Multiple sensor fault detection and isolation of an air quality monitoring ...

Int. J. Adaptive and Innovative Systems, Vol. 1, Nos. 3/4, 2010

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Multiple sensor fault detection and isolation of an air quality monitoring network using RBF-NLPCA model Mohamed-Faouzi Harkat* University Badji Mokhtar – Annaba, Faculty of Engineering Sciences, Department of Electronic, P.O. Box 12, Annaba, 23000, Algeria E-mail: [email protected] *Corresponding author

Yvon Tharrault, Gilles Mourot and José Ragot Centre de Recherche en Automatique de Nancy (CRAN), Nancy University, CNRS UMR 7039, 2 avenue de la forêt de Haye, F-54516 Vandoeuvre-Lès-Nancy, France E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] Abstract: This paper presents a data-driven method based on non-linear 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. Keywords: multiple fault detection and isolation; non-linear PCA; radial basis functions; RBF; reconstruction; air quality monitoring network. Reference to this paper should be made as follows: Harkat, M-F., Tharrault, Y., Mourot, G. and Ragot, J. (2010) ‘Multiple sensor fault detection and isolation of an air quality monitoring network using RBF-NLPCA model’, Int. J. Adaptive and Innovative Systems, Vol. 1, Nos. 3/4, pp.267–284. Biographical notes: Mohamed-Faouzi Harkat received his BEng in Automation from Annaba University (Algeria) in 1996 and his PhD from Intitut National Polytechnique de Lorraine (France) in 2003. He is now an Assistant Professor in the Department of Electronic at Annaba University, Algeria. His main research interests include fault diagnosis, process monitoring, multivariate statistical approaches and neural networks. Yvon Tharrault received his BEng in Automation from the Ecole Suprieure des Sciences et Technologies de l’Ingnieur de Nancy (French School of Engineering) in 2004. He obtained his PhD from the Institut National Polytechnique de Lorraine (Nancy, France) in 2008. His research interests include fault diagnosis, process monitoring, multivariate statistical analysis. Copyright © 2010 Inderscience Enterprises Ltd.

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M-F. Harkat et al. Gilles Mourot received his PhD in Electrical Engineering in 1993 at the Institut National Polytechnique de Lorraine (INPL), Nancy, France. His research activities, within Automatic Control Research Center of Nancy (CRAN), include process diagnosis, fault detection and isolation, modelling and identification of complex systems. José Ragot received his Engineer’s degree with specialisation in Control from Ecole Centrale de Nantes (France) in 1969. Then, he joined the University of Nancy where he received in 1980 the ‘Diplôme de Doctorat-es-Science’. He moved up to Professor in 1985 at the Institut National Polytechnique de Lorraine. Presently, he is a Professor (with highest degree ‘Classe Exceptionnelle’). He is a Researcher in the Centre de Recherche en Automatique de Nancy where he was the Head of the group diagnosis for 12 years. His major research fields include data validation, process diagnosis, fault tolerant control, and the links of diagnosis with identification.

1

Introduction

Recently, fault detection and process monitoring using principal component analysis (PCA) were studied intensively and largely applied to industrial processes. PCA is the optimal linear transformation with respect to minimising the mean squared prediction error (SPE). PCA has been used in the statistical process control area such as process monitoring (Kresta et al., 1991), gross error detection (Tong and Crowe, 1995), sensor fault identification (Dunia et al., 1996). However, PCA is a linear method, and most engineering systems are non-linear. To overcome this problem, we use a non-linear model to generate residuals for fault detection and isolation (FDI). Several non-linear principal component analysis (NLPCA) models have been proposed in recent years (Hastie and Stuetzle, 1989; Kramer, 1991; Dong and McAvoy, 1994; Webb, 1996). Hastie and Stuetzel (1989) proposed a principal curve methodology to provide a non-linear summary of an m-dimensional data set. However, this approach is non-parametric and can not be used for continuous mapping of new data. To overcome the parametrisation problem, an auto-associative neural networks have been used (Kramer, 1991; Dong et al., 1994; Tan and Mavrovouniotis, 1995). Kramer (1991) proposed a NLPCA based on a five-layer auto-associative neural network for uncovering linear and non-linear relationships among the variables. These networks have linear first (input) and fifth (output) layers, and sigmoidal non-linearities on the second and fourth layer nodes. However, the presence of multiple local minima in the optimisation cost function can lead to stability problem in NLPCA. Dong and MacAvoy et al. (1994) developed a non-linear principal components based on the Hastie’s principal curves algorithm and two three-layer neural networks. A supervised training is used for determining the weights of each network, where the desired output of the first network and the input of the second one represent principal components given by the Hastie’s algorithm; it will be noted that this algorithm is biased. Tan and Mavrovouniotis (1995) have proposed a non-linear dimensionality reduction technique based on optimisation of the neural network inputs. The basic idea is to reduce the five-layer auto-associative network to a three layer network by simultaneously optimising the network parameters and network inputs which represent the principal components by minimising the output error. This approach only considers the demapping section of the NLPCA and

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implements a non-linear search for the calculation of the non-linear principal components but it requires more computation in implementation due to the necessity of an online nonlinear optimiser. For such networks, however, the training problem becomes a complicated non-linear optimisation task, defined in a multidimensional space. For this reason, Webb (1996) proposed an approach to non-linear principal components analysis using radial basis function (RBF) networks. Two RBF networks are used in this approach, the first one project the data set to reduced dimension using a non-linear transformation whose parameters are determined by the solution of a generalised symmetric eigenvector equation by maximising the variance of its outputs. The inputs of the second RBF network are the outputs obtained from the first one and his desired outputs represent the data set. The parameters of this second network are determined by minimising the squared error of its outputs. The problem of this approach is the complicated task of training the first network. RBF neural networks provide a powerful technique for generating multivariate nonlinear mapping. Their topological structure being simple and the training of the network is rapid. Based on the idea developed by Dong and MacAvoy (1994), Harkat et al. (2003b) proposed an NLPCA model combining principal curve algorithm (Hastie and Stuetzle, 1989) and two three layer RBF-networks where the non-linear 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 non-linear principal components. A new training procedure of the RBF-NLPCA model is presented in Harkat et al. (2003a). 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 RBF-NLPCA reconstruction method. The variable reconstruction consists in estimating a variable from others plant variables using the PCA model, i.e., using the redundancy relations between this variable and the others. The reconstruction accuracy is thus related to the capacity of the NLPCA model to reveal the redundancy relations among the variables. This approach reduces considerably the number of reconstruction combinations and allows to determine replacement values for the faulty sensors. The proposed paper will be organised as follows. The Section 2 presents a description of linear PCA. Section 3 describes the non-linear principal component problem. The RBF-NLPCA model is described in Section 4. 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 5. Section 6 gives description and application results of the proposed multiple sensor FDI approach of an air quality monitoring network. The last section gives conclusions.

2

Linear PCA

PCA 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 variables (Gertler and McAvoy, 1997). Let us consider T

x (k ) = ⎡⎣x1 (k ) x 2 (k )… x m (k ) ⎤⎦

time instant k .

the vector formed with m observed plant variables at

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Define the data matrix X = ⎡⎣ x (1) x ( 2 )… x ( N ) ⎤⎦ ∈ R N ×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 = [t1t 2 …tm ] ∈ R N ×m , where the vectors ti are called scores or principal T

components and the matrix P = [ p1p2 … 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 :

∑ P ΛP

T

with PPT = PT P = I m

(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 = ⎡Pˆ Pˆm − ⎤ , T = ⎡Tˆ Tˆm − ⎤ ⎣ ⎦ ⎣ ⎦

(3)

will be defined below. According to this partitioning, equation (1) can be where rewritten as: X = Tˆ Pˆ T + Tm − PˆmT− = Xˆ + E

(4)

Xˆ = XCˆ with E = XCˆm −

(5)

with

where the matrices Cˆ = Pˆ Pˆ T and Cˆm − = I m − Cˆ constitute the PCA model. The matrices Xˆ et E represent, respectively, the modelled variations and non-modelled variations of X based on components ( < m ) . The first eigenvectors Pˆ ∈ Rm × constitute

the

representation

space

whereas

the

last

(m − )

eigenvectors

Pm − ∈ Rm ×m − constitute the residual space. It should be noticed that in the particular case = m , then Xˆ = X and X = 0.

3

Non-linear principal component analysis

Non-linear PCA is an extension of linear PCA. Whilst PCA identifies linear relationships between process variables, the objective of non-linear PCA is to extract both linear and non-linear relationships. This generalisation is achieved by projecting the process variables down onto curves or surfaces instead of lines or planes. Figure 1 illustrates the concept of linear PCA. The first principal component minimises the sum of squared orthogonal deviations between the straight line and all the variables. The concept of non-linear PCA is illustrated in Figure 2. The non-linear approach is as for linear PCA,

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except it represents the data by one dimensional smooth curve which is determined by the non-linear relationship between the variables. The curve is defined to minimise the orthogonal deviations between the data and the curve. Figure 1

The linear principal component minimises the sum of squared orthogonal deviations using a straight line

Figure 2

The non-linear principal component minimises the sum of squared orthogonal deviations using a smooth curve

Figure 3

Reduction of data dimensionality

Given an N × m matrix representing N measurements made on m variables, reduction of data dimensionality aims to map the original data matrix to a much smaller matrix of dimension N × ( < m ) , which is able to reproduce the original matrix with minimum distortion through a demapping projection. The reduced matrix describes principal component variables extracted from the original matrix (Figure 3). The mapping model gives from a data matrix X the non-linear principal component T and the demapping model gives estimation Xˆ from principal component T . In this case, the non-linear mapping has the following form

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M-F. Harkat et al. t = G (x)

(6)

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

(7)

where F represents the demapping function. Therefore a data set X including m variables can be expressed in terms of non-linear principal components X = Xˆ + E = F (T ) + E

where T = [T1 ,… ,T

]

(8)

is the matrix of non-linear principal components T = G ( X ) , and

E is the matrix of residuals. The problem is now to identify the non-linear projection functions G and F. To do this, the objective function to minimise is the sum of squared orthogonal deviations: N

min

∑ d =1

x (k ) − xˆ (k )

2

N

= min

∑ x (k ) − F ( G ( x (k ) ) )

2

(9)

d =1

where x (k ) is the ith row of X and xˆ (k ) is its estimation by the NLPCA model. The problem is now to identify the non-linear projection functions. Based on the idea developed by Dong and MacAvoy (1994), a non-linear version of PCA approach combining principal curves algorithm (Hastie and Stuetzle, 1989) and two RBF-networks is proposed (Harkat et al., 2003b). The principal curve has been presented as a non-linear PCA generalisation technique, but the algorithm does not produce a non-linear principal component representation, the model representation is not derived and without a model we can not make applications in process monitoring. The non-linear principal components t, obtained from the principal curves algorithm, are then exploited for training the two RBF networks. The results are a parametric NLPCA model which will be used later for process monitoring. The proposed training procedure involves three steps: 1

Find principal curves by successively applying the principal curve algorithm (Hastie and Stuetzle, 1989) to observed data and residuals. Then in the first step T1 denotes the first non-linear principal component, so: X = F (T1 ) + E1 , where E1 is the

residual. When more than one non-linear principal component is needed we do the same calculation from the residual data (LeBlanc and Tibshirani, 1994). 2

Train an RBF network that maps the original data onto the non-linear principal components (obtained by the principal curves algorithm).

3

Train the second RBF network that maps the non-linear principal components onto the original data.

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273

RBF-NLPCA model

The proposed NLPCA model can be obtained by using two RBF-Networks for mapping and demapping data. Firstly, a three layer RBF-network is used (Figure 4), the hidden layer was composed of radial basis neurons performing a non-linear mapping of the input space onto a lower dimension space, such that the non-linear features are captured. The aim is to use this network to define a transformation G : Rm → R . t = G (x) =

r

∑w φ ( x ) + w i i

0

(10)

i=1

{φi , i = 1,… , r } are RBF 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.

where

The Gaussian basis functions φi are defined as: ⎛

x − ci

⎜ ⎝

2σ i2

φi ( x ) = exp ⎜ −

2

⎞ ⎟ ⎟ ⎠

(11)

where ci and σ i respectively denote the centre and dispersion of the ith basis function. In the first step, the centres ci are initialised with k-means clustering algorithm and the dispersions σ i2

are determined as the distance between ci

c j ( j =/ i , j ∈ {1,…r } ) .

Figure 4

RBF-network for mapping from Rm → R

and the closest

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By preserving the original dimension of the data, the second network tries to perform the inverse transformation from the reduced data (Figure 5). We define the inverse transformation F : R → Rm : xˆ = F ( t ) =

q

∑ v ψ ( t) + v j

j

0

(12)

j =1

for some kernels ψj , ( j = 1,… , q ) . weights V = (v 0,… , vq ) , where q is the number of kernels and vi ∈ R , (i = 0,… , q ) .

Figure 5

RBF-network for mapping from R → Rm

To identify the RBF-NLPCA model, we have to determinate the parameters of the RBF (centres ci , c j and dispersions σ i , σ j ) and the weights wi , v j for the two RBF-networks. The number of PCA to retain in the NLPCA model can be determined by the reconstruction approach (Harkat et al., 2003a). It should be noted that the non-linear 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 non-linear component matrix T by using the principal curve algorithm (Hastie and Stuetzle, 1989). When the matrix T is estimated, each RBF network can be trained separately. So, the training problem is transformed into two classical non-linear regression problems. The training of the two RBF-Networks (mapping and demapping network) and the principal curves algorithm are presented in detail in Harkat et al. (2003b).

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275

Sensor FDI

5.1 Fault detection Abnormal situations that occur due to sensor drifts induce changes in sensor measurements. NLPCA is used to model normal process behaviour and faults are then detected by checking the observed behaviour against this model. Our approach for NLPCA process monitoring involves a SPE chart. At each time instant k, the SPE is given by SPE (k ) =

m

∑e

2 j

(k )

(13)

j =1

with e j (k ) = ( x j (k ) − xˆ j (k ) ) , x j is a process variable and xˆ j is the estimation of x j from the NLPCA model at instant k . Analysis of SPE (k ) provides a way to detect abnormalities in data (Dunia et al., 1996; MacGregor et al., 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 EWMA expression for residual is: e (k ) = ( I − Λ ) e (k − 1) + Λe (k ) ______

SPE (k ) = e (k )

2

(14) (15)

______

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

If SPE (k ) is above the confidence limits, a fault is detected.

5.2 Fault isolation 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. (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 analysing the residuals given by the NLPCA model before and after reconstruction, we can determine the faulty sensor. Moreover this method allows to determine replacement values for the faulty sensors.

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5.2.1 Linear reconstruction The reconstruction using iterative approach was presented by Dunia et al. (1996) for a set of variables. To reconstruct the ith variable of this set, the iterative procedure tries to find a reconstructed value zi closest to its estimation (i.e., zi = zˆi ) zi(

iter )

= ⎡⎢ xT−i z i( ⎣

iter −1) T x +i

⎤c ⎦⎥ i

(16)

0 where ci is the ith column of Cˆ , zi( ) = x i , 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, and iter is the index of iterations.

5.2.2 Non-linear reconstruction For the ith variable, its reconstruction zi is defined, as in the linear case, by an iterative approach, the estimated value zˆi is reestimated by the NLPCA model until convergence (i.e., zi closest to its estimation zˆi ) (Figure 6):

(

zˆi = ξiT F G ( x j )

)

(17)

where xi = [x1x 2 … z i … x m ]

T

and ξi is the ith column of the identity matrix. The

iterative expression given by equation (17) must be started using some better initial value zi( ) . We suggest to use the measurement x i as initial value zi( ) = x i . Note that the reconstruction expression (17) converges quickly (i.e., in one or two iterations), for all the examples that have been treated. From equations (10) and (12), equation (17) can be expressed as: 0

0

⎛ zˆi = ξiT ⎜ ⎜ ⎝

⎞ v j ψj ( ti ) + v0 ⎟ ⎟ j =1 ⎠ q

∑

(18)

where r

ti =

∑w φ (x ) + w l l

i

0

l =1

Figure 6

Depicts schematically an iterative reconstruction for x1

(19)

277

Multiple sensor fault detection and isolation ______

______

SPE j (k ) denotes the index SPE (k ) calculated after reconstruction of the jth variable.

This computation has to be performed successively for all the variables ( j = 1,… , m ) . Therefore, if the ith faulty variable is reconstructed (i = j ) , the index SPEj is in the control limit because the fault is eliminated by reconstruction. If the reconstructed ______

______

variable is not faulty, the index SPE j being always affected by the fault, SPE j is outside its control limit. In summary, when a fault has been detected, all the indices ______

______

SPE j are computed, and if SPE j is under its control limit, the jth 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 two sensors 1 and 4 among 5 ⎡1 0 0 0 0 ⎤ sensors ξ Tj has the form ξ Tj = ⎢ ⎥ . It should be noted that in the ⎣0 0 0 1 0 ⎦

non-linear case of reconstruction we don't have an existence condition 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 et al., 2008)] r ≤ (m − ) 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 m Cmr

reconstructions are

calculated. By analysing the detection index calculated after each reconstruction we can determinate the faulty sensors. However, 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 reconstructions of one up to r simultaneous ⎛ r i ⎞ sensors ⎜ Cm ⎟ , we consider only the reconstruction of r simultaneous possible faulty ⎜ ⎟ ⎝ i =1 ⎠ sensors. It should be noted that the maximum number of reconstruction necessary to isolate s simultaneous faulty sensors (where s ≤ r ) is Cmr . This means that we can only perform

∑

Cmr

reconstructions for isolation of all possible simultaneous s

faults, where

s ∈ {1, 2,…r } . This approach is illustrated through its application to a multiple sensor

FDI of an air quality monitoring network.

6

Application for FDI of air quality monitoring network

6.1 Description of air quality monitoring network The air quality monitoring network AIRLOR working in Lorraine, (France), will be described. The monitoring network consists of twenty stations placed in rural, peri-urban and urban sites. Each monitoring station consists of a set of sensors, dedicated to the

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acquisition of the following pollutants: carbon monoxide CO, nitrogen oxides NO and NO2, sulphur dioxide SO2 and ozone O3. Moreover, seven stations are dedicated to the recording of additional meteorological parameters. The measures are averages calculated over 15 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 non-linear principal components t. Then, the two RBF networks are trained separately and the RBF-NLPCA model is identified (Harkat et al., 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 modelled process, the results obtained are very satisfactory, with the RBF-NLPCA model obtained, even the peaks of NO, O3 and NO2 are well estimated (Figures 4 to 6) which are essential for managing the alarm. In the case of the nitrogen oxides that are more localised pollutants, and more difficult to model, the estimation of these two variables remains correct for weak values as well as for high values. Concerning a prior analysis of fault isolation, for this application, we can reconstruct until 11 sensors at the same time, which help us to isolate 11 multiple faulty sensors. The 11 number of reconstruction combination for this possible faulty sensor is C18 = 31,824. 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 δ i1 , δ i1,i 2 or δ i1 , δ i1,i 2,i 3 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 SPEi1,i 2,i 3 ( SPEi1,i 2,i 3 means that SPE has been computed at time k after reconstruction of the variables vi1 , vi 2 , vi 3 ) 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 SPEi1,i 2,i 3 and the faults δ i1, δ i1,i 2 and δ i1,i 2,i 3 . 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}. Table 1

Table of some fault signatures δ1

δ2

δ3

δ1,3

δ1,4

δ1,3,4

δ2,4,5

δ2,5,6

SPE1,2,3

0

0

0

0

x

x

x

x

SPE1,3,4

0

x

0

0

0

0

x

x

SPE1,4,5

0

x

x

x

0

x

x

x

SPE2,3,4

x

0

0

x

x

x

x

x

SPE2,4,5

x

0

x

x

x

x

0

x

SPE2,5,6

x

0

x

x

x

x

x

0

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It is important to point out that all the signatures are different, that allows to isolate the faulty sensors. Based on the obtained NLPCA model, the indices for detecting and isolating faulty sensors can be calculated online. 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. SPE in Figure 7 almost immediately allows to detect the presence of a fault. To identify the faulty sensors, the reconstructions of 3 sensors among 18 are performed. Figure 8 shows some filtered SPEi1,i 2,i 3 calculated after the reconstruction of each three sensors. Then on Figure 8, we have only indicated, according to the time, the evolution of SPE1,2,3 , SPE 2,3,4 , SPE1,3,4 , SPE 2,4,5 , SPE1,4,2 and 3 SPE 2,5,6 among the C18 possible SPEi1,i 2,i 3 . This result is associated with the column

δ1,3,4 of Table 1; it is clear that the fault is eliminated on the SPE value after the reconstruction of the sensors 1, 3 and 4 which indicates that those sensors are the faulty sensors. In the second fault scenario, sensors 1 and 4 are affected simultaneously by a fault. Figure 9 shows some filtered SPE 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). 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 (Figure 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. Figure 11 shows the fault-free measurements, the faulty measurements and the replacement values obtained by reconstruction for sensors 1, 3 and 4, respectively. It is clear that the reconstructed measurements are good estimations of the fault-free measurements. Figure 7

Measurements and estimations of the ozone concentrations (see online version for colours)

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Figure 8

Measurements and estimations of the NO2 concentrations (see online version for colours)

Figure 9

Measurements and estimations of the NO concentrations (see online version for colours)

Figure 10 Filtered SPE with a fault on v1 , v3 and v 4 (see online version for colours)

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Figure 11 Time evolution of filtered SPE calculated after reconstruction combination of 3 from 18 sensors, with a fault on sensors 1, 3 and 4 (see online version for colours)

Figure 12 Time evolution of filtered SPE calculated after reconstruction combination of 3 from 18 sensors, with a fault on sensors 1 and 4 (see online version for colours)

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Figure 13 Time evolution of filtered SPE calculated after reconstruction combination of 3 from 18 sensors, with a single fault on sensor 1 (see online version for colours)

Figure 14 Reconstruction with a fault on v1 , v3 and v 4 (see online version for colours)

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283

Conclusions

This paper proposes the application of multiple sensor FDI based on RBF-NLPCA method to an air quality monitoring network. The proposed NLPCA model is performed by combining two RBF networks and the principal curves algorithm Hastie and Stuetzle (1989). The RBF-NLPCA model is performed in two steps. The first step consists of using the principal curve algorithm proposed by Hastie and Stuetzle (1989) to find non-linear 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. An extension of the non-linear reconstruction approach is proposed. The variable reconstruction consists in estimating a variable from others plant variables using the NLPCA model, i.e., using the redundancy relations between this variable and the others. This approach is presented as an extension of the reconstruction in the linear case (Dunia et al., 1996) and leads to an iterative expression of the reconstructed measurements. The proposed approach for multiple sensor FDI, 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.

Acknowledgements This work was supported by EGIDE, Partenariat Hubert Currien, CMEP-TASSILI PAI 07 MDU 714, and Agence Universitaire de la Francophonie (AUF).

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