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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,20

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

Monitoring and

Diagnosing Manufacturing Processes Using Fuzzy

Set Theory14.1 Introduction

14.1 Introduction

According to Webster’s New World Dictionary of the American Language, “monitoring,” among severalother 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 cious symptoms while diagnosis determines the cause of the symptoms There are several words and/or

suspi-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.

phrases that have similar or slightly different meanings, such as fault detection, fault prediction, process verification, on-line inspection, identification, and estimation.

in-Monitoring and diagnosing play a very important role in modern manufacturing This is becausemanufacturing 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 ordefects may cause catastrophic consequences Therefore, a great deal of research has been carried out inthe past 20 years Many papers and monographs have been published Instead of giving a partial reviewhere, the reader is referred to two books One by Davies [1998] describes various monitoring and diagnosistechnologies 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 dividedinto 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 monly used in electrical and aerospace engineering The book by Gertler [1988] describes the basics ofmodel-based monitoring The latter uses the features extracted from sensor signals (such as cutting forces

com-in machcom-incom-ing processes and pressures com-in pressured vessels) and can be used com-in various engcom-ineercom-ing 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 theoryfor engineering monitoring and diagnosis The presented method is applicable to almost all engineeringprocesses 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 thediscussions that follow, the readers can see that fuzzy set theory is simple and effective method that isworth exploring

This chapter contains five sections Section 14.2 is a brief review of fuzzy set theory Section 14.3describes how to use fuzzy set theory for monitoring and diagnosing manufacturing processes Section14.4 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 andmonographs on this subject, for example [Baldwin et al., 1995; Klir and Folger, 1988] Hence, this chapteronly gives a brief description of fuzzy set theory for readers who are familiar with the concept but areunfamiliar 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 membershipfunction, µ(A′), varies between 0 and 1, that is µ(A) = [0, 1] In other words, there are cases in whichthe instance of the event x∈A′ can only be determined with some degree of certainty This degree ofcertainty is referred to as fuzzy degree and is denoted as µΑ’(x ∈A′) Furthermore, the fuzzy set is denoted

as x/µA’(x), ∀x∈A′, and µA’(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 formssuch as a discrete tablet,

µ(x): µ(x1) µ(x2) … µ(x n) Equation (14.1)

or a continuous step-wise function,

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,

To demonstrate these operations, a simple example is given below

EXAMPLE 1: Given a discrete space X = {a, b, c, d} and fuzzy events,

µ( )A =1 –µ( )A ,∀ ∈x A

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 eventdescribes the possibility of occurrence of the event while the fuzzy degree describes the uncertainty ofappearance of the event

It is interesting to know, however, that although the fuzzy degree and probability are different, theyare actually correlated [Baldwin et al., 1995] This correlation is through the probability mass function

To show this, let us consider a simple example below

EXAMPLE 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:

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

If P(A) ≤Π(A), ∀ A ∈ 2X, then we have

for i = 2, …, n Equation (14.8a)

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

µ ,

P x k k

n

( )=

=

∑1

1

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

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

′( )

( )=( )=( )=

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 Usethe 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)

EXAMPLE 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.

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(L1∩ M1) = 0, thus m11 = 0 For the element in the second row

and second column, since {a, b} ∩ {b, c} = {b}, card(L2∩ M2) = card({b}) = 1, card(M2) = card({b, c})

= 2, m22 = (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

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 tered: the uncertainty of occurrence and the uncertainty of appearance A typical example is tool conditionmonitoring in machining processes Owing to the nature of metal cutting, tools will wear out Throughyears of study, it is commonly accepted that tool wear can be determined by Taylor’s equation:

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

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 VBmax = 0.5 mm (VBmax is the maximum average flank wear) However, it is alsofound 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 orderwords, 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

Tool lifecurve

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 diagnosingalways takes a similar procedure, as shown in Figure 14.4 In Figure 14.4, the input to a manufacturingprocess 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 arecontrollable while the process conditions may be neither controllable nor directly observable It isinteresting 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 touse 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 beseen Note that both the process and the sensing may be disturbed by noises (an inherited problem inengineering 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 = [x1, x2, …, x n] Note that although the numeric values are mostcommon, the attributes may also be integers, sets, or logic values Owing to the complexity of the processand 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; thefuzzy 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 bedescribed by the following relationship:

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, itcorresponds 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 andinstruction, the learning is a supervised learning To facilitate the discussions, the available learningsamples are organized as shown in Table 14.1

FIGURE 14.4 A unified model for monitoring and diagnosing manufacturing processes.

y

Manufacturingprocess

processing

making

The second phase is classification Given a new sample x, and the relationship R, the corresponding

process condition of the sample y(x) can be determined as follows:

Here again, the operator “•” should not be viewed as simple multiplication Instead, it may mean patternmatching, neural network searching, and fuzzy logic operations depending on the inverse of the rela-tionship

In the following subsection, we will show how to use fuzzy set theory to establish a fuzzy relationshipfunction (Equation 14.19) and how to resolve it to identify the process condition of a new sample(Equation 14.20)

14.3.3 Linear Fuzzy Classification

One of the simplest fuzzy relationship functions is the linear equation defined below [Du et al., 1992]:

where Q represents the linear fuzzy correlation between the classes and the attributes (signal features).

Assuming that there are m different classes and n different attributes, then y is an m-dimensional vector and x is a n-dimensional vector The fuzzy linear correlation function between the classes and the

attributes may take various forms such as a tablet form (Equation 14.1) or a stepwise function (Equation

14.2) For simplicity, let us use the tablet form First, each attribute is divided into K intervals Note that

just like the histogram in statistics, different definitions of intervals may lead to different results As arule of thumb, the number of intervals should be about one tenth of the total number of samples, that is,

Equation (14.22)

and intervals should be evenly distributed For the jth attribute, let

where, k = 1, 2, …, N correspond to the learning samples The width of the interval will be

Equation (14.24)

TABLE 14.1 Organization of the Available Learning Samples

Note: where, y(xj) ∈ Y = {y i , i = 1, 2, , m}, j = 1, 2, , N, represent the process condition and it must

be known in order to conduct learning.

K=N10

∆x

,max– ,min

The intervals would be

where N ijk is the number of samples of the ith class in the kth interval of the jth attribute M ij is the number

of samples of the ith class in the jth attribute From a physical point of view, it represents the distribution

of the training samples about the classes We may also use a similar formula,

Equation (14.29)

where M jk is the number of samples in the kth interval of the jth attribute From a physical point of view,

it represents the distribution of the training samples about the attributes Also, we can use the combination

of both:

Equation (14.30)

where 0 ≤α≤ 1 is a weighting factor As an example, Figure 14.5 illustrates a fuzzy membership function,

in which the attributes X j is decomposed into ten intervals, and the fuzzy membership functions of twoprocess conditions are overlapped

When a new sample, x, is provided, its corresponding process condition can be estimated by

classifi-cation The classification phase starts at checking the fuzzy degree of the new sample Suppose the jth

attribute of the new sample falls into the kth interval I jk, then

q ij (x) = I jk / µ(Iijk) Equation (14.31)

ij

( )=

µI N

M ijk ijk

In other words, the fuzzy degree that the sample belong to jth class is µ(I ijk) Next, using the max–minclassification rule, the corresponding process condition of the sample can be estimated:

Equation (14.32)

The other useful and often better performed classification rule is the max-average rule defined below:

Equation (14.33)

These operations are demonstrated by the example below

EXAMPLE 6: Given the following discrete training samples:

find the linear fuzzy relationship function Furthermore, suppose a new sample x = [a, d] is given,

estimate its class

Solution: There are two (discrete) attributes and three classes (A, B, C), and hence the fuzzy

i

i

ijk j

1µ