On the Use of Principal Component Analysis for Parameter Identification and Damage Detection in Structures

Modal analysis is used extensively for understanding the dynamic behaviour of structures as well as for structural health monitoring or damage detection based on output-only measurements. In this presentation, a different approach based on principal component analysis is considered.
Principal component analysis (PCA), also called proper orthogonal decomposition (POD), is a multi-variate statistical method that aims at obtaining a compact representation of the data. In the present paper, PCA (POD) is used for three purposes, namely damage detection, structural health monitoring and identification of nonlinear parameters.
The key idea of PCA is to reduce a large number of measured data to a much smaller number of uncorrelated variables while retaining as much as possible of the variation in the original data. To this purpose, an orthogonal transformation to the basis of the eigenvectors of the sample covariance matrix is performed, and the data are projected onto the subspace spanned by the eigenvectors corresponding to the largest eigenvalues. This transformation has the property to decorrelate the signal components and to maximize variance.
The first problem to which PCA is applied here is the damage detection problem. When applied to vibration measurements, it can be shown that the basis of eigenvectors (called the proper orthogonal modes) span the same subspace as the mode-shape vectors of the monitored structure. Thus the damage detection problem may be solved using the concept of subspace angle between a reference subspace spanned by the eigenvectors of the initial (undamaged) structure and the subspace spanned by the eigenvectors of the current (possibly damaged) structure.
The second problem concerns structural health monitoring of civil engineering structures when environmental effects (e.g. the influence of the variation of the ambient temperature) have to be removed from the structural changes. In this case, PCA may be applied on identified modal features (e.g. the natural frequencies) to separate the changes due to environmental variations from the changes due to damage sources. This procedure is illustrated on the example of a real bridge located in Luxembourg.
The third problem is related to the estimation of nonlinear parameters using model updating techniques. In this case, the most interesting property of PCA is that it minimizes the average squared distance between the original signal and its reduced linear representation. When applied to nonlinear problems, PCA gives the optimal approximating linear manifold in the configuration space represented by the data. The linear nature of the method is appealing because the theory of linear operators is still available. However, it should be borne in mind that it also exhibits its major limitation when the data lie on a nonlinear manifold.