Comparison of output-only methods for condition monitoring of industrials systems

In the field of structural health monitoring or machine condition monitoring, the activation of nonlinear dynamic behavior complicates the procedure of damage or fault detection. Blind source separation (BSS) techniques are known as efficient methods for damage diagnosis. However, most of BSS techniques repose on the assumption of the linearity of the system and the need of many sensors. This article presents some possible extensions of those techniques that may improve the damage detection, e.g. Enhanced-Principal Component Analysis (EPCA), Kernel PCA (KPCA) and Blind Modal Identification (BMID). The advantages of EPCA rely on its rapidity of use and its reliability. The KPCA method, through the use of nonlinear kernel functions, allows to introduce nonlinear dependences between variables. BMID is adequate to identify and to detect damage for generally damped systems. In this paper, damage is firstly examined by Stochastic Subspace Identification (SSI); then the detection is achieved by comparing subspace features between the reference and a current state through statistics and the concept of subspace angle. Industrial data are used as illustration of the methods. Keywords: KPCA; EPCA; NSA; BMID; SSI subspace; condition monitoring; statistics. 1. Introduction Blind source separation (BSS) techniques allow to recover a set of underlying sources from observations without any knowledge of the mixing process or sources. BSS techniques are shown useful for modal identification [1], for damage detection and condition monitoring [2, 4] from output-only. Among the BSS family, one can cite for example Independent Component Analysis (ICA) [2], Second-Order Blind Identification (SOBI) [5] and an extension of SOBI, called Blind Modal Identification (BMID) [6] which can treat generally damped systems. Principal Component Analysis (PCA) [3] is a linear multivariable statistical methodknown as an efficient method to compress large sets of random variables and to extract interesting features from a dynamical system. However, this method is based on the assumption of linearity. To some extent, many systems show a certain degree of nonlinearity and/or non-stationarity, and PCA may then overlook useful information on the nonlinear behavior of the system. As reported in [7], there are many types of damage that make an initially linear structural system respond in a nonlinear manner. Therefore, detection problem may necessitate methods which are able to study nonlinear systems. Kernel Principal Component Analysis (KPCA) is a nonlinear extension of PCA built to authorize features such that nonlinear dependence between variables. The method is “flexible” in the sense that different kernel functions may be used to better fit the testing data. In the beginning, KPCA has interested many scientists in the domain of image processing [8, 9]. These researchers showed that KPCA may be more advantageous than other techniques such as PCA or Wavelet Transform etc. in encoding image structure. In the last five years, KPCA has been introduced in other fields of research (e.g. biological treatment process [10], machine monitoring [11, 12, 13]) and has shown its ability in the monitoring of nonlinear process. A main drawback of BSS techniques cited above is the need of several sensors. If the number of sensors is too small, modal identification and/or damage detection may not be performed in good conditions. An alternative PCA-based method named Null Subspace Analysis (NSA), using block Hankel matrices was proposed to detect damages in bearings [14] and on an airplane mock-up [15]. Furthermore, the Hankel matrices were also exploited to enhance some other detection methods namely KPCA, BMID, called EKPCA and EBMID [16]. With the data generated by mean of block Hankel matrices, those methods have been proven to be efficient when the number of available sensors is small or even reduced to one sensor only [15-17]. The focus of this paper is the application of output-only health monitoring techniques to detect damaged mechanical components in industrial environment. First the PCA method is described briefly as it constitutes the background of the EPCA and KPCA methods. Then the definition of the block Hankel matrices is recalled to introduce the EPCA methods. KPCA and BMID are next presented as well as their enhanced versions. Two detection indicators are used which are based on the concept of subspace angle and on statistics. The methods are illustrated on two applications which consist in detecting damage in a rotating device and in controlling quality of welded joints in a steel processing plan. In both cases, only one sensor signal is exploited. 2. Principal Component Analysis (PCA) Let us assume that a dynamical system is characterized by a set of vibration features collected in the matrix , where m is the number of sensors and N is the number of samples. In general, PCA involves a data normalization procedure, which leads to a set of variables with zero-mean and unitary standard deviation. This method, also known as Karhunen-Loève transform or Proper Orthogonal Decomposition (POD) [3], provides a linear mapping of data from the original dimension m to a lower dimension p. The dimension p represents the physical order of the system or the number of principal components which affect the vibration features. In practice, PCA is often computed by a Singular Value Decomposition (SVD) of matrix , i.e. (1) where and are orthonormal matrices, the columns of define the principal components (PCs). The order p of the system is determined by selecting the first p non-zero singular values in which have a significant magnitude (“energy”) as described in [3]. A threshold in terms of cumulated energies is often fixed to select the effective number of PCs that is necessary for a good representation of the matrix . In practice, a cumulated energy of 75% to 95% is generally adequate for the selection of the active PCs. 3. Block Hankel matrices The block Hankel matrices play an important role in subspace system identification [18]. Those matrices characterize the dynamics of the analyzed signals and have been used for modal identification and damage detection [15-17]. The covariance-driven block Hankel matrix is defined as: where r, c are user-defined parameters ( in this paper) and Δ represents the output covariance matrix defined by: The data-driven Hankel matrix is defined as: where 2i is a user-defined number of row blocks, each block contains m rows (number of measurement sensors), j is the number of columns (practically 2 1, N is the number of sampling points). 4. Enhanced Principal Component Analysis (EPCA) EPCA is basically a principal component analysis of the covariance-driven block Hankel matrix [15]. The smallest singular values of the matrix which correspond to components of low “energy” are actually associated to noise or to weak dynamics that may be neglected. Hence, the matrix is usually factorized in two subspaces, namely the active ( and the null subspace ( ) [15]. The SVD decomposition of the matrix becomes: S 0 0 S " (5) As stated in paragraph 2, the number of components in the active subspace is user-defined. It should be chosen such that the dynamics of the signal is accurately modeled without accounting for the background noise. The size of the Hankel matrix is also a user-defined parameter. 5. Kernel Principal Component Analysis The key idea of KPCA is first to define a nonlinear map #$ % Φ #$ with #$ , k 1, ... , N which defines a high dimensional feature space F, and then to apply PCA to the data in space F [8]. Let us define the kernel matrix + of dimensions such that: ,-#. , #/0 Φ #. 1Φ #/ (6) 2,3 4 Δ Δ Δ Δ 5 Δ6 Δ67 8 9 8 Δ: Δ:7 5 Δ:767 ; c = r (2) Δ 1 N i . A #$7 BC $D . #$ 0 E i E N 1 (3) , . FGG GGG GGH I I I IJ 5 IK IK7 8 9 8 IK IK7 5 I 7K7 IK7 IK7 IK7 IK7J 5 I 7K I 7K7 8 9 8 I I 7 5 I 7KC LM MMM MMM N O P Q — S T O "VWXY" "Z[Y[ \" (4) The following kernel functions may be used: • polynomial kernel function, ,-#. , #/0 -# #K 10] (7) where d is a positive integer • radial basis function (RBF), ,-#. , #/0 expa # #K a /2σ 0 (8) where 2σ d is the width of the Gaussian kernel It is worth noting that in general, the above kernel functions give similar results if appropriate parameters are chosen [16]. The last function may present some advantages because of its flexibility in the choice of the associated parameter. For example, the width of the Gaussian kernel can be very small (<1) or quite large. Contrarily the polynomial function requires a positive integer for the exponent. KPCA may be effectively enhanced by using the covariance-driven Hankel matrices which improves the sensitivity of the detection [16] and also the computation cost. The combined method will be called Enhanced KPCA (EKPCA) in the following. 6. Blind Mode Identification BMID is a blind source separation technique (BSS) based on the Second-Order Blind Source Identification (SOBI). The principle of the method is to apply SOBI on an augmented and pre-treated dataset. Compared to SOBI, its main advantage for the identification problem, is to deal with generally damped system. SOBI considers the observed signals as a noisy instantaneous linear mixture of source signals. In many situations, multidimensional observations are represented as [5]: # Y e Y f Y gh Y f Y (9) where # Y I Y , ... , I Y 1 is an instantaneous mixture of source signals and of noise. h Y iX Y , ... , Xj Y k1 contains the signals issued from p narrow band sources, p<m. e Y l Y , ... , l Y 1 contains the source assembly at a time t. g is the transfer matrix between the sources and the sensor, called the mixing matrix. Under certain conditions, the mixing matrix identifies to the modal matrix of the structure and the sources correspond to normal coordinates [1]. f Y is the noise vector, modeled as a stationary white, zero-mean random process. Furthermore it is assumed to be independent of the sources. The BMID technique, proposed by McNeil and Zimmerman [6], consists in applying SOBI to a augmented and pretreated dataset formed by the original data, denoted #m Y , and their Hilbert transform pairs #nm Y . The mixing problem becomes double-sized: # g j h j (10) o#m #nm p g j ohm hnm p (11) BMID can also be enhanced by the Hankel matrices, particularly when only one sensor is used for the detection [16]. Like EPCA and EKPCA, the number of blocks of the Hankel matrix is a user-defined parameter. 7. Damage detection using the concept of angle between subspaces The concept of subspace angle was introduced by Golub and Van Loan [19]. This concept can be used as a tool to quantify the spatial coherence between two data sets resulting from observation of a vibration system [3,16]. Given two subspaces (each with linear independent columns) q j and r s (V = t), the procedure is as follow. Carry out the QR factorizations q uvwv , uv x V (12) r uywy , uy t The columns of uv and uy define the orthonormal bases for subspaces q and r are computed from singular values associated with the product The largest singular value is thus related with the largest angle characterizing the geometric difference between two subspaces. The change in the system dynamics may be detected by monitoring the angular coherence between subspaces estimated from a reference observation set and from the observation set of a current state of the system. A state is consider as reference if the system operates in normal conditions (damage does not exist). active subspace built from two principal Figure 1 Angle z formed by active subspaces according to the reference and current states, due to a In the case of EPCA, the considered subspaces by the kernel principal components. And finally, i 8. Damage detection using statistics The second type of indicator is known as “Novelty Analysis”. In the particular case of EPCA, the first columns features into the space characterized for a cu assessed by re-mapping the projected data back to the original space: The residual error matrix E is estimated as From the prediction error vector where w { | } is the covariance of the features. In the case of BMID, the projection is For KPCA, the projection is realized on the first kernel principal components. squared prediction error (SPE), more details can be found in [10]. 9. First application 9.1 Experimental setup This industrial application concerns the case of electro the assembly line has to be assessed. A set of nine rotating devices w the flank of the component, and one mono them are known to be healthy (referenced OK q and r respectively. The angles uvuy: uv uy vy vy vy vy ~ W cos z. , i 1, ... , q Figure 1 shows a 2D example in which an vectors which characterize the dynamic behavior of system. dynamic change [3 are the active subspaces . Based on KPCA, the subspaces are built n BMID, the subspace is built by the first columns of the mixing matrix. based on the computation of errors in the reconstruction of the data. This method of may be used rrent state of the system. The loss of information in this projection can be ƒ „ „} … ƒ …$, the Novelty Index (NI) is defined using the Mahalanobis norm: based on the active subspace defined by the first columns of the mixing matrix. For the last case, the statistic -mechanical devices for which the overall quality at the end of as instrumented with two accelerometers: one triaxial accelerometer axial on the top as illustrated in Figure 2. Among this set -0 OK-4) and the other four are faulty (NOK-1 NOK †‡ˆ ‰ …‡w C„…‡ z. between the