Tài liệu Monitoring and Diagnosing Manufacturing Processes Using Fuzzy Set Theory pptx

Du, R. et al "Monitoring and Diagnosing Manufacturing Processes Using Fuzzy Set Theory"
Computational Intelligence in Manufacturing Handbook
Edited by Jun Wang et al
Boca Raton: CRC Press LLC,2001

©2001 CRC Press LLC

14

Monitoring and
Diagnosing
Manufacturing
Processes Using Fuzzy

Set Theory

14.1 Introduction

14.2 A Brief Description of Fuzzy Set Theory

14.3 Monitoring and Diagnosing Manufacturing
Processes Using Fuzzy Sets

14.4 Application Examples

14.5 Conclusions


Abstract

Monitoring and diagnosis play an important role in modern manufacturing engineering. They help to

detect product defects and process/system malfunctions early, and hence, eliminate costly consequences.
They also help to diagnose the root causes of the problems in design and production and hence minimize
production loss and at the same time improve product quality. In the past decades, many monitoring
and diagnosis methods have been developed, among which the fuzzy set theory has demonstrated its
effectiveness. This chapter describes how to use the fuzzy set theory for engineering monitoring and
diagnosis. It introduces various methods such as fuzzy linear equation method, fuzzy C-mean method,
fuzzy decision tree method, and a newly developed method, fuzzy transition probability method. By
using good examples, it demonstrates step by step how the theory and the computation work. Two
practical examples are also included to show the effectiveness of the fuzzy set theory.

14.1 Introduction

According to

Webster’s New World Dictionary of the American Language,

“monitoring,” among several
other meanings, means checking or regulating the performance of a machine, a process, or a system.
“Diagnosis” means deciding the nature and the cause(s) of a diseased condition of a machine, a process,
or a system by examining the performance or the symptoms. In other words, monitoring detects suspi-
cious symptoms while diagnosis determines the cause of the symptoms. There are several words and/or

R. Du*

University of Miami

Yangsheng Xu

Chinese University of Hong Kong

*This work was completed when Dr. Du visited The Chinese University of Hong Kong.

©2001 CRC Press LLC

phrases that have similar or slightly different meanings, such as fault detection, fault prediction, in-
process verification, on-line inspection, identification, and estimation.
Monitoring and diagnosing play a very important role in modern manufacturing. This is because
manufacturing processes are becoming increasingly complicated and machines are much more auto-
mated. Also, the processes and the machines are often correlated; and hence, even small malfunctions or
defects may cause catastrophic consequences. Therefore, a great deal of research has been carried out in
the past 20 years. Many papers and monographs have been published. Instead of giving a partial review
here, the reader is referred to two books. One by Davies [1998] describes various monitoring and diagnosis
technologies and instruments. The reader should also be aware that there are many commercial moni-
toring and diagnosis systems available. In general, monitoring and diagnosis methods can be divided
into two categories: a model-based method and a feature-based method. The former is applicable where
a dynamic model (linear or nonlinear, time-invariant or time-variant) can be established, and is com-
monly used in electrical and aerospace engineering. The book by Gertler [1988] describes the basics of
model-based monitoring. The latter uses the features extracted from sensor signals (such as cutting forces
in machining processes and pressures in pressured vessels) and can be used in various engineering areas.
This chapter will focus on this type of method.
More specifically the objective of this chapter is to introduce the reader to the use of fuzzy set theory
for engineering monitoring and diagnosis. The presented method is applicable to almost all engineering
processes and systems, simple or complicated. There are of course many other methods available, such
as pattern recognition, decision tree, artificial neural network, and expert systems. However, from the
discussions that follow, the readers can see that fuzzy set theory is simple and effective method that is
worth exploring.
This chapter contains five sections. Section 14.2 is a brief review of fuzzy set theory. Section 14.3
describes how to use fuzzy set theory for monitoring and diagnosing manufacturing processes. Section
14.4 presents several application examples. Finally, Section 14.5 contains the conclusions.

14.2 A Brief Description of Fuzzy Set Theory

14.2.1 The Basic Concept of Fuzzy Sets

Since fuzzy set theory was developed by Zadeh [1965], there have been many excellent papers and
monographs on this subject, for example [Baldwin et al., 1995; Klir and Folger, 1988]. Hence, this chapter
only gives a brief description of fuzzy set theory for readers who are familiar with the concept but are
unfamiliar with the calculations. The readers who would like to know more are referred to the above-
mentioned references.
It is known that a crisp (or deterministic) set represents an exclusive event. Suppose

A

is a crisp set
in a space

X

(i.e.,

A



⊂



X

), then given any element in

X

, say

x

, there will be either

x



∈



A

or

x



∉

A

.
Mathematically, this crisp relationship can be represented by a membership function,

µ

(

A

), as shown in
Figure 14.1, where

x



∉

(b,c). Note that

µ

(

A

) = {0, 1}. In comparison, for a fuzzy event,

A

′

, its membership
function,

µ

(

A

′

), varies between 0 and 1, that is

µ

(

A

) = [0, 1]. In other words, there are cases in which
the instance of the event

x

∈



A

′

can only be determined with some degree of certainty. This degree of
certainty is referred to as

fuzzy degree

and is denoted as

µ

Α

’

(

x

∈



A

′

). Furthermore, the fuzzy set is denoted
as

x

/

µ

A

’

(

x

),

∀



x

∈



A

′

, and

µ

A

’

(

x

) is called the

fuzzy membership function

or the

possibility distribution

.
It should be noted that the fuzzy degree has a clear meaning:

µ

(

x

) = 0 means

x

is impossible while

µ

(

x

) = 1 implies

x

is certainly true. In addition, the fuzzy membership function may take various forms
such as a discrete tablet,

x

:

x

1

x

2

…

x

n

µ

(

x

):

µ

(

x

1

)

µ

(

x

2

)…

µ

(

x

n

) Equation (14.1)
or a continuous step-wise function,

©2001 CRC Press LLC

Equation (14.2)
where

a

,

b

,

c

, and

d

are constants that determines the shape of

µ

(

x

). This is shown in Figure 14.1.
With the help of the membership functions, various fuzzy operations can be carried out. For example,
given

A

,

B

⊆



X

, we have
(a) union:

µ

(

A

∪

B

) = max{

µ

(

A

),

µ

(

B

)},

∀



x



∈



A

,

B

Equation (14.3)
(b) intersection:

µ

(

A

∩

B

) = min{

µ

(

A

),

µ

(

B

)},

∀

x



∈



A

,

B

Equation (14.4)
(c) contradiction:
Equation (14.5)
To demonstrate these operations, a simple example is given below.
E

XAMPLE

1: Given a discrete space

X

= {

a

,

b

,

c

,

d

} and fuzzy events,

f

=

a

/ 1 +

b

/ 0.7 +

c

/ 0.5 +

d

/ 0.1

g

=

a

/ 1 +

b

/ 0.6 +

c

/ 0.3 +

d

/ 0.2
find

f

∪

g

,

f

∩



g

and

f

.
Solution: Using Equations 14.2 through 14.4, it is easy to see

f

∪



g

=

a

/ 1 +

b

/ 0.7 +

c

/ 0.5 +

d

/ 0.2

f

∩



g

=

a

/ 1 +

b / 0.6 + c / 0.3 + d / 0.1
f = a / 0 + b / 0.3 + c / 0.5 + d / 0.9
FIGURE 14.1 Illustration of crisp and fuzzy concept.
µ
x
xa
xa
ba
axb
bxc
dx
dc
cxd
dx
()
=
≤
<≤
<≤
<≤
<












0
1
0
–
–
–
–
µµ
AAxA
()
=
()
∀∈1 –,
©2001 CRC Press LLC
14.2.2 Fuzzy Sets and Probability Distribution
There is often confusion about the difference between fuzzy degree and probability. The difference can
be demonstrated by the following simple example: “the probability that a NBA player is 6 feet tall is 0.7”
implies that there is an 70% chance of a randomly picked NBA player being 6 feet tall, though he may
be just 5 feet 5. On the other hand, “the fuzzy degree that an NBA player is 6 feet tall is 0.7” implies that
a randomly picked NBA player is most likely 6 feet tall (70%). In other words, the probability of an event
describes the possibility of occurrence of the event while the fuzzy degree describes the uncertainty of
appearance of the event.
It is interesting to know, however, that although the fuzzy degree and probability are different, they
are actually correlated [Baldwin et al., 1995]. This correlation is through the probability mass function.
To show this, let us consider a simple example below.
E

XAMPLE
2: Given a discrete space X = {a, b, c, d} and a fuzzy event f ⊆ X,
f = a / 1 + b / 0.7 + c / 0.5 + d / 0.1,
find the probability mass function of Y = f.
Solution: First, the possibility function of f is:
µ
(a) = 1,
µ
(b) = 0.7,
µ
(c) = 0.5,
µ
(d) = 0.1
This is equivalent to:
µ
({a, b, c, d}) = 1,
µ
({b, c, d}) = 0.7,
µ
({c, d}) = 0.5,
µ
({d}) = 0.1
Assuming P(ƒ) ≤
µ
(ƒ), and
P(a) = p
a
, P(b) = p
b
, P(c) = p

c
, P(d) = p
d
it follows that
p
a
+ p
b
+ p
c
+ p
d
= 1
p
b
+ p
c
+ p
d
≤ 0.7
p
c
+ p
d
≤ 0.5
p
d
≤ 0.1
p
i

≥ 0, i = a, b, c, d
Solving this set of equations, we have:
0.3 ≤ p
a
≤ 1
0 ≤ p
b
≤ 0.7
0 ≤ p
c
≤ 0.5
0 ≤ p
d
≤ 0.1
Therefore, the probability mass function of f is
m(a): [0.3, 1], m(b): [0, 0.7], m(c): [0, 0.5], m(d): [0, 0.1]
or
m = {a}: 0.3, {a, b}: 0.2, {a, b, c}: 0.4, {a, b, c, d}: 0.1
In general, suppose that A ⊆ X is a discrete fuzzy event, namely
©2001 CRC Press LLC
A = x
1
/
µ
(x
1
) + x
2
/
µ

(x
2
) + … + x
n
/
µ
(x
n
) Equation (14.6)
Then, the fuzzy set A induces a possibility distribution over X:
Π(x
i
) =
µ
(x
i
)
Furthermore, assume (a)
µ
(x
1
) = 1, and (b)
µ
(x
i
) ≥
µ
(x
j
) if i < j, then:

Π({x
i
, x
i+1
, …, x
n
}) =
µ
(x
i
) Equation (14.7)
If P(A) ≤ Π(A), ∀ A ∈ 2
X
, then we have
for i = 2, …, n Equation (14.8a)
Equation (14.8b)
Solving Equation 14.8 results in
1 –
µ
(x
2
) ≤ P(x
1
) ≤ 1 Equation (14.9a)
0 ≤ P(x
i
) ≤
µ
(x
i

), for i = 2, …, n Equation (14.9b)
Finally, from the probability functions, the probability mass function can be found:
m
f
= { {x
1
, …, x
i
}:
µ
(x
i
) –
µ
(x
i+1
), i = 1, …, n } with
µ
(x
n+1
) = 0
Equation (14.10)
It should be noted that, as shown in Equation 14.9, a fuzzy event corresponds to a family of probability
distributions. Hence, it is necessary to apply a restriction to form a specific probability distribution. The
restriction is to distribute the mass function with a focal element. For example, given a mass function
m = {a, b, c}: 0.3, then there are three focal elements {a, b, c} and its value is 0.3. Hence, applying the
restriction, we have m = (3, 0.3). In general, under the restriction a mass function can be denoted as m
= (L, M), where L corresponds to the size of the focal elements and M represents the value. In the example
above, L = 3 and M = 0.3.
Also, it shall be noted that the mass function assignment may be incomplete. For example, if f = a /

0.8 + b / 0.6 + d / 0.2, X = {a, b, c, d}, then the mass assignment would be
m
f
= a: 0.2, {a, b}: 0.4, {a, b, c}: 0.2; ∅: 0.2
In this case, we need to normalize the mass assignment by using the formula:
µ
(x
i
) =
µ
(x
i
) /
µ
(x
1
), i = 2, 3, .., n Equation (14.11)
and then do the mass assignment. For the above example, the normalization results in f* = a / (0.8/0.8)
+ b / (0.6/0.8) + d / (0.2/0.8) = a / 1 + b / 0.75 + d / 0.25, and the corresponding mass assignment is
m
f*
= a: 0.25, {a, b}: 0.5, {a, b, c}: 0.25
Px x
k
k
n
i
()
≤
()

=
∑
1
µ ,
Px
k
k
n
()
=
=
∑
1
1
©2001 CRC Press LLC
It can be shown that the normalized mass assignment conforms the Dempster–Shafer properties
[Baldwin et al., 1995]:
(a) m(A) ≥ 0,
(b) m(
∅
) = 0,
(c)
14.2.3 Conditional Fuzzy Distribution
Similar to condition probability, we can define the conditional fuzzy degrees (conditional possibility
distribution). There are several ways to deal with the conditional fuzzy distribution. First, let g and g′
be two fuzzy sets defined on X, the mass function associated with the truth set of g given g′, denoted
by m
(g / g′)
, is another mass function defined over {t, f, u} (t represents true, f represents false, and u
stands for uncertain). Let m

g
= {L
i
: l
i
} and m
g′
= {M
i
: m
i
} and form a matrix
where Equation (14.12)
Then, the truth mass function m
(g / g′)
is given below:
Equation (14.13)
where, l
i
.m
j
denotes the element multiplication. The following example illustrates how a conditional mass
function is obtained.
E
XAMPLE
3: Let
g = a/1 + b/0.7 + c/0.2
g
′
= a/0.2 + b/1 + c/0.7 + d/0.1

be fuzzy sets defined on X = {a, b, c, d}. Find the truth possibility distribution, m
(g / g′)
.
Solution: First, using Equation 14.10, it can be shown that
m
g
= {a}: 0.3, {a, b}: 0.5, {a, b, c}: 0.2
m
g′
= {b}: 0.3, {b, c}: 0.5, {a, b, c}: 0.1, {a, b, c, d}: 0.1
Hence, a matrix is formed (enclosed by the single line):
{b}
0.3
{b,c}
0.5
{a,b,c}
0.1
{a,b,c,d}
0.1
{a} ffuu
0.3 0.09 0.15 0.03 0.03
{a,b} t uuu
0.5 0.15 0.25 0.05 0.05
{a,b,c} tttu
0.2 0.06 0.1 0.02 0.02
mA
AF
()
=
∈

()
∑
1
X
MTLMlm
ijij
=
()
{}
/:., TL M
tML
fML
u
ij
ji
ji
/
()
=
⊆
∩=











if
if
otherwise
O
M
tlm
flm
ulm
gg
ij
ijTL M t
ij
ijTL M f
ij
ijTL M u
ij
ij
ij
/
,, /
,, /
,, /
.
.
.
′
()
()
=

()
=
()
=
=






















∑
∑
∑

:
:
:
©2001 CRC Press LLC
The element of the matrix (enclosed by the bold line) may take three different values: t, f, and u, as
defined by Equation 14.11. Take, for instance, the element in the first row and first column, since {a} ∩
{b} = 0, it shall take a value f. For the element in the second row and first column, since {b} ⊆ {a, b}, it
shall take a value of t. Also, for the element in second row and second column, since neither {a, b} ∩ {b,
c} nor {a, b} ⊆ {b, c}, it shall take a value u. Finally, using Equation 14.12, it follows that
m = t: (0.3)(0.3) + (0.2)(0.3) + (0.5)(0.2) + (0.1)(0.2)
= 0.15 + 0.06 + 0.1 + 0.02
= 0.33
f: 0.09 + 0.15 = 0.24
u: 0.25 + 0.03 + 0.05 + 0.03 + 0.05 + 0.02 = 0.43
If we are concerned only about the point value for the truth of g/g ′, there is a simple formula. Use
the notations above to form the matrix
Equation (14.14)
where, “card” stands for cardinality
*
. Then, the probability P(g/g ′) is given below:
Equation (14.15)
E
XAMPLE
4: Following Example 3, find the probability for the truth of g/g ′.
Solution: From Example 3, it is known that
m
g
= {a}: 0.3, {a, b}: 0.5, {a, b, c}: 0.2
m
g′

= {b}: 0.3, {b, c}: 0.5, {a, b, c}: 0.1, {a, b, c, d}: 0.1
The following matrix can be formed:
*
The cardinality of a set is its size. For example, given a set A = [a, b, c], card(A) = 3.
{b}
0.3
{b,c}
0.5
{a,b,c}
0.1
{a,b,c,d}
0.1
{a}
0 0 0.01 0.000750.3
{a,b}
0.15 0.125 0.0333 0.0250.5
{a,b,c}
0.06 0.1 0.02 0.0150.2
M =
{}
=
∩
()
()











m
LM
M
lm
ij
ij
j
ij
card
card
Pg g m
ij
ij
/
,
′
()
=
∑
©2001 CRC Press LLC
Note that the matrix is found element by element. For example, for the element in the first row and
first column, since {a} ∩ {b} = 0, card(L
1
∩ M
1
) = 0, thus m

11
= 0. For the element in the second row
and second column, since {a, b} ∩ {b, c} = {b}, card(L
2
∩ M
2
) = card({b}) = 1, card(M
2
) = card({b, c})
= 2, m
22
= (1/2)(0.5)(0.5) = 0.125. The other components can be determined in the same way. Based on
the matrix, it is easy to find P(g/g′) = 0 + 0 + 0.01 + … + 0.015 = 0.53980.
We can also determine the fuzzy degree of g given g′. It is a pair: the possibility of g/g′ is defined as
Π(g/g
′
) = max(g ∩ g
′
) Equation (14.16)
and the necessity of g/g′ is defined as
π(g/g
′
) = 1 – Π(g/g
′
) Equation (14.17)
This is analogous to the probability support pair and provides the upper and lower bounds of the
conditional fuzzy set.
E
XAMPLE
5: Following Example 3, find its possibility support pair.

Solution: Since
g = a/1 + b/0.7 + c/0.2
g
′
= a/0.2 + b/1 + c/0.7 + d/0.1
it is easy to see
g ∩ g
′
= a/0.2 + b/0.7 + c/0.2
Π(g ∩ g
′
) = 0.7
Furthermore,
g = b/0.3 + c/0.8 + d/1
g
∩ g
′
= b/0.3 + c/0.7 + d/0.1
π(g ∩ g
′
) = 1 – Π(g ∩ g
′
) = 0.3
Hence, the conditional fuzzy degree of g/g′ is [0.3, 0.7].
14.3 Monitoring and Diagnosing Manufacturing Processes Using
Fuzzy Sets
14.3.1 Using Fuzzy Systems to Describe the State of a Manufacturing Process
For monitoring and diagnosing manufacturing processes, two types of uncertainties are often encoun-
tered: the uncertainty of occurrence and the uncertainty of appearance. A typical example is tool condition
monitoring in machining processes. Owing to the nature of metal cutting, tools will wear out. Through

years of study, it is commonly accepted that tool wear can be determined by Taylor’s equation:
VT
n
= C Equation (14.18)
where V is the cutting speed (m/min), T is the tool life (min), n is a constant determined by the tool
material (e.g., n = 0.2 for carbide tools), and C is a constant representing the cutting speed at which the
©2001 CRC Press LLC
tool life is 1 minute (it is dependent on the work material). Figure 14.2 shows a typical example of tool
wear development, and the end of tool life is determined at VB = 0.3 mm for carbide tools (VB is the
average flank wear), or VB
max
= 0.5 mm (VB
max
is the maximum average flank wear). However, it is also
found that the tool may wear out much earlier or later depending on various factors such as the feed,
the tool geometry, the coolant, just to name a few. In other words, there is an uncertainty of occurrence.
Such an uncertainty can be described by the probability mass function shown in Figure 14.3. As shown
in the figure, the states of tool wear can be divided into three categories: initial wear (denoted as A),
normal tool (denoted as B), and accelerated wear (denoted as C). Their occurrences are a function of time.
On the other hand, it is noted that the state of tool wear may be manifested in various shapes depending
on various factors, such as the depth of cut, the coating of the cutter, the coolant, etc. Consequently,
even though the state of tool wear is the same, the monitoring signals may appear differently. In order
words, there is an uncertainty of appearance. Therefore, in tool condition monitoring, the question to
be answered is not only how likely the tool is worn, but also how worn is the tool. To answer this type
of problem, it is best to use the fuzzy set theory.
FIGURE 14.2 Illustration of tool wear.
FIGURE 14.3 Illustration of the tool wear states and corresponding fuzzy sets.
t
VB = 0.3
m

A
(t) m
B
(t)
m
C
(t)
Tool life
curve
©2001 CRC Press LLC
14.3.2 A Unified Model for Monitoring and Diagnosing Manufacturing
Processes
Although manufacturing processes are all different, it seems that the task of monitoring and diagnosing
always takes a similar procedure, as shown in Figure 14.4. In Figure 14.4, the input to a manufacturing
process is its process operating condition (e.g., the speed, feed, and depth of cut in a machining process).
The manufacturing process itself is characterized by its process condition, y ∈ Y = {y
i
, i = 1, 2, …, m}
(e.g., the state of tool wear in the machining process). Usually, the process operating conditions are
controllable while the process conditions may be neither controllable nor directly observable. It is
interesting to know that the process conditions are usually artificially defined. For example, as discussed
earlier, in monitoring tool condition the end of tool life is defined as the flank wear, VB, exceeding
0.3 mm. In practice, however, tool wear can be manifested in various forms. Therefore, it is desirable to
use fuzzy set theory to describe the state of the tool wear.
Sensing opens a window to the process through which the changes of the process condition can be
seen. Note that both the process and the sensing may be disturbed by noises (an inherited problem in
engineering practice). Consequently, signal processing is usually necessary to capture the process condi-
tion. Effective sensing and signal processing is very important to monitoring and diagnosing. However,
it will not be discussed in this chapter. Instead, the reader is referred to [Du, 1998].
The result of signal process is a set of signal features, also referred to as indices or attributes, which

can be represented by a vector x = [x
1
, x
2
, …, x
n
]. Note that although the numeric values are most
common, the attributes may also be integers, sets, or logic values. Owing to the complexity of the process
and the cost, it is not unusual that the attributes do not directly reveal the process conditions. Conse-
quently, decision-making must be carried out. There have been many decision-making methods; the
fuzzy set theory is one of them and has been proved to be effective.
Mathematically, the unified model shown in Figure 14.4, as represented by the bold lines, can be
described by the following relationship:
y • R = x Equation (14.19)
where R is the relationship function, which represents the combined effect of the process, sensing, and
signal processing. Note that R may take different forms such as a dynamic system (described by a set of
differential equations), patterns (described by a cluster center), neural network, and fuzzy logic. Finally,
it should be noted that the operator “•” should not be viewed as simple multiplication. Instead, it
corresponds to the form of the relationship.
The process of monitoring and diagnosing manufacturing processes consists of two phases. The first
phase is learning. Its objective is to find the relationship R based on available information (learning from
samples) and knowledge (learning from instruction). Since the users must provide information and
instruction, the learning is a supervised learning. To facilitate the discussions, the available learning
samples are organized as shown in Table 14.1.
FIGURE 14.4 A unified model for monitoring and diagnosing manufacturing processes.
y
Manufacturing
process
Sensing
Signal

processing
Decision-
makin
g
noise noise
process
operating
condition
x
R