4. General Fuzzy Systems
A fuzzy system is a static nonlinear
mapping between its inputs and outputs
(i.e., it is not a dynamic system).
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Universe of Discourse
The “universe of discourse”
for ui or yi since it provides
the range of values
(domain) of Ui or Yi that
can be quantified with
linguistic and fuzzy sets.
An “effective” universe of
discourse [, ].
Width of the universe of
discourse:   
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 2 , 2 
 4 , 4 
 20,20
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Linguistic Variables
Linguistic expressions are needed for the inputs and
outputs and the characteristics of the inputs and
outputs.
Linguistic variables (constant symbolic descriptions
of what are in general time-varying quantities) to
describe fuzzy system inputs and outputs.
Linguistic variables: u~i is to described the input ui
~
yi is to described the output yi
for example, u~1 “position error”
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Linguistic Values
~
~
Linguistic variables ui and yi take on “linguistic
values” that are used to describe characteristic of
the variables.
~j
Let Ai denote the jth linguistic value of the
linguistic variable u~i defined over the universe of
~
~j
discourse Ui.  A
i  { Ai : j  1,2,..., Ni }
For example, u~1  “speed”
~1
~2
~3
A1 " slow" , A1 " medium" , A1 " fast"
~
~1 ~ 2 ~ 3
A1  { A1 , A1 , A1 }
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Linguistic Rules
The mapping of the inputs to the outputs for a
fuzzy system is in part characterized by a set of
condition  action rules, or in modus ponens (IfThen) form: If premise Then consequent.
Multi-input single-output (MISO) rule:
~j
~k
~l
~
~
~
~
If u1 is A1 and u2 is A2 ,..., and un is An Then ~
yq is Bqp
the ith MISO rule:( j, k ,..., l; p, q)i
Multi-input multi-output (MIMO) rule:
~j
~k
~l
~
~
~
~
~
If u1 is A1 and u2 is A2 ,..., and un is An Then ~
y1 is B1r and ~
y2 is B2s
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Number of Fuzzy Rules
If all the premise terms are used in very rule
and a rule is formed for each possible
combination
of premise elements, then there
n
are  Ni  N1  N 2  ...  N n rules in the rule-base.
i 1
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Fuzzy Quantification of
Rules: Fuzzy Implications
Multi-input single-output (MISO) rule:
~j
~k
~l
~p
~
~
~
~
If u1 is A1 and u2 is A2 ,..., and un is An Then yq is Bq
Define: A1j  {( u1 ,  j (u1 )) : u1  U1}
A1
A2k  {( u2 ,  Ak (u2 )) : u2  U 2 }
2

Anl  {( un ,  Al (un )) : un U n }
n
Bqp  {( yq ,  B p ( yq )) : yq  Yn }
q
These fuzzy sets quantify the terms, in the premise and
the consequent of the given If-Then rule, to make a “fuzzy
implication” (which is a fuzzy relation).
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Fuzzy Implications
A fuzzy implication:If A1j and A2k ,..., and Anl Then Bqp
the fuzzy set A1j quantifies the meaning of the
~j
p
~
linguistic statement “ u1 is A1 “, and Bq quantifies
~p
~
the meaning of “ yq is Bq “.
Two general properties of fuzzy logic rule-bases
1. Completeness whether there are conclusions
for every possible fuzzy controller input.
2. Consistency whether the conclusions that rules
make conflict with other rules’ conclusions.
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Fuzzification
Fuzzification: convert its numeric inputs ui  Ui
into fuzzy sets.
Let U i denote the set of all possible fuzzy sets that
can be defined on Ui. Given ui  Ui, fuzzification
transforms ui to a fuzzy set denoted by Âi fuz defined
on the universe of discourse Ui.
Fuzzification operation: F : U i  U i
where F (ui )  Aˆi fuz
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Singleton Fuzzification
When a singleton fuzzification is used, which
produces a fuzzy set Âi fuz U i with a membership
function defined by
1, x  ui
u Aˆ fuz ( x)  
i
0, otherwise
Any fuzzy set with this form for its membership
function is called a “singleton”.
Other fuzzification methods haves not been used
very much because they are complexity.
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Inference Mechanism
Two basic tasks –
(1) matching: determining the extent to which
each rule is relevant to the current situation as
characterized by the inputs ui, i = 1, 2, …, n.
(2) inference step: drawing conclusions using the
current input ui and the information in the rulebase.
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Matching
Assume that the current inputs ui, i = 1, 2, …, n,
and fuzzification produces Aˆ1fuz , Aˆ2fuz ,..., Aˆnfuz the
fuzzy sets representing the inputs.
Step 1: combine inputs with rule premises
 Aˆ j (u1 )   A j (u1 )   Aˆ fuz (u1 )   Aˆ j (u1 )   A j (u1 )
1
1
1
1
1
 Aˆ l (un )   Al (un )   Aˆ fuz (un )   Aˆ l (un )   Al (un )
n
n
n
n
n
Step 2: determine which rules are on
 i (u1 , u2 ,..., un )   Aˆ (u1 )   Aˆ (u2 )    Aˆ (un )
  A (u1 )   A (u2 )    A (un )
j
1
j
1
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2
k
2
l
n
l
n
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Rule Certainty
We use i (u1 , u2 ,..., un ) to represent the certainty
that the premise of rule i matches the input
information when we use singleton fuzzification.
An additional “rule certainty” is multiplied by i.
Such a certainty could represent our a priori
confidence in each rule’s applicability and would
normally be a number between zero and one.
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Inference Step
Alternative 1: determine implied fuzzy sets
i
B̂
For the ith rule, the “implied fuzzy set” q with
membership function
 Bˆ ( yq )   i (u1 , u2 ,..., un )   B ( yq )
i
q
p
q
Alternative 2: determine the overall implied fuzzy
sets. The “overall implied fuzzy set” B̂q with
membership function
 Bˆ ( yq )   Bˆ ( yq )   Bˆ ( yq )     Bˆ ( yq )
q
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q
2
q
R
q
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Compositional Rule of
Inference
The overall implied fuzzy set:
 Bˆ ( yq )   Bˆ ( yq )   Bˆ ( yq )     Bˆ ( yq )
where  Bˆ ( yq )   i (u1 , u2 ,..., un )   B ( yq )
Sup-star compositional rule of inference:
“sup” corresponds to the  operation, and the
“star” corresponds to the  operation.
 Sup (supremum): the least upper bound
Zadeh’s compositional rule of inference:
maximum is used for  and minimum is used for .
1
q
q
i
q
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2
q
R
q
p
q
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Defuzzification:
Implied Fuzzy Sets
Center of gravity (COG): using the center of are
and area of each implied fuzzy set
R
q
b

i Yq  Bˆ qi ( y q ) dy q
i

1
yqcrisp 
R
i 1 Yq  Bˆqi ( yq )dyq
Center-average: using the centers of each of the
output membership functions and the maximum
certainty of each of the conclusions represented
with the implied fuzzy sets
R
q
b

i sup y q { Bˆ qi ( y q )}
i

1
yqcrisp 
R
i 1 sup yq { Bˆ i ( yq )}
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Defuzzification:
Overall Implied Fuzzy Sets
Max criterion
A crisp output yqcrisp is chosen as the point on the output
universe of discourse Yq for which the overall implied
fuzzy set B̂q achieves a maximum


crisp
yq  arg sup{ Bˆ ( yq )}
q
Yq


“arg supx{(x)}” returns the value of x that results in the
supremum of the function being achieve.
Sometimes the supremum can occur at more than one point
in Yq. In this case you also need to specify a strategy on
how to pick one point for
(e.g., choosing the smallest
yqcrisp
value)
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Defuzzification:
Overall Implied Fuzzy Sets
Mean of maximum (MOM)
A crisp output yqcrisp is chosen to represent the mean
value of all elements whose membership in
B̂q is a maximum.
Define bˆqmax as the supremum of the membership
function of B̂q over Yq. Define a fuzzy set Bˆ q  Yq
with the following membership function
ˆ max
yq  Bˆ  ( yq )dyq

1
,

(
y
)

b

Y
 Bˆq q
q
q
 Bˆ  ( yq )  
 yqcrisp  q
q


ˆ  ( yq )dyq

0
,
othereise
Y
B

q
q
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Defuzzification:
Overall Implied Fuzzy Sets
Center of area (COA)
A crisp output yqcrisp is chosen as the center of area
for the membership function of the overall implied
fuzzy set B̂q .
For a continuous output universe of discourse Yq,
the center of area output is defined by
y
crisp
q

Y yq  Bˆ ( yq )dyq
q
q
Y  Bˆ ( yq )dyq
q
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Functional Fuzzy Systems
Standard fuzzy system:
~j
~k
~l
~p
~
~
~
~
If u1 is A1 and u2 is A2 ,..., and un is An Then yq is Bq
Functional fuzzy system:
~j
~k
~l
~
~
~
If u1 is A1 and u2 is A2 ,..., and un is An Then bi  gi ()
The choice of the function gi(·) depends on the
application being considered. The function gi(·)
can be linear or nonlinear.
Defuzzification: R b 
y  i R1 i i
i 1 i
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Takagi-Sugeno Fuzzy System
Takagi-Sugeno fuzzy system:
bi  g ()  ai ,0  ai ,1u1  ai , 2u2    ai ,nun
If ai,0=0, then the gi(·) mapping is a linear
mapping.
If ai,00, then the gi(·) mapping is called “affine.”
~1 n = 1, R = 2.
~
Suppose
If u is A Then b  2  u .
1
1
1
1
~
If u~1 is A12 Then b2  1  u1.
b11  b2  2
y
1   2
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Singleton O/P Fuzzy System
If gi= ai,0, then Takagi-Sugeno fuzzy system is
equivalent to a standard fuzzy system that uses
center-average defuzzification with singleton
output membership function at ai,0.
The corresponding fuzzy rule is of the form:
~j
~k
~l
~
~
~
If u1 is A1 and u2 is A2 ,..., and un is An Then bi
where bi is a real number.
b

y
 
R
i 1 i i
R
i 1 i
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Consequent Forms
Type 1: a crisp value (singleton output)
~j
~k
~l
~
~
~
If u1 is A1 and u2 is A2 ,..., and un is An Then bi
Type 2: a fuzzy number (standard fuzzy system)
~j
~k
~l
~p
~
~
~
~
If u1 is A1 and u2 is A2 ,..., and un is An Then yq is Bq
Type 3: a function (functional fuzzy system)
~j
~k
~l
~
~
~
If u1 is A1 and u2 is A2 ,..., and un is An Then bi  gi ()
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Universal Approximation Property
Suppose that we use center-average
defuzzification, product for the premise and
implication, and Gaussian membership functions.
Name this fuzzy system f(u). Then, for any real
continuous (u) defined on a closed and bounded
set and an arbitrary
 > 0, there exists a fuzzy system f(u) such that
sup f (u )  (u )  
u
: Psi,  : Epsilon
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