Post on 14-Jan-2016
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Chapter 9
FUZZY INFERENCE
Chi-Yuan Yeh
GMP and GMT
2
Fuzzy rule as a relation
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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
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Example of Fuzzy implications
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),/()()(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
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),/()()(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
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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
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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
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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
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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
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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
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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
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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
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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
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Mamdani
One singleton input and one fuzzy output
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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
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Larsen
One fuzzy input and one fuzzy output
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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
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Mamdani
MIMO to MISO
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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
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11
1 1
112
11
11
1
iqi1ii
MIMOi
MIMOn
MIMO3
MIMO2
MIMO1
MISOq
MISOk
MISOMISO
n
ikiiMISO
kq
k
MISOk
q
k
n
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
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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
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'
ii'
ii'
i'
CC
]R[A]R[A
)]C (B[B)]C (A[AC
Example
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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
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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
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Mamdani
Example
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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
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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
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Mamdani
Two fuzzy inputs and one fuzzy output
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Mamdani
Example
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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
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Multiple rules
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Multiple rules
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Example
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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
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Mamdani method
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Mamdani method
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Mamdani method
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Larsen method
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Larsen method
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Larsen method
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Larsen method
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Tsukamoto method
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Tsukamoto method
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TSK method
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TSK method
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Thanks for your attention!