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8/9/2019 Stochastic Subspace Identification for Output-only Modal Analysis: Accuracy and Sensitivity on Modal Parameter Esti
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Stochastic Subspace Identification for Output-only Modal Analysis:
Accuracy and Sensitivity on Modal Parameter Estimation
Yi-Cheng Liua, Chin-Hsiung Loh*
a
National Taiwan University, Department of Civil Engineering, Taipei 10617, Taiwan
ABSTRACT
In this study an output-only system identification technique for civil structures under ambient vibrations is carried out,mainly focused on the Subspace System Identification (SSI) based algorithms. With the aim of finding accurate and true
modal parameters,a stabilization diagram is constructed by plotting the identified poles of the system withincreasing the
size of data matrix. Comparative study between different approach, with and without Singular Spectrum Analysis to pre-process the data, on determining the model order and selecting the true system poles is examined in this study.
Identification task of the real large scale structure: Guangzhou New TV Tower (GNTVT), a benchmark problem for
structural health monitoring of high-rise slender structures is carried out, for which the capacity of SSI-based algorithm
is demonstrated.
Keywords:State-Space model, Stochastic Subspace Identification, Singular Spectrum Analysis
1. INTRODUCTION
The basic methodology of subspace identification algorithm is recalled through the introductions of two different
categories: stochastic subspace identification (SSI) and subspace identification [1]. The stochastic subspace identification
uses output-only data to identify the system parameters while the subspace identification used both input and output data
to extract the system parameters. For output-only measurements the Stochastic Subspace Identification (SSI) technique is
a well known multivariate identification technique. It was proved by several researchers to be numerically stable, robustto noise perturbation and suitable for conducting non-stationarity of the ambient excitations although its stationary
assumption is violated. There are several varieties of SSI technique such as covariance driven SSI-COV, data driven SSI-
DATA, or combined with other methods like Expectation Maximization technique SSI-EM. Application of SSI-based
algorithm to determine the dynamic characteristics of structures is through the use of ambient vibration measurements. Inoutput-only characterization, the ambient response of a structure is recorded during ambient influence ( i.e. without
artificial excitation) by means of highly-sensitive velocity or acceleration sensing transducers. The stochastic realizationalgorithm was fully enhanced by Van Overschee and De Moor [1,2] along with a set of MATLAB script [2,3].
Application of the SSI-DATA algorithm to investigate the dynamic characteristics of a cable-stayed bridge had been
studied [4]. To investigate the accuracy of SSI-COV method to extract the system dynamic characteristics, in this study,
firstly, a numerical simulation will be performed to determine the accuracy and sensibility of the identified modal
parameters under different proposed scenarios. Comparison with different identification techniques was also discussed.Secondly, identification task of the real large scale structure: Guangzhou New TV Tower (GNTVT), a benchmark
problem for structural health monitoring of high-rise slender structures will be carried out, for which the capacity of SSI-
COV algorithm will be demonstrated.
2. OUTPUT-ONLY COVARIANCE-DRIVEN
STOCHASTIC SUBSPACE IDENTIFICATION (SSI-COV)
Assuming a structure under consideration is being excited by unmeasurable input forces, the discrete time stochasticstate-space-model can be expressed as:
kkk
kkk
x
x
vCy
wAx
+=
+=+1 (1)
*[email protected], phone: +886-2-3366-4248, fax: +886-2-2739-6752
Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2011, edited byMasayoshi Tomizuka, Chung-Bang Yun, Victor Giurgiutiu, Jerome P. Lynch, Proc. of SPIE Vol. 7981, 798149
2011 SPIE CCC code: 0277-786X/11/$18 doi: 10.1117/12.876619
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where 12 nkx is the state vector and1 lky is the measurement vector, wkand vkrepresents the system noise and
measurement noise respectively. The covariance-driven Stochastic Subspace Identification method (SSI-COV) stems
from the need to solve the problem through identifying a stochastic state-space model (matrices Aand C)from output-
only data. The first step is to gather the measurement vectors in a Hankel Data matrix:
=
=
++
++++
+++
++
+
f
p
1N2i12i2i
1Ni3i2i
Ni2i1i
1Ni1ii
1N32
N21
...
............
...
...
...
...............
...
N
1
Y
Y
yyy
yyy
yyy
yyy
yyy
yyy
H (2)
where Ypdenotes the past measurements and Yfdenotes for the future measurements. It can be easily find that the block
Toeplitz matrix can be obtained by a multiplication between future and transpose of past measurements:
( )Tpf
ii
i
YY
RRR
RRR
RRR
T =
=
+
i2212
2i1i
11-i
1/i
...
...............
...
(3)
where the block output covariance with time lag iis defined as Ri:
][ T ikki E = yyR (4)
The Toeplitz matrix can be factorized into the extended observability matrix nlii
2O and the reversed extended
stochastic controllability matrix lini
2 , as shown below:
[ ]GAGGACA
CA
C
OT ......
1
1
1/i
==
i
i
ii (5)
Singular Value Decomposition (SVD) is the tool used to perform the above mentioned factorization:
( ) TT
T
T VSU 1112
11
211/i00
0=
==
V
VSUUUSVT (6)
where liliU and liliV are orthonormal matrices, and Sis a diagonal matrix containing positive singular values indescending order. Comparing (5) and (6), the matrix Oiwhich contains the system matrices (Aand C) can be computed
by splitting the SVD in two parts:2/1
11 UO =i (7)
Ti 1
2/11 VS = (8)
From Oimatrix, the system matrices (A and C) can be obtained easily. In MATLAB notation, the Cmatrix is just the
first block of Oi:
( ):l,:i 1OC = (9)
System matrix Acan be computed by exploiting the shift structure of the extended observability matrix Oi:
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A
CA
CA
C
CA
CA
CA
=
1
2
......ii
(10)
Therefore:
( )( ) ( ):li,:l,:i:l ii 111 += OOA D
(11)
where ( )D denote pseudo-inverse.The modal frequencies and effective damping ratios can be computed by conductingeigenvalue decomposition of the system matrix A, and the corresponding eigenvectors multiplied by the output matrix C
arethe observed mode shapes.
As opposite to the covariance-driven stochastic algorithm, the data-driven method (SSI-DATA) avoids the calculation
of covariance. Instead, the data reduction step is accomplished by projecting the row space of the future outputs into therow space of past outputs. Covariances and projections are closely related, in that they are both intended to eliminate
uncorrelated noise contributions.
3. DISCUSSIONS OF SSI-COV FROM THE SIZE OF TOEPLITZ MATRIX
The first step of SSI-COV method is to form the covariance blocks with correlation of time lag i among themeasurement data, and form the Toeplitz matrix which has the same elements along its diagonal. The Singular Value
Decomposition (SVD) is then used as the principal tool to extract the system information from the Toeplitz matrix. It isimportant to note that if full sensors are used to establish the correlation matrix, the formed covariance block is a square
matrix, but Toeplitz matrix may or may not have the same number of block rows and columns. This depends on different
ways to construct the stabilization diagram. In real world applications, noise is always present at any measurement, and
there is no prior information about the number of modes that can be extracted from the data. Therefore, a stabilization
diagram is used to discriminate between noise or spurious poles and true system poles. There are several ways to buildthe stabilization diagram:
1stversion:Decide first the maximum dimension of the Toeplitz matrix, perform SVD, and let the system matrix order
increases from a lower value till reaching a user defined maximum dimension of the Toeplitz matrix. The advantage of
this version is that only one SVD has to be performed; less time is consumed in the construction of the stabilizationdiagram. The drawback is that, there is no clear criterion to ensure that the chosen maximum dimension is sufficient or
not to reveal true system information. With increasing order of the system matrix A, more noise information which isseparated from the measurement data by SVD will be included in the system matrix Aand consequently, more noise or
spurious poles will appear on the diagram.For this purpose, modal transfer norm [5] was introduced in addition, to clearout the large number of spurious poles at higher orders thus clarifying the stabilization diagram.The concept behind thisversion to construct the stabilization diagram is that, even including more spurious poles in the system matrix A, and the
true modes (frequency, damping, mode shape) extracted from Awill remain stable.
2ndversion:Determine the order of system matrix Aby observing the variation of the singular values, and then, increase
the size of the Toeplitz matrix both rows and columns holding the order of system matrix unchanged. For convenience,this version will called the square Toepliz matrix or original form because it increments both columns and rows, and
the Toeplitz matrix remains squared. The main drawback of this is, first, time consuming, and second, the system matrix
order must be defined previously. For field measurement data, generally there is no clear gap on the distribution of
singular values as it appears in the numerical simulation. The advantage of this version is that, one do not have to try atthe beginning the maximum Toeplitz matrix size, since its required size may vary from case to case. Increase of the
Toeplitz matrix dimension, meaning that more orthogonal components will be decomposed from the signal and therefore
a better separation between signal and noise, means also an increment of the order of extended observability matrix,
since the system matrix Ais extracted by taking advantage of the shift structure of matrix Oias shown in equation (10),clearly one can see that the action of pseudo-inverse of Oito determine the system matrixAhas a least square sense, i.e.,
more information is used to determine the modal parameters and therefore, more and more accurate estimate is expected
to achieved by increasing the Toeplitz matrix order.
3rdversion:This is a modification of the second version. Since Toeplitz matrix has not necessary to be a square matrix,
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an alternative way to construct the stabilization diagram is to increment only block rows of the Toeplitz matrix by
keeping number of block columns and system matrix Aorder constant [12]. For convenience, this version will called therectangular Toeplitz matrix or alternative form because it only increments rows but not columns. The advantage of
this method is that, it conserves the data addition property of the stabilization diagram in a least square sense (it is much
faster than the 2ndversion). The difficulty on the choice of number of block columns constitutes the main drawback ofthis method, since in this version block rows are always greater than columns, but the number of block columns will
determine how many components will be decomposed from the covariances of the signal. If the number of columns ofthe Toeplitz matrix is lower than the required, it will lead to an unstable diagram on the estimation of the lower modes in
the presence of noise which can lead to poor estimation of modal parameters. On the contrary, the use of square Toeplitz
matrix have not to worry about the noise effect and the determination of number of columns, since noise in the
measurement data will only delay the outcome of a stable diagram.
Having reviewed the advantages and drawbacks of each way to construct the stabilization diagram, the use of rectangular
Toeplitz matrix may be recommended for systems because of the prior knowledge about the noise content, but in the first
time identification, the use of square Toeplitz matrix is recommended although it implies more time consuming
As a short summary, SSI-COV technique is robust against noise and signal non-stationarity. The method strongly relies
on SVD, which decompose the projection or covariance of the signal by taking advantage of the orthogonality betweenvectors of the obtained basis U and VT in SVD. Eitherafterprojection (SSI-DATA) or after correlation (SSI-COV), there
will always exist some residuary non-removed noise or non-stationarity signal, particularly in the case of non-white
ambient excitation. This situation may violate the assumptions of SSI algorithms, but its effects can be dropped out byincreasing the size of the projection or the covariance matrix, because more orthogonal components the matrix be
decomposed (by selecting only the subspace corresponding to those theoretically non-zero singular values) , better the
system-related information will be clearly separated from the noise.
3.1 Sensitivity study of the SSI-based algorithms
A comprehensive numerical simulation task was carried out to understand the sensitivity of SSI-based algorithms subjectto different factors and with special attention in their effects on the stabilization diagram. Instead of detail analysis only
the main results and conclusions will be presented here:
a. Noise effect in modal parameters: A 6-DOF shear type building model with well spaced frequencies wasconstructed to simulate structure response subjected to white noise excitation. In this simulation study, if a spatially
white noise was added after the system response was generated (either acceleration or velocity measurements ), there is
not any problem for SSI-based algorithms to identify accurately the modal frequencies since the first step of these
algorithms such as correlation or projection can effectively cancel out the added white noise once at all.Therefore, even
adding 200% of noise in RMS sense (noise to signal ratio), error of the identified frequencies is less than 4% for the 1 stmode and less than 1% for the remaining modes. Even if one adds a noise which is correlated with the output
measurement (such as in acceleration measurements, the external input acceleration can be treated as a measurementnoise), by increasing the order of projection or covariance matrix, frequencies can be accurately identified, the same with
the mode shapes. By observation, lower modes are more affected by the noise effects and take more time to stabilize
meanwhile the matrix order increase. On the contrary of modal frequencies, damping is very sensible to the addition of
any type of noise even it is very small, the error can be too big as 100%.
b. Non-linearity in the signal:Consider a nonlinear SDOF with Duffing model [13]. Results indicate that thelinear SSI-based algorithms are treating the nonlinear response as an equivalent linear system, and its equivalent modal
parameters can be obtained effectively. The nonlinearity of signals do not interfere the stability of the stability diagram.
c. Two closely-spaced frequencies blended with time-varying signals: Although it is not appropriate to use linear
SSI algorithm and stabilization diagram to identify time-varying signals, but the results can be used to check thestabilization diagram of the time-varying system.The closely-spaced frequencies will appear as only one frequency if
only a few model orders (order of system matrix A) were chosen. As the model order increases to a sufficient level, the
two close-spaced frequencies will be split which is a well known phenomena in stabilization diagram.Since there is a
time-varying signal blended in with the same power, it contains, in other words, a lot of frequencies and consequentlythe closely-spaced frequencies cannot be revealed until the system order reaches a certain number. On the other hand, the
time-varying component will appear in the stabilization diagram as a combination of a lot of equivalent frequencies
which will not stabilize at all in the diagram but they follow certain pattern.
d. Noise effect on closely-spaced frequencies: A 2-DOF system with natural frequencies of 7.99 Hz and 8 Hz.
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Even with such closely-spaced frequencies, in the noise free case, there is not any problem for SSI-based algorithms to
identify these two frequencies. However, once the noise is added only one frequency can be identified. If a goodsubspace based pre-processing tool is used, such as SSA which will be discussed in section 4, noise can be filtered out
before entering to the SSI algorithm and therefore, even with noise disturbance, the two closely-spaced frequencies can
be identified correctly.
4. PREPROCESSING FOR SSI-COV ANALYSIS: SINGULAR SPECTRUM ANALYSISSSA is a novel non-parametric technique used in the analysis of time series based on multivariate statistics. This method
was firstly applied to extract tendencies and harmonic components in meteorological and geophysical time series [6].
Except the extraction of tendency, SSA can be applied to smooth a noisy signal, to extract seasonality component, or todetect the singularities. Basically, this method is capable of decomposing the original series into a summation of
principal components, so that each component in this sum can be identified as a tendency, periodic components(stationary), nontationary signal or noise. The SSA procedure consists of four steps: (1) embedding, (2) singular value
decomposition (SVD), (3) grouping, and (4) reconstruction. Detail description of each step can be found in [6].
In this study SSA will be used as a pre-processing tool to extract the principal components from its measurements.
In the beginning, the frequency domain decomposition (FDD) is employed on the original measurements to construct the
distribution of the singular values with respect to frequency which can be served as the reference for identifying the
close-space model frequencies. Since both SSA and SSI-COV depends primary on SVD, and it is reasonable that the
number of singular values be chosen in SSI-COV must be less than that chosen in SSA; in the best case, when thenumber of components chosen in SSA coincide exactly with that of the system, i.e., noise was totally filtered. The main
drawback of the use o SSA as a pre-processing tools before SSI-COV is that, the obtained damping ratio estimate is notreliable since it seems to be always much lower than the true damping ratio. The reason is that, SSA can only extract the
sufficient information to obtain an accurate estimate of the frequency, but a great portion of the signal mixed with noise
was filtered out loosing in this way the possibility to obtain a good damping ratio estimate.
Based on the numerical study the summary of the SSA-SSI-COV procedure is listed as follows:
1. Assemble Hankel Data matrix with number of block rows as large as possible (this usually depends on the memorycapacity of the computer and the time available to conduct the SVD), and the number of columns will determine the
available data point to construct the covariance matrix (Toeplitz matrix). Usually the number of columns is much
larger than the number of block rows, and this latter will determine the number of principal components the signal to
be decomposed.2. After performing SVD to the Hankel Data matrix in the step of SSA, from the plot of singular values one can
preliminary select a set of principal components to be used to reconstruct the signal.3. Reconstruct the signal. (Repeat it for different set of choice)4. Construct the covariance or Toeplitz matrix by doing correlation with the reconstructed signal.
5. Conduct SVD to the covariance matrix and plot the variation of singular values. (The size of the covariance matrix
will be the largest number of block rows that will be reached in the stabilization diagram)
6. Repeat step 4 and step 5 for different choice of SV from the result of SSA.
7. Selecting the one which the change of slope is remarkable in the SV distribution and check if the required informationis included through the plot of Fourier spectrum.
8. Decide the system order which will be within the start and end of the change of slope, and construct the stabilization
diagram.
5. APPLICATION: IDENTIFICATION OF GUANGZHOU NEW TV TOWER (GNTVT)
The Guangzhou New TV Tower (GNTVT) is located at Guangzhou, China. It is a super high-rise tube-in-tube structure
with a height of 610 m, as shown in Figure 1. This structure comprises a reinforced concrete inner tube and a steel outertube with concrete-filled tube (CFT) columns. There are 37 floors connecting the inner tube and the outer tube. The outer
tube consists of 24 CFT columns, uniformly spaced in an oval while inclined in the vertical direction. The inner tube is
an oval shape but with constant dimension of 14m by 17m in plan, but its centroid differs from the centroid of the outer
tube. The Hong Kong Polytechnic University is in charge of the implementation of the long-term SHM both during theconstruction as in the service stage. More details can be found in references [7-9]. The data were recorded from 18:00
pm on 19 January 2010 to 18:00 pm on 20 January 2010, lasting 24 hours. I The acceleration, wind direction, wind speed
and ambient temperature were measured during the period.
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Twenty uni-axial accelerometers (Tokyo Sokushin AS-2000C) were employed for vibration measurements, the
frequency range is DC-50 Hz (3dB), amplitude range 2 g, and the sensitivity 1.25 V/g. They were installed at eightlevels as shown in Figure 1, the 4 th level and the 8th level were equipped with four uni-axial accelerometers, two for
measurement of the horizontal acceleration along the long-axis of the inner structure and the other two for the short-axis.
At the other six levels, each section was equipped with two uni-axial accelerometers, one along the long-axis of the innerstructure and the other along the short-axis of the inner structure. Figure 1 also shows a plan of the section and the
measurement directions of acceleration. The sensors were fixed to the shear wall of the inner structure via a steel angle.The sampling frequency of the acceleration and wind data was 50 Hz. Figure 2 shows the acceleration measurement at
the first minutes of the 1st and 20thsensor which are located at a lower and a higher place of the tower respectively.
Signals of the 20thsensor appear to contain a long period signal (about 2 seconds) as compared to the measurement of 1st
sensor.
5.1 Frequency Domain Decomposition (FDD)
Before applying the SSI analysis, the FDD was used to identify the possible number of modes contain in the response
measurements. The FDD spectrum was calculated using 131072 points, Welchs periodogram [10] was used to estimate
the power spectrum density function with a window length of 8192 points and an overlap of 4096 points which leads to apower spectrum estimate obtained by averaging a total 31 Fourier spectrums. Since the sampling rate is 50 Hz, the used
of window length of 8192 points leads to a frequency resolution of 0.006 Hz. The result of FDD spectrum is shown in
Figure 3.
Figure 1: Locations of accelerometers along the GNTVT. A floor section shows the position of accelerometers.
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Figure 2: Acceleration measurements at the first minutes of the record for a) 1stsensor and b) 20
thsensor.
Figure 3: Plot the results of Frequency Domain Decomposition (0 to 3 Hz) from GNTVT measurements.
5.2 Covariance Driven Stochastic Subspace Identification (SSI-COV)
In the beginning, to construct the stabilization diagram using SSI-COV, 18000 points were used in the correlationfunction, and the size of square Toeplitz matrix increases from 5 to 300 block rows, all 20 sensors were used in the
computation. By observing the variation of singular values, one can note that there is not any significant gap between the
extracted singular values. It is hard to decide the system order. As an extra help, one can count the number of peaks that
appear in FDD spectrum multiplied by two to get an estimate of the number of singular values. In our case, 90 singular
values were chosen. The initial result is shown in Table 1 (1 strow). The results from SSI-DATA are also shown in thistable for comparison (2ndrow).
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F E M M 0 d e N u m b e r
I 2 3
4
5
C 7
F r e q
w i n u w i n a n . 1 1 b b
b . 1 5 9 b
u . 3 4 T h u . 3 b b u
u . 4 n n n u . 4 b l b b . 4 b 5 n
m m I - c o y I C b b b p n i n l n
3 m m m w
F r e q
n . n 4 n 4 n . n 4 n n n . n n n 2 b . 1 3 n 2 n . 3 e 5 2 b . 4 2 4 3 b . 4 7 5 2 b . S n e n
D a m p
m . 2 b b 5 I . m m m m m . m e n a m m 1 3 5 - m . m m m 4 - m . m m m e m . m m I C
m . m m I C
m m I - D A T A e m m m p m i n l m
2 C m m w
F m e q n . m 1 W m . m 3 3 a m . 1 3 7 7 m . 3 C 5 5 m . 4 4 m 2 m . 4 7 7 4
D a m p
I . m m m m m . m 7 2 7 m . m m e m - m . m m I a m . m m 4 e
- m . m m m 2
m m A - m m I - C o y
C a m y m m a e m C m
I 7 m w
F m e q m . m 3 7 e m . m 5 7 2 m . 1 3 8 8 m . 3 8 4 4
m . 4 2 3 8
m . 4 7 5 7 m . 5 m 8 3
D a m p
m . 2 4 1 2
m . 3 8 8 1 n . m 1 2 2 m . m m 4 2 m . m m a n m . m m m e m . m m a I
m m A - m m I - C D V
I a e m v m m a e m I 2 m
I 4 m m m w F m e q m . m 3 4 5 m . m 4 8 5
m . m 8 5 8 m . 1 3 8 1 m . 3 8 4 8 m . 4 2 3 2
m . 4 7 5 8 m . 5 m 8 3
D a m p m 2 3 8 8 m 1 8 7 8
m m 7 m m
m m I m e
m m m 2 4 m m m m c m m m m c m m m 2 4
T h e o r e t i c a l M o d e N u m b e r
8 9 1 0 1 1 1 2 1 3 1 4 1 5
T h e o r e t i c a l F r e q 0 . 7 3 8 0 0 . 9 0 2 0 0 . 9 9 7 0 1 . 0 3 8 0 1 . 1 2 2 0 1 . 2 4 4 0 1 . 5 0 3 0 1 . 7 2 6 0
S S I - C O V 1 8 0 0 0 p o i n t s
3 0 0 r o w
F r e q
0 . 5 2 2 3
0 . 7 9 8 2 0 . 9 6 4 9 1 . 1 5 0 7 1 . 2 0 3 1
1 . 2 5 2 5
1 . 3 8 9 1 1 . 6 4 0 1 1 . 9 4 6 3
D a m p 0 . 0 0 0 0 0 . 0 0 4 1 0 . 0 0 1 3 0 . 0 0 0 2 0 . 0 0 5 8 0 . 0 0 2 2
0 . 0 0 3 6
0 . 0 0 2 2 0 . 0 0 5 3
S S I - D A T A 8 0 0 0 p o i n t s
2 8 0 r o w
F r e q
0 . 5 1 9 3
0 . 7 9 7 7 0 . 9 6 5 4 1 . 1 7 9 4 1 . 2 2 8 1
1 . 3 8 4 8
1 . 6 3 9 7 1 . 9 4 2 8
D a m p - 0 . 0 0 1 7 0 . 0 0 0 9 - 0 . 0 0 0 4 0 . 0 0 1 7 0 . 0 0 1 0
0 . 0 0 0 8
0 . 0 0 0 3 0 . 0 0 1 4
S S A - S S I - C O V
9 5 S V o r d e r 9 0
1 7 0 r o w F r e q
0 . 5 2 2 6
0 . 7 9 8 6 0 . 9 6 5 3 1 . 1 5 1 2 1 . 2 5 1 7
1 . 3 8 9 9
1 . 6 4 0 7 1 . 9 4 4 6
D a m p - 0 . 0 0 0 8 0 . 0 0 2 2 0 . 0 0 0 8 0 . 0 0 0 7 0 . 0 0 1 5
0 . 0 0 2 0
0 . 0 0 1 2 0 . 0 0 3 1
S S A - S S I - C O V
1 3 6 S V o r d e r l 2 O
l 4 O r o w
F r e q
0 . 5 2 2 6
0 . 7 9 8 6 0 . 9 6 5 2 1 . 1 5 0 9 1 . 1 9 3 2
1 . 2 5 1 8 1 . 3 8 9 9
1 . 6 4 0 7 1 . 9 4 4 5
D a m p
- 0 . 0 0 0 8 0 . 0 0 2 0 0 . 0 0 0 6 0 . 0 0 0 6 0 . 0 0 0 8 0 . 0 0 1 3
0 . 0 0 1 8
0 . 0 0 1 0 0 . 0 0 2 6
Table 1:Identified system natural frequencies using four different approaches.
Note: Size of Hankel data matrix in SSI-DATA is 280 block rows, 8000 columns, and with System order: 90.Size of Toeplitz covariance matrix in SSI-COV is 300 block rows, 300 block columns, and with system order: 90.
A total of 18000 data points were used to do correlation calculation.
5.3 SSI-COV with SSA as data pre-processing
Application of SSA, which acts as a filter to filter out the undesired noise from measurement, to extract the majorprincipal components of signals can be proved as an improvement to the stabilization of SSI analysis. Application of this
SSA to the measurements of GNTVT 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 15000columns. The variation of singular values is shown in Figure 4. It is difficult to select a suitable number of singular
values to be extracted from this figure because there is no significant gap on the distribution of singular values. Selection
of percentage of singular values becomes an important issue to generate the stability diagram in SSI analysis.
In performing the SSA-SSI-COV there are two parameters that have to be determined. One is the number of Singular
Value to be chosen in conducting the SSA, and the other is the system order to be determined in the SSI-COV analysis.
From the experience working on the data of GNTVT, a specific number of singular values in SSA step can leads to achange of slope in the distribution of SV when conducting SSI-COV (one can call it a first critical number of
components). If one continue reducing the selected number of SV in the step of SSA, up to a second critical point the
change of slope will
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Figure 4: Singular value decomposition of Hankel matrix with 340 block rows and 15000 columns.
Figure 5:Plot the distribution of SV from SSI-COV analysis; (a) using data from 95 SV in SSA.(b) using data from 136 SV in SSA,
become very sharp, almost a vertical jump (as in the case of 95SV and order 90). In this second critical point, usually the
number of SV in the SSA step will be very closer to the system order. From the experience, the second critical point
gives the best identification results, but those are not well excited modes or those highly contaminated will be alsofiltered out.
Based on the recorded acceleration data from GNTVT, in the beginning, 312 SVs were chosen from the SSA and thejump of Singular Values in SSI-COV analysis is almost imperceptible and nothing is stable in the stability diagram.
When one try with smaller number of SV, e.g., 154 SV in SSA step, the change of slope was a little more remarkable
(this can be considered as approximately the first critical point), and the stabilization diagram was improved. Finally, twodifferent sets of analysis were performed. First, 136 principal components (larger singular values) were extracted from
SSA to reconstruct the response measurements. The Toeplitz matrix was constructed based on the reconstructed data.The order of the system matrix Afor this case is set to 120 and this value is fixed to construct the stabilization diagram(with increasing the size of the covariance matrix or Toeplitz matrix). The choice of 136 SV is more subjective because
one has to try several times and plot SV in SSI-COV step to check if a change of slope appears. Figure 5b shows the plot
of the distribution of singular values of SSI-COV for case of using 136 SV in SSA. With 95 SV in SSA step leads to the
best solution because one can note that the change of slope in the SV plot in SSI-COV shows a clear jump, as shown in
Figure 5a.
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The Fourier Spectrum of the response data can be used to check if all the peaks in the Fourier spectrum are identified. In
the GNTVT case, actually with 95 Singular Values the major peaks are covered, but the peak corresponding to the firstmode in Fourier Spectrum is very fussy, therefore, one has to increase the number of components to be extracted in SSA
so as to conserve the component corresponding to 1st mode. This means that the 1 stmode is quite contaminated with
noise (or with wind force frequency) and cannot be clearly identified. To find the frequency of the 1st mode, one have toincrease the number of SV in SSA step, and also increase the system order in SSI-COV step. Figure 6 shows the
comparison on the stability diagram (between 0.0 Hz and 1.0Hz) from SSA-SSI-COV result. Figure 6a shows the resultusing 95 singular values from SSA and with order of 90 in SSI, and Figure 6b shows the result using 136 singular values
from SSA and with order of 120 in SSI. Comparison on the stability diagram, between 1.0 Hz and 5.0Hz, from SSA-SSI-
COV result is also shown in Figure 7. The identified system natural frequencies are shown in Table 1.
Figure 6:Comparison on the stability diagram (between 0.0 Hz and 1.0Hz) from SSA-SSI-COV result;
(a) using 95 singular values from SSA and with order of 90 in SSI,
(b) using 136 singular values from SSA and with order of 120 in SSI.
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Figure 7:Comparison on the stability diagram (between 1.0 Hz and 5.0Hz) from SSA-SSI-COV result;
(a) using 95 singular values from SSA and with order of 90 in SSI,
(b) using 136 singular values from SSA and with order of 120 in SSI.
From Figure 6 one can conclude that a quite stable diagram has been obtained. The major deficiency of using of 95 SV is
the inability on the identification of the first mode. This may due to the fact that this first mode is not well excited in
addition to the interference of wind dominant frequency, and consequently its corresponding signal component appears
after using 95 SV and thus, was filtered out by SSA. Besides, the mode corresponding to approximately 0.12 Hz was
filtered out by SSA.
Table 1 also show the identified system natural frequency from using SSA-SSI-COV with 95 SV (90 order) and 136 SV.As compare to the numerical study the first two identified frequencies, 0.0345 Hz and 0.0465 Hz, are the external
loading frequencies ( Figure 8 plot the identified poles in complex plane using the above mentioned two differentquantity of SV. It is clear the first two plots of complex pane (or the first two modes) are in relating the wind excitation
frequencies. As for the structural modes, the first structural mode can not be clearly identified if the procedure of SSA-
SSI-COV with 95 SV (90 order) was used. Therefore, SSA-SSI-COV with 136 SV (120 order) is used to extract the firststructural fundamental mode. Comparison on the model frequencies with the FEM was also shown in Table.
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Figure 8: Plot of poles of the first 4 modes identified by SSA-SSI-COV with 95 SV and 136SV in complex plane.
Through this study it is confirmed that the filtering result using 95SV in SSA step is better than using 136 SV. Although
the fundamental mode found by using 136 SV, unlike the other modes which appears almost in a straight line (meaningthat the structure has almost-proportional damping), this 1stmode is very unique which has a diverse distribution of the
complex poles. Although its frequency can be identified, the modal quality is not good due to the before mentioned
reasons.
From numerical studies, one has concluded that by using SSA as a preprocessing tool to SSI-based algorithms it onlyworks with SSI-COV but not with SSI-DATA. This is because the projection used in SSI-DATA can be explained as a
geometric-statistical tool to find the best fit, in a least square sense, of the future measurements in terms of the past data
(it is similar to the AR model fitting in a sense). Since SSA can only recover the principal components but not the
clean signal, and the reconstructed signals provided a bad fitting in the projection step, and generally the lower modesare the most affected.
CONCLUSIONS
The changes of features in a structural system may due to the change of environmental loading pattern, the nonlinearinelastic response of structure or structural damage when subjected to severe external loading. The detection of the
change of features or damage in large structural system, such as buildings and bridges, can improve safety and reduce
maintenance costs. Therefore, feature extract and damage detection from vibration structures are the goals of SHM. Inthis study an output-only system identification technique for civil structures under ambient vibrations is carried out, mainly focused on the Subspace System Identification (SSI) based algorithms. With the aim of finding accurate and true
modal parameters, a stabilization diagram is constructed by plotting the identified poles of the system with the SVD
truncation order. Comparative study between different approach, with and without using Singular Spectrum Analysis to
pre-process the data, on determining the model order and selecting the true system poles is examined in this study.Identification task of the real large scale structure: Guangzhou New TV Tower (GNTVT), a benchmark problem for
structural health monitoring of high-rise slender structures is carried out, for which the capacity of SSI-based algorithm
is demonstrated.
Through the study on the field response data the following conclusions are drawn:1. The difficulty in using SSI-DATA to identify these frequencies may be explained based on the conclusions obtained
from previous simulation studies which indicate that closely-spaced frequencies are difficult to be identified when the
measurements are noisy and generally only one equivalent frequency can be identified. Although SSI-COV shows to be
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able to separate these close frequencies, but there is doubt due to its near zero damping ratio and complex mode shapes
lying only in real or complex axis.2. The use of SSA as a pre-processing tool for SSI-COV is a great help to improve the stabilization diagram and to
extract rapidly the identifiable modes from the measurements. In the case of using 95 SV from SSA, most of all modes
are stabilizes starting from about the 25throw or even earlier. On the contrary, by using SSI-COV without applying SSAfor pre-processing the stabilization start approximately from 100 to 125 rows. This results in significant savings of the
computation time.3. Conventional analysis using SSI-DATA and SSI-COV without pre-processing the measurements was also conduct.
Figure 8 shows that stability diagram from these two approaches. It is clear that, from the observation of stability
diagram, to reach a stable result of identification large number of block rows may be required. Large number of block
rows may require more significant computation time. Table 1 also shows the comparison of the identified system naturalfrequencies among different methods.
4. For noisy measurements, as mentioned above, the modes contaminated by the noise will delay in stabilization, thus,more criterion must be used to discriminate true system poles from numerical poles, for example, the criteria of using
physical meaning of damping ratio and mode shape need to be considered [11].
Figure 9: (a) Stabilization diagram constructed using SSI-COV. The stability criteria is: 1% for frequency, 3% for
mode shape and 5% for damping ratio, (b) Stabilization diagram constructed using SSI-DATA, rangingfrom 0 Hz to 1 Hz.
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ACKNOWLEDGEMENTS
The authors would like to thank Prof. Y.Q. Ni of Hong Kong Polytech University to provide the data for analysis.
Support from National Science Council of the Republic of China, Taiwan (under Contract No. NSC 99-2221-E-002 -088-MY3) and the Research Program of Excellency of National Taiwan University (under Contract No. 99R80805) on the
development of the theory are also acknowledged.Acknowledgements are also given to University of Costa Rica for the
scholarship and support provided to Y.C. Liu during his two years of study at National Taiwan University.
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