Online process monitoring for complex systems with dynamic weighted principal component analysis☆

Zhengshun Fei*,Kangling Liu

1 School of Automation and Electrical Engineering,Zhejiang University of Science and Technology,Hangzhou 310023,China

2 State Key Lab of Industrial Control Technology,Institute of Industrial Control Technology,College of Control Science and Engineering,Zhejiang University,Hangzhou 310027,China

1.Introduction

Advanced manufacturing systems rely on an efficient process monitoring to increase the quality,efficiency and reliability of existing technologies[1,2].Manufacturing process is usually highly complicated and lacks accurate models,which makes the model-based methods[3–5]unsuitable.However, floods of data can be obtained on-line through sensors embedded in the process.This situation facilitates the development of multivariate statistical process monitoring method based on principal component analysis(PCA)[6]that utilizes process data and requires no explicit process knowledge.PCA is widely used in many applications because of its advantage of handling high dimensional and correlated process variables[7–10].For process monitoring,PCA partitions the process data space into a principal component subspace and a residual subspace,and uses T2and Q statistics to monitor the two subspaces respectively.

On the other hand,manufacturing applications are generally dynamic processes and process variables exhibit auto-correlations because of controller feedback and disturbances.Here,auto-correlation means that current observation is correlated with previous ones.As a result,conventional multivariate statistical methods,which rely on assumptions that(1)the process is time invariant and(2)variables are serially uncorrelated,have the tendency to generate false alarms or missed detection[11].This mismatch suggests that a dynamic method analyzing serial correlations is needed[11–14].Some speech recognition approaches,such as hidden Markov model[15]and dynamic time warping[16],were developed for off-line diagnosis.These approaches rely heavily on known fault information,obviously,it is often not complete since we cannot ensure that all possible faults are pre-defined in complex systems.Ku et al.[11]proposed a dynamic PCA(DPCA)that constructs singular value decomposition on an augmented data matrix containing time lagged process variables,which increases the size of variable set and has difficulty in model interpretation[17–19].With the similar idea,some subspace methods based on canonical variate analysis[20]and consistent DPCA[17]were proposed.Bakshi[12]introduced a multi-scale PCA that integrates PCA with wavelet analysis,which is an effective tool to monitor auto-correlated observations without matrix augmentation.Multi-scale PCA first decomposes process data into several time-scales using the wavelet analysis and then establishes PCA on wavelet coefficients for different scales,and a moving window technique is used for online monitoring.A further analysis on multiscale PCA was provided by Misra et al.[14].Yoon[21]pointed out that MSPCA puts equal weights on different scales regardless of the scale contribution to overall process variation and then unreasonably increases the small contribution of high-frequency scales.Recently,Li and Qin[22]proposed a dynamic latent variable(DLV)model to extract auto-correlation and cross-correlations.In particular,some probability methods were developed for dynamic process monitoring[23–25].Choi[23]constructed a Gaussian mixture model based on PCA and discriminant analysis for representing the distribution underlying dynamic data.Li and Fang[24]proposed an increasing mapping based on hidden Markov model for large-scale dynamic processes.Zhu and Ge[25]extended hidden Markov model to characterize the timedomain dynamics.

Inspired by these approaches,we propose a new monitoring method called dynamic weighted PCA(DWPCA),with the advantages that it is dynamic data driven and can detect faults in an automatic manner.The proposed method designs a hybrid correlation structure that simultaneously containsauto-and cross-correlation information of processes.The design includes two tiers.The first tier is to use the PCA method to extract the cross-correlation structure among process data,expressed by independent components,and the second tier is to estimate the auto-correlation structure among the extracted components as autoregressive(AR)models.For online monitoring,we incorporate a weight approach into PCA.Actually,the weight approach is not new and has many applications such as correspondence search[26],face recognition[27]and process monitoring[28,29].To the best of our knowledge,the weight method developed on a two-tier hybrid correlation structure is new for process monitoring.In this work,we use the weight approach to give different weights on different directions of components based on their contributions to a fault.Assume that fault information is associated with online estimation errors of AR models,a weight function is defined based on estimation errors for each component to take emphasis on directions of components,and its essential is that the directions are given high weight values if they have large estimation errors.The weight values are automatically computed when a new observation becomes available.Then,the computed weights can be used to dynamically partition the process data space into two new subspaces,namely an important component subspace and a remaining component subspace,and two new statistics are calculated to monitor them,with similar motivations of conventional PCA monitoring.But the differences are that(1)the proposed method makes use of online process operating information to actively perform subspace partition,(2)two new statistics take both auto-and crosscorrelations into account while T2and Q statistics only consider crosscorrelations,and(3)the contributions of component directions of the proposed method are not at the same degree while those of PCA are with the same value of 1.

The rest of this work is organized as follows.The conventional PCA is introduced brie fly in Section 2.A simple process simulation is provided to illustrate problems of PCA monitoring based on T2and Q statistics.This gives rise to the motivations of DWPCA.In Section 3,DWPCA for process monitoring is detailed,including two new monitoring statistics.Tennessee Eastman process is employed to demonstrate the process monitoring performance of the proposed method in Section 4.The results show that the proposed method outperforms conventional PCA.Finally,Section 5 concludes the work.

2.Principal Component Analysis Monitoring

2.1.Principal component analysis

Suppose that a normal data setcollecting N samples of J variables is scaled to have zero means and unit variances.The principal component analysis(PCA)decomposition is developed as X=Hererepresents the i th component,and its directionand varianceeigenvector and eigenvalue of covariance matrixcomponents are in the order of variance decrease,i.e.λ1≥λ2≥,… ,≥λJ.

The first l components retained span a principal component subspace(PCS)and the remaining J-l components represent a residual subspace(RS).Theare component and direction matrices in the PCS,respectively,andcorrespond to the RS.To determine l,the cumulative percent variance(CPV)is widely used for its simplicity.For a particular observationand Q statistics are established for monitoring the two subspaces.In theis a diagonal matrix,and in theA fault is detected when the monitoring statistics violate their control limits

2.2.Problems of PCA monitoring

Control limits for both statistics can be calculated from an F or weighted χ2distribution[30]with a confidence α,typically set α =95%or 99%.In other words,a fault is detectable by PCA when its statistics must violate their corresponding control limits more than(1-α)?100%times.The essential of PCA monitoring lies in detecting changes in the cross-correlation structure among components.PCA monitoring neglects dynamic information hidden in the data and it may be insensitive to changes in the component auto-correlation structure under the condition formulated in Fig.1.In the PCS,T2is computed based on axesthat represent the directions p1and p2with maximum variances of λ1and λ2,and in the RS,Q is determined according to axes t3and t4along the directions p3and p4with minimum variances of λ3and λ4.Normal sample space lies within the circle and the ellipse.Obviously,auto-correlation structures of components t2and t3change from samples x0→xkto samples xk→xk′,and this change is undetectable by PCA since their statistics are still within the circle and ellipse.

Fig.1.Schematic illustration of problems of PCA monitoring.

Fig.2.PCA monitoring results for the fault.

The problems of PCA monitoring are illustrated by a simulated simple process involving four variables zT=(z1,z2,z3,z4)as

Fig.3.Influence of each component in the fault case.

Fig.4.T2 using components 2 and 4,and Q using components 1 and 3.

Here,of which each elementandof zero mean possesses a variance of 4,2,0.9 and 0.1.We produce 600 observations for modeling(normal case)and generate another 600 samples(fault case)in which z2is set to 2.5 after sample 200.In the PCA modeling,components 1,2,3 and 4 have a variance of λ1=2.2817,λ2=1.1981,λ3=0.4549 and λ4=0.0652,respectively.Components 1 and 2 with a total variance contribution of 87%are retained to compute T2statistic and the remaining components are used to determine Q statistic.The statistic monitoring results using PCA are shown in Fig.2,and the fault is significantly under-reported by PCA.Fig.3 reveals that the four components are not of the same influence degree to the occurrence of the fault.We can find huge changes in the auto-correlation structures of components 2 and 4,which contain most important information of the fault in the time region.However,components 1 and 3 are rarely affected,which provide little fault information for monitoring.The reason of the high under-report rate in the PCA monitoring is probably that important information of components 2 and 4 is submerged by the computation of T2and Q statistics,respectively.The motivation of DWPCA is to take emphasis on directions of components that carry most fault information in component auto-and cross-correlation structures.Fig.4 shows that the fault can be successfully detected when we use components 2 and 4 to compute T2statistic.The missing detection rates are reduced significantly compared with those in Fig.2.

3.Dynamic Weighted PCA

The proposed method combines time series technique and PCA,with the purpose to design a hybrid of auto-and cross-correlation structures in processes.This hybrid design of correlation structure includes two tiers.The first tier is to use the PCA method to extract the cross correlation structure among process data,expressed by independent components,and the second tier is to estimate the auto-correlation structure among the extracted components as auto-regressive(AR)models.Based on the estimated AR models,different weights are determined on different component directions automatically and dynamically,and a component direction is given a high weight value if its component has large estimation error.In this way,the DWPCA method considers the dynamic information in the processes.As a result,the DWPCA method is a dynamic method and can be effectively applied in dynamic systems for process monitoring.The new method produces two new statistics,and Qw,with a similar interpretation to the T2and Q statistics described in the PCA monitoring.

The rest of this section is organized as follows.Section 3.1 determines weights on component directions based on estimated AR models,and the weights are automatically updated when a new observation becomes available.Theand Qwstatistics are developed for online process monitoring in Section 3.2.

3.1.Determination of weights on component directions

Assume that the cross-correlation structure is expressed by independent components using the PCA decomposition.To evaluate the importance of each component i in the auto-correlation structure,for?i=1,2,…,J,we set a weight value wion its component direction piand initially wi=1.Then,the weighted direction isThe next step is the design of the learning algorithm for updating the weights.Let ei（k）?）be the estimation error,where k is an observation index andis an estimation value based on an AR model,i.e.）=In which,αs,i(s=1,2,…,d)is the s th AR coefficient and d is the model order.The multi-variable least squares(MLS)algorithm is applied in αs,iestimation and Akaike information criterion(AIC)is used to determine d.The learning algorithm based on ei(k)forthe online-updating weights is developed as an extended exponential function:

Table 1 Process monitoring with DWPCA

Fig.5.Tennessee Eastman process.

with constants γ ＞1 and δi＞ 0,where γ denotes the maximal bound of weights,wi(k) ∈ [1,γ).The dead-zone operator D[·]prevents the adaption of the weights when the modulus of estimation error|ei(k)|does not exceed its bound δi,thereby reducing false alarms caused by noise.The dead-zone operator D[·]is defined as

The dead-zone bound is determined based on the|ei(k)|(i=1,2,…,J)under normal operating conditions by the kernel density estimation(KDE)method[31,32].KDE is an effective tool to estimate the distribution of data,and a univariate kernel function is defined as

Table 2 Variables for monitoring in the TE process

where z is the data point under consideration;z(i)is an observation value from the data set;h is the window width or the smoothing parameter;n is the number of observations.The kernel function K determines the shape of the smooth curve under the conditionsK(z)≥0.Usually,a Gaussian function is chosen for K.Theδiis obtained bywith a given confidence α=95%or 99%.

3.2.Online process monitoring scheme

We partition the observations into an important component subspace(ICS)and a remaining component subspace(RCS).The ICS is constructed by components that carry most important fault information in the hybrid correlation structure,and the remaining componentscomprise the RCS.The importance of information that component i carries to a fault is given byThe value ofmay change with different observations,which can be written as a function of observation index k,i.e..For a particular observation x(k)∈,components are rearranged in the decreasing order ofand the setis sorted asThe firstcomponents are retained to construct the ICS and the remaining J-lw(k)components comprise the RCS,and lw(k)is determined by the CPV method,

Table 3 Process disturbances in the TE process

Table 4 Fault missing detection rates in the TE process

Similarly,corresponding component directions after weighted are rearranged astwo direction matrices comprised of the first lw(k)directions and the last J-lw(k)directions,are given by

Furthermore,the followingand Qwstatistics can be defined in the ICS and RCS as

whereis a diagonal matrix andThe control limitsandare determined based on normal process data via KDE since their statistic distributions is complicated and KDE has superior ability in dealing with this situation.DWPCA-based process monitoring includes off-line modeling and on-line monitoring as summarized in Table 1.

Fig.6.Monitoring results of fault 5 using DWPCA in the TE process.

Fig.7.Monitoring results of fault 5 using PCA in the TE process.

Remark 1.With the proposed method,the components are in the decreasing order ofthat takes both the information in the auto-and cross-correlation structure into account,whereand λicalculate the contribution of auto-and cross-correlation information,respectively.In contrast,PCA only considers the information in the cross-correlation structure and its components are in the order of λidecrease.

Remark 2.Generally,the time complexity of the conventional PCA is O(NJ2).We can see from Table 1 that the DWPCA method introduces a few additional steps for online monitoring as compared to conventional PCA.The added steps are steps 2,3 and 4,whose running time are O(dJ),O(Jlg J)and O(J).Then,the total of added time complexities is O(max(dJ,Jlg J)).We have max(dJ,JlgJ)＜ ＜NJ2,the time complexity of DWPCA is the same as PCA,O(NJ2).

Theorem 1.Projections onto all componentsare orthogonal to each other andisthe variance ofprojection onto

Proof.From the above introduction to the PCA method,we know that,Incorporatinggives rise toThis illustrates that projection on everyis orthogonal to each other.One the other hand,we haveMoreover,is the variance of componentdenotes the expectation function.Hence,The proof is complete.

Theorem 2.DWPCA reduces to PCA when weights on component directions are of the same value of 1,in other words,PCA is a special case of DWPCA.

Proof.If the weight values equal to 1,then?i=1,2,…,J and wi=1.Sinceandwe haveThen,which means that the order of components remains unchanged,soWe have CPV（lw）=,then choosing CPV(lw)=CPV(l)gives rise to lw=l.In this case,important components that construct the RCS of DWPCA are exactly principal components that comprise the PCS of PCA,similarly,the RCS and the RS are identical.

Fig.8.Monitoring results of fault 5 using DPCA in the TE process.

Fig.9.Monitoring results of fault 5 using DLV in the TE process.

Moreover,,similarly,Qw=Q.The proof is complete.

4.Case Study on Tennessee Eastman Process

Tennessee Eastman(TE)process[34]is widely used for process monitoring[35].It consists of five major operations:reactor,product condenser,vapor–liquid separator,recycle compressor and a product stripper,as shown in Fig.5.The process has 41 measured variables(22 continuous and 19 compositions)and 12 manipulated variables.The 22 continuous measurements and 11 manipulated variables are used for monitoring as listed in Table 2.The plant-wide control structure recommended by Lyman and Georgakis[36]is used in this case study.A total of 22 data sets are collected in different modes(one normal and 21 fault modes),and each data set contains 960 samples of the 33 variables.In each fault mode,the fault is introduced after sample 160.The detailed description of the 21 faults is provided in Table 3.

Conventional PCA,DPCA[11]and DLV[11]and the proposed DWPCA method are illustrated based on the collected data sets.Fault missing detection rate is considered for evaluating the monitoring performance,which denotes the percentage rate of samples under the control limits when a fault is introduced.In this study,the number of principal components of PCA,DPCA,DLV and DWPCA is determined by the CPV with 85%variation,and their control limits are calculated by KDE with 99%con fi dence.The KDE methods are detailed in Section 3.1.In the DWPCA method,we set γ=5 in Eq.(2).This application of the proposed method follows the procedure of Table 1 and more analytical details are provided in Section 3.The specific monitoring results of the proposed method are listed in Table 4 and those of conventional PCA,DPCA[11]and DLV[11]are given for comparison.The lowest fault missing detection rate for each fault is highlighted in bold.Note that both the two methods have high missing detection rate for faults 3,9 and 15,and the three faults are difficult to be detected since they have almost no effect on the variation and the mean.Table 4 shows that the proposed DWPCA method can efficiently reduce the missing detection rate for faults 5,10,16,19 and 20,as compared to conventional PCA,DPCA and DLV.The results of other faults are almost at the same degree.

Fig.10.Weights on component directions for fault 5 in the TE process.

Fig.11.Influence of variables 17 and 33 for fault 5 in the TE process.

4.1.Case study on fault 5

Fault5 is a step change in the condenser cooling water inlet temperature.Once this fault is introduced,a step change happens to the flow rate of condenser cooling water(variable 33)and this change propagates to other variables.As time goes on,the control system tends to tolerate and compensate this fault,thus most variables attain to their steady states again.The monitoring results using DWPCA,PCA,DPCA and DLV are shown in Figs.6–9,respectively.Figs.7–9 show that PCA,DPCA and DLV can detect this fault at the beginning stages,but fails to detect it after sample 340.However,the DWPCA method can detect this fault during the whole process as shown in Fig.6.As compared to PCA,DPCA and DLV,DWPCA is much more sensitive to this fault.The DWPCA method takes emphasis on components with large estimation errors,as a result of high weight values as shown in Fig.10.We can see from Fig.10 that component31 have high weights,so it is still affected after sample 340,and this helps the fault detection using the DWPCA method.Actually,variables 17 and 33 have largest contributions,0.7039 and 0.7027,respectively,to the direction of component 31.Fig.11 shows the influence of variables 17 and 33,in which,variable 33 has a significant step change,then we can determine it as the root of this fault.This isolation result is in agreement with the above analysis.

4.2.Case study on fault 10

Fig.12.Monitoring results of fault 10 using DWPCA in the TE process.

Fig.13.Monitoring results of fault 10 using PCA in the TE process.

Fault 10 involves a random variation in C feed temperature(stream 4),which provides inlet feed for the stripper.Then,this fault firstaffects the stripper temperature(variable 18)and then propagates the influence to other variables.Most variables are able to remain around their steady points and behave similarly as normal.This makes the fault detection rather challenging.Monitoring performances of fault 10 based on DWPCA,PCA,DPCA and DLV are shown in Figs.12–15,respectively.The missing detection rate of Qwis reduced signi fi cantly using DWPCA as compared to the missing detection rates of Q and T2using PCA and DPCA and ofand Qrusing DLV.Fig.16 shows that weight values on components 26,27 and 28 are high.Then,DWPCA can facilitate the fault isolation by narrowing down the faulty variables to variables with large contribution on these components.

5.Conclusions

We have shown that conventional PCA has difficulty in monitoring dynamic processes since it neglects dynamic information underlying process data.To solve this problem,we have proposed a DWPCA method with hybrid correlation structure design for online process monitoring in this work.The main contributions can be summarized as follows.

(1)We have evaluated the monitoring performance of conventional PCA on dynamic processes,based on the idea that online operating information contained in process auto-correlation structures should be used to detect incipient faults with the purpose to reduce the fault missing detection rate.To this aim,we have designed a two-tierhybrid correlation structure thatconsiders both auto-and cross-correlations.

(2)We have introduced the new monitoring scheme that makes use of online operating information to dynamically partition the process data space into the important and remaining component subspaces,and the partition step is based on a contribution indexdefined with variance λiand direction weight wiof each component i.To dynamically monitor the two new subspaces,we have produced two new statistics.

Fig.14.Monitoring results of fault 10 using DPCA in the TE process.

Fig.15.Monitoring results of fault 10 using DLV in the TE process.

Fig.16.Weights on component directions for fault 10 in the TE process.

(3)We have demonstrated the monitoring performance of the proposed DWPCA method in the application of TE process.The monitoring results have shown that DWPCA can obtain a higher accuracy as compared to conventional PCA,DPCA and DLV.Moreover,the results with DWPCA could aid process operators to narrow down the root cause of faults.

Extensions of concepts of the proposed method are recommended for further research.Further research could include the introduction of nonlinear behaviors and uncertainties in processes,and as a result improved monitoring schemes based on the proposed method can deal with process problems that are more practical and close to real word.We can also extend the proposed method for fault detection in discrete event systems or hybrid systems.

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