Expert Systems with Applications 36 (2009) 8103–8106

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Expert Systems with Applications

journal homepage: www.elsevier .com/locate /eswa

Fault diagnosis of turbine based on fuzzy cross entropy of vague sets

Jun Ye *

Department of Mechatronics Engineering, Shaoxing College of Arts and Sciences, 508 Huancheng West Road, Shaoxing, Zhejiang Province 312000, PR China

a r t i c l e i n f o

Keywords:Vague setsCross entropyVague cross entropyTurbineFault diagnosisVibration fault

0957-4174/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.eswa.2008.10.017

* Tel.: +86 575 88327323.E-mail address: yejnn@yahoo.com.cn

a b s t r a c t

The fuzzy cross entropy of vague sets, so-called vague cross entropy, is introduced by analogy with thecross entropy of probability distributions. And then a new method of the fault diagnosis is proposedon the basis of the vague cross entropy and is applied to the fault diagnosis of turbine. The vague crossentropy between a testing sample and the knowledge of system faults is evaluated in the fault diagnosisof the turbine vibration. If the cross-entropy value is small, the testing sample is near to a type of faultknowledge. Then, the type of vibration fault is determined according to the minimum cross-entropyvalue. The fault-diagnosis example of the turbine demonstrates that the proposed method cannot onlydiagnose the main fault types of the turbine, it can also detect useful information for future trends andmulti-fault analysis.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The technique of fault diagnosis has produced the huge eco-nomic benefits by scheduling preventive maintenance and pre-venting extensive downtime periods caused by extensive failure.Therefore, it has become a research hotspot. In the past, variousfault-diagnosis techniques have been proposed, including expertsystems (Wang & Yang, 1996), neural networks (Chen, Li, & Orady,1996), and fuzzy approaches (Wang, 2004; Ye, Qiao, & Wei, 2005).The expert system can take human expertise, and has been suc-cessfully applied in this field. However, there are some intrinsicshortcomings for the expert system, such as the difficulty ofacquiring knowledge and maintaining a database. These may varyfrom utility to utility due to the heuristic nature of the method andno general mathematical formulation can be utilized. Neural net-works can directly acquire experience from training data and exhi-bit highly nonlinear input–output relationships. This can overcomesome of the shortcomings of the expert system. However, thetraining data must be sufficient and compatible to ensure propertraining. A further limitation of the approach of neural networksis its inability to produce linguistic output, because it is difficultto understand the content of network memory.

Entropy is very important for measuring uncertain information.The fuzzy entropy was first introduced by Zadeh (1965), Zadeh(1968). Latter, De Luca and Termini (1972) presented the axiomswith which the fuzzy entropy should comply and defined the en-tropy of a fuzzy set based on Shannon’s function. The starting pointfor the cross-entropy approach is information theory as developed

ll rights reserved.

by Shannon (1948). Kullback and Leibler (1951) proposed a mea-sure of the ‘‘cross-entropy distance” between two probability dis-tributions. In this paper, we will introduce a fuzzy cross-entropyof vague sets (VSs) (Zhang & Jiang, 2008) by analogy with the crossentropy of probability distributions and apply it to the fault diag-nosis of turbine. We investigate problems of the fault diagnosisaccording to the cross entropy of vague sets. The feasibility andrationality of the method are validated by the fault diagnosisexample of the turbine.

2. Cross entropy

Let P = {p1, p2, . . . , pn} be a probability distribution on X, wherepi e [0,1],

Pni¼1pi ¼ 1, then the entropy of probability distribution

P (Shannon, 1948) is defined by

HðPÞ ¼ �Xn

i¼1

pilog2pi ð1Þ

and let Q = {q1, q2, . . . , qn} be the other probability distribution on X,where qi e [0,1],

Pni¼1qi ¼ 1, then the entropy of probability distri-

bution Q is defined by

HðQÞ ¼ �Xn

i¼1

qilog2qi: ð2Þ

The cross-entropy between these two probability distributionsP and Q is the Kullback–Leibler distance (Kullback & Leibler,1951) that is defined as follows:

HðP;QÞ ¼Xn

i¼1

pilog2pi

qið3Þ

8104 J. Ye / Expert Systems with Applications 36 (2009) 8103–8106

which is not a ‘‘distance” in the formal sense since it is neither notsymmetric nor satisfies the triangle inequality.When n = 2, we canassume that P = {p, 1 � p} and Q = {q, 1 � q}, thus Eq. (3) can be ex-pressed by

HðP;QÞ ¼ plog2pqþ ð1� pÞlog2

1� p1� q

: ð4Þ

Now, assume that A = (A(x1), A(x2), . . . , A(xn)) and B = (B(x1),B(x2), . . . , B(xn)) are two fuzzy sets in the universe of discourse,X = {x1, x2, . . . , xn}, xi e X. According to formula (4), the cross entropyof A(xi) from B(xi) can be defined as follows:

MðAðxiÞ;BðxiÞÞ ¼ AðxiÞlog2AðxiÞBðxiÞ

þ ð1� AðxiÞÞlog21� AðxiÞ1� BðxiÞ

ð5Þ

and the fuzzy cross-entropy of A and B can also be defined by

MðA;BÞ ¼Xn

i¼1

MðAðxiÞ;BðxiÞÞ ð6Þ

which indicates the fuzzy information in favor of A against B. Thefuzzy cross-entropy of A(xi) and B(xi) satisfies the followingproperties:

(1) M(A(xi), B(xi)) P 0, xi e X;(2) M(A(xi), B(xi)) = 0 if A(xi) = B(xi), xi e X; and(3) M(Ac(xi), Bc(xi)) = M(A(xi), B(xi)), xi e X.

Here, Ac(xi) and Bc(xi) are the complement of fuzzy sets A(xi) andB(xi), respectively. The above properties can be easily proved inaccordance with Shannon’s inequality (Lin, 1991).

When B(xi) = 0 and A(xi) – 0, M(A(xi), B(xi)) is undefined, so, it isnecessary to modify it as follows:

HðAðxiÞ;BðxiÞÞ ¼ AðxiÞlog2AðxiÞ

12 ðAðxiÞ þ BðxiÞÞ

þ ð1� AðxiÞÞlog21� AðxiÞ

1� 12 ðAðxiÞ þ BðxiÞÞ

ð7Þ

and then

HðA;BÞ ¼Xn

i¼1

AðxiÞlog2AðxiÞ

12 ðAðxiÞ þ BðxiÞÞ

þð1� AðxiÞÞlog21� AðxiÞ

1� 12 ðAðxiÞ þ BðxiÞÞ

!; ð8Þ

where A(xi), B(xi) e [0,1].

3. Fuzzy cross-entropy of VSs

Gau and Buehrer(1993) extended fuzzy sets to vague sets (VSs),which are defined as follows.

Definition 1. A VS A in X, X = {x1, x2, . . . , xn}, is given by

A ¼Xn

i¼1

½lAðxiÞ;1� mAðxiÞ�=xi; xi 2 X;

where lA(x): X ? [0, 1] and mA(x): X ? [0, 1], with the condition0 6 lA(x) + mA(x) 6 1. The numbers lA(x) and mA(x) represent,respectively, a truth-membership function and a false-membershipfunction of the element x to the set A.

In the following, we let A and B be two VSs in the universe ofdiscourse X = {x1, x2, . . . , xn}. VSs can be transformed into fuzzy setsto structure a cross-entropy of VSs by means of lF

AðxiÞ = (lA(xi) +1 � mA(xi))/2. Then, according to the above definition of fuzzycross-entropy (Zhang and Jiang, 2008), the vague cross-entropy be-tween VSs A and B can be defined as

DðA;BÞ ¼Pni¼1

lAðxiÞþ1�mAðxiÞ2 log2

lAðxiÞþ1�mAðxiÞ12 ðlAðxiÞþ1�mAðxiÞÞþðlBðxiÞþ1�mBðxiÞÞ½ �

þPni¼1

1�lAðxiÞþmAðxiÞ2 log2

1�lAðxiÞþmAðxiÞ12 ð1�lAðxiÞþmAðxiÞÞþð1�lBðxiÞþmBðxiÞÞ½ �

; ð9Þ

which also indicates discrimination degree of the VS A from B.According to Shannon’s inequality (Lin, 1991; Zhang and Jiang,2008), one can easily prove that D(A, B) P 0, and D(A, B) = 0 whenlA(xi) = lB(xi) and mA(xi) = mB(xi), xi e X. Moreover, we can easily seethat D(Ac, Bc) = D(A, B), where Ac and Bc are the complement of theVSs A and B, respectively.

However, D(A, B) is not symmetric, so it should be modified to asymmetric form as

D�ðA; BÞ ¼ DðA; BÞ þ DðB;AÞ: ð10Þ

The larger the difference between A and B, the larger D*(A, B).

4. Fault-diagnosis method based on the vague cross entropy

In this section, we will apply the vague cross-entropy to faultdiagnosis. Essentially, the technique of equipment diagnosis is apattern recognition problem. In other word, the operating statusof the machine is divided into normal and abnormal status. Furtherspeaking, the signal sample of the abnormality belongs to whichtype on earth; this is a pattern recognition problem again.

4.1. Fault-diagnosis principle

Assume that there exist m fault patterns (knowledge of faultsamples), which are represented by a VS Fi (i = 1, 2, . . . , m), andthere is a testing sample to be recognized which is representedby a VS Ft. Then diagnosing result Fk should be the nearest one toFt, i.e.

D�ðFt ; FkÞ ¼ Min16i6m

fD�ðFt ; FiÞg;

where D*(Ft, Fi) expresses the discrimination degree of the VS Ft

from Fi. The value of D*(Ft, Fi) is calculated by using Eqs. (9) and(10). Then, we decide that the testing sample Ft should belong tothe fault pattern Fk, where k ¼ arg Min16i6mfD�ðFt; FiÞg .

The cross entropy of VSs provides the strong means for this faultdiagnosis. It can realize the classification and identification of thefault, i.e., we can compare the cross-entropy values by calculatingthe vague cross entropy between a diagnosing sample and knowl-edge of system faults, and then confirm the fault type according tothe minimum cross entropy. The fault-diagnosis process using thevague cross entropy is shown in Fig. 1.

5. Application of the vague cross entropy to fault diagnosis ofturbine

An example with the steam turbine-generator set demonstratesthe effectiveness of the fault diagnosis of the turbine by using theproposed fault-diagnosis method. The vibration of huge steam tur-bine-generator sets suffers the influence of a lot of factors, such asmechanical structure, load, vacuum degree, hot inflation of cylin-der body and rotor, fluctuation of network load, temperature of lu-bricant oil, and ground. In generator sets, and interaction effects inthe factors show the vibration of the generator sets. In vibrationfault diagnosis of the generator sets, we have established the rela-tion between the cause and the fault symptom of the turbine. Now,we investigate the fault-diagnosis problems by means of the vaguecross entropy.

The ten kinds of familiar fault types in rotating machines, suchas unbalance, offset center, and oil-membrane oscillation, are used

Knowledge of system faults Required diagnosis-testing sample

Vague cross-entropy calculation

Vague cross-entropy collating

Minimum vague cross entropy

Fault type

Fig. 1. Block diagram of fault diagnosis using the vague cross entropy.

J. Ye / Expert Systems with Applications 36 (2009) 8103–8106 8105

as the knowledge of fault samples. By use of nine ranges of differ-ent frequency spectrum, the power spectrum of the vibration sig-nals from some generator sets is referred to VSs. We use thetypical fault samples as the fault knowledge as shown in Table 1(Ye et al., 2005).

In the diagnosis process, at first, we establish the knowledgedatabase of fault types, and then calculate the vague crossentropy between a fault-testing sample and fault knowledgesamples.

Ten fault knowledge samples in Table 1 are expressed by vaguesets:

F1 = [0.00, 0.00]/x1 + [0.00, 0.00]/x2 + [0.00, 0.00]/x3 + [0.00,0.00]/x4 + [0.85, 1.00]/x5 + [0.04, 0.06]/x6 + [0.04, 0.07]/x7 + [0.00,0.00]/x8 + [0.00, 0.00]/x9,

F2 = [0.00, 0.00]/x1 + [0.28, 0.31]/x2 + [0.09, 0.12]/x3 + [0.55,0.70]/x4 + [0.00, 0.00]/x5 + [0.00, 0.00]/x6 + [0.00, 0.00]/x7 + [0.00,0.00]/x8 + [0.08, 0.13]/x9,

F3 = [0.00, 0.00]/x1 + [0.00, 0.00]/x2 + [0.00, 0.00]/x3 + [0.00,0.00]/x4 + [0.30, 0.58]/x5 + [0.40, 0.62]/x6 + [0.08, 0.13]/x7 + [0.00,0.00]/x8 + [0.00, 0.00]/x9,

F4 = [0.09, 0.11]/x1 + [0.78, 0.82]/x2 + [0.00, 0.00]/x3 + [0.08,0.11]/x4 + [0.00, 0.00]/x5 + [0.00, 0.00]/x6 + [0.00, 0.00]/x7 + [0.00,0.00]/x8 + [0.00, 0.00]/x9,

F5 = [0.09, 0.12]/x1 + [0.09, 0.11]/x2 + [0.08, 0.12]/x3 + [0.09,0.12]/x4 + [0.18, 0.21]/x5 + [0.08, 0.13]/x6 + [0.08, 0.13]/x7 + [0.08,0.12]/x8 + [0.08, 0.12]/x9,

F6 = [0.00, 0.00]/x1 + [0.00, 0.00]/x2 + [0.00, 0.00]/x3 + [0.00,0.00]/x4 + [0.18, 0.22]/x5 + [0.12, 0.17]/x6 + [0.37, 0.45]/x7 + [0.00,0.00]/x8 + [0.22, 0.28]/x9,

Table 1Knowledge of system faults.

Fault samples Frequency range (f: operating frequency)

0.01– 0.39f 0.40– 0.49f 0.50f 0.51– 0.99f

Unbalance [0.00, 0.00] [0.00, 0.00] [0.00, 0.00] [0.00, 0.00]Pneumatic force couple [0.00, 0.00] [0.28, 0.31] [0.09, 0.12] [0.55, 0.70]Offset center [0.00, 0.00] [0.00, 0.00] [0.00, 0.00] [0.00, 0.00]Oil-membrane oscillation [0.09, 0.11] [0.78, 0.82] [0.00, 0.00] [0.08, 0.11]Radial impact friction of rotor [0.09, 0.12] [0.09, 0.11] [0.08, 0.12] [0.09, 0.12]Symbiosis loose fault [0.00, 0.00] [0.00, 0.00] [0.00, 0.00] [0.00, 0.00]Damage of antithrust bearing [0.00, 0.00] [0.00, 0.00] [0.08, 0.12] [0.86, 0.93]Surge [0.00, 0.00] [0.27, 0.32] [0.08, 0.12] [0.54, 0.62]Looseness of bearing block [0.85, 0.93] [0.00, 0.00] [0.00, 0.00] [0.00, 0.00]Non-uniform bearing stiffness [0.00, 0.00] [0.00, 0.00] [0.00, 0.00] [0.00, 0.00]

F7 = [0.00, 0.00]/x1 + [0.00, 0.00]/x2 + [0.08, 0.12]/x3 + [0.86,0.93]/x4 + [0.00, 0.00]/x5 + [0.00, 0.00]/x6 + [0.00, 0.00]/x7 + [0.00,0.00]/x8 + [0.00, 0.00]/x9,

F8 = [0.00, 0.00]/x1 + [0.27, 0.32]/x2 + [0.08, 0.12]/x3 + [0.54,0.62]/x4 + [0.00, 0.00]/x5 + [0.00, 0.00]/x6 + [0.00, 0.00]/x7 + [0.00,0.00]/x8 + [0.00, 0.00]/x9,

F9 = [0.85, 0.93]/x1 + [0.00, 0.00]/x2 + [0.00, 0.00]/x3 + [0.00,0.00]/x4 + [0.00, 0.00]/x5 + [0.00, 0.00]/x6 + [0.00, 0.00]/x7 + [0.08,0.12]/x8 + [0.00, 0.00]/x9,

F10 = [0.00, 0.00]/x1 + [0.00, 0.00]/x2 + [0.00, 0.00]/x3 + [0.00,0.00]/x4 + [0.00, 0.00]/x5 + [0.77, 0.83]/x6 + [0.19, 0.23]/x7 + [0.00,0.00]/x8 + [0.00, 0.00]/x9.

Suppose that the VSs of fault-testing samples are as follow:Ft1 = [0.00, 0.00]/x1 + [0.00, 0.00]/x2 + [0.10, 0.10]/x3 + [0.90,

0.90]/x4 + [0.00, 0.00]/x5 + [0.00, 0.00]/x6 + [0.00, 0.00]/x7 + [0.00,0.00]/x8 + [0.00, 0.00]/x9,

Ft2 = [0.39, 0.39]/x1 + [0.07, 0.07]/x2 + [0.00, 0.00]/x3 + [0.06,0.06]/x4 + [0.00, 0.00]/x5 + [0.13, 0.13]/x6 + [0.00, 0.00]/x7 + [0.00,0.00]/x8 + [0.35, 0.35]/x9.

The cross-entropy values of vague sets are calculated by use ofEqs. (9) and (10) as follows:

D*(Ft1, F1) = 3.3350, D*(Ft1, F2) = 0.5996, D*(Ft1, F3) = 2.3888,D*(Ft1, F4) = 2.5056, D*(Ft1, F5) = 1.8955, D*(Ft1, F6) = 2.7526, D*(Ft1,F7) = 0.00001, D*(Ft1, F8) = 0.5350, D*(Ft1, F9) = 3.2067, D*(Ft1,F10) = 3.0681;

D*(Ft2, F1) = 2.6888, D*(Ft2, F2) = 1.5356, D*(Ft2, F3) = 1.8875,D*(Ft2, F4) = 1.6071, D*(Ft2, F5) = 0.8436, D*(Ft2, F6) = 1.3131, D*(Ft2,F7) = 2.3599, D*(Ft2, F8) = 1.7330, D*(Ft2, F9) = 1.1981, D*(Ft2,F10) = 1.9373.

f 2f 3–5f Odd times of f High frequency >5f

[0.85, 1.00] [0.04, 0.06] [0.04, 0.07] [0.00, 0.00] [0.00, 0.00][0.00, 0.00] [0.00, 0.00] [0.00, 0.00] [0.00, 0.00] [0.08, 0.13][0.30, 0.58] [0.40, 0.62] [0.08, 0.13] [0.00, 0.00] [0.00, 0.00][0.00, 0.00] [0.00, 0.00] [0.00, 0.00] [0.00, 0.00] [0.00, 0.00][0.18, 0.21] [0.08, 0.13] [0.08, 0.13] [0.08, 0.12] [0.08, 0.12][0.18, 0.22] [0.12, 0.17] [0.37, 0.45] [0.00, 0.00] [0.22, 0.28][0.00, 0.00] [0.00, 0.00] [0.00, 0.00] [0.00, 0.00] [0.00, 0.00][0.00, 0.00] [0.00, 0.00] [0.00, 0.00] [0.00, 0.00] [0.00, 0.00][0.00, 0.00] [0.00, 0.00] [0.00, 0.00] [0.08, 0.12] [0.00, 0.00][0.00, 0.00] [0.77, 0.83] [0.19, 0.23] [0.00, 0.00] [0.00, 0.00]

8106 J. Ye / Expert Systems with Applications 36 (2009) 8103–8106

5.1. Diagnosis result 1

The fault-diagnosis result is as follows:F7 ? F8 ? F2 ? F5 ? F3 ? F4 ? F6 ? F10 ? F9 ? F1.According to the diagnosis order of the above result, we can

think that the vibration fault of the turbine is firstly resulted fromdamage of antithrust bearing, next surge, and then pneumaticforce couple, and so on. By actual checking, we discover that oneof antithrust bearings is damage. Consequently it causes the vio-lent vibration of the turbine.

5.2. Diagnosis result 2

The fault-diagnosis result is as follows:F5 ? F9 ? F6 ? F2 ? F4 ? F8 ? F3 ? F10 ? F7 ? F1.According to the diagnosis order of the above results, we can

think that the vibration fault of the turbine is firstly resulted fromradial impact friction of rotor, next looseness of bearing block, andthen symbiosis looseness, and so on. By actual checking, we dis-cover the friction between the rotor and cylinder body in the tur-bine, and then the vibration values of four ground bolts of thebearing between the turbine and the gearbox are very difference.We also discover that the gap between the nuts and the bearingblock is oversize. Consequently the looseness of the bearing blockcauses the violent vibration of the turbine.

From the above fault diagnostic results, we can see that the pro-posed diagnosis method is effectiveness. Therefore, it can offereffective and reasonable diagnosis information for multi-faultdiagnoses.

6. Conclusion

This paper introduce a fuzzy cross entropy of vague sets and afault-diagnosis method based on the vague cross entropy, and then

it is applied to the fault diagnosis of the turbine. It can be imple-mented easily by PC software, as the calculation of the proposeddiagnosis algorithm is very simple. Diagnosis results for the tur-bine show that the proposed method cannot only diagnose themain fault types of the turbine, it can also detect useful informa-tion for future trends and multi-fault analysis. This approach mer-its more attention, because the cross-entropy theory deservesserious consideration as a tool in this field. The author hopes thispaper will lead to further investigation.

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