Methodology for Recursive Computation

To consider the recursive identification procedure, if the new sampling data with the length of are added to the block Hankel matrix , the old data with equivalent length will be eliminated. The updated block Hankel matrix and its LQ decomposition is re-defined as:

(32) How to compute the new decomposition by using the new sampling data and the old decomposition results, , is the crucial issue in recursive identification.

The Givens rotations actually decouples the LQ decomposition of that the first columns of is returned to the original form of block Hankel matrix

. Remove from , Eq.(32) implies the remains can be represent as:

(33) where is orthogonal, and is close to be a lower triangular matrix. To accommodate the recursive procedure, the new data set is appended to the remains:

(34) where is orthogonal, and is close to be a lower triangular matrix. To make Eq.(34) become a complete LQ decomposition the Givens rotations is used again to transform the

into a real lower triangular matrix :

(35) For recursive identification with p-shift data point, Figure A1 show the summary of the related equations which were used in two consecutive time window. In summary, the first Givens rotations decoupled the old LQ decomposition so that the old data can be deleted. After appending the new data, the second Givens rotations as applied which make the temporary decomposition become a real LQ decomposition.

Moreover, a forgetting factor μ can also be used to improve the convergence of the recursive subspace identification by multiplying it to the past data sets in Eq.(34):

(36) The implementation of forgetting factor is to use the concept of fading memory by decreasing the weight of data points which were away from the current data. The idea of fading memory on the previous data can be used especially to detect the abrupt change of system modal parameters. For example, if the length of moving window is

assumed as and the shifting length is

(r and p indicate the number of data point), and the forgetting factor can be determined from with μ=0.9330, and then the weighting factor can be calculated and applied to the data set of the specified time window, as shown in Figure A2.

Figure A1: Correlation of Hankel matrix in recursive formulation from Data Set 1 to Data Set 2 (with shift p step).

Figure A2: Weighting factor for window length of 6 sec. applied on data between 15 sec and 21 sec.

REFERENCES

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Administration, Hydraulic Engineering Circular No.23, 2001.

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Theory-implementation and application, Kluwer Academic Publisher, Dordrecht, The Netherlands:, 1996.

[8] De Cock K., Mercre G., De Moor B. “Recursive subspace identification for in-flight modal analysis of airplanes,” Proc. Of the Inter. Conf. on Noise and Vibration Engineering (ISMA 2006), Leven, Belbium, Sept., 2006, pp:1563-1577.

[9] Chin-Hsiung Loh, Jian-Huang Weng, Yi-Cheng Liu, Pei-Yang Lin and

Shieh-Kung Huang, “Structural Damage Diagnosis Based on On-line Recursive Stochastic Subspace Identification,” Accepted for publication in J. Smart

Materials and Systems 2011.

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[18] I. Goethals, L. Mevel, A. Benveniste, and B. De Moor, Recursive output only subspace identification for in-flight flutter monitoring, Proc. of the 22^nd International Modal Analysis Conference (2004), Dearborn, Michigan.

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[20] J.N. Juang, Applied system identification (1994). Englewood Cliffs, NJ, USA:

Prentice Hall.

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Fig.1: Sketch and dimension of the bridge test specimen

Fig. 2: Photos of the bridge test site and the test specimen, (a) before scouring test, (b) under scouring test, (c) after scouring test.

Fig. 3: Recorded velocity response from node 1 and node 9 from the bridge scouring test on date 2011-01-26. The observation of scouring depth from each pier is also plotted for comparison.

Fig. 4: Time-frequency analysis from data at node 1 and node 9 by using STFT

Fig. 5: Wigner-Ville transformation of signals from record at sensor Node 2 and Node 9

Node 2

Node 9 Node 2

Node 9

Fig. 6: The relationship between the data in moving time window and the required computation time for each data set.

Fig. 7: Identified time-varying system natural frequencies using RSSI algorithm (Window length = 5 (sec) OMAC=0.98, SVD=0.15, N_row=50).

Time-sec (Data point) 1^st Data Set

2^nd Data Set

3^rd Data Set

t r

pt pt

Computation Time (sec)

0

Computation time for 1^st data set

Computation time for 2^nd data set

pt pt

Computation time for 3^rd data set

……

Delay Time (Computation time

vs. Data run time)

Fig. 8: Plot the correlation coefficient with respect to sensor location and time (use sensor 1 data as the reference)

Fig.9: Plot of CVAC with respect to time; (a) consider sensor node No.1 as a reference, (b) consider sensor node No.6 as a reference

Fig. 10a: Calculated 1^st eigenmode from Proper Orthogoanl Decomposition.

Fig. 10b: Root-mean-square value of the difference between reference eigenmode and the eigenmode calculated from different time window.

Fig.11a: Calculated 2^nd eigenmode from Proper Orthogoanl Decomposition.

Fig. 11b: Root-mean-square value of the difference between reference eigenmode and the eigenmode calculated from different time window.

Fig. 12: Mean value of time-varying Euclidean Norm;

(a) response at Node 2, and (b) response at Node 9.

Fig. 13a: Difference between the 1^st and 2^nd eigenvalue-ratio from Singular Spectrum Analysis on each measurement (for Nodes 3, 6, and 9).

Fig. 13b:Difference between the 1^st and 2^nd eigenvalue from Singular Spectrum Analysis on all set of measurements.

Fig. 14: Plot of RMS error between the measurement and the prediction using the reconstruction (from the two largest eigen values) wave forms of SSA.

Figure 15: Wireless data communication setup for field ambient vibration measurements.

Figure 16: The sensor locations along the bridge deck are also shown (in transverse direction)

Figure 17: Photos of the Nu-Dow old bridge before and during the typhoon period.

Figure 18: Identified time-varying system natural frequencies from sensor nodes of D05 and D14. Before and during the typhoon period.

Figure 19: Plot the re-arrange the recorded data from sensor node 5 (by putting the data during and after typhoon period back to back).

Figure 20: Result from STFT and WVD analyses on the re-arranged data from Figure 18.

Figure 21: (a) Plot of time-varying Euclidean norm from data at Node 5 and Node 14.

(b) Plot of time-varying differences between 1^st and 2^nd singular spectrum, (c) Plot of RMS error at node 5 and node 14 from singular spectrum

analysis.

Figure 22. Location of sensors in the New Nu-Dow bridge during flood (on 2011-10-3)

Figure 23. Recorded acceleration on the new Nu-Dow bridge on 2011-10-2 (flood period with high water level) and on 2011-10-18 (normal water level)

Figure 24. Stability diagram of the identified structural dominant frequencies from measurements of two different periods (normal vs. flood period).