Status
B. Peeters, G. De Roeck, “The Performance of Time Domain System Identification Methods
Applied to Operational Data”, Proceedings of DAMAS 97, Structural Damage Assessment Using
Advanced Signal Processing Procedures, pp. 377-386, University of Sheffield, Sheffield, UK,
30 June - 2 July 1997.
THE PERFORMANCE OF TIME DOMAIN SYSTEM
IDENTIFICATION METHODS APPLIED TO
OPERATIONAL DATA
B. Peeters and G. De Roeck
Department of Civil Engineering
Katholieke Universiteit Leuven, BELGIUM
ABSTRACT
In this paper, the performance of time domain system identification
methods is studied. More specifically, the stochastic realization algorithm
and the stochastic subspace identification method, and their use in modal
analysis, are reviewed. It is shown that a vibrating structure, excited by
an unknown force, can be modelled as a stochastic state space model.
Finally, the system identification methods are applied to experimental
data.
INTRODUCTION
The aim of vibration monitoring in civil engineering is to detect possible
damage and to determine the structure’s safety and reliability. The
damage assessment strategy is based on changes of the dynamic
behaviour of the structure expressed in terms of the modal parameters.
So, an important issue is the estimation of these modal parameters from
measured dynamic data. There exist several well-known both frequency
domain and time domain parameter estimation methods. Most of these
methods are using both input (dynamic forces) and output (accelerations)
measurements. For the frequency domain methods, the time histories are
(Fourier) transformed to Frequency Response Functions (FRF’s) and
most of the time domain methods are using impulse responses (i.e.
inverse Fourier transforms of FRF’s).
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However, in civil engineering it is often difficult and expensive to
artificially excite a structure. “Natural” dynamic excitation on the other
hand is freely available but it must be noticed that this ambient excitation
(e.g. wind load, traffic load, wave impact, ...) is stochastic and quite
difficult, if not impossible, to measure. Therefore, mainly identification
methods that can estimate the modal parameters using only the response
measurements will be considered in this paper. These methods are
typically time domain identification methods which assume a polynomial
or a state space model of the system. Some of these methods and their
application to modal analysis are explained and experimental data is used
to study their performance. Different types of dynamic forces have been
applied to a laboratory concrete beam in order to generate the
experimental data. The identification results for shaker and impact
measurements are compared and also validated with a classical FRFbased parameter estimation technique.
VIBRATING STRUCTURES
The dynamic behaviour of a discrete mechanical system, consisting of n2
masses connected through springs and dampers, is described by the
following matrix differential equation:
(1)
M Ü(t) C U(t) K U(t) P(t)
n ×n
2
2
where M, C, K are the mass, damping and stiffness matrix (
); P(t)
is the excitation force and U(t) is the displacement vector at continuous
time instant t . For systems with distributed parameters (e.g. civil
engineering structures), this equation is obtained as the finite element
approximation of the system with only n2 degrees of freedom left. With
following definitions:
Ac =
0
U(t)
I
-M -1 K -M -1 C
, x(t) =
,
U(t)
(2)
0
Bc =
M -1B2
,
P(t) = B2 u(t)
Equation (1) can be transformed into the state equation:
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x(t) = Ac x(t) + Bc u(t)
(3)
where Ac n×n (n=2n2) is the state matrix and Bc n×m is the input
matrix. In practice, not all the DOF’s are measured. When it is assumed
that the measurements are executed at only l sensor locations, the
observation equation is:
y(t) = C x(t) + D u(t)
(4)
where yt l×1 the response data (accelerations, velocities, displacements),
C l×n the output matrix, D l×m the direct transmission matrix. To
make it more realistic, the so-called continuous time state space system
(Equations (3) and (4)) are sampled and noise terms added, resulting in:
xk+1 = A xk + B uk + wk
(5)
y k = C x k + D u k + vk
with A=exp(Ac t) , B=[A I ] Ac 1 Bc , t the sampling period, t=k t .
wk n×1 is the process noise vector, due to disturbances and modelling
inaccuracies, while vk l×1 is the measurement noise, due to sensor
inaccuracy. They are unmeasurable vector signals, both assumed to be
zero mean, white, Gaussian, with covariance matrix:
w
Q S
.
E [ p wqT vqT ] =
(6)
vp
S T R pq
(E is the expected value operator and pq the Kronecker delta.)
The eigenvalue decomposition of the discrete state matrix A is:
-1
A =
(7)
with the diagonal matrix of discrete eigenvalues ( 1, . . . , n ). These
relate to the eigenvalues ( µ i ) of the mechanical system (Equation (1)) as
follows:
1
µi =
ln ( i)
(8)
t
The mode shapes at the sensor locations are the columns of
= C
l×n
:
(9)
In the case of unmeasurable inputs acting on a structure, a stochastic
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model is more suitable: uk is put equal to zero in Equation (5); or more
precise: the terms with uk are implicitly modelled by the noise terms.
SYSTEM IDENTIFICATION
The aim of system identification is to find a mathematical (black box)
model that accurately describes a dynamic system. The determination of
the parameters of the model is based on input-output measurements on
the system. In the previous section, it was shown that a stochastic state
space model is a good description of a linear vibrating structure excited
by unknown forces. In this section will be explained how the system
matrices of Equations (5) (with uk=0 ) and (6) can be recovered, up to
within a similarity transformation, using only response measurements.
Properties of Stochastic Systems
The stochastic process is assumed to be stationary with zero mean
( E [ xk ]=0 ,
E [ xk xkT ] , where the state covariance matrix
is
independent of the time k ). wk and vk are independent of the actual states
E [ xk vkT ]=0 , E [ xk wkT ]=0 . The output covariance matrices are defined as
T
i E [ yk+i y k ] , and finally the next state - output covariance matrix is
defined as G E [ xk+1 ykT ] . With these definitions and properties, it is
straightforward to prove:
= CA i-1G
(10)
i
Classical Stochastic realization
This last observation, Equation (10), indicates that the output covariances
can be considered as Markov parameters (i.e. impulse responses) of the
deterministic linear time invariant system A, G, C, 0 . Therefore, the
classical realization theory applies, which goes back to Ho and Kalman
[1]. First, the output covariance matrices are estimated:
j-1
ˆ = 1
yk+i ykT
i
j k=0
(11)
and gathered in a Hankel matrix. By a Singular Value Decomposition
(SVD) of this Hankel matrix, the system order n is determined ( n equals
the number of non-zero singular values) and the Hankel matrix
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decomposes into its factors (Equation (10)): the extended observability
and controllability matrices. The system matrices are easily recovered
now (Akaike [2], Aoki [3]). The performance of the stochastic realization
algorithm is for example demonstrated for response data of a simulated
test on a fixed offshore platform (Hoen et al. [4]). An equivalent method,
but for deterministic systems (measured input), is the Eigensystem
Realization Algorithm (ERA) by Juang and Pappa [5,6].
Stochastic Subspace Identification
A second method is the stochastic subspace algorithm, which is one class
of the recently developed subspace algorithms for state space system
identification (Van Overschee and De Moor [7]). Subspace algorithms are
data driven, contrary to the covariance driven realization algorithms. By
the application of robust numerical techniques as QR-factorization and
SVD to a semi-infinite block Hankel matrix, containing the output data,
an estimate of the states ( xk in Equation (5)) is obtained. Once the states
are known, the system matrices are recovered from Equation (5) in a least
square sense. Details can be found in literature (Van Overschee and De
Moor [8]). In Peeters at al. [9], the modal parameters of a bridge are
estimated using the stochastic subspace algorithm. In Abdelghani [10],
general combined deterministic-stochastic subspace algorithms are
successfully applied to modal analysis of structures and its performance
is compared with the well-known ERA-method (see previous section).
APPLICATION
Experimental Setup
A reinforced concrete beam of 6m length and with rectangular cross
section (200×250mm2) is suspended with very flexible springs, connected
at the theoretical nodal points of the first bending mode (distance from the
end section: 0.224 L ). The experimental setup is depicted in Figure 1. A
dynamic force is generated either by an electromagnetic exciter (MB
MODAL 50A) or an impulse hammer (PCB GK291B20). The vertical
force is applied in an outer point of the end section to excite vertical
bending and torsional modes. Accelerations are measured every 0.2m at
both sides of the beam with accelerometers PCB 338A35 and 338B35
(sensitivity ±100mV/g). In 1996, a similar beam was tested more
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extensively (Peeters et al. [11]). Also, the beam of the present paper will
be tested more extensively then herein described.
Figure 1. Experimental setup.
Excitation
A pseudo random signal was chosen to drive the shaker. The signal is
generated in the frequency domain with a flat magnitude spectrum and
random phase distribution (Heylen et al. [12]). The bandwidth was set at
the range 0-500Hz. Some typical shaker measurements are represented in
Figure 2. Another classical dynamic source is the hammer impact. Based
on Figure 3, the frequency content of the impact seems to be in the range
0-1000Hz. For both types of excitation, 12 accelerations and 1 force
signal were simultaneously measured. Each channel consists of 12288
data points, sampled at 5000Hz.
System Identification
Before identification, the data was filtered by a digital lowpass filter (8th
order Chebyshev I, with a cut-off frequency of 250Hz) and resampled at
a lower rate (625Hz). In the remaining frequency range 5 modes should
be identified. The stochastic subspace method is applied to the
preprocessed data. All 12 output channels were considered and the
number of data points used for identification was 1500. Based on a
singular value plot, the system order was chosen to be 20 for both
excitation types and all measurement configurations (It took 6
measurement configurations with 12 accelerometers to scan the whole
beam surface at intervals of 20cm on both sides). A system order n=20
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means that n2=n/2=10 (Equation (3)) modes are identified, while only 5
physical modes are expected in the considered frequency range. Therefore
5 spurious, numerical modes are identified but their damping values were
much too high. So they could easily be rejected. In the case of stochastic
identification, where the force signals are omitted, it is not possible to
identify absolutely scaled mode shapes. To obtain the right relative modal
amplitudes at least, it is important that some accelerations are measured
simultaneously. It is not necessary to use as many accelerometers and
acquisition channels as measurement locations, as long as there is a “wellplaced” reference accelerometer in common to every measurement
configuration. “Well-placed” means that the acceleration record contains
information of every mode. One of the end points of the free-free beam
is such a good place. The reason why the force signals were measured
was to compare the stochastic methods with more classical methods based
on input and output measurements. Hereto the impact measurements were
Fourier transformed to obtain Frequency Response Functions (FRF’s).
These FRF’s are used to estimate the modal parameters with a nonlinear
least squares frequency domain method (Balmès [13]). The method tries
to minimize the following quadratic cost function:
model
J = Hjk
( j l ) - Hjkmeas ( j l )
2
(12)
j,k,l
The cost function expresses the difference in modelled and measured FRF
evaluated for every sensor-actuator ( j,k ) pair at every frequency ( l ).
Results and conclusions
In Table 1, the 3 identification results are compared. For every mode, the
eigenfrequency (f) and damping factor ( ) are given together with their
standard deviations ( f, > ). As expected the eigenfrequencies of the
shaker measurements are somewhat higher. Also the damping results are
different. Comparing the hammer identification results shows that the
correspondence between the time-domain and FRF-based identification
method is very good (mode 4 is an exception; explanation follows). This
may come as a surprise since at first sight both methods have very little
in common. The first method directly works on the recorded time signals
and finds in a linear way a system description (Equation (5)). An
eigenvalue decomposition of the main matrix A of the system description
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yields the modal parameters. The second method transforms the time
records into the frequency domain and tries, with a nonlinear algorithm,
to fit the measured FRF. The fitted FRF is an analytical function of the
modal parameters.
Table 1
mode
1
2
3
4
5
Identification Results
modal
parameters
f ( f) [Hz]
( >) [%]
f ( f) [Hz]
( >) [%]
f ( f) [Hz]
( >) [%]
f ( f) [Hz]
( >) [%]
f ( f) [Hz]
( >) [%]
pseudo random
(time series)
22.51
(0.04)
0.3
(0.1)
62.53
(0.04)
0.23
(0.04)
120.1
(0.1)
0.24
(0.02)
175.3
(0.3)
0.53
(0.06)
198.5
(0.3)
0.34
(0.08)
hammer
(time series)
22.34
(0.01)
0.43
(0.01)
62.43
(0.02)
0.38
(0.01)
119.7
(0.0)
0.38
(0.04)
174.8
(0.2)
0.5
(0.2)
198.1
(0.1)
0.43
(0.01)
hammer
(FRF)
22.36 (0.02)
0.52 (0.02)
62.47 (0.02)
0.37 (0.02)
119.7 (0.1)
0.37 (0.01)
175.7 (0.4)
0.13 (0.04)
198.1 (0.1)
0.43 (0.01)
At the 4th mode (Table 1), the estimated FRF fits not at all to the
measured one. The peak is too narrow (which explains the low damping
factor, compared to the time domain methods). The reason is that at
around 175Hz multiple peaks occurred in the FRF, although there was
only one mode present. Also the time domain methods found multiple
modes around 175Hz. But all but one could easily be rejected as nonphysical ones. Figure 4 represents the identified mode shapes obtained
with the stochastic subspace method applied to the hammer data. The
pseudo-random modes are almost identical. The 4th mode is a torsional
mode; the mode shapes at both sides of the beam surface have opposite
sign.
Stochastic subspace identification is a linear method that works directly
on the time data. Especially in civil engineering, it is a useful technique,
since it can identify vibrating structures, excited by ambient loading. The
modal parameters are determined very accurately. From the comparison
with the FRF-based technique, it seems to be that excluding the input
signal from the identification procedure does not influence the accuracy
of the estimated modal parameters.
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REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Ho, B. L. and Kalman, R. E. “Effective Construction of Linear State-Variable
Models from Input/Output Data”, Regelungstechnik, 1966, Vol. 14, 545-548.
Akaike, H., “Stochastic Theory of Minimal Realization”, IEEE Transactions
on Automatic Control, 1974, 19, 667-674.
Aoki M., State Space Modelling of Time Series, Springer Verlag, Berlin,
1987.
Hoen, C., Moan, T. and Remseth, S., “System Identification of Structures
Exposed to Environmental Loads”, Proceedings of Eurodyn ’93, Balkema,
Rotterdam, 1993, 835-845.
Juang, J.-N. And Pappa, R. S., “An Eigensystem Realization Algorithm for
Modal Parameter Identification and Model Reduction”, Journal of Guidance,
Control and Dynamics, 1985, Vol. 8, No. 5, 620-627.
Juang, J.-N., Applied System Identification, PTR Prentice Hall, Englewood
Cliffs, New Jersey, 1994.
Van Overschee, P. and De Moor, B., Subspace Identification for Linear
Systems: Theory-Implementation-Applications, Kluwer Academic Publishers,
Dordrecht, The Netherlands, 1996.
Van Overschee, P., De Moor, B., “Subspace Algorithms for the Stochastic
Identification Problem”, 30th IEEE Conference on Decision and Control,
Brighton, UK, 1991, 1321-1326.
Peeters, B., De Roeck, G., Pollet, T. and Schueremans, L., “Stochastic
Subspace Techniques Applied to Parameter Identification of Civil
Engineering Structures”, Proceedings of New Advances in Modal Synthesis
of Large Structures: Nonlinear, Damped and Nondeterministic Cases, Lyon,
France, 1995, 151-162.
Abdelghani, M., Identification Temporelle des Structures: Approche des
Algorithmes Sous-Espace dans l’Espace Etat, PhD thesis, Laboratoire de
Mécanique et Génie Civil, Université Montpellier II, France, 1995.
Peeters, B. et al., “Evaluation of Structural Damage by Dynamic System
Identification”, Proceedings of ISMA21, vol. 3, Leuven, Belgium, 1996,
1349-1361.
Heylen, W., Lammens, S., Sas, P., Modal Analysis Theory and Testing, PMA,
Department of Mechanical Engineering, K.U.Leuven, Belgium, 1995.
Balmès, E., Structural Dynamics Toolbox: For Use with MATLAB, Scientific
Software Group, Sèvres, France, 1995.
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pseudo random
force spectrum
4
30
10
20
2
10
F [N]
10
0
0
10
−10
−2
10
−20
−30
0
0.02
t [sec]
0.04
10
a [m/sec^2]
acceleration
−4
0
10
1
10
0
10
−1
10
200
300
f [Hz]
400
500
600
500
600
response spectrum
4
2
−2
0
100
2
0
−2
0.02
t [sec]
0.04
10
−4
0
100
200
300
f [Hz]
400
Figure 2. Pseudo random shaker signals. Top: force; bottom: acceleration
impact force
3000
force spectrum
1
10
0
2000
10
−1
F [N]
10
1000
−2
10
0
10
−3
−4
a [m/sec^2]
10
−1000
0.01 0.02 0.03 0.04 0.05
0
t [sec]
acceleration
0
150
10
500
1000
f [Hz]
response spectrum
1500
−1
100
10
50
10
0
10
−50
10
−100
10
−2
−3
−4
−5
−6
10
−150
0.01 0.02 0.03 0.04 0.05
0
t [sec]
500
1000
1500
f [Hz]
Figure 3. Hammer impact signals. Top: force; bottom: acceleration
mode 1, f=22.3 Hz, xi=0.43%
mode 2, f=62.4 Hz, xi=0.38%
mode 4, f=175 Hz, xi=0.48%
mode 3, f=120 Hz, xi=0.38%
mode 5, f=198 Hz, xi=0.43%
Figure 4. Identified mode shapes. Application of stochastic subspace.
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