terms as follows:
Ωk−1 = [Qk−1T
Qk−1+Qk−1ΓTk−1+Qk−1T
Γk−1+Γk−1ΓTk−1] (3.12)
From equation 3.12, it is clear that Ωk−1 can be understood as a sum of QTQ and the first order error terms [88]. AsN → ∞, QTQ is approximately diagonal for systems having mild to moderate damping under sufficiently broadband excitations. Eqn. 2.49 forms the basis of establishing diag- onal dominance of the Ωk−1 matrix. Gershgorin’s theorem can now be applied on the diagonally dominant matrix which provides recursive eigen space updates using perturbation techniques at each point in time. For dynamical systems of different order (such as chemical systems), [88] the above equations would not hold true and the concept of diagonal dominance has to be enforced upon, for the application of Gershgorin’s theorem. Hence for a structural system, the recursive eigen space update is obtained using a first order perturbation (FOEP) approach [133, 135], which provides a less computationally intensive algorithm in a recursive framework for the eigen value decomposition (TkΛkTTk) of the term (k−1)Ωk−1+βkβkT, yielding the following iterative update equations:
Wk =Wk−1Tk
Ωk= Λkk
(3.13)
Eqn. 5.14 provides an iterative relation between eigen spaces at consecutive time instants. On using the FOEP approach, the recursive eigen vectors obtained at each time instant are not ordered in the same sequence as the previous time instant, thus presenting the problem of permutation ambiguity [68]. This can be resolved by arranging the basis vectors according to the decreasing order of the corresponding eigenvalues in Ωk.
referred to as condition indicators that can detect damage online. Several damage detection and SHM techniques have been proposed in the literature [1–3] that involve use of specific CIs whose changes signify damage to the system. The difficulty in choosing and utilizing condition indicators for the present framework arises from the fact that most of the traditional CIs are not amenable to online implementation. In this research, a set of recursive CIs are developed that are responsive towards online implementation. These indicators are based on the change in the pattern of the eigenspace due to damage which is manifested through alteration of recursively updated eigen-vectors before and after damage. In the following section, temporal recursive residual errors (χRR), spatial recursive residual errors (εRR), eigen-vector change (EVC) and scatter plots have been presented.
3.4.1 Recursive residual error (RRE)
The key condition indicator chosen for the present framework is recursive residual error (RRE). It will be shown later that RRE is the most robust and least outlier prone of all the discussed CIs.
RRE comes in two flavors: temporal or global RRE (χRR) and spatial or local RRE (εRR). The main motivation for this CI is derived from the use of residual error as a criterion in quantification of nonlinear behavior [94] which presents its use as a measure of distortion of subspaces for a nonlinear system with increase in levels of excitation. The RRE as utilized in the present work is derived as follows.
LetWk be a matrix of eigenvectors computed atkth time instant andΛk be the corresponding diagonal matrix of eigenvalues at that instant. The eigenvalues are structured in descending order of magnitude, and the corresponding eigenvectors are re-arranged. Let, Wk= [W1kW2k] such that, W1k is said to represent the least number of eigenvectors whose corresponding eigen values explain more than 90% of the variance. Considering a damage at the end of (k−1)th instant, the subspace spanned by the updated eigenvector W1k deviates in comparison to the subspace spanned by eigen vectors at the previous time stamp Wk−11 . Apart from the instances of damage, or, the initial few seconds, it can be safely assumed that W1k ∼= W1k−1, as there is no significant deviation in the eigenspace otherwise. Based on this assumption, the RREs due to projection of the response at a
TH-1989_156104031
particular time instant k ontoW1k−1 is evaluated as:
X∗(k) = W1k−1∗Wk1T ∗X(k) (3.14)
Based on the above concepts, for detecting the instant of damage, the RREs proposed here are χRR−1 and χRR−2 which are expressed by the following equations:
χRR−1 =
X∗(k)−W1kT ∗X(k)
2
χRR−2 =kX∗(k)−X(k)k2
(3.15)
From the Eqn. 3.15, χRR−1 can be interpreted as the distance metric between the transformed response and its projection on the subspace at the previous time stamp. Similarly,χRR−2 provides a measure of the difference between projections of the transformed response and the original response data.
3.4.2 Recursive eigen vector change
From the precursory discussions, it is certain that there occurs a significant distortion in the eigenspace characteristics at the onset of damage to a system. Hence, tracking the change in the eigenvector updates are expected to show deviations that might correspond to the instant of damage.
Pivoted around this key concept, a new CI, eigenvector change (EVC) is proposed in the context of real time damage detection of structures that tracks the eigenspace updates at each instant of time and provides the accurate damage instant when a significant distortion in the eigenspace is encountered. In the due course of rigorous inspection over a varied range of sample numerical sim- ulations and experimental verifications, it was later found that this CI did not provide successful representation of the damage instant for quite a number of instances. Unlike the RREs, EVC is highly susceptible to false detections especially when a change in eigenvector occurs even due toam- bient noise or some other disturbances from the excitation data, that sometimes become inevitable during experimentation. It was thus kept in consideration for a certain number of cases where it
TH-1989_156104031
actually showed some definite results for the damage instant.
EV C = q
diag([W1k−W1k−1]∗[Wk1−W1k−1]T) (3.16)
3.4.3 Outlier detection using correlation coefficient (ρ)
Outliers can be defined as an observation (or subset of observations) which appears to be inconsistent with the remainder set of data. They can arise due to mechanical faults, changes in system behavior, human error or instrument error. In the present context, although RRE is an efficient CI for online damage detection, it is susceptible to mild levels of false detections. This can be resolved by the use of scatter plots in conjunction with the RREs as shown in Fig. 3.13. Scatter plots between two transformed responses and the corresponding correlation coefficients (ρ) are calculated in the neighborhood of damage (small windows before and after the instant of damage). It is observed that there is considerable directional change in the scatter plots (or a change in the sign of ρ) only at the instant of damage, which serves as a viable indicator for removal of false detection in the algorithm.
It is essential to note that there is no change in the sign of the correlation coefficient due to the presence of outliers. The change in the sign of ρ takes place at the window where the damage has actually occurred and has been aptly illustrated in the following sections.
3.4.4 Local damage detection
The aspect of addressing damage instant as well as locating the damage in a single framework is more involved, especially if attempted in a recursive construct for online damage detection. In the present research, the proposed RPCA-RRE algorithm provides information on the damage location alongside the instant of damage in the same framework with a reasonable degree of accuracy. The key entitlement incorporated in this work is the simultaneous real time damage detection of structures in a single framework, that has previously not been reported in the contextual literature. To address damage localization, certain formulations in the RRE needs to be modified. First, RREs need to be expressed for each DOF (εRR) which varies from the temporal RREs (χRR) that were originally defined for the system as a whole. The key assumption is that when a column of an MDOF structure TH-1989_156104031
is damaged, its effect will be pronounced in the neighboring DOFs, which in turn is manifested as a change in the local RREs for that particular DOF. While finding out the instant of damage (Eqn.
3.15), the concept of spatiality is lost as only global RREs are defined. This motivates the need for a new set of RREs (local or spatial RREs (εRR)) that preserves the individual contribution of the responses, utilized to find the exact location of damage. From Eqn. 3.14 it is clear that the projec- tions of the transformed responsesX∗(k) and the actual responsesX(k) can be expressed as vectors with each individual element corresponding to a degree of freedom (m : total number of DOFs) ac- cording to : X∗(k) = [x∗1(k), x∗2(k), . . . , x∗m(k)]T and X(k) = [x1(k), x2(k), . . . , xm(k)]T where k represents the time instant at which RRE is estimated. Consider each element of X∗(k) and X(k) be represented as xi∗(k) and xi(k) respectively, where (i) corresponds to a particular DOF. This yields a time series of RRE (labeled asεRR−Yi) corresponding to the response for each DOF, which can be expressed as
εRR−Yi =
x∗i2(k)−x2i(k)
(3.17)
Another useful quantity in this connection is the average RRE for ith response for K data points, which can be estimated according to,
hεRR−Yii=
K
P
k=1
x∗i2(k)−x2i(k)
K (3.18)
The local RRE averaged over K samples is shown above. This is particularly useful, as shown later, in quantifying the percentage change in RRE (an indirect measure of loss of stiffness) pre and post damage.