Model-based approaches for fault detection

CHAPTER 6

Model-based approaches for fault detection Contents 6.1. Introduction 6.2. State estimation 6.2.1 State estimation problem formulation 6.2.2 State estimation techniques 6.2.2.1 Extended Kalman filter (EKF) 6.2.2.2 Unscented Kalman filter (UKF) 6.2.2.3 Particle filter (PF) 6.3. Fault detection-based state estimation approaches 6.3.1 Fault detection using multiscale EWMA chart 6.3.1.1 EWMA chart 6.3.1.2 Multiscale EWMA chart 6.3.2 Application to wastewater treatment plant 6.3.2.1 State estimation results 6.3.2.2 Fault detection results 6.4. Fault detection-based state estimation approach 6.4.1 Fault detection using optimized weighted SS-DEWMA chart 6.4.2 Optimized WSS-DEWMA and application to fault detection 6.4.2.1 Application 1: synthetic example 6.4.2.2 Application 2: Cad System in E. coli (CSEC) 6.5. Conclusions References

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6.1 Introduction Fault detection is usually the first step in process monitoring. For example, in biological systems (like wastewater treatment plants (WWTP) and Cad System in E. coli (CSEC)), where the state variable cannot be measured directly, state estimation-based fault detection techniques need to be developed to enhance the monitoring of these systems. Model-based state estimation is one of the most widely applied methods for estimation of unmeasured states based on available process model. The state estimation technique can be applied for residual generation, which is used for fault detection (FD) purposes. For instance, proper operation of wastewater treatment plants requires good understanding of their behavior and tight monitoring of their key Data-Driven and Model-Based Methods for Fault Detection and Diagnosis https://doi.org/10.1016/B978-0-12-819164-4.00015-7

Copyright © 2020 Elsevier Inc. All rights reserved.

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Figure 6.1 Plots of samples of normal and faulty signals.

variables to achieve the desired effectiveness of operation and to ensure maintaining the desired safety standards and protocols. Therefore the main objective of this chapter is developing an improved model-based fault detection technique that aims at enhancing the monitoring of industrial systems. The objectives of this chapter are twofold. Firstly, a state estimation technique that can accurately estimate the state variables in such systems will be developed, and secondly, a new fault detection chart will be proposed. To deal with scenarios where the process model is available and has a pre-defined structure (obtained using material and energy balances), the state variables are estimated using state estimation techniques. The state estimation techniques include extended Kalman filter (EKF) [1,2], unscented Kalman filter (UKF) [3,4], central difference Kalman filter (CDKF) [5], square-root unscented Kalman filter (SRUKF) [6], square-root central difference Kalman filter (SRCDKF) [7], and particle filtering (PF) [8]. The PF has shown good improvement and provides a significant advantage over Kalman filter-based techniques and can be applied to nonlinear models with non-Gaussian errors. After estimating the unknown variables and computing the monitored residuals using the state estimation methods, the sensor fault detection problem will be addressed, in which we assume that there is no change in the relations between pairs of variables (no fault in the process). The term sensor fault states a disparity between the ideal value that a sensor should indicate under normal operation conditions and the value that it actually states. This does not necessarily mean that the sensor itself is faulty. This difference may be caused by an abrupt fault (i.e., offset or bias), incipient or evolutive fault (i.e., drift), or change in the precision (i.e., change in the variance). An example of each fault appears in Fig. 6.1.

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The model-based fault detection methods, mainly the Shewhart chart [9], exponentially weighted moving average (EWMA) chart [10,21], cumulative sum (CUSUM) chart [22], and generalized likelihood ratio test (GLRT) chart [23,24], have been used to improve the fault detection (FD) capabilities. The model-based FD methods generally depend upon the system dynamic structure. Thus the selected measurements are compared with mathematical model under fault-free conditions. For important industrial parameters, estimation is necessary for nonmeasurable quantities before being able to apply monitoring. Whereas the Shewhart chart considers solely the present data sample to evaluate performance, the CUSUM and EWMA charts consider a weighted sum of past observations. The CUSUM chart provides same weight for all past observations, whereas the EWMA chart gives more importance to the more recent observations [25–27]. Both CUSUM and EWMA charts perform almost equally in detecting small mean shift, but the EWMA chart is somewhat easier to set up and operate. Moreover, since EWMA statistic is a weighted average of all previous and present observations, it is less sensitive to the normality assumption [25,28]. Thus, in this chapter, we propose two enhanced control charts based on EWMA to address the problem of fault detection. The EWMA chart usually has two parameters, the control width L and the smoothing parameter λ, which must be specified by the user so that it would be optimal at detecting a specific change size. Thus we will propose an optimized EWMA based on the best selection of L and λ. The multiobjective optimization (MOO) is addressed using three objective functions: i) missed detection rate (MDR), ii) false alarm rate (FAR), and iii) ARL1 values. The developed chart provides quick and good detection. The idea behind the developed EWMA is computing a new chart that takes into account the current and the previous data information by giving more weight to the more recent data. In addition, the multiscale nature of the data provides a representation that can be made robust to noises and errors and has a great impact on the quality of fault detection. Hence, we propose to combine the EWMA chart with wavelet-based multiscale representation to improve the monitoring effectiveness. The first novel chart is the so-called multiscale EWMA (MS-EWMA) chart. The developed MS-EWMA strategy is addressed so that the state variables are estimated using PF method and the faults are detected using MS-EWMA chart.

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Another improved EWMA chart, called Max-Double EWMA (MDEWMA), was proposed in the literature and has shown higher detection performances than the classical EWMA chart in detecting minor and moderate shifts in the mean and/or variance [29]. The M-DEWMA chart considers the highest of the absolute values for two EWMA statistics, one controlling the mean, and the other for the variance. It has been presented that the M-DEWMA chart performed higher than the Max-EWMA chart in detecting shifts in the mean and/or variance. The authors of [30,31] developed an enhanced single chart named Sum of Squares-DEWMA (SS-DEWMA) chart, which aims at detecting shifts of all sizes in the mean and/or variance. It has been shown that the SSDEWMA chart performed higher than the M-DEWMA chart in detecting shifts in the mean and/or variance, and both of them outperformed the classical EWMA [30,31]. Therefore the second proposed chart, which considers the sum of weighted squared values for EWMA charts and improves the classical SSDEWMA chart, will be developed. The proposed detection chart consists of developing a weighted version that makes tradeoffs between the two EWMA statistics. However, the weighted SS-DEWMA (WSS-DEWMA) chart has two tuning parameters, the weight α and smoothing parameter λ, which should be optimized. To do that, an enhanced WSS-DEWMA that optimizes the two parameters will be proposed. The second improved chart is the so-called optimized WSS-DEWMA (OWSS-DEWMA) chart. The proposed OWSS-DEWMA approach is addressed so that the state variables are estimated using PF method and OWSS-DEWMA is applied for fault detection. The detection performances of the proposed strategies are compared to those using the classical techniques in terms of missed detection rate (MDR) and false alarm rate (FAR). The monitoring capability is assessed using a synthetic example and two biological systems: wastewater treatment plants (WWTPs) and Cad System in E. coli (CSEC). When the simulated CSEC model is used, the developed chart is aimed to detect single and multiple faults through monitoring the key variables (cadaverine, transport proteins, enzymes, lysine, and regulatory proteins), whereas the performance of the WWTP is monitored through monitoring six state variables, which are the chemical oxygen demand, dissolve oxygen concentration, active heterotrophic biomass, ammonia concentration, nitrate concentration, and active autotrophic biomass.

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6.2 State estimation 6.2.1 State estimation problem formulation Here the objective is estimating the state vector xk , given the measurements vector yk and the dynamic model (6.1). Many techniques have been developed to solve this state estimation problem, which include the extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filter (PF), and others. In this section, we propose to use the PF approach to estimate the nonlinear state variables. The PF has shown good enhancement and provides significant benefits over EKF and UKF techniques and can be used for estimation of nonlinear processes with non-Gaussian noises. A brief introduction of EKF, UKF, and PF techniques is presented in next section. Here the state estimation problem is formulated for a general system model. Let a nonlinear state space model be described as follows: x˙ = F (x, u, θ, w ), y = R(x, u, θ, v), w ∼ N (0, Q), v ∼ N (0, R),

(6.1)

where x ∈ Rn is a vector of the state variables, u ∈ Rp is a vector of the input variables, θ ∈ Rq is an unknown parameter vector, y ∈ Rm is a vector of the measured variables, w ∈ Rn is a white noise vector with covariance Q, v ∈ Rm is a measurement noise vector with covariance R, and F and L are nonlinear functions. Discretizing the state space model (6.1), the discrete model can be written as follows: xk = F(xk−1 , uk−1 , θk−1 , wk−1 ), yk = L(xk , uk , θk , vk ),

(6.2)

which describes the state variables at any time step k in terms of their values at the previous time step k − 1. The process and measurement noise vectors are assumed to have the following properties: E[wk ] = 0, E[wk wkT ] = Qk , E[vk ] = 0, and E[vk vkT ] = Rk , and F and L are nonlinear functions. Here the objective of this state estimation problem is estimating the state vector xk , given the measurement vector yk and the dynamic model (6.1). Many techniques have been developed to solve this state estimation problem, which include the extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filter (PF), and others. In this section, we propose to

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use the PF approach to estimate the nonlinear state variables. The PF has shown good enhancement and provides significant benefits over EKF and UKF techniques and can be used for estimation of nonlinear processes with non-Gaussian noises. A brief introduction to EKF, UKF, and PF techniques is presented in the next section.

6.2.2 State estimation techniques 6.2.2.1 Extended Kalman filter (EKF) As the name indicates, EKF is an extension of the Kalman filter (KF) [32], where the model is linearized to estimate the covariance matrix of the state vector. As in KF, the state vector xk is estimated by minimizing a weighted covariance matrix of the estimation error, i.e., E[(xk − xk )M(xk − T  xk ) ], where M is a symmetric nonnegative definite weighting matrix. If all the states are equally important, M can be taken as the identity matrix, xk )(xk − xk )T ]. Such a which reduces the covariance matrix to P = E[(xk − minimization problem can be solved by minimizing the following objective function:  1  J = Tr E[(xk − xk )(xk − xk )T ] 2

(6.3)

subject to the model defined in Eqs. (6.1). To minimize the above objective function (6.3), EKF estimates the state vector using a two-step algorithm: prediction and estimation (or update), which are described next (see Algorithm 1).

6.2.2.2 Unscented Kalman filter (UKF) The unscented Kalman filter (UKF) is an extension of the Kalman filter to deal with nonlinear systems. It utilizes the unscented transformation, which is a method for calculating statistics (such as the mean and covariance matrix) of a random variable that undergoes a nonlinear mapping. Assume that a random variable x ∈ RL with mean x¯ and covariance Px is transformed by a nonlinear function y = f (x). To find the statistics of y, define 2L + 1 sigma vectors as follows: X0 = x,



Xi = x + ( (L + λ)Px )i , i = 1, . . . , L , Xi

 = x − ( (L + λ)Px )i ,

i = L + 1, . . . , 2L ,

(6.4)

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Algorithm 1 Extended Kalman filter algorithm. Initialization step:  x0 = E[x],

Px0 = E[(x − x0 )(x − x0 )T ]. Prediction step:  xk|k−1

= F( xk−1|k−1 , uk−1 ),

 yk|k−1

= L( xk|k−1 , uk ).

Estimation (Update) step: Pk|k−1 = Ak−1 Pk−1|k−1 + Gk−1 QPTk−1 , Kk = Pk|k−1 CTk (Ck Pk|k−1 CTk + Hk RHTk )−1 , Pk|k = (I − Kk Ck )PTk|k−1 ,  xk|k

=  xk|k−1 + Kk ( yk|k−1 − yk ).

Return the augmented state estimation  xk . ∂F ∂F ∂L Here A ≈ ∂ x |x , C ≈ ∂ x |x , G ≈ ∂ w |x , and H ≈ linearized system model at every time step.

∂L |x ∂v 

are the matrices of the

where λ = e2 (L + k) − L is a scaling parameter, L is the dimension of the √ state z, ( (L + λ)Px )i denotes the ith column of the matrix square root (L + λ)Px , and the constant 10−4 < e < 1 determines the spread of the sigma points around x. The constant k is a secondary scaling parameter, which is usually set to zero or to 3 − L [33]. These sigma points are then propagated through the nonlinear function, that is, Yi = f (Xi ), i = 0, . . . , 2L ,

(6.5)

and the mean and covariance matrix of y can be approximated as the weighted sample mean and covariance of the transformed sigma points of Yi as follows: y ≈

2L  i=0

Wi(m) Yi ,

(6.6)

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and

2L 

Px ≈

Wi(c) (Yi − y)(Yi − y)T ,

i=0

where the weights are given by Wi(m) = W0(c) = and,

λ , λ+r λ + (1 − e2 + ξ ), λ+r

Wi(m) = Wi(c) =

(6.7)

1 , i = 0, . . . , 2L . 2(λ + r )

The parameter ξ is utilized to incorporate prior knowledge about the distribution of x. It has been found that for Gaussian and non-Gaussian variables, the unscented transformation results in approximations that are precise up to the third and second orders, respectively [33]. The UKF algorithm includes two steps, prediction and update. In the prediction step the predicted state estimate xˆ −k and the predicted covariance matrix estimate Pk are calculated. Then in the update step the updated state estimate xˆ k and the updated covariance matrix estimate Pk (after calculating the innovation residual Pxk yk and the optimal Kalman gain Kk ) are calculated. The UKF algorithm does not always provide a satisfactory performance, especially for highly nonlinear and high-dimensional processes. In such cases, more powerful techniques, such as particle filters, are used. An introduction to particle filtering is presented next.

6.2.2.3 Particle filter (PF) The advantage of the PF is that it is not restricted by the linear and Gaussian assumptions of the system and data, which makes it applicable to a wide range of systems. The PF basic form is simple; however, it may be computationally expensive. Its objective is to compute the conditional probability (posterior distributions) of the state variables, given some noisy and possibly partial observations. Given process observations (measurements) and a dynamical system describing the evolution of the state variables to be estimated, the state estimation problem can be dealt with as an optimal filtering problem [34], within a Bayesian context, in which the posterior distribution p(xk |y0:k ) is recursively estimated. The PF method is used to approximate a posterior distribution, in which the dynamical model is defined as an observation

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model p(yk |xk ) and a state evolution model p(xk |x0:k−1 ) = p(xk |xk−1 ). In a Bayesian framework the state estimation phase can be performed by recursively estimating the filtering distribution function p(xk |y0:k ) and the predictive distribution function p(xk |y0:k−1 ) as 

p(xk |y0:k−1 ) =

p(xk |xk−1 )p(xk−1 |y0:k−1 )dxk−1 ,

Rn

(6.8)

p(yk |xk )p(xk |y0:k−1 ) , p(yk |y0:k−1 )

p(xk |y0:k ) = 

where p(yk |y0:k−1 ) = Rx p(yk |xk )p(xk |y0:k−1 )dxk . The nonlinear form of the model can lead to intractable integrals when calculating the state distribution function p(xk |xk−1 ). Thus Monte Carlo approximation is used to solve the filtering problem, where the joint posterior distribution function p(x0:k |y0:k ) is approximated by a set of particles (weighted samples) {x(0i:)k , (ki) }N i=0 as follows [35]: pˆ N (x0:k |y0:k ) =

N 

(ki) δx(i) (d x0:k )/ 0 :k

i=0

N 

(ki) ,

(6.9)

i=0

where (ki) are the importance weight of the kth sample, δx(i) (d x0:k ) is the 0 :k Dirac delta function of the corresponding sample, and N is the total number of samples. Using the same set of samples, the posterior distribution p(xk |y0:k ) of interest can be approximated as [36] pˆN (xk |y0:k ) =

N 

(ki) δx(i) (d xk )/

i=0

N 

k

(ki) .

(6.10)

i=0

To avoid the problem of degeneracy in the PF technique, resampling is used [36], and the state estimate  xk can be obtained using a Monte Carlo methodology as follows [35]:  xk =

N 

(ki) x(ki) ,

(6.11)

i=0

where (ki) is given by [36]: (ki) ∝

p(y0:k |x(0i:)k )p(x(0i:)k ) . p(x(0i:)k |y0:k )

(6.12)

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The state and the measurement residuals are computed as Rsk = xk − xk and Rmk = yk −  yk , respectively.

6.3 Fault detection-based state estimation approaches 6.3.1 Fault detection using multiscale EWMA chart 6.3.1.1 EWMA chart The EWMA chart was established by Roberts in 1959 and named as Geometric Moving Average (GMA) chart [37]. Later, the GMA chart became popularly referred to as the EWMA chart [38]. Like CUSUM chart [22], EWMA chart is capable of detecting smaller fault shifts in the mean if compared to the Shewhart chart [9]. The EWMA chart is computed using the m sensor residuals evaluated from state estimation technique. In EWMA chart, we have m vectors Zj , j = 1, . . . , m. They are computed using [39] Zi,j = λXi,j + (1 − λ)Zi−1,j , i = 1, . . . , N and j = 1, . . . , m,

(6.13)

where N is the number of process variables, m is the number of sensor measurements, λ denotes smoothing parameter between 0 and 1, which changes the memory of the detection statistic, and X is the data matrix of N process variables and n measurements. The control limits for jth chart (UCLj is the upper control limit, whereas LCLj is the lower control limit) are calculated as [40]

UCLj = μ0 + L σj

LCLj = μ0 − L σj

λ [1 − (1 − λ)2i ], 2−λ λ

2−λ

[1 − (1 − λ)2i ],

(6.14)

(6.15)

where L represents the control width of the EWMA chart, and σj is the in-control standard deviation of Xj , the initial value Z0,j is set equal to the process in-control mean (or target value) μ0,j of the jth chart. Since X is mean centered, Z (Eq. (6.13)) is also centered, and thus the chart Zj , j = 1 . . . m, has mean equal and variance approximately equal, the control limits (UCL and LCL) are computed as UCL =

1 UCLj , m j

(6.16)

Model-based approaches for fault detection

LCL =

1 LCLj . m j

231

(6.17)

When the EWMA statistic Z is between the control limits (UCL and LCL), there is no fault, and if EWMA statistic Z exceeds control limits, then fault is declared in the system.

6.3.1.2 Multiscale EWMA chart Here, the multiobjective optimization is the problem of choosing a most preferred solutions (Lˆ and λˆ ) when three objective functions i) missed detection rate (MDR), ii) false alarm rate (FAR), and iii) Average Run Length (ARL1 ) are to be simultaneously minimized. A central difficulty in such problems is that, unlike in single-objective maximization problems, there is no obvious or simple way to define the concept of a most preferred solution. The multiobjective optimization problem is defined by a set of two decision variables x = (x1 = L , x2 = λ) and a set of three objective functions f = (f1 = MDR, f2 = FAR, f1 = ARL1 ). The objective functions are functions of the decision variables. The aim of optimization is to minimize y = f (x) = (f1 (x), f2 (x), f3 (x)). To compare two candidate solutions in the multiobjective optimization problems, there have been defined the concepts of Pareto dominance and Pareto optimal solutions. Based on the Pareto dominance we can also introduce the optimality criterion for the problem to be solved. In the current work, after selecting the Pareto optimal or nondominated solutions, we choose the solution (Lˆ and λˆ ) that tradeoffs FAR, MDR, and ARL1 . The optimized EWMA statistic is computed using the optimal values Lˆ and λˆ . The general flow-chart of the multiobjective optimization process is shown in Fig. 6.2. The optimized EWMA algorithm is presented in Algorithm 2. The EWMA chart assumes that the process residuals evaluated in the training phase follow a normal distribution. In practice, measured data may not necessarily follow a normal distribution. Modeling errors in the model residuals can also be a source of non-Gaussian errors. Multiscale decomposition is able to help address the issue of non-Gaussian errors as it provides detail signals that are closer to normal (Gaussian) at different scales. Thus wavelet-based multiscale representation of data will be applied to enhance the performance of EWMA.

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Figure 6.2 General flow-chart of the multiobjective optimization process.

Algorithm 2 Optimized EWMA algorithm. 1. 2. 3. 4. 5. 6. 7.

Set L = 0.1, 0.2, . . . , 3 and λ = 0.05, 0.1, . . . , 1. Apply the EWMA chart for all L and λ. Compute MDR, FAR and ARL1 for all L and λ. Generate the Pareto front, Extract the nondominated solution. Compute the corresponding Lˆ and λˆ . Compute the optimized EWMA chart and its control limits LCL and UCL using Lˆ and λˆ .

The proposed multiscale EWMA monitoring technique is performed in two phases. In the first phase, fault free training data are normalized so that they have zero mean and unit variance and are then decomposed at multiple scales using wavelet-multiscale decomposition. Then the EWMA chart is used for the detail signals at different scales and to the last scaled signal. The control limits are calculated at all scales and used then to threshold the wavelet coefficients of detailed signals. If any wavelet coefficient violates the control limits at a certain scale, all the wavelet coefficients at that scale are retained (see Fig. 6.3). If no violation of the limits occur at a certain scale, then all wavelet coefficients at that scale are ignored. The retained detail signals and the last scaled signals are then reconstructed to get the final reconstructed signal. Finally, EWMA is applied on the reconstructed signal to obtain the final multiscale EWMA detection statistic and the control limits. In the second phase, the testing data are decomposed at multiple scales via the same wavelet filters applied in the first phase after normalizing the data using the same mean and standard deviation obtained in training. The con-

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Figure 6.3 Multiscale EWMA strategy.

trol limits obtained from the training phase are then applied to the detailed signals of the testing data at the respective scales and also to the last scaled signal. At any scale, the wavelet coefficients that violate the control limits are retained, whereas others that do not violate the limits are ignored. Then a reconstructed signal from all the retained coefficients is found. Finally, the previously obtained control limits from the reconstructed training data are applied on the EWMA statistic of the reconstructed testing data to detect possible faults (see Fig. 6.3). The multiscale EWMA algorithm is illustrated schematically in Fig. 6.3. The validation of the developed technique is done using a simulated COST wastewater treatment BSM1 model.

6.3.2 Application to wastewater treatment plant The idea behind the developed multiscale optimized EWMA algorithm is incorporating the advantages brought forward by PF state estimation technique with those of EWMA chart, multiobjective optimization, and multiscale representation. The algorithm of the proposed technique is presented in Algorithm 3 and Fig. 6.4. The objective of our contribution is to combine PF state estimation technique with the multiscale representation-based EWMA chart

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Figure 6.4 PF-based MS-EWMA fault detection strategy.

to detect the faults in the residual vector Rm obtained from the PF using a simulated COST wastewater treatment BSM1 model. Algorithm 3 PF-based MS-EWMA fault detection algorithm. Input: yk • Training Data – Estimate the state variables using PF – Compute the measurement residual Rm – Compute the MS-EWMA chart Z and its control limits LCL and UCL • Testing Data – Estimate the state variables using PF – Compute the measurement residual Rm – Compute the MS-EWMA chart Z – If LCL < Z < UCL, then the process is operating under normal operating conditions. Else, a fault is declared.

6.3.2.1 State estimation results The validation of the developed state estimation-based fault detection technique is done using a well-defined simulated activated sludge wastewater treatment plant model. The used benchmark has been first developed by the International Association of Water Quality (IAWQ) and then modified by the European Co-Operation in the field of Scientific and Technical Research (COST) [41–43].

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To perform this illustrative example, wastewater treatment data are needed, which are generated using the simulated dynamic model shown in [44]. In this dynamic model, the influent data associated with the dray conditions are used (see the dry influent data files in [41], where the influent flow rate varies in the range 15000–35000 m3 /d. The performance of the plant is monitored through monitoring the six state variables, which are the chemical oxygen demand (XCOD ), dissolve oxygen concentration (SO ), active heterotrophic biomass (XB,H ), ammonia concentration (SNH ), nitrate concentration (SNO ), and active autotrophic biomass (XB,A ). Thus the data generated by the wastewater benchmark model contain [44]: • Inputs: 6 state variables: XCOD , SO , XB,H , SNH , SNO , and XB,A . • Outputs: 4 sensor measurements: XCOD , SO , SNH , and SNO . The dynamic model is used to generate 1400 observations for each variable, which were initially assumed to be noise-free. Then all variables were contaminated with zero-mean Gaussian noise having a signal-to-noise ratio of 20 to simulate real measurements. The noisy data set was then split into two subsets, training and testing, each consisting of 700 observations. To estimate the six state variables of the COST WWTP, the performances of UKF and PF methods are compared. The state vector X, measured output vector y, and input vector u used in this example are [44] X = [XDCO , SO , XB,H , SNH , SNO , XB,A ]T , y = [XDCO , SO , SNH , SNO ]T ,

(6.18)

u = [XDCO,in , Qa , Qin ]T , all six state variables (XDCO , SO , XSNH , XSNO , XBH , and XBA ) are estimated using the dynamic model and measurements (XDCO , SO , XSNH , and XSNO ) of only three variables, where Qa is the bioreactor air, and Qin refers to the influent flow rates; in addition, all parameters are assumed to be constant and known (given in [41]). The state variables are simulated by changing the input in the dry influent data, as described in the files [41], and using initial values of [20, 10, 20, 6, 650, 160]. Then all state variables were contaminated with zero-mean Gaussian noise (having a standard deviation of 5% of the standard deviation of each state variable) to simulate real measurements. The results of the state estimations using UKF and PF are indicated in Figs. 6.5 and 6.6, and the mean square errors (MSE) between the noise-free and the estimated state variables are summarized in Table 6.1. The MSE is defined as follows:

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Figure 6.5 State estimation of the variables (A) XDCO , (B) SO and (C) XBH using UKF and PF.

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Figure 6.6 State estimation of the variables (A) SNH , (B) SNO and (C) XBA using UKF and PF.

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Table 6.1 Comparison of the MSE for the UKF and PF techniques. Technique XDCO SO XB,H SNH SNO XB,A UKF 0.7012 0.7144 0.7194 0.7463 0.7162 0.7067 PF 0.6541 0.6716 0.6677 0.7098 0.6739 0.6805

MSE = E



Xˆ − Xnf

2  ,

(6.19)

ˆ and E are the noise-free and estimated state variable vectors where Xnf , X, and expectation operator, respectively. The results presented in Figs. 6.5 and 6.6 show that the PF outperforms the UKF. These advantages of the PF technique will be used to improve monitoring of WWTP.

6.3.2.2 Fault detection results Each variable in the model has 1400 observations, which were initially assumed to be noise-free. Then all variables were contaminated with zeromean Gaussian noise having a signal-to-noise ratio of 20 to simulate real measurements. The data were then divided into two subsets, training and testing, each composed of 700 observations. The performance of the proposed detection chart MS-EWMA is tested using simulated data, which are generated using the simulated COST wastewater treatment BSM1 model, and is compared to Shewhart and EWMA charts. In this section, we study two types of faults, abrupt fault (i.e., offset or bias) and incipient or evolutive fault (i.e., drift). Here a bias fault of magnitude equal to 2 of the standard deviation of the second variable SO is added in the testing set between samples 500 and 700, Zj , j = 1, 2, 3, 4, present the four statistics computed using the sensor residuals obtained from the PF method. Figs. 6.7(A), 6.7(B), and 6.7(C) and Table 6.2 show the FD comparison between Shewhart, EWMA, and MS-EWMA methods. We can see from Figs. 6.7(A), 6.7(B), and 6.7(C) and Table 6.2 that the MS-EWMA method shows improved FD results over EWMA and Shewhart methods in terms of false alarm and missed detection rates because of the advantages of the multiscale application. Next, a drift fault with slope value 0.005 is injected in the second state SO at [500 to 700] sample times. The resulting figures show a good improvement of the MS-EWMA chart in terms of MDR and FAR values

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Figure 6.7 Monitoring a bias fault in SO using (A) Shewhart, (B) EWMA, and (C) MSEWMA methods.

(see Fig. 6.8(C)), when compared to EWMA chart (see Fig. 6.8(B)) and Shewhart method (see Fig. 6.8(A)) in terms of MD and FA rates. Table 6.3 presents the detection comparison between the five charts. We can also show from this table that the proposed MS-EWMA chart

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Table 6.2 Summary of MDR (%) and FAR (%) for bias fault. Chart/Fault Detection Metric MDRs (%) FARs (%) Shewhart 62 5.2 EWMA 10 10.20 MS-EWMA 2.5 2

outperforms the classical detection charts in terms of false alarm and missed detection rates. Here we investigate the effect of shift in the mean fault on the detection results of the proposed MS-EWMA chart where the mean shift s varies from 0.5 to 3 and consider different fault sizes for different values using a Monte Carlo simulation of 1000 runs. Table 6.4 shows that as s increases, the MDR, and FAR decreases, and the proposed technique still provides good detection results.

Effect of change rate parameter value on the detection performance The change rate of the parameter value is slow compared to the system dynamics. We can approximate the drift term Zi = a ∗ (i − i0 ), i ≥ i0 , where a is a constant parameter that defines a linear rate of change, and i0 is the time point at which the incipient fault first occurs. To study the effect of change value of parameter a on the detection performance of MS-EWMA method, the FAR and MDR are computed using a Monte Carlo simulation of 1000 runs where a varies from 0.001 to 0.01. Table 6.5 shows that the developed MS-EWMA technique has good detection results and provides small FAR and MDR values under different simulation conditions.

6.4 Fault detection-based state estimation approach 6.4.1 Fault detection using optimized weighted SS-DEWMA chart The SS-DEWMA single-variable chart was developed by Teh et al. [30,31]. It is a superior alternative to M-DEWMA chart for monitoring shifts in both process mean and variance simultaneously [30,31]. The SS-DEWMA chart is simple to build and understand. Suppose that the underlying process comprises of a sequence of individual observations for samples of size j = 1, 2, . . . , ni taken at time i = 1, 2, . . . and following a normal distribution such that Xij ∼ N (μ0 + aσ0 , b2 σ02 ), where μ0 and σ0 are respectively

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Figure 6.8 Monitoring a drift fault in SO using (A) Shewhart, (B) EWMA, and (C) MSEWMA methods.

the in-control mean and in-control standard deviation of the process. If a = 0 and b = 1, then the process is statistically in-control. If not, then is ¯ i and S2 represent the mean and varia shift or alternation. Assume that X i ance of sample i, respectively. Both statistics are independent of each other.

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Table 6.3 Summary of MDR (%) and FAR (%) for drift fault. Chart/Fault Detection Metric MDRs (%) FARs (%) Shewhart 10.75 6.2 EWMA 8.375 4.8 7.125 2.8 MS-EWMA Table 6.4 Summary of MDR (%) and FAR (%) for different values of s. s/Fault Detection MDRs (%) FARs (%) Metric 0.5 12.49 3.81 8.15 2.8 1 1.5 3.85 1.6

0 0 0

2 2.5 3

1 0.75 0

Table 6.5 Summary of MDR (%) and FAR (%) for different values of a. a/Fault Detection FARs (%) MDRs (%) Metric 0.001 47.26 9.02 0.002 25.87 10.22 26.37 12.83 0.003 11.94 9.22 0.004 19.90 6.81 0.005 9.45 3.01 0.006 0.007 10.45 3.01 0.008 9.95 4.01 5.47 4.61 0.009 6.97 3.21 0.01

¯ are independent normal random variables such that The sample means X 2 2 2 ¯ i ∼ N (μ + aσ, b σ ), whereas (ni − 1) 2Si 2 are independent chi-square ranX ni b σ dom variables with ni − 1 degrees of freedom (DoFs). In the structure of the SS-DEWMA chart, the following couple of independent statistics is defined:

Ui =

X¯ i − μ √ σ/ ni

(6.20)

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and Vi =

−1



 (ni − 1)Si2 F ; ni − 1 , σ2

(6.21)

where −1 denotes the inverse of the standard normal distribution function, and F(w ; v) denotes the chi-square distribution function with v DoFs [45]. When the process is in-control, both Ui and Vi in Eqs. (6.20) and (6.21) are independent standard normal random statistics. The distributions of both Ui and Vi do not depend on the sample size ni . The two plotting statistics of EWMA chart are exponentially weighted combinations of the current and past observations, which are computed from Ui and Vi , each one for mean and variance, as follows: Yi = (1 − λ)Yi−1 + λUi for i = 1, 2, . . .

(6.22)

xi = (1 − λ)xi−1 + λVi for i = 1, 2, . . . ,

(6.23)

and

where λ represents smoothing constant such that 0 < λ ≤ 1. When the process is in-control, initial values of Yi and xi are usually set as Y0 = 0 and x0 = 0. Both Yi and xi are independent as Ui and Vi are independent. Using Eqs. (6.22) and (6.23), we can compute the below statistics: Wi = (1 − λ)Wi−1 + λYi , for i = 1, 2, . . .

(6.24)

Qi = (1 − λ)Qi−1 + λxi , for i = 1, 2, . . .

(6.25)

and

Similarly, if process is in-control, then both Wi and Qi are initially set to zero. Based on the two DEWMA statistics in Eqs. (6.24) and (6.25), Khoo et al. [29] defined the M-DEWMA statistic as Mi = max(|Wi |, |Qi |) for i = 1, 2, . . . .

(6.26)

Teh et al. [30] have proposed an improved univariate and single chart called Sum of Squares Double EWMA (SS-DEWMA) defined as Li = W2i + Q2i for i = 1, 2, . . . .

(6.27)

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Data-Driven and Model-Based Methods for Fault Detection and Diagnosis

The SS-DEWMA chart has shown better detection performance when compared to the M-DEWMA chart [30] in detecting minor and moderate shifts in the mean and/or variance. Thus, in the current work, we will apply the SS-DEWMA chart for fault detection purposes. Note that Li consists of nonnegative values because of the approach in adopting the maximum absolute values of the two DEWMA statistics. To improve the detection abilities of SS-DEWMA, an optimized weighted SS-DEWMA (OWSS-DEWMA) is developed. Its chart is given by Oi = α W2i + (1 − α)Q2i for i = 1, 2, . . . ,

(6.28)

where α is the weighted parameter. The two parameters α and λ are to be jointly optimized using a multiobjective optimization scheme (presented in Algorithm 4). Algorithm 4 Optimized WSS-DEWMA algorithm. 1. Set α = 0, 0.2, . . . , 1 and λ = 0, 0.2, . . . , 1. 2. Compute the WSS-DEWMA chart and its control limit for all α and λ. 3. Compute MDR, FAR, and ARL1 for all α and λ. 4. Generate the Pareto front. 5. Extract the nondominated solutions. 6. Compute the corresponding αˆ and λˆ . 7. Compute the optimized WSS-DEWMA chart and its control limit UCL (UCLsd ) using αˆ and λˆ .

The OWSS-DEWMA chart requires solely the higher value UCL, which is defined as [30] 

UCLsd = E(Oi ) + Ksd V (Oi ),

(6.29)

where E(Oi ) and V (Oi ) are the mean and variance of Oi , respectively, given that the process is in-control, whereas Ksd is a constant controlling the width of UCLsd .

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A formula providing a quick computation of UCLsd for the initial state SS-DEWMA chart based on Ksd values is expressed as [30,31] UCLsd = 2(1 + Ksd ) × 

λ4

1 − (1 − λ)2

3 Kλ,i ,

(6.30)

where Kλ,i = 1 + (1 − λ)2 − (i2 + 2i + 1)(1 − λ)2i + (2i2 + 2i − 1)(1 − λ)2i+2 − i2 (1 − λ)2i+4 .

Algorithm 5 illustrates the main steps of the proposed OWSS-DEWMA fault detection chart. Algorithm 5 Proposed fault detection algorithm. Input: Data matrices X and Y. • Modeling Phase 1. Generate residuals using particle filter (PF). • Training Phase 1. Compute the OWSS-DEWMA chart. 2. Compute the OWSS-DEWMA control limit UCL (UCLsd ). • Testing Phase 6. Compute the new residuals for new sample time. 7. Compute the OWSS-DEWMA chart. 8. If the computed chart violate its threshold, then the process is considered out of control, and a fault is declared. 9. Else, there is no fault, and the process is operating under normal operating conditions.

6.4.2 Optimized WSS-DEWMA and application to fault detection The validation of the developed technique is done using two applications: the first application is a synthetic example, and the second one is a simulated Cad System in E. coli (CSEC) model.

6.4.2.1 Application 1: synthetic example The efficiency of the OWSS-DEWMA chart is first validated through a numerical example and compared to SS-DEWMA and EWMA charts.

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Data-Driven and Model-Based Methods for Fault Detection and Diagnosis

Consider the following simulation example with n = 1000 measurements. The monitored variable is described by the following relation: X = bσ + μ + aσ × randn(1, n).

(6.31)

As training data, samples are generated under normal conditions. To evaluate the fault detection performance, two types of faults are simulated. The first one is a mean shift or bias with magnitude of a of the standard deviation σ of the variable X, and the second one is a variance change with magnitude of bσ introduced in X from sample 100 to 300. In case where a = 0 and b = 1, there is no fault if the process has no fault. Figs. 6.9(A) to 6.9(C) show the time evolution of EWMA, SSDEWMA, and OWSS-DEWMA charts where a = 0.5 and b = 1.1. From these figures it is clear that the OWSS-DEWMA presents better detection performances when compared to the SS-DEWMA chart in terms of false alarm rate (FAR), and both of them outperform the classical EWMA chart. Table 6.6 gives a summary of FAR and missed detection rate (MDR) for the three approaches. Next, the validation of the developed state estimation-based fault detection technique is done using a simulated CSEC model.

6.4.2.2 Application 2: Cad System in E. coli (CSEC) Applying particle filter to estimate state variables

Here the state variables are estimated from noisy measurements using PF method. Its performances are evaluated and compared to the classical EKF and UKF methods using CSEC model. The CSEC entails cytoplasmic protein CadA and the transmembrane proteins CadB and CadC, where CadA is the decarboxylase that converts Lys (lysine) into Cadav (cadaverine) in a reaction that consumes the intracellular H + resultant in the consumption of a cytoplasmic proton, CadB is the protein that exports the Cadav and imports the Lys, and CadC is the positive regulator of cadBA that detects the external conditions, which also ensures that the enzyme CadA and the protein CadB are made only under normal conditions of lysine abundance and low pH [46,47]. The qualitative model of CSEC (simplified) is illustrated in Fig. 6.10. The qualitative dynamic model that shows the relationship between the variables enzyme CadA, transport protein CadBA, lysine Lys, and cadaverine Cadav is given

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Figure 6.9 Monitoring a fault in X using (A) EWMA, (B) SS-DEWMA, and (C) OWSSDEWMA charts.

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Table 6.6 MDR (%) and FAR (%) evaluation. Chart/Fault Detection Metric MDRs (%) EWMA 3.9801

SS-DEWMA OWSS-DEWMA

0 0

FARs (%) 13.3917 3.7509

0

Figure 6.10 Qualitative model of the CSEC (simplified).

by d[CadA] dt d[cadBA] dt d[Cadav] dt d[Lys] dt

= α1 [Cadav]g13 − β1 [CadA]h11 , = α2 [CadA]g21 − β2 [cadBA]h22 , = α3 [cadBA]g32 − β3 [Cadav]h33 [Lys]h34 ,

(6.32)

= α4 [CadA]g41 − β4 [Lys]h44 ,

where the model parameters are presented in Table 6.7. For a description of CSEC in more detail, see [47–49]. Such algorithms are already addressed by applying them to simulated time-series metabolic data as concentrations of four metabolites related to the branched pathway given in Fig. 6.10, that is, the enzyme CadAk , the

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Table 6.7 CSEC parameters. Parameter α1 α2 α3 α4

Value 12 8 3 2

Parameter β1 β2 β3 β4

Value 10 3 5 6

Parameter g13 g21 g32 g41

Value −0.8 0.5 0.75 0.5

Parameter h11 h22 h33 & h34 h41

Value 0.5 0.75 0.5& 0.2 0.8

Figure 6.11 Estimation of state variables using various state estimation techniques.

transport protein CadBk , the regulatory protein CadCk , and lysine Lysk . Hereafter we will assume that the state vector to be estimated is 

xk = CadAk CadBk CadCk Lysk

T

,

and all model parameters (α1 , α2 , α3 , α4 , β1 , β2 , β3 , β4 , g13 , g21 , g32 , g41 , h11 , h22 , h33 , h44 ) are considered to be known. The simulation outcomes for estimating the four states CadA, CadB, CadC, and Lys by EKF, UKF, and PF are exposed in Figs. 6.11(A, B, C). It obvious from Fig. 6.11 that EKF provides the lowest performance with respect to other approaches. This is due to the restricted capability of EKF to precisely estimate the mean and covariance matrix of the estimated states through linearization of the nonlinear process model. The figures depict as well that the PF delivers a substantial enhancement compared to the UKF, which is due to the fact that, by using UKF, linearizing the process

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Data-Driven and Model-Based Methods for Fault Detection and Diagnosis

Table 6.8 RMSE of estimated states using EKF, UKF, and PF methods. CadBk CadCk Lysk Technique CadA EKF 0.067 0.12 0.12 0.031 UKF 0.06 0.094 0.11 0.02 PF 0.00074 0.0014 0.0009 0.0013

model does not necessarily provide good estimates of the mean of the state vector and the covariance matrix of the estimation error utilized in state estimation. From Table 6.8 we see that PF method is sufficient in the estimation of CSEC state variables. Although EKF and UKF are the most widely used estimations in biological systems, the linearization and Jacobian matrix computation can lead to some limitations. The PF has shown good improvement and provides a significant advantage over EKF and UKF techniques and can be applied to nonlinear models with non-Gaussian errors. The comparison of the performance RMSE of EKF, UKF, and PF is illustrated in Table 6.8, which shows that PF is more accurate in estimation and provides a lower RMSE value when compared to EKF and UKF. Applying OWSS-DEWMA to detect faults

The time evolutions of the four variables CadA, CadBA, Lys, and Cadav can be obtained using the dynamic system (6.32). The generated data has four variables and 800 measurements, the training data contains 400 samples (used for state estimation), and the testing data has 400 samples (used to fault detection). The amount of fault in mean shift is aσ of the relevant value, and the amount of variance change is bσ of relevant value. When applying the multiscale representation, it is necessary to choose the best decomposition depth to get good detection with lower MDR and FAR. In the current work the best decomposition depth is equal to 3. In this section, we use the simplest wavelet transform, the so-called Haar wavelet transform. In this case, faults (mean shift of 1.5σ and variance change of 0.5σ ) at [200 to 400] sample times, respectively, are introduced in the cadaverine Cadav. The fault detection results are shown in Table 6.9 and Figs. 6.12(A) to 6.12(C). SS-DEWMA chart (see Fig. 6.12(B)) shows good improvement with respect to EWMA chart (Fig. 6.12(A)). We can also see that the developed OWSS-DEWMA chart delivers better results with respect to

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Figure 6.12 Monitoring a multiple faults in cadaverine Cadav using (A) EWMA, (B) SSDEWMA and (C) OWSS-DEWMA charts.

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Data-Driven and Model-Based Methods for Fault Detection and Diagnosis

Table 6.9 MDR (%) and FAR (%) evaluation. Chart/Fault Detection Metric MDRs (%) EWMA 5.4726

SS-DEWMA OWSS-DEWMA

0 0

Table 6.10 MDR (%) and FAR (%) evaluation. Chart/Fault Detection Metric MDRs (%) EWMA 52.2388 SS-DEWMA 0.21 OWSS-DEWMA 0.035

FARs (%) 18.5930 7.55 1.2

FARs (%) 23.5678 16.24 2.23

SS-DEWMA chart (Figs. 6.12(B) and 6.12(C)). The two previous charts provide a performance that is superior to EWMA (Fig. 6.12(A)) charts based on FAR and MDR indicators. At this point, one fault with 2σ of variance change at [200 to 400] is added to the cadaverine Cadav. Various methods of fault detection (EWMA, SS-DEWMA, and OWSS-DEWMA) are compared in Table 6.10 and Figs. 6.13(A)–6.13(C). We can easily conclude that SSDEWMA chart provides a better monitoring with respect to EWMA chart in terms of FAR and MDR indicators. Additionally, we can observe from these illustrated results that the OWSS-DEWMA (Fig. 6.13(C)) provides the best performance with respect to SS-DEWMA (Fig. 6.13(B)) and EWMA (Fig. 6.13(A)). Now we consider a mean shift of size 1σ and a variance change of size 2σ at [150 to 250] (in Cadav) and [300 to 400] (in Lys). The obtained monitoring results using the same four methods are illustrated in Table 6.11 and Figs. 6.14(A) to 6.14(C). The OWSS-DEWMA chart (Fig. 6.14(C)) shows better detection improvements than the SS-DEWMA chart. The SSDEWMA chart (Fig. 6.14(B)) provides a better performance with respect to the classical EWMA chart (Fig. 6.14(A)). More details on the results obtained are presented in [11–21,50,51]. State estimation results using Bayesian approaches are obtained in [52–56].

6.5 Conclusions In this chapter, we proposed an enhanced fault detection-based EWMA techniques to improve monitoring of biological processes. To do that, novel

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Figure 6.13 Monitoring a fault in Cadav using (A) EWMA, (B) SS-DEWMA and (C) OWSSDEWMA charts.

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Figure 6.14 Monitoring a multiple faults in Cadav and Lys using (A) EWMA, (B) SSDEWMA and (C) OWSS-DEWMA charts.

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Table 6.11 MDR (%) and FAR (%) evaluation. Chart/Fault Detection Metric MDRs (%) FARs (%) EWMA 1.4950 20.2020 SS-DEWMA 0.21 13.1 0 2.34 OWSS-DEWMA

statistical strategies combining the benefits of EWMA chart and state estimation technique were developed. To deal with scenarios where the process model is available and has a predefined structure, the monitored residuals were computed using the state estimation technique. Then the improved fault detection-based EWMA charts were applied for fault detection. In the first part, we proposed an enhanced fault detection technique to improve monitoring of biological Cad System in E. coli (CSEC) process. The developed statistical strategy, called the optimized weighted sum of squares double EWMA (OWSS-DEWMA) statistic, was developed. In the second part a multiscale EWMA chart was applied to monitor wastewater treatment plant (WWTP) process and detect different types of faults including bias fault and drift fault. The detection performances of the proposed strategies are compared to those using the classical techniques in terms of missed detection rate (MDR) and false alarm rate (FAR). The monitoring capability is assessed using a synthetic example and two biological applications, biological WWTP and CSEC processes. When the simulated CSEC model is used, the developed chart is aimed to detect single and multiple faults through monitoring the key variables (cadaverine, transport proteins, enzymes, lysine, and regulatory proteins), whereas the performance of the WWTP is monitored through monitoring the state variables, which are the chemical oxygen demand, dissolve oxygen concentration, active heterotrophic biomass, ammonia concentration, nitrate concentration, and active autotrophic biomass.

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