Supervised Mode

Suppose that datasets are available from several states of the structure, which include the healthy state, the damaged state and other states of interest. In this case, TF poles from all the states can be plotted upon the z-plane in order to investigate the effect of damage or varying EOCs on the TF pole locations. Furthermore, several criteria can be used to classify the TF poles of new datasets as healthy or damaged. Firstly, the appropriate model order na of the output-only ARX model has to be determined. For this purpose, models of different orders are built and the mean squared errors (MSEs) of their predictions are calculated for a training dataset. As the model order increases, the MSE will be exponentially reduced. The calculated models, which were fitted to the training dataset, are then used to predict the system’s response for a validation dataset. The MSE of the validation dataset decreases exponentially, but starts to increase at higher model orders for which the model starts to overfit the training dataset by capturing its noise processes. The MSEs of a training dataset and a validation dataset are depicted in Figure 3.3 for a series of model orders. A range of optimal model orders can be obtained by comparing the MSE curve of the training set with that of the validation set. More specifically, the optimal model order lies around the model orders for which the MSE of the training set keeps dropping, while the MSE of the validation set starts to increase. The order of the exogenous part nb can be set equal to the model order

MSE

model order

training dataset validation dataset range of optimum model orders

Figure 3.3: Model mean squared error vs model order. Interval of orders with reducing

MSE of the training set and increasing MSE of the validation set define the range of optimal model orders.

na, because, for the output-only ARX model, there is no delay between the input and the output (since they are both structural responses).

After the appropriate model order has been selected, the TF poles of the baseline state are calculated and plotted upon the z-plane. The number of resulting TF poles is dependent on the selected model order. TF poles usually appear in conjugate pairs but the solutions might occasionally contain poles located on the real axis. These TF poles are ignored because their locations are not consistent for different datasets and because they are expected to create complications at the subsequent step of clustering. Conjugate and real TF poles which lie outside the unit circle constitute nonminimum-phase zeros. As mentioned in section 2.1.6, nonminimum-phase zeros imply that the system response is subjected to undershoot, overshoot or zero crossings and reaches the target response value with some delay. Therefore, these TF poles are not considered in the further analysis, either.

If more datasets from the baseline state are analyzed, their TF poles can be superimposed on the same plane. TF poles with similar frequency and damping content aggregate in groups. Subsequently, clustering algorithms may be employed to build clusters of TF poles characterized by similar frequency and damping. In this work, two clustering algorithms are used in order to cluster TF poles on the z-plane: k-means clustering and affinity propagation. k-means clustering is a partitional clustering algorithm, i.e., it requires a predefined number of clusters. This number of clusters is maintained throughout the clustering process. The algorithm aims at finding an assignment of data points to clusters, so that the sum of the squared distances between each data point and its closest cluster centroid is minimum [22]. The implementation of k-means for the clustering of TF poles of different states allows for the definition of a certain number of clusters which remains the same for all

states. Affinity propagation (AP), on the other hand, is a clustering algorithm which uses measures of similarity between pairs of data points and exchanges information between them in order to build clusters. These measures of similarity are dubbed preferences. The algorithm considers all data points as potential exemplars and, therefore, does not require a predefined number of clusters [37]. The choice between k-means clustering and AP is made, depending on their efficiency on the exanimed data.

A certain number of clusters with TF poles and their centroids are obtained, regardless of the clustering algorithm employed. These clusters can be described by a normal bivariate distribution by calculating the mean value and standard deviation. For instance, level curves, which are essentially ellipses, can be plotted using the mean value (µ) and three times the standard deviation of each pole cluster (3σ). The same procedure can be followed for the datasets of the damaged state.

Figure 3.4 shows an example of level curves for a set of healthy and damaged TF pole clusters. The TF poles are obtained from the modal test data of a 34 m wind turbine rotor blade. TF poles of the healthy state are represented by the blue points and the cyan level curves, while TF poles of the damaged state are shown by the green points and the red ellipses. In this example, the locations of all five TF pole clusters can be simultaneously observed for damage identification. The clusters of the healthy and damaged state can be exploited (i) to describe and quantify pole cluster migration due to structural changes and (ii) to evaluate the TF poles of new datasets by classifying them as healthy or damaged.

In order to describe and quantify pole migration, three simple metrics can be defined: (i) the Euclidean distance d between the cluster means of two states (µh for the healthy state and µc for the current state), (ii) the difference between the standard deviation both in the major and in the minor direction of the ellipse (∆σmajor = σ1c − σ1h and ∆σminor = σ2c − σ2h) and (iii) the rotation of the ellipse (∆θ = θc− θh), i.e., the difference between θc and θh, which are the angles between the horizontal axis and the ellipse major direction for the current state and the healthy state, respectively. The components, based on which these metrics are composed, are shown in Figure 3.5 for one pole cluster.

Furthermore, the TF pole clusters of the healthy and damaged states serve as the training data of a classification algorithm. Hence, TF poles of new datasets are evaluated based on whether they belong to the healthy or the damaged pole cluster. For this purpose, more complex metrics are required. For instance, in [67], transfer function poles are assessed by a damage index which is built as a weighted sum of three indices. More specifically, these indices are based on three different types of classifiers: (i) a classifier based on the nearest cluster mean, (ii) the perceptron classifier and (iii) the mean separation distance criterion. However, achieving a high detection accuracy and quantification in the supervised mode is beyond the scope of the present thesis, which focuses on employing TF pole migration for detecting structural changes in unsupervised mode.

-1 -0.5 0 0.5 1 Re(z) 0 0.2 0.4 0.6 0.8 1 Im(z) model 5 - n=12 - damage 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1:/T 1:/T 0.9:/T 0.8:/T 0.7:/T 0.6:0.5/T:0.4/T:/T 0.3:/T 0.2:/T 0.1:/T healthy damage

Figure 3.4: Healthy and damaged TF poles obtained from the modal test data of a rotor

blade plotted upon thez-plane. TF poles clustered with k-means clustering. Cyan ellipses describe the TF poles of healthy datasets and red ellipses describe the poles of damaged datastes.

μ

σ

σ

σ

σ

μ

d

healthy state current state

Figure 3.5: Ellipses describing the TF poles of the baseline state (subscript h) and the

current state (subscript c). The figure depicts: the cluster means (µhandµc), the standard deviations in the principle axis direction (σ1hand σ2h) and in the perpendicular direction (σ1candσ2c), the angle of the ellipse principle direction with respect to the horizontal axis (θhandθc) and the Euclidean distanced between the two means.