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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,Cd∈Rr×m

select the respective type of the output at the measurement DOFs.

When deﬁning the states x(t)=[q(t)T˙q(t)T]T∈R2m, 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

c∈R2m×2m, observation matrix Csys ∈Rr×2m, process noise w(t)∈R2mand

output noise v(t)∈Rrare

Asys

c=

0I

−M−1K −M−1C

, Csys =hCd−CaM−1KCv−CaM−1Ci, w(t) =

0

M−1

f(t),

and v(t) = CaM−1f(t) + ˜v(t), where the model order is n= 2m. When sampled at discrete time instants

t=kτ , 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(kτ )∈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

su∈Rmbe 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

M−1

su∈R2m,d=CaM−1su∈Rr.

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, ωi∈Rare (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=bsh∈R2m×2hand Cper =dsh∈Rr×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=kτ 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=In∈Rn×n. Then, the recursive ﬁlter equations are

Kk= (APkCT+S)(R+CPkCT)−1,(16)

ˆxk+1 = (A−KkC)ˆxk+Kkyk,(17)

Pk+1 =APkAT+Q−Kk(APkCT+S)T,(18)

where Kk∈Rn×ris the gain matrix, which converges to the steady state gain K∈Rn×rfor increasing k.

Consequently, deﬁning the innovations ek=yk−Cˆxk∈Rr, 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

V∈Rn×nthe linear transformations

ˆxV

k=V−1ˆxk, AV=V−1AV, CV=CV, KV

k=V−1Kk(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

hi∈C2(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 +N−1of a discrete signal ak∈Rb×1be given with i≤j. Then

the corresponding block Hankel matrix Ai|jis deﬁned as

Ai|j=

aiai+1 . . . ai+N−1

ai+1 ai+2 . . . ai+N

.

.

..

.

.....

.

.

ajaj+1 . . . aj+N−1

∈R(j−i+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|p−1,Y+

raw =1

√NYp|2p−1.(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|p−1,Y+

per =1

√Nˆ

Yper

p|2p−1.(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|p−1,E+=1

√NEp|2p−1.

Finally, the extended observability matrix of the combined system writes in the modal basis as

Γ =

CV

CVAV

.

.

.

CV(AV)p−1

,

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 KE−and 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

. . . . . . . . . . . . . . .

CAp−2K C Ap−3K C Ap−4K . . . Ir

,Kl=

Ir0 0 . . . 0

CKlIr0. . . 0

CAKlC Kl+1 Ir. . . 0

. . . . . . . . . . . . . . .

CAp−2KlC Ap−2Kl+1 C Ap−3Kl+2 . . . Ir

,

where the l-th columns of E−

Kand E+

Kare, respectively,

[E−

K]l= (Kl−1− K)[E−]l,[E+

K]l= (Kl−1+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 ΓX−and 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 ΓX−and 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)I−Q12 LT

12,12 L12,12QT

12Q12 LT

12,12−1L12,12 QT

12

= (L34,12QT

12 +L34,34QT

34)I−Q12 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 r−2hredundant 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 r≤2h, 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

T−1

Asys Ab

0Aper

T=

Asys 0

0Aper

,with T=

I T12

0I

, T −1=

I−T12

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 [(I2h⊗Asys)−(AperT⊗I2m)]vec(T12) = −vec(Ab), where ⊗denotes

the Kronecker product and vec(·) the column stacking vectorization operator. Pre-multiplying (14) by T−1

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

=T−1

xsys

k

xper

k

=

xsys

k−T12xper

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

D2−→ 0. 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, E−X−

per

T→0 is shown to simplify the third term, and second, E−

KX−

per

T→0

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

T→0.

28

1. Proof of E−X−

per

T→0

First a few mathematical notations: Consider the σ−algebra Fkgenerated by the observations y1, . . . , yk.

The collection of σ−algebra Fkis increasing, i.e., Fk⊂ Fk+1. The innovation ekat time kis Fkmeasurable,

whereas ˆxkis Fk−1measurable. Since E(ek| Fk−1) = E(ek|Y1, . . . , Yk−1) = 0, the innovations are

uncorrelated with the past Kalman states (and also with the past outputs) [14], i.e.,

E(ekˆxT

k−j)=0,∀j≥0,(C.2)

which holds since E(ekˆxT

k−j) = E(E(ekˆxT

k−j| Fk−1)) = E(E(ek| Fk−1)ˆxT

k−j) = E(0.ˆxT

k−j) = 0. Then,

considering supkE(kekk4+kˆxkk4)< C < ∞, by Theorem 2.8 in [36],

E−X−T=1

N

e0e1

.

.

.eN−1

e1e2

.

.

.eN

.

.

..

.

..

.

..

.

.

ep−1ep

.

.

.ep+N−2

ˆxT

0

ˆxT

1

.

.

.

ˆxT

N−1

=

1

NPelˆxT

l

1

NPel+1 ˆxT

l

.

.

.

1

NPel+p−1ˆxT

l

−→ 0

with probability 1, in particular E−X−

per

T=o(1). It follows E−Y−

per

T=E−X−

per

TΓperT=o(1). Since

(Y−

perY−

per

T)†= (ΓT

per)†(X−

perX−

per

T)−1(Γper)†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−− KE−Y−

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

T→0

With (28) it holds

E−

KX−

per

T=1

N

N−1

X

l=0

(Kl− K)

el

el+1

.

.

.

el+p−1

ˆxper

l

T=1

N

N−1

X

l=0

p−1

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 j≥0,

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

T→0

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=V−1

cAVc= diag(Λ,Λ). Then, the system and periodic parts of the states

ˆxVc

k=V−1

cˆxkrelate to ˆxV ,sys

kand ˆxV,per

kby

ˆxV,sys

k=T−1

mˆxVc,sys

k,ˆxV,per

k=T−1

hˆxVc,per

k, Ta=1

2

Ia−iIa

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

T→0 follows with (C.5).

From the state equation it follows

1

N

N−1

X

k=0

ˆxVc,sys

k+1,i ˆxVc,per

k+1,j =1

N

N−1

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

N−1

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

N−1

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

N−1

X

k=0

λsys

iˆxVc,sys

k,i eT

kKVc,per

k,j

T+1

N

N−1

X

k=0

λper

jˆxVc,per

k,j eT

kKVc,sys

k,i

T+1

N

N−1

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(Lk−νk)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)

NPN−1

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+

per⊥is 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

T†Y−

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

k−j) =

0∀j > 0, since E(ekyT

k−j) = E(E(ek| Fk−1)yT

k−j) = 0 and it can be proved that E+Y−

rawT→0 and

E+

KY−

rawT→0 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

T→0 and E+

KY−

per

T→0, and that E+Y−

raw

T→0 and E+

KY−

raw

T→0. 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 +KE−E−TKT+E−

KE−

K

T+L+LT+ (cross terms) ×o(1) (D.2)

where L= ΓsysX−

sysE−TKT+ Γsys X−

sysE−

K

T+KE−E−

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 +KE−E−TKT+o(1).

The ﬁrst matrix is positive semi-deﬁnite, and since E−E−Tis 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

T−1Y−

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 E−Y−

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−

sys(Γsys 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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