Institute for Advanced Management Systems Research
Department of Information Technologies
Faculty of Technology, Åbo Akademi University
What is fuzzy logic and fuzzy ontology?
Robert Fullér
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• Table of Contents
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c 2010
March 4, 2010
rfuller@mail.abo.fi
Table of Contents
1. Triangular and trapezoidal fuzzy numbers
2. Operations on fuzzy sets
3. Material implication
4. Fuzzy implications
5. The theory of approximate reasoning
6. Crisp relations
7. Fuzzy relations
Table of Contents (cont.)
3
8. Simplified fuzzy reasoning schemes
9. Tsukamoto’s fuzzy reasoning scheme
10. Sugeno’s fuzzy reasoning scheme
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Section 1: Triangular and trapezoidal fuzzy numbers
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1. Triangular and trapezoidal fuzzy numbers
A fuzzy set A of the real line R is defined by its membership function (denoted also by A) A : R → [0, 1]. If x ∈ R then A(x) is interpreted as the
degree of membership of x in A.
Figure 1: Membership functions for ”small” and ”big”.
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Section 1: Triangular and trapezoidal fuzzy numbers
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If x0 is the amount of the salary then x0 belongs to fuzzy set
A1 = small
with degree of membership
A1 (x0 ) =
and to


1−
 0
x0 − 2000
4000
if 2000 ≤ x0 ≤ 6000
otherwise
A2 = big
with degree of membership


1


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6000 − x0
A2 (x0 ) =
1−

4000

 0
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if x0 ≥ 6000
if 2000 ≤ x0 ≤ 6000
otherwise
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Section 1: Triangular and trapezoidal fuzzy numbers
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Definition 1.1. A fuzzy set A is called triangular fuzzy number with peak
(or center) a, left width α > 0 and right width β > 0 if its membership
function has the following form

a−t



if a − α ≤ t ≤ a
1−


α

t−a
A(t) =
1−
if a ≤ t ≤ a + β



β



0
otherwise
and we use the notation A = (a, α, β). The support of A is (a − α, b + β).
A triangular fuzzy number with center a may be seen as a fuzzy quantity
”x is close to a” or x is approximately equal to a”.
Definition 1.2. A fuzzy set A is called trapezoidal fuzzy number with tolerance interval [a, b], left width α and right width β if its membership function
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If A is not a fuzzy number then there exists an γ ∈ [0, 1] such that [A]γ
not a convex
of R. fuzzy numbers
Section 1: isTriangular
andsubset
trapezoidal
1
a-!
a
a+"
Fig. 1.2. Triangular fuzzy number.
Figure 2: A triangular fuzzy number.
has the
Definition 1.1.4 A fuzzy set A is called triangular fuzzy number with peak
(or center) a, left width α > 0 and right width β > 0 if its membership
function has form
the following form
following
 
a−t




1 −a − t if a − α ≤ t ≤ a



α

if a − α ≤ t ≤ a
1
−


tα
−a

A(t)
=

1−
if a ≤ t ≤ a + β



 1
β

if a ≤ t ≤ b


0
otherwise
A(t) =

t−b


β

[A]γ = 
[a − (1 − γ)α, a + (1 − γ)β], ∀γ ∈ [0, 1].


0
otherwise

and we use the notation
if aeasily
≤ tbe≤verified
b + that
β
 A1=−(a, α, β). It can
The support of A is (a − α, b + β). A triangular fuzzy number with center a
may the
be seen
as a fuzzy
and we use
notation
Aquantity
= (a, b, α, β).
”x is approximately equal to a”.
A fuzzy set AJ
is called trapezoidal
fuzzy
numberJ
with
toler- Doc I
JJ 1.1.5 II
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Section 1: Triangular and trapezoidal fuzzy numbers
8
and we use the notation
A=
b, α, β).
A trapezoidal fuzzy number may
be(a,seen
as a fuzzy quantity
(1.1)
It can easily be shown that
[A]γ = [a − (1 − γ)α, b + (1 − γ)β], ∀γ ∈ [0, 1].
”x is approximately in the interval [a, b]”.
The support of A is (a − α, b + β).
1
a-!
a
b
b+"
Fig. 1.3. Trapezoidal fuzzy number.
Figure 3: Trapezoidal fuzzy number.
A trapezoidal fuzzy number may be seen as a fuzzy quantity
”x is approximately in the interval [a, b]”.
Definition 1.1.6 Any fuzzy number A ∈ F can be described as
 %
&

a−t

L
if t ∈ [a − α, a]



α



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2. Operations on fuzzy sets
The classical set theoretic operations from ordinary set theory can be extended to fuzzy sets. Let A and B are fuzzy subsets of a crisp set X. The
intersection of A and B is defined as
(A ∩ B)(t) = min{A(t), B(t)} = A(t) ∧ B(t),
The union of A and B is defined as
(A ∪ B)(t) = max{A(t), B(t)} = A(t) ∨ B(t),
The complement of a fuzzy set A is defined as
(¬A)(t) = 1 − A(t),
for all t ∈ X.
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Section 3: Material implication
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3. Material implication
Let p =0 x is in A0 and q =0 y is in B 0 are crisp propositions, where A and
B are crisp sets for the moment.
The full interpretation of the material implication p → q is that: the degree
of truth of p → q quantifies to what extend q is at least as true as p, i.e.
1 if τ (p) ≤ τ (q)
τ (p → q) =
0 otherwise
τ (p)
τ (q)
1
0
0
1
1
1
0
0
τ (p → q)
1
1
1
0
Table 1: Truth table for the material implication.
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Section 4: Fuzzy implications
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4. Fuzzy implications
Consider the implication statement
if pressure is high then volume is small
Figure 4: Membership function for ”big pressure”.
The membership function of the fuzzy set A, big pressure, can be interpreted
as
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Section 4: Fuzzy implications
12
• 1 is in the fuzzy set big pressure with grade of membership 0
• 2 is in the fuzzy set big pressure with grade of membership 0.25
• 4 is in the fuzzy set big pressure with grade of membership 0.75
• x is in the fuzzy set big pressure with grade of membership 1, x ≥ 5

1



if u ≥ 5
5−u
A(u) =
1−
if 1 ≤ u ≤ 5

4


0
otherwise
The membership function of the fuzzy set B, small volume, can be interpreted as
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Section 4: Fuzzy implications
B(v) =
13

1



1−



0
v−1
4
if v ≤ 1
if 1 ≤ v ≤ 5
otherwise
• 5 is in the fuzzy set small volume with grade of membership 0
• 4 is in the fuzzy set small volume with grade of membership 0.25
• 2 is in the fuzzy set small volume with grade of membership 0.75
• x is in the fuzzy set small volume with grade of membership 1, x ≤ 1
If p is a proposition of the form ’x is A’ where A is a fuzzy set, for example,
big pressure and q is a proposition of the form ’y is B’ for example, small
volume then we define the implication p → q as
A(x) → B(y)
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Section 4: Fuzzy implications
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Figure 5: Membership function for ”small volume”.
For example,
x is big pressure → y is small volume ≡ A(x) → B(y)
Remembering the full interpretation of the material implication
1 if τ (p) ≤ τ (q)
p→q=
0 otherwise
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Section 4: Fuzzy implications
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We can use the definition
A(x) → B(y) =
1
0
if A(x) ≤ B(y)
otherwise
4 is big pressure → 1 is small volume
A(4) → B(1)
0.75 → 1 = 1
The most often used fuzzy implication operators are listed in the following
table.
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Section 4: Fuzzy implications
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Name
Definition
Early Zadeh
Łukasiewicz
Mamdani
Larsen
x→y
x→y
x→y
x→y
Standard Strict
x→y
Gödel
x→y
Gaines
x→y
Kleene-Dienes
Kleene-Dienes-Łukasiewicz
Yager
x→y
x→y
x→y
= max{1 − x, min(x, y)}
= min{1, 1 − x + y}
= min{x, y}
= xy
1 if x ≤ y
=
0 otherwise
1 if x ≤ y
=
y otherwise
1
if x ≤ y
=
y/x otherwise
= max{1 − x, y}
= 1 − x + xy
= yx
Table 2: Fuzzy implication operators.
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5. The theory of approximate reasoning
In 1979 Zadeh introduced the theory of approximate reasoning. This theory
provides a powerful framework for reasoning in the face of imprecise and
uncertain information.
Entailment rule:
Conjuction rule:
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Mary is very young
very young ⊂ young
Mary is young
pressure is not very high
pressure is not very low
pressure is not very high and not very low
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Section 5: The theory of approximate reasoning
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pressure is not very high
or
Disjunction rule:
pressure is not very low
pressure is not very high or not very low
Projection rule:
(x, y) is close to (3, 2)
(x, y) is close to (3, 2)
x is close to 3
y is close to 2
How to make inferences in fuzzy environment?
<1 :
if
observation
pressure is BIG then
pressure is 4
conclusion
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volume is SMALL
volume is 2
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Section 5: The theory of approximate reasoning
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Figure 6: BIG(4) = SMALL(2) = 0.75.
<1 :
if
observation
pressure is BIG then
pressure is 2
conclusion
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volume is SMALL
volume is 4
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6. Crisp relations
A classical relation can be considered as a set of tuples, where a tuple is an
ordered pair. A binary tuple is denoted by (u, v), an example of a ternary
tuple is (u, v, w) and an example of n-ary tuple is (x1 , . . . , xn ).
Definition 6.1. Let X1 , . . . , Xn be classical sets. The subsets of the Cartesian product X1 × · · · × Xn are called n-ary relations. If X1 = · · · = Xn
and R ⊂ X n then R is called an n-ary relation in X. Let R be a binary
relation in R. Then the characteristic function of R is defined as
(
1 if (u, v) ∈ R
χR (u, v) =
0 otherwise
Example 6.1. Let X be the domain of men {John, Charles, James} and Y
the domain of women {Diana, Rita, Eva}, then the relation ”married to”
on X × Y is, for example
{(Charles, Diana), (John, Eva), (James, Rita) }
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7. Fuzzy relations
Definition 7.1. Let X and Y be nonempty sets. A fuzzy relation R is a fuzzy
subset of X × Y . If X = Y then we say that R is a binary fuzzy relation in
X.
Let R be a binary fuzzy relation on R. Then R(u, v) is interpreted as the
degree of membership of (u, v) in R.
Example 7.1. A simple example of a binary fuzzy relation on U = {1, 2, 3},
called ”approximately equal” can be defined as
R(1, 1) = R(2, 2) = R(3, 3) = 1,
R(1, 2) = R(2, 1) = R(2, 3) = R(3, 2) = 0.8,
R(1, 3) = R(3, 1) = 0.3.
The membership function of R is given by
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Section 7: Fuzzy relations
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
1



0.8
R(u, v) =



0.3
if u = v
if |u − v| = 1
if |u − v| = 2
In matrix notation it can be represented as

1
R = 0.8
0.3
0.8
1
0.8

0.3
0.8
1
Fuzzy relations are very important because they can describe interactions
between variables.
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8. Simplified fuzzy reasoning schemes
Suppose that we have the following rule base
<1 :
also
<2 :
if
x is A1 then
y is z1
if
y is z2
<n :
if
x is A2 then
............
x is An then
y is zn
x is x0
fact:
y is z0
action:
where A1 , . . . , An are fuzzy sets.
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Section 8: Simplified fuzzy reasoning schemes
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Suppose further that our data base consists of a single fact x0 . The problem
is to derive z0 from the initial content of the data base, x0 , and from the
fuzzy rule base < = {<1 , . . . , <n }.
<1 :
if
salary is small then
loan is z1
<2 :
if
salary is big then
loan is z2
also
fact:
salary is x0
loan is z0
action:
A deterministic rule base can be formed as follows
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<1 :
if
2000 ≤ s ≤ 6000 then
loan is max 1000
<3 :
if
s ≥ 6000 then
loan is max 2000
<4 : if
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s ≤ 2000 then
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no loan at all
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Section 8: Simplified fuzzy reasoning schemes
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Figure 7: Discrete causal link between ”salary” and ”loan”.
The data base contains the actual salary, and then one of the rules is applied
to obtain the maximal loan can be obtained by the applicant.
In fuzzy logic everything is a matter of degree.
If x is the amount of the salary then x belongs to fuzzy set
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Section 8: Simplified fuzzy reasoning schemes
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• A1 = small with degree of membership 0 ≤ A1 (x) ≤ 1
• A2 = big with degree of membership 0 ≤ A2 (x) ≤ 1
Figure 8: Membership functions for ”small” and ”big”.
In fuzzy rule-based systems each rule fires.
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Section 8: Simplified fuzzy reasoning schemes
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The degree of match of the input to a rule (wich is the firing strength) is
the membership degree of the input in the fuzzy set characterizing the antecedent part of the rule.
The overall system output is the weighted average of the individual rule
outputs, where the weight of a rule is its firing strength with respect to the
input.
To illustrate this principle we consider a very simple example mentioned
above
<1 :
if
salary is small then
loan is z1
<2 :
if
salary is big then
loan is z2
also
salary is x0
fact:
loan is z0
action:
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Section 8: Simplified fuzzy reasoning schemes
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Then our reasoning system is the following
• input to the system is x0
• the firing level of the first rule is α1 = A1 (x0 )
• the firing level of the second rule is α2 = A2 (x0 )
• the overall system output is computed as the weghted average of the
individual rule outputs
α1 z1 + α2 z2
z0 =
α1 + α2
that is
A1 (x0 )z1 + A2 (x0 )z2
z0 =
A1 (x0 ) + A2 (x0 )
A1 (x0 ) =
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1 − (x0 − 2000)/4000
0
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if 2000 ≤ x0 ≤ 6000
otherwise
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Section 8: Simplified fuzzy reasoning schemes
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Figure 9: Example of simplified fuzzy reasoning.
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Section 8: Simplified fuzzy reasoning schemes
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
 1
1 − (6000 − x0 )/4000
A2 (x0 ) =

0
if x0 ≥ 6000
if 2000 ≤ x0 ≤ 6000
otherwise
It is easy to see that the relationship
A1 (x0 ) + A2 (x0 ) = 1
holds for all x0 ≥ 2000.
It means that our system output can be written in the form.
z0 = α1 z1 + α2 z2 = A1 (x0 )z1 + A2 (x0 )z2
that is,
6000 − x0
x0 − 2000
z1 + 1 −
z2
z0 = 1 −
4000
4000
if 2000 ≤ x0 ≤ 6000.
And z0 = 1 if x0 ≥ 6000. And z0 = 0 if x0 ≤ 2000.
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Section 8: Simplified fuzzy reasoning schemes
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Figure 10: Input/output function derived from fuzzy rules.
The (linear) input/oputput relationship is illustrated in figure 10.
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9. Tsukamoto’s fuzzy reasoning scheme
All linguistic terms are supposed to have monotonic membership functions.
The firing levels of the rules, denoted by αi , i = 1, 2, are computed by
α1 = A1 (x0 ) ∧ B1 (y0 ), α2 = A2 (x0 ) ∧ B2 (y0 )
In this mode of reasoning the individual crisp control actions z1 and z2 are
computed from the equations
α1 = C1 (z1 ),
α2 = C2 (z2 )
and the overall crisp control action is expressed as
α1 z1 + α2 z2
z0 =
α1 + α2
i.e. z0 is computed by the discrete Center-of-Gravity method.
If we have n rules in our rule-base then the crisp control action is computed
as
Pn
αi zi
,
z0 = Pi=1
n
i=1 αi
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Section 9: Tsukamoto’s fuzzy reasoning scheme
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where αi is the firing level and zi is the (crisp) output of the i-th rule, i =
1, . . . , n
Example 9.1. We illustrate Tsukamoto’s reasoning method by the following
simple example
<1 :
also
<2 :
if x is A1 and y is B1 then z is C1
if x is A2 and y is B2 then z is C2
fact :
x is x̄0 and y is ȳ0
consequence :
z is C
Then according to the figure we see that
A1 (x0 ) = 0.7,
B1 (y0 ) = 0.3
therefore, the firing level of the first rule is
α1 = min{A1 (x0 ), B1 (y0 )} = min{0.7, 0.3} = 0.3
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Section 9: Tsukamoto’s fuzzy reasoning scheme
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and from
A2 (x0 ) = 0.6,
B2 (y0 ) = 0.8
it follows that the firing level of the second rule is
α2 = min{A2 (x0 ), B2 (y0 )} = min{0.6, 0.8} = 0.6,
the individual rule outputs z1 = 8 and z2 = 4 are derived from the equations
C1 (z1 ) = 0.3,
C2 (z2 ) = 0.6
and the crisp control action is
z0 = (8 × 0.3 + 4 × 0.6)/(0.3 + 0.6) = 6.
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Section 9: Tsukamoto’s fuzzy reasoning scheme
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Figure 11: Tsukamoto’s inference mechanism.
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10. Sugeno’s fuzzy reasoning scheme
Sugeno and Takagi use the following architecture
<1 :
if x is A1 and y is B1 then z1 = a1 x + b1 y
<2 :
if x is A2 and y is B2 then z2 = a2 x + b2 y
also
fact :
x is x̄0 and y is ȳ0
cons. :
z0
The firing levels of the rules are computed by
α1 = A1 (x0 ) ∧ B1 (y0 ), α2 = A2 (x0 ) ∧ B2 (y0 )
then the individual rule outputs are derived from the relationships
z1∗ = a1 x0 + b1 y0 , z2∗ = a2 x0 + b2 y0
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Section 10: Sugeno’s fuzzy reasoning scheme
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and the crisp control action is expressed as
z0 =
α1 z1∗ + α2 z2∗
α1 + α2
Figure 12: Sugeno’s inference mechanism.
If we have n rules in our rule-base then the crisp control action is computed
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as
Pn
αi zi∗
z0 = Pi=1
,
n
i=1 αi
where αi denotes the firing level of the i-th rule, i = 1, . . . , n
Example 10.1. We illustrate Sugeno’s reasoning method by the following
simple example
<1 :
if x is BIG and y is SMALL then
fact :
x0 is 3 and y0 is 2
<2 :
z1 = x + y
if x is MEDIUM and y is BIG then z2 = 2x − y
conseq :
z0
Then according to the figure we see that
µBIG (x0 ) = µBIG (3) = 0.8,
µSM ALL (y0 ) = µSM ALL (2) = 0.2
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Section 10: Sugeno’s fuzzy reasoning scheme
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therefore, the firing level of the first rule is
α1 = min{µBIG (x0 ), µSM ALL (y0 )} = min{0.8, 0.2} = 0.2
and from
µM EDIU M (x0 ) = µM EDIU M (3) = 0.6,
µBIG (y0 ) = µBIG (2) = 0.9
it follows that the firing level of the second rule is
α2 = min{µM EDIU M (x0 ), µBIG (y0 )}
= min{0.6, 0.9} = 0.6.
the individual rule outputs are computed as
z1∗ = x0 + y0 = 3 + 2 = 5, z2∗
= 2x0 − y0 = 2 × 3 − 2 = 4
so the crisp control action is
z0 = (5 × 0.2 + 4 × 0.6)/(0.2 + 0.6) = 4.25.
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Figure 13: Example of Sugeno’s inference mechanism.
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