IEEE Computational Intelligence Magazine - February 2023 - 81

AIeXplained
Dongrui
Wu and Ruimin Peng
Huazhong University ofScience and Technology, CHINA
Jerry M. Mendel
University ofSouthern California, USA
Type-1 and Interval Type-2 Fuzzy Systems
Abstract
F
uzzy sets are suitable for modeling
human linguistic uncertainties,
and the resulting fuzzy
systems can then be used to perform
inferences based on a linguistic rulebase.
They have been successfully used in
numerous applications, particularly controls.
This short paper introduces the
basics of type-1 and interval type-2
fuzzy sets and systems, particularly for
newcomers. The online full paper at
IEEE Xplore provides detailed examples
for better understanding, and interactive
components for a hands-on
exploration.
IF the distance is close AND your relative
speed is fast, THEN your acceleration
should be negative;
and so on. A human driver can
adjust the acceleration intuitively, without
using any mathematics.
Then, how about designing a car
following controller for this task? How
would you transform the above knowledge
(rules) into mathematical formulas,
so that the controller can calculate the
appropriate acceleration based on the
distance and speed?
Fuzzy systems are in this regard very
I. Introduction
IMAGINE you and your friend are
driving on a highway in two separate
cars, and you want to follow your
friend's car at about 50 meters behind.
You can observe your distance to your
friend's car (e.g., far, close, or okay),
and your relative speed to it (e.g., fast,
slow, or okay). Then, how would you
adjust your acceleration (gas pedal), so
that the distance is kept at roughly 50
meters?
Some intuitive rules would apply,
e.g.,
IF the distance isfar AND your relative
speed is slow, THEN your acceleration
should be positive;
Digital Object Identifier 10.1109/MCI.2022.3223496
Date ofcurrent version: 13January 2023
suitable tools. They have been used in
numerous applications, particularly controls,
from cement kiln in Denmark to
Sendai Subway in Japan, from rice
cookers to air conditioners. This short
paper introduces the basics of fuzzy sets
and systems.
II. Type-1 Fuzzy Sets and Fuzzy
Systems
Type-1 (T1) fuzzy set was first introduced
by Zadeh [1] in 1965 to model
linguistic uncertainties. They can be
used to construct T1 fuzzy systems.
A. T1 Fuzzy Sets
For a crisp set, the membership of an
elementinitiseither1(completely
in) or 0 (completely out), and there is
nothing in-between. However, for a
T1 fuzzy set, the membership can be
any value in [0,1], i.e., there can be
partial memberships. This is more
Corresponding author: Dongrui Wu (e-mail: drwu09@
gmail.com.)
intuitive in modeling linguistic uncertainties.
For the car following example,
if we define any distance longer
than or equal to 55 meters as Far,
i.e., any x 55 has membership 1 in
the T1 fuzzy set Far,then x ¼ 54
may have membership 0.8, i.e., it is
also far to a large degree.
Definition 1. A T1 fuzzy set X is comprised
ofa domain DX of real numbers,
together with a membership function
(MF) uX
ðxÞ : DX !½0; 1, i.e.,
X ¼
in which
R
Z
ðxÞ.
uXðxÞ=x;
DX
denotes the collection of all
points x 2 DX with associated membership
grade uX
Some example T1 fuzzy sets are
shown in Fig. 1(a).
B. T1 Fuzzy Systems
Fig. 2(a) shows the diagram of a typical
T1 fuzzy system. The fuzzifier transforms
crisp inputs into T1 fuzzy sets.
The inference engine performs inference
on the T1 fuzzy sets, combining
the outputs of the rules in the rulebase
into a new T1 fuzzy set. The defuzzifier
converts the mapped T1 fuzzy set into a
crisp output.
Rules in a T1 Takagi-Sugeno-Kang
(TSK) fuzzy system are:
Rk : IFX1 is Xk;1 AND ANDXd is Xk;d;
THEN ykðxÞ¼
Xd
i¼1
ak;ixi þ bk;
k ¼ 1; ...; K
FEBRUARY 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 81
(1)
https://orcid.org/0000-0002-7153-9703
IEEE Computational Intelligence Magazine - February 2023

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