IEEE Computational Intelligence Magazine - February 2023 - 82

variable; and, the secondary grade of X~
equals 1 for 8x 2 D~X and 8u 2½0; 1.
Some example IT2 fuzzy sets are
FIGURE 1 (a) T1 and (b) IT2 fuzzy sets for Distance in the car following
example.
shown in Fig. 1(b). Unlike a T1 fuzzy
set, whose membership for each x is a
number, the membership of an IT2
fuzzy set is an interval (the membership
of a general type-2 fuzzy set is a T1
fuzzy set).
FIGURE 2 (a) T1 and (b) IT2 fuzzy systems.
where d is the input dimensionality, K
is the number ofrules, Xk;i is a linguistic
term (modeled by a T1 fuzzy set) for
linguistic variable Xi (i ¼ 1; ...; d)in
the kth rule, xi is the crisp input for Xi,
and ak;i and bk (k ¼ 1; ...; K) are adjustable
regression coefficients. Many times
all ak;i are set to 0, i.e., each rule consequent
is a constant.
For the car following example, d ¼ 2,
X1 denotes Distance, X2 denotes Relative
Speed, K ¼ 9 if both Distance and Relative
Speed have three linguistic terms (T1
fuzzy sets), and x1 and x2 are the current
input values of Distance and Relative
Speed, respectively.
For a particular input x ¼ðx1; ...;
xdÞ, the operations are:
1) Fuzzification: Compute uXk;i
ðxiÞ,the
membership grade ofxi on Xk;i.
2) Inference: Compute the firing level of
each rule:
fkðxÞ¼ uXk;1
yðxÞ¼
ðx1Þ uXk;d
k¼1fkðxÞ ykðxÞ
PK
k¼1fkðxÞ
ðxdÞ:
The inference engine output is:
PK
:
(2)
3) Defuzzification: Since the inference
output is already a crisp number, no
defuzzification is needed in a T1
A. Interval Type-2 Fuzzy Sets
Interval type-2 (IT2) fuzzy sets [3] are
currently the most widely used type-2
fuzzy sets due to their simplicity.
Definition 2. An IT2 fuzzy set
Xi~
defined as:
Xe ¼
Z
x2D~X
Z
FkðxÞ¼½uXk;1
ðx1Þ uXk;d
uXk;1 ðx1Þ uXk;d
½fk; fk:
s
Inference is performed by:
YðxÞ¼
[ PK
1=ðx; uÞ; (3)
u2½0;1
where x, called the primary variable,isin
domain D~X; u 2½0; 1 is the secondary
82 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | FEBRUARY 2023
fk2FkðxÞ
yk2YkðxÞ
Pk¼1fkyk
K
k¼1fk
ðxdÞ;
ðxdÞ
(4)
TSK fuzzy system. The inference
output yðxÞ is the final output.
Defuzzification is necessary in a
Mamdani T1 fuzzy system.
III. Interval Type-2 Fuzzy Sets and
Fuzzy Systems
A T1 fuzzy set is certain in the sense that its
membership grades are crisp values. Type2
fuzzy sets [2], characterized by membership
grades that are themselves fuzzy, may
be used to model more uncertainties. For
example, different drivers may have different
understanding of the linguistic term
Far, and hence may provide different T1
fuzzy sets for it. Type-2 fuzzy sets can be
used to incorporate these inter-personal
uncertainties.
B. IT2 Fuzzy System
Fig. 2(b) shows the schematic diagram of
an IT2 fuzzy system. It is similar to its T1
counterpart, with the major difference
being that at least one of the fuzzy sets in
the rulebase is an IT2 fuzzy set. Hence, the
outputs of the inference engine are IT2
fuzzy sets, and a type-reducer is needed to
convert them into a T1 fuzzy set before
defuzzification can be carried out.
Assume the rulebase of an IT2 TSK
fuzzy system consisting ofK rules in the
following form:
R~k : IFX1 is eXk;1 AND ANDXd is eXk;d;
THENYkðxÞ¼½yk; yk;
k ¼ 1; ...; K
where eXk;i ði ¼ 1; ... ; dÞ are IT2 fuzzy
sets, and YkðxÞ¼½yk; yk is an interval,
which can be understood as the centroid
[3] ofa consequent IT2 fuzzy set.
For a particular input x ¼ðx1; ...;
xdÞ, the output ofthe IT2 TSK fuzzy system
is computed as:
1) Fuzzification: Compute the membership
interval ½uXk;i
ofxi on each eXk;i.
ðxiÞ; uXk;i
ðxiÞ
2) Inference: Compute the firing interval
ofthe kth rule:
:
(5)
(5) is a direct extension of (2),
when all fkðxÞ and ykðxÞ in (2)
become intervals.
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