fuzzy inference system1
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FUZZY INFERENCE SYSTEM
It is a popular computing framework based on
fuzzy set theory, fuzzy if then rules , and fuzzy
reasoning.
3 components:
1. Rule Base: a selection of fuzzy rules
2. Database (or Dictionary): defines the MFs
3. Inference Engine: a reasoning mechanism
which performs the inference procedure
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Cont..
It can take either fuzzy inputs or crisp inputs
but the output it produces are always fuzzy
sets.
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MODELS OF FUZZY INFERENCE SYSTEM
Three types
Mamdani fuzzy model
Sugeno fuzzy model
Larsan fuzzy model
They differ in the consequences of their fuzzy
rules and defuzzification methods
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MAMDANI FUZZY MODEL
Proposedd to control a steam engine and
boiler combination
Mamdani rule base can model the system
using rules that have a high correctness.
Correctness-measure of how close our model
is to the real one
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Cont..
Mamdani model is a crisp model of a system.
It can model a real system where the relationbetween the inputs and outputs are known.
2 common operators that we use are T-normand T-conorm operators.
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MAMDANI RULE BASE
Can be broken down into 4 part
Fuzzification
Determining the output of each rule given its fuzzy
antecedent.
Aggregation
defuzzification
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Fuzzification
Mamdani rule base models a crisp system,it
has crisp inputs and outputs.
Fuzzifier converts the crisp input into fuzzy
variables.
The membership of each fuzzy input variable
is evaluated for a given crisp input.
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EVALUATING THE RULES
Rules are evaluated based on the membership
values.
The rule base let the user to determine which
type of operation to use like minimum ,
maximum, product
If we use min for T-norm and T-conorm
(implication )operators respectively and use
maxmin for composition then the resulting
fuzzy reasoning is as,
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max-min T-conorm/norm
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AGGREGATING THE RULES
The output of the rule base should be the
maximum of the output of each rule.
Can use any one of the operators defined on
fuzzy sets like maximum , algebraic sum or
min.
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DEFUZZIFICATION
a method to extract a representative crispvalue from a fuzzy set.
5 methods
1. Centroid of areazCOA :
zCOA =ZA(z) z dz /ZA(z) dz
Most widely used strategy.
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Cont.
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Cont..
2.Bisector of areazBOA :It generates the value z0 which partitions
the area into two area with same area.
zBOA A(z) dz =zBOA A(z) dz
3.Mean of maximumzMOM :
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BOA
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CONT
Is the average of the maximizing z at which the MF reaches a
maximum .
IfA(z) has a single maximum at z=z then ZMOM=z
IfA(z) reaches its maximum whenever z[zleft,zright] then
ZMOM=(ZLEFT+ZRIGHT)/2.
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CONT.
4.SMALLEST OF MAXIMUM ZSOM
It is the minimum of Z of the maximizing .
5.LARGEST OF MAXIMUM ZSOM
ZSOM is the maximum z of the maximizing z
ZSOM and LOM are not used as often as the other 3 defuzzificationmethods.
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Example: Mamdanis Fuzzy Model
Single-input single-output Mamdani fuzzy
model
If X is small then Y is small.
If X is medium then Y is medium.If X is large then Y is large.
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ADVANTAGES AND
DISADVANTAGES
Advantages
Its well suited to human input
Rules are having high correctness
Disadvantage
Defuzzification requires a large amount of
mathematical calculations.
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II.LARSEN MODEL
Product operator for a fuzzy implication
Max-product operator for the composition
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Contd..
The output of Larsen model are also fuzzy
sets.
Need defuzzification methods to get final
output.
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II.SUGENO FUZZY MODELS
Also known as the TSK fuzzy model
Introduced in 1984 by
T.Takagi
M.Sugeno and
K.T.Kang
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Motivation of TSK
To reduce the number of rules required by the
Mamdani model.
For complex and high dimensional problems
Develop a systematic approach to generate fuzzy
rules from a given input-output data set.
TSK model replaces the fuzzy set (then part) of
mamdani rule with function(equation) of theinput variables.
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TSK fuzzy rule
If x is A and y is B then z=f(x,y)
Where A and B are fuzzy sets in the antecedent
,and
Z=f(x,y) is a crisp function in the consequences .
Usually f(x,y) is a polynomial in the input variables x
and y.
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First order TSK fuzzy model
F(x,y) is a first order polynomial
Example: a two-input one-output TSK If x is Aj and y is Bk then z= px+qy+r
The degree the input matches ith rule is typically
computed using min operator:
wi=min(Aj(x), Bk(y))
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Cont..
Each rule has a crisp output
Overall output is obtained via weighted
average (reduce computation time of
defuzzification required in a Mamdani model)
Z=i wi zi/ i wi
to further reduce computation ,weighted
sum may be used. i.e
Z= i wi zi
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TSK fuzzy model
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example
Example: a single input TSK fuzzy model
can be expressed as,
If X is small then Y = 0.1X + 6.4 If X is medium then Y = - 0.5X + 4
If X is large then Y = X2.
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Example-2
Two-input single-output Sugeno fuzzy model
If X is small and Y is small then z=-x+y+1.
If X is small and Y is large then z=-y+3.
If X is large and Y is small then z=-x+3.
If X is large and Y is large then z=x+y+2
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Zero order TSK
When f is a constant , we have a zero order
TSK fuzzy model.
It has minimum computation time.
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summary
Overall output via either weighted average or
weighted sum is always crisp.
Without the time consuming defuzzification
operation the TSK fuzzy model is by the far
most popular candidate for sample-data-
based fuzzy modeling.
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Advantages
Computationally efficient
Works well with linear techniques
Has continuity of the output surface
Well suited to mathematical analysis.
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Reference
Neuro-fuzzy and soft computingJ.Jang,C.Sun
and E.Mizutani, Prentice Hall 1997
System modelling using a Mamdani Rule Base-
Bryan Davis,University of Florida
Fuzzy systems toolbox,M.Beale and
H.Demuth.
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