Principal Component Analysis

Component Analysis

(1. State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, Liaoning Province, China PR; 2.College of Information Science and Engineering , Northeastern University , Shenyang , Liaoning 110819)

Abstract: A process monitoring approach based on improved multi-scale dynamic principal component analysis (IMSDPCA) is proposed to handle the multi-scale and dynamic characteristics of industrial process data. Augmented data matrix is structured by using the “time lag shift” method, and then multi-scale measurement characteristics of the wavelet is used to analyze the measuring variable augmented matrix in multiplicity scale. So the problem that the measuring variables have the dynamic and the multi-scale characteristics of industrial process data is not only resolved, but the shortcomings that the number of principal components is too much is overcome, on the basis of this, the monitoring index based on T2 and SPE is improved using . The simulation results of the TE chemical process faults simulation and the strip breaking fault of rolling process based on MSDPCA monitoring method show that the IMSDPCA algorithm is feasible and effective, the process monitoring performance is improved compare with PCA and DPCA approach.

Key words: Process monitoring, DPCA, Wavelet transform, TE process, Fault diagnosis, Strip breaking fault of rolling process

1. Introduction

In order to ensure the production process running safety and produce high quality products,all kinds of abnormal conditions which may affect product quality and safety equipment needed for process monitoring . A great number of highly correlated variables measured exist in the industrial production process,these variable values sampled at each time instant contains whether production process is normal, the product quality is qualified and other information.PCA as the representative for the Multivariate statistical method ,using the method of dimension reduction to deal with process measurement data,which does not need the accurate mathematical model and effectively eliminate the redundant information that exist in process data,Also, PCA reduced the dimension of the data.it has been widely used in Condition monitoring and fault diagnosis during industrial process[1].

As a result of the existence of various random noise and interference,in the actual collected process data ,especially the closed-loop control will spread the effect of interference to each input and output variable,causing process variable contains not only the strong mutual correlation also contains a self correlation,which Called dynamic.The PCA method does not consider the characteristics of dynamic data,The main element cannot reflect the real change of characteristic data[2, 3].In order to overcome these shortcomings, Ku and others first proposed the dynamic PCA method,this method constitute the new augmented data matrix by ntroducing the observation values of process variables, then the PCA modeling of the augmented data matrix,thus the sequence correlation of the data can be extracted.After that, the method of fault diagnosis based on DPCA has been widely researched and used[4, 5].But the DPCA method make the first S moments observations extended to the augmented matrix construction, greatly increased the number of the main

element ,and reduced the computational efficiency.

In addition, the industrial process data is multi scale In essence，which mainly embodies as follow:（1）Events Occurring at different location ,With different local characteristics in time and frequency；（2）The energy spectrum of the stochastic process changes with the time or frequency；（3）Variables measured with different sampling rate or variable contains missing data，Nearly all the data of actual process is multi scale.In order to monitor the fault on different scales,Many scholars have put forward a method of fault detection and diagnosis that based on the combination of multiscale analysis and multivariate statistics [6].Documents [7]First proposed the multi-scale fault detection method of wavelet PCA ,the method is based on PCA decomposition of wavelet coefficient on every time scale and then integrated in one scale, thus realize the fault detection on different scales.

This article combines the advantages of wavelet transform and DPCA method,Put forward a process monitoring method based on improved multiple dynamic principal component analysis (IMSDPCA).

The method uses an improved threshold function wavelet transform algorithm to decompose the constructed dynamic augmented data matrix ,reconstructed the augmented matrix,considered the dynamic and multi scales of the data of industrial field ；New data matrix changed the internal structure of the augmented matrix of old data,Also the main element vector and the number of principal element changed.；Significantly improved Tand

2SPE monitoring metrics,

2. The wavelet transformation theory and improved DPCA method

2.1 wavelet transformation

Wavelet transform is a time-frequency localization analysis method that time windows and frequency windows can be changed.Because of its good time-frequency locality, which can effectively extract information from signal,to analysis functions or signals from Multi scale by translation and dilation[8].

Set M dimensional model：

X(t)=f(t)+(t), t=1,2,......,n. (1)

TheX(t)、f(t)&(t)are1mdimension vector，and(t)is a centralized white gaussian noise ,covariance is unknown.f(t) is useful information need to be retained.The elements ofX(t) are as follows,

Among them：1im：

Xi(t)=fi(t)+ei(t),t=1,2,......,n.

Document [9] proposed a simple method to estimate the wavelet coefficientwj,k, The threshold function of which is widely used.But in the hard threshold function, wj,kis not continuous in ,Signal reconstructed by wj,k may have some oscillation;the wj,k estimated by soft threshold function ,although the whole succession is good,but if

wj,k,there is always some deviation between the wj,k and wj,k,directly affects the degree of

approximation between the reconstructive information and the rea information.

This paper adopts a new threshold function[10]：

wj,ksgn(wj,k)wj,kαλ+02λα(1-α),exp(wj,k),wj,kwj,k (3)

in which，0,1。When  is equal to 0 , the above formula is hard threshold function;When  is equal to 1, the above formula is soft threshold function.For a general valueα0,1，the wavelet coefficient wj,kestimated by this threshold function among the results estimated by hard threshold function and soft threshold function.Through adjusting the size of the parameters ,can appropriatly reduce the deviation of the original signal,it can also overcome the disadvantages of soft, hard threshold estimation to some extent. In this article, the algorithm to determine the wavelet threshold λas follows:

λ=σ2lg(n) (4)

In practice，the standard deviation of the noise is unknown，Usually we need to estimate

，we can get the σ from the following formula：

ˆ=median(w1,k)/0.6745，kHH1σ (5)

In which,w1,kis the first level of wavelet decomposition coefficients.

2.2 The improved DPCA method

DPCA introduced time delay data augmented data matrixIn in the analysis of data. improved the traditional PCA method and made up the shortage of traditional PCA method in capturing data on dynamic relationship,this paper adopts the method automatically determine lag S that described in [11] document.set up the augmented matrixas follows：

xtTTxXD(S)t1TxtSnxtT1xtT2xtTSn1xtTSxtTS1xtTn

The Modeling steps of DPCA are as follows:

(1)Given a training aggregate as

X[x1T,xT,2xT]n, where n is the observed value, m as process

variables.Then standardized the training aggregate.

(2)Determine the lag S,extend the observed value of the first S moments to the constructed augmented matrix.

(3)Work out the sample covariance matrix of the augmented matrix and decompose characteristic value.

CXE[(XDE(XD))(XDE(XD))T]VDVT. (7)

In which,Dcontains decreasing nonnegative eigenvalue which along the main diagonal . (12m)，VTV1，and vi called the load vector which is the eigenvector corresponding to

the eigenvaluei.

(4)Reduced-order

The training data after feature value decomposed,the part that corresponds to the larger singular values depict the most System state changes generated in the process.However, the part that corresponds to the smaller singular values depict the random noise.

System changes can be decoupled from the random change by appropriate to determine the number of load vector remained in the PCA model, Thus the two types of changes can be monitored separately,to select the main element number a described as follows:

：

CPV(a)i/i100%CLi1i1an (8)

In which,CL is the set value.

(5)Set up the dynamic PCA model

In order to obtain the the best data variation,choice load matrixPRma，get the dynamic principal component model：

TXDP (9)

(6)Improve monitoring index

When using DPCA for fault monitoring,the traditional statistical index have two：

2T2t1tTtTHotelling-&Square prediction error（Squared prediction error，SPE）。In which,( is

2the main element，is a diagonal matrix made up by the first a eigenvalues )，Because of

T2SPEXX。

and SPE statistics reflect the data characteristics of two complementary

space,the proportion of them is not completely consistent in actual monitoring,

，The former more reflected the information that variables contained in the projected scale，however,the latter focused the information about the variables correlation[12].

In order to better balance the two statistics and simplify the monitoring process in the premise of not affecting the performance of the fault detection,this paper cites reconstruction statistics proposed by Yue and Qin[13] as the monitoring index of DPCA,

The monitoring index is mainly composed of

T2and SPE statistics weighted,

the calculation of reconstruction statistic is as follows：

SPE(X)T2(X)RCS=φ+2XTX2δαχα (10)

Among which,

PΛ-1PT2χαIPPT2δα。

using the approximate distribution to calculate control limits of reconstruction statistics，

reconstruction statistics approximately obey the following distribution:

φ=XTXgχ2(h) (11)

tr(S)2gtr(S)Among which,

，

[tr(S)]2htr(S)2，and,

tr(S)d2χαmΣi=d+1λi2δα，

tr(S)2d4χαmΣi=d+1λi24δα。

2γαWe can obtain the Control limits of the upper limit100% after get g and h by

calculation,

2γαWhen the reconstruction statistics value of sampling point is larger than ，We think

the failure occurred,

3. Process Monitoring Method Based On IMSDPCA

This paper presents a process monitoring method based on IMSDPCA.The method first extend measurement data matrix to the augmented data matrix,then using the multiscale character of wavelet multiscale analysis of the augmented matrix,It can also solve the problem that measured variables both dynamic and multi scale,But also overcome the shortcomings that DPCA method select the number of principal components too much,Improved the calculation efficiency of DPCA method,

And introduce the reconstruction of statistics as dynamic multiscale process state monitoring index.

Process monitoring method based on IMSDPCA as shown in Figure 1 ,

The specific steps of off-line training and on-line monitoring are as follows.

原始数据初始化 X确定S构建增广矩阵离线建模过程小波分解改进阈值函数的小波变换小波逆变换重构数据XPCA建模求出主元模型和RCS控制限采集新样本x并初始化求出其动态主元和RCS统计量2?Yes在线监测过程No报警，故障识别

Figure 1.The block diagram of monitoring method based on IMSDPCA process

(1)off-line training

The training steps of wavelet transform and improved IMSDPCA method are as follows:

Step 1：given a training collection

X[x1T,xT,2xT]xiRmn,.among which,”N” is observed

values.”M” is process variables。standardized training collection。

Step2：Determine the lag S,extend the observed value of the first S moments to the constructed augmented matrix.

Step3：Using the method of wavelet transform to decompose the augmented matrix in different scales, seek out wavelet coefficient.

Step4：reconstruct the augmented matrix according to the decomposed wavelet

coefficient.

Step5：work out the covariance matrix of the augmented matrix, principal component scores

and choose the suitable main element numbers , establish dynamic main element model,TXDP,in which,

Pis load matrix after Dimension reducted.

Step6：calculat reconstruction statistics and control limits according to the formula (10) and (11)

(2)on-line monitoring

The on-line monitoring method based on IMSDPCA are as follows:

Step1：standardized the sampled dataXnew of one moment,such as moment k.

Step2：construct augmented matrix

newXH.

Step3：Using the method of wavelet transform to decompose the new augmented matrix in different scales, seek out wavelet coefficient.

Step4：calculate the dynamic principal component and RCS statistics

。

Step5：judge whether RCS statistic is overrun, if not，there is no failure；Otherwise,there have something wrong happened。

Step6： Diagnose the variables which caused failure according to the contribution of FIG.

。

4. Simulation research

This section will applied the method based on the IMSDPCA to the fault monitoring of typical fault of TE chemical process and belt breakage of rolling process respectively,to verify the validity of the method.

4.1Application Research Based on IMSDPCA method in the TE process monitoring.In order to verify the performance of the fault diagnosis method based on IMSDPCA，In this paper, using the typical faults of TE process to study on the method of simulation.TE process simulation model is internationally recognized which proposed by employees from Eastman Chemical Company Downs J and Vogel E F ,the purpose is to provide a practical industrial process for the evaluation of process control and monitoring method,Detailed description of the process flow diagram and process can be find in document [14],fifty-two process variables was contained in this process,also twenty-one pre-set fault was included.The training set and test set data containing

m52 variables

(Do not include agitation speed，because there is no control of it)。 There are 480 observations of the training set are actually used to construction process monitoring model.In

test set there are 960 observations,and introduced step fault 1 in these observations(A/C feed flow ratio changes，content of component B remained constant.(flow4))。In this paper,using fault 1as an example,compared the monitoring effect of process monitoring method based on IMSDPCA with condition monitoring method based on PCA and DPCA..The monitoring results of fault 1 based on PCA and DPCA as shown in Figure 2 and figure 3.

Figure2. The monitoring results of T2&SPE based on PCA method

Figure3.The monitoring results of T2&SPE based on DPCA method

As can be seen from the chart，although the monitoring effect of SPE statistics based on DPCA and PCA method is very good,

，but the SPE statistic based on DPCA method is doing better in large amplitude exceeds the control limits aspects.

This indicates that the DPCA method considering variable auto correlation played a positive role

。 T and SPE monitoring index of DPCA monitoring method are beyond the control

2limits in the 163th sampling time and the 1th sampling time , T and SPE monitoring

2index of PCA monitoring method are beyond the control limits at the same point 1th sampling time.The process monitoring effect of the two methods approximately the same. but the number of principal components of DPCA (55) are increased nearly one times of the PCA number of principal components (27),in the aspectaspect of extracting principal component .The calculation efficiency of offline modeling based on DPCA is lower than the PCA method, monitoring results of IMSDPCA based process monitoring method for fault 1 as shown in Figure 4。

Figure 4 RCS statistics monitoring effect based on IMSDPCA

From Figure 4 we can see the reconstructed statistics monitoring map have two false points before the 160th sampling points,same as the false alarm rate of the TStatistics based

2on DPCA method,

But compared with SPE statistics less than 6 false alarm points.and the reconstructed Statistics based on IMSDPCA method are beyond the control limits at the 161th sampling times,with better sensitivity to faults than T&SPE monitoring index of DPCA method.in the

2aspect of principal component extract,owing to the extraction of the main element number of IMSDPCA process monitoring method is 10, which greatly improved the calculate efficiency of modeling.the reconstructed monitoring index this paper cited can reflect the change of T

2and SPE, so，In process monitoring, only need to monitor the control chart of RCS,in the case of ensure the original monitoring performance saved a control chart,Reduced the amount of monitoring，simplified The complexity of the monitoring visualization.

In order to identified the cause lead to failure and compared the capability of fault identification of the method proposed in this paper and PCA, DPCA two methods

The chart of method based on PCA、based on IMSDPCA、 and based on DPCA contribut to fault 1 was given in figure 5.

(a) (b) (c)

Figure 5 The chart of method based on PCA、 IMSDPCA and DPCA contribut to fault 1

Which can be seen from figure 5 ，the Contribution figure based on PCA method not well gived the important variable affecting the fault produce,While the DPCA method based on improved a lot ,It also shows that the advantages of DPCA method of dynamic data processing .In the contribution figure based on IMSDPCA method, the variables1 and variables 44 made the greatest contribution

While variables1 is A feed rate in the process of measuring and variables 44 is A feed rate of control variables,The two is a direct variable affecting the fault 1 happened,So conclusions can be drawn：IMSDPCA based process monitoring method can accurately identify the variable that caused the fault mode .

4.2 The application research of process monitoring method based on IMSDPCA used in the rolling process of broken belt fault

In the process of rolling, rolling mill broken belt fault always occurred from time to time,rolling process will not be go smoothly once the fault zone occured.Even caused casualties and huge economic losses.The broken belt fault always keep a very short time,Therefore, it is an important preconditionis to monitor the relevant variables which caused broken fault zone with whole process

for ensure the normal operation of rolling .

This paper choosed 38 process variables to model training from the variables impacted the broken fault zone in rolling process.Extracted 38 process variables from the rolling data,Including looper current, speed, tension and so on.Collection 3000 sampling data for

offline modeling under normal conditions,made another group of belt broken fault data as a test set,1200 sampling data

Included , time interval every 20ms.

T2 and SPE monitoring effect chart of the process monitoring method based on DPCA for

broken fault zone in rolling process as shown in figure 6,RCS diagram and variable

contribution plot of process monitoring method based on IMSDPCA for broken fault zone as shown in figure 7.

What can be known from the comparison of figure 6 and figure 7 is that T and SPE

2monitoring index of DPCA monitoring method are beyond the control limits in the 1024th sampling time and the 1034th sampling time, RCS based on IMSDPCA beyond the control limits in the 1013th sampling time,and the monitoring index before the 1013th sampling times trends to beyond the control limits.This shows that the IMSDPCA based process monitoring method have a better sensitivity,

And the contribution map is more obvious identified variables 7 and variables 9 ,

While the variables 7 and variable 9 respectively stand for the tension and rolling speed of the strip steel,Obviously,the most important factors of the occurrence of the broken fault zone is the abnormal between the two,In the selection of principal component，DPCA based method selected 29 principal components,while the IMSDPCA based method slected 3，verified the effectiveness of the proposed method again.

Figure 6: T and SPE monitoring effect chart of the process monitoring method based on

2DPCA for broken fault zone

Figure 7: RCS diagram and contribution plot of process monitoring method based on IMSDPCA for broken fault zone

5. Conclusion

To overcome shortcomings that low computational efficiency of DPCA based method,this paper presents a method of fault diagnosis based on IMSDPCA,to efficiently monitor time related process data of rolling process.The method used wavelet transform algorithm to process the data of constructed augmented matrix,Greatly improved the computational efficiency of DPCA method

。Shown this method has better fault detection performance and computational efficiency

through the simulation research,by compared the fault detection performance of statistics shown the possibility of introduce reconstructed Statistics.

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