Robust principal component pursuit for fault detection in a blast

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Robust principal component pursuit for fault detection in a blast furnace process YIJUN PAN, ChunJie Yang, RUQIAO AN, and Youxian Sun Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03338 • Publication Date (Web): 11 Dec 2017 Downloaded from http://pubs.acs.org on December 23, 2017

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Industrial & Engineering Chemistry Research

Robust principal component pursuit for fault detection in a blast furnace process Yijun Pan,†,‡ Chunjie Yang,∗,† Ruqiao An,† and Youxian Sun† Department of Control Science and Engineering, Zhejiang University, Hangzhou, China, and Department of Statistics, The University of California, Davis, Davis, USA E-mail: [email protected]

Abstract Since blast furnaces are generally controlled by operators, the minor faults regarded as disturbances might be contained in the collected data matrix. This can severely affect sample distributions, which leads to arbitrary fault detection results using traditional data-driven methods. In this paper, a novel fault detection method named robust principal component pursuit (PCP) to handle minor faults is proposed. The minor faults are separated from columns and rows respectively in the training matrix via two matrix norms. By applying the proposed robust PCP method, a low rank matrix containing important process information, as well as explicit variable relationships, and a block sparse matrix containing minor faults are derived. Moreover, the convergence of the proposed method is discussed. Hotelling’s T 2 statistic is potentially useful for online process monitoring in the low rank matrix. Finally, to evaluate the decomposition capacity of the proposed method for a matrix containing minor faults, a comparison of the proposed method with other robust method is presented. To test the effectiveness of the proposed method for fault detection, a numerical simulation is adopted at first. Finally, the power of the proposed method is illustrated in a real blast furnace process. ∗ To

whom correspondence should be addressed University ‡ Yijun Pan is a visiting scholar in UC, Davis † Zhejiang

1

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Introduction Blast furnace processes are complicated and large scale, making it difficult to analyze their production principles and build accurate models.1 The occurrence of anomalous events in blast furnaces results in enormous economic losses. To ensure the safety and reliability of blast furnace processes, fault detection is necessary.1 Generally speaking, three primary fault detection methods are used: analytic model methods, expert system methods and data-driven methods. More details about fault detection methods can be found in the literatures.2−3 Data-driven methods, one of the fault detection methods, are widely adopted in blast furnace processes.1,4−6 Blast furnaces are controlled subjectively by operators. The parameter adjustments and operation schemes may be different for different operators according to their different experiences. Sometimes, minor faults may be regarded as disturbances that do not need to be controlled by some operators. Therefore, the collected process data may contain minor faults. For traditional data-driven methods, faults and outliers in the training matrix would lead to arbitrary process monitoring results.7 As a result, robust data-driven methods are necessary in the blast furnace processes.8 Previously, several literature reports have been published on the issue of a collected data matrix with corrupted observations. Chen et al. developed a robust and efficient algorithm for the matrix completion problem with corrupted columns.9 Zhang et al. presented a matrix completion method involving corrupted and missing data with respect to a more general basis.10 They also suggested a universal choice of the regularization parameter in the matrix completion problem and reduced the computational cost. Liu et al. studied the problem of recovering the authentic samples that lie on a union of multiple subspaces from their corrupted observations.11 Rahmani et al. proposed a new robust PCA method for considering both element-wise and column-wise corruptions.12 In blast furnace processes, minor faults might be regarded as disturbances and are thus ignored by operators. Unlike noise and outliers in processes, the collected data matrix with minor faults has new characteristics. A minor fault is a continuous process occurring in some variables in a period of time, which is shown in a data matrix as some corrupted columns and rows. There are some robust data-driven methods that address noise and outliers;13−14 however, they cannot be 2


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used for addressing minor faults directly, and they do not consider the block sparse characteristic of a matrix with minor faults. A minor fault only exists in some columns and rows in a small part of the data matrix. The noise exists throughout processes with small values, and one outlier occurs for a variable for a certain observation.7 Thus, it is promising to consider robust fault detection method to address minor faults in the training matrix. Recently, some robust methods have been developed for fault detection. Candes et al. introduced a state-of-the-art robust PCA method named principal component pursuit (PCP) that decomposes a data matrix into a low rank part and a sparse part.15 In 2011, the PCP method was used for process monitoring for the first time by Isom et al.;16 they illustrated that PCP is robust to outliers and that it can detect and isolate faults simultaneously by observing the obtained sparse matrix. A new standardized method and a residual generator suitable for PCP generated process models were developed by Cheng et al.17 Pan et al. proposed a new mean-correlation statistic suitable for online process monitoring based on the PCP method.18 A coordinate descent algorithm based on PCP and its convergence proof that directly utilized a Lyapunov approach were also presented by Cheng et al.19 A process monitoring model and an online monitoring statistic with stable PCP were developed by Yan et al.20 Pan et al. introduced a novel IPCP method derived from low rank representation (LRR) and PCP methods, and an online process monitoring statistic was also developed.21 However, to the best knowledge of the authors, the issue of a training matrix containing minor faults in blast furnace processes has rarely been discussed. In this paper, a novel robust PCP method to handle minor faults in a training matrix is proposed. Compared with the robust matrix completion method introduced in the literature,9 the proposed method considers both columns and rows simultaneously. The proposed method divides the training matrix into two parts by solving an convex optimization model: a low rank matrix containing important process information, as well as explicit variable relationships, and a special sparse matrix containing minor faults. Since in a blast furnace process, to improve safety and reliability, some variables are used for measuring one state, the useful process information data matrix is arguably of low rank.2 Blast fur-

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naces are automation processes controlled by operators. Some minor faults such as small changes in temperature, raw material feeding in and parts hanging might be regarded as disturbances by operators, which exist in the collected data matrix. Moreover, a minor fault occurs at a certain period of time and includes certain variables, implying that the matrix containing minor faults has a block sparse characteristic. The blast furnace data matrix consists of a process-information part and a minor-fault part. Although the special sparse matrix is optimized to have few whole rows and columns that are non-zero, the characteristic of minor faults leads to the element in a special sparse matrix to be zero if the variable at that observation is faultless. Based on the data characteristics, the obtained special sparse matrix in a blast furnace process is block-sparse. Therefore, the proposed method is suitable for fault detection in blast furnace processes. The convergence proof of the proposed method is also presented in this paper. Next, Hotelling’s T 2 statistic is adopted for online process monitoring in a low rank matrix, and the method to build the T 2 statistic is the same as that in the PCA method except for the principal component selection. In section 2, the data definition, assumptions and problem formulation are given. The proposed fault detection method that considers minor faults in the training matrix is introduced in section 3. Section 4 illustrates the power of the proposed method via two numerical simulations and a blast furnace process. Finally, the conclusion of this paper is presented in section 5.

Problem Formulation Suppose that there is a data matrix X ∈ Rn×m , in which each row is an observation and each column is a variable. Among these n rows and m columns, some minor faults exist in parts of the data matrix. A minor fault is a continuous process that occurred in some variables during a period of time with small variations. Under the above setup, X is divided into three parts,

X = A + E + F,

4


(1)

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where A is a low rank matrix with useful process information as well as explicit variable relationships. E + F is a sparse matrix with minor faults. The problem is considered under the following two assumptions: Assumption 1. The process data without faults are noiseless and of low column rank. Assumption 2. A minor fault exists in some variables and continuous observations; i.e., the corrupted data are contained in certain columns and rows. The assumption of noiseless is mild, since the order of magnitude of noise is smaller than that of minor faults in normal condition. For noise, a simple method to relax the constraint condition is the use of a noise matrix N. Thus, the new constraint condition is X = A + E + F + N, where N is a matrix containing noise; more details can be found in the literature.22 Since the solutions and conclusions of the two constraint conditions are similar, noise is ignored in this paper for simplification. Moreover, an outlier can be regarded as a special fault. Because an outlier exists at an observation and for one variable, it does not require individual processing. In this paper, there is no restricted condition regarding faults except for the number of faults.

Fault detection with minor faults in the training matrix Robust matrix recovery with corrupted columns Consider a data matrix X ∈ Rn×m , for which each row is a sample, and each sample has m variables. For a robust matrix recovery problem with corrupted columns, a low rank matrix A and a corrupted columns matrix E can be solved by the following model: min kAk∗ + λ kEk1,2

(2)

s.t. X = A + E, where kAk∗ is the nuclear norm of matrix A. kEk1,2 is the l1,2 norm of matrix E. The parameter λ is used for balancing the two terms in kAk∗ + λ kEk1,2 , and the choice λ is universal, based on

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a formula that can be found in the literature.9 In the relevant literature, the mathematical analyses and theoretical proofs regarding matrix completion with corrupted columns and missing data are discussed; these works are similar to the robust matrix recovery shown in this subsection. The l0 norm is adopted for addressing random corruptions in the PCP method15 and the l1,0 norm is introduced to handle column corruptions.23 kEk0 is the number of non-zero elements in matrix E, and kEk1 is the magnitude sum of all elements in matrix E. The value of kEk1,0 is the number of non-zero l2 norms of each column in matrix E, and kEk1,2 is the sum of l2 norms of each column in matrix E. These variables are defined by the following mathematical equations. Since the solution of kEk0 minimization is equal to minimization of kEk1 , the kEk1,2 norm could be adopted for calculating kEk1,0 .23 kEk0 = the number o f {Ei, j 6= 0} kEk1 = ∑ |Ei, j | i, j

(3) kEk1,0 = the number o f {kE:, j k2 6= 0} s m

kEk1,2 =

n

∑ ∑ ([E]i j )2.

j=1

i=1

Robust principal component pursuit with minor faults As mentioned above, the low rank matrix A is clean without corrupted columns. However, in blast furnace processes, a minor fault occurs during a period of time and for several variables. The collected data matrix is corrupted in some columns and rows. Therefore, a novel model should be developed for addressing the corrupted rows simultaneously. Thus, the improved model is min kAk∗ + λ kEk1,2 + β kFk2,1

(4)

s.t. X = A + E + F,

where

kFk2,1 = ∑ni=1

q

1 .15 max(n,m)

∑mj=1 ([F]i j )2 and λ = β = √

For the robust PCP model shown in equation (4), the inexact augmented Lagrange multiplier

6


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(IALM) method is applied.24 The augmented Lagrangian function is

L(A, E, F,Y, µ) = kAk∗ + λ kEk1,2 + β kFk2,1 + hY, X − A − E − Fi +

µ kX − A − E − Fk2F . (5) 2

The method to solve the robust PCP problem is presented in Algorithm 1. Since Ek and Fk indicate the minor faults in columns and rows respectively, in each iteration Ek is used for calculating Fk , not Ek+1 . The convergence is demonstrated in the next subsection. Algorithm 1 The inexact augmented Lagrange multiplier method to solve robust PCP problem (3) Input: X ∈ Rn×m , parameters λ , β Initialize: A0 = 0, E0 = 0, F0 = 0, Y0 = 0, µ0 = 10−8 , ρ = 1.1, maxµ = 1010 , ε = 10−6 While not converged Do Ak+1 = argmin µ1k kAk k∗ + 12 kAk − (X − Ek − Fk + Yµkk )k2F Ek+1 = argmin µλk kEk k1,2 + 12 kEk − (X − Ak+1 − Fk + Yµkk )k2F Fk+1 = argmin µβk kFk k2,1 + 21 kFk − (X − Ak+1 − Ek + Yµkk )k2F Yk+1 = Yk + µk (X − Ak+1 − Ek+1 − Fk+1 ) µk+1 = min(ρ µk , maxµ ) convergence condition: kX − Ak+1 − Ek+1 − Fk+1 k∞ < ε End Output solutions: (Ak , Ek , Fk )

Note that although calculating matrices Ak , Ek and Fk are convex optimization processes, Ak , Ek and Fk have closed-form solutions. The matrix Ak is solved by the singular value thresholding operator described in Lemma 3.1. Lemma 3.1 Suppose there is a matrix X ∈ Rn×m with rank r and a parameter τ; the singular value thresholding operator Dτ (X) is computed by Dτ (X) = USτ [Σ]V T ,

(6)

where Sτ [Σ] = diag((σi − τ)+ ), and USV T is singular value decomposition (SVD) of matrix X, Σ = diag({σi }1≤i≤r ).25

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Note that according to Lemma 3.1, matrix Ak can be calculated by

Ak+1 = D 1 (X − Ek − Fk + µk

Yk ). µk

(7)

Matrices Ek and Fk are solved by Lemma 3.2.26 Lemma 3.2 For any α, β > 0 and t ∈ Rq , the minimizer of

minq αksk +

s∈R

β ks − tk2 2

(8)

is given by s(t) = max{ktk −

t α , 0} , β ktk

(9)

where we follow the convention 0 · (0/0) = 0.

The convergence theorem In the literature,27 the ALM method is proposed for solving the optimization function,

min f (X), sub ject to h(X) = 0,

(10)

where f : Rn → R and h : Rn → Rm . The augmented Lagrangian function of the optimization model is L(X,Y, µ) = f (X) + hY, h(X)i +

µ kh(X)k2F . 2

(11)

µ is a parameter with a positive value. When parameter µk is nondecreasing in the iteration process and functions f (X) and h(X) are continuously differentiable functions, the ALM method converges to the optimal solution under some mild conditions.27 However, the object function in the robust PCP method is not a continuously differentiable function, the above result cannot be applied here directly. The convergence property of Algorithm 1 should be proved. Since matrices E and F contain minor faults in columns 8


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and rows respectively, the matrix E + F can be regarded as a matrix containing minor faults. The optimal solution E, F should be added to consider. Therefore, the training matrix would be decomposed into a low rank matrix A and a sparse matrix E + F. Theorem 1 In the robust PCP method, if parameter µk is nondecreasing in the iteration process +∞

and ∑ µk−1 = +∞, then there exists an optimal solution (Ak , Ek + Fk ) solved by Algorithm 1. k=1

Proof. Defining (A∗ , E ∗ , F ∗ ,Y ∗ ) as a saddle point of the following Lagrangian function, then A∗ + E ∗ + F ∗ = X.

L(A, E, F,Y ) = kAk∗ + λ kEk1,2 + β kFk2,1 + hY, X − A − E − Fi.

(12)

Y ∗ ∈ ∂ kA∗ k∗ ,Y ∗ ∈ ∂ (kλ E ∗ k1,2 ),Y ∗ ∈ ∂ (kβ F ∗ k2,1 ).

(13)

Thus,

Define a new sequence Ybk = Yk−1 + µk−1 (X − Ak − Ek−1 − Fk−1 ). According to Algorithm 1, X − Ak+1 − Ek+1 − Fk+1 = µk−1 (Yk+1 − Yk ), the following equation could be concluded; this equation is used for proving some sequences that are nondecreasing in the iteration process. kEk+1 − E ∗ + Fk+1 − F ∗ k2F + µk−2 kYk+1 −Y ∗ k2F = kEk − E ∗ + Fk − F ∗ k2F + µk−2 kYk −Y ∗ k2F − kEk+1 − Ek + Fk+1 − Fk k2F − µk−2 kYk+1 −Yk k2F − 2µk−1 (hYk+1 −Yk , Ek+1 − Ek i + hYk+1 −Yk , Fk+1 − Fk i + hAk+1 − A∗ , Ybk+1 −Y ∗ i + hEk+1 − E ∗ ,Yk+1 −Y ∗ i + hFk+1 − F ∗ ,Yk+1 −Y ∗ i). (14) Based on Lemma 1 in the literature,24

Ybk+1 ∈ ∂ kAk+1 k∗ ,Yk+1 ∈ ∂ (kλ Ek+1 k1,2 ),Yk+1 ∈ ∂ (kβ Fk+1 k2,1 )

(15)

are derived. Moreover, if a function is convex, then its subgradient is a monotone operator.24 The

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following formula hAk+1 − A∗ , Ybk+1 −Y ∗ i ≥ 0 hEk+1 − E ∗ ,Yk+1 −Y ∗ i ≥ 0 hFk+1 − F ∗ ,Yk+1 −Y ∗ i ≥ 0

(16)

hEk+1 − Ek ,Yk+1 −Yk i ≥ 0 hFk+1 − Fk ,Yk+1 −Yk i ≥ 0 are concluded. Therefore, according to (14) and (16), kEk+1 − E ∗ + Fk+1 − F ∗ k2F + µk−2 kYk+1 −Y ∗ k2F ≤ kEk − E ∗ + Fk − F ∗ k2F + µk−2 kYk −Y ∗ k2F (17) is obvious. Since µk+1 ≥ µk , −2 kEk+1 − E ∗ + Fk+1 − F ∗ k2F + µk+1 kYk+1 −Y ∗ k2F ≤ kEk − E ∗ + Fk − F ∗ k2F + µk−2 kYk −Y ∗ k2F (18)

could be derived. The following equation could be inferred by transforming equation (14), which is used for obtaining a solution of the model shown in equation (4). µk−2 kYk+1 −Yk k2F = kEk − E ∗ + Fk − F ∗ k2F + µk−2 kYk −Y ∗ k2F − kEk+1 − E ∗ + Fk+1 − F ∗ k2F − µk−2 kYk+1 −Y ∗ k2F − kEk+1 − Ek + Fk+1 − Fk k2F − 2µk−1 (hYk+1 −Yk , Ek+1 − Ek i + hYk+1 −Yk , Fk+1 − Fk i + hAk+1 − A∗ , Ybk+1 −Y ∗ i + hEk+1 − E ∗ ,Yk+1 −Y ∗ i + hFk+1 − F ∗ ,Yk+1 −Y ∗ i) ≤ kEk − E ∗ + Fk − F ∗ k2F + µk−2 kYk −Y ∗ k2F − kEk+1 − E ∗ + Fk+1 − F ∗ k2F − µk−2 kYk+1 −Y ∗ k2F −2 ≤ kEk − E ∗ + Fk − F ∗ k2F + µk−2 kYk −Y ∗ k2F − kEk+1 − E ∗ + Fk+1 − F ∗ k2F − µk+1 kYk+1 −Y ∗ k2F (19)

As a result, +∞

∑ µk−2kYk+1 −Yk k2F < +∞.

k=1

10


(20)

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Thus, −1 kYk −Yk−1 kF → 0. kX − Ak − Ek − Fk kF = µk−1

(21)

Hence, (Ak , Ek + Fk ) is a feasible solution for the robust PCP problem. Next, define f ∗ = kA∗ k∗ + λ kE ∗ k1,2 + β kF ∗ k2,1 as an optimal value. As mentioned above, Ybk+1 ∈ ∂ kAk+1 k∗ , Yk+1 ∈ ∂ (kλ Ek+1 k1,2 ) and Yk+1 ∈ ∂ (kβ Fk+1 k2,1 ) based on Lemma 3 in the literature,24 the following formula is concluded: kAk k∗ + λ kEk k1,2 + β kFk k2,1 ≤ kA∗ k∗ + λ kE ∗ k1,2 + β kF ∗ k2,1 − hYbk , A∗ − Ak i − hYk , E ∗ − Ek i − hYk , F ∗ − Fk i = f ∗ + hY ∗ − Ybk , A∗ − Ak i + hY ∗ −Yk , E ∗ − Ek i + hY ∗ −Yk , F ∗ − Fk i − hY ∗ , A∗ − Ak + E ∗ − Ek + F ∗ − Fk i. (22) The next goal is to prove that there is a sequence that satisfies kAk k∗ + λ kEk k1,2 + β kFk k2,1 ≤ f ∗ , i.e., exactly one optimal solution exists. Similar to the deduction of (20), +∞

−1 (hAk − A∗ , Ybk −Y ∗ i + hEk − E ∗ ,Yk −Y ∗ i + hFk − F ∗ ,Yk −Y ∗ i) < +∞ ∑ µk−1

(23)

k=1

is concluded. +∞

−1 = +∞, a subsequence (Ak j , Ek j , Fk j ) that satisfies hAk j − A∗ , Ybk j − Y ∗ i + hEk j − As ∑ µk−1 k=1

E ∗ ,Yk j − Y ∗ i + hFk j − F ∗ ,Yk j − Y ∗ i → 0 exists. Moreover, (Ak , Ek + Fk ) is a feasible solution for the robust PCP problem, and Ak + Ek + Fk = X = A∗ + E ∗ + F ∗ . From (22), it can be seen that lim kAk j k∗ + λ kEk j k1,2 + β kFk j k2,1 ≤ f ∗

j→+∞

(24)

Hence, (Ak j , Ek j + Fk j ) is approximately equal to the optimal solution (A∗ , E ∗ + F ∗ ) of the robust PCP problem. Since µk → +∞ and Yk are bounded, it can be seen that {kEk j − E ∗ + Fk j − F ∗ k2F + µk−2 kYk j −Y ∗ k2F } → 0. j

11


(25)


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Moreover, as mentioned above, −2 kEk+1 − E ∗ + Fk+1 − F ∗ k2F + µk+1 kYk+1 −Y ∗ k2F ≤ kEk − E ∗ + Fk − F ∗ k2F + µk−2 kYk −Y ∗ k2F , (26)

so it could be concluded that {kEk − E ∗ + Fk − F ∗ k2F + µk−2 kYk −Y ∗ k2F } → 0.

(27)

lim Ek + Fk = E ∗ + F ∗ .

(28)

In other words, k→+∞

Since lim X − Ak − Ek − Fk = 0 and X = A∗ + E ∗ + F ∗ , it can be seen that lim Ak = A∗ . Thus, k→+∞

k→+∞

the convergence of the robust PCP problem is proven.

Fault detection using the proposed robust PCP method In blast furnace processes, the data are collected from production processes directly. The data may contain some minor faults that are not controlled during operating. The proposed robust PCP method shown in equation (4) is effective for considering minor faults, as illustrated in the next section. Hence, before using a training matrix to build a model, it should be preprocessed to remove minor faults. As mentioned above, the training matrix is decomposed into a process information data matrix A and a minor faults data matrix E + F. Either E or F is meaningless if considered individually. The low rank matrix A is a clean training matrix without minor faults and Hotelling’s T 2 statistic is used for online process monitoring. Given a training matrix X ∈ Rn×m and a testing matrix Z ∈ p×m , there are n observations in the training matrix, and p is the number of observations in the testing matrix with m measurement variables. The steps using robust PCP for fault detection are described as follows: Step 1: Matrix decomposition Divide the training matrix X into two parts by using the proposed robust PCP method: a low

12


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rank matrix A with useful process information and a block sparse matrix E + F with minor faults. min kAk∗ + λ kEk1,2 + β kFk2,1

(29)

s.t. X = A + E + F Step 2: Statistic calculation Calculate the mean values and the standard deviations of the variables in matrix A and singular value decomposition of low rank matrix A.28

[U, Λ, P] = svd(A)

(30)

where Λ is the singular value matrix and P is the loading matrix. Step 3: Hotelling’s T 2 statistic threshold The threshold Tα2 of the T 2 statistic calculated based on the training matrix is given as follows: Tα2 =

(n − 1)m × [Fα (m, n − m)] n−m

(31)

where Fα (m, n − m) could be found in the F-distribution chart with significance level (α = 0.05). m and (n − m) are the degrees of freedom.29 Step 4: Normalization The normalization formula of testing matrix Z is given as follows:17

di j =

zi j − a j sj

(32)

where di j is the i j th element of normalized testing matrix D. zi j is the i j th element of the original testing matrix Z. a j and s j are the mean and standard deviation, respectively, of the j th variable calculated in step 2. Step 5: T 2 statistic

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Calculate the online Hotelling’s T 2 statistic by Ti2 = di × P × Λ−2 × PT × diT

(33)

where di is the i th row of normalized testing matrix D.18 Step 6: Online process monitoring If the T 2 statistic calculated in step 5 is greater than the normal threshold Tα2 , then there exists an anomalous event.

Simulation In this section, the simulation results of the proposed robust PCP method are discussed. There are three parts: first, a numerical simulation is used for demonstrating the decomposition capacity of the proposed method for a matrix containing minor faults; second, a numerical simulation and a real blast furnace process are used for illustrating the power for fault detection by the proposed method.

Numerical simulation In this subsection, a numerical simulation is used for demonstrating the decomposition capacity of the proposed robust PCP method for a matrix containing minor faults. In the literature,9 a robust method given by equation (2) is proposed for handling corrupted columns. Since both robust methods could preprocess minor faults, a comparison between the two methods is demonstrated. A low rank matrix is constructed by A = TW T ∈ Rn×p with T ∈ Rn×r , W ∈ R p×r , which have independent and identically distributed entries N(0, 1).15 The matrix H ∈ Rnγ×pξ denoting minor faults is constructed by random variables drawn from N(1, 1). The fraction of fault observations is shown by γ, and the fraction of faulted variables is shown by ξ . There is only one minor fault in this simulation. In this part, n = 500, r = 30, p = 100, γ = 0.3, ξ = 0.1, λ = β = 0.56.

14


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1 ,15 max(n,m)

The parameters λ and β are first calculated by λ = β = √

and are then adjusted by

experience. The simulation results of the two methods are shown in Table 1. Table 1: Comparison between two methods rank (A∗ ) kE ∗ + F ∗ − E − Fk2F Robust PCP (4) 31 123.6802 Robust Matrix Recovery (2) 58 127.2352

convergence time 43.99 53.51

From Table 1, the proposed method is found to obtain a low rank matrix and a sparse matrix. It can recover the useful process information and has a high convergence speed. Similar to the robust matrix recovery model shown in equation (2), the proposed method can also consider minor faults and has a better decomposition capacity. The rank of the obtained low rank matrix A∗ is almost the same as that of the original matrix A. In other words, the robust PCP method can efficiently remove minor faults in the data matrix and recover a goal matrix containing useful process information.

Fault detection in a numerical simulation In the previous subsection, the decomposition capacity of the proposed method for a matrix containing minor faults is demonstrated. In this part, the RPCA method1 introduced by Hubert et al.14 is used for comparison of the power for fault detection with the proposed robust method. More details regarding the RPCA method can be found in the literature.14 The purpose of the numerical simulation is to illustrate that the proposed RPCP method is useful for removing minor faults in the training matrix. The minor faults in a blast furnace process are mainly from small changes in temperature, few raw material feeding in and parts hanging. Since these changes reflected in the collected data matrix are observation variations that could approximatively be denoted by meanshift of variables, the training matrix is the same as the one in the last subsection. Since in a blast furnace process, to improve safety and reliability, some variables are used for measuring one state, it is arguable that the useful process information data matrix is low rank. The training matrix, 1 The

code is derived from part of LIBRA: the Matlab Library for Robust Analysis, which is available at: http://wis.kuleuven.be/stat/robust.html.

15



which is constructed by a low rank part and a minor faults part in the numerical simulation, is consistent for a blast furnace process. The validation matrix used for adjusting the normal condition threshold is constructed by 50 normal observations. There are 50 normal observations and (n − 50) fault observations in the testing matrix. Since the fault data are rare in the blast furnace processes, the threshold is only adjusted by normal observations at significance level α = 0.05. The 100 normal observations are constructed by A = TW T ∈ R100×100 with T ∈ R100×r and W ∈ R100×r , which have independent and identically distributed entries N(0, 1). The fault observations are constructed by random variables drawn from N(3, 0.8). Since this part is used for illustrating the power of the proposed method for removing minor faults in the training matrix, a fault in the testing matrix is just a different distribution from the training matrix, which is not important. The 50 normal observations in the testing matrix are used for monitoring the false alarm rate. Here, n = 450, r = 30, λ = β = 0.56. For the RPCA method, eleven PCs are chosen in the principal component space based on the 90% accumulative contribution rate derived from the RPCA model.12 The online process monitoring results of the proposed method and the RPCA method are shown in Fig. 1-Fig. 2. To evaluate the effectiveness of the proposed robust PCP method, the fault detection rate (FDR) and the fault alarm rate (FAR) are presented in Table 2. The FDR indicates that a fault observation is identified as a fault, and the FAR indicates that a normal observation is identified as a fault. From Fig. 1-2 and Table 2, it can be concluded that the proposed method is more efficient for fault detection than RPCA. Compared with RPCA, the proposed method focuses on recovering a low rank matrix and obtaining a minor fault matrix, not just finding outliers, which is suitable for the data in this part. The proposed method obtains a low rank matrix, reducing the influences of minor faults. The statistic values of fault observations are obviously greater in the proposed method than in RPCA. The FDR of the RPCP method is equal to 1, i.e., the proposed method can detect almost all fault observations in this numerical simulation, and the FAR is 0.02, which is also reasonable. The proposed method can separate the minor faults from the original data matrix, and the recovered low rank matrix contains important information as well as represents

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the explicit variable relationships of the process. From the simulation results, the power of the proposed method for fault detection is illustrated in a numerical example. Since in blast furnace processes, the minor faults might be regarded as disturbances, which are contained in the collected data matrix, a suitable robust method to preprocess minor faults is necessary. 30

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Fault detection for a blast furnace process The production statuses are recorded every time in a blast furnace process. In the recorded chart, the normal situations, fault types and time are collected. Moreover, there are some production technology indices, such as the silicon content, that should be in a certain range under normal

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50 observations threshold

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Figure 2: The simulation result based on the RPCA method for the numerical simulation with Tα2 = 10.1690. conditions. Based on recorded production statuses and production technology indices, the normal and fault data matrices are collected.21 Blast furnaces are continuous production processes with fuel, ores and limestone supplied from the top parts while hot air is blown into the bottom parts. The productions are molten metals and slag.4 Blast furnaces are complicated, making it hard to analyze production principles and build accurate models. Some minor faults such as small changes in temperature, raw material feeding in and parts hanging might be regarded as disturbances by operators, as a result, minor faults can exist in the collected training matrix. In blast furnace processes, traditional data-driven methods such as PCA and partial least squares (PLS) have been adopted widely.30−31 More data-driven methods for fault detection in blast furnace processes could be found in references.4−6,32 However, using these methods directly for online process monitoring might lead to poor performance if minor faults exist. Therefore, it is necessary to develop robust data-driven methods that could preprocess minor faults.7 Since blast furnaces are continuous processes that are controlled by operators all the time, anomalous events and fault data are rare. Therefore, in 400 observations, one normal data matrix and six fault data matrices are found. The six faults are low stock line, cooling, heating, chimney, hanging and slip, as listed in Table 1 in the supporting information. The normal data matrix con-

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tains 100 observations, and each fault type contains 50 observations.33 Here, normal observations are chosen based on the recorded production statuses and the production technology indices that might contain minor faults. Each observation has the 18 variables. The details of the 18 variables can be found in Table 2 in the supporting information. The 100 normal observations are divided into three parts. Fifty normal observations are used for building a training matrix and 25 normal observations are used for establishing a validation matrix to adjust the normal condition threshold. The other 25 normal observations and 50 fault observations constitute each testing matrix. Here, 1 ,15 max(n,m)

λ = β = 0.96. The parameters λ and β are first calculated by λ = β = √

and are then

adjusted by experience. Four PCs are retained in the score space based on the 90% accumulative contribution rate in the RPCA method. The simulation results based on the proposed method and the RPCA are shown in Fig. 3-6. The FDR and FAR of the two methods are shown in Table 3. 30

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Figure 3: The simulation result based on the proposed method for fault 3 in a blast furnace process with Tα2 = 9.3075 × 1028 From Fig. 3-6 and Table 3, the proposed method is found to have a better performance than the RPCA for fault detection. It can remove the influences of minor faults in the training matrix effectively, demonstrating that this method is suitable for a blast furnace process. Fig. 3 shows the 19



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Figure 5: The simulation result based on the proposed method for fault 5 in a blast furnace process with Tα2 = 9.3075 × 1028

Table 3: The FDR and FAR of two methods in a blast furnace process 1 2 3 4 5 6

FDR-Robust PCP FAR-Robust PCP FDR-RPCA FAR-RPCA 0.86 0.04 0.24 0.12 1 0.04 0.66 0.12 1 0.04 0.98 0.12 1 0.04 0.90 0.12 1 0.04 0.56 0.12 0.98 0.04 1 0.12

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0.09 observations threshold

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Figure 6: The simulation result based on the RPCA method for fault 5 in a blast furnace process with Tα2 = 1.1 × 10−3 simulation result of a heating fault. The T 2 statistic chart indicates that the proposed robust PCP method could identify the fault at the 26th observations, demonstrating the power of the proposed method. Heating is a common fault in blast furnace processes that leads to increases in the coke ratio cost and reduces the blast furnace life. Fault 5, hanging, refers to the situation in which the raw materials do not descend smoothly.1 Fig. 5 shows the online process monitoring result. The fault occurred at the 26th observation, which is detected efficiently. However, the RPCA method could only detect the fault at the 63rd observation, which has a delay. The FDR based on the RPCP method is greater than that based on the RPCA method, while the FAR is smaller, which means the proposed method could detect most fault observations in a blast furnace process with a low FAR guarantee. From the simulation results, the raw training matrix with minor faults is found to build an arbitrary model and leads to poor fault detection results. Thus, in this part, the power of the proposed method for fault detection in a blast furnace process is demonstrated.

Conclusion In this paper, a novel robust PCP method was proposed for online process monitoring in a blast furnace process. The proposed method recovers an important process information matrix from a

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raw training matrix with convergence guarantee. Hotelling’s T 2 statistic is used for fault detection. The novel method obtains a fault-less training matrix via calculation of an optimization model; the novel method was found to be much better than other robust methods. The decomposition capacity for a matrix containing minor faults was demonstrated via a numerical simulation. The power of proposed method was tested on a numerical simulation and a blast furnace process. In this paper, the parameters λ and β were chosen by a formula and adjusted by experience; an improved selection method should be developed. In further work, choosing parameters via appropriate formulas should be studied.

Acknowledgement We thank Prof. Xiaodong Li from UC, Davis for helpful guidance. Thanks to China Scholarship Council for funding. This work was supported by the 863 Program (Grant 2012AA041709) and the National Natural Science Foundation of China (Grant No. 61290321).

Supporting Information The variables used for fault detection in a blast furnace process; The six kinds of faults in a blast furnace process

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TOC graphic Fault 3 in a blast furnace process

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