• Basic Fuzzy Mathematics

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    • Abstract: Basic Fuzzy Mathematicsfor Fuzzy Controland Modeling1.1. INTRODUCTIONFuzzy control and modeling use only a small portion of the fuzzy mathematics that isavailable; this portion is also mathematically quite simple and conceptually easy to under-

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Basic Fuzzy Mathematics
for Fuzzy Control
and Modeling
Fuzzy control and modeling use only a small portion of the fuzzy mathematics that is
available; this portion is also mathematically quite simple and conceptually easy to under-
stand. In this chapter, we introduce some essential concepts, terminology, notations, and
arithmetic of fuzzy sets and fuzzy logic. We include only a minimum though adequate amount
of fuzzy mathematics necessary for understanding fuzzy control and modeling. To facilitate
easy reading, these background materials are presented in plain English and in a rather
informal manner with simple and clear notation as well as explanation. Whenever possible,
excessively rigorous mathematics is avoided. The materials covered in this chapter are
intended to serve as an introductory foundation for the reader to understand not only the fuzzy
controllers and models in this book but also many others in the literature.
1.2.1. Limitation of Classical Sets
In traditional set theory, membership of an object belonging to a set can only be one of
two values: 0 or 1. An object either belongs to a set completely or it does not belong at all. No
partial membership is allowed. Crisp sets handle black-and-white concepts well, such as
"chairs," "ships," and "trees," where little ambiguity exists. They are not sufficient, however,
to realistically describe vague concepts.
In our daily lives, there are countless vague concepts that we humans can easily
describe, understand, and communicate with each other but that traditional mathematics,
including the set theory, fails to handle in a rational way. The concept "young" is an example.
For any specific person, his or her age is precise. However, relating a particular age to
"young" involves fuzziness and is sometimes confusing and difficult. What age is young and
what age is not? The nature of such questions is deterministic and has nothing to do with
stochastic concepts such as probability or possibility.
2 Chapter 1 • Basic Fuzzy Mathematics for Fuzzy Control and Modeling
1 Young
Age (year)
0 35
Figure 1.1 A possible description of the vague concept "young" by a crisp set.
A hypothetical crisp set "young" is given in Fig. 1.1. This set is unreasonable because
of the abrupt change of the membership value from 1 to 0 at 35. Although a different cutoff
age at which membership value changes from 1 to 0 may be used, a fundamental problem
exists. Why is it that a 34.9-year-old person is completely "young," while a 35.1-year-old
person is not "young" at all? No crisp set can realistically capture, quantitatively or even
qualitatively, the essence of the vague concept "young" to reasonably match what "young"
means to human beings. This simple example is not meant to discredit the traditional set
theory. Rather, the intention is to demonstrate that crisp sets and fuzzy sets are two different
and complementary tools, with each having its own strengths, limitations, and most effective
application domains.
1.2.2. Fuzzy Sets
Fuzzy set theory was proposed by Professor L. A. Zadeh at the University of California
at Berkeley in 1965 to quantitatively and effectively handle problems of this nature [277]. The
theory has laid the foundation for computing with words [285][287]. Fuzzy sets theory
generalizes 0 and 1 membership values of a crisp set to a membership function of a fuzzy set.
Using the theory, one relates an age to "young" with a membership value ranging from 0 to 1;
0 means no association at all, and 1 indicates complete association. For instance, one might
think that age 10 is "young" with membership value 1, age 30 with membership value 0.75,
age 50 with membership value 0.1, and so on. That is, every age/person is "young" to a
certain degree. By plotting membership values versus ages, like the one shown in Fig. 1.2, we
generate a fuzzy set "young." The curve in the figure is called the membership function of the
fuzzy set "young." All possible ages, say 0 to 130, form a universe of discourse. From this
example, a definition of fuzzy sets naturally follows.
Fuzzy set: A fuzzy set consists of a universe of discourse and a membership function
that maps every element in the universe of discourse to a membership value between 0 and 1.
Unless otherwise stated, we always use a capital letter and tilde (e.g., A) to represent a
fuzzy set in this book. If an element is denoted by x e X, where X is a universe of discourse,
the membership function of fuzzy set A is mathematically expressed as JUJJ(JC), \X~A, or simply
[i. We will use all three representations in the book; the decision of which one to use depends
Section 1.2. • Classical Sets, Fuzzy Sets, and Fuzzy Logic 3
Age (year)
0 10 30 50 70 90
Figure 1.2 A possible description of the vague concept "young" by a fuzzy set.
on the circumstance. For the above age example, X = [0, 130]. Letting A denote fuzzy set
"young," we can represent its membership function by jU^(x), where x eX.
People have different views on the same (vague) concept. Fuzzy sets can be used to
easily accommodate this reality. Continue the age example. Some people might think age 50
is "young" with membership value as high as 0.9, whereas others might consider that 20 is
"young" with membership value merely 0.2. Different membership functions can be used to
represent these different versions of "young." Figure 1.3 shows two more possible definitions
of the fuzzy set "young." Not only do different people have different membership functions
for the same concept, but even for the same person, the membership function for "young"
can be different when the context in which age is addressed varies. For instance, a 40-year-old
president of a country would likely be regarded as young, whereas a 40-year-old athlete
would not. Two different fuzzy sets "young" are needed to effectively deal with the two
These examples show that (1) fuzzy sets can practically and quantitatively represent
vague concepts; and (2) people can use different membership functions to describe the same
vague concept. We now introduce some definitions needed to describe fuzzy controllers and
Age (year)
0 10 30 50 70 90
Figure 1.3 Two more possible descriptions of the vague concept "young" by fuzzy sets.
4 Chapter 1 • Basic Fuzzy Mathematics for Fuzzy Control and Modeling
0 x Figure 1.4 An example of the membership
3.5 function of a singleton fuzzy set.
Continuous fuzzy sets: A fuzzy set is said to be continuous if its membership function
is continuous.
Most fuzzy controllers and models nowadays use continuous fuzzy sets.
Singleton fuzzy sets: A fuzzy set that has nonzero membership value for only one
element of the universe of discourse is called a singleton fuzzy set. Figure 1.4 exhibits a
singleton fuzzy set whose membership value is 0 everywhere except at x = 3.5 where the
membership value is 1.
The majority of typical fuzzy controllers and models employ singleton fuzzy sets in the
consequent of fuzzy rules, as will be shown later in this book.
Support of a fuzzy set: For a fuzzy set whose universe of discourse is X, all the
elements in X that have nonzero membership values form the support of the fuzzy set.
As an illustrative example, the support for the fuzzy set "young," shown in Fig. 1.2, is
Height of a fuzzy set: The largest membership value of a fuzzy set is called the height
of the fuzzy set.
For instance, the height of the fuzzy set "young" in Fig. 1.2 is 1. The height of the
fuzzy sets used in fuzzy controllers and models is almost always 1.
Normal fuzzy set and subnormal fuzzy set: A fuzzy set is called normal if its height
is 1. If the height of a fuzzy set is not 1, the fuzzy set is said to be subnormal.
The fuzzy sets in Figs. 1.2 and 1.3 are normal fuzzy sets, whereas the fuzzy set in Fig.
1.5 is a subnormal one. Subnormal fuzzy sets are rarely used in fuzzy controllers and models.
Center of a fuzzy set: We need to define this concept for four different situations. If
the membership function of a fuzzy set reaches its maximum at only one element of the
universe of discourse, the element is called center of the fuzzy set (Fig. 1.6a). If the
membership function of a fuzzy set achieves its maximum at more than one element of the
universe of discourse and all these elements are bounded, the middle point of the element is
the center (Fig. 1.6b). If the membership function of a fuzzy set attains its maximum at more
than one element of the universe of discourse and not all of the elements are bounded, the
largest element is the center if it is bounded (Fig. 1.6d); otherwise, the smallest element is the
center (Fig. 1.6c).
Section 1.2. • Classical Sets, Fuzzy Sets, and Fuzzy Logic 5
Figure 1.5 An example of a subnormal fuzzy
set. o x
Convex fuzzy sets: Fuzzy set A, whose universe of discourse is [a, b], is convex if and
only if
li^kxx + (1 - X)x2) > mm[ti~A(xx\ti~A(x2)l Vxl9x2 e [a,b] and V2 e [0,1],
where min() denotes the minimum operator that uses the smaller membership value of the two
memberships as the operation result.
The fuzzy set illustrated in Fig. 1.7 is convex, whereas the one shown in Fig. 1.8 is not.
To avoid possible confusion, it is important to note that the definition of convex fuzzy sets
does not necessarily imply that the membership functions of convex fuzzy sets are convex
functions. Nevertheless, the definition requires membership functions to be concave. Of
course, according to the definition of convex fuzzy sets, if the membership function of a fuzzy
set is convex, the fuzzy set is convex. Typical fuzzy controllers and models employ convex
fuzzy sets.
K*) U(*)
1 1
0 X 0 X
center center
(a) (b)
M« M(*)
1 1
0 x 0 x
center center
(c) (d)
Figure 1.6 A definition of the center of a fuzzy set for four different cases.
According to the definition of fuzzy sets, any function, continuous or discrete, can be a
membership function as long as its value falls in [0,1]. The discrete type is uncommon,
6 Chapter 1 • Basic Fuzzy Mathematics for Fuzzy Control and Modeling
0 x Figure 1.7 An example of a convex fuzzy set.
Figure 1.8 An example of a nonconvex fuzzy
0 x set.
however. Indeed, one of the key issues in the theory and practice of fuzzy sets is how to define
the proper membership functions of fuzzy sets. Fuzzy control and modeling are no exception.
Primary approaches include (1) asking the control/modeling expert to define them; (2)
using data from the system to be controlled/modeled to generate them; and (3) making them
in a trial-and-error manner. Each different approach has its benefits and drawbacks. In more
than 25 years of practice, it has been found that the third approach, though ad hoc, works
effectively and efficiently in many real-world applications.
Numerous applications have shown that only four types of membership functions are
needed in most circumstances: trapezoidal, triangular (a special case of trapezoidal),
Gaussian, and bell-shaped. Figure 1.9 shows an example of each type. All these fuzzy sets
are continuous, normal, and convex. Among the four, the first two are more widely used. In
the figure, we purposely use asymmetric membership functions to make the illustration more
general. More often than not, however, symmetric functions are used.
1.2.3. Fuzzy Logic Operations
In classical set theory, there are binary logic operators AND (i.e., intersection), OR (i.e.,
union), NOT (i.e., complement), and so on. The corresponding fuzzy logic operators exist in
fuzzy set theory. Fuzzy logic AND and OR operations are used in fuzzy controllers and
models. Unlike the binary AND and OR operators whose operations are uniquely defined,
their fuzzy counterparts are nonunique. Numerous fuzzy logic AND operators and OR
operators have been proposed, some of them purely from the mathematics point of view. To a
large extent, only the Zadeh fuzzy AND operator, product fuzzy AND operator, the Zadeh
Section 1.3. • Fuzzification 7
li(x) Pix)
l 1
o x 0 x
(a) (b)
H(x) |I(X)
1 1
0 X 0 X
(c) (d)
Figure 1.9 Examples of four commonly used input fuzzy sets in fuzzy control and
modeling: (a) trapezoidal, (b) triangular, (c) Gaussian, and (d) bell-shaped.
Note that they are all continuous, normal, and convex fuzzy sets.
OR operator, and the Lukasiewicz OR operator have been found to be most useful for fuzzy
control and modeling [79]. Their definitions are as follows:
Zadeh fuzzy logic AND operator: ^^(x) = min(/i^(x), /i^(x))
product fuzzy logic AND operator: ^ n ^ W = ftfa) ^W
Zadeh fuzzy logic OR operator: ^ ^ ( x ) = maxQifa), /*]*(*))
Lukasiewicz fuzzy logic OR operator: fi^ix) = min(/i^(x) + ^(x), 1)
where max() and min() are the maximum operator and minimum operator, respectively.
As a concrete demonstration, suppose that a specific age, say 30, is "young" (a fuzzy
set) with a membership value of 0.8 and is "old" (another fuzzy set) with a membership value
of 0.3. Then, the membership value for the age being "young and old" (a newly formed fuzzy
set) is 0.3 if the Zadeh fuzzy AND operator is used or 0.24 if the product fuzzy AND
operation is applied. By the same token, the membership value for the age being "young or
old" (another newly formed fuzzy set) is 0.8 if the Zadeh fuzzy OR operator is utilized, or 1 if
the Lukasiewicz fuzzy OR operation is involved.
Fuzzy control and modeling always involve a process called fuzzification at every sampling
time. Fuzzification is a mathematical procedure for converting an element in the universe of
discourse into the membership value of the fuzzy set. Suppose that fuzzy set A is defined on
[a,b]; that is, the universe of discourse is [a,b]; for any x e [a,b], the result of fuzzification is
simply fi^ix). Figure 1.10 shows an example in which the fuzzification result for x = 7 is 0.4.
8 Chapter 1 • Basic Fuzzy Mathematics for Fuzzy Control and Modeling
0 Figure 1.10 An example showing how fiizzifi-
7 cation works.
A fuzzy controller or model uses fuzzy rules, which are linguistic if-then statements involving
fuzzy sets, fuzzy logic, and fuzzy inference. Fuzzy rules play a key role in representing expert
control/modeling knowledge and experience and in linking the input variables of fuzzy
controllers/models to output variable (or variables).
Two major types of fuzzy rules exist, namely, Mamdani fuzzy rules and Takagi-Sugeno
(TS, for short) fuzzy rules [202].
1.4.1. Mamdani Fuzzy Rules
A simple but representative Mamdani fuzzy rule describing the movement of a car is:
IF Speed is High AND Acceleration is Small THEN Braking is (should be) Modest,
where Speed and Acceleration are input variables and Braking is an output variable. "High,"
"Small," and "Modest" are fuzzy sets, and the first two are called input fuzzy sets while the
last one is named the output fuzzy set.
The variables as well as linguistic terms, such as High, can be represented by
mathematical symbols. Thus, a Mamdani fuzzy rule for a fuzzy controller involving three
input variables and two output variables can be described as follows:
IF xx is A AND x2 is B AND x3 is C THEN ux is D, u2 is E, (1.1)
where x{, x2, and x3 are input variables (e.g., error, its first derivative and its second
derivative), and ux and u2 are output variables (e.g., valve openness). In theory, these variables
can be either continuous or discrete; practically speaking, however, they should be discrete
because virtually all fuzzy controllers and models are implemented using digital computers.
A, B, C, Z), and E are fuzzy sets, and AND are fuzzy logic AND operators. "IF x\ is A AND
x2 is B AND x3 is C" is called the rule antecedent, whereas the remaining part is named the
rule consequent.
The structure of Mamdani fuzzy rules for fuzzy modeling is the same. The variables
involved, however, are different. An example of a Mamdani fuzzy rule for fuzzy modeling is
IF y(n) is A AND y(n - 1) is B AND y(n - 2) is C AND u(n) is D
AND u(n - 1) is E THEN y(n + 1) is F ,
Section 1.4. • Fuzzy Rules 9
where A, B, C, D, E, and F are fuzzy sets, y(n), y(n - 1), and y(n - 2) are the output of
the system to be modeled at sampling time n, n — 1 and n — 2, respectively. And, w(«) and
M(« - 1) are system input at time n and n - 1, respectively; >, (1.4)
where fy represents singleton fuzzy set Wj that is nonzero only at Zj = /?..
1.4.2. TS Fuzzy Rules
Now, let us look at the so-called TS fuzzy rules. Unlike Mamdani fuzzy rules, TS rules
use functions of input variables as the rule consequent. For fuzzy control, a TS rule
corresponding to the Mamdani rule (1.1) is
IF xx is A AND x2 is B AND x3 is C THEN ux =/(*!,x 2 ,x 3 ) 9 u2 = g(x{ ,x2,x3),
where/() and g() are two real functions of any type. Similarly, for fuzzy modeling, a TS rule
analogous to the Mamdani rule (1.2) is in the following form:
IF y(n) is A AND y(n -l)isB AND y(n - 2) is C AND u(n) is D
AND u(n - 1) is E THENX« + 1) = F(y(n),y(n - l),y(n - 2), u(ri), u(n - 1)),
where FQ is an arbitrary function. In parallel to the general Mamdani fuzzy rule (1=3), a
general TS rule for both fuzzy control and fuzzy modeling is
IF Vl is Sx AND.. .AND vM is SM
THENz! = / 1 ( v 1 , . . . , v M ) , . . . , zP =fP(v{,..., vM).
In theory, JjQ can be any real function, linear or nonlinear. It seems to be appealing to
use nonlinear functions for all the rules or to use a combination of linear and nonlinear
functions as rule consequent (i.e., linear functions for some rules and nonlinear ones for the
remaining). In this way, rules are more general and can potentially be more powerful.
Unfortunately, this idea is impractical, for properly choosing or determining the mathematical
formalism of nonlinear functions for every fuzzy rule is extremely difficult, if not impossible.
This difficulty is fundamentally the same as those encountered in classical nonlinear control
and modeling theory. It is well known that there is no general nonlinear control or modeling
theory because general nonlinear system theory has not been, and most likely will not be,
established. For these reasons, linear functions have been employed exclusively in theoretical
research and practical development of TS fuzzy controllers and models. We call a TS rule
employing a linear (nonlinear) function TS fuzzy rule with linear (nonlinear) rule consequent.
10 Chapter 1 • Basic Fuzzy Mathematics for Fuzzy Control and Modeling
In this book, we focus only on frizzy controllers and models that use the linear TS rule
Fuzzy inference is sometimes called fuzzy reasoning or approximate reasoning. It is used in a
fuzzy rule to determine the rule outcome from the given rule input information. Fuzzy rules
represent control strategy or modeling knowledge/experience. When specific information is
assigned to input variables in the rule antecedent, fuzzy inference is needed to calculate the
outcome for output variable(s) in the rule consequent.
Mamdani fuzzy rules and TS fuzzy rules use different fuzzy inference methods.
For the general Mamdani fuzzy rule (1.3), the question about fuzzy inference is the
following: Given v,- = a p for all i, where af are real numbers, what should zj be? For fuzzy
control and modeling, after fuzzifying vi at a, and applying fuzzy logic AND operations on
the resulting membership values in the fuzzy rule, we attain a combined membership value, fi,
which is the outcome for the rule antecedent. Then, the question is how to compute "THEN"
in the rule. Calculating "THEN" is called fuzzy inference. Specifically, the question is: Given
fi, how should Zj be computed? Since, mathematically, the computation is the same for
different output variables, we use z and W to represent, respectively, z; and Wj in the following
discussion on fuzzy inference methods.
A number of fuzzy inference methods can be used to accomplish this task (e.g., [163]),
but only four of them are popular in fuzzy control and modeling and we will use them only in
this book [157]). They are the Mamdani minimum inference method, the Larsen product
inference method, the drastic product inference method, and the bounded product inference
method. We denote them by RM, RL, RDP, and RBP, respectively. The definitions of these
methods are given in Table 1.1, where \i^(z) is the membership function of fuzzy set W in
fuzzy rule (1.3) and \x is the combined membership in the rule antecedent.
For a better understanding, we graphically illustrate the definitions in Fig. 1.11. The
results of the four fuzzy inference methods are the fuzzy sets formed by the shaded areas.
Obviously, the resulting fuzzy sets can be explicitly determined since the formulas describing
the shaded areas can be derived mathematically. Among the four methods, the Mamdani
method is used most widely in fuzzy control and modeling.
TABLE 1.1 Definitions of Four Popular Fuzzy Inference Methods for Fuzzy
Control and Modeling: (a) Mamdani minimum inference, (b) Larsen product
inference, (c) drastic product inference, and (d) bounded product inference.
Fuzzy Inference Method Definition3
Mamdani minimum inference, RM min(^, j%(z)), for all z
Larsen product inference, RL / i x ji%(z), for all z
Drastic product inference, RDP
Bounded product inference, RBP
{ fi,
for fifv(z) = 1
Hw(z), for fx = 1
for n < 1 and \i^{z) < 1
liw(z), for+ Hwiz) ~ 1.0)
maxO jn = 1
0, for ft < 1 and \i^(z) < 1
Bounded product inference, RBP(1.3) is utilized inmax(/i + ^^(z) — 1,0) is the
General Mamdani fuzzy rule (1.3) is utilized in the definitions. \i^{z) is the
membership function of fuzzy set W representing Wj in the rule consequent,
whereas \i is the final membership yielded by fuzzy logic AND operators in the
rule antecedent.
Section 1.6. • Defuzzification 11
W w w w
0 z
Figure 1.11 Graphical illustration of the definitions of the four popular fuzzy inference
methods whose mathematical definitions are provided in Table 1.1: (a) the
Mamdani minimum inference method, (b) the Larsen product inference
method, (c) the drastic product inference method, and (d) the bounded
product inference method.
w w w w
Figure 1.12 For Mamdani fuzzy controllers and models using singleton fuzzy sets in the
rule consequent, the outcome of using the four different inference methods is
As stated above, typical Mamdani fuzzy controllers and models employ singleton
output fuzzy sets as the rule consequent (see rule (1.4). Under this condition, the four
different inference methods produce the same inference result, as shown in Fig. 1.12.
For TS fuzzy rules, fuzzy inference is simpler and only one method exists. For general
TS fuzzy rule (1.5), the result of the fuzzy inference is fi xfj(yl9..., vM) for Zj. Instead of
viewing this as a fuzzy inference result, one may also think of it as the rule consequent being
weighted by the combined membership value from the rule antecedent.
Defuzzification is a mathematical process used to convert a fuzzy set or fuzzy sets to a real
number. It is a necessary step because fuzzy sets generated by fuzzy inference in fuzzy rules
must be somehow mathematically combined to come up with one single number as the output
of a fuzzy controller or model. After all, actuators for control systems can accept only one
value as their input signal, whereas measurement data from physical systems being modeled
are always crisp.
12 Chapter 1 • Basic Fuzzy Mathematics for Fuzzy Control and Modeling
Every fuzzy controller and model uses a defuzzifier, which is simply a mathematical
formula, to achieve defuzzification. For fuzzy controllers and models with more than one
output variable, defuzzification is carried out for each of them separately but in a very similar
fashion. In most cases, only one defuzzifier is employed for all output variables, although it is
theoretically possible to use different defuzzifiers for different output variables.
Different types of defuzzifiers are suitable for different circumstances; below, we
present some of the more popular ones. Since most fuzzy controllers and models use
singleton fuzzy sets in the fuzzy rule consequent, our presentation will concentrate on
singleton output fuzzy sets. Nonetheless, extending the discussion to nonsingleton fuzzy sets
is straightforward.
1.6.1. Generalized Defuzzifier
The generalized defuzzifier represents many different defuzzifiers in one simple
mathematical formula [64].
Assume that the output variable of a fuzzy controller or model is z. Suppose that
evaluating T Mamdani fuzzy rules using some fuzzy inference method produces N member-
ship values, / i 1 ? . . . , fiN, for N singleton output fuzzy sets in the rules (one value for each
rule). Let us say that these fuzzy sets are nonzero only at z = jSj, . . . , fiN. The generalized
defuzzifier produces the following defuzzification result:
z=~ N . (1-6)
where a is a design parameter.
Continue the above case, but assume that the fuzzy controller or model uses TS rules
instead. Let us say that the rule consequents in the T fuzzy rules are V
gk(v{,..., vM), k = 1 , . . . , N; then defuzzification outcome is achieved using the general-
ized defuzzifier
£A4 x &t0i,...,v M )
*=-—N • a-7>
1.6.2. Centroid Defuzzifier, Mean of Maximum Defuzzifier,
and Linear Defuzzifier
Different types of defuzzifiers are realized using different a values in the generalized
defuzzifier, where 0 < a + oo. When a = 1, the most widely used centroid defuzzifier is
obtained. The defuzzifier is of the centroid type because it computes, in a sense, the centroid
of the singleton fuzzy sets from different rules. The occasionally used mean of maximum
defuzzifier is realized when a = oo.
Exercises 13
A few studies in the literature use a linear defuzzifier. When Mamdani fuzzy rules are
involved, the defuzzification result is
z=£/4x]8*. (1.8)
On the other hand, for TS fuzzy rules, we get
= E4 x ^i»-.v M ).
The difference is obvious: A linear defuzzifier does not have the denominator.
We will use the centroid defuzzifier and generalized defuzzifier only in this book
because of their popularity.
This chapter introduces the concept of fuzzy sets and their advantages over the classical sets.
Also presented are concepts and notations of different types of fuzzy sets and fuzzy logic
operations. The common building blocks of typical fuzzy controllers and models are
described. They include fuzzification, fuzzy rules, fuzzy inference, and defuzzification.
There are a number of introductory textbooks on fuzzy set theory and fuzzy systems (e.g.,
[101][102][242][293]). Fuzzification, fuzzy rules, fuzzy inference, and defuzzification are
basic components of a typical fuzzy system, fuzzy controller, or fuzzy model. More
information on these segments can be found in these books as well. A brief history of
fuzzy sets, fuzzy logic, and fuzzy systems is given in [151].
1. List some concepts in our daily lives that cannot be accurately described by conventional sets
but can be by fuzzy sets.
2. Graphically draw your definitions of continuous fuzzy set "young" in some different
circumstances. Can they be described by mathematical formulas? If not, can you approximate
your definitions by formulas? Do your definitions belong to the four common types of fuzzy
sets mentioned in this chapter?
3. Answer the same questions as in Problem 2 for continuous fuzzy set "middle age."
4. For the fuzzy sets that you defined in the above two problems, what are their supports, heights,
and centers? Are they normal? Are they convex?
5. Derive two new fuzzy sets "young and middle age" and "young or middle age" from the
fuzzy sets established in Problems 2 and 3. Use different fuzzy logic AND and OR operators
discussed in this chapter. Do this exercise graphically and mathematically, if possible.
6. Describe a fuzzy integer 5 using the Gaussian fuzzy set (i.e., use the Gaussian formula in
statistics). How do you use a singleton fuzzy set to represent integer 5?
7. What are the apparent similarities between fuzzy set and probability? What are the funda-
mental differences between them? What are the implications of the differences to application?
14 Chapter 1 • Basic Fuzzy Mathematics for Fuzzy Control and Modeling
8. Make some Mamdani fuzzy rules and TS fuzzy rules of your own. Which type would you
prefer? Why?
9. Is it meaningful to compare the effects of the different defuzzifiers? If yes, how can you
compare them? If no, why?
10. If the same questions as Problem 9 are asked for the different fuzzy inference methods, what
are your answers?

Use: 0.0441