62
Fuzzy Inference and Reasoning
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 thoseintoBin 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
yxyx
yxyxyx
yxyx
yxyxyx
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 "BTA ,high""A
H.h and T t variablesdefine andhumidity, and re temperatuof universe be H and TLet
htht BA
Example of Fuzzy implications
22
),/()()(BAh)R(t, htht BA
ht 20 50 70 9020 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 9020 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 uu A w C R u ww 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 Au A u A u A w Cw 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 Au 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 uu A w C R u ww 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 uu A w C R u ww 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 uu A w C R u ww 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 uu A w C R u ww 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 Ifsets.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
MamdaniFact: is ' and is ' : ( , ) Rule: If is and is then is : ( , , ) Result: is '
u A v B R u vu A v B w C R u v ww 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 Ifsets.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
MamdaniFact: is ' and is ' : ( , ) Rule: If is and is then is : ( , , ) Result: is '
u A v B R u vu A v B w C R u v ww 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 Ifsets.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 Ifsets.fuzzy r triangulaare
(2,3,4)C .5),(0.5,1.5,2A (1,2,3),C (0,1,2),A whereC is z then ,A is x if:RC 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
