )

Result: is ' : ( ) ( ) ( , )

u A R u

u A w C R u w

w C R w R u R u w

Mamdani

One singleton input and one fuzzy output

29

Mamdani

One singleton input and one fuzzy output

30

Fact: is ' : ( )

Rule: If is then is : ( , )

Result: is ' : ( ) ( ) ( , )

u A R u

u A w C R u w

w C R w R u R u w

Larsen

One singleton input and one fuzzy output

31

Larsen

One fuzzy input and one fuzzy output

32

Fact: is ' : ( )

Rule: If is then is : ( , )

Result: is ' : ( ) ( ) ( , )

u A R u

u A w C R u w

w C R w R u R u w

Mamdani

One fuzzy input and one fuzzy output

33

Mamdani

Ri consists of R1 and R2

34

iii C is then w,B is vand A isu If :i Rule

)]}μμ(μ[)],μμ(μmin{[

)]}μμ(,μmin[)],μμ(,μmin{min[max

)]}μμ(),μμmin[(),μ,μmin{(max

)]μμ(),μμmin[()μ,μ(

)μ)μ,μ(min()μ,μ(

)μμ()μ,μ(μ

)CB and (A)B,(AC

CBBCAA

CBBCAA,

CBCABA,

CBCABA

CBABA

CBABAC

iii''

i'

i'

i'

i'

i'

ii''

ii''

ii''

ii''

i'

vu

vu

2i

1i

2i

'1i

'

ii'

ii'

i'

CC

]R[A]R[A

)]C (B[B)]C (A[AC

Example

35

output? then , )(Singleton 1.5 y and 1 input x If

sets.fuzzy r triangulaare (5,6,7)C (1,2,3),B (0,1,2),A where

C is z then B, isy andA is x if:R

00

Two singleton inputs and one fuzzy output

36

Mamdani

Fact: is ' and is ' : ( , )

Rule: If is and is then is : ( , , )

Result: is '

u A v B R u v

u A v B w C R u v w

w C : ( ) ( , ) ( , , )R w R u v R u v w

Two singleton inputs and one fuzzy output

37

Mamdani

Example

38

output? then , )(Singleton 1.5 y and 1 input x If

sets.fuzzy r triangulaare (5,6,7)C (1,2,3),B (0,1,2),A where

C is z then B, isy andA is x if:R

00

Two fuzzy inputs and one fuzzy output

39

Mamdani

Fact: is ' and is ' : ( , )

Rule: If is and is then is : ( , , )

Result: is '

u A v B R u v

u A v B w C R u v w

w C : ( ) ( , ) ( , , )R w R u v R u v w

Two fuzzy inputs and one fuzzy output

40

Mamdani

Two fuzzy inputs and one fuzzy output

41

Mamdani

Example

42

output? then , set)(Fuzzy 3.5) 2.5, (1.5, B' and (1,2,3) A'input If

sets.fuzzy r triangulaare (5,6,7)C (1,2,3),B (0,1,2),A where

C is z then B, isy andA is x if:R

Multiple rules

43

Multiple rules

44

Multiple rules

45

Example

46

output? then , )(Singleton 1 input x If

sets.fuzzy r triangulaare

(2,3,4)C .5),(0.5,1.5,2A (1,2,3),C (0,1,2),A where

C is z then ,A is x if:R

C is z then ,A is x if:R

0

2211

222

111

Mamdani method

47

Mamdani method

48

Mamdani method

49

Mamdani method

50

Larsen method

51

Larsen method

52

Larsen method

53

Larsen method

54

Fuzzy Logic Controller

55

Inference

56

Inference

57

Inference

58

Inference

59

Defuzzification

• Mean of Maximum Method (MOM)

60

Defuzzification

• Center of Area Method (COA)

61

Defuzzification

• Bisector of Area (BOA)

62