Chapter 8 Conclusions
8.2 Recommendations for future work
The long-term goal of the research in SHM is the application of the system identification and damage detection methods to the real scale structures, which in fact are much more complex, and in addition to the environmental factors, both the automatization of the identification process for continuous monitoring, as seeking for
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reliable criterias and indexes to translate the system identification outcome into a safety or warning message are challenges to be overcome. Therefore, further researches and experiences with large scale structures are needed.
The computation speed is another difficult to be overcome in the online application.
In the bridge scouring experiment, all 12 sensors were required in the application of rSSA-SSI-COV to realize accurately the modal frequencies and mode shapes, however, using a modern computer and in terms of the selected rSSA-SSI-COV parameters, the computation consumed about 10 times more than the required timing for online application. Although with reduced number of sensors and reducing the number of block rows (less accuracy) the timing requirement can be satisfied, the tracking result for modal frequencies is more scattered and is not as clear as that obtained with full measurements; furthermore, it is impossible to recover a good mode shape if only a few points are available, and thus, damage location cannot be identified. A feasible solution to increase the computation speed is through the developement of parallel computation algorithms which can exploit the full computation potential of the microcontroller of each sensing unit.
Although the system damage and location can be identified correctly in this study, there is still a lack in this research about damage quantification and the estimation of the remaining service life. Since SSI-based algorithms are able to accurately identify the modal parameters, a possible approach is through the finite element model updating of the mass, dampin and stiffness matrix, therefore the damage can be quantified and the remaining service life can be assessed.
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Appendix A: Frequency Domain Decomposition (FDD)
The Frequency Domain Decomposition can be considered as an SVD-enhanced power sepctrum, and it applies when it is the case of multiple measurements. The procedure starts from estimating the power spectrum density matrix, which is formed applying Discrete Fourier Transform (DFT) to the Covariance matrix shown in (2.22), for time lag k ranging theoretically from minus infinity to infinity:
( ) ∑∞ straight-forward way to compute the spectrum estimated called modified Welch’s periodogram [54], which begins by calculating the DFT of the weighted measured
where wk is a window function to avoid leakage. n is a segment of the total data length N. If n is a power of 2, DFT can be efficiently computed using FFT at the discrete function, but averaging over all available samples of DFTs, i.e., periodogram:
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Overlap is permitted in the sampling and, if higher frequency resolution is required, the sampled segment can be zero-padded.
Finally, SVD is applied to the spectrum matrix (which is a complex valued matrix) for each discrete frequency ωρ. The set of major singular values can be plotted against frequency and a singular spectrum in frequency domain will be obtained. The advantage of using FDD over the traditional power spectrum is that, information of multiple sensors can be gathered and combined in only one outcome, especially if it is the case of closely-spaced frequencies, its effects will be reflected in the singular values, e.g., if there are two close frequencies, weight of the second singular value will become closer to the first one.
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Appendix B: Prediction Error Method through Stochastic Subspace Identification (PEM/SSI)
The objective of this appendix section is to show the similarity that exists between the system realization algorithm based on autoregressive (AR) model, solved through prediction error method/least square (PEM/LS), and SSI-DATA. This enables one to modify the traditional least square PEM to a subspace approach.
Prediction error methods PEMs are very common and widely used system identification methods. The main idea is to identify a system of linear equations in the sense that: based on past inputs and outputs, can predict any output. For the special case of multivariate output-only measurements, these models are known as autoregressive with moving average vector ARMAv [59].
In the output only system identification, the traditional PEM is carried out using a two-stage least squares approach. The autoregressive AR model can be written as:
i
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available data length for least square.
The error sequence obtained from least square fitting can be used as pseudo-inputs of the system and therefore, a pseudo-ARX model (AR model with exogenous input) can be built and again, fitted with least square:
ic matrix coefficients. The AR model coefficient matrices Ak are, therefore, arranged and associated with the state-space model as that is done in the conventional state-space realization algorithm [61]. This two step algorithm is also knwon as the AR-ARX method. However, our target is to compare the PEM/LS algorithm with the orthogonal projection used in SSI-DATA, to later, be able to apply the SSI procedure to PEM.
Actually, the equation (B.2) can be rewritten as a simple least square problem in the notation of SSI:
error
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αAis the matrix containing the AR model coefficient matrices Ak.
The well-known least square solution for αA and, YˆTf/1,1, the best estimate of
where (B.5c) is simply the transpose of (B.5b), and one can realize that (B.5c) is almost the same expression that (2.34), the only difference is that (2.34) gets at once the best estimate for whole future measurement Yf /1,i while only one block row is obtained by (B.5c).
Therefore, the projection matrix obtained in (2.38) can be interpreted as, in fact, the best estimate of the future output in terms of the past outputs in a least square sence, i.e., orthogonal projection of the future data in the past data. Therefore, PEM and SSI-DATA can be combined and stated at the following:
First, obtain the projection matrix, i.e., the best estimates of the future data From (2.37) and (2.38). Then, the error matrix can be obtained by computing the difference between the estimate data matrix and the original data matrix:
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averaged along its antidiagonal as that is done for SSA (section 2.7) and considered as pseudo-inputs. In a similar way than the projection for SI using input-output data [47], and using the LQ decomposition formulated before, the Hankel data matrix can be now re-arranged by inserting the corresponding pseudo-input matrix:
where ic is the order selected for the pseudo-inputs, and since the first i data points were used in the least square fitting, for this second stage the data begins at point i+1, j’ is used for the row length instead of j, since length of this latter is no longer the same.
A new projection matrix can be obtained by performing a similar LQ determined as presented before in the section 2.3.
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In fact, the LQ decomposition of the Hankel matrix shown in (B.7) is the transpose version of the least square problem shown in (B.3):
error simple words, PEM/SSI is actually a 2-step-projection SSI-DATA algorithm, since the orthogonal projection is done twice. The performance of this algorithm is tested and discussed in Chapter 3, where simulation study is carried out.
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Appendix C: Novelty index through Kalman-Filter-based prediction error
If a structure is subject to damage, the system matrix A will be changed, therefore, if the system information is stored at the undamaged state and used for prediction of the structure response, it is expected that, as the damage is accentuated, the prediction will deviate in a higher degree from the measured response. Statistics made from this prediction error serve as useful indexes for damage detection [62]. Discrete-time Kalman filter with unknown inputs is the instrument used to perform this prediction error task.
To make the Kalman Filter more adaptive, the version of Kalman Filter shown in chapter 3 of [60] is used, which consiste of 2 states: prediction state and updating state.
Formulas to be implemented are shown below, detail derivation of each statement can be find in [60].
Firstly assume an initial system state xˆ (0/0) and prediction error covariance P(0/0) and later, using the system matrices (A,C) and the stochastic noise covariances (Q,R) computed by SSI-DATA algorithm shown in section 2.5, the Kalman filter algorithm can be implemented as follows:
(1) Given the state xˆ (k/k) and P(k/k), compute the predicted state:
(
k 1/k)
ˆ( )
k/kˆ x
x + =A (C.1)
where (k+1/k) means the predicted state at k+1 step from step k.
(2) Compute the predicted error covariance matrix:
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( )
AP( )
A QP k+1/k = k/k T+ (C.2)
(3) Compute the Kalman gain matrix:
(
+1) (
=P +1/)
C[
CP(
+1/)
C +R]
−1Kk k k T k k T (C.3)
(4) Compute the updated system state from the measurement y(k+1)
(
1/ 1) (
ˆ 1/) ( ) ( )
1[ 1 ˆ(
1/)
]ˆ k+ k+ =x k+ k + k+ y k+ − x k+ k
x K C (C.4)
where (k+1/k+1) means the updated state at k+1 step from measurement at k+1.
(5) Compute the updated error covariance matrix:
(
k+1/k+1)
=[I−K(
k+1) (
C]Pk+1/k)
[I−K(
k+1)
C]T +K(
k+1)
RKT(
k+1)
P (C.5)
(6) set k = k+1 and return to step (1).
This 2-state Kalman filter differs from the Forward Innovation model [62], because the Kalman Gain of this latter is constant and it is obtained by solving Riccatti equation.
However, the 2-state Kalman Gain is updated at each state by the prediction error covariance.
The prediction error can be computed at the predicted state as:
(
k+1/k+1) ( )
= y k+1 − xˆ(
k+1/k)
e C (C.6)
since xˆ (k+1/k) is determined only by the system matrix A, degree of change in the system will be reflected directly in the computed prediction error. The advantage of the use of the 2-state Kalman filter is that, the update state will correct this deviation at every step, which could be very large when the system was severely changed.
Several statistical indexes can be defined based on the computed prediction error
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or as the Mahalanobis norm:
k sequence being considered.
Moreover, an outlier analysis can be done from the norm sequences. The mean θ′and standard deviation σ′ can be calculated from the undamaged state:
∑
where the prime sign indicates undamaged state. An upper control limit can be defined as a horizontal line:
σ counting how many times the prediction error norms (in % of total samples) are passing over the upper limit, in the given windowed sequence. Additionally, the ratio of the
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mean values and standard variations of NI, between damaged and reference states respectively, can also be used as damage indicators.
A similar strategy can be applied to each sensor individually and (C.7) become a simple Root-Mean-Square of the error sequence. Then, (C.9) and (C.10) can be applied to the sampled data of each sensor individually; comparison between the outcome of each sensor allows to find the damage location, since is expected that the damage occurs in the place where is larger the prediction error.
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Table 2-1 Comparison of identification results of SSI-COV Excitation level velocity measurements (noise free data)
Square
Acceleration measurements (noisy data) Square
* Number of rows for square Toeplitz matrix, means the number of block rows and columns from the beginning to the end of the stabilization diagram. But for rectangular Toeplitz matrix the first value is the number of block columns
* Number of rows for square Toeplitz matrix, means the number of block rows and columns from the beginning to the end of the stabilization diagram. But for rectangular Toeplitz matrix the first value is the number of block columns