is successfully addressed by the hybrid FOEP based RPCA-RSSA approach, which remains one of the major entitlements of this dissertation. Despite the amalgamation of the recursive approaches in the hybrid algorithm, the time complexity for providing eigenspace updates in real time is still a bare minimum, which is reported in the later stages of this chapter. The scope of the present work assesses the detection prowess of the proposed FOEP based real time SHM methods in general and a hybrid method in particular, to address simultaneous temporal and spatial damage of both linear and nonlinear systems in real time. Case studies aimed at detecting real time damage for weakly to strongly nonlinear family of structural systems have been explored in detail. The present work deals with damage detection on numerically simulated systems, complemented with experimental test beds involving the previously described setups and newer trials using vibro-impact systems, from which important conclusions are drawn. Both the methods are successfully applied on real life cases on the vibration data obtained from UCLAFB that concludes the chapter with an extensive comparison of the performance of the real time detection methods developed so far.
time consuming, thereby impeding its practical applicability towards real time damage detection.
This can be alleviated through the use of FOEP approach [134] which provides recursive updates of eigen subspace from the previous eigen-space of the data at a particular time instant. Although this exercise follows the similar notion of the previously conducted theoretical developments on RPCA and RPCA-TVAR based approaches, the subtle differences, in particular, that arise here, are described in detail.
For structural systems dealing with data evolving from zero mean processes, the recursive es- timation of the Hankel covariance matrix (Ck) at any instant k can be structured in terms of the current sample vector (Xk) and the previous covariance estimates (Ck−1) as follows :
Ck = k−1
k Ck−1+ 1
kXkXkT (5.1)
However, for nonstationary data sets, the recursive mean at each instant of time needs to be ac- counted for in the estimation of the covariance matrix. The recursive mean atkthinstant,µk, depends on the mean at the previous instant through the relation: µk = k−1k µk−1+1kXk. The covariance es- timate for cases involving mean shift can be expressed as: C˜k = k−1k Ck−1+1k[Xk−µk] [Xk−µk]T. In SHM problems, the dynamics of the system evolve predominantly from zero mean processes.
Damage detection formulations premised on recursive correction updates of the covariance matrix generally assume to ignore the local non zero means at each time instant. Case studies from pre- viously published works as well as the current work reveal that the aforesaid assumption does not hinder the practical applicability of the online damage detection algorithms. The incorporation of C˜ in the formulation of the covariance estimate is therefore, inconsequential. Henceforth, for the remainder of the chapter, ˜C has been dropped for subsequent formulation and analysis.
The current L dimensional sample vector (Xk = [xk−L+1, xk−L+2, xk−L+3, ...xk]T) comprises of L lagged elements of the one dimensional time series, L being the initial signal length. The objective of pre-selecting a desired initial signal length is to preserve the components with most of the damage information (by selecting components with higher singular values to retain the trend and oscillatory parts) and to eliminate the components of less significance such as noise. The covariance estimate at kthinstant can be written in terms of eigen value and eigen vector matrix at a particular time instant TH-1989_156104031
as Ck = UkΥkUkT with $k = UTk−1Xk, as the projection of the sample vector into the previous eigen subspace. Substituting these expressions in Eqn. 5.1, the following equation is obtained:
Uk(kΥk)UTk = (k−1)Uk−1Υk−1UTk−1+$k$TkUk−1UTk−1
=Uk−1{(k−1)Υk−1+$k$kT}UTk−1
(5.2)
In the above expression, with a finitely large samplesize (k) and low damping estimates [111], the term ((k−1)Υk−1+$k$Tk) generally shows a diagonally dominant behavior. The diagonal dominant structure of this term ensures the application of Gershgorin’s theorem [132, 134], rendering in the application of FOEP approach admissible to obtain the eigen values and eigen vectors. Interested readers could refer [132–136] for details. The application of Gershgorin’s theorem ensures the EVD of the term to be of the form PkΓkPTk, where Pis orthonormal and Γ is diagonal. Substituting the EVD in place of ((k−1)Υk−1+$k$Tk) in Eqn. 5.2, a new formulation is obtained as shown below:
Uk(kΥk)UTk = (Uk−1Pk)Γk(PTkUTk−1) (5.3)
From equation (5.3), recursive updates of the eigen subspace are obtained as a function of the previous eigen space as:
Uk=Uk−1Pk
Υk =Γk/k
(5.4)
Eqn. 5.4 provides an iterative relation between eigen spaces at consecutive time instants. On using FOEP approach, the recursive eigen vectors obtained at each time instants are not ordered in the same sequence as the previous time instant, thus presenting the problem of permutation ambiguity.
This can be resolved by arranging the obtained eigen vectors according to the decreasing order of the corresponding eigen values in Γk. For the working of RSSA algorithm, this is one of the key steps as eigen values of the covariance Hankel matrix need to be ordered before proceeding with the evaluation of PCs and reconstruction of Hankel matrix using them. The contribution factor for a particular ith eigen vector Ui is given by nβ2i
P
i=1
β2i
, where βi2 is the eigen value corresponding to Ui. After obtaining the eigen space updates at a particular time instant from the previous eigen space and current sample vector, principal component values of the time series at a particular time series TH-1989_156104031
can be extracted as below:
ψi(k) = (UiT(k))1×L(Xk)L×1 (5.5) Using the above equation, the ith principal component value at particular time instant is obtained.
Depending upon the relative contribution of the eigen vectors, the number of PCs required for reconstruction can be automated. For the present work, at any instantk, the PCs of eigenvectors that explain more than 90% of the system’s variance is used for reconstruction. In the reconstruction step, the ith PCs are projected back into its original subspace to obtain last column of the corresponding elementary matrix.
Ri(k) = (Ui(k))L×1×(ψi(k))1×1 (5.6) It is to be noted here that the last entry of the Ri(k) vector, denoted here asri(k) is used to obtain the value of the reconstructed time series entry at the kth instant of time as shown below:
x(k) = 1 n
n
X
i=1
ri(k) (5.7)
where n denotes the number of eigen values explaining more than 90% of the variance. The re- constructed signal values as per equation (5.7) is recursively obtained at each instant of time as and when the vibration data streams in. The importance of this reconstructed signal is that it has simpler components, making it amenable towards a lower AR order model, deemed sufficient [51]
to capture the dynamics of the structure and more sensitive towards singular events like damage.
TVAR modeling is subsequently performed on this reconstituted signal.