00_1 - Slide Pelengkap (dari Buku Neuro Fuzzy and Soft Computing).ppt

Definition 2.2 Support
The support of fuzzy set A is the set of all points x in X such that
A(x) > 0:
Support (A) = {x A(x) > 0}.
Definition 2.3 core
The core of a fuzzy set A is the set of all points x is X such that
A(x) =1:
Core(A) = {x A(x ) =1}.
Definition 2.4 Normality
A fuzzy set A is normal if its core is nonempty. In other words,
we can always find a point x  X such that A(x) =1.

Definition 2.5 Crossover points
A crossover point of a fuzzy set A is a point x  X at which
A(x)= 0.5:
Crossover(A)={x A(x) = 0.5}
Definition 2.6 Fuzzy singleton
A fuzzy set whose support is a single point in X with A(x) = 1 is
called a fuzzy singleton.

Definition 2.7 -cut,strong -cut
The -cut or -level set of a fuzzy set A is a crisp set defined by
A’
 = {x  A (x)  }.
Using the notation for a level set, we can express the support
and core of a fuzzy set A as
Support(A) = A’
0’
And
Core(A)=A1’
Respectively.

Definition 2.9 Fuzzy numbers
A fuzzy number A is a fuzzy set in the real line (R) that satisfies
the conditions for normality and convexity.
Most (noncomposite) fuzzy sets used in the literature satisfy the
conditions for normality and convexity, so fuzzy numbers are the
most basic type of fuzzy sets.

Definition 2.10 Bandwidths of normal and convex fuzzy sets
For a normal and convex fuzzy set, the bandwidth of width is defined
as the distance between the two unique crossover points:
Width(A) = x2 – x1,
Where A(x1) = A(2) = 0.5.
Definition 2.11 Symmetry
A fuzzy set A is symmetric if its MF is symmetric around a
certain point x = c, namely,

Definition 2.12 Open left, open right, closed
A fuzzy set A is open left if limx A (x) =1 and limx+ A(x) =
0; open right if limx- A (x) = 0 and limx+A(x) = 1; and
closed if limx-A (x) = limx+A (x) = 0.

Law of contradiction A  = 
Law of the excluded middle A  = X
Idempotency A  A = A, A  A = A
Involution
= A
Commutativity A  B = B  A, A  B = B  A
Associativity (A  B)  C = A  (B  C )
(A  B)  C = A  ( B  C )
Distributivity A  (B  C ) = (A  B)  (A  C )
A  (B  C) = (A  B )  (A  C )
Absorption A  ( A  B ) = A
A  ( A  B ) = A
Absorption of
Complements
A  (  B) = A B
A  (  B) = A B
DeMorgan’s laws = 
= 
A
A
B
A
B
A 
B
B
A
A
A
A
A

Cylindrical Extension
Definition 2.24 Cylindrical extensions of one-dimensional fuzzy sets
If A is a fuzzy set in X, then its cylindrical extension in X x Y is a fuzzy
set c(A) defined by

 XxY y
x
x
A
c A ).
,
/(
)
(
)
( 

Projection
Definition 2.25 Projection of fuzzy sets
Let R be a two-dimensional fuzzy set on X x Y. Then the projections of
R onto X and Y are defined as












X
x
y
x
R
y
X
R /
)
,
(
max













Y
y
y
x
R
x
Y
R /
)]
,
(
max


Four of the most frequently used T-norm operators are
Minimum Tmin(a, b) = min (a,b) = a  b.
Algebraic product Tap (a, b) = ab.
Bounded product Tbp(a, b) = 0  (a + b-1)
a, if b= 1.
Drastic product Tdp(a,b) = b, if a = 1.
0, if a,b  1.

Corresponding to the four T-norm operators in the previous example,
we have the following four T-conorm operators.
Maximum : S(a, b) = max(a, b) = a  b.
Algebraic sum : S(a, b) = a + b – ab.
Bounded sum : S(a, b) = 1  (a + b ).
a, if b = 0.
Drastic sum: S(a,b) = b, if a = 0.
1, if a, b > 0.

Yager [9]: For q > 0,
Ty (a,b,q) = 1-min{1,[(1-a)q + (1-b)q]1/q},
Sy(a,b,q) = min{1,(aq +bq)1/q}.
Dubois and Prade [4]: For   [0,1],
TDP(a,b,) = ab/max {a,b,}
SDP(a,b,) = [a+b – ab – min{a, b, (1-)} / max {1-a, 1 – b,}
Hamacher [6]: For  > 0,
TH (a, b, ) = ab / [+ (1-)(a + b – ab)],
SH (a,b, ) = [a + b + ( -2) ab] / [1 + ( - 1) ab].

Frank [5]: For s > 0,
TF (a, b, s) = logs,[1 + (sa – 1)(sb –1)/(s –1)],
SF (a, b, s) = 1 – logs[1 + (s1 – a –1) (s1-b-1)/(s-1)].
Sugeno [8]: For   -1,
TS(a, b, ) = max{0, (+ 1)(a + b –1) - ab},
Ss (a, b, ) = min{1, a + b - ab}.
Dombi [2]: For  > 0,
TD (a, b, ) =
SD(a, b, ) =
,
/
1
1
1
]
)
1
(
)
1
[(
1
1






 

b
a
,
/
1
1
1
]
)
1
(
)
1
[(
1
1


 







 b
a

0
0.5
1
(a) Two fuzzy sets A and B
A B
0
0.5
1
(b) T-norm of A and B
0
0.5
1
(c) T-conorm (S-norm) of A and B
Figure. Schweizer and Sklar’s parameterized T-norms and T-conorms: (a)
membership functions for fuzzy set A and B; (b) Tss(a,b,p) and (c) Sss (a, b, p)
with p = ∞ (solid line), 1 (dashed line) and –1 (dash-dotted line).

Definition 3.1 Extension principle
Suppose that function f is a mapping from an n-
dimensional Cartesian product space X1 x X2 x Xn to a one-
dimensional universe Y such that y = f (x1,…,xn), and suppose
Ai,…,An are n fuzzy sets in X1,…,Xn, respectively. Then the
extension principle asserts that the fuzzy set B induced by the
mapping f is defined by
max [mini Ai(xi)], if f-1(y)0.
B(y) = (x1,…,xn), (x1,…,xn) = f-1(y)
0, if f-1(y) =0.

Example 3.1 Application of the extension principle to fuzzy sets with discrete
universes
Let
A = 0.1/-2+0.4/-1+0.8/0 + 0.9/1 +0.3/2
And
f (x) = x2 -3.
Upon applying the extension principle, we have
B = 0.1/1+0.4/-2+0.8/-3+ 0.9/ -2+ 0.3/1
= 0.8/ -3+ (0.4 V 0.9) / -2 + (0.1 V 0.3)/1
= 0.8/ -3 + 0.9/-2+ 0.3/1
where V represents max. Figure 3.1 illustrates this example.

0 0.111 0.200 0.273 0.333
R = 0 0 0.091 0.167 0.231 ,
0 0 0 0.077 0.143
y -x
R (x,y) = x+y+2 if y > x.
0, if y  x.
Example 3.3 Binary fuzzy relations
Let X = Y =R+ (the positive real line) and R = “y is much greater than x”, The
MF of the fuzzy relation R can be subjectively defined as
if X = {3,4,5} and Y = {3,4,5,6,7}, then it is convenient to express the fuzzy
relation R as a relation matrix:
where the element at row I and column j is equal to the membership grade
between the ith element of X and jth element of y.

Example 3.4 Max-min and max-product composition
Let
R1 = “x is relevant to y”
R2 = “y is relevant to z”
Be two fuzzy relations defined on X x Y and Y x Z, respectively, where X = {1,
2, 3}
Y = {, , , }, and Z = {a, b}. Assume that R1 and R2 can be expressed as the
following relation matrices:











2
.
0
3
.
0
8
.
0
6
.
0
9
.
0
8
.
0
2
.
0
4
.
0
7
.
0
5
.
0
3
.
0
1
.
0
1
R













2
.
0
7
.
0
6
.
0
5
.
0
3
.
0
2
.
0
1
.
0
9
.
0
2
R

Now we want to find R1R2, which can be interpreted as a derived fuzzy
relation “x is relevant to z” based on R1 and R2. For simplicity, suppose that
we are only interested in the degree of relevance between 2 ( X) and a (
Z). If we adopt max-min composition, then
On the other hand, if we choose max-product composition instead, we have
).
min
max
(
7
.
0
)
7
.
0
,
5
.
0
,
2
.
0
,
4
.
0
max(
)
7
.
0
9
.
0
,
5
.
0
8
.
0
,
2
.
0
2
.
0
,
9
.
0
4
.
0
max(
)
,
2
(
2
1
n
compositio
by
a
R
R










).
min
max
(
63
..
0
)
63
.
0
,
40
.
0
,
04
.
0
,
36
.
0
max(
)
7
.
0
9
.
0
,
5
.
0
8
.
0
,
2
.
0
2
.
0
,
9
.
0
4
.
0
max(
)
,
2
(
2
1
n
compositio
by
x
x
x
x
a
R
R







Extension principle on fuzzy sets with discrete
universes
0.3
0.9
0.8
0.4
0.1
X Y
f (x )
1
0
-1
-2
-3
2
1
0
-1
-2

Composition of fuzzy relations
1
2
3




a
b
X Y Z
R1
R2
0.9
0.2
0.5
0.7
0.4
0.2
0.8
0.9

From the preceding example, we can see that the term set
consists of several primary terms (young, middle aged, old)
modified by the negation (“not”) and/or the hedges (very, more
or less, quite, extremely, and so forth), and then linked by
connectives such as and, or, either, and neither. In the sequel,
we shall treat the connectives, the hedges, and the negation as
operators that change the meaning of their operands in a
specified, context-independent fashion.

Definition 3.6 Concentration and dilation of linguistic values
Let A be a linguistic values characterized by a fuzzy set with
membership function A(). Then AK is interpreted as a modified
version of the original linguistic value expressed as
In particular, the operation of concentration is defined as
CON (A) = A2,
While that of dilation is expressed by
DIL(A) = A0.5.
.
/
)]
(
[ x
x
A k
A
x
k

 

Following the definition in the previous chapter, we can interpret
the negation operator NOT and the Connectives AND and OR
as
NOT(A) = ¬A =
A AND B AB = 
A OR B = AB = 
Respectively, where A and B are two linguistic values whose
meanings are defined by A() and B().
 
x
A x
x ,
/
)]
(
1
[ 
x
A x)
(
[ ,
/
)]
( x
x
B

x
A x)
(
[ ,
/
)]
( x
x
B


Example 3.6 Constructing MFs for composite linguistic terms
Let the meanings of the linguistic terms young and old be
defined by the following membership functions:
young (x) = bell(x,20,2,0)=
old(x) = bell(x,30,3,100)=
where x is the age of given person, with the interval [0,100] as
the universe of discourse. Then we can construct MFs for the
following composite linguistic terms:
,
4
20
1
1








 x
,
6
30
100
1
1











 x

 more or less old = DIL (old) = old0.5
 not young and not old = ¬young ¬old
 young but not too young =young ¬young2
 extremely old
,
6
30
100
1
1
1
4
20
1
1
1 x
x
x
x




































































.
6
30
100
1
1 x
X
x














.
2
20
1
1
1
4
20
1
1 x
x
x
x











































































 






Definition 3.8 Orthogonality
A term set T = t1,…,tn of a linguistic variable x on the universe X
is orthogonal if it fulfills the following property:
where the ti’s are convex and normal fuzzy sets defined on X
and these fuzzy sets make up the term set T.





n
i
X
x
x
ti
1
,
,
1
)
(


If we interpret AB as A coupled with B, then
R = AB = AxB =
where is a T-norm operator and AB is used again to
represent the fuzzy relation R. On the other hand, is AB is
interpreted as A entails B, then it can be written as four different
formulas:
 
XxY
B
A y
x
y
x ),
,
/(
)
(
~
)
( 


~

A entails B
• Material implication:
• Propositional calculus:
•Extended propositional calculus:
• Generalization of modus ponens:
.
B
A
B
A
R 




).
( B
A
A
B
A
R 





.
)
( B
B
A
B
A
R 






}
1
0
)
(
*
~
)
(
sup{
)
,
( 


 c
and
y
c
x
c
y
x B
A
R 



Suppose that we adopt the first interpretation, “A coupled with B,” as the
meaning of AB. Then four different fuzzy relations AB result from
employing four of the most commonly used T-norm operators.
 Rdp = A x B = XxY A(x) B(y)/(x,y), or
F (a,b) = a b =
.̂
a if b = 1.
b if a = 1.
0 Otherwise.
This formula uses the drastic product operator for conjunction
.̂

When we adopt the second interpretation, “A entails B,” as the
meaning of AB, again there are number of fuzzy implication
functions that are reasonable candidates. The following four
have been proposed in the literature:
• Ra =AB = XxY1 (1-A(x)+B(y)) /(x,y), or a(a,b) = 1(1-
a+b). This is Zadeh’s arithmetic rule, which follows
Equation (3.19) by using the bounded sum operator for .
• Rmm =A(AB) = XxY (1-A(x))(A(x)B(y))/(x,y), or
m(a,b) = (1-a)  (ab). This is Zadeh’s max-min rule,
which follows Equation (3.20) by using max for .

 Rs=AB=XxY ( 1-A(x))  B(y), or s(a,b) = (1-a)  b. This is
Boolean fuzzy implication using max for .
· R =XxY ( A(x B (y))/(x,y),where

 .
1
.
/
~ b
a
if
b
a
if
a
b
b
a




This is Goguen’s fuzzy implication, which follows Equation (3.22) by using
the algebraic product for the T-norm operator.

~

Specially, let A, c(A), B, and F be the MFs of A, c(A), B, and F,
respectively, where c(A) is related to A through
c(A) (x,y) = A(x).
Then
c(A)F(x,y) = min [c(A) (x,y), F(x,y)]
= min [A (x), F(x,y)]
by projecting c(A)  F onto the y-axis, we have
B (y) = maxx min [A(x), F(x,y)]
= Vx[A(x)  F(x,y)]
This formula reduce to the max-min composition (see Definition 3.3 in Section
3.2) of two relation matrices if both A (a unary fuzzy relation) and F (a binary
fuzzy relation) have finite universes of discourse. Conventionally, B is
represented as
B = A F,

Definition 3.9 Approximate reasoning (fuzzy reasoning)
Let A, A’, and B be fuzzy sets of X, X, and Y, respectively, Assume that the
fuzzy implication AB is expressed as a fuzzy relation R on X x Y. Then the
fuzzy set B induced by “x is A’” and the fuzzy rule “if x is A then y is B” is
defined by
μB’(y) = max min[μA’(x),μR(x,y)]
= Vx[μA’(x)  μR(x,y)],
or, equivalently,
B’= A’ R = A’ (AB).

The fuzzy rule in premise 2 can be put into the simpler form “A x B C.”
Intuitively, this fuzzy rule can be transformed into a ternary fuzzy relation Rm
Based on Mamdani’s fuzzy implication function, as follows:
The resulting C’ is expressed as
C’ = (A’ x B’)  (A x BC).
Thus
μC’(z) = Vx,y[μA’(x)  μB’(y)]  [μA(x)  μB(y)  μC(z)]
= Vx,y{[μA’(x)  μB’(y)  μA(x)  μB(y)]}  μC(z)
= {Vx[μA’(x)  μA(x)]}  {Vy[μB’(y)  μB(y)]}  μC(z)
= (w1  w2)  μC(z),
firing
strength
where w1 and w2 are the maxima of the MFs of A  A’ and B B’,
respectively.
).
,
,
/(
)
(
)
(
)
(
)
(
)
,
,
( z
y
x
z
c
y
B
x
A
XxYxZ
xC
AxB
C
B
A
m
R 

 





Theorem 3.1 Decomposition method for calculating B’
C’ = (A’ x B’)  (A x B  C)
= [A’ (AC)]  [B’  (BC)]
Proof:
μC’(z) = V x,y[μA’(x)  μB’(y)]  [μA(x)  μB(y)  μC(z)]
= V x {μA’(x)  μB (y)  μC(z)  Vy[μB’(y)  μB(y)  μC(z)]}
= μA’(x)  μB(y)  μC(z)  μB’ (BC)(y).

To verify this inference procedure, let R1 = A1 x B1C1 and R2 = A2 x B2C2.
Since the max-min composition operator  is distributive over the  operator,
it follows that
C’= (A’x B’)  (R1  R2)
= [(A’x B’)  R1]  [ (A’x B’)R2]
= C’1  C’2,
where C’1 and C’2 are the inferred fuzzy sets for rules 1 and 2, respectively.

In summary, the process of fuzzy reasoning or approximate reasoning can
be divided into four steps:
•Degrees of compatibility Compare the known facts with the antecedents
of fuzzy rules to find the degrees of compatibility with respect to each
antecedent MF.
•Firing strength Combine degrees of compatibility with respect to
antecendent MFs in a rule using fuzzy AND or OR operators to form a firing
strength that indicates the degrees to which the antencedent part of the rule
is satisfied.
•Qualified (induced) consequent MFs Apply the firing strength to the
consequent MF of a rule to generate a qualified consequent MF. (The
qualified consequent MFs represent how the firing strength gets propagated
and used in a fuzzy implication statement.)
•Overall output MF Aggregate all the qualified consequent MFs to obtain an
overall output MF.

x is A1
y is B1
x is A2
y is Br
x is Ar
x is Br
Rule 1
Rule 2
Rule 3
Aggregator Defuzzifier y
(Fuzzy)
(Fuzzy)
(Fuzzy)
(Fuzzy)
w1
w2
w3
(Crisp)
Figure Block diagram for a fuzzy inference system.

•Centroid of area zCOA:
zCOA = ,
)
(
)
(


z dz
z
A
z zdz
z
A


•Bisector of area zBOA: zBOA satisfies
 

zBOA
zBOA
dz
z
A
dz
z
A

 
 ,
)
(
)
(
Mean of maximum zMOM: zMOM is the average of the maximizing z at which
the MF reach a maximum μ*. In symbols,
zMOM =
where Z’ = {z A(z) = *}. In particular, if A(z) has a single maximum at z =
z*, then zMOM =z*. Moreover, if μA(z) reaches its maximum whenever z 
[zleft’zright] (this is the case in Figure 4.4), then zMOM = (zleft + zright)/2. The mean
of maximum is the defuzzification strategy employed in Mamdani’s fuzzy
logic controllers [6].
,
'
'


z dz
z zdz

• Smallest of maximum zSOM:zSOM is the minimum (in terms of magnitude) of
the maximizing z.
• Largest of maximum zLOM:zLOM is the maximum (in term magnitude) of the
maximizing z. Because of their obvious bias, zSOM and LOM are not used as
often as the other three defuzzification methods.

Example 4.1 Single-input single-output Mamdani fuzzy model
An example of a single-input single-output Mamdani fuzzy model with three
rules can be expressed as
If X is small then Y is small.
If X is medium then Y is medium.
If X is large then Y is large.

-10 -8 -6 -4 -2 0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
X
Membership
Grades
small medium large
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
Y
Membership
Grades
small medium large
-10 -8 -6 -4 -2 0 2 4 6 8 10
0
1
2
3
4
5
6
7
8
9
10
X
Y
Figure Single-input single-output Mamdani fuzzy model in Example 4.1: (a)
antecendent and consequent MFs; (b) overall input-output curve. (MATLAB file:
mam1.m)

Example .4.2 Two-input single-output Mamdani fuzzy model
An example of a two-input single-output Mamdani fuzzy model with four rules
can be expressed as
If X is small and Y is small then Z is negative large.
If X is small and Y is large then Z is negative small.
If X is large and Y is small then Z is positive small.
If X is large and Y is large then Z is positive large.

-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.5
1
X
Membership
Grades
small large
-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.5
1
Y
Membership
Grades
small large
-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.5
1
Z
Membership
Grades
large negative small negative small positive large positive
-5
0
5
-5
0
5
-3
-2
-1
0
1
2
3
X
Y
Z
Figure Two-input single-output Mamdani fuzzy model in Example 4.2(a)
antecendent and consequent MFs; (b) overall input-output surface. (MATLAB file:
mam2.m)

Therefore, to completely specify the operation of a Mamdani fuzzy inference
system, we need to assign a function for each of the following operators:
• AND operator (usually T-norm) for calculating the firing strength of a
rule with AND’ed antecendents.
• OR operator (usually T-conorm) for calculating the firing strength of a
rule with OR’ed antecendents.
• Implication operator (usually T-norm) for calculating qualified
consequent MFs based on given firing strength.
• Aggregate operator (usually T-conorm) for aggregating qualified
consequent MFs to generate an overall output MF.
• Defuzzification operator for transforming an output MF to crisp
single output value.

Theorem 4.1 Computation shortcut for Mamdani fuzzy inference systems
Under sum-product composition, the output of a Mamdani fuzzy inference
system with centroid defuzzification is equal to the weighted average of the
centroids of consequent MFs, where each of the weighting factors is equal
to the product of a firing strength and the consequent MF’s area.
Proof: We shall prove this theorem for a fuzzy inference system with two
rules. By using product and sum for implication and aggregate operators,
respectively, we have
μC’(z) = w1μC1(z) + w2μC2(z).

A1
A2
x
x
x


B1
y

B2
x

Min or
Product
Weighted Average
W1
W2
z1 = p1 x + q1 y + r1
z2 = p2 x + q2 y + r2
2
1
2
2
1
1
w
w
z
w
z
w
z



Figure The Sugeno fuzzy model.

(Note that the precending MF could have values greater than 1 at certain
points.) The crisp output under centroid defuzzification is
where ai (=zCi(z)dz) and zi are the area and centroid of the
consequent MF Ci(z), respectively.



z dc
z
C
z zdz
z
C
COA
z
)
(
'
)
(
'


 

 


dz
z
C
w
dz
z
C
w
zdz
z
C
w
zdz
z
C
W
)
(
2
2
)
(
1
1
)
(
2
2
)
(
1
1




2
2
1
1
2
2
2
1
1
1
a
w
a
w
z
a
w
z
a
W



)
)
(
)
(
(



z dz
z
i
C
z zdz
z
i
C



Example 4.3 Fuzzy and nonfuzzy rule set-a comparison
An example of a single-input Sugeno fuzzy model can be expressed as
If Xis small then y = 0.1X + 6.4.
If X is medium then Y = -0.5X+4.
If X is large then Y = X-2.

-10 -5 0 5 10
0
0.2
0.4
0.6
0.8
1
X
Membership
Grades
small medium large
(a) Antecedent MFs for Crisp Rules
-10 -5 0 5 10
0
2
4
6
8
X
Y
(b) Overall I/O Curve for Crisp Rules
-10 -5 0 5 10
0
0.2
0.4
0.6
0.8
1
X
Membership
Grades
small medium large
(c) Antecedent MFs for Fuzzy Rules
-10 -5 0 5 10
0
2
4
6
8
X
Y
(d) Overall I/O Curve for Fuzzy Rules
Figure Comparison between fuzzy and non fuzzy rules in Example 4.3: (a)
Antecendent MFs and (b) input-output curve for nonfuzzy rules; (c) Antencendent
MFs and (d) input-output curve for fuzzy rules. (MATLAB file:sug1.m)

Example 4.4 Two-input single-output Sugeno fuzzy model
An example of a two-input single-output Sugeno fuzzy model with four rules
can be expressed as
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.

-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
X
Membership
Grades
Small Large
-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
Y
Membership
Grades
Small Large
Figure Two-input single-output Sugeno fuzzy model in Example 4.4: (a)
antecendent and consequent MFs; (b) overall input-output surface. (MATLAB file
sug2.m)

A1
A2
x
x
x


B1
y

B2
x

Min or
Product
C1
z

C2
z

Weighted Average
Z2
Z1
2
1
2
2
1
1
w
w
z
w
z
w
z



Example 4.5 Single-input Tsukamoto fuzzy model
Figure The Tsukamoto fuzzy model.

An example of a single-input Tsukamoto fuzzy model can be expressed as
If X is small then Y is C1
If X is medium then Y is C2
If X is large then Y is C3
Where the antecendent MFs for “small,” “medium,” and “large” are shown in
Figure 4.12(a), and the consequent MFs for “C1,” “C2,” and “C3” are shown in
Figure 4.12(b). The overall input-output curve, as shown in Figure 4.12(d), is
equal to ( ) / ( ), where fi is the output of each rule induced
by the firing strength wi and MF for Ci. If we plot each rule’s output fi as a
function of x, we obtain Figure 4.12(c), which is not quite obvious from the
original rule base and MF plots.
 
3
1 1
1
i
f
w 
3
1 1
i
w

-10 -5 0 5 10
0
0.2
0.4
0.6
0.8
1
X
Membership
Grades
small medium large
(a) Antecedent MFs
0 5 10
0
0.2
0.4
0.6
0.8
1
Y
Membership
Grades
C1 C2 C3
(b) Consequent MFs
Figure Single-input single output Tsukamoto fuzzy model in Example 4.4:(a)
antecendent MFs; (b) consequent MFs; (c) each rule’s output curves; (d)
overall input-output curve. ( MATLAB file: tsu1.m)

(a) (b) (c)
Figure Various methods for partitioning the input space: (a) grid partition; (b) tree
partition; (c) scatter partition.

Generally speaking, the standard method of constructing a fuzzy inference
system, a process usually called fuzzy modeling, has the following features:
• The rule structure of a fuzzy inference system makes it easy to
incorporate human expertise about the target system directly into the
modeling process. Namely, fuzzy modeling takes advantage of domain
knowledge that might not be easily or directly employed in other
modeling approaches.
• When the input-output data of a target system is available, conventional
system identification techniques can be used for fuzzy modeling. In
other words, the use of numerical data also plays an important role in
fuzzy modeling, just as in other mathematical modeling methods.

Conceptually, fuzzy modeling can be pursued in two stages, which are not
totally disjoint. The first stage is the identification of the surface structure,
which included the following tasks:
1. Select relevants input and output variables.
2. Choose a specific type of fuzzy inference system.
3. Determine the number of linguistic terms associated with each
input and output variables. (For a Sugeno model, determine the
order of consequent equations.)
4. Design a collection of fuzzy if-then rules.

Specially, the identification of deep structure includes the following
task :
1. Choose an appropriate family of parameterized
MFs(see Section 2.4).
2. Interview human experts familiar with the target systems
to determine the parameters of the MFs used in the
rule base.
3. Refine the parameters of the MFs using regression and
optimization techniques.

Beberapa tipe teknik kendali adaptif berbeda hanya dalam cara
parameter pengendali disesuaikan. Pada dasarnya ada 4
pendekatan berbeda untuk kendali adapatif konvensional, yaitu:
1.Pendekatan Gain Scheduling
2.Kendali Adaptif Referensi Model
3.Pendekatan Self-tuning
4.Pendekatan Kendali Stochastic

Gain
Scheduler
Controller Plant
Auxillary Measurement
Controller
Parameters
r
U Y
Gambar Pengendali Gain Scheduling

Reference Model
Controller Plant
Reference input
(r)
Adaptive laws
Control input (u)
Plant output
Yp
Model output
Ym
Gambar Teknik Kendali Adaptif Referensi Model

Controller Plant
Reference input (r)
Estimator
Controller input (u)
output (Y)
Design criteria
Controller Parameter
Estimated Parameters
Gambar Teknik Self-Tuning

Controller Plant
r
Hyperstate
Caculation
U Yp
Hyperstate
Gambar Pengendali Sthochatic Generik

Controller Plant
Desired
reference
Input Output
Gambar Kendali Inverse

Plant
error
Inverse
Dynamics
Model
T(t)
+
-
)
(t
θ
Gambar Pemodelan Inverse Langsung

Trained
NN (A)
error
Plant
+
-
NN (B)
Controller
U(k)
Yd(k+1)
(B = copy of A)
Y(k+1)
Gambar Arsitektur Pembelajaran tidak Langsung

Plant
+
-
Controller
U output, Y
Input, r e
model NN
Predicted control signal
Gambar Struktur Kendali Imitasi

NN
Emulator
Z
U(k)
Plant
A(X(k), U(k))
X(k)
U(k)
U(k)
X(k+1)
X
1))
(k
X
1)
(X(k
e 


 ˆ
Gambar Proses Pengarahan Emulator NN

C C
T T
U(0)
Back - Propagation
C
T
X (0) X (1) U(1) X(NT-1)
U(NT-1)
X(NT)
Xd(NT)
eNT
+
-
Gambar Training Pengendali NN

Control NN Model NN
Plant
Critic NN
Input
output
.
Gambar Struktur Kendali BAC

Critic
Action
X(t)
U(t)
J'(t)
Gambar Adaptasi Jaringan Aksi dengan struktur ADAC

Model
Reference
Optimization
NNc
NNi
Plant
Yr
r
U
U
d'
Yp
Ym
Gambar Struktur Kendali Prediktif

Inverse
Dynamics
Model
K Plant
)
(t
Ti
)
(t
d
 )
(t

)
(t
T




error
)
(t
Tf
Gambar Pembelajaran Error Umpan Balik

1 0 0 0 1 0
1 1 1 0 1 1
1 0 1 0 1 1
1 1 0 0 1 0
+ +
Crossover site
Gambar Crossover algoritma genetik standar: Dua mating individual (kiri) dan
keturunan (kanan)

max
max
]
max[
))
exp(
1
/(
1
1
]
max[
max
]
max[
1
)
(
k
k
jika
j
seluruh
untuk
atau
k
k
jika
k
i
j
untuk
x
k
jika
l
i
j
untuk
x
k
jika
k
i
j
untuk
x
f















Modifikasi NN digunakan Yoshiaki dan Toshiyuki mendefinisikan fungsi aktifasi
sebagai berikut:

Model NN
Delay
Plant
Predicted Output
Output
Input
5

t
to
t
U
t
U
)
6
( 
t
m
Y
)
6
( 
t
p
Y


Gambar Struktur Tapped-Delay-Line

Delay Delay
)
3
(
1 
t
Y )
3
(
2 
t
Y
)
2
(
1 
t
Y )
2
(
2 
t
Y )
(
1 t
U )
(
2 t
U )
1
(
1 
t
U )
1
(
2 
t
U )
2
(
1 
t
U )
2
(
2 
t
U
Gambar Arsitektur Umpan balik Output

)
3
(
1 
t
Y )
3
(
2 
t
Y
)
1
(
1 
t
U )
1
(
2 
t
U )
1
(
3 
t
U
Output Layer
Hidden Layer
Context Layer
Input Layer
Gambar Arsitektur Jaringan Recurrent Dasar

Controller
TDL
Plant
TDL
Reference Model
r
U
e
m
Y
l
Y


Gambar Kendali Adaptif Langsung

Linear Feedback
Sliding
Netw ork
Modulate
Plant
)
(t
xd
)
(t
xn
d
)
(t
x
)
(t
U pd
)
(t
U gl
)
(t
Uad
)
(t
x



Gambar Pengendali Adaptif Langsung dikemukakan oleh Slotine dan Sanner

G
G
G
G
)
(
1 t
x
)
(
2 t
x
1
W 2
Ŵ
1
ˆ
A
h
1
ˆ
A
b
)
(
1 t
a

ad
t
u )
(




Gambar Struktur komponen adaptif dari aturan kendali untuk plant gain
nonunity

Controller NNC Nonlinear Plant
TDL
TDL
TDL
TDL
Identif ication
Model
Ref erence Model
Z
U
i
e
m
Y
i
e
p
Y
r
c
e




Gambar Kendali Adaptif tidak Langsung dengan Menggunakan Jaringan Neural

Ng
Nc g z-1
f
z-1
Nf
-Nf
0.6
Reference Model


 



)
(k
r


 
 




 )
(k
ec
)
(k
e
identification Model
)
(
ˆ k
Y
Gambar Pendekatan Narendra

Inverse
Dynamics
Model
Plant
Forward
Dynamics
Model
error in Control Command
error
in
Trajectory
+
-
)
(t
d
 )
(t

)
(t
Ti
Gambar Pemodelan Forward dan Inverse

)
(
1
1 k
X
)
(
1
2 k
X
1
11
S
1
11
V
3
12

3
1
S
Memory Neuron
Network Neuron
S dan V ----- Weight
Gambar Struktur Jaringan Neural Memori

Inverse
Dynamics
Model
Plant
Forward
Dynamics
Model
Reference
+
-
MNN1
z-1
Input
r(k)
U(k)
Yp(k)
+
-
ec
e1
Gambar Struktur Model Kendali Adaptif Referensi dengan Memori
Jaringan Neural

Research Topics
 Type 2 Fuzzy Logic Systems
– Jerry M. Mendel dkk.

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