Hidden Markov Model Based Adaptive ... - ACS Publications

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Hidden Markov Model Based Adaptive Independent Component Analysis Approach for Complex Chemical Process Monitoring and Fault Detection Mudassir M. Rashid and Jie Yu* Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7 ABSTRACT: For complex chemical processes with multiple operating conditions and inherent system uncertainty, conventional multivariate process monitoring techniques such as principal component analysis (PCA) and independent component analysis (ICA) are ill-suited because they are unable to characterize shifting modes and process uncertainty. In this article, a novel hidden Markov model (HMM) based ICA approach is proposed for process monitoring and fault detection. First the hidden Markov model is built from measurement data to estimate dynamic mode sequence. Further the localized ICA models are developed to characterize various operating modes adaptively. HMM based state estimation is then used to classify the monitored samples into the corresponding modes, and the HMM based I2 and SPE statistics are established for fault detection. The effectiveness of the proposed monitoring approach is demonstrated through the Tennessee Eastman Chemical process. The comparison of monitoring results shows that the proposed HMM-ICA approach is superior to the conventional ICA method and can achieve accurate detection of various types of process faults with minimized false alarms.

1. INTRODUCTION Effective monitoring and diagnosis of chemical processes is important in process systems engineering as it is essential to ensure the stable operation of chemical plants, maintain product quality at desired grades, optimize production profit, and improve plant safety and environmental sustainability. Notable advancements of computing and information technology over the past decades have led to powerful process sensors, instruments, and data collection devices as well as a huge amount of historical data. The large number of correlated variables being measured from a multitude of sensors across the plant cause the failure of the traditional univariate process monitoring techniques such as statistical process control (SPC) charts (e.g., Shewart charts, CUSUM charts, or EWMA charts).1 Multivariate statistical process monitoring (MSPM) techniques such ad principal component analysis (PCA) and partial least-squares (PLS) have been widely applied to chemical process monitoring with some successes.2−6 This type of method projects the highly correlated and noisy data onto a lower dimensional subspace that best characterizes the variations of the multiple process variables. The lower dimensional subspace is defined by a number of orthogonal latent vectors that capture the feature information from the process variables. Further, test statistics such as T2 and SPE are defined to monitor the time-series observations with respect to the latent variable subspaces.7−12 An inherent assumption for the mathematical derivation of the confidence limits of T2 and SPE statistics is that the data obey a multivariate Gaussian distribution approximately. Therefore, non-Gaussian processes may not be effectively monitored using the conventional MSPM methods.13 Moreover, regular PCA and PLS models assume that the observations at a time are statistically independent from previous observations and thus ignore the time-varying process dynamics. Nevertheless, chemical process © 2012 American Chemical Society

operation may not always remain under steady state conditions and the process variables are often autocorrelated due to inherent dynamics. Time-lag shift methods are used in dynamic PCA (DPCA) and dynamic PLS (DPLS) to explore the nonsteady-state behavior in the process.9 However, an estimate of the number of time lags is needed in order to expand the observation matrix. Meanwhile, supervised classification techniques such as Fisher discriminant analysis (FDA) and support vector machine (SVM) are used to identify out-of-control periods of plant operation but require labeled output samples to build the models.14−16 More recently, Gaussian mixture model (GMM) approach is used to monitor non-Gaussian processes with multiple operating modes.17,18 It decomposes the non-Gaussian distribution into various multivariate Gaussian density functions and then a Bayesian inference based probabilistic index is derived to monitor process operation. Nevertheless, the measurement data from each operating mode are assumed to follow multivariate Gaussian distribution. Alternately, independent component analysis (ICA) has been applied to nonGaussian process monitoring and fault diagnosis. It extracts process features based on higher-order statistics and thus can ensure mutual independence among latent variables.19−22 However, the conventional ICA technique searches for the latent variables with maximized negentropy index, which does not account for the system multimodality due to shifting operating conditions in processes. Hidden Markov model (HMM) is a machine learning technique for estimating the probability distributions of state transitions and the probabilities of measured outputs in a dynamic sequence of process Received: Revised: Accepted: Published: 5506

January 23, 2012 March 20, 2012 March 27, 2012 March 27, 2012 dx.doi.org/10.1021/ie300203u | Ind. Eng. Chem. Res. 2012, 51, 5506−5514

Industrial & Engineering Chemistry Research

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observations.23 HMM assumes that the time-varying observations are generated from the underlying process with discrete hidden states. Then the measured process variables are treated as realizations of the underlying stochastic process. The abrupt changes in operating conditions, complex dynamics, and process uncertainty all pose great challenges for the existing monitoring approaches. In contrast, the strong stochastic and inferential features of HMM make it an excellent candidate for monitoring chemical processes with inherent dynamics and system uncertainty. In this study, the advantages of HMM to account for system dynamics and uncertainty are combined with the merit of ICA for non-Gaussianity handling. The integrated HMM-ICA approach is proposed to characterize the process uncertainties and random transitions of various operating conditions in chemical plants. The estimated state outputs of HMM model are mapped to various operating conditions in a dynamic manner and then the multiple localized ICA models are built within all the identified operating states from HMM. Further, the monitored samples can be categorized into the corresponding operating conditions so that the appropriate ICA models may be adaptively selected for fault detection through the localized SPE and I2 indices. The remainder of this article is organized as follows. Section 2 reviews the conventional ICA based monitoring technique. Then the novel HMM-ICA based process monitoring approach is developed in Section 3. Section 4 demonstrates the effectiveness of the proposed HMM-ICA method using the Tennessee Eastman Chemical process and the monitoring results of HMM-ICA approach are compared with those of regular ICA method. The conclusions of this work are summarized in Section 5.

In ICA, the Euclidean norm based leading independent components are selected for process monitoring purpose and the I2 and SPE statistics can be further defined.20,22 First, the obtained independent components are sorted in descending order according to the Euclidean norms. Then, the trend plot of the percentages of Euclidean norms can be observed to determine the break line between the first few components and the remaining ones so that the percentage norms of the top components are much larger than those of the rest. In this way, the number of independent components can be determined for ICA models.22 For a monitored sample xt ∈ 1×K, we have ŝt = Wxt. Then the I2 index is expressed as

I 2 = st̂ stT̂

which explains the process variation within the independent component subspace. On the other hand, the SPE statistic describes the variation within the residual subspace and is defined as follows SPE = etT et = (xt − xt̂ )T (xt − xt̂ ) xt̂ = Q−1Bst = Q−1BWxt

3. INTEGRATED HMM-ICA APPROACH FOR PROCESS MONITORING Industrial processes often have different operating conditions and production strategies, non-Gaussian and dynamic behaviors, and process and measurement uncertainties, which make it a challenging task to monitor plant operation and detect process faults precisely. The conventional PCA based monitoring method is ill-suited for processes with strong non-Gaussianity. Though the regular ICA approach can handle non-Gaussian process, it may not effectively account for the multimodality, systems dynamics, and process uncertainty. In this study, the hidden Markov model is integrated with independent component analysis for complex process monitoring and abnormal event detection. Hidden Markov model is used to quantitatively characterize the dynamic system in terms of parametric random process with unobserved states.23 Within HMM, a set of finite hidden states are connected through transition probabilities and the outputs corresponding to each state are measurable. Moreover, the future state only depends on the present instead of the past ones under Markov chain property. A HMM involves the following key elements: • the total number of states, NS, in the model • the number of output variables, NO, corresponding to each state • the state transition probability distribution, AH ∈ Ns×Ns • the emission probability, BH ∈ 1×Ns • the initial state distribution, π = {πi|πi = P(qt=1 = Si)} where qt represents the hidden state variable at sampling instant t and Si (1 ≤ i ≤ NS) denotes any possible state in the process. Consider a state sequence Q = {q1, q2, ...qt, ...qT} with qt ∈ {Si}, the transition from qt to qt+1 is determined by the state transition probability matrix AH = {ai,j} with ai,j = P(qt+1 = Sj|qt

(1)

(2)

where W is the inverse matrix of A if M = K. Through the initial step of whitening, the transformed data matrix Z can be expressed as Z = QX = BS

(3)

where Q is the whitening matrix and B = QA is an orthogonal matrix. By combining eqs 2 and 3, the demixing matrix W can be obtained as W = BT Q

(7)

To monitor multivariate processes, the I2 and SPE confidence limits are needed and can be computed through Gaussian kernel density estimation (KDE).

where A ∈ K×M is the unknown mixing matrix, S ∈ M×N is the loading matrix of independent components, and E is the residual matrix. Then the ICA objective is to calculate a demixing matrix W such that S = WX

(6)

with the prediction, x̂t, being calculated as

2. PRELIMINARIES ICA is an alternate multivariate statistical process monitoring technique and it extracts the statistically independent components so that the negentropy index is maximized along the latent variable directions. Originally ICA is developed to handle the blind source separation (BSS) problem for recovering independent source signals.24,25 Let X ∈ K×N be the normalized plant data matrix with N samples of K process variables. In the ICA algorithm, the K normalized process variables are expressed as linear combinations of M independent components as X = AS + E

(5)

(4) 5507

dx.doi.org/10.1021/ie300203u | Ind. Eng. Chem. Res. 2012, 51, 5506−5514


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= Si) for 1 ≤ i,j ≤ NS. Further there is an emission probability matrix BH = {bj(k)} with bj(k) = P(yk|qt = Sj) for 1 ≤ k ≤ T and 1 ≤ j ≤ NS to define the conditional probability from state Sj to output yk. Finally, the initial state distribution is denoted by π = {πi|πi = P(q1 = Si)}. Hence the above three probabilities, λ = f{AH, BH, π}, can determine the dynamic sequences of a hidden Markov model. In this work, a HMM model is used to characterize dynamic chemical processes with various operating modes and process uncertainty.26 Each operating mode that may occur in the plant is treated as one of the possible states as follows S1: The 1st operating mode S2: The 2nd operating mode ⋮ SNS : The NS−th operating mode

(8)

The HMM model, λ = f{AH, BH, π}, is calculated from the following sequence of measurements

{

Y = Y

(t = 1)

,Y

(t = 2)

, ..., Y

Figure 1. Illustration of HMM based operation mode identification and localized ICA modeling.

}

(t = T )

T ⎡ ⎧ ⎤T = ⎨⎡⎣y1(t = 1) , y2(t = 1) , ..., ⎤⎦ , ⎣y1(t = 2) , y2(t = 2) , ..., yN(t = 2)⎦ , ..., O ⎩ T ⎡ (t = T ) (t = T ) (t = T )⎤ ⎫ , y2 , ..., yN ⎦ ⎬ ⎣y1 O ⎭

where QL*, BL*, and WL* denote the whitening matrix, orthogonal matrix, and demixing matrix, respectively, of the L*-th localized ICA model. The control limits of I2HMM and SPEHMM indices can be adaptively adjusted within each ICA model through kernel density estimation. KDE is a kind of nonparametric density estimation technique and it uses kernel function to estimate an arbitrary probability density distribution from data.30 In this research, the abnormal operation events can be detected by using the HMM-driven localized ICA monitoring indices and the corresponding control limits. A schematic diagram of the proposed HMM-ICA monitoring approach is shown in Figure 2 and the detailed implementation procedure is summarized below. (1) Build hidden Markov model using training data and the Baum−Welch algorithm and obtain λ = f(AH, BH, π); (2) Use the constructed HMM and the Viterbi algorithm to estimate the state sequence of training data and classify all the samples into L different operating modes; (3) Develop the L localized ICA models corresponding to different operating modes and estimate the control limits within each local ICA model; (4) For every monitored sample, identify the operating mode to which it belongs by using the Viterbi algorithm with the maximized likelihood; (5) Compute the HMM driven localized monitoring statistics I2HMM and SPEHMM for the monitored sample; (6) Use the localized monitoring statistics and the corresponding adaptive control limits to determine if the process operation is normal or faulty.

(9)

Given the HMM model trained by the Baum−Welch algorithm and the observed output sequence, the corresponding optimum state sequence Q* = {q1, q2, ..., qT} is determined by maximizing the conditional probability P(Q|Y, λ) through dynamic programming based Viterbi algorithm27−29 Q * = arg max P(Q |Y , λ) Q

(10)

where Q* represents the optimum state sequence for the observed measurements Y in the training set. After the HMM is used to estimate the sequences of operating modes for the training set, the localized ICA models can be built within each hidden state that corresponds to a specific operating mode. Further any monitored sample Ym can be classified into an appropriate operating mode by maximizing its likelihood as follows L* = arg max P(Ym|SL) 1 ≤ L ≤ NS

(11)

With the identified operating modes from HMM, the local ICA model can be adaptively selected on each monitored sample for process monitoring and fault detection. An illustration of the proposed HMM-ICA monitoring approach is shown in Figure 1. Then HMM based localized monitoring statistics can be defined as 2 IHMM = WL *YmYmTWLT*

4. APPLICATION EXAMPLE 4.1. Tennessee Eastman Chemical Process. The Tennessee Eastman process (TEP)31 is used in this research to evaluate the effectiveness of the proposed HMM-ICA monitoring method. The process flow diagram of the TEP is shown in Figure 3. It consists of five major unit operations, which include a chemical reactor, a condenser, a recycle compressor, a stripper, and a vapor/liquid separator. Four gaseous reactants are fed into the reactor to produce two products along with a byproduct and an inert. The products

(12)

and SPE HMM = (Ym − Ym̂ )T (Ym − Ym̂ )

(13)

with Ym̂ = Q L−*1BL *WL *Ym

(14) 5508



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Table 1. Six Different Operating Modes of Tennessee Eastman Chemical Process operating mode

G/H mass ratio

production rate (stream 11)

1 2 3 4 5 6

50/50 10/90 90/10 50/50 10/90 90/10

7038 kg/h G and 7038 kg/h H 1408 kg/h G and 12669 kg/h H 10000 kg/h G and 1111 kg/h H maximum maximum maximum

Table 2. Process Faults of Tennessee Eastman Chemical Process fault ID

fault description

IDV(l) IDV(2) IDV(3) IDV(4) IDV(5) IDV(6) IDV(7)

step in A/C feed ratio, B composition constant step in B composition, A/C ratio constant step in D feed temperature (stream 2) step in reactor cooling water inlet temperature step in condenser cooling water inlet temperature A feed loss (step change in stream 1) C header pressure loss-reduced availability (step change in stream 4) random variation in A+C feed composition (stream 4) random variation in D feed temperature (stream 2) random variation in C feed temperature (stream 4) random variation in reactor cooling water inlet temperature random variation in condenser cooling water inlet temperature slow drift in reaction kinetics sticking reactor cooling water valve sticking condenser cooling water valve

IDV(8) IDV(9) IDV(10) IDV(11) IDV(12) IDV(13) IDV(14) IDV(15)

Figure 2. Schematic diagram of the proposed HMM-ICA monitoring approach.

mixture stream is separated through the downstream operations. Overall 41 measured output variables and 12 manipulated variables are involved in the process. Moreover, there are six modes of process operation, as listed in Table 1. A list of possible faults and disturbances of TEP is given in Table 2. Because the process is essentially open-loop unstable, various types of control strategies have been developed. In our study, the decentralized control design is adopted for the closed-loop process operation.32 The 22 continuous measurements among

the 41 output variables are selected as monitored variables, which are listed in Table 3. The sampling time is 0.05 h, and all the six operating modes may occur during normal plant operation with significant uncertainty applied. The monitoring results of the proposed HMM-ICA approach are compared to those of the ICA method. A training data set is generated with 3000 observations consisting of all six operating

Figure 3. Process flow diagram of Tennessee Eastman chemical process. 5509



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Table 3. Monitored Variables of Tennessee Eastman Chemical Process

Table 4. Four Test Cases of Tennessee Eastman Chemical Process

variable no.

variable description

base case value

units

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

A feed (stream 1) D feed (stream 2) E feed (stream 3) A and C feed (stream 4) recycle flow (stream 8) reactor feed (stream 6) reactor pressure reactor level reactor temperature purge rate (stream 9) separator temperature separator level separator pressure separator underflow (stream 10) stripper level stripper pressure stripper underflow (stream 11) stripper temperature steam flow compressor work reactor coolant temperature condenser coolant temperature

0.251 3664 4509 9.35 26.90 42.34 2705 75 120.4 0.337 80.1 50 2634 25.16 50 3102 22.95 65.73 230 341 94.6 77.3

kscmh kg/h kg/h kscmh kscmh kscmh kPa % °C kscmh °C % kPa m3/h % kPa m3/h °C kg/h kW °C °C

case no. case 1

case 2

case 3

case 4

modes and random process uncertainty. Furthermore, four test cases with different kinds of prespecified operating modes and faulty scenarios are designed, as shown in Table 4. 4.2. Comparison of Process Monitoring Results of ICA and HMM-ICA Approaches. In the first test case, the plant is operated under Mode 2 or 3 with step error in reactor cooling water inlet temperature during the period from the 246th to the 342nd samples. As shown in Figure 4, the ICA based monitoring method does not perform well in process fault detection. It can be readily observed that a vast majority of the faulty samples cannot be captured from either the I2 or SPE plots. As opposed to the poor monitoring results of the ICA method, the HMM-ICA (Figure 5) approach accurately triggers the alarms on the faulty samples while the normal points remain below the control limits with very few false alarms. Furthermore, the HMM method can accurately identify two different operation states (Modes 2 and 3). In the corresponding two localized ICA models, the first five and four ICs are selected, respectively. As listed in Table 5, the fault detection and false alarm rates of HMM-ICA method are 97.1% and 4.2%, respectively. It should be noted that the rates in this table are the average values of I2/SPE statistics for ICA and HMM-ICA methods. In comparison, the fault detection rate of the ICA method is as low as 11.6%, which is much worse than that of HMM-ICA approach. On the other hand, the false alarm rate of ICA is 18.0%. Though the ICA method can deal with the non-Gaussianity in general, it is not well suited for the process multimodality in this case. Moreover, the process uncertainty poses another challenge for the regular ICA monitoring method. However, the proposed HMM-ICA approach has inherent capabilities to account for dynamic changes of multiple operating modes as well as the uncertainty in the stochastic process. Thus its monitoring results are superior to those of the conventional ICA method. For the second case, the process operation may change between Modes 2 and 4 along with a slow drift error in reaction

mode

test scenario

period

2 2 3

normal operation IDV(4): step change in reactor coolant inlet temperature normal operation

4 4

normal operation IDV(13): slow drift in reaction kinetics

2

normal operation

4 4 3

normal operation IDV(2): step change in B composition, A/C ratio constant normal operation

3

IDV(13): slow drift in reaction kinetics

1

normal operation

6 6 1

normal operation IDV(l): step change in A/C feed ratio, B composition constant normal operation

1

IDV(13): slow drift in reaction kinetics

2

normal operation

2

IDV(14): sticking reactor cooling water valve

3

normal operation

first−245th 246th− 342nd 343rd− 640th first−104th 105th− 199th 200th− 689th first−104th 105th− 199th 200th− 348th 349th− 444th 445th− 641st first−124th 125th− 219th 220th− 416th 417th− 512th 513th− 757th 758th− 854th 855th− 1003rh

Figure 4. ICA based monitoring results of test case 1 in Tennessee Eastman chemical process. 5510



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Figure 6. ICA based monitoring results of test case 2 in Tennessee Eastman chemical process.

Figure 5. HMM-ICA based monitoring results of test case 1 in Tennessee Eastman chemical process.

Table 5. Comparison of Quantitative Results of ICA and HMM-ICA Monitoring Methods for The Four Test Cases of Tennessee Eastman Chemical Process case no. case 1 case 2 case 3 case 4

monitoring method

fault detection rate (%)

false alarm rate (%)

ICA HMM-ICA ICA HMM-ICA ICA HMM-ICA ICA HMM-ICA

11.6 97.1 12.5 95.8 48.2 95.8 25.9 95.7

18.0 4.2 18.5 1.4 17.8 4.2 12.6 6.4

kinetics from the 105th to the 199th samples. As shown in Figure 6, the ICA based monitoring statistics and control limits are unable to reliably identify the process faults. In the ICA based I2 plot, it can be seen that a vast majority of the faulty samples fall below the control limit line. Though some of the faulty points can be captured in the ICA based SPE plot, it has a long delay to alarm the drifting fault. The HMM-ICA based monitoring statistics as shown in Figure 7, however, can precisely detect the abnormal points while minimizing the false alarms during the normal operation periods. The fault detection and false alarms rates of HMM-ICA approach are 95.8% and 1.4%, as opposed to the low detection rate of 12.5% and high false alarm rate of 18.5% of ICA method. The mode switches and dynamic uncertainty of the plant operation cause the failure of the conventional ICA monitoring method. In contrast, the localized ICA models in HMM-ICA approach can adaptively characterize the mode shifts and the HMM based state estimation is able to tackle the dynamics and uncertainty of the process. A more complex test scenario is considered in Case 3, in which there are two types of process faults of both step error


and slow drift. Likewise, the ICA method is unable to distinguish between normal operation and faulty event as shown in Figure 8. The I2 statistic remains below the control limit throughout the plant operation, while the SPE index values exceed the control limit for most of the normal and faulty samples. The average fault detection and false alarm rates of ICA monitoring method are 48.2% and 17.8%. The dynamic changes of various operating modes and random process 5511



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faults including step change, slow drift, and valve stiction. Similar to the previous cases, the ICA method does not detect process faults effectively. As observed in Figure 10, the ICA


uncertainty, however, are well identified by the proposed HMM-ICA approach. The monitoring results of the HMMICA method in Case 3 are shown in Figure 9. It is evident that the I2HMM and SPEHMM statistics accurately capture the different types of process faults with negligible false alarms. In the last test case, the normal process operation is among Modes 1, 2, 3, and 6 and mixed with three types of process


based I2 index is very insensitive to the process faults so that most of the faulty samples cannot be detected. Although the SPE index has higher fault sensitivity, a significant number of normal samples are identified as faulty ones incorrectly. Nevertheless, the HMM-ICA approach consistently demonstrates superior performance on fault detection. In Figure 11, the faulty segments are clearly above the control limit in both the I2HMM and SPEHMM plots while the normal data rarely trigger the false alarms. Its fault detection and false alarm rates are 95.7% and 6.4%, which are substantially improved over those of the ICA method. The conventional ICA monitoring method relies upon a global independent component model and thus cannot handle complex mode changes in process operation. Furthermore, it is lacking the capability to account for the system dynamics with uncertainty. As a comparison, the proposed HMM-ICA approach overcomes the above drawbacks through localized ICA models as well as the hidden Markov state estimations.

5. CONCLUSIONS A novel hidden Markov model based adaptive ICA approach is developed for process monitoring and fault detection in this article. Aimed at complex chemical processes with shifting operation modes and dynamic uncertainty, the proposed HMM-ICA method employs hidden Markov model to identify operating mode sequences and determine the specific mode corresponding to each monitored sample. Moreover, the process dynamics and uncertainty can be accounted for through the hidden Markov estimation. Then the multiple localized ICA models are developed for various operating modes, and the HMM based I2 and SPE statistics are derived to

Figure 9. HMM-ICA based monitoring results of test case 3 in Tennessee Eastman chemical process. 5512



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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

REFERENCES

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monitor process operation and detect abnormal events. Compared to the regular GMM based multimode process monitoring method, the presented HMM-ICA approach does not rely upon the assumption that the process data from each individual operating mode follows a multivariate Gaussian distribution approximately. Hence, it can handle more complicated applications with strong non-Gaussianity in each or some of the operating modes. However, it should be noted that the computational load of the HMM-ICA based monitoring method is potentially higher than that of GMM method. This indicates that the GMM method may be preferable for some industrial implementations where the Gaussianity within individual operating modes does hold. The presented HMM-ICA monitoring approach is applied to the challenging Tennessee Eastman chemical process. The comparison of monitoring results demonstrates that the HMMICA approach is superior to the conventional ICA method in terms of higher fault detection rates and lower false alarm rates. In addition, it is shown to be consistently effective in handling multiple operating modes with process uncertainty as well as detecting different process faults. Since the proposed HMMICA method is essentially an unsupervised monitoring technique, it only requires normal operating data for model training and thus can detect both known and unknown process faults. For unmeasured disturbances, they will be captured as faults only if they cause significant process upsets or abnormality on the measurement variables. In industrial practice, process operation knowledge and operator feedback can be incorporated into the data-driven monitoring systems to better differentiate between process faults and unmeasured disturbances. Future research may focus on the fault identification and diagnosis aspect of complex processes with multimodality, dynamics, and uncertainty. 5513



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