Download - Mamdani-Sugeno
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Fuzzy Inference Systems
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Content The Architecture of Fuzzy Inference Systems Fuzzy Models:
– Mamdani Fuzzy models– Sugeno Fuzzy Models– Tsukamoto Fuzzy models
Partition Styles for Fuzzy Models
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Fuzzy Inference Systems
The Architecture of
Fuzzy Inference Systems
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Fuzzy Systems
Fuzzy Knowledge base
Input FuzzifierInference
EngineDefuzzifier Output
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Fuzzy Control Systems
Fuzzy Knowledge base
FuzzifierInference
EngineDefuzzifier Plant Output
Input
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FuzzifierFuzzy
Knowledge baseFuzzy
Knowledge base
I nput FuzzifierI nference
EngineDefuzzifier OutputI nput Fuzzifier
I nferenceEngine
Defuzzifier Output
Converts the crisp input to a linguistic
variable using the membership functions
stored in the fuzzy knowledge base.
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FuzzifierFuzzy
Knowledge baseFuzzy
Knowledge base
I nput FuzzifierI nference
EngineDefuzzifier OutputI nput Fuzzifier
I nferenceEngine
Defuzzifier Output
Converts the crisp input to a linguistic
variable using the membership functions
stored in the fuzzy knowledge base.
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Inference EngineFuzzy
Knowledge baseFuzzy
Knowledge base
I nput FuzzifierI nference
EngineDefuzzifier OutputI nput Fuzzifier
I nferenceEngine
Defuzzifier Output
Using If-Then type fuzzy rules converts the fuzzy input to the fuzzy output.
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DefuzzifierFuzzy
Knowledge baseFuzzy
Knowledge base
I nput FuzzifierI nference
EngineDefuzzifier OutputI nput Fuzzifier
I nferenceEngine
Defuzzifier Output
Converts the fuzzy output of the
inference engine to crisp using
membership functions analogous to the
ones used by the fuzzifier.
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Nonlinearity
In the case of crisp inputs & outputs, a fuzzy inference system implements a nonlinear mapping from its input space to output space.
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Fuzzy Inference Systems
Mamdani
Fuzzy models
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Mamdani Fuzzy models
Original Goal: Control a steam engine & boiler combination by a set of linguistic control rules obtained from experienced human operators.
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The Reasoning Scheme
Max-Min Composition is used.
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The Reasoning Scheme
Max-Product Composition is used.
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DefuzzifierFuzzy
Knowledge baseFuzzy
Knowledge base
I nput FuzzifierI nference
EngineDefuzzifier OutputI nput Fuzzifier
I nferenceEngine
Defuzzifier Output
Converts the fuzzy output of the inference engine to crisp using membership functions analogous to the ones used by the fuzzifier.
Five commonly used defuzzifying methods:– Centroid of area (COA)– Bisector of area (BOA)– Mean of maximum (MOM)– Smallest of maximum (SOM)– Largest of maximum (LOM)
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DefuzzifierFuzzy
Knowledge baseFuzzy
Knowledge base
I nput FuzzifierI nference
EngineDefuzzifier OutputI nput Fuzzifier
I nferenceEngine
Defuzzifier Output
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DefuzzifierFuzzy
Knowledge baseFuzzy
Knowledge base
I nput FuzzifierI nference
EngineDefuzzifier OutputI nput Fuzzifier
I nferenceEngine
Defuzzifier Output
( )
,( )
A
ZCOA
A
Z
z zdz
zz dz
( )
,( )
A
ZCOA
A
Z
z zdz
zz dz
( ) ( ) ,BOA
BOA
z
A A
z
z dz z dz
( ) ( ) ,BOA
BOA
z
A A
z
z dz z dz
*
,
{ ; ( ) }
ZMOM
Z
A
zdz
zdz
where Z z z
*
,
{ ; ( ) }
ZMOM
Z
A
zdz
zdz
where Z z z
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Example
R1 : If X is small then Y is smallR2 : If X is medium then Y is mediumR3 : If X is large then Y is large
X = input [10, 10]Y = output [0, 10]
Overall input-output curve
Max-min composition and centroid defuzzification were used.
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Example
R1: If X is small & Y is small then Z is negative largeR2: If X is small & Y is large then Z is negative smallR3: If X is large & Y is small then Z is positive smallR4: If X is large & Y is large then Z is positive large
X, Y, Z [5, 5]
Overall input-output curve
Max-min composition and centroid defuzzification were used.
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Fuzzy Inference Systems
Sugeno
Fuzzy Models
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Sugeno Fuzzy Models
Also known as TSK fuzzy model – Takagi, Sugeno & Kang, 1985
Goal: Generation of fuzzy rules from a given input-output data set.
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Fuzzy Rules of TSK Model
If x is A and y is B then z = f(x, y)
Fuzzy Sets Crisp Function
f(x, y) is very often a
polynomial function w.r.t. x
and y.
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Examples
R1: if X is small and Y is small then z = x +y +1
R2: if X is small and Y is large then z = y +3
R3: if X is large and Y is small then z = x +3
R4: if X is large and Y is large then z = x + y + 2
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The Reasoning Scheme
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Example
R1: If X is small then Y = 0.1X + 6.4R2: If X is medium then Y = 0.5X + 4R3: If X is large then Y = X – 2
X = input [10, 10]
unsmooth
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Example
R1: If X is small then Y = 0.1X + 6.4R2: If X is medium then Y = 0.5X + 4R3: If X is large then Y = X – 2
X = input [10, 10]
If we have smooth membership functions (fuzzy rules) the overall input-output curve becomes a smoother one.
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Example
R1: if X is small and Y is small then z = x +y +1R2: if X is small and Y is large then z = y +3R3: if X is large and Y is small then z = x +3R4: if X is large and Y is large then z = x + y + 2
X, Y [5, 5]
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Fuzzy Inference Systems
Tsukamoto
Fuzzy models
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Tsukamoto Fuzzy models
The consequent of each fuzzy if-
then-rule is represented by a fuzzy
set with a monotonical MF.
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Tsukamoto Fuzzy models
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Example
R1: If X is small then Y is C1
R2: If X is medium then Y is C2
R3: if X is large then Y is C3
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Fuzzy Inference Systems
Partition Styles for Fuzzy Models
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Review Fuzzy Models
If <antecedence> then <consequence>.
The same style for• Mamdani Fuzzy models• Sugeno Fuzzy Models• Tsukamoto Fuzzy models
Different styles for• Mamdani Fuzzy models• Sugeno Fuzzy Models• Tsukamoto Fuzzy models
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Partition Styles for Input Space
GridPartition
TreePartition
ScatterPartition