FAULT DIAGNOSIS OF ROTATING MACHINERY USING KNOWLEDGE-BASED FUZZY NEURAL NETWORK

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1 Applied Mathematics and Mechanics (English Edition), 2006, 27(1): c Editorial Committee of Appl. Math. Mech., ISSN FAULT DIAGNOSIS OF ROTATING MACHINERY USING KNOWLEDGE-BASED FUZZY NEURAL NETWORK LI Ru-qiang(oXr), CHEN Jin(?), WU Xing(Î() (The State Key Laboratory of Vibration, Shock and Noise, Shanghai Jiaotong University, Shanghai , P. R. China) ψψψ (Communicated by ZHU Jin-fu) Abstract: A novel knowledge-based fuzzy neural network (KBFNN) for fault diagnosis is presented. Crude rules were extracted and the corresponding dependent factors and antecedent coverage factors were calculated firstly from the diagnostic sample based on rough sets theory. Then the number of rules was used to construct partially the structure of a fuzzy neural network and those factors were implemented as initial weights, with fuzzy output parameters being optimized by genetic algorithm. Such fuzzy neural network was called KBFNN. This KBFNN was utilized to identify typical faults of rotating machinery. Diagnostic results show that it has those merits of shorter training time and higher right diagnostic level compared to general fuzzy neural networks. Key words: rotating machinery; fault diagnosis; rough sets theory; fuzzy sets theory; generic algorithm; knowledge-based fuzzy neural network Chinese Library Classification: TP Mathematics Subject Classification: 82C32; 93C42 Digital Object Identifier (DOI): /s Introduction There are crucial needs for checking rotating machine operation status precisely since they are mostly under harsh environments such as severe shocks, vibration, heat, friction, dusts, etc. Consequently, fault defection, fault identification and diagnosis of rotating machines have become a vigorous area of work. Presently, one of the widely used and effective methods for fault detection and diagnosis of rotating machines is vibration analysis [1]. Attempts have been made towards classification of the most common type of rotating machinery problems, through the following four steps [2], namely, vibration measurement, signal processing, feature extraction and fault identification. Both time domain and frequency domain approaches can be used to analyze vibration signals. Even though time approach provides insight into the physical nature of vibration, this approach becomes practically impossible in presence of multi-tone vibration signals. On the other hand, frequency domain approaches, including both the amplitude and power spectrum to be identified, are more useful for the vibration analysis. Most diagnostic methods are based on pattern matching or pattern classification on the basis of spectral analysis. Research has been extended to implement fault diagnosis using artificial intelligent techniques, of which artificial neural networks (ANNs) and fuzzy neural networks (FNNs) [3] have appealed more attention. Recently, rough sets theory (RST) [4] has also emerged as a tool for fault diagnosis and gotten application successfully in power system [5], vehicle transmission system [6], and fuel injection system of diesel engine [7]. Some integrations, such as integrating RST with ANNs or FNNs, have also been applied for fault diagnosis. Received Oct.11,2003; Revised Aug.23,2005 Project supported by the National Major Science and Technology Foundation of China during the 10th Five-Year Plan Period (No.2001BA204B05-KHK Z0009) Corresponding author LI Ru-qiang, Lecturer, Doctor, rqli163@163.com

2 100 LI Ru-qiang, CHEN Jin and WU Xing ANNs have remarkable capability of predominant nonlinear pattern classification, powerful selforganization, self-studying, memory recalling, parallel information disposal and generalization of knowledge. Fuzzy sets theory (FST) is capable of handing linguistic variables and vague data and thus diagnosis based on FST, to some certain degree, simulate human thinking mode and involve fuzzy conception to make use of multifactor for diagnosis, but there are still many disappointing points for real application such as the difficulty of exactly formulation of fuzzy membership function, fuzzy rules and fuzzy reasoning algorithm. Even though FNNs have its excellent merits of both FST and ANNs, some FNNs using fuzzy value as their inputs and outputs have the same problem as ANNs of unfixed neuron number of hide layers; meanwhile, the selection of fuzzy parameters still involves human operation, which, in some degree, has certain influence on identification level in diagnosis based on FNNs. RST has the advantage of knowledge acquisition, but it is also its disadvantage for diagnosis based on rules as it is a fact that, knowledge come from RST without redundancy since current RST only deal with discrete data has low generalization ability for fault diagnosis. Genetic algorithm (GA) [8], based on the survival of the fittest, is a probability-searching algorithm. It has its inherent parallel information disposal ability and the performance of excellent global search capability. Thus, in this paper, we can well take advantages of those theories and offset their shortages to implement fault diagnosis for rotating machinery by hybridizing RST, FST, ANNs and GA. 1 Rough Sets Theory 1.1 Rough sets theory and rules acquisition RST [4], founded on the mechanism of classification of conception, is a mathematical theory for data analysis proposed by Poland mathematician Pawlak Z. Formally, a knowledge system, S can be seen as a system S =(U, A, V, f) where U is a nonempty finite set called domain of discourse and A a nonempty finite set of attributes, A = C D, with subsets C and D the condition and decision attribute sets, respectively, and C D = Φ. V = U a A V a is a set of attribute values and V a is the range of attribute a A. For x U,a A there exist information function f : U A V a such that f(x, a) V a hold. This type of formulation of knowledge system makes it easy to describe the system using a two dimensions table, which is called decision table (DT). Indiscernibility of B A in the domain U is defined as IND (A) = { (x, y) U 2 a A, f(x, a) =f(y, a) }. Let U = {x 1,x 2,,x m }, C = {a 1,a 2,,a n } in DT S, then every item of an m m decernibility matrix [9] is { a, for f(x i,a) f(x j,a)andf(x i,d) f(x j,d), a C, c ij = i, j =1, 2,..., m (1) φ, for f(x i,a) f(x j,a)andf(x i,d)=f(x j,d), a C, Let the attribute values of a i be v i, i =1, 2,...,n, then decernibility function is f(v 1,v 2,..., v n ; D) = { c ij :1 j<i m, c ij Φ}, (2) where and are operators of disjunction and conjunction, respectively. As far as the classification of knowledge system is concerned, not all the attributes are necessary. Reduction of knowledge system is to try to find the simplest dependency relation of decision attributes with respect to condition attributes. Consequently, rules acquisition based

3 Fault Diagnosis of Rotating Machinery using Fuzzy Neural Network 101 on RST can be regarded as a procedure of knowledge expression, attributes reduction, attribute values reduction and rules acquisition. In this paper, we implement heuristic reduction of the knowledge system based on Ref. [10] to obtain decision rules. 1.2 Dependency factors and antecedent coverage factors of rules To fit in the situation of rules requisition of fault diagnosis, suppose that all the decision attribute values in the DT S are different with each other, that is, D = {d t }, f(x, {d i }) f(x, {d j }), where t =1, 2,..., m, 1 i, j m, m denotes the number of objects. Construct decernibility matrix and decernibility function according to the equations (1) and (2); and for the i object x i,1 i m of DT, we get decision rule R i : if f(x, a 1 )=v i 1 and f(x, a 2)=v i 2,..., f(x, a j)=v i j then f(x, D) =d i k,c= H i, (3) where v i j,di k V a, i, k =1, 2,..., m; j =1, 2,...;1 j n; c istherulestrengthandh i is the value of c, H i [0, 1]. Let B i = {b i 1,bi 2,..., bi j } be condition attribute set of the rule of Ri,and then the dependency factor of decision attribute {d i } with regard to condition attribute set B i is defined as γ i = card (pos B i({d i})), (4) card (U) where card ( ) is the cardinality of set, pos B i({d i })isthe{d i }-positive region with regard to condition attribute set B i. We define function NB i = i Bi, 1 i n, (5) where denote the number of the symbol of, NB i is the number of condition attributes i B i of the rule R i, then the antecedent coverage factor of the rule R i in term of b i j,1 j n is defined as follows O i (B i,b i j)= γi NB i, 1 i m, 1 j n. (6) i 2 Construction, Configuration and Optimization of ANNs 2.1 ANNs A five-layer-neural-network, as shown in figure 1, is selected, which is a forward network and the Ith and IIth layers are the input and fuzzification layers, respectively; the antecedent and consequential parts of the rules are described by the IIIth and IVth layers; and the Vth layer is layer of unfuzzification. A π-function [11] with range [0,1] is selected as the fuzzy membership function of the fuzzification layer, which is represented by 2(1 F i C /λ) 2, λ/2 F i C λ, π(f i,c,λ)= 1 2( F i C /λ) 2, 0 F i C λ/2, (7) 0, others, where λ is the radius and C the central point of linguistic variables or vague data as depicted in Fig.2. Suppose there are l output neurons in the output layer. For the input vector of the c k th sample data, the target output vector is defined as O k =[o k,1,o k,2,..., o k,k 1,o k,k,o k,k+1,..., o k,l ], where o k,j =0,j k; o k,j =1,j = k. LetO p =[o p,1,o p,2,..., o p,l ] be the actual training output vector of neural network, the distance between O p and O k is expressed as d pk = l (o p,i o k,i ) 2. (8) i=1

4 102 LI Ru-qiang, CHEN Jin and WU Xing The membership of the training output vector O p in class c k, lying in the range [0,1], is defined as μ k (O p )=1/(1 + (d pk /f d ) fe ), (9) where positive f d and f e are adjustable fuzziness factors. Fig.1 Structure of ANNs Fig.2 Fuzzy membership function 2.2 Crude rules acquisition Let n-dimensional input vector I t =[F t,1,f t,2,,f t,n ]bethetth input vector out of m input vectors, then the m n-dimensional input vectors form an m n input matrix. For the data of the ith column vector of matrix F i, which characterizes the ith diagnostic feature, order them asendingly. Let M m,i, F max,i, F min,i be the mean, maximum and minimum values of this column, respectively, and M l,i, M h,i the mean values of data located in the ranges [F min,i,c m,i ) and (C m,i,f max,i ], respectively, then the centers and radii of the three linguistic property sets of the feature F i are defined as C m,i = M m,i, C l,i = M m,i, C h,i = M h,i, λ l,i = C m,i C l,i, λ h,i = C h,i C m,i, λ m,i =(C h,i C m,i )/2. (10) Then an m 3n fuzzy matrix is presented after calculating the three segment fuzzy values of input vectors using Eqs.(7) and (10). Here we make use of the m 3n fuzzy matrix and the m fault values to acquire rules based on RST. Let the fault number and the number of vectors of the fault type c k be N F, k m respectively, then the k m vectors come into being a k m 3n fuzzy matrix. Threshold T ck is

5 Fault Diagnosis of Rotating Machinery using Fuzzy Neural Network 103 utilized to discretize the matrix, that is, when an item of the matrix is more than T ck, replace it with 1, otherwise 0, thus a k m 3n discrete matrix is obtained. Let d k be the value of the fault type c k, then if we take those k m 3n-dimensional discrete values as the ones of condition attributes of DT and those k m fault values d k as the ones of decision attributes of DT, then we acquire decision table T k. It must be noted that the number of objects in T k is k m and the values of all decision attributes are same. For decision table T k,letc k be condition attributes; then we obtain the equivalent class of objects [x i ] ck,1 i k m. If x m (1 m k m )satisfy the formulation max(card([x m ] Ck )), where card( ) refers to the cardinality of a set and max( ) means to get maximum, we select object x m as a representative of decision table T k. For the training sample of every fault type, we can get its representative object of DT in the same way as described above. Then we put those objects in one DT, and this DT is the very DT from which we will extract rules. Let C be the condition attributes of DT and {d k } the decision attributes, the heuristic reduction method, as described in Ref.[10], of RST is used to reduce DT and the reduced decision table {T r } is obtained. From this DT, the rules are acquired and their formulation is described by Eq.(3), and at the same time we calculate the dependency factors and antecedent coverage factors of the rules according to Eqs.(4), (5) and (6). 2.3 Construction and configuration of ANNs We configure the ANN shown in figure 1 as follows: The neuron numbers in the input layer and output layer equal to the one of diagnostic features and faults, respectively. The function of the IIth layer is to fuzzicate the values transferred from the Ith layer by high, medium and low three fuzzy segments. The neuron number of the IIIth layer is equal to the one of the rules and the neuron number of the IVth layer is same as that of the Vth layer, and equals to the count of faults. We use the connecting initial weights between the second and third layer to describe the antecedent parts of the rules and that between the third and fourth layer to describe the consequential parts of the rules. Let use the ith neuron of the third layer to characterize the ith rule. The output of the ith neuron of the third layer exclusively connects to one neuron in the fourth layer, this connection means the consequential part of this rule R i, and the connection initial weight is the dependency factor γ i. In the second layer, only those attributes appearing in the rule R i respectively connect to the neurons of the third layer, with the condition coverage factors being their connection initial weights. In DT, if the discrete value of one certain attribute b i j Bi is 1, then the connection weight between the connection of the neuron corresponding to this attribute and the ith neuron of the third layer is preset O i (B i,b i j ), otherwise, Oi (B i,b i j ), 1 i m, 1 j n. Other connection weights of neurons between the second and third layer, those between the third and fourth layer are initialized as random values ranged [0,0.01]. Thus, we complete the configuration of the ANN, which is called KBFNN, based on the rules and factors from RST. 2.4 Optimization of adjustable fuzziness factors GA [8] is chosen to optimize the parameters, f d and f e, of the outputs of KBFNN. An 8-bit string encodes the above parameters, respectively. Thus the string length is 16 for an individual. We select the number of initial population, crossover probability for single point crossover and mutation probability as 50, 0.8, and 0.01, respectively. The fitness function is obtained through Eq.(9). 3 Experiment and Results 3.1 Fault sample acquisition The typical faults of rotating machinery, such as unbalance, radial rubbing, oil whirl and rotor crack, are considered in our problem. Here, we also take normal condition as a fault for convenient description. The fault samples are obtained by simulating corresponding faults on Bently Rotor Kit at certain constant speeds. In horizontal direction and vertical directions,

6 104 LI Ru-qiang, CHEN Jin and WU Xing respectively, 100, 100, 240, 160 and 200 groups of data are obtained for the faults of normal, unbalance, radial rubbing, oil whirl and rotor crack, respectively, with sample number of 4096 in one group and sample frequency of khz. For every fault sample, half are chosen randomly as training samples and the alternative as test samples. 3.2 Selection and acquisition of symptom It is the investigative result that the frequency spectrum of vibration signals reflects sensitively the typical faults of rotating machines. So symptoms from frequency spectrum have widely applied for fault diagnosis. The frequency spectrum includes amplitude spectrum and power spectrum and both of them have been utilized in fault diagnosis. Here we take into account the symptoms from amplitude spectrum and by considering the symptom table of Sawyer [12], select eleven features as symptom of fault diagnosis. We firstly sum up all the amplitudes of each frequency range, denoting S i,i =1, 2,, 11; then the sum of all amplitudes ranged from 0 to 10 time frequency is calculated as S A ; finally the ratio S R = S i /S A,i =1, 2,, 11 and be obtained, denoting one symptom of this frequency range. Of course, all the considerations above are based on certain frequency resolution according to sample frequency and sample data length. Table 1 gives a set of symptoms of training sample about above five faults, where each column is one symptom corresponding to the frequency range of the column. Figure 3 illustrates the relationship of amplitude sum of the eleven frequency ranges and we can notice that the eleven points are not separable linearly. Fig.3 Table 1 Relationship of a set of symptoms about five faults A set of symptoms of training sample of above five faults fr fr 0.5 fr fr 1 fr 1.5 fr Normal Unbalance Radial rubbing Oil whirl Rotor crack fr 3fr 3 5fr oddfr 5 10 fr Normal Unbalance Radial rubbing Oil whirl Rotor crack

7 Fault Diagnosis of Rotating Machinery using Fuzzy Neural Network 105 Table 2 Decision table of input feature after discretization Normal Unbalance Radial rubbing Oil whirl Rotor crack Normal Unbalance Radial rubbing Oil whirl Rotor crack Rules acquisition based on RST and construction of an ANN For the training sample of every fault, we obtain the representative object by setting T ck, 1 k 5, 0.7. Because there are four different representative objects for the fault type unbalance, we get four distinct DTs. One of them is shown in Table 2. Reduce the DT of Table 2 employing heuristic reduction method and we obtain the reduced DT as shown in Table 3. The indiscernibility functions of individual objects in DT are defined as Table 3 Reduced decision table (1)L 1 (10)L 4 (17)M ( 6) {d i } x x x x x f 1 (x 1, {d 1 })=L 1 L 4, (11) f 2 (x 2, {d 2 })=L 4 M 6, (12) f 3 (x 3, {d 3 })=M 6 L 1 + M 6 L 4, (13) f 4 (x 4, {d 4 })=M 6 L 1 + M 6 L 4, (14) f 5 (x 5, {d 5 })=L 1 M 6. (15) For objects x 3 and x 4 we get two rules, respectively, f1 3 (x 3, {d 3 })=M 6 L 1, (16) f2 3 (x 3, {d 3 })=M 6 L 4, (17) f1 4 (x 4, {d 4 })=M 6 L 1, (18) f2 4 (x 4, {d 4 })=M 6 L 4. (19) For Eqs.(11), (12), (15) (19), after calculating the dependency factors and antecedent coverage factors of the rules, we obtain the following: γ 1 =0.2, O 1 (L 1 L 4,L 1 ) = 0.1, O 1 (L 1 L 4,L 4 ) = 0.1, γ 2 =0.6, O 2 (L 4 M 6,L 4 )= 0.3, O 2 (L 4 M 6,L 6 )= 0.3, γ1 3=0.6, O3 1 (M 6 L 1,M 6 )= 0.3, O1 3(M 6 L 1,L 1 )= 0.3, γ2=0.6, 3 O2(M 3 6 L 4,M 6 )= 0.3, O2(M 3 6 L 4,L 4 )= 0.3, γ1=0.6, 4 O1(M 4 6 L 1,M 6 )= 0.3, O1(M 4 6 L 1,L 1 )=0.3, γ2 4=0.6, O4 2 (M 6 L 4,M 6 )= 0.3, O2 4(M 6 L 4,L 4 )=0.3, γ 5 =0.6, O 5 (L 1 M 6,L 1 )=0.3, O 5 (L 1 M 6,M 6 )= 0.3. Then, we can construct and configure our ANN based on the number of the rules, dependency factors and antecedent coverage factors of the rules, as shown in Fig.4.

8 106 LI Ru-qiang, CHEN Jin and WU Xing Fig.4 Construction and configuration of ANNs 3.4 Experimental result We compare training time and classification ability of our KBFNN with that of the general back propagation FNNs (GFNNs), which have same structure as one shown in Fig.1. Table 4 lists the classification level for all the faults mentioned above with GFNNs and KBFNNs of different structures. In Table 5, we give the diagnostic level for every fault with KBFNN of structure with initial knowledge obtained by attributes (L 1,L 4,M 6 ) and GFNNs of structure In this table, the training time refers to the time for training process in running the procedure developed by c++ written by us on the computer of Pentium IV/2.4GHz/256MB. Table 4 shows the comparison of classification level for five faults using KBFNNs and GFNNs with different structures. For the GFNNs only, the fault diagnostic level of GFNNs with structure of is higher than those of others. For the KBFNNs only, the identification Table 4 Comparison of classification level for five faults using KBFNNs and GFNNs with different structures Epoch GFNNs(%) KBFNNs(%) times L 1,L 2,L 5 L 1,L 2,L 4 L 1,L 4,M 6 M 1,L 2,L

9 Fault Diagnosis of Rotating Machinery using Fuzzy Neural Network 107 level of KBFNNs with initial knowledge created by attributes (L 1,L 4,M 6) is comparatively higher and shows more stability under different epoch times. With the same epoch time, the performances of KBFNNs with different initial knowledge are generally better than those of GFNNs with distinct structures. From Table 5 we can conclude that, KBFNN has remarkable higher classification level than GFNNs in diagnosing the faults of normal and unbalance, compared with other faults. For the fault of oil whirl, KBFNN has excellent capability to identify it, whilst only on some situations can GFNNs do. It is obvious that KBFNNs are superior to GFNNs in the performance of classification under the same training epoch times for different faults. In general, KBFNNs have 3% higher classification level than GFNNs, with the highest classification level of 13.75%; Besides, there is shorter training time for KBFNNs comparatively. Table 5 Comparison of performance using KBFNNs and GFNNs KBFNNs( )(%) GFNNs( )(%) Epoch times Normal Unbalance Radial rubbing Oil whirl Rotor crack Level Train time(s) Conclusion One kind of KBFNNs, based on hybrid techniques of RST, FST, ANNs and GA, are proposed, which employ the advantages of those techniques while eliminate their disadvantages. The KBFNNs were designed to diagnose five typical faults, including normal, unbalance, radial rubbing, oil whirl and rotor crack of rotating machinery. Diagnostic classification results show that, from the training time aspect, KBFNNs use shorter time than GFNNs, and from the aspect of classification capability, they have higher identification level, compared with GFNNs. At the same time the KBFNNs make the design of ANNs no blindness, which depend on the rules obtained based on RST. References [1] Renwick J T. Vibration analysis a proven technique as a predictive maintenance tool [J]. IEEE Transactions on Industry Application, 1985, 21: [2] Toshio T, Tornoya N, Peng C. Failure detection and diagnosis of rotating machinery by orthogonal expansion of density function of vibration signal [C]. In: Proceedings. EcoDesign 99: First International Symposium On Environmentally Conscious Design and Inverse Manufacturing. Tokyo, Japan, 1999, [3] Yao Hongxing, Zhao Lindu, Sheng Zhaohan. The application of fault diagnosis based on fuzzy neural networks [J]. Turbing Technology, 2000, 42(5): (in Chinese). [4] Pawlak Z. Rough sets[j]. International Journal of Information and Computer Science, 1982,11(5) [5] Ou Jian, Sun Caixin, Bi Weimin, et al. A steam turbine-generator vibration fault diagnosis methodbasedonroughset[c].in: Proceedings of IEEE International Conference on Power System Technology. Kunming, China, 2002, 3,

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