Fuzzy Inference and Reasoning
description
Transcript of Fuzzy Inference and Reasoning
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Fuzzy Inference and Reasoning
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Proposition
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Logic variable
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Basic connectives for logic variables
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(1)Negation
(2)Conjunction
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(3) Disjunction
(4)Implication
Basic connectives for logic variables
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Logical function
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Logic Formula
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Tautology
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Tautology
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Predicate logic
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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].
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Fuzzy Propositions
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p : temperature (V) is high (F).
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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
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p : Age (V) is very(S) young (F).
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Representation of Fuzzy Rule
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Representation of Fuzzy Rule
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Fuzzy rule as a relation
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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 "BTA ,high""A
H.h and T t variablesdefine andhumidity, and re temperatuof universe be H and TLet
htht BA
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Example of Fuzzy implications
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),/()()(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
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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 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
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Representation of Fuzzy Rule
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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
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Representation of Fuzzy Rule
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Multiple rules
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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'
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n'
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AuAuAu
AuAuAu'
'
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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:
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Representation of Fuzzy Rule
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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
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One singleton input and one fuzzy output
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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
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One singleton input and one fuzzy output
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Mamdani
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One singleton input and one fuzzy output
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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
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One singleton input and one fuzzy output
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Larsen
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One fuzzy input and one fuzzy output
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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
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One fuzzy input and one fuzzy output
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Mamdani
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Ri consists of R1 and R2
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)μμ()μ,μ(μ)CB and (A)B,(AC
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Example
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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
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Two singleton inputs and one fuzzy output
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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
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Two singleton inputs and one fuzzy output
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Mamdani
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Example
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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
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Two fuzzy inputs and one fuzzy output
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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
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Two fuzzy inputs and one fuzzy output
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Mamdani
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Two fuzzy inputs and one fuzzy output
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Mamdani
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Example
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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
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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 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
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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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Fuzzy Logic Controller
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Inference
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Inference
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Inference
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Inference
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Defuzzification
• Mean of Maximum Method (MOM)
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Defuzzification
• Center of Area Method (COA)
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Defuzzification
• Bisector of Area (BOA)
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