Interval PCA Based Fault Detection and Isolation With ...

International Conference on Automatic control, Telecommunications and Signals (ICATS15) University BADJI Mokhtar - Annaba - Algeria - November 16-18, 2015

Interval PCA Based Fault Detection and Isolation With New Interval SPE Statistic Tarek AIT IZEM, Wafa BOUGHELOUM, Mohamed Faouzi HARKAT, Messaoud DJEGHABA Badji-Mokhtar, Annaba University Department of Electronics P.O.Box 12, 23000 Annaba, Algeria ait [email protected], [email protected], [email protected], [email protected] literature. Cazes et al [3], and Chouakria et al [4], proposed the first adaptations, known as the centers PCA (C-PCA) and the vertices PCA (V-PCA) methods. The centers method computes the principal components (PCs) using the interval centers, whereas the vertices method computes the PCs using the vertices of the observed hyper-rectangles. Other interval PCA methods include the midpoints-radii PCA (MR-PCA) introduced by Palumbo and Lauro [5], treating midpoints and the interval ranges as two separate variables to enhance C-PCA by incorporating radius. DUrso and Giordani [6], introduced an alternative approach using least squares for MRPCA. Different from these methods, Gioia and Lauro [7], put forward an analytical interval PCA based on an interval-valued covariance matrix, and LeRademacher and Billard [8], employed symbolic covariance to extend the classical PCA. A new interval PCA method with an enhanced covariance matrix calculation, and which tends to employ all the information in hypercubes is proposed by Huiwen et al. [9], and is called the completeinformation principal component analysis (CIPCA). Based on the different interval PCA methods found in the literature, it is most likely possible to apply such approaches for monitoring of uncertain systems [10],[15], by modeling the sensors uncertainties in the form of intervals. However, several notions of SPM are to be adapted for the new interval model, that is, fault detection and localization techniques. In this Paper, the late CIPCA method is employed for modeling of uncertain systems, upon which the generation of interval residuals is performed, the detection of aberrant information in the data is realized using a new Interval SPE control chart that is better suited for the task than the classical PCA based SPE, and the extended interval reconstruction principle is used for the isolation of faults. An application on a simulated example is presented to illustrate the performance of the presented techniques.

Abstract—One way of enhancing the accuracy of principal component analysis (PCA) model for a better process monitoring is to take into account the uncertainties on the measurements provided by the process sensors. A suited solution consists in using interval datasets, where unlike classical single point value, the data may take interval values depending on the variable uncertainty. This obviously requires different approaches for the determination of the PCA model, as well as the techniques for system monitoring. This paper extends the methodology of PCA based statistical process monitoring (SPM) to that for interval-valued data, this includes using the so-called complete information principal component analysis (CIPCA) method for process modeling, the generation of interval residuals for fault detection, where a new Shewhart type Interval SPE control chart is introduced for a more accurate detection in interval dataset case, and the application of the extended interval reconstruction principle for fault isolation. Keywords—Principal Component Analysis, Interval Data, Complete-Information PCA, Reconstruction Principle, Fault detection and Isolation, SPE statistic.

I.

I NTRODUCTION

Principal Component Analysis (PCA) is a widely used technique for sensors fault detection and isolation [1], and more generally for the detection of aberrant information [2]. PCA provides, implicitly, a model of the system and reveals linear relationships between its variables without making an explicit model, while reducing the dimensionality of the used dataset into one of less dimension in order to facilitate the visualization and extraction of the main trends in a highdimensional dataset, this obtained model can be subsequently used to monitor the system behavior or its components [2]. The PCA model identfication consists in estimating the process structure and parameters by an eigen-decomposition of the covariance matrix of data which are assumed to be collected under normal operating conditions. However, in real life case these data are but approximate values given by the process sensors, and are generally stained with uncertainties due to different factors, and in order to obtain a guaranteed state reconstruction, the ifluence of these uncertainties have to be taken into account. A contemporary representation of sensors uncertainties is the interval model approach which consists in providing a nominal model of the process influenced by uncertainties assumed to be represented by intervals with known bounds. The determination of PCA model in this case requires using new approaches adapted for the interval type of the data. Several extensions of PCA to interval-valued data exist in the

II.

P RINCIPAL C OMPONENT A NALYSIS

Principal component analysis is a vector space transformation often used to transform multivariable space into a subspace which preserves maximum variance of the original space in minimum number of dimensions. The measured process variables are usually correlated to each other. PCA can be defined as a linear transformation of the original correlated data into a new set of uncorrelated data that explain the trend of the process. Consider a data matrix X ∈ Rn×m containing n samples of m process variables collected under normal operation. This matrix must be normalized to zero mean and unit variance 1

International Conference on Automatic control, Telecommunications and Signals (ICATS15) University BADJI Mokhtar - Annaba - Algeria - November 16-18, 2015

hyper-rectangle of 2m vertices, the length of its segments are given by the intervals of every description variable. The complete information principal component analysis CIPCA, employed in this paper for modeling of interval data, discriminates itself from various well-established methods, e.g., VPCA, CPCA, and MR-PCA [9] in that it can capture the complete information in interval-valued observations. Taking a hypercube view with ifinitely dense points uniformly distributed within the hypercubes, CIPCA defines the inner product of interval-valued variables, and transforms the PCA modeling into the computation of some inner products in the covariance matrix. In particular, CIPCA provides an efficient and effective way for conducting PCA for large-scaled numerical data, and lend the meaningful structure information hidden in massive data. We first define basic operators and notions of the CIPCA method. For an interval-valued variable [Xj ] =    
0 x1j , x1j , x2j , x2j , ..., xnj , xnj , the mean is given by

with the scale parameter vectors of mean and variance. PCA determines an optimal linear transformation of the data matrix X in terms of capturing the variation in the data T = XP , X = T P T

(1)

With T ∈ < being the principal component matrix, and the matrix P ∈