Multiple sensor fault detection and isolation of an air quality monitoring network using RBF-NLPCA model

Proceedings of the 7th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes Barcelona, Spain, June 30 - July 3, 2009

Multiple sensor fault detection and isolation of an air quality monitoring network using RBF-NLPCA model ⋆ M.-F. HARKAT ∗ G. MOUROT ∗∗ J. RAGOT ∗∗ ∗

Badji-Mokhtar University, Facult´e des Sciences de l’Ing´enieur, D´epartement d’Electronique, BP. 12 ANNABA, ALGERIA ( e-mail: [email protected]). ∗∗ Centre de Recherche en Automatique de Nancy (CRAN), Nancy University, CNRS 2 avenue de la forˆet de Haye, F-54516 Vandoeuvre-L`es-Nancy (e-mail: gilles.mourot, [email protected])

Abstract: This paper presents a data-driven method based on nonlinear principal component analysis to detect and isolate multiple sensor faults. The RBF-NLPCA model is obtained by combining a principal curve algorithm and two three layer radial basis function (RBF) networks. The reconstruction approach for multiple sensors is proposed in the non linear case and successfully applied for multiple sensor fault detection and isolation of an air quality monitoring network. The proposed approach reduces considerably the number of reconstruction combinations and allows to determine replacement values for the faulty sensors. 1. INTRODUCTION Recently, fault detection and process monitoring using principal component analysis (PCA) were studied intensively and largely applied to industrial process. PCA is the optimal linear transformation with respect to minimizing the mean squared prediction error. PCA has been used in the statistical process control area such as process monitoring [Kresta, 1991], gross error detection [Tong, 1995], sensor fault identification [Dunia, 1996]. However, Principal component analysis is a linear method, and most engineering problems are nonlinear. To overcome this problem, we use a nonlinear model to generate residuals for fault detection and isolation. Several nonlinear principal component analysis (NLPCA) models have been proposed in recent years [Hastie, 1989, Kramer, 1991, Dong, 1994, Webb, 1996]. Harkat [2003b] proposed an NLPCA model combining principal curve algorithm [Hastie, 1989] and two three layer RBF-networks where the nonlinear principal components are the outputs of the first network. The second network tries to perform the inverse transformation by reproducing the original data from the nonlinear principal components. A new training procedure of the RBFNLPCA model is presented in Harkat [2003b]. Our presentation is devoted to the problem of multiple sensor fault isolation in data. In this paper we propose a multiple sensor fault isolation approach based on RBFNLPCA reconstruction method. Which reduces considerably the number of reconstruction combinations and allows to determine replacement values for the faulty sensors. ⋆ This work was supported by EGIDE, Partonariat Hubert Currien, CMEP-TASSILI PAI 07 MDU 714, and Agence Universitaire de la Francophonie (AUF).

978-3-902661-46-3/09/$20.00 © 2009 IFAC

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The proposed paper will be organized as follows. The section 2 presents a description of linear PCA. The RBFNLPCA model is described in section 3. Based on the RBF-NLPCA model, the test for detection of faulty sensors and the isolation approach based on the reconstruction principle are presented in section 4. Section 5 gives description and application results of the proposed multiple sensor fault detection and isolation approach of an air quality monitoring network. The last section gives conclusions. 2. LINEAR PCA Principal component analysis is one of the most popular statistical methods, for extracting information from measured data, which finds the directions of significant variability in the data by forming linear combinations of T variables. Let us consider x(k) = [x1 (k) x2 (k) . . . xm (k)] the vector formed with m observed plant variables at time instant k. T

Define the data matrix X = [x(1) x(2) . . . x(N )] ∈ ℜN ×m with N samples x(k)(k = 1, . . . , N ) which is representative of normal process operation. PCA determines an optimal linear transformation of the data matrix X in terms of capturing the variation in the data: T = XP

and

X = TPT

(1)

with T = [t1 t2 . . . tm ] ∈ ℜN ×m , where the vectors ti are called scores or principal components and the matrix P = [p1 p2 . . . pm ] ∈ ℜm×m , where the orthogonal vectors pi , called loading or principal vectors, are the eigenvectors associated to the eigenvalues λi of the covariance matrix (or correlation matrix) Σ of X :

10.3182/20090630-4-ES-2003.0305

7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009

Σ = P ΛP T

with

P P T = P T P = Im

(2)

where Λ = diag(λ1 . . . λm ) is a diagonal matrix with diagonal elements in decreasing order. The partition of eigenvectors and principal components matrices gives: P = Pˆℓ P˜m−ℓ , 



T = Tˆℓ T˜m−ℓ 



(3)

where ℓ will be defined below. According to this partitioning, equation (1) can be rewritten as: ˆ +E X = Tˆℓ Pˆ T + T˜m−ℓ P˜ T = X (4) ℓ

with

ˆ = X Cˆℓ X

m−ℓ

with

E = X C˜m−ℓ

(5)

where the matrices Cˆℓ = Pˆℓ PˆℓT and C˜m−ℓ = Im − Cˆℓ constitute the PCA model. ˆ et E represent, respectively, the modeled The matrices X variations and non modeled variations of X based on ℓ components (ℓ < m). The first ℓ eigenvectors Pˆℓ ∈ ℜm×ℓ constitute the representation space whereas the last (m − ℓ) eigenvectors P˜m−ℓ ∈ ℜm×m−ℓ constitute the residual space. It should be noticed that in the particular case ˆ = X and E = 0. ℓ = m, then X

3.1 Problem settings Nonlinear PCA is an extension of linear PCA. Whilst PCA identifies linear relationships between process variables, the objective of nonlinear PCA is to extract both linear and nonlinear relationships. This generalization is achieved by projecting the process variables onto curves or surfaces instead of lines or planes. The nonlinear approach is like linear PCA, except it represents the data by one dimensional smooth curve which is determined by the nonlinear relationship between the variables. The curve is defined to minimize the orthogonal deviations between the data and the curve. The NLPCA model can be represented by two sub-models (Mapping and Demapping model). The mapping model gives from a data matrix X the nonlinear principal component T and the demapping model ˆ from principal component T . In this gives estimation X case, the nonlinear mapping has the following form (6)

where x and t are rows of X and T respectively, G is the mapping function. The demapping model gives estimation ˆ of x from the nonlinear principal component t and has x the form ˆ = F(t) x

min

N X

k=1

2

ˆ (k)k = min kx(k) − x

N X

k=1

2

kx(k) − F (G (x(k)))k

(9)

th

ˆ (k) is its estimation where x(k) is the i row of X and x by the RBF-NLPCA model. 3.2 RBF-NLPCA model The proposed nonlinear principal component analysis model can be obtained by using two RBF-Networks for mapping and demapping data. Firstly, a three layer RBFnetwork is used (Fig. 1), the hidden layer was composed of radial basis neurons performing a nonlinear mapping of the input space onto a lower dimension space, such that the nonlinear features are captured. The aim is to use this network to define a transformation G : ℜm → ℜℓ . r X wi φi (x) (10) t = G(x) = i=1

3. RBF-NLPCA MODEL

t = G(x)

The problem is now to identify the nonlinear projection functions G and F. To do this, the objective function to minimize is the sum of squared orthogonal deviations:

(7)

where F represents the demapping function.

Therefore a data set X including m variables can be expressed in terms of ℓ nonlinear principal components ˆ + E = F (T ) + E X=X (8)

where T = [T1 , ..., Tℓ ] is the matrix of nonlinear principal components T = G(X), and E is the matrix of residuals.

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where {φi , i = 1, . . . , r} are radial basis functions of x ∈ ℜm , r is the number of kernels and {wi , i = 1, ..., r} denotes a set of weight parameters of the output layer to be determined. The gaussian basis functions φi are defined as: ! 2 kx − ci k φi (x) = exp − (11) 2σi2

where ci and σi respectively denote center and dispersion. In the first step, the centers ci are initialized with K-means clustering and the dispersions σi2 are determined as the distance between ci and the closest cj (j 6= i, j ∈ {1, ..., r}).

By preserving the original dimension of the data, the second network tries to perform the inverse transformation from the reduced data (Fig. 2). We define the inverse transformation F : ℜℓ → ℜm : k X ˆ = F(t) = vj ψj (t) + v0 (12) x j=1

for some kernels ψj , (j = 1, ..., k), weights V = (v0 , ..., vk ), where k is the number of kernels and vi ∈ ℜℓ , (i = 0, . . . , k).

To identify the RBF-NLPCA model, we have to determinate the parameters of radial basis functions (centers and dispersions) and the weights for the two RBF-networks. The number of principal component analysis to retain in the NLPCA model can be determined by the reconstruction approach [Harkat, 2003a]. It should be noted that the nonlinear principal component matrix T being unknown, the training of the two RBF-networks separately is then impossible. To overcome this problem we suggest to estimate this nonlinear component matrix T by using the principal curve algorithm [Hastie, 1989]. When the matrix T is estimated, each RBF network can be trained separately. So, the training problem is transformed into two classical nonlinear regression problems.

7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009

1

and faults are then detected by checking the observed behavior against this model.

φ1

Our approach for NLPCA process monitoring involves a Squared Prediction Error (SPE) chart. The SP E is given by m X e2j (k) (13) SP E(k) =

w11 t1

x1 x2

. . .

φ2

xm

t2

. . .

j=1

with ej (k) = (xj (k) − x ˆj (k)), xj is a process variable and x ˆj is the estimation of xj from the NLPCA model. Analysis of SP E(k) provides a way to detect abnormalities in data [Dunia, 1996, MacGregor, 1996]. To reduce false alarms, exponentially weighted moving average (EWMA) filter can be applied to the residuals but it introduces a delay in detecting faults. The general EWMA expression for residual is:

tℓ wr1 φr

Fig. 1. RBF-Network for mapping from ℜm → ℜℓ .

¯(k) = (I − Λ)¯ e e(k − 1) + Λe(k)

1 ψ1

v01

2

v11

SP E(k) = k¯ e(k)k

x ˆ1

t1 t2

. . .

. . .

tℓ

(15)

¯(k) and SP E(k) are the filtered residuals and where e SP E(k) respectively. Λ = γI denotes a diagonal matrix whose diagonal elements are forgetting factors for the residuals.

x ˆ2 ψ2

(14)

If SP E(k) is above the confidence limits, a new event is found in the data, which is not described by the process model.

x ˆm

vmk

4.2 Fault isolation

ψk

Fig. 2. RBF-Network for mapping from ℜℓ → ℜm 3.3 RBF-NLPCA training The proposed training procedure involves three steps : (1) Find principal curves by successively applying the principal curve algorithm [Hastie, 1989] to observed data and residuals. Then in the first step T1 denotes the first nonlinear principal component, so: X = F1 (T1 ) + E1 , where E1 is the residual. When more than one nonlinear principal component is needed we do the same calculation from the residual data [LeBlanc, 1994]. (2) Train an RBF network that maps the original data onto the nonlinear principal components (obtained by the principal curves algorithm). (3) Train the second RBF network that maps the nonlinear principal components onto the original data. The training of the two RBF-Networks (mapping and demapping network) is presented in detail in Harkat [2003b]. 4. SENSOR FAULT DETECTION AND ISOLATION 4.1 Fault detection Abnormal situations that occur due to sensor drifts induce changes in sensor measurements. Nonlinear Principal component analysis is used to model normal process behavior

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After the presence of a fault has been detected, it is important to identify the fault and apply the necessary corrective actions to eliminate the abnormal data. The variable reconstruction approach for sensor fault isolation was proposed by Dunia et al. [Dunia, 1996] in the linear case. This approach assumes that each sensor may be faulty (in the case of a single fault) and suggests to reconstruct the assumed faulty sensor using the NLPCA model from the remaining measurements. By examining the residuals given by NLPCA model before and after reconstruction, we can determine the faulty sensor. Moreover this method allows to determine replacement values for the faulty sensors. Linear reconstruction The reconstruction using iterative approach was presented by Dunia [1996] for the set of variables. To reconstruct the ith variable, the iterative procedure tries to find a reconstructed value zi closest to its estimation (i.e. zi = zˆi ). h i (iter−1) T (iter) x+i ci = xT−i zi zi (16) where ci is the ith column of Cˆℓ , zi = xi and the subscripts −i, +i are used to indicate a vector formed by the first i − 1 and the last m − i elements of the original vector, respectively. (0)

Nonlinear reconstruction For the ith variable, its reconstruction zi is defined, as in the linear case, by an iterative approach, the estimated value zˆi is re-estimated by the NLPCA model until convergence (Fig. 3).

7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009

(17)

where xi = [x1 x2 ...ˆ zi ...xm ]T and ξi is the ith column of the identity matrix. The iterative expression given by equation (0) (17) must be started using some better initial value zi . We suggest to use the measurement xi as initial value (0) zˆi = xi . Note that the reconstruction expression (17) converges quickly (i.e. in one or two iterations), for all the examples that have been treated. zˆ1

x1

x

G

F

ˆ x

Fig. 3. Depicts schematically an iterative reconstruction for x1 SP E j (k) denotes the index SP E(k) calculated after reconstruction of the j th variable. Therefore, if the ith faulty variable is reconstructed (i = j), the index SP E j is in the control limit because the fault is eliminated by reconstruction. If the reconstructed variable is not faulty, the index SP E j being always affected by the fault, SP E j is outside its control limit. In summary, when a fault has been detected, all the indices SP E j are computed, and if SP E j is under its control limit, the j th sensor is considered as a faulty one. For multiple sensor reconstruction, ξiT is not a vector but a matrix containing all possible direction of faults, for example to reconstruct twosensors 1 and  4 among 5 1 0 0 0 0 . It should be sensors ξjT has the form ξjT = 0 001 0 noted that in the nonlinear case of reconstruction we have not an existence conditions for reconstruction. The only reconstruction condition is related to the convergence of the reconstruction algorithm (17). The maximum number of reconstructed sensors r is (as in the linear case, [Tharrault, 2008]) r ≤ (m − ℓ) which is the dimension of the residual subspace. For multiple sensor fault isolation, the same reasoning (as in the linear case) can be used, we can reconstruct several sensors simultaneously. If we assume that r sensors can be faulty at the same time, the combination of r among r m (Cm ) reconstructions are calculated. By analyzing the detection index calculated after each reconstruction we can determinate the faulty sensors. The classical reconstruction approach (used for fault isolation) leads to combination explosion of faulty scenarios related to multiple faults to consider. Instead of considering the reconstruction of r P i ), we consider one up to r simultaneous sensors ( Cm i=1

only the reconstruction of r simultaneous possible faulty sensors.

It should be noted that the maximum number of reconstruction necessary to isolate s simultaneous faulty sensors r (where s ≤ r) is Cm . This means that we can only perform r Cm reconstructions for isolation of all possible simulta-

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neous s faults, where s ∈ {1, 2, ..., r}. This approach is illustrated through its application to a multiple sensor fault detection and isolation of an air quality monitoring network. 5. APPLICATION FOR FDI OF AIR QUALITY MONITORING NETWORK 5.1 Description of air quality monitoring network The air quality monitoring network AIRLOR working in Lorraine, (France), will be described. The monitoring network consist of twenty stations placed in rural, periurban and urban sites. Each monitoring station consists of a set of sensors, dedicated to the acquisition of the following pollutants: carbon monoxide CO, nitrogen oxides NO and NO2 , sulfur dioxide SO2 and ozone O3 . Moreover, seven stations are dedicated to the recording of additional meteorological parameters. The measures are averages calculated over fifteen minutes. In this work, only six stations are considered. The purpose is to detect sensor failures mainly those which record ozone concentration (O3 ) and primary pollutants like nitrogen oxides (NO and NO2 ). The matrix X contains 18 variables, v1 to v18 , corresponding, respectively, with ozone O3 and nitrogen dioxide (NO2 and NO) of each station. Data set is chosen to have different concentration levels to show the performance of the proposed method. By applying the principal curve algorithm we can calculate the nonlinear principal components t. Then, the two RBF networks are trained separately and the RBF-NLPCA model is identified [Harkat, 2005]. The three following figures present, respectively, measurements and estimation of O3 , NO2 and NO levels for one station. The estimations are given by the RBF-NLPCA model (12). For our application, six components was retained in the model which explain 95% of the variance of data. By taking into account the nature of the modeled process, the results obtained are very satisfactory, with the RBFNLPCA model obtained, even the peaks of NO, O3 and NO2 are well estimated (Figure 4 to 6) which are essential for the alarm procedures. In the case of the nitrogen oxides that are more localized pollutants, and more difficult to model, the estimation of these two variables remains correct for weak values as well as for high values. 160

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zi = ξiT F(G(xi ))

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Fig. 4. Measurements and estimations of the Ozone concentrations Concerning a prior analysis of fault isolation, for this application, we can reconstruct until 11 sensors at the

7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009

of 3 sensors among 18 are performed. Fig. 8 shows some filtered SP Ei1,i2,i3 calculated after the reconstruction of each three sensors. Then on Fig. 8, we have only indicated, according to the time, the evolution of SP E1,2,3 , SP E2,3,4 , SP E1,3,4 , SP E2,4,5 , SP E1,4,2 and SP E2,5,6 among the 3 possible SP Ei1,i2,i3 . This result is associated with C18 the column δ1,3,4 of Table 1; it is clear that the fault is eliminated on the SP E value after the reconstruction of the sensors 1, 3 and 4 which indicates that those sensors are the faulty sensors.

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Fig. 6. Measurements and estimations of the NO concentrations same time, which help us to isolate 11 multiple faulty sensors. The number of reconstruction combination for this 11 possible faulty sensor is C18 = 31824. It is impossible to represent in this paper all faults signatures. So, we just establish (for examples given below for multiple sensor fault isolation), a table of some fault signatures in Table. 1. Some possible faults noted δi1 , δi1,i2 or δi1,i2,i3 in the first row, those affecting sensors 1, 2, 3, and those affecting the couples of sensors {1, 3}, {1, 4}, {1, 3, 4}, {2, 4, 5}, {2, 5, 6}. The first column relates to the indices SP Ei1,i2,i3 (SP Ei1,i2,i3 (k) means that SP E has been computed at time k after reconstruction of the variables vi1 , vi2 , vi3 ) calculated after the reconstruction of some combinations of 3 among 18 sensors, {1, 2, 3}, {1, 3, 4}, {1, 4, 5}, {2, 3, 4}, {2, 4, 5}, {2, 5, 6}. This table provides a correspondence between the symptoms SP Ei1,i2,i3 and the faults δi1 , δi1,i2 and δi1,i2,i3 . For example, the fault δ1,3 affect all the indices except those established without components 1 and 3, {1, 2, 3} and {1, 3, 4}. SP E1,2,3 SP E1,3,4 SP E1,4,5 SP E2,3,4 SP E2,4,5 SP E2,5,6

δ1 0 0 0 x x x

δ2 0 x x 0 0 0

δ3 0 0 x 0 x x

δ1,3 0 0 x x x x

δ1,4 x 0 0 x x x

δ1,3,4 x 0 x x x x

δ2,4,5 x x x x 0 x

δ2,5,6 x x x x x 0

In the second fault scenario, sensors 1 and 4 are affected simultaneously. Fig. 9 shows some filtered SP E after the reconstruction of each three sensors (combination of 3 from 18 sensors). Indices, for which variables 1 and 4 are reconstructed simultaneously, are under their control limits which indicates that those variables are the faulty ones (this result is associated with the column δ1,4 of table 1). Filtered SPE2−3−4

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Table 1. Table of some fault signatures Based on the obtained NLPCA model, the indices for detecting and isolating faulty sensors can be calculated on line. To illustrate the sensor validation method, a bias fault is introduced simultaneously on sensors 1, 3 and 4 (O3 and NO of the first station, O3 for the second station) between samples 800 and 1080. The magnitude of the fault amounts to 20% of the range of variation of each variable. SP E in Fig. 7 almost immediately allows to detect the presence of a fault. To identify the faulty sensors, the reconstructions

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In the last case, a single fault is considered on the sensor 1. After calculation of all reconstruction combinations, the indices for which the sensor 1 is reconstructed, are under their control limits (Fig. 10) (this result is associated with the column δ1 of table 1), so the sensor 1 is considered as the faulty one. For the first fault scenario, sensors 1, 3 and 4 being identified as the faulty sensors, then we can reconstruct them in order to give replacement values for the faulty measurements. Fig. 11 show the fault-free measurements, the faulty measurements and the replacement values obtained by reconstruction for sensors 1, 3 and 4, respectively. It

7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009

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method to an air quality monitoring network. The proposed NLPCA model is performed in two steps. The first step consists of using the principal curve algorithm proposed by Hastie [1989] to find nonlinear principal components. Then, two cascade RBF-networks are trained with a three phase procedure. A NLPCA model is built, using data obtained when the process is under normal condition. The proposed approach for multiple sensor fault detection and isolation, using reconstruction method, is presented and successfully applied to an air quality monitoring networks in Lorraine, (France). The proposed approach allows to reduce the number of reconstruction combinations and also gives replacement values of faulty measurements.

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REFERENCES

Fig. 9. Time evolution of filtered SP E calculated after reconstruction combination of 3 from 18 sensors, with a fault on sensors 1 and 4 Filtered SPE

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