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Chapter 9 FUZZY INFERENCE Chi-Yuan Yeh
Chapter 9
FUZZY INFERENCE
Chi-Yuan Yeh
GMP and GMT
2
Fuzzy rule as a relation
3
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Bin andA in of degrees membership theing transformof
task theperforms ,function"n implicatiofuzzy " is f where
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function membership dim-2set with fuzzy a considered becan ),R(
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)B( then ),A( If
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B is then A, is If
R
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yx
yxyx
yx
yxyx
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yxyx
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BA
Fuzzy implications
4
Example of Fuzzy implications
5
),/()()(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
6
),/()()(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
7
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
Compositional rule of inference
8
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
9
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
10
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'
'
Representation of Fuzzy Rule
11
Fact: is '
Rule: If is then is
Result: is '
u A
u A w C
w C
fuzzy set
inputFuzzy
Singleton
Singleton
Representation of Fuzzy Rule
12
Fact: is '
Rule: If is then is
Result: is '
u A
u A w C
w C fuzzy set
fuzzy set with a monotonic function
crisp function
Consequence:
fuzzy set fuzzy set with a
monotonic function
crisp function
Representation of Fuzzy Rule
13
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
14
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
15
Mamdani
One singleton input and one fuzzy output
16
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
17
Larsen
One fuzzy input and one fuzzy output
18
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
19
Mamdani
MIMO to MISO
20
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])[( where}{
}])[({
]})[(,],)[(],)[({
})()({
}{
:
D is z , ,C is z then , B isy and A is x If
:rule therepresents R where
}R , ,R ,R ,{R R
21
11
1 1
112
11
11
1
iqi1ii
MIMOi
MIMOn
MIMO3
MIMO2
MIMO1
MISOq
MISOk
MISOMISO
n
ikiiMISO
kq
k
MISOk
q
k
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ikii
n
iqii
n
iii
n
iii
n
iqii
n
i
MIMOi
MIMOi
RBRBRBRB
zBARBRB
zBA
zBAzBAzBA
zzBA
RR
R
Ri consists of R1 and R2
21
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''
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i'
vu
vu
2i
1i
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i'
CC
]R[A]R[A
)]C (B[B)]C (A[AC
Example
22
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
23
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
24
Mamdani
Example
25
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
26
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
27
Mamdani
Two fuzzy inputs and one fuzzy output
28
Mamdani
Example
29
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
30
Multiple rules
31
Multiple rules
32
Example
33
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
34
Mamdani method
35
Mamdani method
36
Mamdani method
37
Larsen method
38
Larsen method
39
Larsen method
40
Larsen method
41
Tsukamoto method
42
Tsukamoto method
43
TSK method
44
TSK method
45
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Thanks for your attention!
