Hybrid FOEP based RPCA-RSSA framework

The damage detection strategies based on FOEP methods such as RPCA-RRE, RPCA-TVAR and RSSA-TVAR, have shown that the FOEP approach can be easily improvised for systems involving various complexities. Problems in dynamical systems involving single and multi-channel inputs, linear and nonlinear systems with changes in linear stiffness and /or nonlinear force term, quantified as damage [1–3], can be addressed using the FOEP based framework, to an accurate degree of identification in real time. In this work, a novel hybrid framework is proposed which serves as an ideal paradigm as to how FOEP technique can be improvised to create a hybrid approach towards a SHM problem. It can be understood that the FOEP approach is an efficient method to utilize the eigenspace updates obtained in real time for damage detection purposes. It is worth mentioning the fact that the proposed algorithm accommodates the use of a low model order for TVAR modeling. A recently established real time damage detection strategy has shown significant promise in detecting structural damage of the order of around 15%. In comparison, the main advantage of the hybrid method is thefiner detectability of structural damage in real time, that has been successfully reported to be of the order of 10%, shown in the later stages of this chapter. In the backdrop of FOEP approach, the estimated eigenspace retains necessary information pertaining to the spatial and temporal patterns of damage, that is tracked through the DSFs for identifying damage in real time.

The proposed framework involves the sequential use of RPCA and RSSA as an application of the FOEP technique on the raw vibration response in order to obtain a transformed response online as and when the data streams in. The physical response matrix is provided as input to the hybrid FOEP algorithm where the response is first transformed using the RPCA scheme. The RPCA algorithm provides a set of transformed response (X_tr), updated covariance estimates and recursive eigenspace updates at each time stamp, which can be obtained by carefully scrutinizing the FOEP derivations as previously shown. The transformed response obtained at each instant of time is then provided as input to the RSSA module, where the following recursive update equation gives the covariance estimate:

C_k= k−1

k Ck−1+ 1

kX_tr¹X_tr^1T (5.17)

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whereX_tr¹ is the top floor acceleration obtained after RPCA transformation of the physical response matrix. Following the FOEP approach, the recursive eigenspace updates can be easily obtained from Eqn. 5.4. Following similar lines of development, the reconstructed signal (Y_R¹) can be obtained by projecting the PCs back to the original subspace and averaging it over the eigen values that explain more than 90% of the variance.

Based on the aforesaid perception of spatio-temporal damage detection carried out in a single framework, the hybrid approach also strives to similarly meet the requirements of an efficient recur- sive algorithm that provides eigenspace updates at each instant of time. The overall methodology is outlined in Fig. 5.2. The crucial point of contrast for the implementation of the hybrid method is that while the RPCA and RSSA modules work intandem to provide eigenspace updates successively, the individual RPCA or the RSSA based procedures utilize only one set of iterative equations to provide the eigenspace in real time. The hybrid algorithm commences by processing the raw accel- eration data using the RPCA-RSSA module, that provides a set of transformed responses. TVAR modeling is carried out on the transformed response, yielding TVAR coefficients at each instant of time that are tracked online for any major and minor changes. The final form of Eqn. 4.15 therefore, becomes:

Y_R¹(k) =a₁(k)Y_R¹(k−1) +a₂(k)Y_R¹(k−2) +V(k) (5.18) The changes in the mean level of the plots of the TVAR coefficients indicate the exact instant of damage to the system. DSFs such as RRE are employed to further corroborate the instant of damage.

Considering the damage at the end of k^th instant, the subspace spanned by the eigenvector G_k+1 deviates in comparison to the subspace spanned by the eigen vectors at the previous time stamp G_k. To identify global damage to the structure, the temporal RRE can therefore, be evaluated as:

χRR−1 =

G¹_k∗G¹_k+1^T ∗Y_R¹(k)−G¹_k+1^T ∗G¹_k∗G¹_k+1^T ∗Y_R¹(k)

2

(5.19)

Once the damage instant is detected, the spatial module is invoked that further resolves the location of damage. Following the similar lines of development, the time series corresponding to each DOF, can be expressed as:

εRR−Y_i(t) = y^1∗

2

R (k)−y_R¹²(k)

(5.20)

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Concluding, the spatial RRE used for localizing the damage can be expressed, according to:

hεRR−Y_i(t)i=

K

P

k=1

y^1∗

2

R (k)−y¹_R²(k)

K (5.21)

A summary of the key steps involved in the proposed FOEP based example are enumerated below, for simplification:

1. First, batch PCA is applied on some initial points (around 100 in number) in order to obtain the initial eigenvector and eigenvalue estimates. It should be noted that the number of data points chosen here is arbitrary and considered only to stabilize the algorithm to facilitate subsequent real time damage detection. Similarly, batch SSA is also employed on approximately 100 sample points to get the initial eigen estimates for the RSSA algorithm.

2. The RPCA algorithm then operates online on the real time input of the streaming data.

This provides the set of transformed response at each instant of time. The sample covariance update at the present time instant is derived using the covariance estimate at the preceding time instant, employing the use of the recursive gain depth parameter. From the recursive updates shown in Eqn. 5.14, the eigenvector and the eigenvalue matrices are updated using the FOEP approach and the transformed responses (principal components) are obtained using the RPCA algorithm.

3. The transformed response obtained from the previous steps are now provided as input to the RSSA algorithm, that generates a Hankel matrix out of the set of the input responses. Using the FOEP approach, the recursive updates of the eigenvector and the eigenvalue matrices are updated at each time instant and a new set of transformed responses are obtained, evident from the Eqn. 5.4.

4. During the reconstruction phase, an approximate time series is obtained according to the relative order of significance as given by the decreasing order of the corresponding eigen values.

The proposed time series models are generated based on the approximated time series and a TVAR model is fit. The DSFs are tracked real time in order to extract the changes in the TH-1989_156104031

model coefficients, thereby revealing the faults in the system, facilitating real time temporal damage detection.

5. On identifying the damage instant, the algorithm shifts on to the next module where the spatial damage detection takes place. DSFs are tracked online recursively to capture the spatial effect of the damage. To further validate the instant and location of damage, local RREs are tracked recursively to provide information about the exact location of damage in the structure.

Initial Response

SPATIAL MODULE TEMPORAL MODULE

Y

=

W_k=W_k-1H_k

TVAR

t

_d Damage

Identify Instant

Estimate

Estimate where is significant

Local damage region

RPCA

Intial Covariance Estimate

RR RR

W0 ₀W₀^T

R0

Batch SSA on initial

samples

RSSA U_k=U_k-1P

k

Recursive DSFs RRE

First set of transformed response

Final set of transformed response

Yes

Previous eigenspac e

No

Figure 5.2: Basic framework of the proposed algorithm

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