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 [...]... York Davies, A (Ed.), 1998, Handbook of Condition Monitoring, Techniques and Methodology, Chapman & Hall, London Du, R., Elbestawi, M A and Li, S., 1992, Tool Condition Monitoring in Turning Using Fuzzy Set Theory, Int J Mach Tools Manuf., vol 32, no 6, pp 781-796 Du, R., Elbestawi, M A and Wu, S M., 1995, Computer Automated Monitoring of Manufacturing Processes: Part 1, Monitoring Decision-Making Methods,... seen the fuzzy set theory is effective The reader should be aware that, like other monitoring and diagnosis methods such as neural networks and expert systems, fuzzy set theory is not a universal solution that can plug -and- play Instead, successful applications rely on a complete understanding of the process, the sensors, and the sensor signals Use of effective sensors is often the most effective and costeffective... 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... used to establish the fuzzy relationship, and three of them are discussed in detail, including the fuzzy linear equation method, fuzzy C-mean classification method, and the fuzzy transition method For the faults such as wear and fatigue, the fuzzy transition method takes the effect of time into consideration, and hence, provides more effective and reliable results This method is new and may outperform... engineering processes This usually consists of two steps: (i) establish a fuzzy relationship between the sensor signal features and the process condition classes, and (ii) classify the new sensor signal using the fuzzy relationship Note that the definitions of process condition classes are often imprecise (e.g., tool wear may be manifested into various forms), hence, the use of a fuzzy set is meaningful and. .. use fuzzy set theory to establish a fuzzy relationship function (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]: x=Q•y Equation (14.21) where Q represents the linear fuzzy correlation between the classes and. .. be used to form the fuzzy linear equation method described in Section 14.3.3 The resulting fuzzy sets are ts1: x / µA(x), x / µ B(x), x / µC(x) ts2: x / µA(x), x / µ B(x), x / µC(x) …… tsK: x / µA(x), x / µ B(x), x / µC(x) Next, we can calculate the mass functions and conditional mass functions using Equations 14.10 and 14.14 Finally, the fuzzy transition probability and limiting fuzzy transition probability... One-Step Fuzzy Transition Without Time Intervals in Self-Classification Test A A B C B C 0.4845 0 0 0.4890 0.8851 0 0.0265 0.1149 1 TABLE 14.12 The Probabilities of One-Step Fuzzy Transition Without Time Intervals in Prediction Test A A B C B C 0.4929 0 0 0.4786 0.8936 0 0.0265 0.1064 1 14.5 Conclusions In this chapter, we briefly described how to use fuzzy set theory for monitoring and diagnosing manufacturing. .. 121-132 Gertler, J., 1988, Fault Detection and Diagnosis in Engineering Systems, Marcel Dekker, New York Klir, G J and Folger, T A., 1988, Fuzzy Sets, Uncertainty, and Information, Prentice-Hall, Englewood Cliffs, New Jersey Klyele, R M and de Korvin, A., 1998, Constructing One-Step and Limiting Fuzzy Transition Probabilities for Finite Markov Chains, J Intelligent Fuzzy Systems, no 6, pp 223-235 Quinlan,... tool wear is somewhat fuzzy, and hence the fuzzy transition method is used For each tool, three transition steps are used corresponding to the three states of tool wear (A: ts1 = [0, 0.3], B: ts2 = [0.3, 1.1], C: ts3 = [1.1, 3]) The monitoring signal feature is the averaged spindle motor current and the feed motor current The fuzzy membership functions are first determined using the fuzzy linear equation . 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. R. et al " ;Monitoring and Diagnosing Manufacturing Processes Using Fuzzy Set Theory& quot; Computational Intelligence in Manufacturing Handbook Edited