Process monitoring based on mode identification for multi-mode process with transitions

Chemometrics and Intelligent Laboratory Systems 110 (2012) 144–155

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Chemometrics and Intelligent Laboratory Systems journal homepage: www.elsevier.com/locate/chemolab

Process monitoring based on mode identification for multi-mode process with transitions Fuli Wang, Shuai Tan ⁎, Jun Peng, Yuqing Chang 131Box, College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning Province, 110004, PR China

a r t i c l e

i n f o

Article history: Received 20 April 2011 Received in revised form 30 September 2011 Accepted 23 October 2011 Available online 28 October 2011 Keywords: Mathematical modeling Process monitoring Multi-mode continuous process Mode identification

a b s t r a c t Some industrial processes frequently change due to various factors, such as alterations of feedstocks and compositions, different manufacturing strategies, fluctuations in the external environment and various product specifications. Most multivariate statistical techniques are under the assumption that the process has one nominal operation region. The performance of it is not good when they are used to monitor the process with multiple operation regions. In this paper, we developed an effective approach for monitoring multi-mode continuous processes with the following improvements. 1). Offline mode identification algorithm is proposed to identify (i) stable modes, (ii) transitional modes between two stable modes, and (iii) noise. 2). According to the data distribution, proper multivariate statistical algorithm is selected automatically to realize fault detection for each mode. 3). When online monitoring, the right model is chosen based on Mode Transformation Probability (MTP), which makes full use of the empirical knowledge hidden in offline data. This method can enhance realtime performance of online mode identification for continuous process and timely monitoring can be further realized. The proposed method is illustrated by application in furnace temperature system of continuous annealing line. The effectiveness of mode identification and fault detection is demonstrated in the results. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Statistical process control forms the basis of performance monitoring and detection for process malfunctions. With huge amounts of data collected from the real production, there has been increasing interest in pursuing methods that are capable of extracting the underlying correlation from these correlated data. The correlation information is hidden inside the data in the form of combinations of variables: the latent structures [1]. The methodology projects the process data down onto a low dimensional subspace, through the definition of latent structures, and in this way the major sources of variability associated with the process are summarized. Many methods discovering the latent structures of the data, such as PCA (Principal Component Analysis, PCA), ICA (Independent Component Analysis, ICA), are employed for data compression and major information extraction [2]. So far, these statistical projection methodologies have been applied to diverse processes. Nomikos and MacGregor [3,4] proposed Multiway PCA to realize fault detection for batch process. Ku et al. [5] used dynamic PCA to include process dynamics in a PCA model. Lee et al. [6] used kernel PCA to detect faults for nonlinear process. Li et al. [7] developed Recursive PCA for adapting process changes. Unfortunately, these methods generally tend to behave unsatisfactorily when process operates multiple modes resulting from alterations

⁎ Corresponding author. Tel.: + 86 24 83687434. E-mail address: [email protected] (S. Tan). 0169-7439/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemolab.2011.10.013

of feedstock and manufacturing strategies. In most continuous industries, the production mode of manufacturing process frequently changes due to various factors, such as alterations of feedstock and compositions, different manufacturing strategies, fluctuations in the external environment and various product specifications. Furthermore, the operating conditions, such as set points of reactor temperature and pressure, compositions of catalysts, etc., are sometimes adjusted to meet the different production specifications. Generally, the different modes have different process characteristics. If one model is used to describe data with different characteristics, the fault will be mistaken as normal, because the confidence range for monitoring statistics might cover abnormal operating region. In order to solve this problem, some research effort has been reported to approach the multimode process monitoring issue. Hwang and Han [8], Lane et al. [9] proposed a united model to monitor multiproduct processes. In 1994, Kosanovich et al. [10] point out that a division of the data into sets that correspond to the major chemical phenomena will provide clarity and allow for interpretation based on process understanding. Based on this, Bhagwat et al. [11], Nga et al. [12], Doan et al. [13], Chen et al. [14], and Zhao et al. [15] developed multiple models for multi-mode process monitoring. The above PCA-based approaches can perform well for the multi-mode process, where each mode follows the Gaussian distribution. Gaussian mixture model, which is a mixture of finite weighted Gaussian components, is also explored in multi-mode process monitoring recently. Choi et al. [16], Yoo et al. [17], Camacho et al. [18], Thissen et al. [19], Hyndman et al. [20], and Yu et al. [21] have studied deeply on this problem. Jie

F. Wang et al. / Chemometrics and Intelligent Laboratory Systems 110 (2012) 144–155

Yu and S. Joe Qin [21] estimated Bayesian posterior probabilities of each monitored sample belonging to the multiple components and derived an integrated global probabilistic index for fault detection of multimode processes. The above methods did not focus on transitional mode between two stable modes, which may lead to loss of production time, off-grade materials, and lack of reproducibility of product grades. In recent years, Lu et al. [22], Zhao et al. [23], and Yao et al. [24] proposed a transitional model expressed as a weighted sum of two stable phases nearby this transitional phase. Zhao et al. proposed a transitional model expressed as a weighted sum of two stable phases nearby this transitional phase. The monitoring effect of this model depends mainly on the characteristic similarity between transitional data and stable model. In industries, stable mode is the main process which occupies the most production time. In order to yield high productivity, production schedule should avoid transitional modes as possible as can be. If there does not have enough modeling data of transitional phase, the robustness of weighted transitional model is bad. Furthermore, an important point has been just mentioned or even ignored in previous research: online mode identification, which is selecting the proper model for online samplings. For batch process, each batch is repetitive process. Online mode identification is easy to realize according to time indication. However, besides the method of selecting model according to minimum SPE value, there are few researches to guide online mode identification during continuous process. In this paper, we propose an improved modeling method for multimode continuous process involving offline mode identification and online mode identification. Firstly, stable modes, transitional modes and noise are identified by analyzing the duration and the characteristics of different modes. Secondly, the proper modeling method is automatically chosen to extract process feature based on data distribution in different modes. Thirdly, the online work provides a high real-time solution based on Mode Transformation Probability (MTP), which picks up the empirical knowledge hidden in offline data to guide online model identification. At last, practical application clearly demonstrates the effectiveness and feasibility of the proposed method. 2. Mode identification algorithm for modeling data For multi-mode process, how to divide data with different characteristics into different modes is important for monitoring. We propose a two-step mode identification algorithm used to achieve division of stable mode and transitional mode. The first step is preliminary division for modeling data based on improved clustering algorithm. The second step is further mode identification based on underlying process behaviors.

that x and v are contrary to each other. Based on this, each offline data is classified into the clustering center corresponding to the maximum similarity degree D. S cluster centers, which are selected from data points, can not well represent the center of data distribution. Combining K-means clustering algorithm, the cluster centers are updated using P ðnewÞ their average values. vs ¼ n1s xi ; ðs ¼ 1⋯ SÞ, where ns is the number xi ∈s

of data belonging to s cluster [26]. This process is repeated until the distance between new center vs(r) and the former center vs(r − 1) is small enough to satisfy the convergence condition: ‖vs(r) − vs(r − 1)‖ b γ, where, γ is the threshold. Another threshold parameter β named “minimum modeling number” can be set as 2 to 3 times of the number of process variables according to the modeling experience of multivariate statistical regression methods mentioned in reference [27]. If the sampling number of one class is less than β, that is to say, the data of this class is not enough to cover statistical characteristics. This class can be ignored and the data of which are reclassified into another class corresponding to the maximum similarity degree D. The improved clustering algorithm can be summarized as shown in Fig. 1. In summary, the improved clustering algorithm combines the merits of both subtractive clustering algorithm and K-means clustering algorithm. It can find initial clustering centers automatically. Meanwhile, the class without enough modeling data is deleted.

2.2. Mode identification In multi-mode industries most production process is in stable mode which yields high productivity. The transitional mode between neighboring stable processes, such as grade change, startup, and shutdown, does not play the primary role. In our paper, the stable mode is defined as the constant process with the stable characteristics which can be described using one model. And the transitional mode is the process with time-correlation information which needs a series of sub-models to grasp more accurate details. Regularly, these transitional sub-modes show the gradual changeover between two neighboring stable modes. At the beginning they have the similar underlying characteristics which more belong to the previous stable mode, gradually, it is more similar to the next one.

2.1. Clustering algorithm The data with the same characteristics can be described using one model. Grouping offline data for modeling according to similarity of characteristic is a reasonable solution way. There are many measurements for data similarity, such as: the Euclidean distance between PCA loading matrices [23], principal angle of PCA loading vectors [15] and so on. In this article, distance between samplings is used to describe data similarity considering the large amount of offline data. If the measurement of data similarity, which is not the key point of our research, is replaced by other formulas and it will not affect the main idea of monitoring algorithm for multimode process. Clustering algorithm based on the distance is used to realize preliminary division of different modes. The initial clustering centers are estimated using subtractive clustering algorithm which is an efficient method to estimate the number of clusters and their center values [25]. We assume that S centers are obtained using the sub-clustering method. The similarity degree D = exp(− ‖x − v‖2) is used as the distance measurement of sampling x and center v. D ∈ [0, 1], when D = 1 means x = v and D = 0 means

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Fig. 1. Flow chart of clustering algorithm.

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We summarize the characteristics of multi-mode process as follows: ①. The stable mode is the long duration process with the same statistical characteristics; ②. Transitional mode is only in the process between two different stable modes; ③. The characteristics of transitional mode change gradually from one mode to another mode. According to these three points, we propose a new algorithm for mode identification. Firstly, we define H as the minimum stable mode duration. If the length of one clustering segment exceeds H, this segment can be considered as a stable mode. Conversely, if the clustering segment is less than H, this segment is marked as “transient mode”, which may be two kinds of possible modes: transitional mode or interference noise which is abnormal and should be neglected. Here, H is minimum duration during which the process data can cover the characteristics of stable mode. Secondly, the noise segment between stable modes is identified. If two neighboring stable modes of “transient mode” are the same one, the “transient mode” between them is noise segment. If not, they are transitional modes. Thirdly, the noise segment between transitional modes is identified. As mentioned above, the characteristics of transitional mode change gradually from the previous stable mode to the next stable mode. Then the similarity degree D of transitional mode from the previous stable mode decreases gradually. The segment which does not follow the decline rule is considered as noise. To help understand the details of mode identification, we simply take an ideal multi-mode process with 500 samplings containing two stable modes and one transitional mode as example. Fig. 2 shows the training result, 500 samplings are divided into 8 segments which are clustered into 5 classes respectively. Arranging these samplings with class labels in time sequence and 8 segments is displayed by order in Fig. 2. The clustering result is shown in Fig. 2. We can recognize that this process is composed with 8 segments. The segment with the same statistical characteristics, whose duration length is longer than H (Here, H = 50), is a stable mode. Then three segments: segment1

(1–111), segment3 (115–190) and segment8 (290–500) are considered as stable modes. Segment2, segment4, segment5, segment6 and segment7 are “transient modes”. As segment1 and segment3 belong to the same class, then segment2 is not the transitional mode but noise segment, which should be deleted. Segment4, segment5, segment6 and segment7 are in the area between two different stable
is used to support the mode modes. The average similarity degree D    
¼ 1 P exp −x −v2 ; ðs ¼ 4; 5; 6; 7Þ, identification for them. D s

ns

s

xs ∈s

where v is the clustering center of segment1, ns is the number of sam
bD
>D
. Seg
>D plings collected in segment s. It is clear that D 4 5 6 7 ment6 is considered as abnormal process influenced by noise. The final results of mode identification are shown as Fig. 3. Segment I and segment V are two stable modes, and segment II, segment III, and segment IV are three transitional sub-modes between them.

2.3. Modeling data construction Assume that there are M stable modes, and mode m has Cm(m = 1, 2 ⋯ M) segments. The modeling matrix of each segment is (m)

(m)

(m)

(m)

XCm (NCm × J), where NCm is the sampling number of matrix XCm , J is the number of variables. Each row is the process variables at one sampling time, and each column is the trajectory of one variable over all time within this segment. Cm segments can be described using 2 one model. Then modeling data of stable mode m is 3 ðmÞ X1 Cm P 6 7 ðmÞ X ðmÞ ¼ 4 ⋮ 5 , where NðmÞ ¼ Ni . ðmÞ i¼1 X Cm ðmÞ N

J

Transitional mode is a process changing gradually from one stable mode to another one. The variables' response to noise is so sensitive that it is difficult to control transitional process following the set trajectory. In this article, we use a series of transitional sub-modes to grasp more details of transitional process. It is quite possible that variable fluctuation will cause different identification results of transitional sub-modes. More steps are needed to construct modeling data for transitional mode. We simply take two transitional trajectories from mode A to mode B as example. One trajectory is classified into four transitional sub-modes, such as shown in Fig. 4 Case 1, and the other one is classified into three transitional sub-modes, such as shown in Fig. 4 Case 2.

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We select a case as standard. Here refers to Case 1. Because the case with more sub-modes can describe process details better. The transitional sub-modes of Case 2 are matched with standard submodes on the basis that two segments with the maximum similarity degree have the most similar characteristic. The average similarity ð2Þ degree between X~ k (k = 1, 2, 3) and X~ ðgstanÞ (g = 1, 2, 3, 4) is defined as 0  2 1    P P B ~ð2Þ ðstanÞ  1 ~ ð2Þ
ð2Þ ¼ 1ð2Þ  C ~ x exp − x − D @ ð stan Þ g  k  A. X k is the k;g Nk Ng ð2Þ ð2Þ ðstanÞ ~ ðstanÞ   ∈X g x~ ∈X~ x~ g k

k

matrix of transitional sub-mode k in Case 2. X~ ðgstanÞ is the matrix of transitional sub-mode g in standard Case 1. Nk(2) and Ng(stan) are the ð2Þ ð2Þ sampling number of matrix X~ and X~ ðstanÞ respectively. X~ is classik

g

k

fied into the transitional sub-mode g corresponding to the maximum
. X~ and X~ ðstanÞ can be described using one model. Then the value D g k;g k ð2Þ

ð2Þ

modeling data for transitional sub-mode g can be expressed as  3 2 X~ ðgs tanÞ Nðgs tanÞ  J ~ 4 5   Xg ¼ . ð2Þ ð2Þ X~ k Nk  J ðNðgs tanÞ þNðk2Þ ÞJ

3. Offline modeling based on multivariate normality test Different modes may have different process distributions. In general, a variable affected by small random factors can be considered as Gaussian distribution. For example, under the production conditions without changes, most probability distribution of variables can be approximated by Gaussian distribution. In order to extract the underlying process-correlation characteristics of each mode, we develop a mode-representative modeling way to built proper statistical model for each mode.

3.1. Multivariate normality test It is well known that independent component analysis (ICA) is greatly developed to recover hidden, statistically independent components (ICs) underlying non-Gaussian process measurements, which are reported to have more interpretable characteristics and will benefit process comprehension and analysis [28]. ICA provides more meaningful knowledge extraction and reliable monitoring

Fig. 4. Two clustering results of transition from A to B.

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performance using higher-order statistics compared with PCA. Distinguishing from ICA, PCA is limited to only deal with second order statistics, which are often expressed in terms of covariance matrices and, thus, may fail to reveal enough information involved in the nonGaussian processes. When the case has mixed distributions including Gaussian and non-Gaussian property, any single-feature extraction technique is not competent to reveal the hidden information adequately enough. The way combining ICA and PCA drew out by Manabu et al. [29,30] is introduced to ensure valuable process feature all included. Here an improved multivariate process monitoring algorithm based on normality test is proposed. It can choose an appropriate algorithm according to test result automatically. A distribution test method based on principal components (PCs) is used to realize multivariate normality test. Assume that J vectors measured at N time are organized as a two-way matrix XN × J = (x1, x2, ⋯ xJ). Then the normalized matrix X is decomposed as X = T ⋅ P T, where TN × J = (t1, t2, ⋯ tJ) is the principal component matrix,  PJ× J = ~ ⇔ (p1, p2, ⋯ pJ) is the loading matrix. X ~ NJ(μ, Σ) ⇔ T eNJ μ~ ; Σ tj(j = 1 ⋯ J) ~ N(μj, Σj) is deduced using probability theory [31]. That is to say, if each principal component is Gaussian distribution, then X is a J-dimensional Gaussian distribution. Otherwise, anyone of the principal components does not obey Gaussian distribution. X is not multivariate Gaussian distribution. Based on this, complex multivariate normality test is simplified into one-dimensional normality test, which realized adopting skewness and kurtosis test. And the right modeling method is selected to extract data statistic characteristics. 3.2. Modeling strategy based on data distribution As mentioned above, PCA is selected to realize fault detection for the Gaussian mode. Monitoring algorithm based on ICA is chosen for the non-Gaussian mode. We determine the remaining ICs by calculating iteration error γ(i) = ‖wi − wi − 1‖ between decomposing vectors wi and wi − 1. If the error is sufficiently small, we think there are not valuable non-Gaussian characteristics to be extracted in residual matrix. kX−X^ k If the error of estimated model ε¼ kX k is sufficiently small, that is to say, the process features can be captured completely using whether PCA or ICA. If not, the process contains both Gaussian and non-Gaussian property. ICA–PCA two-step analysis is deduced as the modeling method. The modeling steps are shown in Fig. 5:

traditional statistics T 2 and SPEPC and their confidence limits, which can be used for fault detection, are obtained as follows [32,33]:   ðmÞ ðmÞ  −1 A N −1 ð m Þ ð m ÞT O t T 2ðmÞ ¼ t e N ðmÞ −AðmÞ F AðmÞ ;NðmÞ −AðmÞ ;δ    ðmÞ ðmÞ ðmÞT ðmÞ ðmÞ ðmÞ ðmÞ T ðmÞ ðmÞ2 X −X^ PC SPEPC ¼ e e ¼ X −X^ PC e g χ h;δ : ðmÞ

Where, t (m) is the row of T (m); A (m) is the number of principal ðmÞ components; O (m) is the covariance matrix of T (m); X^ PC ¼ ðmÞ 2 2 m ð m Þ ð Þ ðmÞ v X ðmÞ P ðmÞ P ðmÞT is the reconstruction of X (m);g ðmÞ ¼ 2m ¼ vðmÞ ; ðmÞ ;h m (m), v (m) are respectively the average and corresponding variance (m) of SPEPC . For ICA model, statistics I 2 and SPEIC based on ICs give an expression for the non-Gaussian information: ðmÞ

I 2ðmÞ ¼ s

  ðmÞ −1 ðmÞT M s

   ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ T X −X^ IC SPEIC ¼ X −X^ IC :

ð2Þ

Where, s (m) is the row of independent component matrix S (m); ðmÞ ðmÞ is the covariance matrix of S (m); X^ IC ¼ AC SðmÞ is the reconM (m) struction of X (m); AC is mixing matrix. In ICA, the restriction of IC's non-Gaussianity makes it impossible to calculate the control limits of statistics in a way similar to that in PCA. Here, kernel density estimation [34] is employed to define the normal coverage of statistics I 2 and SPEIC. With enough training samplings, a reliable density fitting can be well-obtained in theory. (m)

Step1: Assume that J variables are measured at N (m) time instances in the mode m. Then collected modeling data can be organized as the matrix X (m)(N (m) × J). Step2: Standardize X (m) and test the distribution of X (m). If X (m) is multivariate Gaussian distribution, then PCA method is used to extract A (m) principal components from X (m). Otherwise, if X (m) is not multivariate Gaussian distribution, Ei = X, (i = 1). Step3: ICA is used to extract one independent component si = wiEi. Where, s is one independent component and w is the decomposing vector of decomposing matrix WC. Step4: Ei+ 1 = Ei − aisiT. Where, a is the corresponding vector of mixing matrix AC. Step5: Calculation of γ (i) = ‖wi − wi − 1‖. If γ (i) is greater than threshold α, algorithm returns back to Step3. While, if γ (i) is less than the threshold α, (i − 1) ICs are picked up and the algorithm flow goes into the next step. k Step6: Calculation of ε¼ kX−AS kX k . If the error of estimated model ε is greater than threshold β, ICA is deduced as the modeling method. While, PCA is used on the residual matrix and (i − 1) ICs have been picked up. Two-step analysis ICA–PCA is introduced. 3.3. Monitoring control limit development In PCA monitoring system, the loading P (m) and score matrix T (m) are calculated, which preserve the major relations of mode m. Two

ð1Þ

Fig. 5. Illustration of offline modeling based on multivariate normality test.

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Fig. 6. Illustration of online monitoring for multi-mode process.

If two-step analysis ICA–PCA is introduced to measure the complete process information, three statistics I 2, T 2 and SPEIC − PC are used to monitor process operation: "

ðmÞ

ðmÞT

ðmÞ

S ¼W X  −1 ðmÞT I2ðmÞ ¼sðmÞ M ðmÞ s

" EðmÞ ¼ X ðmÞ −AðmÞ SðmÞ C ð Þ ð Þ ð Þ T m ¼ E mP m  −1 ðmÞT T 2ðmÞ ¼t ðmÞ OðmÞ t 2    ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ T ðmÞ ^ X ðmÞ −X^ IC −X^ PC SPEIC−PC ¼ X −X IC −X^ PC 6 6 ^ ðmÞ 4 X IC ¼ AðCmÞ SðmÞ ðmÞ ðmÞ ðmÞT X^ PC ¼ T P :

PCs follow the Gaussian distribution. This provides the basis that T 2 is F-distribution, whose confidence limit can be calculated using empirical formula as formula (1). In residual subspace, the representative confidence limit of SPEIC − PC can be approximated by a weighted Chisquared distribution, as formula (1). Meanwhile, the confidence limit of I2 should be obtained using kernel density estimation [35,36].

4. Online monitoring based on mode transformation probability ð3Þ

When online monitoring, proper monitoring model should be selected for the current sampling. There are few related previous researches for reference of the online mode identification. The most popular way is to choose the model yielding the minimum SPE among all models. Exhaustive method will lead to great amount of

Fig. 7. Flow diagram of the annealing furnace.

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Fig. 8. Process variables for continuous annealing line.

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Transformation Probability P ðBjAÞ ¼ numAB =numA−all as the transition probability from mode A to mode B. numA − all is the number of all mode transitions beginning with A; numAB is the number of mode transitions from A to B. The MTP is introduced as the guidance for online mode identification. The transition corresponding to the maximum MTP can be considered as the most possible one. Here, we assume that the mode type at time (k − 1) is a stable one and the mode type has been known, where k is the current sampling time. Firstly, monitoring the current sampling using the control limits at time (k − 1). If the sampling does not go beyond the control limits,

computations. It will decrease the responsiveness and effectiveness of online fault detection. Here, Mode Transformation Probability (MTP) is proposed to support online model matching. A new method based on MTP is intended to save online computing time and enhance real-time performance of process monitoring. Mode Transformation Probability is a probability parameter describing the frequency of transition between two stable modes. If offline data is abundance enough to represent the whole operation process, mode transition behaviors are hidden within the collected data. Assume that A and B are two stable modes. We define Mode

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it is concluded that the current state is normal. Otherwise, there are two possible cases:①. It turns into a transitional mode from the current time;②. It is abnormal at the current time, that is to say, it is either a fault or an un-modeled mode whose characteristic is not included in modeling data. In order to distinguish the two situations, we start hypothesis testing from case ① based on MTP. Assume that the stable mode at time (k − 1) is A. P(Z|A) shows the frequency of transition from stable

mode A to other stable modes, where Z represents all stable modes that appeared in process. It is easy to understand that the stable mode Z (max) corresponding to the most MTP Pmax(Z (max)|A) is the most possible mode turned into. The k-time sampling is remonitored using the first sub-model of all transitional modes from the most possible one to the least one. If all sub-models can not fit the process characteristics, it may be a fault or an un-modeled mode. The detail steps are shown as Fig. 6:

Fig. 12. 23 variables of the modeling set from the 108000th to 112000th samplings.

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heating, soaking, cooling (cooling section includes: slow cooling, reheating, fast cooling, over aging, and final cooling), as shown in Fig. 7 [37]. The inner structure of strip successively experienced the stages of grain recovery, recrystallization, grain growth, carbide precipitation and etc. The inner quality can be well improved by the stages mentioned above. As strip temperature will influence internal crystal structure directly, the temperatures of each furnace section must be controlled within confidence ranges of normal production. Here, we choose 23 process variables as the input variables, which are shown in Fig. 8.

Fig. 13. Times of different transitions.

5.2. Offline modeling based on mode identification

Fig. 14. MTP for online mode identification.

The online monitoring for multi-mode process can be summarized as followed: Step 1: Monitoring the current sampling xk using the confidence limits of the time (k − 1). Step 2: If the statistics are below the confidence limits, the current sampling is considered to be normal. And the mode of k-time is consistent with the one at the time (k − 1). Otherwise, the process may turn into transitional mode from k-time. Step 3: Re-monitoring the k-time sampling using all possible models according to Mode Transformation Probability in descending order. Step 4: If the statistics can be all reduced to below the normal region of one model, the transitional mode is determined and process monitoring is continued using this transitional model. Otherwise, none of transitional models can fit this sampling, it is considered as either a fault or an un-modeled mode. 5. Application results and discussion 5.1. Annealing furnace system Annealing furnace is an important part of continuous annealing line. The main processes in annealing furnace include pre-heating,

Blank plates with different hardness need different operating conditions. The furnace temperature and pressure, conveyer speed, etc., have to be adjusted to meet the production specifications. Collecting a total of N (N = 1.04 × 10 7) samples for 23 variables within four months, the modeling data matrix X(N × 23) is generated. It includes three stable modes (T-3CA, T-4CA, and T-5CA) and the transitions between them. We draw out the trajectories of 23 modeling variables Xtest(172219 × 23) in Fig. 9. Firstly, using the mode identification algorithm mentioned in Section 2, three stable modes and several transitional sub-modes between them are identified. Here, 172,219 test samplings Xtest (172219 × 23) with mode label are collected to test the validity of mode identification. The result is shown in Fig. 10. It is clear that three stable modes are distinguished accurately. The transitional mode has more details than the stable mode. Then the transitional regions are divided into three or four sub-modes, which show the gradual changeover between two neighboring stable modes. For comparison, we amplify two Tinplate Hardness processes (‘T-4CA’ and ‘T-5CA’) and the transitional processes between them (transition from ‘T-4CA’ to ‘T-5CA’). The results of mode identification are as shown in Fig. 11: 77977–109277 (stable mode T4), 109278–109539 (transitional sub-mode T45-1), 109540–109939 (transitional sub-mode T45-2), 109940–110439 (transitional submode T45-3), 110440–110644 (transitional sub-mode T45-4), and 110645–133563 (stable mode T5). The 23 modeling variables during this time are as shown in Fig. 12. It can be observed that the beginning time and ending time of stable modes identified using our algorithm are almost the same with the real situation.

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When online monitoring, another 4000 sample data Xonline(4000×23) is collected including two stable modes and the transitions between them: stable mode ‘T-4CA’; stable mode ‘T-5CA’; transition from ‘T-4CA’ to ‘T-5CA’. A fault is introduced from the 3300th sampling time to simulate a gradual temperature drift of the first furnace area in 1OA. The fault occurring time locates in the transition from ‘T-4CA’ to ‘T-5CA’. To illustrate the advantages of online mode identification based on MTP, the traditional methods by matching the current sampling with all offline models is compared with our method. The traditional method is based on “minimum SPE principle”. The model corresponding to the minimum SPE value is considered as the proper model for the current sampling. 21.0 seconds are cost to identify mode for 4000 samplings using this method. Comparing with it, the computing time of our method based on MTP is decreased to 6.6 s, which is one third of the old method. Fig. 15 shows results of online mode identification using our method. The mode changes at the 1680th sampling. It is clear that the transitional mode is well identified in Fig. 15.

Proper feature extraction algorithms for each mode are selected automatically based on data distribution. Most stable modes can be described using PCA method. Transitional sub-modes are modeled using PCA–ICA two-step modeling method. This shows that the data of stable modes can be approximated by Gaussian distribution and the data of transitional modes have both Gaussian and non-Gaussian information. 5.3. Online monitoring based on mode transformation probability Mode Transformation Probability can be calculated from modeling data X(N × 23), N = 1.04 × 10 7. There are three stable modes (T-3CA, T-4CA, and T-5CA) and six transitional modes between them. By analyzing modeling data, the times of different transitions are counted as shown in Fig. 13. In Fig. 14, the MTP values are calculated using P ðBjAÞ ¼ numAB =numA−all . The larger value represents the higher possibility of transformation.

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Transitional mode can be considered as a repetitive process. In order to compare monitoring results, another two accredited methods: “stage-based sub-PCA modeling method” proposed by Lu and “soft-transition multiple PCA method” proposed by Zhao, are compared with our method. The two methods start work from the 1680th sampling. The error is added from the 3300th sampling. Figs. 16 and 17 respectively show the monitoring results of the two methods. From the monitoring results shown in Fig. 15, the abnormality is detected at 3314th sampling. The fault is warned timely. The T 2 and SPE in Fig. 16 go beyond the control limit at 3660th and 3367th sampling times respectively. The alarm time is delayed for a longer time. In Fig. 17, there are too many false alarms for the normal. The model is so sensitive that the false alarm rate is increased. The above simulations show that the online monitoring based on mode identification can choose proper monitoring model automatically and realize fault detection timely. 6. Conclusions A new monitoring method for multi-mode continuous process based on mode identification is proposed in this article. Firstly, clustering algorithm partitions the modeling data into different modes according to the similar characteristics. In order to extract the different underlying process behaviors of each mode, we propose a modeling strategy based on data distribution, which can choose the proper statistical method automatically. When online monitoring, Mode Transformation Probability can make full use of the empirical knowledge hidden in offline data. This can save online computing time and enhance real-time performance of online mode identification. Process monitoring based on mode identification can realize fault detection timely. The illustration results demonstrate that the proposed method is more reliable and effective for multi-mode continuous process monitoring. Acknowledgment This work was supported by the National Science Foundation of China under Grant 61074074 and project 973 under Grant 2009CB320601. References [1] S. Wold, K. Esbensen, P. Geladi, Principal component analysis, Chemometrics and Intelligent Laboratory Systems 2 (1987) 37–52. [2] J.E. Jackson, A User's Guide to Principal Components, Wiley, New York, 1991. [3] P. Nomikos, J.F. MacGregor, Monitoring batch processes using multiway principal component analysis, AICHE Journal 40 (1994) 1361–1375. [4] P. Nomikos, J.F. MacGregor, Multivariate SPC charts for monitoring batch processes, Technometrics 37 (1995) 41–59. [5] W. Ku, R.H. Storer, C. Georgakis, Disturbance detection and isolation by dynamic principal component analysis, Chemometrics and Intelligent Laboratory Systems 30 (1995) 179–196. [6] J.-M. Lee, C.K. Yoo, S.W. Choi, P.A. Vanrolleghem, I.-B. Lee, Nonlinear process monitoring using kernel principal component analysis, Chemical Engineering Science 59 (2004) 223–234. [7] W. Li, H. Henry Yue, S. Valle-Cervantes, S. Joe Qin, Recursive PCA for adaptive process monitoring, Journal of Process Control 10 (2000) 471–486. [8] D.-H. Hwang, C. Han, Real-time monitoring for a process with multiple operating modes, Control Engineering Practice 7 (1999) 891–902. [9] S. Lane, E.B. Martin, R. Kooijmans, A.J. Morris, Performance monitoring of a multi-product semi-batch process, Process Control 11 (2001) 1–11.

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