Assessment of a Multiple Model Based Parametric Method for Output-Only Vibration-Based Damage Detection for a Population of Like Structures

This study focuses on the problem of vibration-based damage detection for a population of like structures. Although nominally identical, like structures exhibit variability in their characteristics due to variability in the materials and manufacturing. This inevitably leads to variability in the dynamics, which may be so significant as to mask deviations due to damage. Damage detection via conventional vibration-based methods, using a common threshold in the decision making mechanism thus becomes highly challenging. The study presents a detailed assessment of a recently introduced Multiple Model (MM) based AutoRegressive (AR) model parameter method aiming at addressing this problem. The assessment is based on high numbers of experimental test/inspection cases using composite beams damaged via impact, as well as comparisons with the corresponding conventional (single model based) method. The results confirm significant improvement over the method's conventional counterpart. A sensitivity analysis additionally indicates that the method is relatively insensitive to the model order, but sensitive to the specific beams selected as baseline (training) ones; in fact their selection may lead to excellent results.


Introduction
When dealing with a population of like structures there are two possibilities for vibration-based Structural Health Monitoring: Either a common threshold is adopted in the decision making mechanism [1] for all structures, or an individual threshold is established for each individual structure. The former approach has some obvious advantages in terms of set-up and cost, and is of high interest. Yet, the problem is that like structures are never identical due to inevitable variability associated with materials and manufacturing, plus of course the fact that they may operate under somewhat different conditions. This inevitably leads to variability in the dynamics, which may be so significant as to mask deviations due to damage.
Unlike other uncertainty factors, such as those associated with environmental and operational variability [2], the handling of the variability associated with a population of like structures and its effects on vibration-based SHM is, thus far, effectively unexplored. In two recent studies [3,4] by the present authors, a Multiple Model (MM) vibration-based framework for SHM for a population of like structures (composite beams subject to delamination) was introduced, and very interesting results and favourable comparisons with corresponding conventional methods were obtained. The main idea behind this framework is on the use of vibration data obtained from several -instead of a single -like structures for modelling the healthy dynamics during the methods' training (baseline phase) and then decision making based on the simultaneous use of all baseline (healthy) models based on a minimal distance type criterion. The results of these studies have illustrated the poor performance of conventional methods and the significantly improved performance of the Multiple Model methods. Yet, the assessments presented are only preliminary and not comprehensive, in the sense that neither the sensitivity of the methods with respect to the specific structures selected for inclusion in the baseline, nor that referring to the selected model order have been studied.
Thus the aim of the present study is exactly to address these issues and provide a comprehensive assessment of one of the methods -the MM based AutoRegressive (AR) model parameter method. The laboratory-scale composite beams subject to delamination and other types of damage induced by impact, used in the previous studies [3,4], are presently employed as well, along with a high number of test/inspection cases. This is made possible by "rotating" the baseline beams within the set of all healthy beams. Moreover, the sensitivity of the method with respect to the number and specific structures selected for inclusion in the baseline phase, as well as with respect to model order employed, are studied in detail.
The rest of this article is organized as follows: The experimental set-up and the damage scenarios are briefly reviewed in section 2, the MM based AutoRegressive (AR) model parameter method is elaborated upon in section 3, the assessment of the method is performed in section 4, and the conclusions are summarized in section 5.

The experimental set-up and the damage scenarios
The study is based on 26 nominally identical, composite beams consisting of several layers of woven and unidirectional (UD) fabric. They have square hollow cross section, uniform along their length with dimensions 600x65x65 mm (LxWxH), wall thickness of 3mm and corner radius of 8mm (figure 1). Each beam represents the topology of the main part of a commercial UAV boom. In the experimental setup each beam is tightly clamped at one end, simulating its connection to the fuselage, while its free end is attached to an aluminum mass representing the aircraft tail (figure 1). The beam is excited at its free end with a random, approximately white, Gaussian force applied vertically at Point X via an electromechanical shaker, while vibration acceleration response signals are measured at point Y1 via a lightweight accelerometer. Three damage scenarios (table 1) that may include a combination of delamination, small cracks and broken fibres are induced in 3 of the 26 beams at point D (figure 1) by using a pendulum type impact hammer with distinct energy level for each damage. Welch-based FRF magnitude estimates [5, pp 329, 338] for the healthy and damaged beams, revealing significant variability in the dynamics  figure 2. For a detailed presentation of the experimental set-up, the employed equipment and the damage scenarios the reader is referred to [3].

The Multiple Model (MM) based AR parameter method
The method is of the response-only type, based on the parametric AR modelling [6] of the random vibration response signal(s) -the excitation is not required. A scalar AR model is of the form: with [ ] designating the random vibration response signal, the model order, the -th AR parameter, and [ ] the model residual that is a white Gaussian zero-mean with variance 2 sequence. stands for Normally Independently Distributed. The employed characteristic quantity is the AR model parameter vector ( = ) [7].
The main idea behind the method is that AR models corresponding to several healthy structures, instead of just one (as in conventional methods), are used to effectively represent the healthy structural dynamics while taking variability into account. As a vibration-based method [7][8][9], the MM based AR parameter method is applied in two phases: (a) The baseline (training) phase, and (b) the inspection (damage detection) phase.
Baseline phase. Inspection phase. In this phase vibration response signal(s) are obtained from the current structure under inspection (which is in unknown state). The objective is to decide whether the current structure's dynamics are represented by the set of healthy AR models obtained in the baseline phase, in which case the current structure is declared as healthy -if this is not the case, then the structure is declared as damaged.
In doing so, let [ ] designate the current response signal(s) and the respective AR model (the subscript u is used to designate unknown state). Then a proper distance metric in the method's characteristic quantity space (AR model parameters) is employed, and the distances between and each one of the ,1 , … , , used as baseline are obtained.
The KL divergence [10] is the distance employed, which for , , , each followingdimensional Gaussian distribution with means u , o and covariance matrices , , , respectively, is defined as (capital/lower case bold face symbols represent matrix/vector quantities): with tr( •) designating trace and det(•) determinant of the indicated quantity.
Once the distances between the current AR model and those of the baseline are available, damage detection is based on the following decision making mechanism that takes all distances to the baseline models into account through an overall distance metric : Healthy structure (2) Otherwise  Damaged structure with lim designating a user defined (based on the baseline signals) threshold.

Assessment of the MM based AR parameter method
The assessment of the MM based AR parametric method is based on damage detection tests with an increasing number of beams used in the baseline phase ( = {1,4,8,12,16}). Note = 1 designates the corresponding conventional method. The testing procedure for each number of beams in the baseline phase is based on the following algorithm: Step 1: All possible combinations among the 23 healthy beams in sets of beams are considered.
Step 2: If the total number of combinations is very large (for = {4,8,12,16}), a random subset of combinations is employed, so that each one of the 23 healthy beams is used in at least one combination. Otherwise, all possible combinations are employed (these are referred to as the baseline combinations).
Step 3: The first baseline combination and one experiment per beam (baseline experiments) are used in the baseline phase of the method. At present the remaining experiments with the baseline beams are not subsequently used.
Step 4: The remaining healthy beams (23 − ) are used in the inspection phase, and along with the damaged beams, formulate the set of inspection beams. Damage detection is performed for all the experiments associated with the inspection beams (inspection experiments).
Step 5: Steps 3 and 4 are repeated until all baseline combinations are exhausted. At the end, a total number of damage detection tests for all the inspection experiments and all the baseline combinations is obtained, forming the set of inspection cases.
This algorithm is a modified cross validation eliminating the effects of baseline beam selection, plus leading to a drastically higher number of inspection cases to assess the method with. For each

Baseline phase
The estimation of the parametric AR models is based on = 10 000 sample long (≈ 2.15s) data records using Ordinary Least Squares with QR implementation [11, pp 318-320]. The model order selection is based on the Bayesian Information Criterion (BIC) and the Residual Sum of Squares / Signal Sum of Squares (RSS/SSS) [11, pp 498-514]. The selected model order for all models is presented in table 3, along with a summary of estimation details (the performance characteristics correspond to an individual baseline model).

Inspection phase
Damage detection results for = {1,12} and the corresponding number of inspection cases (table 2) are presented in figure 3. The distance metric D from the current model to the baseline model set is provided per inspection case. As depicted, for = 1 (conventional approach) several distance metrics related to damaged beams are lower than those related to their healthy counterparts, thus rendering damage detection ineffective. Improved results are obtained by the MM based method as is increased to 12.  Despite the significant improvement, the method is still not effective as shown in figure 4, which is reasonable as the fundamental difficulty stemming from the high variability in the healthy beam dynamics cannot be fully overcome. Then a sensitivity analysis is performed based first on the AR model order and secondly on the selection of specific baseline beam combinations. Thus, tests with 5 different AR orders ( = {27,37,47,57,67}) and 12 baseline beams are made, examining whether the selection of a different model order affects detection performance. The results of these tests are presented in figure 5 in terms of ROC curves, depicting limited sensitivity of the MM based method.  Subsequently, several baseline beam combinations are tested in order to examine whether some specific combinations may lead to effective detection. Indicative results for 50 properly selected combinations with = 12, are presented in figure 6 in terms of the ROC curve. It is evident that with proper selection of the baseline beams the method's performance improves significantly; in fact reaching excellence.

Concluding remarks
A detailed assessment of a recently introduced Multiple Model (MM) based AutoRegressive (AR) model parameter method for Structural Health Monitoring of a population of like structures using a single, common, threshold in the decision making mechanism has been presented. The assessment has been based on high numbers of experimental test/inspection cases using composite beams subject to damage via impact. Comparisons with the corresponding conventional (single model based) method have been also made. The results achieved demonstrate the MM method's significant improvement over that of the corresponding conventional method, but also that it is not possible to generally (that is in all cases, irrespectively of the beams selected as baseline) and fully overcome the fundamental difficulty associated with significant variations in the healthy population dynamics. A sensitivity analysis has additionally indicated that the method is relatively insensitive to the selected model order, but sensitive to the specific beams selected as baseline (training) ones, with proper selection potentially leading to excellent results.