Multimode Continuous Processes Monitoring Based on Hidden Semi

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Multimode Continuous Processes Monitoring Based on Hidden Semi-Markov Model and Principle Component Analysis Zhijiang Lou, and Youqing Wang Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b01721 • Publication Date (Web): 31 Oct 2017 Downloaded from http://pubs.acs.org on November 5, 2017

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Industrial & Engineering Chemistry Research

Multimode Continuous Processes Monitoring Based on Hidden Semi-Markov Model and Principal Component Analysis Zhijiang Lou1, Youqing Wang*2,1 1. College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China 2. College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China *Email: [email protected] Abstract: Several studies have applied hidden Markov model (HMM) in multimode process monitoring. However, because the inherent duration probability density of HMM is exponential, which is inappropriate for the modeling of multimode process, the performance of these HMMbased approaches are not satisfactory. As a result, hidden semi-Markov model (HSMM), which integrated the mode duration probability into HMM, is combined with principal component analysis (PCA) to handle with the multimode feature, named as HSMM-PCA. PCA is a powerful monitoring algorithm for unimodal process, and HSMM specializes in mode division and identification. HSMM-PCA inherits the advantages of these two algorithms and hence it performs much better than the existing HMM-based approaches do. In addition, HSMM-PCA can detect the mode disorder fault which challenges the most multimode approaches. Key words: process monitoring; multimode process; hidden Markov model (HMM); hidden semi-Markov model (HSMM); principal component analysis (PCA)

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1 INTRODUCTION To ensure the process safety and deliver of high quality consistent product, process monitoring is critical importance for chemical industry. In recent years, with the fast development of computer and information technologies, a massive amount of process history data is available for abnormal conditions detection. As a result, multivariate statistical process control (MSPC) methods1, 2, which only require historical process data, are becoming more and more popular in large-scale chemical industries. As one of the most popular MSPC methods, principal component analysis (PCA)3-5 has been successfully applied in online continuous process monitoring, and a great quantity of advanced PCA approaches have been put forward in literature6-9. One basic assumption for PCA is that the normal data should come from a single operation region, i.e. the process data obeys unimodal Gaussian distribution. In reality, due to various factors, such as the alterations of feedstock and compositions, the changing of manufacturing strategies, and the disturbance in the external environment, usually the industry processes works at multiple operating modes and hence this assumption is invalid. As a result, the process mode is not fixed and it shifts from one mode to another, which confines the application of the traditional PCA approaches. To address the multimode problem, three kinds of approaches have been put forward: (1) global model approach, which builds one uniform model to fit all modes10; (2) the multimode approach, which divides the entire process into different stages by using the clustering algorithms and then builds the corresponding sub-PCA model for each stage separately11, 12; (3) the mixture model approach, which combines different PCA models for process approximation1315

. In recent years, many research efforts have been done based on these approaches, and a lot of

papers have been published16, 17.


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All these above-mentioned approaches have one common drawback, i.e., they do not take the mode shifting probability into consideration. For most chemical industry processes, usually their operating sequences are fixed and hence the current operation mode probabilistically depends on the last operation mode. Without the restriction of mode shifting probability, in the offline modeling stage, these multimode approaches may falsely divide the operation modes and build the wrong PCA model for each mode; and in the online monitoring stage, they may obtain the wrong mode localization result. Hidden Markov model (HMM)18-20, which was first developed by Baum and his co-workers, has gained great success in a wide range of fields, such as speech recognition and bioinformatics. HMM contains finite number of hidden states and each hidden state emits one observation; moreover, each hidden state shifts to other states with a certain probability, called as state transition probability. For HMM, the hidden states can be regard as the operation modes in multimode process and the state transition probability can be used to describe the operation mode shifting probability, so several papers have adopted HMM to address the multimode problem in recently years21-25. For these papers, the major weakness is that the inherent duration probability density of HMM is exponential26, whereas the duration of operation modes in real chemical processes usually fluctuates around a nonzero value, so HMM is inappropriate for description of multimode processes. Another drawback of these HMM-based approaches is that they are invalid for detecting mode disorder fault. The mode disorder fault means that data in each mode is in normal condition whereas the mode shifting order is wrong. For example, for a process where mode A can only shift to mode B, when a mode shifting disorder fault occurs (mode A directly shift to mode C), these HMM-based approaches just adopt the most suitable mode model (mode C) to monitor the process data. Without the restriction that mode A cannot shift to mode C


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directly, these methods cannot detect this abnormal condition, because the data in mode C is normal. As an extended model of HMM, hidden semi-Markov model (HSMM)27-29 introduces the state duration probability matrix to describe the time spent in a given operation mode, so HSMM has better ability to describe multimode processes. In papers30, 31, Chen and Jiang first combined HSMM with two improved PCA approaches, and hence achieved MPCA-HSMM and DMPCAHSMM for monitoring multimode batch process. The main idea of MPCA-HSMM and DMPCA-HSMM is similar: preprocessing the whole process data by using PCA and then monitoring the uncorrelated principal components with HSMM. However, PCA is a unimode algorithm, so it may fail in extracting the uncorrelated principal components in different operation modes32. To address this issue, this article exchanged the order of HSMM and PCA, and proposed a new combination of PCA and HSMM (HSMM-PCA). In HSMM-PCA, HSMM is used for dividing and identifying the process modes first, and then PCA is adopted for monitoring the data in each operation mode. The greatest advantage of HSMM-PCA is that it adopts the mode affiliation information of the historical data, the mode shifting probability, and the mode duration probability for mode identification, and hence it can detect the mode disorder fault in multimode process, which is validated by the simulation in the modified TE process. The characteristics of the traditional multimode approaches, the existing HMM-based approaches, and HSMM-PCA are summarized as in Table 1. The remainder of this paper is structured as follows. The HSMM and PCA are briefly introduced in Section 2; in Section 3, HSMM-PCA based process monitoring framework is discussed; then the parameter selection problem and sensitivity analysis is studied in Section 4; Section 5 reports the simulation studies of the proposed approach using the modified Tennessee


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Eastman (TE) simulation process33 and its comparison with the existing multimode approaches; finally, conclusions are drawn in Section 6. Table 1. Characteristics of various multimode approaches. Information of mode shifting probability Traditional multimode approaches Existing HMM-based approaches The proposed HSMM-PCA

Duration distribution

Detect mode disorder fault

Not been used

Incapable

Only been used in mode division Been used in both mode division and identification

Restricted to exponential distribution

Incapable

No restriction

Capable

2 PRELIMINARIES This section briefly reviews two algorithms, HSMM and PCA, as a prelude to the proposed approach. 2.1. Hidden Semi-Markov Model HSMM is an extension of HMM, and generally speaking, HSMM can be regard as a HMM with explicit state duration probability distributions. HSMM represents a stochastic sequence, where the states are unobservable, but each state generates a sequence of observations according to a probability distribution, so these states are called as hidden states. The key elements in HSMM are as follows: (1) The discrete hidden states S = {S1 , S2 ,L, S M } , where M denotes the number of possible hidden states in the whole process. (2) The

state

transition

probability

matrix

A = {aij }

(

1 ≤i, j ≤ M

),

where

aij = P ( q (t + 1) = S j | q (t ) = S i ) denotes the transition probability from state S i to the state S j

, and q (t ) ∈ {S1 , S2 ,L , S M } denotes the hidden state at time t .


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(3) The initial state probability vector π = {πi } , where πi = P ( q(1) = Si ) . (4) The state duration probability distribution matrix

Di (T ) = P ( q(t + 1) = Si ,L , q(t + T − 1) = Si , q(t + T ) ≠ Si | q(t ) = Si , q(t − 1) ≠ Si ) , where T ∈ {1, 2,L , Tmax } denotes the duration of the hidden state, and Tmax is the maximum amount of time spent in each state. (5) The observations o (t ) ( t = 1, 2,L , Ttotal ), where Ttotal denotes the total number of observations in the process. (6) The observations probability distribution Bi ( o(t ) ) = P ( o(t ) | q (t ) = Si ) , which indicates that the output at time t depends only on the corresponding hidden state qt . Take λ = ( A, π, { Bi } , { Di }) as the set of whole model parameters, and they can be estimated by the forward-backward algorithm, which can be referred in34 for details. Then one can calculate the joint probability as follows:

ς i (t ) = P ( o(1), o(2),L , o(Ttotal ), q (t ) = Si λ ) ,

(1)

Further, the state affiliation probability γ i (t ) can be obtained as

γ i (t ) = P(q (t ) = Si λ , o(1), o(2),L , o(Ttotal ))  P(o(1), o(2),L , o(Ttotal ), q (t ) = Si λ )    P ( o(1), o(2),L , o(Ttotal ) )   , = M    ∑ P(o(1), o(2),L , o(Ttotal ), q (t ) = S j λ )   j =1  P ( o(1), o(2),L , o(Ttotal ) )       ς (t ) = M i ∑ ς j (t )

(2),

j =1


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And the maximum a posteriori (MAP)35 estimation of the hidden states can be obtained as follows: qˆ(t ) = arg max ς i (t ) = arg max γ i (t ) , Si ,1≤i ≤ M

(3)

Si , 1≤i ≤ M

2.2. Principal Component Analysis for process monitoring PCA is a dimension reduction method that compresses the original high dimensional data into a set of lower dimensional latent variables, or called the principal components (PCs). Before applying PCA, the process data should be normalized to zero means and unit variance by subtracting the average of the training data and dividing by the standard deviation of the training data. Then the normalized process data matrix X ∈ R n×s (where n is the number of samples and

s is the number of variables) can be decomposed as X = TP T + E ,

(4)

where T ∈ R n× ρ represents the score matrix, P ∈ R s× ρ refers to the loading matrix, and E ∈ R n× s is the matrix of residuals. Then PCA constructs the T 2 and SPE indexes

36

to monitor the principal component space

and residual space spanned by T and E respectively. Statistic T 2 is a measure of the principal components and statistic SPE is a measure of the approximation error. Given a normalized monitoring vector y ∈ R1×s , the T 2 and SPE statistics can be computed as follows: T 2 = yP ( Λ ) −1 PT y T ,

(5)

SPE = y (I − PPT )(I − PPT ) y T ,

(6)

where I is the identity matrix; Λ = diag (λ1 L λρ ) ∈ R ρ × ρ is the estimated covariance matrix of principal component scores. The control limit of T 2 index for a Gaussian process is


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δ T2 =

( n − 1)( n + 1) ρ Fα ( ρ , n − ρ ) , where Fα ( ρ , n − ρ ) is an F-distribution with ρ and (n − ρ ) n(n − ρ )

degrees of freedom with the level of significance

δ = θ1[ 2 Q

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Cα h0 2θ2

θ1

1

θ h (h ) + 1 + 2 0 0 ]h , where θi = θ1 0

s

∑ρ λ

i j

α ; the threshold of Q is

， h0 = 1 −

j = +1

2θ1θ3 ， and Cα is the normal 3θ 22

deviate corresponding to the ( 1 − α ) percentile.

3 PROPOSED METHODOLOGY The last section has presented the preliminaries of HSMM and PCA. Because PCA is a widely acknowledged algorithm in unimodal process monitoring, and HSMM specializes in mode division and identification, this section combines them for multimode process monitoring. As shown in Figure 1, HSMM divides and identifies the process modes, and then PCA is adopted for monitoring the data in each operation mode. Before the detail introduction to each steps in HSMM-PCA scheme, the process to calculate the observations probability distribution { Bi } should be illustrated firstly. Assume that data in each operation mode obeys unimodal Gaussian distribution, i.e. xi ~N ( µi , Σi ) ( i = 1,2,L, M ), where x i denotes the data belonging to mode i , and µ i and Σ i are the corresponding

% = diag ( Σ (1,1), Σ (2, 2),L , Σ ( s, s)) , then expectation vector and covariance matrices. Set Λ i i i i % −1 and hence the corresponding principal component data x i can be normalized as x% i = (xi − µi ) Λ i

% −1P ~N ( 0, Λ ) ( i = 1,2,L, M ), where matrix P is the loading scores ti = x% i Pi = (xi − µi ) Λ i i i i matrix of mode i and Λ i is the estimated covariance matrix of t i . As a result, the observations probability distribution { Bi } can be calculated as


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Bi ( x ) =

f ( ( x − µ i ) Λ i−1Pi 0, Λ i )

∑ f ((x − µ M

j

−1 j

) Λ P j 0, Λ j

j =1

)

,

(7)

where f ( • 0, Λ i ) is the Gaussian probability density function with expectation 0 and variance Λ i . According to equation (7), Bi ( x) is calculated in the subspace of principal components rather than in the subspace of original data, so it is robust to the process noise.

Figure 1. Flow chart of HSMM-PCA.

The detail operation in each steps are as follows:

Offline modeling: Available Data: historical normal training data set X ∈ R n×s . Step 1: Set the phase number M and the maximum duration Tmax for HSMM.


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Step 2:Preset the initial values as λ = ( A, π, { Bi } , { Di }) . Vector π i ∈ R 1×T

max

M

satisfies the condition

∑π

i

( i = 1,2,L, M )

= 1 , matrix A ∈ R M ×M can be set as a random matrix satisfying

i=1

M

∑ A(i, j ) = 1 and

A(i, i) = 0 for all i = 1,2,L, M , and vector Di ( i = 1,2,L, M ) can be set as

j =1

Tmax

random vectors satisfying the condition

∑ D (T ) = 1 . Equation (7) indicates that observations i

T =1

probability distribution Bi can be described by four parameters: µ i , Λ i , Pi , and Λ i . To obtain the initial values of these four parameters, the training data should be clustered into M groups by k-means algorithm firstly, and then calculate the corresponding µ i , Λ i , Pi , and Λ i of each group as the initial values of each parameter. Step 3: Use forward-backward algorithm to re-evaluate the model parameters A , π , and

{ Di } ,and then calculate the state affiliation probability γ i (t ) . The detailed steps of the forwardbackward algorithm are as follows: Step 3.1. Calculate the forward variable α i (t,T ) = P ( o(1), o(2),L, o(t ), q(t ) = Si ,τ t = T ) using

α i (1, T ) = πi Bi ( o(1) ) Di (T )  ,    α ( t , T ) = α ( t − 1, T + 1) B o ( t ) + α ( t − 1,1) a B o ( t ) D ( T ), t > 1 ( ) ( )   ∑ i i i j ji i i   j ≠i  

(8)

where τ t denotes the remaining time of the current state q (t ) . Calculate the backward variable

βi (t,T ) = P ( o(t + 1), o(t + 2),L , o(Ttotal ), q(t ) = Si ,τ t = T ) with formula


10

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   βi (t ,1) = ∑ aij Bi ( o(t + 1) )  ∑ DS j (T )β j (t + 1, T )  , t < T total j ≠i  T ≥1    , βi (t , T ) = Bi ( o(t + 1) ) βi (t + 1, T − 1), T > 1, t < T total β (T , T ) = 1  i total 

(9)

Step 3.2. Calculate the following joint probabilities:

ξt (i, j ) = P ( o(1), o(2),L , o(Ttotal ), q (t − 1) = Si , q (t ) = S j )   = α i (t − 1,1) aij B j ( o(t ) )  ∑ D j (T )β j (t , T )   T ≥1 

,

(10)

ηi (t,T ) = P ( o(1), o(2),L , o(Ttotal ), q (t − 1) ≠ Si , q (t ) = Si ,τ t = T ) = ∑ α j (t − 1,1)a ji Bi ( o(t ) ) Di (T ) βi (t,T)

,

(11)

j ≠i

and

ς i (t ) = P ( o(1), o(2),L , o(Ttotal ), q (t ) = Si ) ς i (t + 1) + ∑ ( ξt +1 (i, j ) − ξt +1 ( j , i ) ) ,  j ≠i = ς i (Ttotal ) = ∑ α i (Ttotal , T )  T ≥1

(12)

Step 3.3. Re-estimate parameters {ai, j } , {πi } , and { Di } with the respective equations: Ttotal

M Ttotal

t =1

j ′=1 t =1

aî , j = ∑ ξt (i, j ) / ∑∑ ξt (i, j′) ,

(13)

M

πˆ i = ς i (1) / ∑ ς j (1) ,

(14)

j =1

Ttotal

Tmax Ttotal

t =1

T ′=1 t =1

Dˆ i (T ) = ∑ηi (t,T ) / ∑∑ηi (t , T ′) ,

(15)

Step 3.4. Re-estimate parameters µi and Σi using the following35 Ttotal

µ i = ∑ γ i (t ) X(t ) ,

(16)

t =1


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Ttotal

Σi = ∑ γ i (t )( X(t ) − µ i )T ( X(t ) − µ i ) ,

(17)

t =1

% = diag ( Σ (1,1), Σ (2, 2),L , Σ ( s, s)) . Re-calculate the variance of each variable as Λ i i i i Step 3.5. Repeat steps 3.1-3.4 until all parameters converge. Step 4: Recalculate the loading matrices {Pi } and the variance matrices {Λi } for all phases

i = 1,2,L, M . Also, compute the control limits of T 2 and SPE statistics for each mode i , 2 termed as Tl imit (i ) and SPElimit (i ) .

Online monitoring: Available Data: real time test data y (t ) ∈ R1×s . Step 1: Estimate the mode affiliation probability γ i (t ) . In online monitoring stage, one cannot

obtain the future data y(t ′) ( t ′ > t ) for calculation of ς i (t ) , and hence the forward variable α i (t , T ) , which only demands the historical data before time t , is used to replace ς i (t ) in

calculation of mode affiliation probability: Tmax

γî (t ) = T

∑ α (t , T ) i

T =1 max M

∑∑ α

, j

(18)

(t , T )

T =1 j =1

Equation (8) indicates that, α i (t , T ) is calculated based on α i (t − 1, T ) , A , and { Di } , which means that the mode affiliation information of the historical data, the mode shifting probability, and the mode duration probability are used in online mode identification. Step 2: Calculate T 2 and SPE statistics as M

T 2 (t ) = ∑ γî (t )T 2i (t ) ,

(19)

i =1


12

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M

SPE (t ) = ∑ γî (t ) SPEi (t ) ,

(20)

% −1 (y (t ) − µ )T P ( Λ ) −1 P T (y (t ) − µ ) Λ % −1 , Ti 2 (t ) = Λ i i i i i i i

(21)

% −1 (y (t ) − µ )T (I − P P T )(I − P P T )(y(t ) − µ ) Λ % −1 , SPEi (t ) = Λ i i i i i i i i

(22)

i =1

where

Similarly, the control limit of T 2 and SPE statistics at time t can also be approximated as M

2 T 2limit (t ) = ∑ γî (t )Tlimit (i) ,

(23)

i =1

M

SPElimit (t ) = ∑ γî (t ) SPElimit (i ) ,

(24)

i =1

Equations(19)-(24) indicate that HSMM-PCA calculates the mode affiliation probability γˆt (i ) of each mode, and then weighted sums all delectation results from various operation modes, so as the corresponding control limits. When the data is within the normal condition, the statistics for the relevant modes are very small and the statistics for the irrelevant modes alarm the fault. However, because the affiliation probability values of the irrelevant modes are close to 0, so the final statistics T 2 (t ) and SPE (t ) are still within the normal range. When a traditional fault occurs in the process, the statistics for both the relevant and irrelevant modes are larger than their control limits, so the final weighted summation statistics will also alarm the fault. When a mode disorder fault occurs, for example, mode A directly shifts to mode C (mode A can only shift to mode B), then TC2 (t ) and SPEC (t ) are within the normal range and the other statistics are abnormal. However, as mode A cannot directly shift to mode C, which means that γˆC (t ) ≈ 0 , then TC2 (t ) and SPEC (t ) are not summed in equations (19) and (20), and hence the final statistics still

alarm the fault.


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Step 3: Combine the two statistics into one to simplify the fault detection task:

T 2 (t ) / T 2 limit (t ) + SPE (t ) / SPElimit (t ) CI (t ) = , 2

(25)

which incorporates the SPE and T 2 statistics in a balanced way. When CI (t ) > 1 , the process is diagnosed as faulty. The above content indicates that: firstly, HSMM just identify mode affiliation probabilities for each data sample, rather than localizing the sample in a specific mode; secondly, HSMM-PCA adopts the mode affiliation probability and state duration probability for the mode identification in online monitoring stage, and all the relevant mode’s PCA models are adopted for process monitoring, so as the control limits; Moreover, HSMM can also extract useful information from data of the transitional mode and use them for each mode’s PCA modeling. As a result, HSMMPCA can successfully handle with multimode process.

4 STUDY ON PARAMETER SELECTION AND SENSITIVITY ANALYSIS

Compared with PCA, HSMM-PCA introduces additional two parameters: the mode number

M and the maximum duration Tmax . The following mathematical model is constructed to study the function of these two parameters:


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 1   1  0  2  x1 ( t )    −1    x t ( )  2  =  1  x3 ( t )    0     x t  4 ( )    −2  5  1   0  2

2 0   w1 ( t ) + 5    mode 1 1   2 w2 ( t ) + 3  1 2 0   w1 ( t ) + 10    mode 2 1   2 w2 ( t ) + 2   1 1 0   2 w1 ( t ) + 1  mode 3  1   w2 ( t ) + 1   5

where w1 ( t ) and w2 ( t ) are two random Gaussian variables. This is a three-mode process, and the mode shifting probability matrix for three modes is set as  0.00 1.00 0.00  A =  0.00 0.00 1.00  ,  0.50 0.50 0.00 

which indicates that mode 1 can only shift to mode 2, mode 2 can only shift to mode 3, and mode 3 have equal probabilities shifting to mode 1 and mode 2. The duration for each operation mode

i follows Gaussian distribution N ( µi ,100 ) , where µ1 = 40 , µ 2 = 50 , and µ3 = 60 , respectively. About 4000 samples of normal data are generated as training data for HSMM-PCA, and Figure 2 shows the mode identification result of the first 1000 samples, where Tmax is fixed as 100 and

M varies from 2 to 5. When M = 2 , which is smaller than the true mode number, HSMM does not have enough modes to describe the process and hence the mode division is unreasonable. Figure 2(c) indicates than when M equals to the true mode number, HSMM can successfully identify the process modes. Because there are only 3 modes in this process, when M > 3 , some


15


independent modes will be divided into several sub-modes by HSMM. In Figure 2 (d), when

M = 4 , HSMM has the same mode division result as the situation M = 3 , excepting that the original mode 2 in Figure 2 (c) is divided into two sub-modes (mode 2 and 3) in Figure 2 (d); as for situation M = 5 , both original mode 2 and mode 3 in Figure 2(a) are divided to two submodes (mode 2 and 5 for original mode 2, and mode 3 and 4 for original mode 3).The above results indicates that when M is smaller than the true mode number, HSMM fails to divide the process; when M is larger than the true mode number, HSMM can successfully identify the

mode

mode

mode

mode

process modes and divide some mode into several sub-modes.

mode

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Figure 2. The maximum likelihood modes identified with different M : (a) the true mode

affiliation; (b) M = 2 ; (c) M = 3 ; (d) M = 4 ; (e) M = 5 Figure 3 shows the function of Tmax , where M is fixed as 3. If Tmax is too short ( Tmax = 40 ), when the true mode duration exceeds Tmax , HSMM will allocate the variables to other modes even there is no mode shifting that time. However, when interval [0, Tmax ] has cover the variation


16

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of true mode duration, the adjustment of Tmax will not affect the mode division result anymore, as

mode

mode

mode

mode

a result, Figure 3(c)-(e) has the same result.

mode

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 3. The maximum likelihood modes identified with different Tmax : (a) the true mode

affiliation; (b) Tmax = 40 ; (c) Tmax = 70 ; (d) Tmax = 100 ; (e) Tmax = 150 To test the fault sensitivity of HSMM-PCA with different M and Tmax , another 200 samples data are generated, whose mode affiliation is the same as those of the first 200 sample in Figure 2(a), and two types of fault are introduced to the process: Fault 1: a step change occurs in the process at sample time 101, whose amplitude is ψ ; Fault 2: the process shift to mode 1 at sample time 51. The definition of fault sensitivity is different for each fault: for Fault 1, we tune the fault amplitude ψ to find the minimum ψ min can be detected by HSMM-PCA , and take ψ min as the measure of the fault sensitivity, which means the smaller fault amplitude ψ min represents the better fault sensitivity. As for Fault 2, the mode disorder shifting fault, we just check whether


17


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HSMM-PCA can detect this fault. If can, HSMM-PCA is sensitive to this fault. The control limits of HSMM-PCA is calculated based on a confidence limit of 99%. As shown in Table 2, when M < 3 , HSMM-PCA performances much worse than the other situations in Fault 1, and it fails to detect the mode disorder shifting fault. However, when M ≥ 3 , HSMM has enough mode to describe the operation phases in process, and hence achieves almost the same performance in both faults with different M .Parameter Tmax has the similar character to M : when interval [0, Tmax ] has covered the variation of true mode duration ( Tmax ≥ 70 ), HSMM-PCA can successfully handle with the duration feature of each mode and successfully detect the two faults. Table 2. Fault sensitivity of HSMM with different M and Tmax .

ψ min for Fault 1 Tmax

M 2 3 4 5

40

70

100

150

35.8 15.0 12.6 12.4

14.5 9.5 8.5 8.4

15.5 7.9 8.0 8.4

15.5 8.1 8.0 7.9

Detect Fault 2

Tmax

M 2 3 4 5

40

70

100

150

Incapable Incapable Incapable Incapable

Incapable Capable Capable Capable



Theoretically, once M and Tmax has exceeded their corresponding lower bound values, e.g. 3 and 70 in this test, adjustment of them will not improve the monitoring performance anymore. The reasons are as follows: on the one hand, the number of operation modes in a process is fixed, and hence the redundant hidden states in HSMM cannot extract any new data structure


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information from the process; one the other hand, the probability of the redundant duration values are 0 and they are useless in mode description. Considering that the larger M and Tmax requires more computation at the same time, both two parameters should be chosen within the computation capacity. Usually the engineers or workers are familiar with the chemical process and hence the reference value of M and Tmax can be obtained from them.

5 ILLUSTRATIONS AND RESULTS

In this section, the Tennessee Eastman (TE) process simulation, whose codes can be downloaded from http://depts.washington.edu/control/LARRY/TE/download.html, is used to evaluate the monitoring performance of HSMM-PCA.TE process simulates an industrial process in Tennessee Eastman Chemical Company, which consists of five major unit operations: a reactor, a product condenser, a vapor–liquid separator, a recycle compressor, and a product stripper. Overall 41 measured output variables and 12 manipulated variables are present in the process, and this section adopts 33 of them (as listed in Table 3) to test the proposed HSMMPCA, Mixture Bayesian PCA (MBPCA)14, and hidden state probability integration (HSPI)23. MBPCA is an extension of the probabilistic PCA14 in multimode processes, and HSPI is the application of HMM in process monitoring. Compared with MBPCA and HSPI, on the one hand, HSMM-PCA can be regard as an improved MBPCA with additional state transition probability and state duration probability distribution in model description; on the other hand, HSMM-PCA also can be regard as an improvement of HSPI, which replaces HMM with HSMM and then introduces PCA to improve the process monitoring performance.


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The traditional TE process is a unimode process working in steady state. To test the multimode approaches, three operation modes are introduced to this process, as listed in Table 4. The mode shift probability matrix for three modes is set as  0.00 1.00 0.00  A =  0.00 0.00 1.00  ,  0.50 0.50 0.00 

which indicates that mode 1 can only shift to mode 2, mode 2 can only shift to mode 3, and mode 3 have equal probabilities shifting to mode 1 and mode 2. The duration for each operation mode

i follows Gaussian distribution N ( µi ,100 h 2 ) , where µ1 = 80h , µ 2 = 70h , and µ3 = 60h , respectively. Set the sampling time as 0.5 h and 4000 samples (2000 h) normal data are generated as training data. Table 3. Monitored variables in TE process.

Variables 1 A feed (stream 1) 2 D feed (stream 2) 3 E feed (stream 3) 4 Total feed (stream 4) 5 Recycle flow (stream 8) 6 Reactor feed rate (stream 6) 7 Reactor pressure 8 Reactor level 9 Reactor temperature 10 Purge rate (stream 9) 11 Product separator temperature 12 Product separator level 13 Product separator pressure

18 Stripper temperature 19 Stripper steam flow 20 Compressor work 21 Reactor cooling water outlet temperature 22 Separator cooling water outlet temperature 23 D feed flow valve (stream 2) 24 E feed flow valve (stream 3) 25 A feed flow valve (stream 1) 26 Total feed flow valve (stream 4) 27 Compressor recycle valve 28 Purge valve (stream 9) 29 Separator pot liquid flow valve (stream 10) 30 Stripper liquid product flow valve (stream 11)

14 Product separator under flow 31 Stripper steam valve (stream 10) 15 Stripper level 32 Reactor cooling water flow


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


16 Stripper pressure 33 Condenser cooling water flow 17 Stripper underflow (stream 11)

Table 4. Three process operation modes in TE process

Mode 1 2 3

Product separator level setpoint 40 50 50

Mole % G in product. 50/50 40/60 50/50

Production rate setpoint 22.89 22.89 18.40

Because the reference value of M and Tmax can be obtained from the engineers or workers, in this test, M = 3 and Tmax = 150h (300 samples) are assumed as the prior-knowledge. The mode shifting probability matrix estimated by HSMM-PCA and HSPI are as follows (MBPCA does not have mode shifting probability matrix):

A HSMM-PCA

A HSPI

 0.00 1.00 0.00  =  0.00 0.00 1.00   0.75 0.25 0.00 

 0.99 0.00 0.01 =  0.00 0.99 0.01  0.00 0.01 0.99 

Because HSMM is flexible enough to describe the time spent on a given state, it has a good ability to describe the multimode feature. Compare matrix A HSMM-PCA with A , one knows that the first two lines of A HSMM-PCA is equal to those in matrix A , but A HSMM-PCA ( 3,1) and

AHSMM-PCA ( 3, 2 ) are not exactly the same as the true values. This is reasonable, because the limited training data cannot support enough mode shifting information for the precise estimation


21


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of mode shifting probability. However, the mode identification result of HSMM-PCA (Figure 4 (a) and (d)) indicates that HSMM-PCA can successfully identifying the mode affiliation in the training data even with biased mode shifting probability matrix A HSMM-PCA .For HMM, because its inherent duration probability density is exponential, which is inappropriate for the modeling of multimode process data, so A HSPI deviates a lot from A .In Figure 4 (c), HSPI falsely classifies mode 1 and 2 into a single mode and treats the transitional stage from mode 1 to mode 2 and from mode 2 to mode 3 as another mode, which also shows that HMM is not suitable to describe the multimode process. As for MBPCA, without the constraint of state transition probability and state duration probability distribution, it fails in obtaining a reasonable mode affiliation result: in Figure 4 (b), mode 3 is missing and the rest two modes are disordered.

Figure 4. Mode identification result for training data: (a) the true mode affiliation; (b) the

maximum likelihood modes identified by MBPCA; (c) the maximum likelihood modes identified by HSPI; (d) the maximum likelihood modes identified by HSMM-PCA


22

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Another 600 samples (300 h) of data is generated for testing, whose mode affiliation is shown as in Figure 5 (a). Figure 5 (b) shows the mode affiliation probability identified by HSMM-PCA and it indicates that HSMM-PCA can successfully identify the mode affiliation in the process. In addition, Figure 5 (b) also demonstrates that the data in transitional stages has features belonging to more than one phases, and it cannot strictly follow a certain phase’s distribution. About 22 faults are introduced to test the performance of three methods, which occur in different modes and continue until the end (as listed in Table 5). For fault 20-22, data of each mode is in normal condition, but the mode shifting order is wrong, e.g. in fault 20, the process shifts from mode 1 to mode 3 directly. The control limits of the three algorithms are calculated based on a confidence limit of 99%. (a) True mode

M ode

3

2

1 0

50

100

150

200

250

300

Time（ h） (b) Identified mode probability 1

Probability

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.8

Mode 2 Mode 3 Mode 1

0.6 0.4 0.2 0 -0.2 0

50

100

150

200

250

300

Time（ h）

Figure 5. True mode affiliation of the test data and the identified mode probability.

The detection rates and false alarm rates of three methods are listed in Table 6, and the best results are marked in bold and underline. For MBPCA, without the restriction of mode shifting probability and mode duration probability, it falsely divides the process modes and obtains the


23


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Page 24 of 31

wrong PCA models for each mode. As a result, MBPCA achieves the worst false alarm rate and fault detection rate for all faults. For the rest two methods, HSMM-PCA has a litter larger false alarm rate than HSPI, but it achieves much better fault detection rate in most faults, especially in fault 4, 9, 18, 20, 21, and 22. Table 6 also indicates that the last three faults, which are mode shifting disorder fault, can only be detected by HSMM-PCA. The reasons for these results are as follows: firstly, as mentioned before, HSMM can successfully identify the mode affiliation of each data sample and HMM falsely divides the modes, so HSMM-PCA can obtain precise PCA model for each mode and has a better ability to detect the abnormal condition in different modes; secondly, HSMM-PCA calculates observations probability distribution

{Bi }

by using the

Mahalanobis distance in the subspace of principal components rather than in the subspace of original data as HSPI, so HSMM-PCA is more robust to the noise; last but not least, HSMM adopts both mode shifting probability and mode duration probability for offline training and online monitoring, and hence HSMM-PCA is able to detect the mode disorder fault. The first two reasons contribute to the superiority performance of HSMM-PCA in faults 1-19, and the third reason makes HSMM-PCA successfully detects fault 20-22. Table 5. Fault Descriptions for TE Process

No.

Description

Type

Fault occurs time

1 2 3 4 5 6 7 8 9 10 11 12 13

Feed ratio of A/C, composition constant of B (stream 4) Composition of B, ratio constant of A/C (stream 4) Feed temperature of D (stream 2) Inlet temperature of reactor cooling water Inlet temperature of condenser cooling water Header pressure loss of C—reduced availability (stream 4) Feed composite of A, B, and C on (stream 4) Feed temperature of D (stream 2) Feed temperature of C (stream 4) Inlet temperature of reactor cooling water Inlet temperature of condenser cooling water Reaction kinetics Valve of reactor cooling water

Step Step Step Step Step Step Random variation Random variation Random variation Random variation Random variation Slow drift Sticking

100 h (in mode 2) 100 h (in mode 2) 100 h (in mode 2) 100 h (in mode 2) 100 h (in mode 2) 100 h (in mode 2) 180 h (in mode 3) 180 h (in mode 3) 180 h (in mode 3) 180 h (in mode 3) 180 h (in mode 3) 180 h (in mode 3) 180 h (in mode 3)


24

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


14 15-19 20 21 22

Valve of condenser cooling water Unknown Mode shifts from mode 1 to mode 3 Mode shifts from mode 1 to mode 3 Mode shifts from mode 2 to mode 1

Sticking Unknown Step Step Step

250 h (in mode 1) 250 h (in mode 1) 250 h (in mode 1) 50 h (in mode 1) 100 h (in mode 2)

Table 6. Detection rates (%) of HSMM-PCA and HSPI in TE process.

Methods Index Fault 1 Fault 2 Fault 3 Fault 4 Fault 5 Fault 6 Fault 7 Fault 8 Fault 9 Fault 10 Fault 11 Fault 12 Fault 13 Fault 14 Fault 15 Fault 16 Fault 17 Fault 18 Fault 19 Fault 20 Fault 21 Fault 22 False alarm rate

MBPCA HSPI HSMM-PCA 2 T SPE NLLP CI 2.00 99.50 100.00 100.00 20.25 65.75 97.25 98.50 2.00 1.50 0.75 3.50 2.25 1.50 76.25 99.75 3.00 1.50 1.50 3.75 3.00 95.75 100.00 100.00 15.00 59.80 96.25 98.33 3.33 2.50 2.92 4.58 2.50 2.50 15.42 53.75 2.08 3.75 80.83 97.50 2.50 2.50 2.50 12.08 72.80 91.25 96.25 97.92 2.08 2.50 78.33 95.42 2.00 6.00 1.00 9.00 2.00 6.00 1.00 9.00 3.00 9.00 71.00 89.00 6.00 10.00 57.00 77.00 1.00 6.00 17.00 83.00 29.00 70.00 97.00 97.00 5.00 5.00 2.00 92.00 2.28 12.60 1.00 62.80 2.25 7.75 6.00 100.00 2.00

0.05

0.03

0.07

To further demonstrate the features of HSMM-PCA, the monitoring charts for Faults 9, 18, and 20 are shown in Figure 6, 7, and 8, respectively. One common phenomenon in all of the three


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Page 26 of 31

figures is that, HSMM-PCA is more sensitive to the transitional stages (interval [ 60h, 70h ] and

[140h,150h ] )

than HSPI and MBPCA, and hence HSMM-PCA achieves a little larger false

alarm rate. According to the result of Figure 4, for HSMM-PCA, the transitional stages are not divided as an independent mode but they are presented by the combination of the existing steady modes; for HSPI, the transitional stages are treated as one independent mode, and hence they are monitored by a specific model; for MBPCA, it fails to obtain a reasonable mode division result and it is not sensitive to the abnormal conditions. Sometimes the transitional stages cannot be presented as the weighted summation of the steady modes, so HSMM-PCA may achieve larger false alarm rate than HSPI and MBPCA in transitional stages. However, as HSMM-PCA successfully divides the operation modes and builds the PCA model for each mode, HSMM-PCA is more sensitive to the faults and has better fault detection performance in Figure 6 and 7. In Figure 8, when the process directly shifts from mode 1 to mode 3, both HSPI and MBPCA just locate the process to mode 3 without concerning the restriction of mode shifting probability. As the data in mode 3 is normal, both HSPI and MBPCA fails in detecting the mode shifting disorder fault. According to the mode shifting probability, one knows that mode 1 can only shift to mode 2 or remain at mode 1, so HSMM-PCA just monitoring the process data with the PCA models in mode 1 and 2, as a result, HSMM-PCA can successfully detect this abnormal condition in time.


26

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(a) MBPCA

1.5

(b) HSPI

fault introduced time

T2

NLLP

1


10 2

0.5

10 1

0 0

50

100

150

200

250

300

0

50

100

Time（h）

150

200

250

300

200

250

300

250

300

250

300

Time（h） (c) HSMM-PCA

1.5 fault introduced time

10 1

CI

SPE

1 fault introduced time 10 0

0.5

10-1

0 0

50

100

150

200

250

300

0

50

100

Time（h）

150

Time（h）

Figure 6. Monitoring chart for Fault 9.

(a) MBPCA

1.5

(b) HSPI



10 2

T2

NLLP

1

0.5

10 1

0 0

50

100

150

200

250

300

0

50

100

Time（h）

150

200


1.5 fault introduced time 1


CI

SPE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10 0

0.5

0 0

50

100

150

200

250

300

0

50

100

Time（h）

150

200

Time（h）

Figure 7. Monitoring chart for Fault 18.


27


(a) MBPCA

1.5

(b) HSPI


10 2


T2

NLLP

1

0.5

10 1

0 0

50

100

150

200

250

300

0

50

100

Time（h）

150

200

250

300

250

300


10 1

1.5 fault introduced time 1


CI

SPE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 31

10 0

0.5

0 0

50

100

150

200

250

300

10-1

0

50

100

Time（h）

150

200

Time（h）

Figure 8. Monitoring chart for Fault 20. 6 CONCLUSIONS

In this study, HSMM is combined with PCA to address the multimode problem in processes monitoring. HSMM is flexible enough to describe the time spent in a given state, so it is adopted for mode division and identification. HSMM-PCA takes use of the mode affiliation information of the historical process data, mode shifting probability, and mode duration probability for the mode identification in online monitoring, so it can detect the mode disorder fault which challenges the previous multimode approaches. The test result in the TE process shows that HSMM-PCA is good at monitoring the data in steady mode, but it cannot successfully handle with the data in the transitional stages, which may contain nonlinear, dynamic, or non-Gaussian features simultaneously. Fortunately, many novel PCA methods have been put forward to address these problems3, 6, 37, 38 and these methods could be integrated into HSMM-PCA, which will be studied in the near future. ACKNOWLEDGEMENT


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This study was supported by the National Natural Science Foundation of China under Grant 61374099and Research Fund for the Taishan Scholar Project of Shandong Province of China. REFERENCES

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Multimode Continuous Processes Monitoring Based on Hidden Semi

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