Chapter 4 Application of SSI to the identification of Canton Tower
4.3 SSA-SSI-COV
4.3.1 Implementation
In this section, SSA will be used as a pre-processing tool in the sense of a
“subspace filter”, to extract first the principal components from the measurements, thus, to enhance the stability of SSI-COV. The SSA-SSI-COV procedure is listed as follows:
1. Assemble Hankel Data matrix (100~200 block rows are recommended). The number
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of columns will determine the available data point to construct the oncoming Toeplitz matrix. Usually the number of columns should be much larger than the number of block rows. The number of rows will determine the number of principal components the signal to be decomposed.
2. Perform SVD to the Hankel Data matrix in the step of SSA, from the singular spectrum (plot of the singular values) obtained in SSA, a preliminary set of principal components can be selected to reconstruct the signal.
3. Reconstruct the signal and repeat it for the set of choices of SV.
4. Construct Toeplitz matrix from the reconstructed signals, as it is done in SSI-COV.
5. Conduct SVD to the Toeplitz matrices and plot the singular spectrum (The size of Toeplitz matrix could be the largest number of block rows that will be reached in the stabilization diagram. 100~200 block rows is recommended for field noisy measurements).
6. Repeat step 4 and step 5 for the set of choices of SV from SSA.
7. Go from large to small number of components (SV) selected from SSA, and seek for the one whose singular spectrum in SSI-COV has a remarkable change of slope.
8. The best system order will be within the start and end of the change of slope, and the stabilization diagram can be constructed for pole discrimination.
The introduction of SSA before SSI-COV enables the determination of system order, which is totally subjective if SSI algorithms are used alone. The above described procedure will be demonstrated in the following Canton Tower identification task.
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4.3.2 Canton Tower identification through SSA-SSI-COV
Application of SSA-SSI-COV to the measurements of Canton Tower was studied.
In performing the SSA, 20 sensors measurements in form of vector were placed at once in Hankel data matrix with the following dimensions: 340 block rows (totally 6800 rows) by 15000 columns. The outcome singular spectrum is shown in Figure 4-8. It is difficult to select a suitable number of singular values from this figure because there is not any gap on the distribution of singular values obtained by SSA.
In the implementation of SSA-SSI-COV there are two parameters to be determined:
the first one is the number of Singular Values (SV) to be chosen by conducting the SSA, and the other is the system order to be determined in the SSI-COV analysis. From the experience gathered by working on the data of Canton Tower, a specific number of SV in SSA step leads to a change of slope in the singular spectrum obtained in SSI-COV (one can call it the first critical number of components). This is shown in Figure 4-9 b).
If the selected number of SV in the step of SSA continues decreasing, up to a second critical point the change of slope will become very sharp, almost a vertical jump, as in the case of 95SV shown in Figure 4-9 d). This latter phenomenon will remain as the number of SV chosen in SSA continues decreasing as that shown in e) and f). In this second critical point, usually the number of SV in the SSA step will be very closer to the system order, where is an almost vertical jump. From experience, the second critical point gives the best identification results, but those not well excited or highly contaminated modes will be also filtered out.
Different stabilization diagrams were made for comparison purpose. Figure 4-10 shows the result for different choices of SV, ranging from 0 Hz to 1 Hz. In the beginning, 312 SV were chosen from SSA, the jump in the singular spectrum of
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SSI-COV is almost imperceptible and only a few modes appear in the stability diagram.
When a smaller number of SV is selected, e.g., 136 SV in SSA step, as shown in Figure 4-9 b), the change of slope is a little more remarkable (this can be considered as approximately the first critical point), and the stabilization diagram shown in Figure 4-10 c) was improved. If one keep reducing the selected SV in SSA step, the diagram become more stable and starts earlier. However, in the case of Figure 4-10 d) when 95 SV were chosen, the 1st mode has been filtered out although a stable diagram starts even earlier than Figure 4-10 c) in which 136 SV were selected. Finally, as discussed above in the simulation section, although the number of SV selected are fewer than the required as in the case of Figure 4-10 e) and f), certain modes will not be discarded, but the diagram is totally stable just at the beginning, i.e., correct answers were found at a few block rows; in other words, these totally stable modes are free from noise perturbation after the pre-processing with SSA.
To understand the absence of the 1st mode when 95 SV are extracted from SSA.
The Fourier Spectrum of the response data is shown in Figure 4-11. Title of the figure indicates sensor number. In the case of 95 SV, the major peaks are covered, but the peak corresponding to the first mode was almost totally filtered out, which is originally very small comparing to others and looks fuzzy and blended with the noise frequency. By increasing the number of selected SV to 136 (Figure 4-12), in this case, the peak corresponding to the 1st mode is conserved, however, the noise filtering is not as good as it is the case of 95 SV. Therefore, 95 SV is slightly better in terms of stability diagram tan 136 SV. Comparing Figure 4-11 and Figure 4-12, one can note that there is another peak at 1.2 Hz just filtered out by SSA using 95 SV.
The stabilization diagram with frequency ranging from 1 to 5 Hz is shown in
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Figure 4-13 a) and b), for the cases of 136 and 95 SV respectively. Finally, complex mode shapes together with the modal frequency and damping ratio are shown in Figure 4-14 and Figure 4-15 respectively. A summary and comparison of the identified system modal frequencies with the Finite Element Model of the structure is shown in Table 4-1.
Similar to the outcome from SSI-COV, the first two identified frequencies by SSA-SSI-COV: 0.0345 Hz and 0.0465 Hz (using 95 SV), which are probably wind frequencies, whose mode shapes plotted in complex plane appear without any regularity.
Although the fundamental mode was found by extracting 136 SV, unlike the other modes which appear almost in a straight line (meaning that the structure has almost-proportional damping) this 1st mode is the unique which has the lowest value of R excepting the wind modes, which is equal to 0.7. Here one can conclude that, from the several choices of SV from SSA, the use of 95 SV leads to the best stability. But to achieve a better identification quality of the 1st mode shape, a larger Toeplitz matrix size may be needed.
4.3.3 Canton Tower identification through SSA-SSI-DATA
A unique remaining question is that, if SSA can be combined with other SSI algorithm as a pre-processing tool, such as SSI-DATA? The stabilization diagram using SSA-SSI-DATA is shown in Figure 4-16. From this result one can conclude that SSA serves as a preprocessing tool only in conjunction with SSI-COV but not for SSI-DATA.
The result obtained by SSA-SSI-DATA is worse than that applying directly SSI-DATA.
This may be explained by the fact that, the orthogonal projection used in SSI-DATA is trying to find the best fit by least square of the future measurements in terms of the past data. However, the principal components recovered by SSA used for reconstruction of
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the signal may provide a bad fitting in the projection and worse results were obtained.