Incipient Fault Detection Based on Fault Extraction and Residual

Article pubs.acs.org/IECR

Incipient Fault Detection Based on Fault Extraction and Residual Evaluation Wenshuang Ge, Jing Wang,* Jinglin Zhou, Haiyan Wu, and Qibing Jin College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China S Supporting Information *

ABSTRACT: Process variables can be classified into three stages: normal operation, incipient fault, and significant fault stage. A two-step incipient fault detection strategy was proposed for monitoring the complex industrial process. The first step aims at the significant fault detection using the traditional multivariate statistical process monitoring methods. Then a method combining the wavelet analysis with the residual evaluation was carried out for monitoring the incipient fault. Wavelet analysis aims at extracting the incipient fault features from process noise. The residual generation is optimization based on the robustness and sensitivity index, which can be realized directly using the test data. An improved kernel density estimation based on signal to noise ratio is proposed to adaptively determine the detection threshold. The proposed incipient fault detection scheme is tested on a numerical example and the Tennessee Eastman process. Compared to other traditional fault detection methods, good monitoring performances, such as higher fault detection rate and lower false alarm rate, are obtained. Kalman filter8 and wavelet transforms.9 Carneiro et al.10 used the wavelet transforms to identify and characterize the transient phenomena in the time and frequency domains for the incipient fault detection of motor-operated valves. Harmouche et al.11 used the Kullback−Leibler Divergence to identify the small difference in probability density of latent scores. This method is sensitive to the incipient fault, which is a new feature extraction approach. D’Angel et al.12 proposed a two-step fuzzy/Bayesian formulation for detecting the change point, which was used to treat a fault detection problem in dynamical systems. Artificial neural networks (ANN)13 is a popular tool for extracting the incipient characters. For the second category, Zhang et al.14 presented work to transform the original system into two subsystems which can separate the uncertainties and faults. Zhang et al.15 used a bank of isolation estimator with the adaptive residual threshold for the abrupt and incipient faults diagnosis in uncertain dynamic systems. The adaptive detection observer16 and the functional observers17 were proposed to generate residual signals for the detection and isolation of the component or actuator faults. However, these methods are mostly applied to the specific system which has the analytical model, or only the improved data-driven methods were utilized. The precise mathematical model is usually difficult to be obtained for the complex chemical process. So the integration of model-based and datadriven-based approaches is an alternative selection. Ding et al.18 proposed a new data-driven fault detection scheme based on the parity space identification, which is different from the typical data-driven techniques and which has been extended recently.19,20 However, the parity space-based method shows

1. INTRODUCTION Fault diagnosis plays a key role on improving the system reliability and reducing the accident rate, so a more accurate and timely health monitoring system is needed for the increasingly complex industrial process. A large number of approaches have been presented for the abrupt (significant) fault and the incipient fault. The multivariate statistical process monitoring (MSPM) is very popular for the abrupt fault detection and diagnosis, such as principal component analysis (PCA) and partial least-squares (PLS).1,2 But the incipient fault detection and diagnosis is also a noteworthy problem. The incipient faults or the potential abnormal change may cause many serious problems such as the decline of product quality and accidents, if they are not monitored timely. The incipient fault usually refers to the fault that has a small amplitude, including early changing and the slow developing.3,4 Recently, different approaches for the incipient fault detection and diagnosis are developed, which can be classified into three main categories: (1) extract the incipient fault feature using some feature extraction methods, diagnose the incipient fault based on some traditional data-driven approaches such as MSPM. (2) Design an appropriate decoupling method to isolate the uncertainty and the incipient fault then incipient fault diagnosis is performed using the common methods. (3) A test signal is acted on the detected system periodically or at critical times for finding the abnormal behaviors. This method is usually called as active fault detection.5,6 The active approach requires that an input or test signal is applied to the system for the fault detection, and a cost function is used to optimize the test signal, which would not disturb the detected system as far as possible. Thus, the active detection is different from these traditional approaches which monitor the system outputs. For the first category, an N-step-ahead moving window PCA7 was presented for improving the capability of detecting the slowly developing faults. The common techniques of feature extraction include different filter methods such as the © 2015 American Chemical Society

Received: Revised: Accepted: Published: 3664

August 25, 2014 March 20, 2015 March 24, 2015 March 24, 2015 DOI: 10.1021/acs.iecr.5b00567 Ind. Eng. Chem. Res. 2015, 54, 3664−3677

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Industrial & Engineering Chemistry Research

Figure 1. Three stages of process variables.

occur in the second stage. It is difficult for the operator to identify the abnormal state. Then the incipient fault changes gradually and exceeds the high threshold, which leads to the significant faults. The fault can be easily observed by sensors or monitoring software at the third stage. When the operator takes some safety prevention, the fault is removed quickly, and the value of the observed variable gradually falls back to the second stage or the first stage. For the plant-level monitoring, especially in a complex industrial chemical process, a large number of sensors should be monitored and their measuring points related to each other. The high thresholds usually are used to judge whether a significant fault occurs. The significant faults are certain to carry weight since they could lead to the product quality decline or a serious accident, while the incipient faults in the second stage are often ignored due to their negligible features. So a sequence step is designed for these two different situations. The significant fault can be detected easily by conventional MSPM method. Then, the incipient fault detection scheme is carried out when the variables are under the second stage in Figure 1. In this work, a new incipient fault detection scheme, which is a hybrid method of wavelet transformation and optimal parity space analysis, is proposed. This approach is more complex than conventional MSPM because it is oriented to the incipient fault diagnosis. Therefore, a two-step incipient fault diagnosis strategy is proposed here to detect online the significant and incipient fault, respectively. First, conventional MSPM method is used to monitor process variable in the third stage in Figure 1 and identify whether a significant fault happened. The typical PCA method is used for decreasing the calculation and complexity of the fault detection system. The basic PCA theory is available in the Supporting Information. The second fault diagnosis step is continued only when no significant faults occur. The fusion of wavelet analysis and improved residual evaluation method is given for monitoring the incipient fault variables in the second stage. The residual is generated by a data-driven method in the parity space, which is abbreviated to PSDD. If an incipient fault is finally detected, the fault alarm is given; otherwise, the diagnosis system returns to the first step for monitoring the process again. The two-step incipient fault diagnosis flowchart is shown in Figure 2.

good performance only in the significant fault detection but not the incipient fault detection. This paper presented a two-step incipient fault detection scheme based on the first category. The first step aims at the significant fault detection. The existing MSPM methods (i.e., PCA) can be used for the purpose of simplification. When a significant fault is not detected, it is necessary to judge whether an incipient fault occurs through the second step. The second step is a hybrid method of wavelet transformation and optimal parity space analysis. The optimal residual generation is proposed based on Robustness and Sensitivity Index, and an improved kernel density estimation index is designed according to the signal to noise ratio (SNR), which aims to solve the trade-off between false alarm rate and fault missing rate. The remainder of this paper is organized as follows. The incipient fault detection problem is described, and the two-step detection strategy is presented in section 2. Section 3 demonstrates the proposed incipient fault diagnosis scheme in detail. Wavelet analysis is reviewed and a novel optimal residual generation method is presented. Then an improved kernel density estimation (IMKDE) index with signal to noise ratio is proposed to obtain new incipient fault evaluation logic. Section 4 is the simulation study of the two-step fault detection strategy. Finally, the conclusions are presented in section 5.

2. PROBLEM FORMULATION Many researchers currently classify the mode of process variables by normal mode or fault mode. However, the transient stage is also very important. Monroy et al.21 considered that the period between the fault occurrence and the fault detection is a latency period, and the dynamical evolution of process variables include latent stage, transient stage, and steady stage. This period is an interesting domain, and our study is motivated by this. Here a new classification about process variables is presented in Figure 1, which also has three stages. As shown in Figure 1, the first stage is the normal operating phase and the variables that be monitored are under the low threshold. Assuming that the process is changed due to some unknown reasons, the process variable value exceeds the low threshold and reaches the second stage. The main characteristic of this stage is that the variable amplitude is still small and not conducive to observe, therefore, the incipient faults generally 3665

DOI: 10.1021/acs.iecr.5b00567 Ind. Eng. Chem. Res. 2015, 54, 3664−3677

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Figure 2. Two-step scheme of incipient fault detection system.

Figure 3. Integrated detection scheme of incipient faults.

3. HYBRID DETECTION STRATEGY FOR INCIPIENT FAULT

based residual generation schemes can be realized directly using test data. Since SIM is a data-driven method, PSDD is an abbreviation of the parity space identification and data-driven method in this paper. The feature of incipient fault is usually mixed with the disturbance or noise, so how to identify the fault feature is very important. Here, we apply wavelet analysis for extracting the potential feature of incipient fault, by which the fault feature and the noise can be isolated. The detail scheme is shown in Figure 3. The novel optimal residual is generated based on the robustness and sensitivity index, and new residual evaluation

In this section, a hybrid method of wavelet analysis and residual evaluation is proposed for the incipient fault detection. The residual generator is directly identified from the test data based on subspace identification method (SIM) in the parity space. Comparing to the traditional MSPM methods such as PCA and PLS, the main advantage of such optimal residual generation scheme is to deal with process dynamics, robustness issues by using the well-established observer theory, and the parity space3666


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vector u(k) and initial state vector x(0). System matrices A, B, C, and D and system order n are generally unknown. First, a data vector is defined as θ(k) ∈ ℜζ, which can represent x(k), u(k), y(k), respectively, and the corresponding dimension ζ is n, l, and m. Some new notations are introduced,

logic is given for the incipient fault detection based on an improved kernel density estimation (IMKDE) method. 3.1. Features Extraction Based on Wavelet Analysis. Wavelet analysis is a kind of variable window technique, which can use a signal in the time domain to analyze its frequency components. It has favorable local time-frequency characteristics. Continuous wavelet transform (CWT) and discrete wavelet transform (DWT) are generally expressed22 as follows, respectively. CWTf (b , τ ) =

⎛

⎞

∫ f (t )ψ *⎝ t −b τ ⎠ dt

1 b

⎜

DWTf (m , n) = 2−m /2

⎡ θ(k − s + 1)⎤ ⎢ ⎥ s·ζ ⋮ θs(k) = ⎢ ⎥ ∈ ℜ , Θs = [θs(k) ⎢ ⎥ θ(k) ⎣ ⎦

⎟

∫ f (t )ψ *(2−mt − n) dt

··· θs(k + N − 1)] ∈ ℜs·ζ × N

(1)

where s and sample number N are selected by users such that s ≥ n and N is large enough. The standard framework of process model for subspace identification can be obtained by iterating eq 4,

(2)

where ψ*(t) is the conjugate function of ψ(t), b and τ are scaling and shift parameters, respectively, and m and n are integers. Many scholars used wavelet decomposition and reconstruction for extracting fault features, which is the famous Mallat algorithm. Ruqiang Yan23 elaborated the theory of extracting different kinds of features, including incipient feature by multiresolution analysis of wavelet. The main process is that the wavelet is decomposed into the approximation space and the detail space. Then a signal is reconstructed according to the low-pass and high-pass filter analysis on the original signal. The detailed decomposition process can be found in the literature.22,23 Many studies reported that a wavelet threshold filter is effective for enhancing the fault information.24 The popular threshold selection method is soft threshold, ⎧ sign(W ) ·(|W | − thr ), |W | > thr Ŵ = ⎨ |W | ≤ thr ⎩ 0,

Ys = ΓsX k + H u, sUs + H w, sWs + Vs

rs(k) = W0[Ys(k) − Hu , sUs(k)],

(3)

x(k + 1) = Ax(k) + Bu(k) + w(k)

⎡ ω1 ⎤ ⎢ ω ⎥ 2 ⎥ W0 = ⎢ ⎢ ⋮ ⎥ ⎢ ⎥ ⎣ ωms ‐ n ⎦

(4)

where x ∈ ℜ , u ∈ ℜ , y ∈ ℜ are the state variable vectors, input signals and measured output signals. w ∈ ℜn, v ∈ ℜm are process noise and measurement noise with zero-mean value and normal distribution. They are independent of the input n

l

W0 ∈ Γ⊥s

(7)

where Γ⊥s is denoted as the left null space of Γs (i.e., Γ⊥s Γs = 0). Residual can be generated from eq 7 with the identification of Γ⊥s and Γ⊥s Hu,s from process data. The data-driven based identification approach was presented25,26 to obtain the identification Γ⊥s and Γ⊥s Hu,s. This approach is based on subspace identification via principal component analysis, and the detail description is presented in the Supporting Information. Furthermore, some new fault detection and isolation (FDI) strategies for nonlinear systems were presented by Mhaskar et al.27 and Du et al.28 On the basis of the general nonlinear system, the sensor and actuator fault isolation scheme was proposed, and this extends the theory of analytical redundancy to the nonlinear system. Here, we did not considered the residual generation directly from a nonlinear system but a linear system eq 4. In fact, nonlinear model or linear model is difficult to be identified in an actual process. Although the residual eq 7 is derived from a linear system, it is also a general representation since all the identification matrices are obtained directly from process input and data. All the nonlinear characteristics are shown in process data, and it is not related to the actually system structure. So it can be used to detect faults for a nonlinear system.19 The residual generator can be obtained from eq 7 which is directly identified from the input and output data (i.e., PSDD approach). But the primary residual cannot guarantee it has the maximum sensitivity for each fault, especially in the case of incipient fault. So the identification result cannot be used directly. In this section, new schemes are proposed for obtaining the enhanced residual. The primary residual vector is given by eq 7, and the design of optimal residual vector is equivalent to finding a matrix W0. Here, the parity matrix W0 is rearranged as follows,

where Ŵ is the reconstructed signal, W is the original signal, and thr is the threshold. The threshold has many selection criteria, such as Stein’s based threshold selection rule.24 Here, the soft threshold technique in eq 3 is used for isolating the incipient fault characteristics from the noise or disturbance. Wavelet basis function has important effects on the result of feature extraction. Inappropriate wavelet function will result in the failure of fault detection, especially for the incipient fault. When the orthogonal wavelet function is selected, the signals in approximation space and detail space are independent, which help to improve the accuracy of feature extraction by concentrating the feature information. Thus, the orthogonal wavelet functions from orthogonal wavelet family are selected for extracting the incipient feature in this work. 3.2. Optimal Residual Generation. Residual evaluation or residual generation has different kinds of techniques. Here parity space and subspace identification method is used to generate residual.18 The current results of the parity space and data-driven (PSDD) method is explained simply in this section. Consider a process system,

y(k) = Cx(k) + Du(k) + v(k)

(6)

with Γs is the extended observability matrix and Hu,s, Hw,s is the lower triangular block Toeplitz matrix. The residual generator based on the parity space is as follows

⎪ ⎪

(5)

m

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where ν is the eigenvector and λ is the eigenvalue. The value of the performance index is the maximal eigenvalue, and the corresponding eigenvector is ωi that is used to select an optimal parity vector. This is one of the commonly used robust residual generation methods.20,29 Thus, the optimal parity vector is as follows,

where ωi ∈ ℜ1×ms, i = 1,2,..., ms−n, is the so-called parity vector. From eq 8, it is known that W0 spans a subspace and allows only the selected signals to have an influence on the residual rs(k). If a vector is selected from W0, the residual is optimal in one direction of the subspace spanned by W0. The fault detection can be invalid when an incipient fault or significant fault is in other directions. Therefore, the parity matrix is chosen to ensure the residual can catch the change when one fault happens. It is worth noting that the robustness to the sensitivity may be only suboptimal in some directions.29 The approach based on structured residual vector (SRV)30 is used for obtaining the optimal residual, whereas the SRV approach is a global optimal strategy and it cannot ensure each direction has the maximum sensitivity. On the basis of this, an optimal residual generation scheme is presented. A typical idea is that every direction has the equal influence on the residual, and this is the average index. 3.2.1. Average Index (AI). Different design method for the selector W0 can have a different effect on the primary residual. In this scheme, we first consider every parity vector has the same impact on the residual in parity matrix W0. The detailed demonstration is as follows. The rows of Γ⊥s have a lot of redundancy information, thus we can arrange this information in different ways. The index is defined as αj =

1 ms − n

ωi = νi Γ⊥s

To ensure the sensitivity to every fault and the robustness to unknown inputs, set ω1 = νminΓ⊥s , ωms−n = νmaxΓ⊥s , and ωi = νiΓ⊥s , νmin < νi < νmax, that is ⊥ ⎡ ω1 ⎤ ⎡⎢ νmin Γ s ⎤⎥ ⎢ω ⎥ ⎢ 2 ⎥ = ⎢ ⋮ ⎥⎥ W0 = ⎢ ⎢⋮ ⎥ ⎢ ⋮ ⎥ ⎢ ⎥ ⎣ ωms ‐ n ⎦ ⎢⎣ ν Γ⊥ ⎥⎦ max s

Γ⊥s (i , j)

i=1

(9)

where j = 1, 2, ..., ms. The average index is ωave = [ α1 α2 ⋯ αms ]

(10)

Then the new optimal residual is obtained, ⎡ ω1 ⎤ ⎡ ωave ⎤ ⎢ω ⎥ ⎢ω ⎥ 2 ⎥ = ⎢ ave ⎥ W0 = ⎢ ⎢⋮ ⎥ ⎢ ⋮ ⎥ ⎥ ⎢ ⎥ ⎢ ⎣ ωms ‐ n ⎦ ⎣ ωave ⎦

⎧ R e is known ⎪ rs(k )R ers(k ), J=⎨ ⎪ ⎩ rs(k) ·rs(k), R e is unknown

ωi

ωi Γ⊥s ·(ωi Γ⊥s )T ωi Γ⊥s Hu , s·(ωi Γ⊥s Hu , s)T

(12)

The maximization problem can be solved to a generalized eigenvalue−eigenvector equation as follows,29 ν(λ[Γ⊥s Hu , s(Γ⊥s Hu , s)T ] − [Γ⊥s (Γ⊥s )T ]) = 0

(16)

where Re is the covariance matrix of the residual and usually is unknown at the data-driven case. Thus, the detection statistic in this paper is calculated by J = rs(k)·rs(k). Another important work is to design the threshold. The common threshold calculation is under the assumption that SWR will follow a central Chi-square distribution with (ms-n) degrees of freedom [i.e., J ∼ χ2(ms − n)]. If the system is in fault status, SWR will not be central Chi-square distributed. Therefore, fault can be detected by simply comparing the SWR index against the confidence limit χ2α(ms − n). When the observed process data are significantly noisy, the detection index J relying on residual vector will lead to false alarms caused by the process noise. How to adjust fault threshold adaptively according noise level plays an important role on the detection and diagnosis of incipient fault. Traditional kernel density estimation (KDE) is a nonparameter method to calculate the fault evaluation threshold. Sometimes it will result in higher probability of false alarm. So an improved kernel density estimation (IMKDE) method is designed, whose fault evaluation logic is designed according to the signal to noise ratio (SNR). Kernel density estimation method is similar to the histogram in that the probability density function is built up of a number of individual kernels centered on the sampled data points. Suppose that the probability density function f(x) is defined by

(11)

The residual may have a relativly low amplitude in the normal operating condition, and the important variation of the residual is remarkable when a fault appears. However, the average index aims to ensure each direction of the subspace spanned by W0 has the same influence on the residual. This scheme may ignore the differences among the different directions of residual and cannot catch the fault information effectively. Thus, a robustness and sensitivity index, which is robust to unknown inputs and is sensitive to faults, is proposed to optimize the selector W0. 3.2.2. Robustness and Sensitivity Index (RSI). Here, the main task is to select an optimal matrix W0 from the parity space Γ⊥s to make the residual eq 7 to be nonsensitive to unknown inputs while sensitive to faults. For this purpose, a new performance index is utilized as follows, J = max

(15)

The optimal residual can be generated by eqs 7 and 15, and the main idea of RSI is based on the robust residual generation and the optimal parity matrix. Thus, each direction variation can be caught by the optimal residual. This is helpful to detect incipient fault because the incipient fault may cause fault evolution, that is the fault direction can be changed during the fault appearance. 3.3. Fault Evaluation Design. The fault detection statistic should be designed for the purpose of detection or isolation when the residual is generated. Generally, the squared weighted residual (SWR) index,30 shown in eq 16 is used for fault detection which is a much better performance measure of detection and isolation fault than the widely used squared prediction errors (SPE) index.

ms − n

∑

(14)

(13) 3668


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Industrial & Engineering Chemistry Research f (x ) =

1 nh

n

⎛ x − Xi ⎞ ⎟ h ⎠

λ=

∑ K ⎜⎝ i=1

(17)

⎛ || y ̂ ||2 ⎞ 2 ⎟⎟ SNR = 10 log10⎜⎜ || − y y ̂ ||22 ⎠ ⎝

n →∞

not-negative kernel function such that +∞

K (x) ≥ 0,

∫−∞

K (x ) d x = 1

(18)

⎧ ⎛ a ⎞ 1 ln⎜ − 1⎟ , SNR ≤ 50 dB ⎪1 + ⎝ ⎠ b sigmoid SNR · ⎨ λ= ⎪ SNR > 50 dB ⎩1,

∞

∫x (α) f (x) dx = α

(19)

0

where P{·} represents the probability, x0(α) is the value with a preselected level of significance α(0 < α < 1) (e.g., α = 0.05). The detection threshold is defined by Jth = λx0(α)

(26)

where a > 0, b > 0, sigmoid > 0 is the selected parameter. Their default values are 101,1,1, respectively. Here we assume that the SNR value of process data is lower than 100 dB. The numerical relationship between λ and SNR is simulated under different sigmoid in Figure 4. It is reasonable because the

(20)

where λ is synthetically index coefficient, and the default value is 1. The fault detection logic to decide process status is ⎧ ⎪ J < Jth , ⎨ ⎪ ⎩ J > Jth ,

(25)

where ∥ ∥2 is 2-norm of a matrix, y is the original data, and ŷ is the reconstructed data by wavelet. Now let us consider how to adaptively adjust the detection threshold according to the noise level [i.e., the selecting of predesigned function φ(SNR)]. It is similar to the tuning of PID controller, which selects a proper range by experience for ensuring the convergence of the dynamic process. On the basis of this and motivated by the sigmoid function, we define the 1/ φ(SNR) as follows,

The resulting shape of a KDE estimator is sensitive to the window width and the kernel function. Gaussian function is usually selected as the kernel function. Appropriate choice of window width is thus vital to the success of the KDE method. Mean Integrated Squared Error criterion aimed at optimizing the window width for a given data set is considered.31,32 After the probability density function f(x) is estimated, the quantile x0(α) is calculated as follows, P{x > x0(α)} =

(24)

where φ(SNR) is a predesigned function according to the process operation, SNR is the signal to noise ratio of process data and can be estimated by,

where n is the number of samples, h is the so-called window width, which is similar to the bin width in the case of the histogram. Generally, lim h = 0, lim nh = n → ∞. K() is a n →∞

1 φ(SNR)

fault‐free fault

(21)

Parameter λ aims at improving the accuracy of incipient fault detection system under the noisy condition. It is obviously a trade-off between fault detection rate (FDR) and false alarm rate (FAR). Some literature use missed detection rate (MDR) to substitute the FDR index, since MDR = 1 − FDR. FAR and FDR (or MDR) usually is used to evaluate the performance of the monitoring system under normal and abnormal operation modes, respectively. The higher FDR and the lower FAR mean that the monitoring performance is better. FAR and FDR indices of the fault detection system is defined as follows, FAR = FDR =

the number of samples (J > Jth |fault‐free) total samples (fault‐free) the number of samples(J > Jth |fault) total samples (fault)

× 100%

× 100%

Figure 4. Relationship between parameter λ and SNR. (22)

The threshold Jth has an influence on the indices of FAR/ FDR (i.e., the higher threshold means lower FAR and FDR). It should be adjusted according to the level of noise. In order to improve the FAR/FDR, the threshold should be increased in the case of higher noise (the smaller SNR), Jth =

x0(α) φ(SNR)

higher SNR means the noise level is lower and the threshold keeps the original value (λ = 1). For the case of higher noise (the smaller SNR), the threshold is relaxed with the increasing of parameter λ. The SNR is actually hard to determine in practice. The estimated SNR by eq 25 may be inaccuracy and unreliable, so selecting appropriate φ(SNR) can eliminate the negative effect caused by unknown SNR. For example, if a linear φ(SNR) is considered such as φ(SNR) = ζ·SNR/SNR(ζ ≠ 0), then eq 24 can be converted into

(23)

Here the signal to noise ratio is introduced to select an appropriate λ for improving the threshold, and λ is defined by eqs 20 and 23 as follows, 3669


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Now let us consider the fault strength a1 = 0.5. For the first step fault detection, PCA is utilized to detect the significant fault. PCA model of training data was obtained offline first, and two principle components who account for 85% of total variance were selected according to cumulative percent variance. Then the faulty data are used to verify the monitoring system through the PCA model. The FAR of T2 statistic and SPE statistic for normal data in the faulty set are 0.83% and 3.96% with the level of significant limit 0.01, respectively. FDRs of first PCA monitoring are 2.71% (T2 statistic) and 8.13% (SPE statistic). The detection results are shown in Figure 5a, and it is shown that significant fault is not detected. Thus, the incipient fault detection scheme is used to check whether an incipient fault happens or not. To illustrate the effect of different approaches in second step detection, three residual generation approaches based on SRV,30 AI, and RSI proposed in this work are compared. The sym3 wavelet basis function is selected, and the decomposition scale is 1. The order of parity space is s = 10, and the order of state variables is n = 5. Parameter λ based on SNR is utilized to estimate the adaptive threshold with a = 101, b = 1, and sigmoid = 5. The detection results are shown in Figure 5 (panels b−d). The FARs of SRV, AI, and RSI are 0.63%, 1.67%, and 2.50% with the level of significant limit 0.05. This means that these methods are effective under normal operation conditions. However, Figure 5 (panels b and c) shows that the incipient fault is not detected successfully. The FDRs of SRV method and the proposed AI are 1.46% and 2.29%, respectively. By contrast, the 96.3% FDR is obtained by RSI as shown in Figure 5d, which illustrates that the proposed optimal residual generation scheme RSI is effective for the incipient fault detection and has the maximum sensitivity for the variation on x1. Thus, the RSI-based residual generation is more valid than the SRV method for incipient fault diagnosis. To illustrate the influence of different fault amplitudes, several other values of a1 are changed from 0.15 to 0.5, and the results are shown in Figure 6. Figure 6 demonstrates that the SPE statistic of PCA has the highest FARs but has the lower FDRs in different fault amplitudes. The SRV approach and AI method have the lower FARs and FDRs. For example, the biggest FDR of SRV approach is 6.88%. However, the FDRs of RSI have the linear increase with the average 2% FARs. FDR can reach to 31.7% when the fault amplitude is 0.25, and the lowest FDR is 8.54%, which is more than 5%. Thus, the expecting results show that the proposed RSI is very effective for detecting incipient faults. As mentioned in section 3.3, the IMKDE method has the adaptive threshold but the threshold depends on the parameter λ. Thus, this part is going to check the relation between λ and FAR/FDR. Table 2 shows the detection results of three approaches in different sigmoid values (λ values) under the condition of special SNR. The FDRs of RSI approach are more than 95% under different λ values, but the FDRs of AI and SRV are lower than 8%. This reports that parameter λ does not affect the decision for incipient fault diagnosis. But parameter λ can obtain the better trade-off between FAR and FDR. For example, the FAR of RSI is 6.04% and FDR of RSI is 99.0% when λ = 1. This result means that the higher FDR is at the cost of the higher FAR. However, the IMKDE method may solve the problem, and the lower FAR (under 5%) and higher FDR can be obtained.

(27)

The threshold is irrelevant with SNR if ζ = 1, λ = 1. If the SNR is lower, we can select ζ < 1, λ > 1 in order to decrease FAR. In this section, residual generation and residual evaluation are given, and the incipient fault detection strategy is presented. It is worth noting that there are several parameters that should be selected to obtain optimal settings. The tuning methods are concluded in Table 1, and the detailed demonstration can be Table 1. Tuning Methods of Parameters parameters

target

wavelet basis function

feature extraction

the order of parity vector s the index coefficient λ

residual generation fault evaluation logic

tuning methods orthogonal wavelet function ref Ding et al.33 eq 26 in section 3.3

found in the references. The order of the parity space (s) is selected largely, which means the better monitoring performance based on Ding’s work.33 However, a proper value should be selected based on the situation considering the calculation and the complexity of the monitoring system. The order of the system n is not a necessary parameter in the system design.

4. CASE STUDY 4.1. Numerical Example. To illustrate the effectiveness of the incipient fault detection scheme or the two-step incipient fault detection system, a numerical example34 is given as follows, ⎡ 0.118 −0.191⎤ ⎡1.0 2.0 ⎤ x(k) = ⎢ u(k − 1) ⎥x(k − 1) + ⎢ ⎣ 3.0 −4.0 ⎥⎦ ⎣ 0.847 0.264 ⎦ y(k) = x(k) + v(k)

where u and y are the measured inputs and outputs. The correlated input u is such that ⎡ 0.811 −0.226 ⎤ u(k) = ⎢ ⎥u(k − 1) ⎣ 0.477 0.415 ⎦ ⎡ 0.193 0.689 ⎤ +⎢ ⎥w(k − 1) ⎣−0.320 −0.749 ⎦

where w and v are Gaussian noise such as w ∼ N(0,1) and v ∼ N(0,0.012). The fault condition is considered as follows, and this can be shown as sensor faults. ⎡ y (k) ⎤ ⎡1 + a 0 ⎤⎡ x1(k) ⎤ ⎡ v1(k) ⎤ 1 ⎢ 1 ⎥=⎢ ⎥+⎢ ⎥ ⎥⎢ ⎢ y (k)⎥ ⎣ 0 1 + a 2 ⎦⎢⎣ x 2(k)⎥⎦ ⎢⎣ v2(k)⎥⎦ ⎣ 2 ⎦

where a1 and a2 show the fault strength or amplitudes on system outputs. In this work, we consider y1 is affected by faults (i.e., a1 ≠ 0, a2 = 0) and the range of fault is [0, 0.5]. The normal data set is obtained with 480 observations. The faulty data set is generated in the following 960 samples (i.e., fault happens at the 481th sample). 3670


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Figure 5. Detection results of different methods with a1 = 0.5. (a) PCA monitor for significant fault. (b−d) Incipient fault detection based on SRV, AI, and RSI, respectively.

The other important parameter is the wavelet function within the proposed approach, and the detection results are shown in Table 3. Parameters are the same with Figure 5, except the wavelet function. Table 3 demonstrates that the db3/sym3 wavelet basis function is the best alternative parameter for RSI. Wavelet function has significant impact on the detection performance, although the lowest FDR is 76.9%. Meanwhile, the AI and SRV approaches failed to detect the incipient fault with different wavelet functions. Table 3 is a simple analysis about wavelet functions and detection accuracy. 4.2. Tennessee Eastman Benchmark Process. Tennessee Eastman (TE) process, a realistic simulation program of a chemical plant, is a benchmark for control and monitoring studies. Figure 7 shows the flow diagram of the TE process. The process has five major units, including reactor, condenser, compressor, separator, and stripper. The reaction involves eight components, which are four reactants, two products, an inert, and a byproduct. This paper considers the base operating mode and uses the decentralized control program.35 TE process data sets by N.L. Ricker in our studies can be downloaded from

Figure 6. Monitoring results of different fault amplitudes.

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Industrial & Engineering Chemistry Research Table 2. Detection Results (%) of Different λ Values sigmoid

5

10

20

30

−

λ

1.546

1.273

1.137

1.091

1.000

FAR/FDR

FAR

FDR

FAR

FDR

FAR

FDR

FAR

FDR

FAR

FDR

RSI AI SRV

2.50 1.67 0.63

96.3 2.29 1.46

3.54 3.33 2.29

97.3 3.96 3.96

3.96 4.17 3.33

98.3 5.00 5.21

4.97 4.17 3.75

98.3 5.00 5.63

6.04 4.58 4.58

99.0 5.83 7.92

Table 3. Detection Results (%) of Different Wavelet Functions RSI db3 sym3 coif4 dmey

AI

Table 4. Monitoring Variables in TE Process

SRV

FAR

FDR

FAR

FDR

FAR

FDR

2.50 2.50 3.54 2.71

96.3 96.3 82.5 76.9

1.67 1.67 1.87 1.46

2.29 2.29 0.83 2.71

0.63 0.63 0.83 1.04

1.46 1.46 0.83 1.67

http://depts.washington.edu/control/LARRY/TE/download. html. This TE process has 53 measurements, including 41 process variables and 12 manipulated variables. Similar to ref 36, 16 main process variables and 11 manipulated variables (except reactor agitator speed) are selected for monitoring. The process variables are shown as Table 4. There are 21 process faults in the TE process. Fault 3 is the step change of D feed temperature and fault 9 is the random variation of D feed temperature. Fault 15 is the sticking fault of condenser cooling water valve. The three faults are the incipient

no.

measured variables

no.

measured variables

1 2 3 4 5 6 7 8

A feed D feed E feed A and C feed recycle flow reactor feed rate reactor temperature purge rate

9 10 11 12 13 14 15 16

product separator temperature product separator pressure product separator underflow stripper pressure stripper temperature stripper steam flow reactor cooling water outlet temperature separator cooling water outlet temperature

faults, which mean these faults have small amplitudes and are not easy to be detected.37 Faults 3 and 9 are used for testing the effectiveness of the proposed two-step detection strategy. First two kinds of data sets are obtained from the TE process, which include training and test data. Three training data are collected in training set by 3 runs (including one normal operation, one fault 3 operation, and one fault 9 operation), and each operation is performed for 24 operation hours with a sampling interval of 3 min. Then the size of data matrix is 480

Figure 7. Flow diagram of the TE process. 3672


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Figure 8. Fault 3 detection results. (a) PCA monitor for significant fault. (b−d) Incipient fault detection based on SRV, AI, and RSI, respectively.

× 27 (16 process variables and 11 manipulated variables). Correspondingly, three test data (also consist of normal, faults 3 and 9 operations data) are collected, each is for 48 operation hours, and the size of every data matrix is 960 × 27. All faults were introduced at the eighth hour (i.e., at the 161th observation of the operating time). The detection results about faults 3 and 9 are shown to illustrate the effectiveness of the proposed method. Fault 3 detecting results are shown in Figure 8. Figure 8a is the result based on PCA, and this is the first step fault detection. The PCA model was obtained offline first, and 14 principle components account for 85% of total variance. Then the test data are used to verify the monitoring system through the PCA model. The FAR and FDR of SPE statistic are 2.50% and 10.0%. This illustrates that an ambiguity result is obtained as the lower FDR should not guarantee that the change is caused by faults or disturbances. Thus, the incipient fault detection scheme is utilized. For the second step detection, the dmey wavelet basis function is selected, and the decomposition scale is 5. The order of parity space is s = 13, and the order of state variables is n = 6 though it is not essential. Parameter λ based on SNR is utilized to estimate the adaptive threshold with a = 101, b = 1,

sigmoid = 5. The detection results are shown in Figure 8 (panels b−d) based on SRV, AI, and RSI. The special parameter for SRV is sigmoid = 40, and sym6 wavelet basis function is selected for AI. The FARs of SRV, AI, and RSI are 0, 0, and 3.75% with the level of significance 0.05. Figure 8b shows that the SRV approach failed to catch the incipient fault with FDR = 3.00%, whereas, the AI and RSI approach can detect the incipient fault, as shown in Figure 8 (panels c and d). Their FDRs are 34.3% and 97.2%, respectively, and this is the clear identification for engineers or operators. Furthermore, the proposed RSI approach is better than the SRV method or AI method for incipient fault diagnosis. Thus, RSI approach is recommended for incipient fault diagnosis. Figure 9 demonstrates the detecting results of fault 9. The test data are used to verify the monitoring system through the PCA model. FAR and FDR of SPE statistic are 2.50% and 8.62%, respectively. The significant fault is not detected, so the incipient fault detection scheme is performed. For the second step detection, the parameter is same with fault 3 diagnosis, and the detection results are shown in Figure 9 (panels b−d) based on SRV, AI, and RSI. The FARs of SRV, AI, and RSI are 0, 0, and 3.13% with a level of significance of 0.05. Figure 9b shows that the SRV approach failed to catch the 3673


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Figure 9. Fault 9 detection results (a) PCA monitor for significant fault. (b−d) Incipient fault detection based on SRV, AI, and RSI, respectively.

Figure 10. FARs/FDRs in different sigmoid value (fault 3).

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Industrial & Engineering Chemistry Research incipient fault with FDR = 6.25%. The AI approach is also invalid to detect the incipient fault as shown in Figure 9c (i.e., FDR = 9.88%). The FDR of RSI is 76.1%, and this can be a clear warning for operators, although there are some lags in the monitoring chart. In conclusion, the proposed RSI approach is better than the SRV method for incipient fault diagnosis. Moreover, the RSI approach can detect the incipient fault, and the second step detection is helpful when process variability is not evident or is confused for engineers. To analyze the relationships between parameter λ and FAR/ FDR, different sigmoid values are set for checking the fault detection accuracy. Figure 10a demonstrates that the FAR of RSI is higher than the other two approaches but the increase of FAR is nonlinear. Furthermore, we could find that the FARs is increasing with the sigmoid increasing (parameter λ decreasing). The relationships between the FDR and parameter sigmoid (or parameter λ) are shown in Figure 10b. The FDR of RSI is higher than the AI or SRV approach, and the FDR can reach to 97.1% when sigmoid = 5. The results verify the fact that the FARs or FDRs increase with sigmoid increasing (i.e., parameter λ decreasing) in Figure 10. Thus, the RSI approach has the best monitoring performance, and the FAR of normal conditions is under the 5%. The adaptive IMKDE method can give some helpful instructions and solve the trade-off between FAR and FDR in practice (e.g., sigmoid = 5 is the proper parameter with higher FDR and lower FAR). The detection performances under different wavelet function and the order of parity vector s are compared in Tables 5, 6,

Table 7. Detection Results (%) of Different Wavelet Function and s (SRV) db3

sym6

dmey

db3

sym6

dmey

normal/FAR Fault 3/FDR Fault 9/FDR normal/FAR Fault 3/FDR Fault 9/FDR normal/FAR Fault 3/FDR Fault 9/FDR

s = 11

s = 12

s = 13

s = 14

1.04 50.0 65.9 1.46 52.4 61.9 1.46 87.3 56.5

1.25 55.8 74.8 1.67 59.5 68.4 1.67 96.3 65.4

2.08 59.3 78.8 2.08 60.5 70.8 1.87 96.6 70.9

1.04 58.5 78.0 1.25 68.3 71.6 1.87 97.1 76.1

1.67 59.0 79.5 0 53.9 62.5 2.29 98.1 74.9

Table 6. Detection Results (%) of Different Wavelet Function and s (AI) db3

sym6

dmey


s = 10

s = 11

s = 12

s = 13

s = 14

3.33 19.38 17.37 2.92 12.38 8.38 2.29 3.62 0.37

3.75 7.62 14.12 2.92 4.50 6.25 3.13 17.37 14.62

3.33 3.38 3.25 3.33 8.87 2.00 3.33 0.37 1.87

2.92 1.63 1.50 2.50 59.88 20.87 1.87 14.25 4.25

3.54 11.50 7.62 3.33 5.00 6.12 3.54 1.13 4.00

s = 10

s = 11

s = 12

s = 13

s = 14

3.75 4.75 4.50 2.92 1.63 4.13 4.17 9.25 7.25

2.92 4.13 4.88 3.13 9.25 2.88 3.54 2.75 3.62

2.08 3.25 5.00 1.04 3.75 5.12 1.46 5.12 3.13

3.33 4.25 5.50 3.75 3.50 4.62 3.33 3.13 6.75

3.13 2.13 9.12 3.33 4.00 5.25 4.37 5.50 8.25

performance. For example, the FDR is 58.5%, 68.3%, and 97.1% for db3, sym6, and dmey wavelet functions when s = 13. This FDR detection difference is very obvious, so the wavelet function should be selected carefully. The results based on the AI approach are shown in Table 6. Table 6 demonstrates that the AI approach is effective under some special parameter conditions such as db3 wavelet function/s = 10, dmey function/s = 11, and sym6 function/s = 13. Except for s = 12, the FDRs are more than 10%, whereas the FDRs are less than 10% in Table 7. Table 7 shows that there are no optimal parameters for monitoring faults 3 and 9. Thus, the SRV approach is invalid for monitoring incipient faults. 4.3. Discussion. The problem of the incipient fault diagnosis was explored in this work, and a hybrid method of wavelet transformation and optimal parity space analysis was proposed for this purpose. This study showed that the proposed approach can improve the ability of incipient faults detection. The performance of the proposed two-step detection strategy is based on several factors. First, a successful incipient fault detection demands large numbers of process data. The quality of these process data, whether they can cover the major operation conditions or not, play the key role in results monitoring. For example, different results are presented based on the proposed AI method in the numerical example and TE benchmark process. Figures 5c and 8c (or Tables 3 and 6) show that the AI method is effective in the TE process, but the fail detection result is presented in the numerical example. The intrinsic difference between the two cases is the test data sets. The data that were generated by the numerical example can represent the simple process conditions only, and these data are susceptible to noises or the values of the constant matrices. Thus, the incipient features may be overwhelmed if the matrix W0 was obtained from these data. But the TE process is a realistic simulation program of a chemical plant, its process data are sufficiently informative in many different operation conditions. So the TE process will always be used to test all kinds of fault detection methods. Second, different approaches can have an important impact on the detection results. Let us consider the AI and RSI method, which are conducted from total different views. AI aims to ensure each direction of the subspace spanned by W0 has the same influence on the residual, while the RSI method can catch all the variations in each direction, which are robust to unknown inputs and sensitive to faults. So, case study shows that the RSI method is more effective than the AI method. Last, in order to obtain higher detection accuracy for the proposed approach, the parameter optimization should be discussed deeply, such as the wavelet basis function. Here we

Table 5. Detection Results (%) of Different Wavelet Function and Order of Parity Vector s (RSI) s = 10


and 7. Table 5 shows the results based on the RSI approach. With the order of parity vector increasing, the FDRs of faults 3 and 9 increase widely. This reports that the higher value of s can result in better monitoring performance. The optimal wavelet functions are dmey and db3 for faults 3 and 9, respectively. Different wavelet functions have significant influence on the 3675



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just make a preliminary discussion on this issue in the case study, and further research should be focused on it.

ASSOCIATED CONTENT

S Supporting Information *

Supporting Information can be found in the web version. This material is available free of charge via the Internet at http:// pubs.acs.org.

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REFERENCES

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5. CONCLUSIONS In this paper, we divided the dynamic of process variables into three stages, normal operation, incipient fault, and significant fault. Correspondingly, a two-step incipient fault detection scheme was proposed for the monitoring of complex chemical processes. First, the traditional multivariate statistic monitoring method is used to monitor significant fault. Second, the integration of wavelet analysis and residual evaluation method is used for monitoring the incipient fault. Here, wavelet analysis can extract the incipient fault features, and the optimal residual generation is proposed based on robustness and sensitivity index. Then the improved kernel density estimation (IMKDE) method based on signal to noise ratio is proposed to adaptively determine the residuals detection threshold, which can improve the monitoring system performance, such as FAR and FDR. As the incipient fault has no obvious feature, the traditional multivariate statistic monitoring method will be invalid for the incipient fault detection. Thus, the wavelet decomposition and reconstruction is given to extract the potential fault feature. PSDD approach is one approach of residual generation, and the residual generator can be realized directly using test data. But the primary residual cannot guarantee it has the maximum sensitivity for incipient fault, so the enhancing residual can be obtained by RSI. IMKDE is a nonparametric estimation method to calculate residual evaluation threshold. The adaptive threshold can solve the trade-off between FAR and FDR by introducing the signal to noise ratio of process data. For a numerical example and TE process, the proposed RSI method shows better monitoring performance than the SRV method for incipient fault diagnosis. Finally, the tuning methods of parameters are concluded, and parameter sensibility analysis is carried out.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by National Natural Science Foundation of China (Grant 61174128), Beijing Natural Science Foundation (Grant 4132044), and Fundamental Research Funds for the Central Universities of China (Grant YS1404). Prof. Jing Wang is also deeply indebted to China Scholarship Council (CSC) for sponsoring her research at the University of Delaware. Special thanks should go to Prof. Babatunde A. Ogunnaike for his instructive advice and useful suggestions on the research work. 3676


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