Robust linear and nonlinear structural damage detection using recursive canonical correlation analysis

Mechanical Systems and Signal Processing 136 (2020) 106499

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Robust linear and nonlinear structural damage detection using recursive canonical correlation analysis B. Bhowmik a, T. Tripura b, B. Hazra b,⇑, V. Pakrashi a a Dynamical Systems and Risk Laboratory and School of Mechanical and Materials Engineering; Science Foundation Ireland Marine and Renewable Energy research, development and Innovation (MaREI) Centre, University College Dublin, Ireland b Department of Civil Engineering, Indian Institute of Technology, Guwahati, Assam, India

a r t i c l e

i n f o

Article history: Received 8 July 2018 Received in revised form 1 October 2019 Accepted 31 October 2019

Keywords: Recursive canonical correlation analysis (RCCA) Time-varying autoregressive modeling (TVAR) Damage sensitive features (DSF) First order eigen perturbation (FOEP) Damage detection

a b s t r a c t A novel approach for robust damage detection of linear and nonlinear systems using recursive canonical correlation analysis (RCCA) is proposed in this paper. The method generates proper orthogonal modes (POMs) at each time stamp that are recursively tracked to provide iterative eigenspace updates, using first order eigen perturbation (FOEP) method. The transformed responses obtained are subsequently fit using time varying auto regressive (TVAR) models, in order to aid as damage sensitive features (DSFs) for identifying temporal and spatial patterns of damage. As most of the past work utilize offline algorithms that gather data in windows and operate in batch mode, recursive algorithms that provide iterative eigenspace updates at each instant of time, are missing in the context of structural health monitoring (SHM). This greatly motivates the development of the present work, where the both uni and multi directional structural responses are considered to be available in real time. The transformed responses are obtained by using the proposed algorithm that utilizes FOEP for multi-directional block covariance structure to solve the recursive generalized eigenvalue problem. The TVAR coefficients modeled on the transformed responses obtained in real time using FOEP, and residual error examined in a recursive framework, act as efficient real time DSFs for detecting damage online. Numerical simulations carried out on nonlinear systems and backed by experimental setups devised in controlled laboratory conditions, demonstrate the efficacy of the RCCA algorithm. Application of the proposed algorithm on the combined ambient and earthquake responses obtained from the UCLA Factor Building, demonstrates the robustness of the proposed methodology as an ideal candidate for real time SHM. Ó 2019 Elsevier Ltd. All rights reserved.

Abbreviations: CCA, canonical correlation analysis; PCA, principal component analysis; POC, principal orthogonal component; POM, proper orthogonal mode; EVD, eigen value decomposition; BSS, blind source separation; ICA, independent component analysis; RPCA, recursive principal component analysis; RCCA, recursive canonical correlation analysis; TVAR, time varying auto regressive; DSF, damage sensitive features; RRE, recursive residual error; DOF, degree of freedom; SHM, structural health monitoring; FOEP, first order eigen perturbation; AR, auto regressive; UCLA, University of California Los Angeles; UCLAFB, UCLA factor building. ⇑ Corresponding author. E-mail address: [email protected] (B. Hazra). https://doi.org/10.1016/j.ymssp.2019.106499 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

q

Canonical Correlation Block covariance matrix Covariance matrix using x- and y-directional data Displacement vector Eigen vector matrix ! Principal Orthogonal Coordinate Matrix c Error term Cxx ðkÞ Covariance matrix at kth time instant W Matrix of eigenvectors of covariance matrix e w POC estimate vector Hk Matrix of eigenvectors of the diagonally dominant term at kth instant Kk Diagonal matrix of eigenvalues of the diagonally dominant term at kth instant Time varying AR coefficients a1 ; a2 Pw Covariance matrix of process noise B; C; c Kalman state variables z; Q ; A; b bouc-wen parameters C Cxy X V

1. Introduction Vibration based structural damage involving condition assessment, fault diagnosis and prediction of remaining useful life of the structure, have motivated the development of numerous algorithms aimed at detecting structural damage [1–4]. Damage is defined as the change in the structural parameters, such as mass, stiffness and damping that subsequently alters the modal parameters of the system such as natural frequency, modeshapes and damping ratio. Despite significant advances in the area of structural health monitoring (SHM), the development and deployment of efficient algorithms at the sensor level face several issues such as improper planning, unavailability of adequate resources, thereby impeding a comprehensive real time implementation of SHM [1–3]. As traditional detection schemes are mostly offline, robust strategies that could identify and localize the damage simultaneously in real time, should be considered. Recent work [6,39–41] towards real time damage detection schemes developed around the concepts of first order eigen perturbation (FOEP) techniques provide information on iterative eigenspace updates at each time instant. The application of recursive algorithms address the facets of real time SHM that involves finer levels of detectability and are robust to practical implementations for structural systems with multidirectional dependencies, presence of nonlinearities, degrading events and torsional coupling. Additionally, an overarching framework that could potentially incorporate all the recursive eigen perturbation algorithms in a single conceptual framework is yet to be developed. Towards this, the paper proposes a robust damage detection framework using time varying auto regressive (TVAR) modeling in conjunction with recursive canonical correlation analysis (RCCA), to detect spatio-temporal structural damage in real time. Traditional damage detection strategies encompass a wide class of methods premised on the idea that alterations in the structural modal parameters (such as natural frequencies [8,9], modeshapes [10,11] and damping [1]) are indicative of damage. A significant portion of these methods are model based, i.e., the need for updating the model parameters in order to meticulously determine the damage is an essential requirement [5]. A key shortcoming of these approaches is the associated computational cost due to the parametric dependencies, which makes finer level of damage detection onerous. Response based methods, however, overcome the drawback of performing step-wise finite element (FE) modeling and detect the damaged state by relying on output-only vibration data obtained from the sensors (experimentally or otherwise) [14–16,25]. These methods mostly offer intermittent monitoring of structures to extract damage sensitive features (DSFs) from the recorded responses and utilize advanced signal processing techniques to identify the events corresponding to a damaged state of the structure. An efficient paradigm in this regard involves the use of statistical signal processing techniques that mostly encompass time-frequency analysis methods such as wavelet transform [20], empirical mode decomposition [17,21], blind source separation [20,22,26], to name a few. Although these non-parametric approaches are quite robust indicators of damage, the development of their real time counterparts still pose a major challenge. This motivates the need for developing real time robust damage detection schemes, that are efficient enough for practical SHM. Algorithms based on eigenvalue decomposition (EVD) of the covariance matrix of the measured physical response, such as principal component analysis (PCA) [18,19], have significantly contributed to a multitude of structural dynamics applications. However, PCA based implementations are mostly offline in nature, that impede their applicability in real time condition monitoring. Additionally, PCA based algorithms are not suited for applications involving multi-directional data, as the covariance matrix of the physical responses do not incorporate directional dependencies. In this regard, a powerful statistical technique developed in the recent years, canonical correlation analysis (CCA), utilizes the block covariance matrix of the physical responses in order to measure the underlying correlation between two sets of multidimensional variables [34–38]. The primary objective of CCA is to find projections such that the mutual correlation coefficient is maximized

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[34–36]. In addition to multivariate statistical analysis [35], statistical imaging [36], the underlying concepts of CCA are utilized in finding key patterns of time series [38] for both linear and nonlinear systems, and are also applied in meteorological studies and flood frequency estimation [38]. CCA takes into account the eigenspace characteristics, which could be globally tracked to ascertain the damage occurred to the system [39–41]. It can incorporate a mathematically general structure consistent with theoretical advancements in structural dynamics [31,33] and allows accommodating both damage detection and modal identification within a single framework. The implementation of CCA as a structural damage detection tool is largely missing, which greatly motivates the need of carrying out this work. Based on recent works [6,39–41], the implementation of CCA as a structural damage detection tool can be envisaged from the successful implementation of blind source separation (BSS) towards modal identification [20] and damage detection [22,26]. Techniques involving separation of modes, such as BSS and independent component analysis (ICA), emerge as potential candidates for modal identification algorithms [23–25]. As the family of BSS methods come under the category of CCA based schemes, a direct application of CCA towards modal identification and damage detection can be more efficient [7]. However, to date, BSS based algorithms using CCA have rarely been studied in the context of modal identification literature. Moreover, it is noteworthy that the majority of the aforementioned damage detection schemes are offline, requiring windowing of the data in order to compare the newer set of the response to the baseline values [22]. An indispensable requirement of any detection algorithm to identify fine levels of damage, lies in its ability to compare the recorded data at continuous intervals, from which structural damage can be inferred. Therefore, the algorithms must work online, an aspect that has rarely been reported in the context of damage detection literature involving the use of BSS algorithms [15,22]. As damage is a real time event, it further warrants that the detection algorithm must work online, which motivates the utilization of recursive canonical correlation analysis (RCCA), a direct extension of the traditional CCA approach using the concepts of rank one eigenspace updates through FOEP [6,39–41]. In line with the previous theoretical treatment of the topic [39,40], FOEP technique provides rank-one eigenvector updates at each instant, as and when the vibration data streams in real time [6,39–41]. The use of RCCA as an effective damage detection tool emanates from the conceptualization of the distortion of eigenspace, that can be modeled using TVAR and utilized to identify the spatio-temporal damage in a simultaneous online framework. The implementation of a recently established method, recursive principal component analysis (RPCA), is based on the principles of orthogonal transformation of the data covariance matrix arising from an uniaxial vibration [39,40]. The CCA method, however, generates a block covariance matrix utilizing multi-directional data, that arises due to biaxial orientation of structural systems, such as buildings, which potentially generates torsional coupling, due to asymmetry in plan. The presence of torsional modes renders the identification of the LNMs and/or POMs [29,30], difficult. It may be noted that the RPCA based methodologies are not equipped to perform for multi-directional data generated due to non-orthogonal orientation of a structure, as opposed to the CCA based methods, where the inherent formulation of the technique pivots around building block covariance matrices [39,40]. It can therefore be argued, that the RCCA method is expected to provide better detectability results for multi-directional dependencies and flexible structures with torsionally coupled systems as compared to the previously developed RPCA scheme [39,40], which motivates the present study. The current framework proposes TVAR modeling on the transformed responses obtained from the RCCA algorithm instead of the raw vibration responses, that allows for a relatively low model order time series for identifying damaged states of the system [6,41]. The major contributions of this work are as follows: First, a framework has been provided using RCCA as an vibration based structural damage detection tool that works in real time for continuously streaming data. Second, the use of TVAR modeling on the transformed responses provides damage estimates as fine as 10% in real time. This accounts for a finer level of detection than currently available in the literature, which remains of the order of 15% for nonlinear systems [6,41]. A finer level of detection in real time allows for a data-driven, extensive and informed condition based monitoring, which is important in the context of damage diagnosis and prognosis strategies. Third, identifying spatio-temporal patterns of damage in real time is difficult, especially when attempted in a recursive framework. This is addressed in the present work using both numerically simulated and practical case studies. Finally, the paper discusses application of the proposed algorithm for rank deficit scenarios where the number of instrumented locations is less than the actual degrees of freedom (DOF). A real life case study involving a multi-directional data from the recorded responses of UCLA Factor Building (UCLAFB) [20] is considered, where the change of state of the system from the ambient regime to a damaged state is accurately captured by the RCCA algorithm. In this regard, the theoretical development of the RCCA scheme is discussed first in detail, in the context of real time structural damage detection. The use of DSFs in order to identify the damage from the transformed response obtained from the RCCA module is discussed next, followed by the key detection results obtained from numerical simulations. A comparative study involving the proposed scheme against the recently established RPCA method is also provided to shed a light on the efficacy of the proposed scheme. Finally, the results are presented using experimental setups carried out in a lab environment to demonstrate the efficiency and robustness of the proposed algorithm in practical situations. The experimental results are also complemented through a full-scale study of the UCLA factor building under a combination of earthquake and ambient data.

2. Background CCA is an exploratory method for determining the relationship between two multivariate sets of vectors by extracting information from the cross-covariance matrices. Developed in the late 1930’s by Hoteling, CCA is a standard tool in statistical

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analysis that finds its application in econometrics, meteorology, medical studies and in the fields of signal processing [34,35]. This work utilizes the concepts of CCA as an effective real time damage detection tool through FOEP based formulation. To the best of the knowledge of the authors, the present work is one of the cardinal attempts at integrating CCA into a structural damage detection framework. The method finds two bases, one for each variable, that are optimal with respect to correlations and simultaneously obtains the corresponding correlations. It finds two basis vectors in which the correlation matrix between the variables is diagonal and mutually maximized by projecting the original set of variables onto an optimum subspace [34,35]. An important feature of the method is that the dimensionality of these new bases is equal to or less than the dimensionality of the individual variables. To utilize the concepts of CCA into a structural dynamics framework, consider two column vectors X ¼ ðx1 ; x2 ; . . . ; xn ÞT and Y ¼ ðy1 ; y2 ; . . . ; yn ÞT of random variables with finite moments. Projecting these vari^ x and Y ¼ Y T w ^ y need to be mutually maximized which is given ables onto the basis vectors, the linear combinations X ¼ X T w by:

h i ^ Tx XY T w ^ Ty E w E½XY  wTx Cxy wy q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i h i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2   2  ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wTx Cxx wx wTy Cyy wy  EX EY ^ Tx XX T w ^ Ty YY T w ^x E w ^y E w

ð1Þ

The covariance matrix of the random variables (centered to zero mean) can be expressed as:

 C¼

Cxx

Cxy

Cyx

Cyy

 ð2Þ

The covariance matrix is essentially a block matrix where Cxx and Cyy are the within-set covariance matrices of X and Y respectively and the relation Cxy ¼ CTyx holds true for the cross-set covariance matrices. The aim of this methodology is to obtain the maximized canonical correlations in the optimum subspace between X and Y, that can be easily obtained by solving the following sets of eigen decomposition equations: 1 2^ ^ C1 xx Cxy Cyy Cyx wx ¼ q wx

) ð3Þ

1 2^ ^ C1 yy Cyx Cxx Cxy wy ¼ q wy

^ x and w ^ y represent the where the eigenvalues q2 are the squared canonical correlations and the corresponding eigenvectors w normalized the canonical correlation basis vectors [34]. In practice, Eq. (3) can be recast into a single eigenvalue equation, given by:

^ ^ ¼ qw B1 Aw

ð4Þ

where the matrices involved in the eigenvalue decomposition (EVD) are obtained as:

 A¼

0

Cxy

Cyx

0

 ;

 B¼

Cxx

0

0

Cyy



^ ¼ and w

^x lx w ^y ly w

! ð5Þ

It is worthy noting that a slightly different structure of the covariance matrix will give rise to the traditional PCA based approach, a concept that has consistently exhibited its potential in identifying the spatial and temporal patterns of structural damage, over the recent years [18,19]. PCA, a special instance of CCA, can be formulated on substituting the matrices A and B as: A ¼ Cxx and B ¼ I. PCA is a dimensionality reduction approach that projects the data onto principal subspace, such that the variance of the projected data is maximized. In carrying out the projection, PCA reveals some simplified structures relevant to the dataset that could be obtained by EVD on the covariance matrix. As the evolution of PCA can be considered as a special case of a CCA based implementation, the data-driven nature of CCA can be related to the theoretical development of structural dynamics as well. To tailor the basic CCA into a recursive framework, it becomes essential to revisit certain key theoretical developments that center around covariance estimates and FOP techniques. The utility of CCA as an effective damage detection strategy aids in identifying spatial and temporal patterns of damage even for data gathered from multiple directions and for strongly nonlinear systems as well. In the backdrop of recursive implementation, the detailed formulation of FOEP is presented next in details. 3. First order eigen perturbation technique (FOEP) Recent developments in vibration based damage detection has witnessed the advent of numerous statistical signal processing techniques aimed at extracting key features of damage through the use of certain DSFs. While most of these methods function mostly in batch mode operations, the present research sheds light on to the development of a new class of damage detection strategies that identify the damage in real time, using the key principles of FOEP technique [27,28]. Although there are a host of other algorithms in the domain of adaptive filtering like Kalman filter [12,13] and recursive least squares, that are amenable to online damage detection, FOEP promises to provide a mathematically consistent framework to all the relevant family of algorithms that intend to exploit eigenstructure of the dynamical systems in real time to detect damage.

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FOEP expresses the eigenstructure of the ðk þ 1Þth step of the physical covariance estimate in terms of the eigenstructure at the kth step as the ðk þ 1Þth data streams in [6,39–41]. This is accomplished by expressing the EVD of the symmetric positive definite covariance matrix in terms of the rank one perturbation of eigenvalue and eigenvector matrices [6,39–41]. The scope of this paper lies within the application of FOEP techniques in the context of structural damage detection in real time, towards which recursive algorithms such as RPCA, RSSA and RCCA have been proposed [6,39–41]. From streaming data, the initial covariance estimate for a few samples (around 100 in number) is carried out. The FOEP technique transforms the batch mode operations into recursive implementations with an added advantage that the covariance estimates can be avoided by providing eigenspace updates evolving at each time stamp. In order to intuitively understand FOEP, consider the eigenvalues of a perturbed matrix C þ DC to be of the form K þ aaT , i.e., the rank-one update of the matrix K. Using the following definitions:

CV ¼ KV

ð6Þ

ðC þ DCÞðV þ DV Þ ¼ ðV þ DV ÞðK þ DK Þ

where, DV and DK are the perturbation matrices. The EVD of the diagonally dominant term, can be expanded as follows:

C þ DC ¼ VKVT þ VKDTV þ VDK VT þ DTV KVT þ O

 3



ð7Þ

Recognizing, that C ¼ VKV and invoking the fact that VV ¼ I, the EVD of the perturbed matrix DC by using Eq. (7) and ignoring second order perturbation terms can be written as: T

T

  DC ¼ aaT ¼ VKDTV þ VDK VT þ DTV KVT þ O 3

ð8Þ

The above expressions provide a central idea to the data driven nature of the FOEP approach. The detailed methodologies and theoretical derivations of the FOEP based formulation of CCA are presented in the upcoming sections. 4. RCCA: detailed derivation While CCA can be the basis of a structural damage detection tool, the major drawback of using CCA is that it analyzes data in batches, primarily rendering it an offline approach. In order to detect finer levels of damage, it is essential to estimate the eigenspace recursively, thereby providing iterative updates at each instant of time. The formulation of the basic CCA approach necessitates that the covariance matrix obtained from the set of recorded responses is essentially a block matrix, that needs to be updated for its online implementation. The major challenge towards tailoring basic CCA for an online implementation is to perform EVD on the block covariance matrix, which is particularly cumbersome and memory consuming. This can be alleviated through the use of the FOEP based approach which provides recursive updates of the eigenspace obtained from the previous eigenspace of the data at a particular instant [6,28,39–41]. An important aspect to be considered here is that the FOEP based recursive approach updates the eigenspace characteristics at each time instant, instead of updating the covariance matrix as a whole, thereby reducing the time complexity of the recursive implementation [6,39–41]. The theoretical development of RCCA is premised on the objective of finding a recursive update of the eigenspace characteristics at each instant of time, a central idea envisaged from FOEP improvisation. The individual lower dimensional matrices shown in Eq. (2) needs to be updated at each time stamp, providing recursive eigenspace estimates through the FOEP strategy. In this context, the response covariance matrix Cxx ðkÞ at any instant k can be expressed as a function of the covariance matrix at the previous time stamp, Cxx ðk  1Þ and the response vector X k at the kth instant, according to: Cxx ðk  1Þ þ 1k X k X Tk . Cxx ðkÞ ¼ k1 k As previously mentioned, the FOEP based approach is a versatile idealization that does not take into consideration the nature of the dataset. The applicability of the method for nonstationary datasets involves the consideration of the recursive mean at each instant of time. The recursive mean update at kth instant, lk , depends on the mean at the previous time stamp lk1 þ 1k X k . The covariance estimate for cases involving mean update can be expressed as: through the relation: lk ¼ k1 k

   e xx ðkÞ ¼ k  1 R1 Rk1 C e xx ðk  1ÞRk1 R1 þ R1 Dl DlT R1 þ 1 Xk  l Xk  l T C k k k k k k k k k k

ð9Þ

~kW e xx ðkÞ ¼ W fk! f T . Eq. (9) The eigen decomposition of the covariance estimate shown in Eq. (9) can be written in terms of C k can be recast as:

~kW e xx ðk  1ÞRk1 R1 þ R1 Dl DlT R1 þ 1 Xk  l Xk  l T f k! f T ¼ k  1 R1 Rk1 C W k k k k k k k k k k k

ð10Þ

f T X k . Carrying out proper substitutions in Eq. 10, and ~k ¼ W The POC vector at the kth time instant can be estimated as: w k1 scaling the data to unit variance, one obtains:

h ih iT ~kW ~ k1 W f k! fT ¼ k  1 W f T þ Dl DlT þ 1 Wk1 w f k1 ! ~ k  l Wk1 w ~k  l W k k1 k k k k k k

ð11Þ

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For structural systems in particular, data is assumed to consistently evolve from zero mean processes. Therefore, the above equation can be simplified as:

h i ~kW ~ k1 þ w f k k! fT ¼ W f k1 ðk  1Þ! fT ~T W ~ kw W k k k1

ð12Þ

h i ~ k1 þ w ~ kw ~ T should diagonally dominant, An important conclusion can be drawn from this discussion: the term ðk  1Þ! k for the RPCA algorithm to be stable and robust. Subsequently, as the system consists of very low to moderate levels of damp~ T represents the correlation between the ~ kw ing, the EVD can be evaluated using the Gershgorin’s theorem [27]. The term w k

f T Xk XT W f k1 ) [29,30], can be written ~T ¼ W POC estimates at a particular instant. The covariance between POC estimates (w~k w k k k1 as (to an arbitrary scale factor):

~ T ¼ q qT þ cqT þ cT q þ ccT w~k w k1 k1 k1 k k1

ð13Þ

As far as the dynamics of structural systems are considered, the error term in the Eq. (13) can be neglected as the number of sampling points increases and under moderate to low damping [31,33]. Recognize the similarity of the first term in Eq. (13) that closely resembles the covariance of the normal coordinates at the instant ðk  1Þ. Thus the term qk1 qTk1 represents ~ T can be safely assumed to be diagonally ~ kw a matrix whose diagonal terms dominate its off-diagonal terms; hence, the term w k h i ~ ~ kw ~ T , facilitating straightforward dominant. This in turn, ensures the diagonal dominance of the matrix ðk  1Þ!k1 þ w k application of Gershgorin’s theorem. Hence for a structural system, the recursive eigenspace update is obtained using FOEP approach which provides a less computationally intensive algorithm in a recursive framework. The EVD of the matrix h i ~ k1 þ w ~ T can be substituted as Hk Kk HT into Eq. (12) as, ~kw ðk  1Þ! k

k

h

ih

~k f k k! W

ih

fk W

iT

h

i

h

f k1 Hk ½Kk  W f k1 Hk ¼ W

iT

ð14Þ

yielding the following iterative update equations:

fk ¼ W f k1 Hk W

)

ð15Þ

!k ¼ Kkk

The recursive covariance structure is transformed to obtain the values of Hk and Kk . As previously described, the term ~ k1 þ w ~ T is diagonally dominant, which ensures that the eigen values are the diagonal entries of the matrix. there~ kw ðk  1Þ! k ~ k1 and w ~ 2 , where ki is the ði; iÞ element of ! ~ i is fore, the ith diagonal entry of the term Kk can be represented as ðk  1Þki þ w i

the ith entry of the POC estimate. Once the eigen values are obtained, the corresponding eigen vectors can be found out, leading to Hk . Based on the above discussion, the recursive eigenspace estimate of a single set of response (more particularly, the responses obtained in a single direction) is achieved. The block covariance matrix shown in Eq. (2) consists of the covariance matrices obtained from the responses in the orthogonal direction as well. Further, the block matrix comprises of the cross covariance matrices, that needs to be recursively updated as well. In line with the above derivations, a similar presentation of the recursive eigen estimation can be arrived at, which is repetitive and therefore, omitted here for brevity. Once each of the individual covariance estimates are obtained, the next task is to implement these matrices onto a recursive framework prescribed by the eigenvalue problem shown in Eq. (4). At any time instant k, the fundamental matrices involved in the eigen decomposition of Eq. (4) can be expressed as:



AðkÞ ¼

0

Cxy ðkÞ

Cyx ðkÞ

0





;

BðkÞ ¼

Cxx ðkÞ

0

0

Cyy ðkÞ



ð16Þ

The recursive updates of the covariance estimates can be substituted into the above equation. These updates can be expressed as:

"

AðkÞ ¼

0 k1 Cyx ðk k

k1 Cxy ðk k

 1Þ þ 1k Y k X Tk

 1Þ þ 1k X k Y Tk 0

"

#

and BðkÞ ¼

k1 Cxx ðk k

 1Þ þ 1k X k X Tk k1 Cyy ðk k

0

#

0  1Þ þ 1k Y k Y Tk

ð17Þ The above mentioned equation set can be rewritten in terms of recursive estimates that are updated at each instant of time. Recasting Eq. (17) in a recursive format, one obtains:

AðkÞ ¼

" 0 k1 1 Aðk  1Þ þ k k Y k X Tk

X k Y Tk 0

#

and BðkÞ ¼

" T k1 1 XkXk Bðk  1Þ þ k k 0

0 Y k Y Tk

#

ð18Þ

B. Bhowmik et al. / Mechanical Systems and Signal Processing 136 (2020) 106499

7

An important feature of the FOEP based techniques lies in reduced time complexities by a significant margin. As evident from the literature, the step wise update of the covariance matrix at each iteration is cumbersome [42] and requires efficient techniques to perform the recursive updates. Additionally, performing an iterative estimation in the backdrop of a generalized eigenvalue problem is more involved and complicated. A simplified route that could be envisioned is through the structural modification of the equation under study and then performing the recursive updates. Based on this premise, consider a simplified version of Eq. (4) at the kth time instant, given by:

^ ðkÞ ¼ qðkÞBðkÞw ^ ðkÞ AðkÞw

ð19Þ

From a thorough understanding of the FOEP approach, it is clear that Eq. (19) needs to be updated for every sample. As evident from Eq. (15), the rank-one update of the covariance matrix provides the recursive estimations at each instant. Similar to this notion, consider the rank-one perturbation of the eigen decomposition given by Eq. (19): ^ Þ ¼ ðq þ DqÞðB þ DBÞðw ^ Þ. ^ þ Dw ^ þ Dw ðA þ DAÞðw Considering the fact that the perturbed matrices are of low magnitude, each of the small perturbations correspond to evaluating the matrix at the k þ 1th instant. This translates to evaluating the EVD at the k þ 1th time stamp, which is precisely the basic objective of the FOEP methodology:

^ ðk þ 1Þ ¼ qðk þ 1ÞBðk þ 1Þw ^ ð k þ 1Þ Aðk þ 1Þw

ð20Þ

From the aforementioned discussions, it is obvious that the individual matrices involved in the EVD can be recursively updated using the FOEP technique. A key feature of this update is the recursive estimation of the eiegnspace, which can be readily obtained by following the same methodology developed for the recursive update shown in Eq. (17). This demonstrates that the eigenspace updates can be obtained through the FOEP technique without actually performing EVD on the subsequent data points, thereby reducing the time complexity and memory consumption, to an enormous extent. 5. Recursive damage sensitive features The current framework exploring the concepts of RCCA facilitates online processing of the data and yields the recursive updates of the eigenvalues and the eigenvectors, referred to as the ’eigenspace’, in the present context. The eigenspace by itself is inadequate in exhibiting the deviations inflicted at the onset of the damage. Therefore, certain damage markers, known as DSFs, are employed to ascertain the presence of damage and its location, visually or otherwise. The key characteristics of a good DSF lies in its potential to detect the presence of damage, locate and estimate the severity of damage and effectively distinguish between the damaged and undamaged states of the structure. Further, the auxiliary attributes of these DSFs arise from their ability to function online, in order to identify the change of states in real time [6,39–41]. In this context, TVAR coefficients and the RRE vectors are utilized as appropriate DSFs for identifying the spatio-temporal patterns of damage in real time. 5.1. Time varying auto regressive coefficients In the context of structural damage detection, the instances of using traditional AR modeling as a criterion for novelty detection has been well documented in the literature [26,40]. As the estimation of AR coefficients is frequently accomplished through the use of windowing and baseline data for detecting damage, online implementation becomes difficult [26]. Motivated by these shortcomings, a TVAR modeling based framework for carrying out online damage detection studies is adopted [40,41]. In the recent times, the use of TVAR modeling for detecting structural damages and change of state in real time has been attempted by the authors with successful applications on both linear and nonlinear, weak and strong nonlinear systems [6,40,41]. The FOEP based RCCA approach tracks the TVAR coefficients in real time and indicates the exact instant of damage through a change in the mean level of the plots, prior to and after damage [6,40,41]. The changes that appear in the mean level of the coefficients indicate alterations in the dynamical properties of the system, such as shifts in natural frequencies, changes in modeshapes of the system, etc., induced due to the damage inflicted to the system. In the current work, the transformed response obtained after the RCCA method is modeled using TVAR approach. As previously discussed, the POCs obtained at each instant of time show near resemblance to normal coordinates, which enables the adoption of a low model order for the transformed response. In the proposed framework, a model order of 2 (two) is pre-selected for all cases. Let X ðkÞ represent the transformed response at any instant which captures the maximum kinetic energy of the system and let V ðkÞ denote a zero mean Gaussian white noise with variance r2v . Then the AR model of order p can be represented as:

X ðkÞ ¼

p X ai X ðk  iÞ þ V ðkÞ

ð21Þ

i¼1

To tailor the basic AR model towards an online implementation, the POCs are modeled using the TVAR approach. Further, the use of TVAR model warrants the robustness of the process against the nonstationarities that might be introduced due to practical issues such as earthquake excitations, real time occurrence of damage, etc., thereby making it an potential candidate to address detection for the aforementioned cases as well. In order to implement the TVAR model, the KF technique is utilized to estimate the recursive updates at each instant, which in turn corresponds to establishing the coefficients (ai ðt Þ) for

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the TVAR model. The following equation is the discrete representation of the ai ðtÞ coefficients and W ðkÞ is the process noise with variance r2w and covariance, Pw ¼ Ipp r2w . Both noise measurements V ðkÞ and W ðkÞ are considered mutually independent and uncorrelated. Considering the following set of equations, which can be used to estimate the unknown state vector BðkÞ is expressed as shown:

BðkÞ ¼ Cðk  1ÞBðk  1Þ þ W ðkÞ

ð22Þ

X ðkÞ ¼ CðkÞBðkÞ þ V ðkÞ

 T The state vector BðkÞ is given by: BðkÞ ¼ a1 ðkÞ; a2 ðkÞ; . . . . . . ap ðkÞ . The matrix Cðk  1Þ is an identity matrix (Ipp ). The matrix C k is the observation data with discrete k steps given as: CðkÞ ¼ ½X ðk  1Þ; X ðk  2Þ . . . . . . X ðk  pÞ. The Kalman filter has mainly two processes: one is the stepwise time update (prediction) and the other one is measurement update (correction) of the predicted data. The relevant equations for obtaining the prediction-correction estimates for the KF updates can be found in [26,39–41]. Although the TVAR model facilitates adequate representation of the nonstationary transformed response, however, in order to use it for damage detection, the TVAR model alone is not enough. Consequently, DSFs are applied on the TVAR coefficients to detect damage which run in real time. The basic Eq. (21) therefore, becomes:

X ðkÞ ¼ a1 ðkÞX ðk  1Þ þ a2 ðkÞX ðk  2Þ þ V ðkÞ

ð23Þ

where, X ðkÞ represents the POC at any instant that captures the maximum variance of the system. It can be expected that the TVAR coefficients a1 and a2 might alter mildly at each time instant due to the nonstationarity of the dataset and the drift in the post-damage values from the pre-damage ones, the exact damage instant is characterized by changes in the overall trend of the coefficients, usually depicted as the change in the mean level of the plot. 5.2. Recursive residual errors (RRE) The present work envisions the concept of spatio-temporal damage detection in real time using RREs, which holds a major entitlement of the study. In this context, the temporal detection module identifies the instant of damage through a change in the TVAR plots that are further validated by the RRE plots. To address damage localization, an improvised version of the temporal RRE, known as spatial RRE, is defined and expressed for each DOF (eRR ), which shows deviation at the particular DOF where the damage has occurred. The selection of RRE as a real time DSF in the current work has also been backed through its widespread use in damage identification literature in recent times [39–41]. Considering a damage at the end of ðk  1Þth instant, the subspace spanned by the updated eigenvector W1k deviates in comparison to the subspace spanned by eigen vectors at the previous time stamp W1k1 . As there is no deviation in the eigenspace for the initial few seconds (apart from the instances of damage), it can be assumed that W1k ffi W1k1 . Based on this assumption, the RREs due to projection of the response at a particular time instant k onto W1k1 is evaluated as [39,40]:



2

vRR1 ¼ W1k1  W1k  X ðkÞ  W1k  W1k1  W1k  X ðkÞ T

T

T

ð24Þ 

From Eq. (24) it is clear that the projections of the transformed responses X ðkÞ and the actual responses XðkÞ can be expressed as vectors with each individual element corresponding to a degree of freedom (m: total number of DOFs) accord T T ing to: X ðkÞ ¼ x1 ðkÞ; x2 ðkÞ; . . . . . . ; xm ðkÞ and XðkÞ ¼ ½X 1 ðkÞ; X 2 ðkÞ; . . . . . . ; X m ðkÞ where k represents the time instant at which RRE is estimated. The present work provides information on spatio-temporal patterns of damage in real time, in a single framework, with a reasonable degree of accuracy. Once the instant of damage has been ascertained, the next step is to localize the presence of damage in the system. Towards this, the formulation of RRE for a spatial detection, needs to be modified. The first step is to express the RREs for each DOF. The primary assumption for local damage detection is that when a column of an MDOF structure is damaged, its effect will be pronounced in the neighboring DOFs as well, which in turn is manifested as a distortion in the local RREs for that particular DOF [39,40]. Let each element of Y ðkÞ and Y ðkÞ be represented as yi ðkÞ and yi ðkÞ respectively, where (i) corresponds to a particular degree of freedom. This yields a time series of RRE (labeled as eRR  Y i ) corresponding to the response for each degree of freedom, which can be expressed as: [39,40]





eRRY i ðtÞ ¼

yi ðkÞ  y2i ðkÞ

2

ð25Þ

In this context, a useful quantity that can be computed is the average of ith response for K datapoints, which can be estimated as:





eRRY i ðtÞ ¼

K

X

2

yi ðkÞ  y2i ðkÞ

k¼1

K

ð26Þ

B. Bhowmik et al. / Mechanical Systems and Signal Processing 136 (2020) 106499

9

The above expression provides an estimate of local RRE estimated over K samples. As shown in the following sections, this quantity is particularly useful, in quantifying the percentage change in RRE before and after damage, corresponding to an indirect measure of the loss of stiffness of the structure. 6. Proposed algorithm The comprehensive methodology followed in the present work for identifying and localizing the damage in real time entails two contributing modules: temporal and the spatial module, working simultaneously in a single framework. The primary course of action followed is to first identify the instant of damage, and then proceed on to detecting its location. The first module deals with identifying the instant of damage where the recorded vibration data is provided as input to the RCCA algorithm, in order to obtain the transformed responses. These responses, consequently modeled using TVAR, produces TVAR coefficients which are indicative of damage to the system. On ascertaining the instant of damage, the spatial module is invoked where the spatial RRE is tracked over each DOF to identify the location of damage. A distortion in the RRE plots confirms the exact location of damage in the structure. For an easy comprehension of the detailed process, the basic steps of the algorithm are enumerated as follows: 1. First, a block covariance matrix incorporating the individual auto- and cross-covariance matrices is fabricated from the set of physical responses. Traditional CCA is applied on some initial data points (around 100 in number) in order to estimate the initial eigenspace. The number of data points chosen here is arbitrary and considered only to stabilize the algorithm for subsequent real time damage detection. The batch implementation comprises the EVD updates shown in Eq. (3), ^ yielding the eigenspace represented by q and w. 2. The recursive gain depth parameter is employed to evaluate the covariance estimate at the present time instant using the covariance estimate at preceding time stamp. From these recursive updates, the eigenvector and eigenvalue matrices are updated using FOEP approach and the transformed responses (principal components) are obtained using the RCCA algorithm. While the covariance updates for the matrices are given by Eq. (17), the iterative set of eigenspace updates carried out after the initial batch implementation can be easily obtained through Eq. (15). 3. The transformed responses are fit using TVAR models of appropriate model order. This provides a set of TVAR coefficients that evolve with the progression of time and aid in identifying the exact instant of damage, governed by the Eq. (23). In addition, the temporal RRE plot, given by Eq. (24), further substantiates the damage instant corroborated by the TVAR plots. 4. Once the instant of damage is determined, the algorithm shifts on to the next module where the spatial detection of damage takes place. The mathematical expressions administering its implementation can be clearly obtained from Eq. (26). The local RRE is tracked over the entire system as a whole that generates distortions in the plot, indicative of damage at that particular location. The aforementioned steps for the proposed method are shown in the form of a stepwise flowchart in Fig. 1. Based on the theoretical derivations, the TVAR coefficients and RRE show deviations in the mean level of the plots, that visually aid in identifying the instant of damage. A few important characteristics of the proposed algorithm include: (i) implementation on the data obtained at each time stamp, as and when the data becomes available; signifying that the algorithm essentially operates online (ii) detection of damage instant and its location without ad hoc windowing, making it less reliant on reference data and (iii) absence of parameters controlling the functioning of the algorithm, rendering it parameter independent. These characteristics of the proposed algorithm makes it an ideal candidate for real time structural damage detection framework. 7. Detection results using proposed algorithm Case studies are undertaken for both global and local damage detection using the proposed algorithm in the context numerical simulations and verified using experimental studies. In this context, two numerically simulated systems are considered here: (i) a 5 DOF B-W system and (ii) a 7 DOF B-W system. The source of nonlinearity for both the systems is through the nonlinear force parameter at the base of the models, controlled by the nonlinear force term j. It can be conjectured at this stage that the effect of change of the value of j for the latter case will not be as pronounced as compared to the 5 DOF model, primarily due to the higher number of DOF of the structure, that constraints the nonlinearity propagation throughout the model. The temporal damage detection cases are obtained by changing the value of the nonlinear force term j for the BW system at a specified instant of time, sequentially followed by the cases for real time spatial damage detection. 7.1. 5 DOF B-W model description A numerically simulated 5 storied structure with 4 floors of linear stiffness and a nonlinear B-W isolator lumped at the base is considered for damage detection studies. The model under consideration has been adopted from Krishnan et al. [40]. A lead rubber bearing isolator (LRB) separates the base from the surrounding ground. The governing differential equation of motion of the system can be represented as:

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Fig. 1. Flowchart of the RCCA method.

€ þ CX_ þ KX ¼ F  MIX €g MX

ð27Þ

M, C, and K are the assembled mass, damping, and stiffness matrices, respectively. A simple shear building representation is assumed to arrive at the expressions for M, C, and K as per [40] which can be referred for detail. The state equations for this system subjected to an external excitation vector W can be written as:

X_ ¼ AX þ EW

ð28Þ

Y ¼ BX

Here, the vector X is the vector of states, defined as the smallest possible subset of system variables that can represent the entire state of the system at any given time. The vector Y represents the output vector, which is governed by the B matrix. The system matrix, A, the excitation matrix E and the observation matrix B are given by

"

A¼ 

½I55

½055

#

M1 K M1 C

E ¼ 0 0 0 0 0  m1  m1  m1  m1  m1 " # ½055 ½I55 B¼ M1 K M1 C

T

ð29Þ

Numerical simulations are carried out on a simple 5 DOF mass, spring, and dashpot system. The mass at each of the four floor levels from the top is 7461 kg and at the base is 6800 kg. The damping coefficients for each floor level above the base is 23:71 kNs/m and 3:74 kNs/m for the base. The stiffness coefficients for each of the floors above the base is 11,912 kN/m and that for the base is 232 kN/m [6,40,41]. € g represents the ground acceleration. The vector X represents In Eq. (27), the vector I consists of all elements as unity and X the displacement of each floor and the base, with X 4 being the displacement of the top floor. It should be noted that the forces due to base damping and stiffness terms (kb and cb ) are included in the nonlinear force (F) due to the LRB base isolator, expressed as:

F ¼ jzQ pb þ kb X b þ cb X_ b where, Q pb

ð30Þ

 k ¼ 1  k yield Q y and kb and cb are the stiffness and the viscous damping respectively, in the horizontal direction. initial

The term kinitial is the initial shear stiffness and kyield is the post yield shear stiffness of the LRB. The evolutionary variable z is used to provide the hysteretic component of the horizontal force, Q hyst ¼ zQ pb . The variable z can be obtained by solving the following nonlinear differential equation:

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z_ ¼ cz X_ b

zn1  bX_ b jzn j þ AX_ b

ð31Þ

Here c; b; A and n are the shape parameters of the hysteresis loop [6,40,41]. For the current model, A ¼

kyield kinitial

 , c = b and n = 1.

The yield force Q y is selected as 5% of the total weight of the building which gives Q y = 17,800 kg and pre-yield to post- yield

 kyield ¼ 16. k

stiffness ratio

nitial

The acceleration plots for the 1st; 3rd and 5th DOF for the B-W model is shown in Fig. 2. For the present study of the model, c=b=39:1. In addition to the nonlinearities arising due to the inherent formulation for the system, the first term j in Eq. (30) ensures that the nonlinear parameter is controlled throughout the global damage detection implementation. For instance, a change in j from 1 to 0.3 can be interpreted to be a 70% change in the nonlinear force term of the system. This force induction through a change in the value of j corresponds to a damage in the present context, as it contributes to a change in the nonlinearity of the system. An important aspect of the damage induction through changes in nonlinear force term is that the changes are permanent and do not wear over time, and thus, it can be safely assumed that the damage is certainly not repaired as time progresses. The change in the nonlinear force term, j, takes place at a particular instant of time, closely emulating a temporal damage to the system. Instances of real life occurrences of earthquakes justify variations in the force-displacement characteristics of the system. Under these conditions, the plots of the force-displacement graphs (i.e., hysteresis plots) deviate due to the presence of damage, thereby validating the fact that the change in the nonlinear force term induces a global damage to the structure. Directed towards this study, an empirical damage index (DI) has been adopted from a recent work by the authors [40], that can be expressed as follows:

DI ¼ 1 

  K1 K0

ð32Þ

 K1 K0

is defined as the ratio between the secant modulus (K 1 ) associated with a changed level of nonlinearity (dam  aged state) to the initial secant modulus (K 0 ) of the pristine (undamaged) state. For cases of no damage, the ratio KK 10 in which

becomes unity, thereby making the damage index zero. It can be understood from the above relation that as the percentage of damage increases, the slope of the secant modulus line for the damaged state (K 1 ) decreases. It has been observed that the percentage change in nonlinearity follows a linear relationship with DI to a good approximation. The values of the DI corresponding to various levels of change in nonlinearity along with the corresponding j values can be found in [39]. In a similar fashion, the response vectors for the 7 DOF B-W model can be readily obtained. The mathematical intricacies are arduous and repetitive and therefore, omitted here for brevity. The detection results using both the models subjected to a diverse nature of excitation are discussed thoroughly in detail in the following sections.

Fig. 2. Acceleration plots for the DOFs.

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7.2. Temporal damage detection studies for the B-W systems excited using white noise A brief numerical study on aforementioned systems has been carried out using Gaussian white noise as a excitation. Temporal damage detection cases are studied first by sequentially changing the j corresponding to 15% and 10% changes in nonlinear characteristics respectively. 7.2.1. Temporal damage detection results for the 5 DOF B-W system Case studies using 5 DOF B-W system for 10% and 15% change in nonlinearity are reported in this section. Recent investigation has revealed that damages of the order of 15% have often been reported as a lower bound for real time vibration based damage detection [41]. However, the proposed algorithm provides successful detection results for finer levels of damage, as low as 10%, for the 5 DOF B-W system. From Fig. 3 a damage instant at 31 s can be clearly observed for 15% change in nonlinearity. The sharp peak discerned at the damage instant primarily attributes to the sudden change in the mean level of the TVAR plot, an event that is consistent for both the coefficients. In order to validate these findings, the temporal RRE plot clearly substantiates the instant of damage through a sharp distortion at 31 s. A separate figure dedicated towards real time damage detection for a 10% change in nonlinearity is shown in Fig. 4. From both Fig. 3 and 4 it can be observed that, the plots of the TVAR coefficients show changes in the mean level of the plot around the 31 s mark, indicating a possible event of damage. Further, residual error when examined in a recursive framework, detects the instant of damage at 31 s, clearly observed from the figures, thereby validating the inferences concluded from the TVAR plots. In line with the above findings, it can be very well established that the proposed RCCA algorithm is efficient in determining fine levels of damage in real time, a feat that has only been reported in a recently established hybrid algorithm using an association of the RPCA-RSSA strategies. 7.2.2. Spatial damage detection results for the 5 DOF B-W system The present section deals with the performance of the proposed algorithm towards simultaneously detecting a spatiotemporal damage in real time, which is one of the key entitlements of the current work. The notion of local damage is brought about by a change in linear storey stiffness of a particular DOF for an MDOF system, that forms the key cornerstone for processing online spatio-temporal damage. It is worth noting that the proposed algorithm invokes the spatial module only when the instant of damage has been previously ascertained in the temporal constituent. A through investigation of the recently established real time damage detection literature reveals that detection studies using the RPCA algorithm could be traced back to a lower limit of 25% in real time. Motivated by this key shortcoming, the proposed method focuses on the detection potential for the weakly nonlinear B-W system for a spatial damage of both 10% and 15%, the results of which are presented next in detail. The key consideration to be incorporated here is the fact that the system is made fully nonlinear by scaling up the value of the nonlinear parameter j to unity and the linear storey stiffness of the third floor are reduced by 10% and 15%, respectively, at 31 s from the start of the excitation. From Fig. 5, it can be visualized that a clear damage instant is identified for 15% reduction in storey stiffness through the deviation in the TVAR plot. After the damage instant is established from TVAR plot, the spatial RRE in the vicinity of damage

Fig. 3. Recursive DSF plot for 15% temporal damage for both the B-W systems (using white noise excitation).

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Fig. 4. Recursive DSF plot for 10% temporal damage for the B-W system (using white noise excitation).

Fig. 5. Recursive DSF plot for 15% spatio-temporal damage for the B-W system.

(say 21 s–41 s) is examined. The spatial RRE plots for all the DOFs are portrayed beside the TVAR coefficients clearly indicate a distortion in the 3rd DOF, thereby conforming with the previous consideration of the occurrence of damage at 31 s confined to that floor. Furthermore, a sharp change in the mean level of the TVAR plots at 31 s is clearly evident from Fig. 6, thereby establishing the fact that the proposed method is well suited at identifying the local patterns of damage for as low as 10%, which is clearly a major accomplishment considering the online nature of the algorithm. This is one of the major entitlements of the current study. 7.3. Detection results for the 5 DOF B-W system excited by El Centro ground motion The robustness of the proposed RCCA algorithm towards solving a wide range of damage identification problems can ensured with its application towards nonstationary excitation cases as well. This situation arises primarily due to the inher-

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Fig. 6. Recursive DSFs for 10% spatio-temporal damage for the B-W system (under white noise excitation).

ent formulation of the damage detection techniques which are mostly pivoted around linear systems excited using stationary events such as white noise. It is expected that the proposed method will provide good results for case studies involving nonstationary excitations as well. To this effect, the use of TVAR modeling on the transformed responses substantially aid in the damage detection scheme, by masking the nonstationary effect of the input excitation. To ensure the versatility of the proposed RCCA algorithm, the B-W system is excited using El Centro ground motion and the detection results are presented. The damage to the model is numerically simulated through a 20% change in the nonlinear force term at the base of the model, that eventually contributes to a global damage to the system. Recently established RPCA algorithm provides a real time detection of 30% for the El Centro excitation case, that confirms the superiority of the proposed RCCA based scheme over RPCA. In the subsequent sections, a thorough comparison of the two said methods will be provided, that sheds a light on the efficacy of the proposed algorithm over the recently established schemes. The detection results displayed in Fig. 7 clearly indicate the exact instant of damage at 25 s from the start of the excitation. The temporal RRE plots, in addition to the TVAR coefficients, aptly validates the damage instant for both cases. The local damage is simulated for the B-W system by numerically reducing the linear storey stiffness of the 3rd floor by 20%, at 25 s from the commencement of the excitation. The plots of the TVAR coefficients shown in Fig. 8, indicate a clear instant of damage at 25 s. The spatial module of the algorithm gets invoked at the determination of damage instant, due to which, the spatial RREs are tracked at each of the floor levels. In the neighborhood of the damage instant, the spatial RREs are evaluated for each floor, displayed in Fig. 8, which show distortion at 25 s for the 3rd storey. This confirms the localization of damage at the 3rd floor of the system, corroborated from the spatial RRE plots. Recently established detection schemes such as RPCA has reported a lower limit of detectability of 30% global damage for a nonstationary input excitation [40,6]; hence, a key contribution of the present work is the development of a framework that successfully reports sptiotemporal damage as low as 20% for a nonstationary excitation. 7.4. Effect of different block covariance structure using partial sensor information It is worth noting that the covariance matrix generated from the physical responses obtained from the structure are usually full rank in nature. In practical scenarios, cost, improper accessibility and unavailability of good quality sensors, impede the possible instrumentation of all the physical DOF of the system. This results in a rank-deficient block covariance structure, where the number of instrumented sensors is less than the actual number of DOF. The applicability of any damage detection scheme towards such continuous health monitoring and condition based maintenance lie in the performance assessment of the methods towards effectively identifying the damage patterns for rank deficient cases as well. In this context, the proposed method is examined in a recursive framework to effectively determine the instant and location of damage for such a case. The performance check mainly assesses the functioning of the algorithm to determine damage from a subset of sensors, considered below:

B. Bhowmik et al. / Mechanical Systems and Signal Processing 136 (2020) 106499

15

Fig. 7. Recursive DSF plot for 20% temporal damage for the B-W system under El Centro excitation.

Fig. 8. Recursive DSF plot for 20% spatio-temporal damage for the B-W system under El Centro excitation.

1. Case 1: Generating a block covariance matrix using a subset of sensors instrumented at the 3rd; 4th and 5th floors. 2. Case 2: Generating block covariance matrices using the first set of responses obtained from 3rd; 4th and 5th floors and the other set obtained from 2nd; 3rd and 5th floors 3. Case 3: Generating a block covariance matrix using sensor information from the 2nd and 4th floor, with a faulty sensor on the 3rd floor where the damage has occurred For the aforementioned case studies, the previously described B-W system is considered. The system undergoes a temporal damage of 20% and 15% at 31 s from the start of excitation. Approaching from a theoretical standpoint, the number of instrumented sensors must equal the number of actively participating modes, which is 3 in the present context. Thus the resultant rank deficit system is expected to produce proper orthogonal modes (POMs) and principal orthogonal values (POVs) [24,25,31–33] corresponding to the reduced system order. Once the block covariance matrix is created, the RCCA algorithm operates on the reduced order physical responses to provide a set of corresponding transformed responses, on which the TVAR models are fit. In this context, two sub cases are further considered: (i) a global damage considering 15% change in nonlinearity and (ii) a local damage case considering reduction of the linear storey stiffness by 20%, at 31 s from the start of the excitation.

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Detection results for Case 1: The first case considers the streaming input data obtained from 3rd; 4th and 5th floors that is required for creating a block covariance matrix. Once the transformed responses are obtained, the TVAR modeling is adopted that provides the key DSFs for identification of damage. The TVAR plots shown in Fig. 9 depict the exact instant of damage by indicating a change in the mean level at 31 s. In order to validate these findings, the temporal RRE plot shows a distortion that confirms the damage event occurring at 31 s. It is obvious that the present results are invariant of any instabilities that might arise due to the rank deficit nature of the responses, thereby demonstrating the superior prowess of the proposed scheme over recently established methods such as RPCA, that are prone to instabilities, even for a 20% global change in nonlinearity. Detection results for Case 2: The present case provides an insight into the efficacy of the proposed scheme in determining spatio-temporal damage for a rank deficit case. The first instance considers the block covariance matrix to be formed by the set of the physical responses obtained from 3rd; 4th and 5th floors. The linear stiffness of the 3rd storey is reduced by 20% at 31 s from the start. The transformed responses are modeled using TVAR coefficients, portrayed in Fig. 10. The exact instant of damage is identified at 31 s from TVAR plots. On determining the exact instant of damage, the spatial module of the algorithm is invoked where the local RRE for the reduced number of floors are tracked over time. It is clear from Fig. 10 that the local RRE corresponding to the 3rd floor provides a significant distortion at 31 s. It can therefore be interpreted that the absence of a similar pattern of distortion from the other floor levels validates the exact localization of damage, a phenomenon indeed confined only to the 3rd storey of the system. The second instance considers the inputs two sets of response: the first set obtained from the 3rd; 4th and 5th floor, while the other set is acquired from the 2nd; 3rd and 5th floor response. The block covariance matrix is generated based on these inputs. It can be conjectured that the detectability of the proposed scheme under a non-identical set of responses might be affected, primarily due to the divergent nature of the input processes, which makes the individual covariance matrices dissimilar. The plots of recursive DSFs shown in Fig. 11 provides a clear insight into the matter. Although the exact damage instant of 31 s can be observed from the TVAR plots, the spatial module fails to provide adequate detection results for localizing the damage in real time. It is clearly visible that the distortions in the 3rd DOF are not as noteworthy as the ones reported in the previous detection estimates, thereby rendering the spatial localization scheme inconsequential for this case. A review of the recently established RPCA based detection strategy provides a 20% and 30% lower limit of detectability for the global and local damage cases, respectively. Based on the above inferences, the proposed method successfully identifies comparatively finer levels of spatio-temporal damage, which is a key contribution, considering the real time essence of the algorithm. Detection results for Case 3: In real life situations, issues may arise when a sensor becomes faulty during monitoring. Under such circumstances, a subset of response from the functioning sensors should be able to provide information regarding the state of the system in real time. Instances from literature reveal that most of the established algorithms rely on the output from a complete set of sensor response for detecting potential events [4,5]. Therefore, it is crucial to assess the performance of the proposed method for situations where a sensor turns faulty during experimentation and/or monitoring. In this regard, a subset of information using the 4th and 2nd floor responses are considered, with a faulty sensor on the 3rd floor, where the damage has occurred. The reduction in linear stiffness of the order of 30% and 20% (studied separately) at 31 s provides suitable test beds for examining the robustness of RCCA method in adapting to situations encountering a fault in instrumentation. For maintaining parity, spatial RREs are chosen as the recursive DSFs for identifying the location of damage to the system.

Fig. 9. Recursive DSF plot for case-1, Global damage, 15% change in non-linearity excited by white noise.

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17

Fig. 10. TVAR plot for case-2, first instance.

Fig. 11. TVAR plot for case-2, second instance.

Recently established literature associated with localizing structural damage in real time [6,39–41] suggests that spatial RREs indicate distortions in the DOFs adjacent to the damaged tier. In the present context, the RREs evaluated on the transformed responses obtained from the RCCA algorithm indicate such distortions corresponding to both 4th and 2nd floors. It is evident from Fig. 12(a) that deviations corresponding to 40% damage is more pronounced as reported for 30% and 20% cases. This provides exclusive evidence regarding the sensitivity of the algorithm (and the recursive DSF, in turn) for cases where a faulty sensor is encountered. The dotted lines on Fig. 12(b) presents the behaviour of RREs for a fully functioning sensor on the 3rd floor that suffers damage at 31 s. It is clear that distortions in the DSFs are more pronounced when the percentage of damage encountered is high. This confirms the fact that the proposed approach can be useful for cases where the sensor becomes faulty during monitoring, which is a common occurrence for real world structures inspected over a continuous period of time.

7.5. Case study of a 7 DOF B-W system excited using white noise A numerically simulated 7 DOF B-W system is excited using a Gaussian white noise of 50 s duration, sampled at 100 Hz. An extension of the previously inspected 5 DOF B-W system, the key motivation for using this system is to assess the

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Fig. 12. Spatial RRE with faulty sensor on the damaged DOF.

performance of the method against an increase in the number of DOF. It is well understood that the nonlinear change at the base of the model propagates strongly to the vicinity of damage. Therefore, the nonlinear propagation throughout the model is complicated that affects only a few of the neighboring DOF of the superstructure. In order to clearly understand the aspect of damage for an increased number of DOF, keeping all other factors invariant, the proposed method is applied to the 7 DOF system. In this context, the nonlinear force parameter j is varied by 15% at 31 s from the commencement of excitation. The recursive DSFs shown in Fig. 3 clearly portray the exact instant of damage through a sharp change in the mean level at 31 s. A salient feature of the DSFs can be observed here. A recently established real time damage detection algorithm, RPCA, has provided successful results for a 15% temporal damage case [40]. The study, conducted on a 5 DOF B-W system failed to show significant distortions in the global RRE plots. The robustness of DSFs lie in their ability to determine damage for a wide range of applications, including a varied class of nonlinear systems as well. One of the key features of the proposed scheme is to incorporate similar set of DSFs for a wide class of problems dealt within the scope of this study, without compromising on the time complexity or the efficacy of the proposed methodology. Therefore, the significant distortion in the RRE plots displayed in both the Fig. 3 is of key importance and provides exclusive evidence towards the effectiveness of the proposed methodology and thereby, the DSFs. The key entitlement of the work lies in the ability of the method to identify spatio-temporal damage in real time. Following a similar notion, the proposed RCCA method is examined for a local damage case for the 7 DOF B-W system. From the previous detailed discussions, it is clear that the change in the linear storey stiffness remains confined to a single DOF; therefore, the proposed method is expected to provide credible results for spatio-temporal damage for the current model as well. Hence, it is envisioned that the RCCA scheme will provide successful detection results for a similar percentage of local damage in real time. In this regard, the linear storey stiffness of the 3rd floor is reduced by 15% and 10% and studied extensively in separate case studies. For the first case, the sharp peak of the TVAR plots shown in Fig. 13, indicate a clear damage instant at 31 s. While the spatial RREs of all the other floors fail to indicate a distortion, the significant deviation that occurs in the 3rd storey of the system, validate the exact occurrence of damage, recursively, in a simultaneous framework. Detection results for a 10% linear storey stiffness change is provided in Fig. 14 that clearly brings out the spatio-temporal aspect of the proposed framework. Based on the above discussions, some of the key conclusions that can be drawn are:

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Fig. 13. Recursive DSF plot for 15% spatio-temporal damage for the B-W 7 DOF system under white noise excitation.

Fig. 14. Recursive DSF plot for 10% spatio-temporal damage for the B-W 7 DOF system under white noise excitation.

1. The propagation of the change in nonlinearity for an increased number of DOF for a similar type of system is comparatively less, a result that is discussed in details in the later stages of the draft. Therefore, although the effect of global damage is appropriately captured by the TVAR coefficients for both the cases, the RRE plots provide an rough estimation for the 7 DOF B-W system, which can be considered as a direct implication of the increased number of DOF. 2. The use of RRE as an efficient DSF even for fine level of damage of the order of 10% is a key entitlement of the work, which readily provides an authentication to the efficacy of the proposed algorithm in addition to recognizing the utility of the said DSFs. 3. The local damage cases, however, are expected to show similar results for both the systems in context. A summary of the detection results for the systems can be obtained from Fig. 5 and 13 for a 15% change in the linear storey stiffness. As the proposed method successfully provided consistent detection results for a 10% saptio-temporal damage for the 5 DOF B-W system (Fig. 6), it is expected that the method should present similar results when the model is replaced with an increased number of linear DOF, depicted in Fig. 14. It can therefore be well concluded that due to the confinement of the damage to a single storey, the proposed method provides similar spatio-temporal damage detection results for both the aforesaid models, in real time.

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7.6. Case study for a strongly nonlinear system: a 5 DOF structure modeled with Duffing oscillator on its 3rd floor In this section, the performance of the proposed method is assessed against the damage induced to a strongly nonlinear 5 DOF system, modeled with a Duffing oscillator on its 3rd floor. While the source of nonlinearity is provided by the Duffing parameter a, obtained from the governing differential equation of motion for the system, the other 4 floors are idealized as having linear storey stiffness. The governing equation of motion is modeled as Ito’s stochastic differential equation and subsequently discretized using Taylor’s 1.5 strong scheme considering a step size D ¼ 0:01 s. The governing equation of motion can be written as:

€ þ CX_ þ KX þ WZ ¼ Fðt Þ MX

ð33Þ

where M, C and K follow the similar notations as before. W represents the location of DOF modeled with the Duffing oscillator and its neighboring DOF involved towards the formulation of the equation of motion. The vector Z represents a curtailed version of the operation ðX 2  X 3 Þ3 that arises due to the coupling of the DOF while obtaining the equation of motion. Upon expanding the governing differential equation term by term, the following equations are obtained:

 € 1 þ c1 X_ 1 þ k1 X 1 þ c2 X_ 1  X_ 2 þ k2 ðX 1  X 2 Þ ¼ F 1 ðtÞ m1 X



 € 2 þ c2 X_ 2  X_ 1 þ k2 ðX 2  X 1 Þ þ c3 X_ 2  X_ 3 þ k3 ðX 2  X 3 Þ þ aðX 2  X 3 Þ3 ¼ F 2 ðtÞ m2 X



 € 3 þ c3 X_ 3  X_ 2 þ k3 ðX 3  X 2 Þ þ aðX 3  X 2 Þ3 þ c4 X_ 3  X_ 4 þ k4 ðX 3  X 4 Þ ¼ F 3 ðtÞ m3 X



 € 4 þ c4 X_ 4  X_ 3 þ k4 ðX 4  X 3 Þ þ c5 X_ 4  X_ 5 þ k5 ðX 4  X 5 Þ ¼ F 4 ðtÞ m4 X

 € 5 þ c5 X_ 5  X_ 4 þ k5 ðX 5  X 4 Þ ¼ F 5 ðt Þ m5 X

ð34Þ

The state-space format for the system can be evaluated which is subsequently utilized for formulating the detailed Taylor 1.5 scheme expressions. The basic equation for the response of the system is as follows:

X nkþ1 ¼ X nk þ ank DT þ

m m  o X X 1 @ n:j n n:j bk DW j þ I0 ank DT 2 þ Ij ank DZ j þ bk DW j DT  DZ j 2 @t j¼1 j¼1

ð35Þ

On closely observing the steps, the final form of the response can be expressed as shown:

1 Y ð:; k þ 1Þ ¼ Y ð:;kÞ þ AðkÞDT þ I0 AðkÞDT 2 þ BDW þ I1 AðkÞDZ 1 þ I2 AðkÞDZ 2 þ I3 AðkÞDZ 3 þ I4 AðkÞDZ 4 þ I5 AðkÞDZ 5 2 ð36Þ The spatio-temporal damage is numerically simulated by altering the linear storey stiffness of the second floor at a particular instant of time. The value of the Duffing parameter a remains invariant at 1. For simulation following parameters are adopted: mi = 2 kg, ki = 1 kN=m2 , ci = 2.0 N-s=m2 and ri = 1.0 (i 2 1; 2 . . . 5). The system parameters of The system is excited using Gaussian white noise for a duration of 50 s, sampled at 100 Hz for a total of 1500 Monte Carlo Simulations. The detection results local damage are discussed next in detail. The spatio-temporal damage detection capability of the proposed method is examined in a simultaneous framework. Conforming with the previous notion of change in linear storey stiffness as an indicator of local damage, the 2nd floor of the model undergoes a stiffness change at 31 s from the start. A 15% damage case is studied in detail. The detection result is displayed in Fig. 15, from which the exact instant of damage at 31 s is clearly visible through the TVAR plots. Once the temporal damage is ascertained, the RCCA algorithm shifts to the spatial module where the local RREs for all the floors are tracked recursively. A significant distortion in the 2nd floor RRE and its absence in others, confirms the location of damage, which is confined to the 2nd storey. Considering the strongly nonlinear behavior of the system and simultaneous spatiotemporal damage detection capability of the proposed method, the above findings hold a major entitlement of this research. 7.7. A case study with negligible non-linearity: spatio-temporal damage detection In this section, the potential of the proposed RCCA towards an almost linear system is examined. The previous discussions were mainly centred around nonlinear systems and therefore, it is important to understand the applicability of the method towards a linear system as well. To obtain a nearly linear system, the nonlinear parameter a of the above mentioned model is scaled down to 0:005. The spatio-temporal damage detection case is studied by considering a 20% damage to the system at 31 s from the start, to the 2nd storey of the model. It can be clearly observed from Fig. 16 that the TVAR plots are effective at identifying the temporal damage instant through a sharp peak at 31 s. Once the spatial module gets invoked, the local RREs for the 2nd floor shows a significant distortion that confirms the presence of damage confined to that storey. Hence, it can be concluded that the proposed method is effective in identifying the spatial and temporal patterns of damage for both linear and nonlinear systems to a fair degree to accuracy, as evident from the underlying findings.

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Fig. 15. Spatio-temporal damage detection for strongly nonlinear system for 15% damage.

Fig. 16. Spatial damage detection for negligible non-linearity (a =0.05) corresponding to 20% linear stiffness change.

7.8. Performance check of the proposed method against RPCA: a comparative study In this section, the performance check of the proposed method against a recently established real time damage detection scheme, utilizing the concepts of RPCA, is provided. The performance evaluation of the proposed method against its recently established counterpart takes place in three major case studies: 1. Global and local damage detection case studies for the 5 DOF B-W system excited using white noise 2. Global damage detection studies for the 5 DOF B-W system excited using El Centro ground motion The 5 DOF B-W system previously described is taken into consideration to examine the real time damage detection performance of both RCCA and the RPCA algorithms. The RPCA algorithm is a recently established real time scheme that has been published in the Journal of Mechanical Systems and Signal Processing, and is an important contribution in the context

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of real time damage identification of structures. The basic premise of the method was centered around finding damage for linear to weakly nonlinear systems. Since RPCA and RCCA both consider the EVD of the covariance matrices through their formulations, it is important to understand the performance of both the said methodologies for the same damage conditions. In this context, the global damage cases are first presented, followed by the results of the local damage detection. 7.8.1. Case study for 15% global damage detection using 5 DOF B-W system The global damage is carried out by changing the nonlinear force term j at 31 s by 15%. Detection results obtained using RCCA can be found in Fig. 5. The RPCA algorithm operates in a similar fashion by first creating a covariance matrix out of the physical responses, performing a batch PCA on the initial samples and then using FOEP to estimate the eigenspace updates at each instant of time. Using TVAR coefficients, the transformed responses are modeled and the detection results using both the algorithms are shown Fig. 17(a). It is evident from the plot that the RPCA algorithm shows certain instabilities for such a low percentage of damage in real time, which can be primarily attributed due to the nature of the covariance matrix obtained from the physical responses. In comparison, the RCCA method creates a block covariance matrix that provides eigenspace updates at each time instant, and therefore, devoid of instabilities of any kind. Moreover, the RPCA algorithm is premised mostly linear system theory and its applicability towards a weakly nonlinear system is its direct extension. Therefore, it can be concluded that the RCCA method provides better estimates of detectability even for a fine level of damage in real time as compared to the RPCA scheme. However, it should be noted that both the algorithms provide a reasonable first-hand representation of the instant of damage in real time. 7.8.2. Case study for 25% local damage detection using 5 DOF B-W system In this section, the performance of the FOEP based RPCA and RCCA methods are examined for a 25% spatio-temporal damage. A snapshot of the comparative study is provided in Fig. 17(b). It can be observed from the plot that both the recursive algorithms are able to identify the exact instant of damage at 31 s that occurs in the 3rd storey of the system. However, from a straightforward visual inspection of the figure, it can be understood that the RPCA algorithm is susceptible to certain instabilities that might arise due to the inherent nonlinear nature of the system. Contrary to this, the RCCA method provides smooth detection results and therefore, preferred in identifying damage for a nonlinear system over the recently established RPCA framework. 7.9. Case study for the 5 DOF B-W system excited using El Centro ground motion A review of the recently established real time damage detection literature reveals that a 30% global damage for a nonstationary excitation is reported as the lower limit of detectability. However, the present work has successfully proposed spatiotemporal damage detection studies as fine as 20%, in real time. Based on these discussions, an even comparison with the RPCA method can only be justified at its lower limit of detectability, i.e, at 30% global damage. The detection results provided in Fig. 18 show a clear instant of damage at 31 s using both the algorithms. While the TVAR plots using RCCA provide impeccable detection results, the plots obtained using RPCA is highly susceptible to the nonstationarities that arise due to the input excitation. This creates a zone of significant instability in the TVAR plot, thereby rendering the application of RPCA under nonstationary environment, ineffective. Hence, it can be safely concluded that the proposed RCCA algorithm provides better damage detection results in real time even for a nonstationary excitation for a nonlinear system as well.

Fig. 17. Comparison between RCCA and RPCA for global and local damage using 5 DOF B-W system (under white noise excitation).

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Fig. 18. Comparison between RCCA and RPCA for 30% global damage using 5 DOF B-W system (under El Centro excitation).

The percentage changes in the mean level of the TVAR coefficients are shown in Table 1. It is well understood the wavy modulations indicated by the fluctuations in the TVAR plots obtained from the RPCA transformed responses can be accounted through the recursive standard deviations (SD) pre and post damage. It can be inferred from the statistical descriptions shown in Table 1 that the RPCA algorithm provides a significantly higher percentage change of the recursive SD before and after damage. This can mostly be attributed due to the inconsistent modulations in the TVAR plots obtained from the RPCA algorithm, clearly depicted in Fig. 17 and 18. A change in the mean level of the TVAR coefficients obtained from both the algorithms correspond to the occurrence of damage inflicted to the system at 31 s. The higher percentage change of the statistical parameters reflected by the RPCA-TVAR algorithm characterize the presence of instabilities in the recursive DSF plots. 7.10. Threshold for the proposed algorithm Quantification of a lower limit of detectability for an algorithm is important in understanding its applicability towards desired implementation. In this regard, an adequate threshold of the proposed RCCA based damage detection approach is defined, which is centred around TVAR coefficients as statistical markers of damage. The threshold is based on the premise that distinct peaks shown by the TVAR coefficients are real time indicators of damage suffered by the system. While potential damage states are clearly distinguishable through the peaks in TVAR coefficients, detection of lower percentage damage (of the order of 5–6%) becomes increasingly difficult in an online framework. The present section introduces the concept of threshold in light of both linear and nonlinear case studies and demonstrates the effect of stiffness change as a function of damage [7]. Case studies corresponding to 15%, 10%, 8% and 6% change in storey stiffness for both the linear system and the nonlinear B-W system are considered for setting the threshold. Changes in stiffness for the third storey for both the case studies are illustrated in Fig. 19 using the RCCA-TVAR approach. Damage is introduced in the systems exactly at 31 s from the initial vibration state. The subplots (a)–(d) correspond to the damage detectability for the B-W system, where it can be clearly observed that the absence of indistinct peaks for (d) adequately sets the threshold limit for the algorithm. Again, a pronounced distortion observed in each of the subplots (e)–(g) is clearly missing in (h), which can be interpreted as a threshold for the linear system under study. It can, therefore, be concluded that the threshold for the proposed method is limited at 8% of structural damage in real time. In most practical cases, stiffness changes of the order of 10% and beyond are more

Table 1 Percentage changes in statistical mean of TVAR coefficients for 5 DOF B-W system. Damage condition

Method

Mean (undamaged)

Mean (damaged)

% change

Mean + SD (undamaged)

Mean + SD (damaged)

% change

15% global (WN) 25% local (WN) 30% global (El Centro)

RCCA RPCA RCCA RPCA RCCA RPCA

0.71 0.79 0.68 0.67 0.01 0.13

0.76 0.85 0.69 0.69 0.008 0.005

6.72 7.70 1.47 2.98 20 64.54

0.78 0.85 0.73 0.72 0.009 0.14

0.77 0.86 0.70 0.69 0.005 0.051

0.85 1.06 4.92 4.98 45.56 62.22

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Fig. 19. Threshold for real time damage detection using proposed algorithm.

commonly encountered [7]. Thus, the practical applicability of the proposed RCCA based approach can be consequently, warranted. Considering the purely online nature of implementation of the algorithm, these results provide compelling evidence towards applicability of the algorithm in practical studies where damages of significant low order are encountered. It is interesting to note that the proposed method holds a similar threshold level for both linear and nonlinear systems, thereby asserting the fact that the method is indeed, a purely data driven approach.

8. Experimental verification A crucial feature of any algorithm is to assess its effectiveness by validation through experimental case studies, carried out under controlled laboratory environment. In the recent years, numerous experimental setups have been developed that closely emulate real time damage scenarios. In the present work, the focus has shifted towards detecting the change of state from a linear to nonlinear regime and vice versa, through an extensive experimental setup. To substantiate the robustness of the proposed approach, an experimental setup is devised consisting of a two-storey aluminum shear building prototype placed with a water dispenser at its top floor. The aluminum model is of height 1.5 m with inter-storey distance of 0.74 m fixed on a base plate that is bolted on top of a shake table. The cross section of the base measures 0:3 m  0:3 m  0:02 m placed on the 0:15 m thick base plate and firmly bolted against the shake table. The water dispenser of height 0:26 m is firmly affixed to the roof plate using commercially available epoxy resin. The model is subjected to scaled Gaussian white noise excitation and the acceleration data are collected using QuantumX MX410 HBM Data Acquisition (DAQ) system at a sampling frequency of 400 Hz. The shear building model is instrumented at the floor levels in orthogonal directions using PCB Piezotronics triaxial acceleormeters of sensitivity 100 mV=g, bearing a frequency range of 0.5– 10,000 Hz. The dispenser when filled with water provides nonlinearity to the setup. This comes into effect when the water sloshes against the walls of the dispenser as the model is subjected to scaled vibration excitation. The flow valve of the dispenser emancipates water at the rate of 1 L/30 s during the run-time of the experiment. The experimental trials were conducted at two different alignments of the shear building model (Fig. 20): Alignment A: The model is placed along the axis of the shake table with the local x-axis coinciding with the global x-direction and Alignment B: The model positioned at a non-orthogonal orientation with respect to the global x- and y-axes. For each of these cases, the experiment is carried out by allowing the water to drain out of the dispenser at 0 s and at 20 s in separate trials, that continued for the remainder of the excitation. The time stamp of the trial was captured using a stop watch and the experiment was also recorded in the form of videos, in order to gain an immaculate insight about the dynamics of the process. The streaming acceleration data is recorded and the proposed algorithm is applied in real time that yields the transformed responses. As the water drains out from the system, the sloshing of the water leads to a variable mass loss that accounts for the transition from nonlinear to linear state, which can be distinctly identified using the proposed method, described next in detail. In addition to demonstrating the detection potential of the RCCA algorithm, the performance of the recently established RPCA method is also dealt with in detail. Based on the previous discussions, it can be easily understood that the RPCA algorithm premises on the concept of orthogonal transformations for data gathered from linear to weakly nonlinear systems. Its

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Fig. 20. Details of the experimental setup: (a) Biaxial and (b) Orthogonal directions, 1 & 2. Accelerometer locations for orthogonal setup, 3. Water outlet, 4. Water Dispenser, 5. Accelerometer locations for biaxial setup, 6. Floor mass, 7. Aluminium column, 8. Shake table, ‘1–1’. local axis, ‘X-Y’. Global axis.

performance for a multi-directional case can therefore, be expected to falter as the formulation of the algorithm does not take into consideration the effects of torsionally coupled systems involving multi-directional dependencies. In this regard, the application of both the algorithms on the streaming dataset from the experimental trials generate eigenspace updates at each time instant, from which the transformed responses are modeled using TVAR approach. The comparative study of the performance of these recursive algorithms are discussed in the following section. 8.1. Detection results for the experimental trials: a comparative study with RPCA The behavior of the recursive DSFs for Alignment A are illustrated in Figs. 21 and 22, for the mass loss at 0 s and 20 s, respectively. For the case where the water drains out at 0 s from the start, Fig. 21 shows a gradual change in the mean level of the TVAR coefficients. The draining of water is a continuous event, which necessitates that the recursive damage markers should be able to illustrate a resemblance through their representations. This conceptualization can be exactly inferred from the TVAR coefficients where the consistent changes in the mean level of the plots correspond to a gradual loss of water from the system. This clearly depicts the transition from a nonlinear to a linear state of the model, validated by the DSF plots. Considering the TVAR plots obtained from the RPCA algorithm, the coefficients displayed in Fig. 21 indicate the initial regime, but the inconsistency of the plots gradually impedes its expected utility. The loss in mass due to water drainage is a gradual event, a fact corroborated by the DSFs obtained from RCCA transformed response. In the present case, RPCA algorithm fails to provide any substantial evidence of the gradual mass loss scheme, conducted through the experiment. However, the global change in the mean level from the 0 s mark until the entire experimental duration is indicative of a variable mass loss and can be roughly interpreted as a transition of state for the system. In line with the above findings, the DSFs shown in Fig. 22(a) and (b) clearly identifies the commencement of the water loss from 20 s from the start of the excitation. It can be inferred that the water loss from 20 s represents a gradual event, evident from the consistent shifts in the mean level of the TVAR plot. An interesting feature to observe here is that the graph maintains an apparent horizontal alignment until the commencement of water loss at 20 s, illustrating the fact that the transition from the nonlinear to the linear regime is aptly depicted by the proposed algorithm. Additionally, spatial damage detection in real time is carried out by employing RREs for both the floors, shown in Fig. 22(c). Significant distortion in the top floor at 20 s indicates the start of mass loss from the system which gradually continues till the remainder of the excitation. A small-but noteworthy distortion on the bottom floor implies that the mass loss for the system is biased for the top floor (where the dispenser is glued) in comparison to the bottom floor. These results are in good agreement with the previously reported case study where the algorithm is robust in detecting damage for faulty (or unavailable) sensor on the region of interest (Ref. Section 7.4). The detection results using RPCA are not expected to show consistent changes indicating the phenomena of the mass loss, the results of which are omitted here for brevity.

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Fig. 21. DSFs for mass loss at 0 s, Alignment A.

Fig. 22. DSFs for mass loss at 20 s, Alignment A.

The subsequent trials on the experimental setup were conducted by aligning the model in a non-orthogonal orientation with respect to the shake table (Alignment B). A similar procedure is adopted where the variable mass loss takes place at 0 s and 20 s, for two individual case studies. The main objective of this experimental trial is to assess the efficacy of the proposed algorithm for a multi-directional case, closely emulating a practical scenario, where the ground motions might excite a structure from a non-orthogonal alignment. Motivated by this objective, the RCCA method is applied on to the biaxial (nonorthogonal) datasets obtained from the streaming vibration response. From the experimental trials, it was observed that the sloshing of water due to the excitation in the aligned direction had a greater magnitude compared to the mutually orthogonal oriented cases. Fig. 23 clearly shows that the change in the mean level of the DSF plot takes place from 0 s, thereby conforming to the fact that the water loss commences exactly at the beginning of the excitation. However, it is observed that the shifts in the mean level of the TVAR coefficients is not uniform, which can be attributed to the multidirectional nature of the recorded responses. The torsional component generated during the vibration of the shake table

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Fig. 23. DSFs for mass loss at 0 s, Alignment B.

allowed a rapid sloshing of water with a greater magnitude, compared to the previous case. It can be clearly observed from Fig. 23 that the TVAR coefficients obtained using RPCA algorithm fail to show consistent deviation in the mean level, an occurrence that had immaculately been corroborated by the RCCA algorithm. Furthermore, the gradual transition of state from the 0 s mark is not obvious, attributed primarily due to the global non-orthogonal pairs of vibration input. The comparative study with the recently established RPCA method conclusively indicates that the RCCA algorithm provides better detection results even when the input features are locally orthogonal with respect to each other. A similar case study for the DSFs involving water loss at 20 s is reported in Fig. 24. While the change in the mean level of the plots is pronounced from 20 s, the shifts in the level is not consistent. It can therefore, be concluded that the proposed RCCA based approach provides better identification results when the vibration data streams in from a pair of globally orthogonal sensors, as opposed to the ones obtained from the pair of multi-directional locally orthogonal datasets. 9. Case study for the UCLAFB: a practical problem In order to ensure the robustness of any proposed methodology, it is important to understand its behavior towards real life scenarios that generally involves data sets of large dimensions. The proposed RCCA based approach is now applied to the mutually orthogonal pair of E-W and N-S recorded vibration data of the UCLAFB. Continuous monitoring of the structure provides a rich test bed for blind identification, damage detection and condition based monitoring processes by making the data globally available through a remote database server. The data is sampled at 100 Hz. To examine the efficacy and damage detection potential of the proposed algorithm, floor accelerations due to the combined ambient vibration and earthquake data (with Mw = 6.0, on 28 September 2004, 10:15 AM PDT from Parkfield, CA) are considered. Based on the records, out of the actual earthquake event of 30 s, the strong shaking occurred for a time-span of 10 s. The complete set of response involving multi-directional dependencies is considered. This consists of the responses gathered from pair of mutually orthogonal E-W and N-S directions. Both RCCA and RPCA algorithms are examined to identify the damage events. 9.1. Comparative study for UCLAFB data The set of multi-directional translational responses are provided as inputs to both the algorithms and the transformed responses are obtained at each time stamp on which TVAR models are fit. It can be clearly observed from the TVAR plots shown in Fig. 25, that the RCCA algorithm could effectively distinguish between the ambient regime and the occurrence of earthquake. It is well understood that the ambient regime lasts up to 380 s, following which, the mean values of the DSFs begin to change. The time stamp 380 s marks the onset of the earthquake and the steady change in the mean level continues until 410 s, where the damage to the structure is pronounced. This is certified through the peaks of TVAR coefficients that slowly begin to change after 410 s, marking the steady reduction in the intensity of the earthquake. This signifies the entry into a post-damage ambient zone, clearly demarcated in Fig. 25. UCLAFB has also been recently studied in the purview of damage detection through RPCA [39,40] based algorithm. Previously reported results using RPCA have consistently failed to identify the exact onset of the damage around the 410 s mark, due to the high nonstationary nature of the dataset. Additionally, in the context of RPCA, the TVAR plots alone shown in Fig. 25, could not identify the exact onset of the earthquake at the 380 s mark, a feat clearly achieved by the DSFs using the transformed response obtained from RCCA technique. Moreover, the consistent fluctuations observed in the mean level of the TVAR plots using RPCA do not provide ample evidence of the

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Fig. 24. DSFs for mass loss at 20 s, biaxial orientation.

Fig. 25. A comparative study of the DSFs for UCLAFB.

start of the post-damage ambient zone. It can therefore, be very well concluded that the proposed RCCA method is well adept at identifying the change of states even for a practical system, that greatly enhances its robustness and efficacy. 10. Real time infrastructure monitoring: critical assessment The inability of the proposed approach, or any output-only approach for structural health monitoring, system identification and detection of damage or other features of interest remain not just a limitation of this study, but a major challenge to look forward to, in future. The choice of a representative class of dynamical systems with a broad range of condition indicators can contribute to suitable benchmarks and lead to guidelines for a multitude of SHM applications. A reasonable, yet tedious way to address this, is to create an extensive evidence base by carrying out tests in a large scale and scope, along with consolidation of data from existing benchmark studies - all pertaining to applications and sector-specific performance data. In practical implementations however, many decisions regarding the state of a monitored system need to be quick, as correct as possible and in real time, and therefore, acquiring evidence from a sequence of exhaustive analyses is not always

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possible. In practice, several sensing technologies based on vibrations, strain, perturbation of electromechanical field due to movement, Doppler effect, interferomery, Bragg diffraction, or even acoustics, are now employed for civil infrastructure monitoring. Different condition indicators (CIs) based on the feature space extracted from the streaming response dataset could be utilized for this purpose on the basis of their statistical properties, adaptability to various applications and changes in sensing techniques. Under these circumstances, an extensive performance benchmarking involving scale and background independence of the CIs, receiver operating characteristics (ROC) subject to different applications and elimination of true and false positives or negatives (arising due to instabilities, poor resolution or even sensor noise) seem as appropriate approaches towards setting thresholds for robust damage detection. An investigation of this scope, detail and extent is not present at the moment, but can be considered as a natural but long-term extension of the present work in combination with a class of such methods that have and will be entertaining the dependence on output-only information, especially when the input and the system information are unknown, vague, or significantly incomplete. In future, such an approach will also contribute not just to data and techniques, but also to model imputation as well. 11. Conclusions A novel real time damage detection scheme based on recursive canonical correlation analysis principles, is proposed in this paper. Unlike the recently established RPCA based damage detection scheme that provides proper orthogonal modes from the covariance estimates of an uniaxial vibration data, the RCCA method employs the eigenspace estimates generated from a multi-directional block covariance matrix in order to solve a generalized eigenvalue problem. The proposed method provides transformed response at each time stamp through the utilization of first order eigen perturbation techniques. The method applied to both weak and strongly nonlinear systems provides successful spatio-temporal detection results of the order of finer damages of 10%, in real time; whereas the RPCA method shows up to 15% damage accuracy. From the detailed comparative assessments against RPCA, it has been shown that the proposed RCCA based detection scheme provides better results, and can be extended for both full and partial sensor inputs as well. Case studies considered for rank deficit systems with the block covariance matrix generated using the rank deficit data gathered from the same set of floor responses, have provided finer estimates than those obtained using different set of responses. It is observed that the propagation of the change in nonlinearity for an increased DOF of a similar type of system is comparatively less, thereby detecting global damages limited to the order of 15% for the B-W system. As the change in the linear storey stiffness remained confined to a single floor, the spatio-temporal damage detection module of the RCCA algorithm provided a 10% real time damage, consistent with the results obtained for the 5 DOF B-W model. For the experimental case study, the application of the RCCA method provided successful detection results for both globally and locally orthogonal datasets. In this context, the RPCA algorithm fails to provide consistent results for a gradual mass varying experimental case study and fails to identify events pertaining to the biaxially oriented datasets. Real time damage detection studies on the recorded ambient and vibration response data from the UCLAFB demonstrates the efficacy and robustness of the proposed scheme in determining the transition from undamaged to damaged state at the onset of the Parkfield earthquake. The proposed RCCA method provides recursive DSFs that can efficiently differentiate between the onset of earthquake, duration of strong shaking, and its transition to the postearthquake ambient regime. On the other hand, the DSFs obtained using RPCA could identify only the exact instant of damage to the system and fails to provide information about the commencement of the earthquake. Hence, it can be concluded that RCCA can be successfully used to provide fine levels of detection for nonlinear systems and linear uniaxial as well as multi-directional torsionally coupled systems, in real time. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements Basuraj Bhowmik acknowledges the SEAI-funded WindPearl project (Project Ref. No.: RDD/00263). Vikram Pakrashi acknowledges the EU-funded SIRMA (Strengthening Infrastructure Risk Management in the Atlantic Area) project (Grant No. EAPA\_826/2018) and the European Union’s Horizon 2020 (ERA-NET Cofund MarTERA) FLEXAQUA project (Project Ref. No.: PBA/BIO/18/02). References [1] C.R. Farrar, K. Worden, An introduction to structural health monitoring, Philos. Trans. R. Soc. London A 365 (1851) (2006) 303–315. [2] S.W. Doebling, C.R. Farrar, M.B. Prime, et al, A summary review of vibration-based damage identification methods, Shock Vib. 30 (2) (1998) 91–105, Citeseer. [3] M. Gul, F.N. 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