2.2 Recent trends in damage detection
2.2.1 PCA and structural damage detection
In recent years, PCA has been extensively applied to measure structural dynamic response signals with the purpose of dimensionality reduction studies, to account for changes due to environment and structural damage, for sensor validation, and so on [84–90]. In addition to identifying the presence of damage in a system, PCA can also be used to estimate the severity of the damage through the acquired vibration data from numerical simulations and/or experimental trials. PCA builds a baseline model from the output response obtained from the pristine state of the structure and compares it with the newer acquired set of data in order to identify the presence of damage through certain DIs such as T2, Q, I2 and φ; instances of which, are replete in literature [93].
PCA, can be defined as the orthogonal projection of the data onto a lower dimensional space, known as the principal subspace, such that the variance of the projected data is maximized, and can be applied for dimensionality reduction, feature extraction and data visualization. On reducing the complex data set to a lower dimension, PCA reveals some simplified structures relevant to the data set which can be extracted using EVD on the sample covariance matrix. It is closely related to Karhunen-Lo`eve transform or proper orthogonal decomposition (POD) [111–122]. The basic idea of KL transform is to decompose a general second-order random process using an orthogonal expansion of uncorrelated random variables. Although classically many authors [93] have attempted to explain the connection using an approach that entails solution of integral equations, it is perhaps more instructive to understand PCA in the light of decorrelation and the pathway to achieving it by dint of eigenvalue decomposition of covariance matrix [121, 122].
Towards that, let a multivariate data X be defined such that the individual univariate data are correlated. PCA seeks an orthogonal transformation of the form: T=PX, where T represents the space of transformed variables that are decorrelated. So, PCA can be thought of as a constrained optimization problem [93] where the objective is to diagonalize TTT subject to the constraint PPT =I. Mathematically, it translates to defining an objective function O as under:
O = TTT −Λ(PPT −I)
= PTXTXP−Λ(PPT −I) (2.39)
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The optimal transformation is obtained by setting the partial derivative of O tozero as follows:
∂O
∂PT = 0 =⇒ (XTX)P= Λ(P) (2.40)
Thus, from Eqn. 2.40, it is clear that the optimum orthogonal decorrelating transformation is achieved by the eigen-decomposition of the covariance matrix of the correlated data X.
This decomposition produces the eigen value matrices, expressible as time series, known as the principal orthogonal values (POVs) and eigen vector matrices, also known asproper orthogonal matrices (POMs) [118,119,122]. Eigen values/POVs describe the relative significance of each POM in the response as a whole. The uncorrelated new set of variables produced by the linear combinations of the original variables are known as the principal orthogonal components (POCs) [118, 121]. The new set of variables have an improved potential to detect deviations (such as structural damage) in the system as compared to the original set of variables and hence, finds its way in the field of structural damage detection [84, 85].
Damage detection indices based on PCA
The transformed response obtained after PCA does not provide sufficient information regarding the localization of damage and its severity. As these projections are sometimes not enough, it is necessary to use certain statistical parameters that are considered as DIs. In this context, few DIs and their utility are described next.
Q-statistic (SPE index): This damage index is based on analyzing the residual data matrix to represent the variability of the data projection within the residual subspace. Considering theith row of the matrix E, the Q-statistic for each experimental observation is defined as:
Qi =eieTi =xi(I−PPT)xTi (2.41)
Hotelling’s T2 -statistic (D index): This index is based on the analysis of the score matrix T to check the variability of the projected data in the new space of the PCs. The concept of Eucledian distance comes into effect by utilizing the covariance matrixCx as the normalization factor. For the TH-1989_156104031
ith sample, the DI is expressed according to:
T2i =
r
P
j=1 t2sij
λj =tsiΛ−1tTsi
=xiPΛ−1PTxTi
(2.42)
In the above equation, tsi is the ith row vector of the matrix T, defined as the projection of the experiment xi onto the new space, related by the expression, tsi=xiP
Combined index (φ index): This DI is essentially a blend of the Q index and the T2 index that provided alternatives for merging information into a single value. The mathematical expression defining this DI is given as:
φ index=Q index+T2 index
=xTMφx
=xT(I−PPT +PΛ−1PT)x
(2.43)
I index: This DI has its very roots in its utilization for clinical studies and mainly used for meta-analysis, accounting towards a percentage of heterogeneity. The I index provides variation in study outcomes between experimental trials and can be mathematically defined as:
MI =
Q−(k−1)
Q ×100% f or Q >(k−1)
0 otherwise
(2.44)
where k is the number of experimental trials.
PCA and its relation to structural dynamics
Other than being a purely data driven decorrelating orthogonal transform, PCA can also be in- terpreted in the framework of structural dynamics. The objective of PCA is to find a orthogonal transformation of the form T = PX (where size of T is m×n) that completely diagonalizes the covariance matrix of the data setX.P (also known as POMs) has the orthonormal basis of vectors as rows. Since the structural modes are orthonormal to each other with respect to mass matrix, PCA can be utilized to extract the linear normal modes (LNMs) of vibrating systems which are TH-1989_156104031
closely related to the POMs (i.e. the eigenvectors) of PCA [112–114]. To understand the relation- ship, consider an undamped MDOF system with mass and stiffness matrices M and K undergoing free vibration with x as the displacement vector. The dynamics of the system can be expressed as:
M ¨X+KX= 0 (2.45)
The modal responses are related to the physical responses (as individual functions of t) according to the following expression: X =VQ, where X is a matrix of physical responses arranged column wise, of size M ×N, such that X = [X1,X2,X3, ...,XM]T, M being the number of degrees of freedom (DOF) of the structure and N is the number of sampling points. In this context, V is the eigen vector (mode shape) matrix and the modal ensemble matrix is denoted byQ which comprises of the numerous vectors of the modal coordinates.
The covariance matrix of the modal responses can be identified as: RQ = N1QQT. For undamped free vibration and also for mildly damped system, the covariance matrix of the modal response RQ approximate to a diagonal matrix as the samplesize of the responses increase [111, 119, 121, 122]. For forced vibration under broadband excitation, the evolution of the physical response,xi(t) and modal response qi(t) in time domain can be expressed in terms of the impulse response function h(t) as given by the following expressions: xi(t) =
∞
R
0
h(t−τ)Fi(τ)dτ and qi(t) =
∞
R
0
h(t−τ)fi(τ)dτ, where Fi(τ), and fi(τ) represent the actual and modal forces, related by the equation fi(τ) = ϕTi Fi(τ), where ϕ is the mode shape corresponding to the mode. The individual elements of the covariance matrix RQ can be expressed as:
rijQ= 1 N
N
X
k=1
qi(tk)qj(tk) =
∞
Z
τ=0
∞
Z
θ=0
fi(τ)fj(θ)
"
1 N
X
i,j
hi(t−τ)hj(t−θ)
#
dτ dθ (2.46)
For finitely large N,RQcan be expected to be a diagonal matrix for undamped system and nearly diagonal for light to moderate modal damping. To understand the behavior of rQij in the context of POD, under moderate modal damping and finite sample size, it is instructive to understand the behavior of the term [N1 P
i,j
hi(t−τ)hj(t−θ)] as N → ∞, which can be achieved by expressing the
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POCs (ψ) as a sum of the true linear modal coordinates (qi) and error terms (ε) according to:
ψi(t) =qi(t) +εiT
x(t) (2.47)
To observe the characteristics of rijQ, consider the covariance matrix of the POCs as: RΨ = N1ΨΨT. Each individual element of the covariance matrix formed by the POCs can be obtained from the following steps. Substituting from Eqn. 2.47,
rijψ = 1 N
N
X
k=1
ψi(tk)ψj(tk)
= 1 N
N
X
k=1
[qi(tk)qj(tk) +εTi x(tk)qj(tk) +qi(tk)εTj(tk) +εTi εTjx(tk)2]
(2.48)
Ignoring the second order error terms (i.e., ε2 terms) Eqn. 4.4 can be written as:
rψij = 1 N
N
X
k=1
[qi(tk)qj(tk) +εiT
ϕiqi(tk)qj(tk) +qi(tk)εjT
ϕjqj(tk)]
=
∞
Z
τ=0
∞
Z
θ=0
fi(τ)fj(θ)
"
1 N
X
i,j
hi(t−τ)hj(t−θ)
#
dτ dθ+
∞
Z
τ=0
∞
Z
θ=0
εiTϕifi(τ)fj(θ)
"
1 N
X
i,j
hi(t−τ)hj(t−θ)
#
dτ dθ+
∞
Z
τ=0
∞
Z
θ=0
εjTϕjfi(τ)fj(θ)
"
1 N
X
i,j
hi(t−τ)hj(t−θ)
# dτ dθ
(2.49)
It is clear from Eqn. 2.49, that for practical systems under moderate modal damping and finite sample size, the POC’s provide a good approximation to the true linear modal component which deviate from each other as damping increases. However, the matrix RΨ is still expected to show a diagonally dominant behavior in the limitN → ∞and when the errors are low (i.e., low to moderate damping) [121, 122]. It can now be well understood that PCA decorrelates the dataset obtained from the inputs of multiple sensors. For practical cases, where the system dimension is large, it is not possible to instrument each DOF. Moreover, PCA does not fare well with the nonlinearities associated with datasets. In this context, the shortcomings of the PCA based methods are discussed TH-1989_156104031
next in detail.
Drawbacks of the PCA based approaches
It has been observed that the the PCA model, once built from the data, is time invariant, while most real life processes are time-varying in nature. When a time invariant PCA model is used to monitor processes with the aforementioned normal changes, false alarms often result, which significantly compromise the reliability of the monitoring system [88]. As PCA is a baseline reliant approach that requires data to be processed in batches or windows, it cannot be implemented towards online detection problems. The aim of the present research is to address the shortcomings of PCA in a systematic manner, by proposing certain essential modifications in its formulations and expand the range of applicability of PCA based approaches towards addressing system-specific problems. The key shortcomings of PCA can be summarized as under:
1. Traditional PCA based methods analyze data in a batch mode, offline. As PCA is a baseline reliant approach, the analyzed data are windowed so as to compare it with reference over certain intervals of time. Practical engineering applications require the choice of window parameters such as window size, shift, overlap, etc. to be discrete and problem specific. This issue is addressed through the modifications proposed in the formulation of basic PCA to tailor it towards an online implementation, utilizing the FOEP techniques to develop a novel real time damage detection algorithm, RPCA.
2. A key shortcoming of PCA appears while dealing with problems involving nonlinearities in the datasets. For applying the concept of PCA for incorporating non-linear dependence between variables, a variation of PCA, known as kernel PCA (KPCA) [89] is sometimes utilized to account for the presence of nonlinearities in the data in the context of damage detection.
However, KPCA based approaches relying on the concept of subspace angle is computationally expensive. Moreover, it has been observed that changes in the subspace angle sometimes might appear due to the presence of outliers, which might provide false detections and thereby compromise the reliability of the detection process.
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3. The central idea of any PCA based method is to perform EVD on the sample covariance estimate to obtain the eigenvalues and eigenvectors which constitute the eigenspace for the dataset. Physically, for a real time framework, performing EVD at each instant is cumbersome.
To alleviate this issue, the concepts of RPCA approach provides iterative updates at each instant of time by utilizing the eigenspace and not the covariance estimate as a whole. The details of the RPCA based approach can be found in chapter 3 of this dissertation.
4. The current form of basic PCA is such that it utilizes the streaming vibration data input from multiple sensors. This sometimes become an impediment due to the unavailability of large number of sensors, improper accessibility and cost and other factors. This issue can be addressed by developing a real time damage detection strategy that takes the input from only a single sensor and provides accurate estimation results. Established using the very basics of EVD (as in PCA), the new detection strategy RSSA, towards identifying damage, uses the input from a single sensor only; the details of which are thoroughly discussed in chapter 4.