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The modes of linear time invariant mechanical systems can be estimated from output-only vibration measurements under ambient excitation conditions with subspace-based system identification methods. In the presence of additional unmeasured periodic excitation, for example due to rotating machinery, the measurements can be described by a state-space model where the periodic input dynamics appear as a subsystem in addition to the structural system of interest. While subspace identification is still consistent in this case, the periodic input may render the modal parameter estimation difficult, and periodic modes often disturb the estimation of close structural modes. The aim of this work is to develop a subspace identification method for the estimation of the structural parameters while rejecting the influence of the periodic input. In the proposed approach, the periodic information is estimated from the data with a non-steady state Kalman filter, and then removed from the original output signal by an orthogonal projection. Consequently, the parameters of the periodic subsystem are rejected from the estimates, and it is shown that the modes of the structural system are consistently estimated. Furthermore, standard data analysis procedures, like the stabilization diagram, are easier to interpret. The proposed method is validated on Monte Carlo simulations and applied to both a laboratory example and a full-scale structure in operation.
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Kalman filter-based Subspace Identification for Operational Modal Analysis
under Unmeasured Periodic ExcitationI
Szymon Gre´sa,, Michael D¨ohlerb, Palle Andersenc, Laurent Mevelb
aAalborg University, Department of Civil and Structural Engineering, Thomas Manns Vej 23, 9000 Aalborg, Denmark
bUniversit´e Gustave Eiffel, Inria, COSYS/SII, I4S, Campus de Beaulieu, 35042 Rennes, France
cStructural Vibration Solutions A/S, NOVI Science Park, 9220 Aalborg, Denmark
Abstract
The modes of linear time invariant mechanical systems can be estimated from output-only vibration mea-
surements under ambient excitation conditions with subspace-based system identification methods. In the
presence of additional unmeasured periodic excitation, for example due to rotating machinery, the measure-
ments can be described by a state-space model where the periodic input dynamics appear as a subsystem in
addition to the structural system of interest. While subspace identification is still consistent in this case, the
periodic input may render the modal parameter estimation difficult, and periodic modes often disturb the
estimation of close structural modes. The aim of this work is to develop a subspace identification method
for the estimation of the structural parameters while rejecting the influence of the periodic input. In the
proposed approach, the periodic information is estimated from the data with a non-steady state Kalman
filter, and then removed from the original output signal by an orthogonal projection. Consequently, the
parameters of the periodic subsystem are rejected from the estimates, and it is shown that the modes of
the structural system are consistently estimated. Furthermore, standard data analysis procedures, like the
stabilization diagram, are easier to interpret. The proposed method is validated on Monte Carlo simulations
and applied to both a laboratory example and a full-scale structure in operation.
Keywords: Operational modal analysis, Ambient excitation, Periodic excitation, Non-steady state
Kalman filter, Subspace system identification
1. Introduction
The estimation of modal parameters from output-only vibration measurements is the fundamental task of
Operational Modal Analysis (OMA). Therein, system identification methods are frequently used to estimate
the eigenstructure of a linear system from the accelerations, displacements, velocities or strains recorded on
I<2021>. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.
org/licenses/by-nc- nd/4.0/.
Corresponding author; E-mail address: sg@civil.aau.dk
Preprint submitted to Mechanical Systems and Signal Processing March 22, 2021
the structure during ambient excitation conditions [1]. Often, the ambient excitation is assumed to be white
and stationary, which is sometimes violated, in particular in the presence of periodic movement of rotating
machinery on a structure during its operation. Then both ambient and unmeasured periodic forces act on
the structure, and the outputs of the corresponding system are described by both the structural system
dynamics as well as the dynamics of the periodic excitation. This might render OMA difficult in practice,
since the identified eigenstructure then contains a mix of periodic and structural modes [2]. Moreover, when
structural and periodic modes are close, the correct identification of the structural parameters may become
a problem [3]. The aim of this paper is to develop a subspace identification method for the consistent
estimation of the structural modal parameters while rejecting the influence of the unmeasured periodic
excitation.
Two classes of methods are commonly used for OMA under periodic excitation. In the first class, the
periodic subsignal is separated from the random response signal, and subsequently the modal parameters are
estimated with classical methods. For example, time-synchronous-averaging (TSA) is a method extracting
periodic waveforms from signals by averaging their blocks synchronized in the angular domain. For OMA,
this averaged signal is subtracted from the raw measurements, which results in the removal of the peri-
odic frequencies selected to synchronize the blocks [4]. Angle matching is often achieved with tachometer
measurements, which is not practical in real-life applications and was attempted to be overcome in the
context of TSA in [5]. A family of methods that does not require tachometer measurements is based on the
cepstrum, which is an inverse Fourier transform of the logarithm of spectrum. A number of applications of
cepstral lifters to harmonic removal can be found in [6, 7]. Although the cepstrum is capable to filter the
periodic frequency components out of the spectra of the output data, its empirical premise does not ensure
the consistency of the resulting modal parameter estimates. Another group of methods uses a parametric
approach to estimate periodic subsignals and removes them from the raw measurements, using for example
the Gauss-Newton algorithm [8], or parametric frequency modulation [9].
The second method class comprises techniques that are robust towards intricate input conditions. For
example, some methods relax the assumption of random white noise excitation to arbitrary signals, e.g.,
colored noise, or white noise with periodic components [10, 11]. In [2, 10–13] the authors explore the use
of a combination of transmissibility functions under different loading conditions (location or amplitude) to
estimate the eigenstructure of the system under white noise excitation mixed with a periodic subsignal.
However, the approach proposed therein imposes additional constraints on the system inputs that are not
present in classical system identification methods. For example, distinctive excitation conditions are required
whose number is known and smaller than the number of output locations [11], which cannot always be
ensured in OMA in practice.
A group of classical system identification methods that are well adapted for OMA are stochastic subspace-
based methods [14, 15]. They have been proved to enjoy non-stationary consistency [16], as well as practical
2
aspects like computational efficiency [17] and explicit variance expressions of the identified parameters [18–20]
under the white noise assumption. However, when both random and periodic inputs are present, the system
output is not strictly ergodic and the output covariances of the system depend on the initial conditions
[21, 22]. Since the subspace parameter estimates rely on the output covariances and ergodicity is not given,
the consistency of subspace methods is not evident in this setting. In [23] the authors prove consistency
of both system and oscillatory parameters for estimates from a covariance-based subspace identification
method. This fact can be used to discard the periodic poles of the system based on the consistent estimate
of its eigenstructure, which was illustrated on a theoretical example in [24].
The current paper builds upon this work with the aim to develop a robust subspace method in the context
of OMA for the identification of structural modes while rejecting the periodic contribution of the signals.
The system matrices of the underlying state space model are time-invariant, and the unmeasured periodic
excitation is assumed to be a superposition of sinusoids. The proposed approach operates in the data-driven
output-only stochastic subspace identification framework. The periodic excitation is modeled as a part of
the system states, thus it can be optimally estimated from the data with a non-steady state Kalman filter. A
subspace algorithm is proposed, where the raw output data are projected on the orthogonal complement of
the estimated periodic sequence in order to remove the latter from the raw data. This is particularly useful
when the periodic modes are close to structural modes, or when they are of high energy and then may mask
the system response to the random part of the input. Subsequently, the periodic excitation parameters are
rejected from the resulting estimates of the system matrices in subspace identification based on the projected
signal, and the eigenstructure of the underlying structural system is properly identified.
This paper is organized as follows. The background and modeling of the considered problem is given
in Section 2. The proposed method for the removal of the periodic subsignal and subsequent subspace
identification is derived in Section 3, including a proof of the consistency of the method. The method is
validated on Monte Carlo simulations in Section 4. An application to two experimental cases is reported in
Section 5, namely to a plate in the laboratory and to a full-scale ship in operation.
2. Problem statement
In this section, the vibration model is recalled, and the impact of the periodic excitation on the models
and identified parameters is stated. The latter is illustrated by Monte Carlo simulations of a mass-spring
chain system.
2.1. Stochastic system models and parameters
Assume that the vibration behavior of a viscously damped, linear time-invariant (LTI) structural system
with mdegrees of freedom is described by the differential equation
M¨q(t) + C˙q(t) + Kq(t) = f(t) (1)
3
where tdenotes continuous time, and M,C,K ∈ Rm×mdenote mass, damping and stiffness matrices, re-
spectively. Vectors q(t)Rmand f(t)Rmcontain the continuous-time displacements and the unmeasured
external forces, respectively. Let system (1) be observed by sensors measuring, e.g., accelerations, velocities
or displacements, at rdegrees of freedom (DOF) of the structure, collected in the vector
y(t) = Ca¨q(t) + Cv˙q(t) + Cdq(t) + ˜v(t) (2)
where y(t)Rris the output vector, ˜v(t)Rrdenotes the sensor noise, and matrices Ca,Cv,CdRr×m
select the respective type of the output at the measurement DOFs.
When defining the states x(t)=[q(t)T˙q(t)T]TR2m, the structural system model (1) with output
equation (2) yields the continuous-time state space model
˙x(t) = Asys
cx(t) + w(t),(3)
y(t) = Csysx(t) + v(t),(4)
where the state matrix Asys
cR2m×2m, observation matrix Csys Rr×2m, process noise w(t)R2mand
output noise v(t)Rrare
Asys
c=
0I
−M1K −M1C
, Csys =hCdCaM1KCvCaM1Ci, w(t) =
0
M1
f(t),
and v(t) = CaM1f(t) + ˜v(t), where the model order is n= 2m. When sampled at discrete time instants
t= , where τis the time step and kis an integer, the resulting discrete-time state space model is given
by [25]
xk+1 =Asysxk+wk,(5)
yk=Csysxk+vk,(6)
where xk=x( )R2mare the discrete states and Asys = exp(Asys
cτ)R2m×2mis the state transition
matrix. Note that the index (·)sys indicates here that the respective parameter refers in particular to the
structural system, which is distinguished from the periodic part denoted by (·)per later on in the paper. The
discrete process noise wkand output noise vkare assumed to be zero-mean white noise vectors with finite
fourth order moments. For simplicity, assume in addition that they are Gaussian. Their covariance matrix
is given by
E
wk
vk
hwlvli
=
Q S
STR
δkl 0.
Matrices Asys and Csys are of particular interest since they are used to identify the modal parameters
of the structure. They can be estimated from data {yk}with subspace identification methods, based on
the column space of an adequate projection of the data sequences [14, 20]. The i-th natural frequency fsys
i,
4
damping ratio ζsys
iand mode shape ϕsys
iof the underlying structural system are related to the eigenvalue
λsys
iand eigenvector φsys
iof Asys by
fsys
i=|λsys
ci |
2π, ζsys
i=−<(λsys
ci )
|λsys
ci |, ϕsys
i=Csysφsys
i,(7)
where the i-th eigenvalue λsys
ci of the continuous-time system yields exp(λsys
ci τ) = λsys
i.
2.2. Impact of mixed random and periodic excitation on the state-space model
In this section, the influence of unmeasured periodic excitation on the classical formulation of the state-
space model (5)–(6) and its parameters is developed and illustrated on a numerical example.
Assume that a deterministic periodic force u(t) acts on the system in addition to the random noise
input w(t). For simplicity of notation let this periodic force be one-dimensional, i.e., u(t)R, and let
suRmbe an index vector containing 0’s and 1’s, indicating at which degree(s) of freedom the periodic
force acts on the structure. Denote the resulting states as xsys
k, containing the displacements and velocities
at the DOFs of the structure under both unknown noise and periodic forces. Then, the continuous-time
state space model (3)–(4) becomes
˙xsys (t) = Asys
cxsys(t) + bu(t) + w(t),(8)
y(t) = Csysxsys (t) + du(t) + v(t),(9)
where
b=
0
M1
suR2m,d=CaM1suRr.
Since the periodic force u(t) is unmeasured, it is the goal to eliminate it from the state space model and
include its effects in the system matrices and in the state vector. Assuming that the periodic force contains
hfrequency components of the shape
u(t) =
h
X
i=1
aisin(ωit+gi),(10)
where ai, gi, ωiRare (unknown) amplitude, shift and circular frequencies of the periodic input components,
then these components can become part of a combined state vector in order to eliminate the periodic input
component in model (8)–(9), as follows. Define
xper(t) =
a1sin(ω1t+g1)
a1cos(ω1t+g1)
.
.
.
ahsin(ωht+gh)
ahcos(ωht+gh)
R2h,thus ˙xper (t) =
ω1a1cos(ω1t+g1)
ω1a1sin(ω1t+g1)
.
.
.
ωhahcos(ωht+gh)
ωhahsin(ωht+gh)
.
5
The relationship between ˙xper(t) and xper(t) follows as
˙xper (t) = Aper
cxper(t),where Ap er
c= diag(H1, . . . , Hh) and Hi=
0ωi
ωi0
,(11)
and the relationship between u(t) and xper(t) is given by u(t) = shxper(t), where sh= [1 0 . . . 1 0] R1×2h.
Thus, defining Ab
c=bshR2m×2hand Cper =dshRr×2h, the state space model (8)–(9) containing the
unmeasured periodic input u(t) can be equivalently rewritten as a combined state space model without the
periodic input as
˙xsys (t)
˙xper (t)
=
Asys
cAb
c
0Aper
c
xsys(t)
xper(t)
+
w(t)
0
,(12)
y(t) = hCsys Cperi
xsys(t)
xper(t)
+v(t),(13)
where the model order is n= 2(m+h). Note that due to the upper right block structure of the state matrix,
the eigenvalues of the combined system are the combined sets of eigenvalues of Asys
cand of Aper
c. While the
eigenvectors of the combined state matrix regarding the structural part become [φsys
i
T0]T, the resulting
mode shapes are ϕsys
i, as in system (8)–(9).
Sampling model (12)–(13) at discrete time instants t=yields the combined discrete-time state-space
model
xsys
k+1
xper
k+1
=
Asys Ab
0Aper
xsys
k
xper
k
+
wk
0
(14)
yk=hCsys Cperi
xsys
k
xper
k
+vk,(15)
where the combined system matrix yields
Asys Ab
0Aper
= exp
Asys
cAb
c
0Aper
c
τ
.
Recall that the first 2mcomponents of the process noise are related to the ambient excitation of the system,
while the deterministic periodic excitation is translated into the states xper
kas shown above.
The eigenvalues of Asys and Ap er are denoted by λsys
i, λsys
i,i= 1, . . . , m, and λper
i, λper
i,i= 1, . . . , h,
respectively. The eigenvalues of the structural system yield |λsys
i|<1, while the eigenvalues of the periodic
part of the system are situated on the unitary circle, i.e., |λper
i|= 1. Hence both kinds of modes can be
distinguished in the combined state matrix. Moreover, the periodic excitation (10) corresponds to undamped
modes of the periodic part of the continuous-time system as can be seen in Equations (11) and (12), i.e.,
<(λper
ci ) = 0.
6
These properties are illustrated in the context of a system subjected to mixed random and periodic
excitation in the following section.
2.3. Illustrative example
Consider a 6 DOF mass-spring chain system that, for any consistent set of units, is modeled with spring
stiffness k1=k3=k5= 100 and k2=k4=k6= 200, mass of each element mi= 1/20 and a proportional
damping matrix such that each mode has a damping ratio of ζsys
i= 3%. The system is subjected to
white noise excitation in all DOFs and sampled with a frequency of 50 Hz for 2000 seconds. An additional
sinusoidal excitation with a frequency of 8.69 Hz, close to the third natural frequency of the system, is
applied at all DOFs. This excitation is devised to mimic a periodic input from, e.g., an engine rotating at
a constant speed. The resulting acceleration responses are obtained at DOFs 1, 2 and 5. Gaussian white
noise with 5% of the standard deviation of the output is added to the response at each channel.
The modal parameters of the combined system model including the periodic part are depicted in Table 1.
The resulting eigenvalues of the discrete-time system and the respective continuous-time system are shown
in the complex plane in Figure 1. It can be seen that the periodic and the system poles can indeed be
distinguished in the complex plane. This fact will be used to estimate the periodic states and consequently
the periodic subsignal of the output with the method proposed in this paper.
Furthermore, the presence of periodic poles can be seen in the Power Spectral Density (PSD) of the
data [26]. Figure 2 shows the two largest singular values of the PSD matrix constructed from the structural
Table 1: Exact modal parameters of the chain system, and properties of the periodic excitation.
Natural frequency (Hz) Damping ratio (%)
fsys
1fsys
2fsys
3fsys
4fsys
5fsys
6fper
1ζsys
1ζsys
2ζsys
3ζsys
4ζsys
5ζsys
6ζper
1
1.93 5.62 8.68 14.49 15.85 17.01 8.69 3 3 3 3 3 3 0
Figure 1: Discrete-time and continues-time poles of the system from (12) and (14) respectively.
7
Figure 2: Two largest singular values of PSD matrix from output data of the system subjected to random (left) and mixed
random and periodic (right) excitation.
Figure 3: Estimates of natural frequency and damping ratio from one simulation. Model order 12 (left) and 14 (right).
responses with and without the periodic excitation. It can be observed that the fundamental frequency of
the periodic excitation manifests as a sharp spike in the output power spectra. However, since its frequency
is close to a system pole, both poles cannot be easily distinguished visually from the PSD plot.
The next example illustrates the estimation of the natural frequencies and damping ratios by a Monte
Carlo experiment with 1000 simulations. The output-only data driven subspace-based system identification
with the unweighted principal component (SSI-UPC) [14, 15] is deployed, using 30 time lags for the data
Hankel matrix and estimating the system matrices at model orders of 12 and 14. For both model orders,
the respective sets of modes are tracked in each simulation. The estimates of the natural frequency and
damping ratio from one simulation are depicted in Figure 3. Based on all simulations, Figures 4 and 5
show the histograms of the natural frequencies and the damping ratios of the mode closest to the periodic
frequency for both model orders of 12 and 14.
Recall that model order 12 corresponds to the structural system, and model order 14 corresponds to the
combined system with the periodic mode. When the mode close to the periodic mode is estimated at model
8
Figure 4: Histograms of the natural frequency of the third mode identified with model order 12 (left) and 14 (right).
Figure 5: Histograms of the damping ratio of the third mode identified with model order 12 (left) and 14 (right).
order 12, it can be observed in the left parts of Figures 3–5 that its frequency and damping ratio estimates
are in between the structural mode (at 8.68 Hz and 3% damping) and the periodic mode (at 8.69 Hz and 0%
damping). When estimating the mode at model order 14, the mean values of the histogram of the natural
frequency and the damping ratio in the right parts of Figures 4 and 5 are close to the exact values. This
infers that by augmenting the exact model order to account for the periodic pole, both the system and
periodic parameters are consistently estimated, which agrees with the state-space model proposed in (14)
and (15) in the previous section. This fact is used in the method proposed in the following section.
3. Subspace-based system identification under mixed periodic and random excitation
As shown in the previous section, the exact model order of the structural system can be increased by the
number of periodic poles that are present in the data. Using output-only stochastic subspace identification,
consistent estimates of both structural and periodic poles can be obtained [23], and the latter can be rejected
from the modal estimates as illustrated on the theoretical example in the previous section. However, the
exact model order is unknown in practical applications and the periodic excitation may coincide with a
9
natural frequency of the structure, or it may be of high energy that masks the system response to the
random part of the input. In these cases, it is desirable to discard the periodic excitation from the data and
without additional knowledge of, e.g., tachometer measurements. In this section, a scheme for the removal
of the periodic subsignal is proposed, based on three steps:
1. Estimation of the periodic poles by subspace-based system identification from the raw output data,
2. Estimation of the periodic subsignal using the Kalman filter,
3. Projection of the row space of the raw output data onto the orthogonal complement of the row space
of the periodic subsignal estimate.
Based on the projected signal, any further signal processing can be carried out for the analysis of the
structural system response, where the nuisance from the periodic inputs is removed. In particular, subspace-
based system identification can be used to estimate only the structural system modes, which is detailed in
the remainder of this section.
In the following, it is assumed that the first step has already been carried out and that the periodic
modes are selected, including the modes corresponding to the harmonics, for example based on indicators
developed in [27, 28]. In practical applications, the selection of periodic poles can be done with simple
indicators like kurtosis [27, 28], entropy [29] or damping ratios [30].
3.1. Estimation of periodic subsignal
The periodic subsignal is estimated based on the Kalman filter. Note that the eigenvalues of the periodic
subsystem are situated on the unitary circle, thus the considered combined system (14)–(15) is not a classical
stable system. Moreover, the periodic part does not have process noise. Nevertheless, the Kalman filter can
be applied and is stable, as detailed in [23, 31].
For the estimation of the periodic subsignal the Kalman filter states are retrieved in the modal basis
in order to distinguish the states referring to the periodic poles. For this, the Kalman filter states are
obtained first in an arbitrary basis, namely in the basis corresponding to estimates of the system matrices
{A, C, Q, R, S}of the combined system corresponding to (14)–(15). These system matrices can be estimated
from data using for example the SSI-UPC method described in [14]. Then, the Kalman filter states are
converted to the modal basis, and the modal states corresponding to the periodic modes are selected to
estimate the periodic subsignal. This procedure is detailed in the following.
With the Kalman filter, unbiased and minimum variance estimates ˆxk+1 of the states xk+1 are obtained.
In this work, the non-steady state Kalman filter is used that accounts for the correlation between process
and output noise, based on [32]. The initial state estimate is assumed to be ˆx0= 0 Rn, and the initial
10
error covariance matrix is assumed as P0=InRn×n. Then, the recursive filter equations are
Kk= (APkCT+S)(R+CPkCT)1,(16)
ˆxk+1 = (AKkCxk+Kkyk,(17)
Pk+1 =APkAT+QKk(APkCT+S)T,(18)
where KkRn×ris the gain matrix, which converges to the steady state gain KRn×rfor increasing k.
Consequently, defining the innovations ek=ykCˆxkRr, the Kalman filter states and system outputs
yield
ˆxk+1 =Aˆxk+Kkek,(19)
yk=Cˆxk+ek,(20)
which is the state-space model in innovation form. The states are not unique since for any invertible matrix
VRn×nthe linear transformations
ˆxV
k=V1ˆxk, AV=V1AV, CV=CV, KV
k=V1Kk(21)
yield the state-space model
ˆxV
k+1 =AVˆxV
k+KV
kek,
yk=CVˆxV
k+ek
that is equivalent in terms of outputs, eigenvalues and mode shapes. Without loss of generality, the modal
basis is chosen for the transformation in order to distinguish the states that are corresponding to the periodic
part of the system. More precisely, matrix Vis chosen based on the eigenvectors of Aas follows. Recall that
φsys
iand φsys
i,i= 1, . . . , m and φp er
iand φper
i,i= 1, . . . , h, are the pairs of conjugated complex eigenvectors of
Acorresponding to the structural system and to the periodic part, respectively. Analogously, λsys
iand λsys
i,
i= 1, . . . , m and λper
iand λper
i,i= 1, . . . , h, are the respective eigenvalues. Applying the transformation
defined by
V=h<(Ψ) =(Ψ)i,where Ψ = hφsys
1. . . φsys
mφper
1. . . φper
hiC2(m+h)×(m+h),(22)
yields the state-space model with real-valued system matrices in the modal basis with
AV=
<(Λ) =(Λ)
−=(Λ) <(Λ)
, CV=h<(Φ) =(Φ)i,
where Λ = diag(λsys
1, . . . , λsys
m, λper
1, . . . , λper
h) and Φ = CΨ = [ϕsys
1. . . ϕsys
mϕper
1. . . ϕper
h] contain one
element of each complex conjugated pair of eigenvalues and mode shapes, respectively.
11
Note that it is impossible to obtain the system matrices and the Kalman filter states in the same state
basis as in (14)–(15) when estimated from data. However, the previous transformation of the system matrices
into the modal basis {AV, CV, K V
k}yields a canonical format when the system matrices are identified in an
arbitrary basis. Moreover, the states corresponding to the system and to the periodic parts are decoupled
in the state vector ˆxV
k. Subsequently, the states corresponding to the periodic part can be selected from the
respective entries of ˆxV
kand the periodic output subsignal related to these states estimated. For this, define
a selection matrix Swith
S=
0m×m
Ih
0m×m
Ih
,
where the identity matrices Ihrefer to the entries of the state vector that are related to the periodic modes.
Consequently, the estimation of the output data contribution due to the periodic modes writes as
ˆyper
k=CVSˆxV
k,(23)
which is the desired estimate of the periodic subsignal.
In practice only the estimates of CV,AVand KV
kare available, which are computed on data of finite
length, e.g., after [14]. Consequently, an approximate Kalman state is used in (23) to estimate the periodic
subsignal.
3.2. Removal of the periodic subsignal by orthogonal projection
The estimate of the periodic subsignal can be decoupled from the row space of output data by using
an adequate projection. Different projection methods exist and are often used in the context of subspace-
based system identification, e.g., see [14, 20]. In the following, an orthogonal projection of the raw output
data onto the orthogonal complement of the estimated periodic subsignal is carried out. It is proved that
the resulting signal indeed represents only the structural system, and the periodic part of the combined
state-space system is canceled out.
In these projections, Hankel matrices of the respective signals are used, which are defined as follows.
Definition 1. Let the samples k=i, . . . , j +N1of a discrete signal akRb×1be given with ij. Then
the corresponding block Hankel matrix Ai|jis defined as
Ai|j=
aiai+1 . . . ai+N1
ai+1 ai+2 . . . ai+N
.
.
..
.
.....
.
.
ajaj+1 . . . aj+N1
R(ji+1)b×N.
12
Analogously to subspace methods, Hankel matrices are built from the signals with a ‘past’ and ‘future’
time horizon. Let pbe a parameter that defines the time lags for the past and future horizons. Based on
the raw output data yk(e.g., see (15) or (20)), the past and future data Hankel matrices are defined as
Y
raw =1
NY0|p1,Y+
raw =1
NYp|2p1.(24)
Similarly, the past and future data Hankel matrices of the estimated periodic subsignal ˆyper
k(see (23)) are
defined as
Y
per =1
Nˆ
Yper
0|p1,Y+
per =1
Nˆ
Yper
p|2p1.(25)
Furthermore, denote the past and future block-row matrix of Kalman filter states in the modal basis ˆxV
k
(see (21)) as
X=1
Nˆ
XV
0|0,X+=1
Nˆ
XV
p|p.
The rows of X,X+R2(m+h)×Ncorresponding to the system part are denoted as X
sys,X+
sys R2m×N,
and corresponding to the periodic part as X
per,X+
per R2h×N, respectively. The past and future Hankel
matrices based on the innovations ek(see (19)–(20)) are defined as
E=1
NE0|p1,E+=1
NEp|2p1.
Finally, the extended observability matrix of the combined system writes in the modal basis as
Γ =
CV
CVAV
.
.
.
CV(AV)p1
,
and let Γsys and Γper be its columns corresponding to the system and to the periodic part, respectively.
With this notation, the data Hankel matrices of the raw data and of the estimated periodic subsignal can
be expressed in terms of the states and the innovations by recursion of the innovation state-space model
(19)–(20) as follows.
Corollary 2 (Matrix output-only innovation state-space equations).It holds
Y
raw = ΓX+KE+E
K,Y+
raw = ΓX++KE++E+
K,(26)
Y
per = Γper X
per,Y+
per = Γper X+
per,(27)
where KEand KE +are related to the innovation terms with K ∈ Rpr×pr being defined based on the steady
state Kalman gain K, and E
Kand E+
Kare remainder terms that are related to the difference to the actual
13
non-steady state Kalman gain Kk, with
K=
Ir0 0 . . . 0
CK Ir0. . . 0
CAK C K Ir. . . 0
. . . . . . . . . . . . . . .
CAp2K C Ap3K C Ap4K . . . Ir
,Kl=
Ir0 0 . . . 0
CKlIr0. . . 0
CAKlC Kl+1 Ir. . . 0
. . . . . . . . . . . . . . .
CAp2KlC Ap2Kl+1 C Ap3Kl+2 . . . Ir
,
where the l-th columns of E
Kand E+
Kare, respectively,
[E
K]l= (Kl1− K)[E]l,[E+
K]l= (Kl1+p− K)[E+]l.(28)
These remainder terms converge to zero as lgrows since the non-steady state Kalman gain Klconverges to
the steady state gain Kduring the transient phase, which is necessary due to possible errors in the initial
estimates ˆx0and P0. Note that the transient aspect of the non-periodic part of these remainder terms could
be neglected since the respective part of the gain converges to its steady state limit fast, namely at exponential
rate. However, since the periodic part converges only at a linear rate [33], they cannot be neglected without
further analysis.
Equations (26) and (27) are required for the analysis of projections of the data Hankel matrices. In
particular, the terms ΓXand ΓX+in the raw data in (26) contain both system and periodic parts,
yielding ΓX= ΓsysX
sys + ΓperX
per and ΓX+= ΓsysX+
sys + ΓperX+
per. In the proposed method the raw data
matrix is projected on the orthogonal complement of the data matrix of the estimated periodic subsignal in
order to remove the periodic part from the raw data. To this end, the projection matrices
Y
pro =Y
raw/Y
per=Y
raw − Y
rawY
perT(Y
perY
perT)Y
per,(29)
Y+
pro =Y+
raw/Y+
per=Y+
raw − Y+
rawY+
perT(Y+
perY+
perT)Y+
per (30)
are defined. With these projections, the periodic parts of the terms ΓXand ΓX+in the raw data in (26)
are removed asymptotically, as shown in the following theorem.
Theorem 3. The orthogonal projection of the raw data matrix onto the orthogonal complement of the data
matrix of the estimated periodic subsignal yields the decomposition
Y
pro = ΓsysX
sys +KE+E
K+o(1),
Y+
pro = ΓsysX+
sys +KE++E+
K+o(1),
where o(1) is a matrix whose norm converges almost surely to zero when N→ ∞.
Proof: See Appendix C.
14
Hence, the orthogonal projections (29)–(30) provide a reconstructed output signal where the periodic
system parts are (asymptotically) removed. The samples ˆypro
kof this signal can be recovered from the
block rows of Y
pro or Y+
pro. Based on these signals, the subspace-based system identification can be used to
estimate only the structural system modes, as detailed in the following section.
3.3. Output-only subspace identification of the system part from Ypro
In SSI-UPC [14], a projection of the future output data matrix onto its past yields the factorization into
the observability matrix of the system and a Kalman filter state sequence. From the observability matrix,
the system matrices Aand Care obtained, and subsequently the modal parameters.
In the following, the modal parameters of the structural system are obtained from a projection of the
data matrices Y+
pro and Y
pro analogously to the UPC method, namely
H=Y+
pro/Y
pro.(31)
To investigate the properties of this projection with respect to the identification of the structural system,
define the matrices Y+
sys = ΓsysX+
sys +KE+and Y
sys = ΓsysX
sys +KE. These matrices would contain
the outputs of the structural system without the contribution of the periodic excitation, see Appendix A.
Note that they are not actually computed on data, nor do they contain the contribution of the transient
part of the non-steady Kalman filter since it decays with N. With these definitions, the proposed subspace
procedure in (31) yields the same projection as Y+
sys onto Y
sys, as shown in the following theorem.
Theorem 4. The projection of the future projected data matrix Y+
pro onto its past Y
pro yields the factorization
H=Y+
pro/Y
pro =Y+
sys/Y
sys +o(1) = ΓsysX+
sys/Y
sys +o(1),
from where an estimate of the observability matrix Γsys of the system part can be obtained.
Proof: See Appendix D.
Estimates of the system parameters from matrix Hare shown to be consistent in the following corollary.
Corollary 5. The subspace method using Hin Theorem 4 is consistent for the estimation of the system
matrices and subsequently of the modal parameters of the structural system, i.e., they converge to the true
parameters of the structural system for N→ ∞.
Proof: See Appendix E.
The estimates of Asys and Csys and subsequently the estimates of modal parameters can be computed
from Hin a classical way after Appendix B and (7).
15
Remark 6. Projections are a common tool for system identification with subspace methods. In the proposed
approach, the orthogonal projections (29)(30) were used to reconstruct an output signal, where the periodic
parts are removed (Theorem 3), and which can be used for consistent system identification (Theorem 4 and
Corollary 5). A similar result can be achieved by subtracting the estimated periodic subsignal from the raw
data, instead of the proposed orthogonal projection. Defining Y
diff =Y
raw − Y
per and Y+
diff =Y+
raw − Y+
per
yields a similar decomposition as in Theorem 3 thanks to properties (26)(27), and analogous results as in
Theorem 4 and Corollary 5 can be proven for consistent identification using H=Y+
diff /Y
diff .
3.4. Numerically efficient implementation
The projections to obtain Y
pro,Y+
pro and Hin (29), (30) and (31) may be costly in computational efforts.
An efficient numerical implementation that avoids the explicit computation of these projections is described
in this section. The LQ decomposition of the stacked Yper and Yraw writes
Y
per
Y+
per
Y
raw
Y+
raw
=
L11 0 0 0
L21 L22 0 0
L31 L32 L33 0
L41 L42 L43 L44
QT
1
QT
2
QT
3
QT
4
=
L12,12 0
L34,12 L34,34
QT
12
QT
34
.(32)
The combined projected data matrices Y
pro and Y+
pro from (29)–(30) can also be (asymptotically) expressed
as follows, and plugging in (32) yields
Y
pro
Y+
pro
=
Y
raw
Y+
raw
.
Y
per
Y+
per
= (L34,12QT
12 +L34,34QT
34)IQ12 LT
12,12 L12,12QT
12Q12 LT
12,121L12,12 QT
12
= (L34,12QT
12 +L34,34QT
34)IQ12 QT
12
=L34,34QT
34 =
L33 0
L43 L44
QT
3
QT
4
.(33)
The projection from (31) yields thus
H=Y+
pro/Y
pro = (L43QT
3+L44QT
4)Q3LT
33(L33 LT
33)1L33 QT
3=L43QT
3.
Since the observability matrix Γsys is estimated from the column space of H(see Appendix B) and since
Q3is an orthogonal matrix, L43 can directly be used to estimate Γsys , without explicitly performing the
projection in (31). The proposed scheme is summarized in Algorithm 1.
Remark 7. When the number of sensors is bigger than the number of periodic modes, i.e., r > 2h, the
estimated periodic subsignal from (23) contains r2hredundant responses and rank (Yper) = 2h(p+ 1).
16
Consequently L12,12 becomes rank deficient, while full rank is needed to obtain (33). In such a case, the
periodic states selected directly from the rows of ˆxV
kcan substitute the estimation of the periodic subsignal
from (23) by
ˆyper
k= ˆxV,per
k.(34)
This leads to a reduction of the dimensions of Y
per and Y+
per without changing the projected matrices Y
pro
and Y+
pro, and consequently (33) holds.
4. Numerical validation
In this section the proposed method is deployed first to remove the periodic mode information from
the simulation of the chain system described in Section 2.3 and second to identify its structural modal
parameters.
First, the adequacy of the combined state-space model (14)–(15), containing both system and periodic
parts, is illustrated by comparing the exact system states to their computed Kalman filter counterparts. For
this, output data of the mechanical system are simulated under both white noise and periodic excitation as
well as output noise, using the discrete-time version of model (8)–(9). Then, the exact system and periodic
states of model (14)–(15) are computed and transformed into the real-valued modal basis. To compare them
to their estimates, the system matrices of the combined model are estimated at model order 14 from the
simulated outputs, and the non-steady state Kalman filter states are computed with (16)–(18). To transform
Algorithm 1: Removal of periodic subsignal and identification of structural system
Input : raw data ykof the system under ambient and periodic excitation;
model order n
Output: reconstructed time series ˆypro
kwithout periodic subsignal;
modes of structural system
1build data matrices Y
raw and Y+
raw from ykin (24) and compute ˆ
A,ˆ
Cat selected model order n
with subspace system identification (Appendix B);
2compute Kalman filter states ˆxkin (16)–(18);
3map periodic poles of ˆ
Aand compute similarity transform ˆ
AV,ˆ
CV, ˆxV
kin (21) and (22);
4compute periodic sequence ˆyper
kin (23) if r > 2h, or in (34) if r2h, and fill data matrices Y
per
and Y+
per in (25);
5compute LQ decomposition of the stacked Y
per,Y+
per,Y
raw and Y+
raw in (32);
6reconstructed time series ˆypro
kcan be obtained from Y
pro =L33QT
3(see (33));
7computation of ˆ
Asys,ˆ
Csys from L43 (Appendix B) and modal parameters in (7)
17
them into the modal basis of the exact states, an appropriate scaling of the estimated eigenvectors is needed.
These scaling factors are obtained by relating the identified mode shapes to the theoretical ones.
In Figure 6, the exact states and computed Kalman states corresponding to the third structural mode of
the chain system are shown, which is the mode closest to the periodic mode. In Figure 7, the exact states
and computed Kalman states of the periodic mode are shown. It can be observed that the Kalman states of
the structural mode are close to the exact states after about 5 samples, and after approximately 80 samples
for the periodic mode. This suggests that a good approximation of the system states is obtained after a
transient phase, where the non-steady state Kalman gain from (16) converges. Note that the non-steady
state Kalman filter is able to estimate the states of the periodic mode accurately while the initial condition
of those was not exact. This would not be possible using the converged steady state Kalman gain whose
periodic part is zero.
Next, Algorithm 1 for the removal of the periodic part is applied. In Figure 8 the two highest PSD
singular values are shown that are computed from the raw data yk(left) and from the time series ˆypro
kthat
is reconstructed from the projected data matrix Y
pro after (29) (right). From Figure 8 it can be seen that
Figure 6: Exact states and computed Kalman states corresponding to the structural mode closest to the periodic mode.
Figure 7: Exact states and computed Kalman states corresponding to the periodic mode.
18
Figure 8: Two largest singular values of PSD matrix from raw data yk(left) and from projected data ˆypro
kafter removal of the
periodic subsignal (right).
the sharp peak corresponding to the periodic frequency at 8.69 Hz in Figure 8 (left) is cancelled in Figure 8
(right), and the resultant PSD plot resembles the reference case where no periodic inputs are present in
Figure 2. Therefore it can be conjectured that the periodic information is removed from the raw data.
This is also verified in a Monte Carlo simulations, where the modal parameters are estimated first from
the raw data and second from the projected data after removal of the periodic part with Algorithm 1. In
Figure 9 the plots of the estimated frequencies versus damping ratios are shown for all the simulations. The
periodic mode visible in Figure 9 (left) is identified with a low damping ratio and can easily be distinguished
from modes of the structural system for the removal procedure. It can be observed that the periodic mode is
indeed rejected in Figure 9 (right), and the estimates of natural frequencies and damping ratios are centered
around the exact values from the model. Furthermore, it can be seen that the estimation uncertainties of
the proposed method are similar as in the classical SSI-UPC, since the scattering of the estimated modal
parameters is of the same magnitude before and after the rejection of the periodic subsignal.
5. Application
In this section, two experimental cases are depicted to illustrate the performance of the proposed method.
The first example is a plate subjected to a mix of random and periodic excitation in laboratory conditions.
The second example is a full-scale test of a ship excited by random environmental load with interference
from rotating machinery on-board.
5.1. Plate with harmonics
The experimental setup and the geometry of the plate are shown in Figure 10. Periodic excitation
is applied by a shaker with a sinusoidal signal of 370 Hz continuously throughout the experiment. The
19
Figure 9: Estimates of natural frequencies and damping ratios from raw data yk(left) and from projected data ˆypro
kafter
removal of the periodic subsignal (right) for the complete Monte Carlo simulation.
measurements are sampled with 4096 Hz over a 120 seconds interval. The same experiment is also carried
out without the periodic excitation in order to compare the modal parameter estimates with and without
periodic excitation.
The frequency of the periodic signal is close to the first natural frequency of the plate, which is a
particular challenge for system identification of experimental data [34]. In this context, the stochastic
subspace identification of the raw data containing the responses to mixed random and periodic excitation
is carried out with p= 20 and model orders ranging from nmin = 10 to nmax = 40. In Figure 11 the
resulting stabilization diagram of natural frequencies is shown. It can be seen that the periodic mode at 370
Hz and the close structural mode at 341 Hz cannot be identified below model order 20, since they are not
separated. Other structural modes can already be identified at lower model orders, which suggests that the
first structural mode may be perturbed by the periodic mode.
The frequency and damping ratio alignment of the periodic mode are presented in Figure 12 (left).
It can be observed that its damping ratio is indeed small. The periodic mode estimate at model order
Figure 10: The experimental setup: plate with 16 acceleration channels, shaker, acquisition system (left). The plate model
with 16 acceleration channels in ARTeMIS Modal Pro 6.0 (right).
20
40 is then selected to estimate the periodic subsignal, used in the orthogonal projection in (29)–(30). In
Figure 12 (right) the two largest PSD singular values are shown from the raw, the estimated periodic and
the reconstructed system output data. It can be seen that the peaks of the PSD from the estimated periodic
signal (blue line) match well with the periodic peaks of the PSD from the raw measurements.
Finally, the system identification results with the proposed method are presented after the removal of
the periodic subsignal, corresponding to Algorithm 1. The corresponding stabilization diagram is shown
in Figure 13. It can be observed that the periodic mode is no longer part of the estimated modes. In
addition, the first natural frequency at 341 Hz is better estimated, namely already for much lower model
orders compared to Figure 11, after the periodic part is removed.
A detailed comparison of the modal alignments for the natural frequencies and the damping ratios of
the first mode estimated before and after the removal of the periodic subsignal is shown in Figure 14, where
Figure 11: Stabilization diagram of the natural frequency estimates from the raw measurements of the plate containing both
structural and periodic modes.
Figure 12: Left: Stable modal alignments of the natural frequency and damping ratio estimates of the periodic mode of the
plate. Right: Two largest singular values of PSD from raw measurements (top), and estimated periodic signal and projected
data (bottom).
21
Figure 13: Stabilization diagram of the natural frequency estimates from measurements of the plate after the removal of the
periodic subsignal.
the results are also compared to the reference estimates from the plate experiment with only random and
no periodic excitation. The estimated modal parameters are close to their counterparts estimated from the
random response. While the estimated natural frequency and damping ratio are closer to their equivalent
random response estimates after the removal of the periodic subsignal in Figure 14 (right) than before (left),
the change in frequency towards its reference value is very small (0.05% of the value) and may not be
significant. The change in damping ratio towards its reference value, however, is more significant (14% of
the reference value), indicating a less biased damping estimate of the first mode. For both the frequency
and damping ratio, the alignments of the first mode after the removal of the periodic subsignal are more
stable and start at a lower model order in Figure 14 (right) than before (left).
In Table 2 the alignment means of the first six modes are shown for the different experimental cases.
With respect to the reference estimates from the random response, the frequency estimates before and after
the removal of the periodic subsignal show very small differences that are of the same order for all modes.
The differences in the damping estimates are naturally larger, and it can be seen that the estimates after
the removal are either close to the values before the removal, or closer to the reference values.
From these results it can be concluded that the removal of the periodic subsignal leads to more stable
alignments of the structural mode close to the periodic one, and the damping estimates of some of the modes
(in particular of the mode close to the periodic one) are closer to the reference values.
5.2. Ship in operation
The considered structure is a roll-on roll-off ship on a test trail [35] that is subjected to random wind and
wave loads interfered with periodic excitation from the propellers and the engine. Output accelerations are
measured with 16 channels and are sampled with 128 Hz for 5400 seconds. The geometry of the ship with
the measured degrees of freedom is illustrated in Figure 15. Prior to the analysis the data are decimated to
8 Hz.
22
Figure 14: Modal alignments of the natural frequency and damping ratio estimates for the first structural mode from measure-
ments before (left) and after (right) the removal of the periodic subsignal.
Table 2: Modal parameters of the plate without periodic excitation, and with periodic excitation estimated before and after
removal of the periodic subsignal.
Data type/Mode 1 2 3 4 5 6
random (reference) f[Hz] 341.04 472.12 688.34 837.03 933.62 1382.04
ζ[%] 0.563 0.494 0.757 0.453 0.569 0.971
mixed random and periodic f[Hz] 341.23 472.34 688.80 837.90 933.61 1381.62
ζ[%] 0.459 0.482 0.840 0.406 0.517 0.756
mixed random and periodic after removal
of the periodic subsignal in (29)
f[Hz] 341.06 472.49 688.91 837.79 932.81 1381.08
ζ[%] 0.537 0.466 0.721 0.392 0.503 0.841
Figure 15: The ship at the Flensburg shipyard (left). The geometry of the ship with 16 acceleration channels in ARTeMIS
Modal Pro 6.0 (right).
Similar to the plate example from the previous section, the modal parameters are estimated from the
data subjected to the mixed random/periodic excitation first. For this purpose the UPC algorithm is used
with p= 25 and model orders ranging from nmin = 10 to nmax = 50. The resulting stabilization diagram of
natural frequencies is shown in Figure 16. It can be observed that the periodic frequency at 2.05 Hz is close
23
Figure 16: Stabilization diagram of the natural frequency estimates from the raw measurements of the ship containing both
structural and periodic modes.
Figure 17: Left: Stable modal alignments of the natural frequency and damping ratio estimates of the periodic mode at 2.05 Hz.
Right: Two largest singular values of PSD from raw measurements (top), and estimated periodic signal and projected data
(bottom).
to a structural mode of the ship whose estimation is possibly perturbed by the periodic mode.
A zoom on the modal alignment of the periodic mode is presented in Figure 17 (left). It can be observed
that the damping ratio estimates of the periodic mode are low, which distinguishes it from the structural
modes. Subsequently, the periodic mode at the model order 50 is selected for the removal of the periodic
subsignal with (29). In Figure 17 (right) the two largest PSD singular values are shown of the raw, the
estimated periodic and the reconstructed structural system output data. It can be observed that the peak
of the PSD from the estimated periodic subsignal coincides well with the peak of the periodic mode from
the raw measurements. Moreover the reconstructed output data contain no high energy frequency content
at 2.05 Hz, suggesting that the periodic mode information is removed.
Subsequently, results from the system identification with the proposed method are presented in Figure 18
after the removal of the periodic subsignal. They clearly illustrate that the periodic mode at 2.05 Hz is
24
Figure 18: Stabilization diagram of the natural frequency estimates from measurements of the ship after the removal of the
periodic subsignal.
Figure 19: Modal alignments of the natural frequency and damping ratio estimates for the second structural mode (close to
the periodic mode) from measurements before (left) and after (right) the removal of the periodic subsignal.
successfully removed from the data. A detailed comparison of the modal alignments of the close structural
mode estimated from the measurements before and after the removal of the periodic subsignal is shown in
Figure 19. Deviation of both natural frequencies and damping ratio estimates from the mean values of their
modal alignments is lower when the periodic information is removed, suggesting that the proposed approach
is beneficial in a practical modal analysis application. Moreover, the structural mode is already identified
at lower model orders after the removal.
6. Conclusion
In this paper, a subspace framework has been derived for the estimation of the structural modes of
a mechanical system under both ambient and periodic excitation. This approach consists of three steps,
starting with a classical output-only SSI that provides initial estimates of the system matrices. These
25
matrices are used in the second step to compute a sequence of non-steady Kalman states in the modal basis
in order to estimate the periodic subsignal from the periodic modes. In the final step, the raw output data is
projected onto the orthogonal complement of the estimated periodic signal, yielding a new subspace method
for the identification of the structural modes while rejecting the periodic modes from the data.
Besides the development of the algorithmic procedure, a few theoretical results have been proved. First,
it has been shown that the considered mechanical model under both ambient and periodic excitation is
equivalent up to a similarity transform to the stationary modeling proposed in [23], where the stochastic
and periodic subsystems are decoupled. This validates the first step of the developed method and yields
stability properties of the Kalman state estimates. Second, the rejection of the periodic information from
the raw signal has been explicitly formulated. Using the resulting signal for a UPC-like projection of its
future time horizon onto its past, the factorization into the observability matrix and Kalman states of the
structural system – without any periodic parts – has been proved to hold asymptotically. Consequently,
consistency of the proposed SSI method has been shown.
Both the modeling and the removal of the periodic subsignal have been validated on simulations of a chain
system, and the consistency of the identification was illustrated on Monte Carlo simulations. Furthermore,
the proposed method was demonstrated on experimental data under both ambient and periodic excitation,
namely on an aluminum plate excited by a shaker in the lab and on a ship in operation excited by the
combination of environmental load and the interference from the rotation of engine and propellers. In both
cases the periodic frequency was close to a natural frequency of the structure. The results illustrate that
the periodic subsignal has been removed successfully, and that it led to more consistent modal parameter
estimates of the structural modes with modal alignments that stabilize from a lower model order. Future
work includes the uncertainty analysis of the algorithm.
Acknowledgments
Qinghua Zhang is acknowledged for discussions about the stability properties of the Kalman filter. Niels-
Jørgen Jacobsen from Br¨uel & Kjær is acknowledged for carrying out the experimental tests on the plate
and sharing its results. Sven-Erik Rosenow, Santiago Uhlenbrock and G¨unther Schlottmann from University
of Rostock are acknowledged for sharing the measurements of the ship.
26
Appendix A. Theoretical innovation model and stability of Kalman filter
In the following it is shown that the considered combined state-space model (14)–(15) corresponds to
the theoretical innovation model
zs
k+1
zd
k+1
=
As0
0Ad
zs
k
zd
k
+
K
0
ek,(A.1)
yk=hCsCdi
zs
k
zd
k
+ek.(A.2)
This model was analyzed in [23], where observability of the pair (Cd, Ad) is required, as well as observability
and controllability of the stochastic subsystem. The theoretical innovation ekhas finite fourth order mo-
ments. Under these conditions, stability of the Kalman filter is given, and finite fourth order moments of
the states can be assumed.
First it is shown that there exists an invertible matrix T, such that
T1
Asys Ab
0Aper
T=
Asys 0
0Aper
,with T=
I T12
0I
, T 1=
IT12
0I
.
Matrix T12 R2m×2h(and thus matrix T) is constructed by multiplying out the left expression, leading to
the necessary condition AsysT12 T12 Aper +Ab= 0. This Sylvester equation has a unique solution since
the eigenvalues of Asys (inside the unitary circle) are distinct from the eigenvalues of Aper (on the unitary
circle). The solution is given through [(I2hAsys)(AperTI2m)]vec(T12) = vec(Ab), where denotes
the Kronecker product and vec(·) the column stacking vectorization operator. Pre-multiplying (14) by T1
leads to the transformed state-space model
˜xsys
k+1
˜xper
k+1
=
Asys 0
0Aper
˜xsys
k
˜xper
k
+
wk
0
,(A.3)
yk=hCsys (CsysT12 +Cper )i
˜xsys
k
˜xper
k
+vk,(A.4)
with the transformed states
˜xsys
k
˜xper
k
=T1
xsys
k
xper
k
=
xsys
kT12xper
k
xper
k
.
Note that the process noise term in (A.3) remains the same after the similarity transform. Finally, assuming
that the stochastic subsystem is observable and controllable by the noise, and assuming that the periodic
subsystem is observable, the theoretical innovation model corresponding to (A.3)–(A.4) is indeed given by
(A.1)–(A.2). Thus, following [23], the Kalman filter applied to data from the considered combined state-
space model (14)–(15) is stable, and the fourth moments of the Kalman filter states and innovations are
27
bounded [31]. Due to the stability of the Kalman filter, the same holds also when applying the non-steady
state Kalman gain.
Note that regarding the states, (A.3) shows that there is a state space basis in which the periodic states
xper
k(or equivalently ˜xper
k) are decoupled from the structural states ˜xsys
k, unlike the physical structural states
xsys
kof the mechanical model in (14)–(15). Due to the block diagonal structure of the state transition matrix
in (A.3), this is also the case for the modal basis.
Appendix B. Subspace system identification
Assume the matrix ˆ
Henjoys (asymptotically) the factorization property into H= ΓXwhere Γ
R(p+1)r×nis defined as
Γ =
C
CA
.
.
.
CAp
,
and Xare the system states which are dependent on the chosen projection algorithm [14]. In practice, the
estimate of the observability matrix Γ is computed from the SVD of Hestimated from the measured data
sequences
ˆ
H=hU1U2i
D10
0D2
VT
1
VT
2
,
where an estimate of Γ is taken as ˆ
Γ = U1D1/2
1. Matrices U1and V1are the left and right singular vectors
corresponding to first nnon-zero singular values D1and U2with V2are the left and right kernel of ˆ
Hwhere
D20. The estimates ˆ
Aand ˆ
Ccan be computed in a least-square sense from the shift invariance property
of ˆ
Γ.
Appendix C. Proof of Theorem 3
The first projection in Theorem 3 yields
Y
pro =Y
raw/Y
per=Γsys X
sys + ΓperX
per +KE +E
K/Y
per
= ΓsysX
sys/Y
per+ Γp erX
per/Γp erX
per
| {z }
=0
+KE/Y
per+E
K/Y
per,(C.1)
where the second term cancels out since an orthogonal projection yields A/A= 0 for any matrix A. The
analysis of the last two terms requires a deeper insight into the properties of the estimated innovations and
Kalman filter states. First, EX
per
T0 is shown to simplify the third term, and second, E
KX
per
T0
is shown for the fourth term. For this we suppose that ykand ˆxkhave uniformly bounded fourth order
moments, which is justified in Appendix A. Finally, the first term is simplified by showing X
sysX
per
T0.
28
1. Proof of EX
per
T0
First a few mathematical notations: Consider the σalgebra Fkgenerated by the observations y1, . . . , yk.
The collection of σalgebra Fkis increasing, i.e., Fk⊂ Fk+1. The innovation ekat time kis Fkmeasurable,
whereas ˆxkis Fk1measurable. Since E(ek| Fk1) = E(ek|Y1, . . . , Yk1) = 0, the innovations are
uncorrelated with the past Kalman states (and also with the past outputs) [14], i.e.,
E(ekˆxT
kj)=0,j0,(C.2)
which holds since E(ekˆxT
kj) = E(E(ekˆxT
kj| Fk1)) = E(E(ek| Fk1xT
kj) = E(0.ˆxT
kj) = 0. Then,
considering supkE(kekk4+kˆxkk4)< C < , by Theorem 2.8 in [36],
EXT=1
N
e0e1
.
.
.eN1
e1e2
.
.
.eN
.
.
..
.
..
.
..
.
.
ep1ep
.
.
.ep+N2
ˆxT
0
ˆxT
1
.
.
.
ˆxT
N1
=
1
NPelˆxT
l
1
NPel+1 ˆxT
l
.
.
.
1
NPel+p1ˆxT
l
0
with probability 1, in particular EX
per
T=o(1). It follows EY
per
T=EX
per
TΓperT=o(1). Since
(Y
perY
per
T)= (ΓT
per)(X
perX
per
T)1per)is bounded due to the distinct periodic eigenvalues and the
resulting independence of the periodic state components, the third term in (C.1) yields
KE/Y
per=KE− KEY
per
T(Y
perY
per
T)Y
per =KE +o(1)Y
per.(C.3)
Since the signal has bounded moments of order 4, it holds o(1)Y
per =o(1).
2. Proof of E
KX
per
T0
With (28) it holds
E
KX
per
T=1
N
N1
X
l=0
(Kl− K)
el
el+1
.
.
.
el+p1
ˆxper
l
T=1
N
N1
X
l=0
p1
X
j=0
[Kl− K]jel+jˆxper
l
T
where [Kl− K]jindicates the (j+ 1)-th block column of matrix (Kl− K). Recall that the Kalman gain
is deterministic and independent of the observations. It converges linearly in the periodic part [33] and
exponentially for the other components [37], thus ||Kl− K|| =O(1/l).
Denote ˜el,j = [Kl− K]jel+j. Since the innovations have bounded fourth order moments, this is also the
case for ˜el,j . Analogously to part 1 of the proof, it follows with [36] that 1
NPl˜el,j ˆxper
lT=o(1) for j0,
finally E
KX
per
T=o(1) and thus E
KY
per
T=E
KX
per
TΓperT=o(1). Thus the fourth term in (C.1) yields
E
K/Y
per=E
K− E
KY
per
T(Y
perY
per
T)Y
per =E
K+o(1)Y
per.(C.4)
29
3. Proof of X
sysX
per
T0
The columns of matrices X
sys and X
per are the components of the Kalman states ˆxV
kcorresponding to
the system and periodic parts, respectively, in the real-valued modal basis after similarity transform (21)
with Vdefined in (22). Denote these parts by ˆxV ,sys
kand ˆxV,per
k, respectively.
For simplicity of notation we carry out the proof in the complex-valued modal basis defined by Vc=
hΨ Ψi(cf. (22)), thus AVc=V1
cAVc= diag(Λ,Λ). Then, the system and periodic parts of the states
ˆxVc
k=V1
cˆxkrelate to ˆxV ,sys
kand ˆxV,per
kby
ˆxV,sys
k=T1
mˆxVc,sys
k,ˆxV,per
k=T1
hˆxVc,per
k, Ta=1
2
IaiIa
IaiIa
,(C.5)
where mand hare the number of system and periodic mode pairs (see Section 2.2). Denote the i-th
component of vector ˆxVc,sys
kby ˆxVc,sys
k,i , and the j-th component of vector ˆxVc,per
kby ˆxVc,per
k,j . In the following,
it is shown that 1
NPkˆxVc,sys
k,i ˆxVc,per
k,j 0 for any i, j, from where X
sysX
per
T0 follows with (C.5).
From the state equation it follows
1
N
N1
X
k=0
ˆxVc,sys
k+1,i ˆxVc,per
k+1,j =1
N
N1
X
k=0
(λsys
iˆxVc,sys
k,i +KVc,sys
k,i ek)(λper
jˆxVc,per
k,j +KVc,per
k,j ek)
=1
N
N1
X
k=0
(λsys
iλper
jˆxVc,sys
k,i ˆxVc,per
k,j +λsys
iˆxVc,sys
k,i eT
kKVc,per
k,j
T+λper
jˆxVc,per
k,j eT
kKVc,sys
k,i
T+KVc,sys
k,i ekeT
kKVc,per
k,j
T)
where KVc,sys
k,i is the i-th row of the system part of the Kalman gain KVc
k, and KVc,per
k,j is the j-th row of its
periodic part. Then
(1 λsys
iλper
j)1
N
N1
X
k=0
ˆxVc,sys
k,i ˆxVc,per
k,j +1
NˆxVc,sys
0,i ˆxVc,per
0,j 1
NˆxVc,sys
N,i ˆxVc,per
N,j
=1
N
N1
X
k=0
λsys
iˆxVc,sys
k,i eT
kKVc,per
k,j
T+1
N
N1
X
k=0
λper
jˆxVc,per
k,j eT
kKVc,sys
k,i
T+1
N
N1
X
k=0
KVc,sys
k,i ekeT
kKVc,per
k,j
T.(C.6)
The first term on the right hand side goes to zero since ˆxsys
kand ekare independent. The second term goes
to zero in a similar manner. The third term is a mean of the independent process Lk=KVc,sys
k,i ekeT
kKVc,per
k,j
T
since ekis an independent Gaussian process. Let νk= E(Lk) = KVc,sys
k,i QkKVc,per
k,j
T, where Qkis the innova-
tion covariance at kthat is bounded since the fourth moment of the innovation is bounded. Thus, it holds
PE(Lkνk)2/k2<. Then [38, Theorem 3.7] (or [39, Theorem 5.4.1]) can be applied, yielding 1
NPLk
1
NPνk=o(1), and since
1
NPνk
1
NPkKVc,sys
k,i k kQkk kKVc,per
k,j k ≤ supkkQkkO(1)
NPN1
k=0 O(1/k) =
o(1), the third term on the right hand side of (C.6) goes to zero. The remaining terms on the left hand side
yield 1
NˆxVc,sys
0,i ˆxVc,per
0,j 1
NˆxVc,sys
N,i ˆxVc,per
N,j 0 by the Chebyshev inequality and Borel Cantelli Lemma, since
the moments are bounded. Finally, X
sysX
per
T=o(1) and
X
sys/Y
per=X
sys +o(1)Y
per.(C.7)
30
4. End of the proof
Plugging (C.3), (C.4) and (C.7) into (C.1) yields
Y
pro = ΓsysX
sys +KE+E
K+o(1)Y
per.(C.8)
Analogously, the relation for Y+
pro =Y+
raw/Y+
peris obtained as
Y+
pro = ΓsysX+
sys +KE++E+
K+o(1)Y+
per,(C.9)
To conclude the proof, since Y
perY
per
Tand Y+
perY+
per
Tare bounded, the norm of the remainder term is o(1).
Appendix D. Proof of Theorem 4
Since Y
pro =Y
raw − Y
raw/Y
per, the pro jection in (31) yields
H=Y+
pro/Y
pro =Y+
pro Y
raw − Y
raw/Y
perTY
proY
pro
TY
pro
| {z }
=W1
=
Y+
proY
raw
T
| {z }
=P1
− Y+
proY
per
T
| {z }
=P2
(Y
perY
per
T)Y
perY
raw
T
| {z }
W2
W1(D.1)
1. Preliminaries
The components of Y
raw are observations of the considered linear system. Thus, the moments of Y
raw
derive from the moments of the underlying state and innovations. Analogously to (C.2), it holds E(ekyT
kj) =
0j > 0, since E(ekyT
kj) = E(E(ek| Fk1)yT
kj) = 0 and it can be proved that E+Y
rawT0 and
E+
KY
rawT0 similarly to Appendix C.
2. Asymptotic formulations for P1and P2
Plugging (C.9) into P1and P2yields
P1= ΓsysX+
sysY
raw
T+KE+Y
raw
T+E+
KY
raw
T+o(1)Y+
perY
raw
T
P2= ΓsysX+
sysY
per
T+KE+Y
per
T+E+
KY
per
T+o(1)Y+
perY
per
T
The matrix Y+
perY
per
Tis bounded since it is filled by covariances of the signal Yp er, which are bounded
since the signal has bounded moments of order 4. Thus o(1)Y+
perY
per
T=o(1). The matrix Y+
perY
raw
Tis
bounded too, since a Cauchy Schwartz inequality relates Y+
perY
raw
Tto Y+
perY+
per
Tand Y
rawY
raw
Twhich are
both covariance matrices and thus bounded because the signals Yper and Yraw have moments of order 4.
Recall that E+Y
per
T0 and E+
KY
per
T0, and that E+Y
raw
T0 and E+
KY
raw
T0. Then, P1and P2
simplify to
P1= ΓsysX+
sysY
raw
T+o(1),P2= ΓsysX+
sysY
per
T+o(1).
31
3. Asymptotic expression of the UPC projection
Subsequently the projection in (D.1) yields
H=Y+
pro/Y
pro = ΓsysX+
sys Y
raw
T− Y
per
TW2W1+o(1)W1+o(1)W2W1
Matrix W2is the product of bounded matrix (Y
perY
per
T)(see Appendix C, part 1) and Y
perY
raw
T, which
can also be proved to be bounded similarly to Y+
perY
raw
T. Next we show that Y
proY
pro
Tis of full rank. With
(C.8) it follows
Y
proY
pro
T= ΓsysX
sysX
sys
TΓT
sys +KEETKT+E
KE
K
T+L+LT+ (cross terms) ×o(1) (D.2)
where L= ΓsysX
sysETKT+ Γsys X
sysE
K
T+KEE
K
T. The term E
KE
K
Tin (D.2) is the diagonal covariance
of E
K, whose norm is decreasing as the gain converges, so this goes to zero. Thus, the third term of Lalso
goes to zero by Cauchy Schartz inequality. The first two terms of Lare o(1) analogously to Appendix C.
All the cross terms in (D.2) are bounded, hence
Y
proY
pro
T= ΓsysX
sysX
sys
TΓT
sys +KEETKT+o(1).
The first matrix is positive semi-definite, and since EETis positive definite and Kis of full rank, the
smallest singular value of matrix Y
proY
pro
Tis bounded from below for Nlarge enough. Thus Y
proY
pro
T
is bounded and
H= ΓsysX+
sys Y
raw
T− Y
per
TW2W1+o(1)Y
pro.(D.3)
4. Relation of UPC projection to Y
sys and end of proof
Since Y
pro =Y
raw
T− Y
per
TW2, it follows from (D.3)
H= ΓsysX+
sysY
pro
TY
proY
pro
T1Y
pro +o(1)Y
pro (D.4)
where Y
pro =Y
sys +E
K+o(1)Y
per, and define Y
sys = ΓsysX
sys +KE. Then Y
proY
pro
T= (Y
sysY
sys
T+
o(1))1= (Y
sysY
sys
T)1+o(1) by the matrix inverse sensitivity, and the bounds proved previously. Finally,
X+
sysY
pro
T=X+
sys(Y
sys +E
K+o(1)Y
per)T
=X+
sysY
sys
T+X+
sysE
K
T+X+
sysY
per
To(1) = X+
sysY
sys
T+o(1)
and thus
X+
sysY
pro
T(Y
proY
pro
T)1Y
pro =X+
sysY
sys
T(Y
sysY
sys
T)1Y
sys +O(1)E
K+o(1)Y
per
where E
KE
K
T=o(1). Then it follows with (D.4), since E
KY
per
T=o(1) with Appendix C, and other
cross-terms are bounded,
H= ΓsysX+
sys/Y
sys +o(1) = Y+
sys/Y
sys +o(1),(D.5)
32
since EY
sys
T=o(1), and the proposed method yields the same formulation as the SSI-UPC approach
computed on the so-called virtual structural system, which concludes the proof.
Appendix E. Proof of Corollary 5
The main challenge in the asymptotic study of H– and thus the consistency of Aand Cthat are
estimated from its column space – is the fact that the number of columns of Hgrows with the number
of samples N. By evaluating the asymptotic properties of the “square” matrix HHT, which has the same
column space as Hbut whose dimensions are finite and non-increasing, the consistency of Aand Ccan be
analyzed [16, 20].
Define H1= ΓsysX+
sys/Y
sys. Then it follows from (D.5)
HHT= (H1+o(1))(H1+o(1))T=H1HT
1+ (H1+o(1))o(1)T+o(1)HT
1
=H1HT
1+o(1),
since it can be proved that kH1kis bounded similarly to Appendix D. In consequence, it holds
HHT= ΓsysX+
sys/Y
syssys X+
sys/Y
sys)T+o(1).
This matrix satisfies Condition 1 of [16], and its consistency results from [16], Section 4.A, where the
assumptions of Theorem 1 in [16] have been verified. It follows that the estimates of (A, C ) from HHTare
consistent. Since the estimates of (A, C) from HHTand Hcoincide [20], the estimates of (A, C) from Hare
consistent too. This finishes the proof.
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... Statistical indicators like kurtosis can also be used to isolate harmonics in SSI algorithms. Once the harmonics are identified, the damping estimate can be enhanced using an algorithm like the Kalman-filter-based SSI (KF-SSI) technique developed by Greś et al. (2021). Here, a Kalman filter is used to obtain an estimate of the harmonic component of the system state. ...
... A refined damping estimate can now be obtained by applying classical SSI techniques to the projected data. Greś et al. (2021) demonstrate the advantage of KF-SSI over classical SSI techniques experimentally with measurements from a vibrating plate, excited with a harmonic signal with its frequency close to the first structural frequency of the plate. As seen in Fig. 14, classical SSI is unable to distinguish between the structural frequency and the harmonic for model order below 20, while KF-SSI is able to identify the structural frequency also at low model orders. ...
... Stabilisation diagram of the signal after applying classical SSI (top) and KF-SSI (bottom). For orders lower than 20, SSI fails to distinguish between the structural mode and harmonic and presents a merged version, while KF-SSI exclusively identifies the structural mode (Greś et al., 2021). (Devriendt and Guillaume, 2007). ...
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Operational Modal Analysis (OMA) provides essential insights into the structural dynamics of an Offshore Wind Turbine (OWT). In these dynamics, damping is considered an especially important parameter as it governs the magnitude of the response at the natural frequencies. Violation of the stationary white noise excitation requirement of classical OMA algorithms has troubled the identification of operational OWTs due to harmonic excitation caused by rotor rotation. Recently, a novel algorithm was presented that mitigates harmonics by estimating a harmonic subsignal using a Kalman filter and orthogonally removing this signal from the response signal, after which the Stochastic Subspace Identification algorithm is used to identify the system. Although promising results are achieved using this novel algorithm, several shortcomings are still present like the numerical instability of the conventional Kalman filter and the inability to use large or multiple datasets. This paper addresses these shortcomings and applies an enhanced version to a multi-megawatt operational OWT using an economical sensor setup with two accelerometer levels. The algorithm yielded excellent results for the first three tower bending modes with low variance. A comparison of these results against the established time-domain harmonics-mitigating algorithm, Modified LSCE, and the frequency-domain PolyMAX algorithm demonstrated strong agreement in results.
... Other studies have reported the same scattering trend for the identification of frequency and damping ratio [32]. Moreover, both the conventional and proposed methods exhibited the same variance for the damping ratio. ...
... These estimated semi-active inputs were in agreement with each other excluding their amplitudes because the piecewise bias removal process in the proposed self-sensing method did not use the piezoelectric parameters. The piezoelectric parameters act on the amplitude of the estimated semi-active input shown in Eq.(32).Figure 12(b) shows the three identification results obtained from the estimated input and output. Identification results under the nonnominal cases were a vertical shift of the result under the nominal case. ...
Article
Structural system identification (SSI) that uses many actuators and sensors will be more accessible as a sensing technology if the number of these devices decreases. Recently, a semi-active SSI using a piezoelectric transducer and a unique semi-active input generation circuit was proposed. The semi-active SSI outperforms conventional SSIs with low energy consumption. If the semi-active input and its vibration response are estimated from the voltage generated by a piezoelectric transducer, some sensors will become redundant. Conventional self-sensing methods estimate structural vibrations without using vibration sensors. The semi-active SSI in combination with the self-sensing method will be more accessible than current technologies from the perspectives of a lower energy consumption and fewer number of devices. However, the following two problems exist: (1) A semi-active input generation circuit cannot introduce conventional hardware-based self-sensing methods with additional electric circuits because the electrical components in the additional circuits degrade the performance of the piezoelectric transducer. (2) Software-based self-sensing methods, such as the Kalman filter, cannot be used in system identification scenarios because the electromechanical parameters and models are unknown. To overcome these drawbacks, a novel software-based self-sensing method that does not require electromechanical models is proposed in this paper, and the effectiveness of the proposed method is thoroughly investigated using both simulations and experiments. The identification performance of the proposed method using the estimated input and output is the same as that using directly measured data. The proposed method can especially contribute to the predictive maintenance of structures in isolated environments.
... These algorithms typically assume Gaussian white noise input, whereas atmospheric turbulence is not Gaussian in general [155] and the turbulence spectrum is shaped according to Kolmogorov's law [78]. Another aspect is that the rotor revolution introduces both cyclostationary effects, resulting in a frequency shift of energy [61,98] and specific periodic loads that distort the statistical properties [111]. In addition, the rotor involves intrinsic periodicity [87,235] which is in general not in agreement with the typically involved assumption that the dynamics can be characterized by a Linear Time Invariant (LTI) system. ...
... In Refs. [111,176] a Kalmanfilter approach is used to identify and handle harmonics with frequencies not known in advance. For further reading Ref. [144] provides a detailed review of methods to identify harmonic components. ...
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Aircraft testing is key for certification and the verification of models and assumptions inherently related with the aircraft’s design. However, the in-flight quantification of properties like aeroelastic phenomena can be associated with considerable challenges, since deterministic testing is often beyond the technical limitations. This includes non-measurable inputs and insufficient excitation. A promising alternative is to exploit stochastic perturbations of the system during regular operation. Suitable approaches like Operation Modal Analysis and Blind Source Separation allow to compute eigenmodes from the stochastic structural response only. Based on simulations, this work demonstrates that atmospheric turbulence provides an adequate excitation of the elastic helicopter structure to enable in-flight output-only modal analysis. However, the identification of rotor modes in vicinity of the main-rotor blade-passing frequency is almost impossible, presumably due to the high main-rotor aerodynamic damping, which is supported by literature. This becomes relevant for the identification of airframe modes if the airframe’s impedance at the main-rotor interface is comparably low. In this case, the helicopter tail-section subject to atmospheric turbulence is an important enabler of output-only helicopter modal analysis.
... This Hankel matrix has 2i rows and j columns. In Equation (16), y k is the displacement vector, a k is the acceleration vector, χ is the acceleration integral displacement coefficient, and Y p ∈ R i×j , Y f ∈ R i×j represent the output matrices' 'past' and 'future', respectively. Now, operating the two matrices in Equation (16), the block Toeplitz matrix composed of the output covariance matrix can be expressed as: ...
... In Equation (16), y k is the displacement vector, a k is the acceleration vector, χ is the acceleration integral displacement coefficient, and Y p ∈ R i×j , Y f ∈ R i×j represent the output matrices' 'past' and 'future', respectively. Now, operating the two matrices in Equation (16), the block Toeplitz matrix composed of the output covariance matrix can be expressed as: ...
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A method of modal parameter identification of structures using reconstructed displacements was proposed in the present research. The proposed method was developed based on the stochastic subspace identification (SSI) approach and used reconstructed displacements of measured accelerations as inputs. These reconstructed displacements suppressed the high-frequency component of measured acceleration data. Therefore, in comparison to the acceleration-based modal analysis, the operational modal analysis obtained more reliable and stable identification parameters from displacements regardless of the model order. However, due to the difficulty of displacement measurement, different types of noise interferences occurred when an acceleration sensor was used, causing a trend term drift error in the integral displacement. A moving average low-frequency attenuation frequency-domain integral was used to reconstruct displacements, and the moving time window was used in combination with the SSI method to identify the structural modal parameters. First, measured accelerations were used to estimate displacements. Due to the interference of noise and the influence of initial conditions, the integral displacement inevitably had a drift term. The moving average method was then used in combination with a filter to effectively eliminate the random fluctuation interference in measurement data and reduce the influence of random errors. Real displacement results of a structure were obtained through multiple smoothing, filtering, and integration. Finally, using reconstructed displacements as inputs, the improved SSI method was employed to identify the modal parameters of the structure.
... Among them are subspace identification methods [3], whose aim is to identify system matrices of a linear time-invariant state-space model from input/output or output-only data. During the last decade, the subspace methods found a special interest in engineering applications for modal parameter estimation, see e.g., [4][5][6][7][8][9][10][11][12][13][14][15], due to their favorable statistical and numerical properties e.g., stationary [16] and non-stationary [17] consistency, asymptotic normality [16], and numerical efficiency in identification of large-scale systems [18]. ...
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The uncertainty afflicting modal parameter estimates stems from e.g., the finite data length, unknown, or partly measured inputs and the choice of the identification algorithm. Quantification of the related errors with the statistical Delta method is a recent tool, useful in many modern modal analysis applications e.g., damage diagnosis, reliability analysis, model calibration. In this paper, the Delta method-based uncertainty quantification methodology is validated for obtaining the uncertainty of the modal parameter and the modal indicator estimates in the context of several well-known subspace identification algorithms. The focus of this study is to validate the quality of each Delta method-based approximation with respect to the experimental Monte Carlo distributions of parameter estimates using a statistical distance measure. On top of that, the accuracy in obtaining the related confidence intervals is empirically assessed. The case study is based on data obtained from an extensive experimental campaign of a large scale wind turbine blade tested in a laboratory environment. The results confirm that the Delta method is, on average, adequate to characterize the distribution of the considered estimates solely based on the quantities obtained from one data set, validating the use of this statistical framework for uncertainty quantification in practice.
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System identification in vibrating structures is useful for structural health monitoring. Onsite structural system identification must be achieved with low energy consumption. Accordingly, semi-active inputs should be considered to apply onsite system identification because they can be generated with low energy consumption. However, conventional identifications that use semi-active inputs could not continuously generate inputs suitable for system identification because of switching failure. Switching failure is caused by the mechanism of the semi-active input generation circuit. Due to switching failure, properties of semi-active inputs deviate from the property suitable for identification. Their results deviate from the results obtained using suitable inputs. Moreover, semi-active identification could not be used for structural health monitoring because they cannot determine the cause of significant variations in the identification results (either because of system identification faults from switching failure or because of structural damage). To solve these drawbacks of semi-active identification, we propose a new strategy to generate semi-active inputs that can realize ideal identification. The proposed strategy has two novelties compared to conventional semi-active identification methods: (1) The strategy prevents switching failures and continuously provides semi-active inputs suitable for identification. (2) The strategy is modeled as an optimization problem and provides appropriate circuit control to generate semi-active inputs suitable for identification by solving binary search. Numerical simulations and experiments confirm that the proposed strategy exhibits better identification performance than the conventional strategies. In addition, small variances are observed in the identification results in all time domains. The semi-active inputs generated by the proposed strategy can contribute to precise structural health monitoring using onsite structural identification because it provides accurate identification results with less variation and energy consumption.
Article
The presence of deterministic harmonic excitation, such as that induced by rotating machinery, violates the classical operational modal analysis (OMA) assumption that the system output is strictly ergodic. This paper proves that the presence of harmonic excitation has no effect on the identification of structural modes except in the case where some of the harmonic excitation frequencies are close to the structural frequencies. Therefore, the harmonic component must be removed from the mixed random and harmonic system output before further processing. This paper proposes a Ramanujan subspace projection (RSP) method for harmonic removal, which is realized by projecting the raw system output onto the complex conjugate division (CCD) of the Ramanujan subspace. The novelty of this study is that it reveals the relationship between the frequencies defined in the period and frequency domains, allowing the RSP method to directly extract the harmonic component from the specific CCD without any frequency domain analysis. In addition, an energy indicator is proposed to select the underlying CCDs with the most robust harmonic feature. Using the indicator, one only needs to project the raw system output onto a subset of the CCDs rather than all of them, thereby significantly reducing the computational effort. After removing the harmonic components from the raw output, the remainder can be fed into the covariance-driven stochastic subspace identification (Cov-SSI) method for OMA. The numerical, experimental and field test results show that the proposed RSP method is not only resistant to random noise but also capable of precisely extracting the weak harmonic component from the raw output with less computation. Furthermore, the modal parameters of the structures subjected to mixed random and harmonic excitation can be accurately identified by combining the Cov-SSI and RSP methods.
Article
Operational modal analysis plays an important role in the structural health monitoring and safety diagnosis of arch dam. However, due to background noise, it is difficult to accurately extract the effective characteristics information of arch dam from the vibration responses whose amplitudes are too small under ambient vibration, and the deviation caused by traditional identification methods will directly affect the estimation accuracy for structural modal parameters. Therefore, a novel methodology for modal parameter identification of arch dam based on multi-level information fusion is proposed in this paper. The proposed method is based on multi-sensor data-level fusion to identify the structural natural frequency and damping ratio, which greatly preserves and extracts structural modal properties in the vibration responses. Meanwhile, structural mode shapes are identified based on dynamic feature-level fusion, which significantly improves the identification accuracy. The effectiveness and feasibility of the proposed method are verified by the modal results of digital signals and simulated signals in the 7-DOF system. Prototype engineering case shows that the closely spaced and high-frequency modes can be decomposed and identified by the proposed method, and this method has a higher identification accuracy, which can provide a new idea for modal parameter identification of arch dam.
Thesis
Dans une première phase de travail, l’analyse modale théorique d’une turbine soumise à un débalancement a permis de montrer que les turbines présentent un riche contenu harmonique, et que ces harmoniques sont sélectives, c’est-à-dire qu’elles n’excitent qu’un mode structural de diamètre nodal prédéfini. Il est démontré que ces harmoniques génèrent effectivement des résonances en opération. Dans un deuxième temps, un algorithme d’identification est proposé. La méthode utilise un schéma combiné de suivi d’ordres synchrones pour extraire les résonances harmoniques, et d’un modèle d’analyse modale opérationnel bayésien pour réaliser l’inférence. Enfin, une dernière phase de travail a consisté à améliorer le modèle bayésien existant, qui présente de nombreuses approximations asymptotiques, non raisonnable dans notre cadre où l’on dispose de peu de données. En ce sens, un modèle statistique d’échantillonnage numérique a été développé, basé sur un algorithme de Gibbs avancé. L’outil obtenu est plus fiable pour caractériser des données réduites, et montre d’excellentes propriétés de convergence le rendant compétitif avec les approches existantes.
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The cepstrum has a very long history, since the first paper was published in 1963, two years before the publication of the FFT algorithm. The first application was to the determination of echo delay time, but it was quickly applied to speech analysis for two reasons. Firstly, it was able to detect voiced speech, and measure the voice pitch, and secondly it was able to separate the forcing and transfer function components in speech signals. These two properties apply equally to machine vibration signals, and the cepstrum can be used to detect and remove periodic discrete frequency components (harmonics and modulation sidebands) from the spectrum (and corresponding time signals) and also to extract the modal properties of a structure in the presence of the forcing function. This paper describes the development of these applications to structural modal analysis over many years, primarily to operational modal analysis (OMA), where the modal information is extracted from response signals, often with the structure in its normal operating environment. The cepstrum extracts the modal information in terms of pole/zero models, and requires compensation for the effects of unmeasured out-of-band modes, by an equalisation process, and overall scaling to obtain scaled mode shapes. Steady advances to achieve this have been made over the years. A recent development that has revolutionised the application has been the ability to edit stationary and slowly varying non-stationary signals using the real cepstrum (which does not contain the phase information of the original signals) but regenerating time signals by combining the edited amplitude spectra with the original phase spectra (with negligible error). There are currently two ways in which this can be used in operational modal analysis: 1. Pre-processing of the response signals to remove most excitation components and other disturbances , and enhance the modal properties, before applying standard OMA procedures. 2. Extracting the modal models from the pre-processed signals by curve-fitting pole/zero models to the cepstra, and applying compensation for equalisation and scaling. The paper describes the history, current situation and potential future development of the application of cepstral analysis to structural modal analysis, this seemingly being greatly under-utilised.
Conference Paper
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All vibration-based machine condition monitoring uses response signals, which are a mixture of forcing function and transfer function effects, and often an assumption is made as to whether a change is due to one or the other. Operational modal analysis (OMA) is one way of determining the dynamic properties of a structure or machine from the response signals only, while in operation, and thus provides the potential to separate them from the forcing functions. It then becomes a powerful machine diagnostic tool. The cepstrum has the property that forcing functions and transfer functions are additive, at least for single input, multiple output (SIMO) situations, and moreover they are often located in disparate regions in the cepstrum, allowing them to be separated. In recent years, new cepstral analysis methods have been developed to assist OMA in two ways: 1) As a pre-processing tool to enhance the modal properties and remove other effects, such as forcing functions (often at discrete frequencies) to simplify the application of standard OMA techniques. 2) To perform the OMA by fitting a pole/zero model of the structural dynamic properties to the enhanced response data. The current paper shows the success of both these methods in the case of simulated signals from a variable speed gearbox, by vastly reducing the effects of the very complicated forcing function. An extension from earlier proposed approaches is the determination of zeros (normally masked by noise in response signals) using transmissibilities, measured between pairs of responses, the noise being reduced by averaging. These new results confirm that limited, and only partially successful, earlier results were contaminated by nonlinearities in the support of the test object.
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In this Chapter we treat the subspace identification of purely deterministic systems, with no measurement nor process noise (vk ≡ wk ≡ 0 in Figure 1.4). We treat this problem for two reasons: Most of the conceptual ideas and geometric concepts, which will also be used in the Chapters to follow, are introduced by means of this simple identification problem. We treat the problem from a different point of view as in the literature, which makes it easier to assimilate it as a special case in the Chapters to follow. Similarities between the presented algorithm and the literature are pointed out.
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The dynamic behavior of structures can be studied by the identification of their modal parameters. Classical modal analysis methods are based on the relation between the forces applied to structures (inputs) and their vibration responses (outputs). In real operational conditions it is difficult, or even impossible, to measure the excitation. For this reason, operational modal analysis approaches which consider only output data are proposed. However, most of these output-only techniques are proposed under the assumption of white noise excitation. If additional components, like harmonics for instance, are present in the exciting force, they will not be separated from the natural frequencies. Consequently, this assumption is no longer valid. In this context, an operational modal identification technique is proposed in order to only identify real poles and eliminate spurious ones. It is a method based on transmissibility functions.The objective of the proposed paper is to identify modal parameters in operational conditions in the presence of harmonic excitations. Identification is performed using a method based on transmissibility measurements and then with the classical stochastic subspace identification method, which is based on white noise excitation. These two methods are first applied to numerical examples and then to a laboratory test. Results validate the novel ability of the method based on transmissibility measurements to eliminate harmonics, contrary to the stochastic subspace identification approach.
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An important step in the operational modal analysis of a structure is to infer on its dynamic behavior through its modal parameters. They can be estimated by various modal identification algorithms that fit a theoretical model to measured data. When output-only data is available, i.e. measured responses of the structure, frequencies, damping ratios and mode shapes can be identified assuming that ambient sources like wind or traffic excite the system sufficiently. When also input data is available, i.e. signals used to excite the structure, input/output identification algorithms are used. The use of input information usually provides better modal estimates in a desired frequency range. While the identification of the modal mass is not considered in this paper, we focus on the estimation of the frequencies, damping ratios and mode shapes, relevant for example for modal analysis during in-flight monitoring of aircrafts. When identifying the modal parameters from noisy measurement data, the information on their uncertainty is most relevant. In this paper, new variance computation schemes for modal parameters are developed for four subspace algorithms, including output-only and input/output methods, as well as data-driven and covariance-driven methods. For the input/output methods, the known inputs are considered as realizations of a stochastic process. Based on Monte Carlo validations, the quality of identification, accuracy of variance estimations and sensor noise robustness are discussed. Finally these algorithms are applied on real measured data obtained during vibrations tests of an aircraft.
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In the case of rotating machinery that is operating at constant rotational speed, Operational Modal Analysis is sometimes a challenge due to the presence of significant harmonic peaks. In many cases, the rotating components affect the full frequency range being analysed due to the presence of additional orders. In this paper, a harmonic peak reduction method is presented. The method directly subtracts the harmonic content from the raw time signal. A Gauss-Newton fitting algorithm is used to find the least square reduced fitting of the measurement data to a harmonic deterministic function. The method is tested on two examples. The first is a simple aluminium plate excited with random as well as sinusoidal excitation. In this case, the harmonic peaks are completely removed. The second example consists of noisy measurements from a Prolec GE transformer. In this case, the harmonic reduction method is capable of reducing the harmonic peaks with more than 40 dB on an average.