Average Incremenral Correlarion Analysis Model and Irs Applicarion in Faulr Diagnosis

Liu Jiefang（劉解放），Liu Sifeng（劉思峰），Wu Lifeng（吳利豐），F(xiàn)ang Zhigeng（方志耕）

1.School of Mathematical Science，Henan Institute of Science and Technology，Xinxiang 453003，P.R.China；2.College of Economics and Management，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China

（Received 13 December 2014；revised 4 March 2015；accepted 10 March 2015）

Average Incremenral Correlarion Analysis Model and Irs Applicarion in Faulr Diagnosis

Liu Jiefang（劉解放）1，2*，Liu Sifeng（劉思峰）2，Wu Lifeng（吳利豐）2，F(xiàn)ang Zhigeng（方志耕）2

1.School of Mathematical Science，Henan Institute of Science and Technology，Xinxiang 453003，P.R.China；2.College of Economics and Management，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China

（Received 13 December 2014；revised 4 March 2015；accepted 10 March 2015）

Absrracr：The concept of average incremental correlation degree is put forward.It has been proved that the average incremental correlation model has such properties as parallelism，consistency，affine，affine transformation isotonicity and interference factors independence，and it will not lead to changes of the sequence order relation because of the data transformation.Therefore，the new model keeps good stability.Finally，the incremental average correlation model is applied to failure model analysis of equipment，and an ideal diagnostic effect is obtained.

average increment；grey correlation analysis；fault diagnosis；equipment

0 Inrroducrion

As an important part of grey system theory，the grey correlation analysis is the basis of grey clustering，grey decision and grey prediction［1-5］. Since the concept of grey correlation was proposed，it has been widely used in different domain［6-20］.The basic principle of grey correlation analysis is that the degree of correlation between various factors can be compared through the geometric relationships.Grey correlation analysis is an important technical method to complex system modeling that the influential factors are unclear. Grey correlation analysis can give a quantitative order relation to grey large system，and it can be used to determine the influential factors of uncertain systems.The related theory and knowledge of grey correlation analysis are reviewed in this paper.The concept of average incremental correlation degree is put forward，and its properties are studied.It is proved that it meet with the parallelism，uniformity，affine，order preserving after affine transformation and interference factors independence.Finally，the average incremental correlation model is applied to fault diagnosis of equipment，and good result is achieved.

1 Four Axioms of Grey Correlarion

Assume that the system behavior characteristic sequence is

The system behavior characteristic compared sequence is

Assumeγ（x0（k），xi（k））is a real numbers，if Satisfies

（1）Normative

（2）Integrality

For Xi，Xj∈X=2｝，there is the following result，namely

（4）Proximity

2 Quanrirarive Evaluarion of Gray Correlarion Model

In the grey correlation model，parallelism，uniformity and isotonicity of grey correlation model also have the vital significance besides the grey incidence axioms.Whether the grey correlation model has the above characters often influence the result of the grey correlation analysis.

Definirion 1［5］（Parallelism） Assume the reference sequence is X0=（x0（1），x0（2），…，x0（n）），and the compared sequence is Xi=（xi（1），xi（2），…，xi（n）），when xi（k）=x0（k）+ c，c=const，k=1，2，…，n，the sequences of X0and Xiare parallelism.If the correlation degree satisfies

the correlation degree satisfies the parallelism.

Definirion 2［5］（Uniformity） Assume the reference sequence is X0=（x0（1），x0（2），…，x0（n）），and the compared sequence is Xi=（xi（1），xi（2），…，xi（n）），when xi（k）=αx0（k），α=const，k=1，2，…，n，If the correlation degree satisfies the correlation degree satisfies the uniformity.

Definirion 3［5］（Isotonicity） Assumeγf（X0，Xi）andγ（X0，Xi）are the correlation degree of Xiunder data transform f and original data X0，respectively.For any compared sequence Xt，Xl，whenγf（X0，Xt）＜γf（X0，Xl），there is

the correlation degree satisfies the isotonicity.

Xie et al.［21］has proved that among the existing grey correlation degree，there is seldom one that satisfied the parallelism，uniformity and isotonicity.Therefore，when grey relation analysis is used，there will appear the phenomenon of reverse order due to the data transformation or the interference factors.This affects the effect of the grey correlation analysis.Hence，a new grey correlation degree model is built，which can satisfy the parallelism，uniformity and isotonicity，and their properties are studied in this paper.Finally，the new grey correlation degree model is applied to the weapon system of engine fault diagnosis，and the satisfactory results are achieved.

3 Consrrucrion of Average Incremenral Correlarion Degree Model

Definirion 4Assume that X0and Xiare the same as Eqs.（1，2）.

Thenγ（X0，Xi）is called average incremental correlation degree of X0and Xi.

Theorem 1The average incremental correlation degree satisfies the normative，even symmetry and the proximity axiom，but does not satisfy integrity.

Proof

（1）Normative（3）Proximity Since

Then the smaller the distance of f0（k）and fi（k）is，the greaterγ（x0（k），xi（k））is.

（4）Integrity For

4 Characrers of Average Incremenral Correlarion

Proposirion 1The average incremental correlation degree model satisfies the parallelism

ProofAssume the referential sequence of system is S0=（s0（1），s0（2），…，s0（n）），and the compared sequence is Si=（s0（1）+c，s0（2）+c，…，s0（n）+c）. So the average incremental correlation degree model satisfies the parallelism.

Proposirion 2The average incremental correlation degree model satisfies the uniformity.

ProofAssume the referential sequence of system is S0=（s0（1），s0（2），…，s0（n）），and the compared sequence is Si=（s0（1）+c，s0（2）+c，…，s0（n）+c）.

So the average incremental correlation degree model satisfies the uniformity.

Proposirion 3The average incremental correlation degree model satisfies the affinity.

ProofAssume the referential sequence of system is S0=（s0（1），s0（2），…，s0（n）），and the compared sequence is Si=（s0（1）+c，s0（2）+c，…，s0（n）+c）. So the average incremental correlation degree model satisfies the affinity.

Proposirion 4The average incremental correlation degree model satisfies the characters of parallel transform order preserving.

ProofAssume that

Therefore，the average incremental correlation degree model satisfies the characters of parallel transform order preserving.

Proposirion 5The average incremental correlation degree model satisfies the characters of multiplication transformation order preserving.

ProofAssume that

Therefore，the average incremental correlation degree model satisfies the characters of multiplication transformation order preserving.

Proposirion 6The average incremental correlation degree model satisfies the characters of affine transformation order preserving.

ProofAssume that

Proposirion 7The average incremental correlation degree model satisfies interference factors independence.

ProofFrom the definition of X0and Xi，it can be seen thatγ（X0，Xi）only has relation with X0and Xibut has nothing to do with other factors.Therefore，the average incremental correlation degree model satisfies interference factors independence.

Therefore，the average incremental correlation degree model satisfies the characters of affine transformation order preserving.

Through the description of the characters of the average incremental correlation degree model，we can see that the new correlation model fully excavate the information that the system contains from the perspective of system information change.And it satisfies not only the parallelism，consistency，affine，but also the characters of parallel transform order preserving，the characters of multiplication transformation order preserving，and the characters of affine transformation order preserving.As a result，in the process of modeling with the average incremental correlation degree model，the order correlation will not change because of data transformation.Therefore，the new correlation degree model has goodstability compared with the previous model，and it is suitable to determine the influential factors for a kind of system that need to do affine transformation frequently.

5 Case Srudy

In a certain type of weapon system，the engine is the most important components.Therefore，the normal operation of the engine plays an important role for guaranteeing the reliability of the whole system.When the engine occur failure，it need fast diagnosis for the causes of the problem for the security requirements of combat readiness.The cause of engine failure usually has the following reasons：（1）The cooling water temperature is too high；（2）The torque sensor is opencircuit；（3）The one of the cylinder does not spray；（4）The temperature sensor is damaged；（5）The air flow sensor is damaged.The influence factors are：the torque signal 1，temperature signal 2，the throttle opening signal 3，air flow rate signal 4，rotational speed，and pulse width，etc.For a certain type of weapon system engine，its failure mode of standard signal sequence is obtained as shown in Table 1.In the table，U stands for the engine that participates in the experiment and the symbol V the unit of acquisition sensor signal voltage.Among the table，the failure mode 0 means that the system is normal.

Suppose that there exists a data sequence of a certain type of weapon system of engine fault as shown in Table 2.In the table，S stands for the engine that needs for fault diagnosis.It is required to judge the failure mode of each sample. The average incremental correlation model is used for the diagnosis of fault modes in the below.

Due to the factors of different dimension，in order to calculate the data unity，the data need to make a normalized processing.There are many normalized processing methods.In essence，these methods are a kind of affine transformation.Because the average incremental correlation model satisfies the affine transformation order preservation，the correlation order before and after transformation remains the same.To make full use of the sequence of the maximum and the minimum，the method of interval value treatment normalization is adopted.

Table 1 Srandard parrern sample sequence of cerrain rype of weapon sysrem of engine

After the normalized sequences of each sample are obtained，the average incremental correlation model is used to calculate the grey correlation degree and their average.The result is shown in Table 3，and the specific steps are omitted due to space limitations.In Table 3，Pistands for the degree of association.

Table 3 Average incremenral correlarion of resr sample and srandard sample

The specific results of fault diagnosis is shown in Table 4.

Table 4 Faulr diagnosis based on new grey correlarion analysis

I

t can be seen from Table 4 that the accuracy rate reaches 100%based on the average incremental correlation model.

6 Conclusions

The average incremental correlation model is constructed in this paper based on the incremental data difference degree，and its characters are analyzed.It is proved that the new correlation model not only satisfies parallelism，consistency，and affinity but also parallel transformation order preserving，multiplication transformation order preserving，and affine transformation order preserving.Then，the averageincremental grey correlation model will not appear reversed order because of the data affine transformation，and maintain stability.Finally，the average incremental grey correlation model is applied to the engine fault recognition of a certain type of weapon system，and a satisfactory recognition effect is obtained. And the effectiveness and practicability of the new model are verified.

Acknowledgemenrs

This work was supported by Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme（Grant No.FP7-PIIF-GA-2013-629051），the National Natural Science Foundation of China（No.91324003），and Social Science Foundation of the China（10zd&014，12AZD102）.

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（Executive Editor：Xu Chengting）

N941.5Documenr code：AArricle ID：1005-1120（2015）05-0541-08

*Corresponding aurhor：Liu Jiefang，Lecturer，E-mail：liujf101@126.com.

How ro cire rhis arricle：Liu Jiefang，Liu Sifeng，Wu Lifeng，et al.Average incremental correlation analysis model and its application in fault diagnosis［J］.Trans.Nanjing U.Aero.Astro.，2015，32（5）：541-548.

http：//dx.doi.org/10.16356/j.1005-1120.2015.05.541