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User-friendly method to split up the multiple coherence function into noise,
nonlinearity and transient components illustrated on ground vibration testing of an
F-16 fighting falcon
P. Z. Csurcsia
Department of Engineering Technology (INDI), Vrije Universiteit Brussel
* pcsurcsi@vub.ac.be
ABSTRACT
This paper provides a novel method to split up the multiple coherence function into noise, nonlinear distortion, and transient
components. The method relies on the nonparametric estimation framework called the Best Linear Approximation (BLA) where
vibro-acoustic systems are excited by special so called multisines (pseudo-random noise) signals. Testing of multiple-input,
multiple-output (MIMO) nonlinear systems is very involved, and it usually requires expert users. It is because the design and
processing of experiments, and nonlinear assessment are not trivial questions. In order to cope with these issues, this paper
shows the recent results of a research project addressing the questions related to the user-friendly nonlinear (pre-)processing of
MIMO experiments of vibrating structures. The key idea is the novel analysis of the (co-)variance estimates of the BLA
framework that can be tailored to split up the classical multiple coherence function into noise, nonlinearity, and transient
components. Using the suggested approach, a novice user can quickly determine whether the underlying structure is linear or
not, and how much is the possible gain using nonlinear modeling tools. The proposed approach is demonstrated on, but not
limited to, the analysis of ground vibration testing of a decommissioned F-16 fighting falcon.
Keywords: MIMO systems, nonlinearity, nonparametric estimation, ground vibration testing, multiple coherence function
1 INTRODUCTION
The goal of the paper is to provide a simple yet effective nonparametric nonlinear approach for analyzing multiple-input,
multiple-output (MIMO) real-life (industrial) vibro-acoustic experiments. Most vibro-acoustic systems such as mechanical,
civil, acoustic, electro engineering systems are inherently nonlinear. For understanding, design, and control accurate models
are needed, however, modeling nonlinear systems is very involved due to the fact that there are many types of differently
behaving nonlinearities. This challenge inspired many data-driven modelling techniques. A detailed overview of the nonlinear
system identification issues and challenges can be found in [1] [2] [3] and [4].
To obtain accurate and realistic models, experiments have to be designed. The aim of the vibration (structural dynamic) testing
is to obtain experimental measurement data of the vibrating structure such as road and air vehicles. The structural testing
typically takes place at the end of the development process as it requires a physical prototype to be tested. The vibration testing
methods play a crucial role in the product quality, safety and comfort improvements. The increasing needs for higher accuracy
and faster testing techniques inspirited a lot of international researches [5] [6] [7]. As a result, the classical time-consuming
testing procedures such as the phase resonance method [8] [9] [10] are nowadays fully substituted by so-called phase separation
techniques that find the modes by evaluating the frequency response functions (FRFs). This work provides an additional set of
tools for this FRF evaluation process.
The rapid improvement in computational power allows us to design a large variety of complex input (perturbation) signals that
can be used to experimentally estimate the FRFs. These FRFs are required to obtain parametric models as well (e.g. resonance
frequencies, modes shapes in modal analysis, or e.g. state-space models in control engineering). One of the (best) possible
excitation signals is the special, random phase multisine (known as pseudo random signal as well [11]) because it can avoid
spectral leakage, inconsistency, non-persistency, and they provide a handy, robust solution to build linear models. In addition
to coping with the classical signal processing issues, multisine signals can be also used to detect the level and type of
nonlinearities.
Understanding and using advanced nonlinear model requires expert users and tremendous amount of time. On the other hand,
theory of linear systems is well understood, and the cost of linear modeling is very low. Therefore, many nonlinear systems are
approximated with linear models. However, the loss of precision through this decision is rarely studied. To this end, using the
proposed (user-friendly) approach, it is easy to quantify the loss of precision, thus helping users directly to make a well-balanced
decision. In this work a user-friendly industrial framework is proposed to guide the users (industrial practitioners) using
classical instrumentation setups, well-understood notions.
The state-of-the-art linear approximation framework known as the Best Linear Approximation (BLA) is already available and
well-studied for single-input, single-output (SISO) systems [12] [13] [14] [15] [16]. When a (SISO) system is perturbed by
multisines, it is possible to tell if a linear model is sufficient to model the system. However, for MIMO systems, the design of
experiment is not a trivial question since the input and output channels are not (necessarily) mutually independent. Some earlier
works coped with the design of excitation of MIMO systems [17] [18] verified on simulations only. [19] proposes a possible
solution for MIMO BLA modeling framework with 2 inputs and 2 outputs. [17] proposes a design of experiment for MIMO
systems with an arbitrary large number of input and output channels, but as was pointed out by [19], the procedure does not
allow to measure the covariance matrix of the stochastic nonlinearities in the BLA setup that is needed to separate the noise
and nonlinear distortion components (as elaborated later on). This work extends the BLA methodology to arbitrary number of
input and output channels by using an orthogonalization technique that allows to estimate the covariance matrix of the stochastic
nonlinearities in the MIMO BLA setup. The proposed method is based on the novel generalization of the Best Linear
Approximation (BLA) framework [20] [17] [21].
A further novelty of this work is that using the proposed technique, the multiple coherence (also known as magnitude square
coherence) function can be separated into noise, nonlinearity, and transient components [22] as well.
This work makes use of the multisine excitation based BLA framework applied to systems with an arbitrary number of input
and output channels as an extension of the conceptual work [21]. There are several advantages of the proposed framework,
namely, 1) there is no problem with spectral leakage and/or transient (due to periodical nature of the excitation), 2) excellent
linear models are obtained, and 3) useful information about the types and levels of nonlinearities is made accessible. The latter
one was until now, only manifested as a relative measure of the noise and nonlinear distortion estimates (such as variances, or
signal-to-noise ratio (SNR)/signal-to-nonlinearity ratio (SNLR) estimates).
The proposed technique consists of two main steps.
First step considers the experiment design. The proposed excitation signal consists of independent series of periodic multisines
that are mutually independent over the whole experiment (see details in Section 3).
The second step considers the processing, and analysis of the measurements via the multisine-driven BLA framework. This
stage differs from the classical (industrial) H1 Frequency Response Function/Matrix (FRF/FRM) estimation process. The
classical (SISO) BLA framework uses distortion variance estimates to quantify the noise and nonlinear contributions.
Interpreting these results requires a deeper understanding and hands-on experience. In the other hand, most of the practitioners
are already familiar with understanding the (multiple) coherence function [22]. It provides an aggregated indication to signal
processing issues such as excessive noise, nonlinearities, synchronization problems, etc. In this work we make use of some
statistical features of the framework (see details in Section 4) to provide more detailed information by splitting up the coherence
function into noise, nonlinearity, and transient coherence components.
This method will allow novice users to easily analyze the measurements in detail. The proposed methodology is illustrated on
a multiple input ground vibration testing (GVT) measurement of a decommissioned F16 aircraft. The numeric results of this
work are obtained using the Simplified Analysis for Multiple Input Systems (SAMI) toolbox [23] [24] [25].
The remainder of this paper is structured as follows. Section 2 introduces the basic definitions, assumptions, and the defines
the underlying systems. In Section 3, the multisine driven multiple input design of experiment is elaborated. Section 4
introduces the MIMO BLA framework. In Section 5 the description of the F-16 MIMO ground vibration measurement and the
results of the analysis can be found. Section 6 provides the conclusions.
2 BASICS
2.1 Systems considered
The dynamics of a linear MIMO system can be characterized in the frequency domain by its Frequency Response Matrix (FRM,
a matrix whose elements are FRFs [20]) 𝐺 at frequency index 𝑘, which relates 𝑛 inputs 𝑈 to 𝑛 outputs 𝑌 of 𝑁 measurement
samples as follows:
𝑌[𝑘]=𝐺[𝑘]𝑈[𝑘] (1)
where 𝐺[𝑘]∈ℂ×, 𝑌[𝑘]∈ℂ×,𝑈[𝑘]∈ℂ×, 𝑘=0…
at frequency 𝑓=𝑘𝑓/𝑁 with sampling frequency of 𝑓.
To make the text more accessible, the frequency indices and dimensionalities will be omitted.
The system represented by 𝐺 is linear when the superposition principle is satisfied in steady state, i.e.:
𝑌=𝐺{(𝑎+𝑏)𝑈}= 𝑎 𝐺{𝑈} + 𝑏 𝐺{𝑈}=(𝑎+𝑏) 𝐺{𝑈} (2)
where a and b are scalar values. If 𝐺 is constant, for any a, b (and excitation), then the system is called linear-time invariant
(LTI). On the other hand, when 𝐺 varies with a and b (and the variation depends also on the excitation signal – e.g., level of
excitation, distribution, etc.) then the system is called nonlinear.
Because time-varying systems are often misinterpreted as nonlinear systems, it is important to mention that when G varies over
the measurement time, but at each time instant the principle of superposition is satisfied, then the system is called linear time-
varying (LTV) [26] [27] [28].
The underlying systems are damped, bounded-input, bounded-output stable, time-invariant, nonlinear systems where the linear
response of the system is still present, and the output of the underlying system has the same period as the excitation signal (i.e.,
the system has PISPO behaviour: period in, same period out [29]).
2.2 Measurement and instrumentation
From an instrumentation point of view, it is assumed that the measurement system is perfectly synchronized: the samples at
different channels are acquired at the same time instant, the sampling frequency is kept constant.
The actuator (e.g., shaker, loudspeaker, function generator) of the system is linear, and the excitation signal is measured
accurately resulting in a signal-to-noise ratio (SNR) of at least 20 dB.
The output (denoted by 𝑌 in frequency domain) is measured with additive, i.i.d. Gaussian noise (denoted by 𝐸) such that the
measurement 𝑌 is given by:
𝑌 =𝑌+𝐸=𝐺𝑈+𝐸, 𝐸~𝒩(0,𝜎) (3)
2.3 Coherence function
When estimating transfer functions from measurement, in an industrial context, it is a common practice to estimate the
(magnitude squared) coherence function [30]. It is used to identify the linear relationships between an output signal and an
input signal. Despite the fact that in practice it is often omitted, this technique assumes that the excitation signal is wide sense
stationary (WSS) [31]. The proposed excitation signal (see next section) automatically fulfills this condition.
The ordinary SISO (single-input, single-output) coherence function between the input and output signals is (frequency-wise)
given by:
𝛾
=
(4)
where 0≤ 𝛾
≤1, 𝑆, 𝑆 is the input/output power spectrum, 𝑆 is the input-output cross-power spectrum.
The coherence function 𝛾
takes values between 0 and 1, with 1 indicating 100% linear relationship between the input and
the output, and 0 indicating 0% linear relationship i.e., there is a complete lack of coherence.
In MIMO cases, the multiple coherence function is given for each output channel independently by the so-called multiple
(magnitude squared) coherence function [22] [32]:
𝛾
=𝑆
𝑆
𝑆𝑆
(5)
where 0≤ 𝛾
≤1, 𝑆, 𝑆 is the (all) input/ (one selected) output power spectrum, 𝑆 is the inputs-output cross-power
spectrum, 𝑥 is the Hermitian (complex conjugate) transpose of 𝑥, and 𝑥 is the pseudoinverse of 𝑥. The capital 𝑈 refers to
the fact that all input channels are simultaneously considered, and small 𝑦 refers to the fact that only one output channel is
used.
In practice, the spectra in (4)-(5) should be estimated, most used generic methods are elaborated in [22] [11].
3 DESIGN OF EXCITATION SIGNAL
3.1 Multisine excitation
The evolution in last decades in signal processing allows the usage of an extremely wide variety of excitation signals to test the
underlying structures in a user-friendly, time-efficient manner. [11]. Many industrial users prefer noise excitations, because
they are simple to implement, but there is a possible leakage (transient) error, and the usage for detecting nonlinearities is
limited. Therefore, it is recommended to use periodic excitation signals to 1) easily tackle issues with transient term, 2) avoid
spectral leakage, 3) (nonparametrically) characterize the measurement noise, and 4) to be able to detect nonlinearities.
The proposed technique relies on the easy to generate (periodic) multisines. They are also known as pseudo-random noise
because in time domain they look and behave like white noise, but, off course, they are not noise.
The (typically flat) magnitude characteristics of multisines are set by the user in the frequency domain. The phases (in the
frequency domain) are randomly chosen from a uniform distribution (𝒰) between 0 and 2𝜋. Mathematically, the considered
multisines are a sum of harmonically related sinusoids defined as:
𝑢(𝑡)=∑𝑎cos(𝜔𝑘𝑡+𝜑), 𝜑~𝒰[0,2𝜋[
(6)
where 𝜔 is the fundamental angular frequency (that sets the frequency resolution), 𝑎 is the amplitude of the 𝑘harmonic
(i.e., frequency index 𝑘) set by the user, and 𝑘 is the highest harmonic component considered.
If the multisine contains all/only odd or even harmonics, then it is called full (band)/odd or even multisine. The proposed flat
magnitude (i.e. all 𝑎=1) full band multisine signals are wide sense stationary signals [31]. For a detailed study on excitation
signals we refer to [20] [30].
3.2 Multisines for multiple input experiments
This subsection considers the experiment design for MIMO setups. For sake of simplicity, we illustrate the problems on systems
with two inputs and outputs channels, and one experiment of two independent input signals, see Figure 1.
Figure 1: Illustration of a system with two inputs and two outputs.
The output-input relationship is then given in the frequency domain as follows:
𝑌
𝑌=𝐺 𝐺
𝐺 𝐺𝑈
𝑈 (7)
where the indices refer to the input/output channel.
Equation (7) represent a rank deficient problem resulting in no unique solution: there are four unknown parameters (in 𝐺) and
only two measurements (𝑌−𝑈 equations). In order to solve the algebraic equation (7), it is important to increase the number
of independent experiments (𝑌−𝑈 equations) such that there are at least 4 independent experiments.
Classical literatures suggest using so called Hadamard decorrelation technique [33] (known as +- technique as well [34]) to
achieve unique solution. However, the author recommends using orthogonal random multisine excitation [21]. The proposed
procedure is to generate independent random excitations for every input channel such that we have (more) randomness in the
measurement with respect to the Hadamard’s technique where the same segment of signal is repeated with +1 or -1
multiplication.
The proposed method starts with individually generated multisine sequences that are allocated to every input channel. Then
these initial input sequences are orthogonally shifted with the help of the transformation matrix 𝑊 defined elementwise as:
𝑊 =𝑒()()
(8)
where c refers to the index of the input channel (i.e., the row number in U), n refers to the experiment number (column number
in U), 𝑛 stands for the number of inputs, and 𝑗 stands for the imaginary unit.
In case of two-by-two experiments, the following equation is given for the proposed method:
𝑌
. 𝑌
.
𝑌 𝑌 =𝐺 𝐺
𝐺 𝐺 𝑈
. 𝑈
.
𝑈 𝑈 =𝐺 𝐺
𝐺 𝐺𝑈𝑊 𝑈𝑊
𝑈𝑊 𝑈𝑊=𝐺 𝐺
𝐺 𝐺𝑈𝑈
𝑈−𝑈 (9)
Please note that the two input channels case (𝑛=2) is similar to the result of the classical Hadamard technique where the
baseline equation would be given by:
𝑌 𝑌
𝑌 𝑌=𝐺 𝐺
𝐺 𝐺𝑈𝑈
𝑈−𝑈 (10)
An illustration of the classical Hadamard and the proposed multisine signals are shown in Figure 2.
There are two practical disadvantages of the classical Hadamard decorrelation technique. First, it unnecessarily “stretches” the
structure. For example, think of an airplane testing, where the wings are excited. The “stretch” appears by applying inputs of
opposite sign at certain time instances. Furthermore, the Hadamard technique is less rich in terms of randomness.
For the cases when a higher number of input channels are considered, the differences are even more spectacular (e.g., numerical
conditioning due to the restriction that the order of the Hadamard matrix must be 1, 2, or multiples of 4.), however, the main
advantage of the proposed method is that it generates very rich excitation signals.
Figure 2: Comparison of the classical Hadamard decorrelation technique using noise signal and the proposed orthogonally
shifted pseudo-random noise (multisine) signals for a very simple 2x2 MIMO scenario. The number of data samples is kept
low to allow visual comparison. Observe that the proposed solution offers more randomness, and the magnitude characteristic
is completely flat.
3.3 Periodicity
The proposed framework requires the use multiple periods (blocks) and realizations of the multisine signal. Using multiple
periods, the SNR of the measurement will be improved. To enrich the randomness of the (periodic) multisines, it is needed to
apply multiple random realizations. Increasing the number of random realizations results in more robust nonlinearity estimates.
Because the orthogonal shift, in case of 𝑀 independent random realizations and 𝑛 inputs there are in total 𝑛∙𝑀 realizations
(in other words there are 𝑛-times orthogonal phase-rotated 𝑀 different realizations). Assuming 2 inputs (𝑛=2), 𝑀 different
realizations and 𝑃 periods, the excitation signal is given by:
𝑈=
⎣
⎢
⎢
⎢
⎡
𝑈
()…𝑈
()
𝑈
()…𝑈
()
𝑈
()…𝑈
()
𝑈
()…𝑈
()
⎦
⎥
⎥
⎥
⎤
…
⎣
⎢
⎢
⎢
⎡
𝑈
()…𝑈
()
𝑈
()…𝑈
()
𝑈
()…𝑈
()
𝑈
()…𝑈
()
⎦
⎥
⎥
⎥
⎤
=[𝑈[][]…𝑈[][]… 𝑈[∙][]…𝑈[∙][]
( , ) ] (11)
0 0.5 1 1.5
-2
-1
0
1
2
Hadamard time domain
0 5 10 15
frequency (Hz)
-300
-200
-100
0
0 0.5 1 1.5
time (s)
-2
-1
0
1
2
0 5 10 15
-300
-200
-100
0
frequency domain
0 0.5 1 1.5
-2
-1
0
1
2
Orthogonal multisine time domain
0 0.5 1 1.5
time (s)
-2
-1
0
1
2
0 5 10 15
-300
-200
-100
0
frequency domain
0 0.5 1 1.5
frequency (Hz)
-2
-1
0
1
𝑼
𝟏
𝑼
𝟏
𝑼
𝟐
−
𝑼
𝟐
𝑼
𝟏
𝑼
𝟏
𝑼
𝟏
−
𝑼
𝟏
3.4 Multiple coherence using multisines
The multiple (magnitude squared) coherence function can be calculated via (5). The necessary spectra should be estimated
based on the available measurement data. The traditional power spectrum estimators are typically using (sliding) windowing
and/or correlation-based techniques in order to cope with spectral leakage and transient. However, the usage of multisines
allows us to easier estimate the power spectra in a computational efficient way. For a measurement with 𝑃 periods, the power-
spectra are estimated (frequency-wise) at realization 𝑚 with the simple spectral averaging as follows:
𝑆[][]=
∑𝑈[][]𝑈[][]
(12)
𝑆[]
[]=∑𝑈[][]𝑦
[][]
(13)
𝑆
[]
[]=∑𝑦
[][]𝑦
[][]
(14)
Equation (12) can be further simplified to 𝑆[][]=𝑈[]𝑈[] because it is assumed that the input is known exactly (or in
other words, measured precisely).
4 BEST LINEAR APPROXIMATION
4.1 Introduction
The Best Linear Approximation (BLA) has become very popular to estimate nonparametrically frequency response functions
of nonlinear systems [35]. The essence of BLA is to minimize the mean square error between the measured nonlinear response
of the system and the response of the linear BLA model. In the proposed BLA framework, systems are excited by multiple
periods and realizations of orthogonally shifted random phase multisines. The main components of BLA modeling framework
(for an illustration, see Figure 1) are elaborated next.
Figure 3: The building blocks of the BLA.
𝐺 is the (classical) linear (phase coherent) component of the model.
The nonlinear (non-coherent phase) component of the model is 𝐺. Non-coherent phase relationship means that the varying
phase at the input results in a random phase rotation at the output (unlike the phase coherent component where the output phase
rotates proportionally to the input phase changes). Increasing the number of random realizations of the multisine signal allows
us to tackle the random output phase rotations as a nonlinear noise source. Nonlinear noise source means that the values of 𝐺
vary randomly over the different realizations of the signal, but its values are fixed within the repetitions of the same signal
segment (period).
The classical additive measurement noise is represented by the component 𝐺. Applying multiple periods reduces the effects
of the measurement noise (𝐺). Applying multiple realizations reduces the impact of nonlinear noise (non-coherent
nonlinearities, 𝐺) and measurement noise (𝐺).
𝐺 represents the bias error, i.e., the remaining (coherent) nonlinearities after multiple realizations.
For a detailed explanation of these components, we refer to [20] [21].
4.2 Multi-dimensional averaging
The processing and analysis of the signals differ from the well-known H1 estimation process [20]. In short, H1 FRF estimation
is obtained in one step by multiplying the input-output cross-power estimate with the inverse of the input auto-power estimate,
and to determine the quality of the measurement (and indirectly the model quality) the (multiple) coherence function is
calculated [11].
In case of multisine driven BLA, one can of the statistical features of the excitation signal, and for that reason, the measurement
must be preprocessed carefully, in multiple steps.
The first step is the segmentation of data: retrieving the periods, realizations, and phase rotations. Next, the trends from the
individual segments are removed [11]. For a detailed procedure with algorithms and illustration we refer to [21].
In this proposed framework there are (number of input times phase-rotated) 𝑀 different realization of the multisine excitation
signal, each realization is repeated 𝑃 period times. The considered steady-state model at period 𝑝 and realization 𝑚 at frequency
bin k is given by:
𝐺[][]= 𝑌
[][]𝑈[]
,=(𝐺
[][]
+ 𝐺[]
+ 𝐺[][]
)
,𝑈[] (15)
where 𝐺[][]∈ℂ×, 𝑌
[][]∈ℂ×,𝑈[]∈ℂ×, and the 𝑥 is the generalized (Moore–Penrose) inverse of 𝑥.
In order to estimate the BLA, one has to average over
𝑃
periods of repeated excitation signal (leading to a partial FRM estimate
𝐺[]), and over the 𝑀 different realizations (leading to 𝐺), as illustrated in Figure 4. If the assumptions given in Section 2.1-
2.1 and 4.1 are satisfied, then 𝐺 equals 𝐺, i.e.:
𝐺
=1
𝑀∑1
𝑃∑𝑌𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑
[𝑚][𝑝]𝑈[𝑚]−1
𝑃
𝑝=1
𝑀
𝑚=1
𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑜𝑓 𝑎𝑙𝑙 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝐹𝑅𝐹𝑠 =1
𝑀∑(𝐺
𝐵𝐿𝐴
𝐹𝑅𝐹
𝑓𝑖𝑥𝑒𝑑
𝑀
𝑚=1 + 𝐺
𝑆
[𝑚]
+ 𝐺
𝐸
[𝑚]
)=𝐺
𝐵𝐿𝐴 (16)
Figure 4: Illustration of the multi-dimensional averaging with the corresponding quantities. Each of this block represent a
period of input-output measurement.
4.3 Calculating the variances in a nutshell
The multisine driven BLA proves a handy tool to estimate different (noise, nonlinear) variance quantities.
A partial FRM estimate is obtained via averaging over the repeated periods. If the number of periods is sufficiently large, then
the noise terms 𝐺
[] converges to zero (see (15)-(16). Because the stochastic nonlinear contribution 𝐺[] does not vary over
the same repeated blocks, one has to average over the different realizations as well. If the number of realizations is sufficiently
large, then the nonlinear noise term converges to zero (see (15)-(16).
The estimate of the noise covariance 𝜎 of BLA is obtained by the improved averaged sample variance of partial FRM estimates
as:
𝜎=
∑
[]
with 𝜎[]
=
∑[][][]
(17)
The additional normalization with 𝑃 in the partial noise estimate 𝜎[]
accounts for the variance improvement w.r.t the repeated
blocks in experiment 𝑚 (in other words, SNR improves as the number of averages increases). This term corresponds to 𝐺/𝑃
in the block schematic given in Figure 3. The additional normalization with 𝑀 in the improved noise estimate 𝜎 accounts for
the variance improvement w.r.t the different realizations. This term corresponds to 𝐺/𝑀𝑃 in the BLA block schematic.
The total variance of the FRM 𝜎
is calculated from the sample variance of each different partial BLA estimates 𝐺[]:
𝜎
=
∑[]
(18)
The additional normalization with 𝑀 in the total variance estimate 𝜎
accounts for the variance improvement w.r.t the
different realizations. This term corresponds to 𝐺/𝑀+𝐺/𝑀𝑃 in the BLA block schematic.
The difference between the total variance and the noise variance is an estimate of the variance of the stochastic nonlinear
contributions:
𝜎
≈𝜎
−𝜎=𝐺/𝑀 (19)
For computational and fundamental details, see [21] [35].
4.4 Calculating the coherence components
Industrial practitioners (and many scientists) prefer to complement FRM estimates with the multiple (magnitude squared)
coherence function. As alternative, it is also a common practice to show the classical FRF variance derived from the coherence
function. In [33] it has been shown that the derived classical FRM variance equals the total variance of BLA (i.e., 𝜎
).
The multiple coherence function is used to judge the linearity of the signals between an outputs and inputs [22] [32]. In case of
multisine excitation, for an arbitrary chosen output channel, (5) can be estimated by using (12)-(14). The assessment of the
multiple coherence function estimate is precisely the same as the ordinary SISO coherence function: a value of 1 shows that
the input signals are linearly related to the (selected) output. A value of 0 means that the input signals are not related to the
output.
The lack of coherence can be for many reasons but due to the assumptions applied in this work, the following three causes are
inherently excluded: 1) leakage (periodic excitation), 2) synchronization issues (assumption on instrumentation), 3) time-
varying systems (beyond the scope of this article).
That means, the lack of coherence can only be due to the following reasons:
a) the measurement noise is significant (w.r.t. the ideal signal),
b) the response of the system is (strongly) nonlinear,
c) the response of the periodic excitation contains (significant) transient.
When the transient analysis is correctly performed, and the corresponding periods have been discarded c) can be eliminated
from the list. According to the classical data processing techniques, there is no further possibility to separate the effects of a)
and b). However, using the proposed BLA framework’s multidimensional averaging technique, the coherence function can be
separated into a), b) and c) cases providing the practitioner component-wise coherence information.
Next, the logical reasoning of the previous section will be used. Considering the steady-state data segments, for a given output
and an orthogonal phase rotated realization 𝑙, the coherence function estimate is given by:
𝛾[]
=𝑆[][][..]
𝑆[][]
𝑆[][][..]𝑆[][..][][..]
(20)
In other words, (20) calculates the multiple coherence for each steady-state realization independently. Let us remind to the fact
that the (phase non-coherent) nonlinearities do not vary over the periods within a realization. This means that the only thing
that varies is the (random measurement) noise. The average over all realizations leads to the improved noise component of the
multiple coherence function estimate defined as:
𝛾
=
∙∑𝛾[]
∙
(21)
where 𝑀 is the number of realizations, and 𝑛 is the number of phase rotations (i.e., the number of input channels).
Next, the steady-state multiple coherence function is estimated considering all transient-free measurements at once:
𝛾
=𝑆..[..]..[..]
𝑆..[..]..[..]
𝑆..[..]..[..]𝑆..[..]..[..]
(22)
Similarly to the total (co)variance (18), 𝛾
contains both of the effect of noise (included in 𝛾
) and the nonlinearities, the
difference (similarly to (19)) gives the following (nonlinear) quantity:
𝛾
=𝛾
−𝛾
(23)
where 𝛾
≥𝛾
.
Because we are interested in the coherence component due to the nonlinearity only, one shall consider:
𝛾
=1−𝛾
=1−(𝛾
−𝛾
)=𝛾
(1−𝛾
) (24)
If the system is linear, then 𝛾
and 𝛾
provide the same result (𝛾
=𝛾
), therefore 𝛾
=0 resulting 𝛾
=1.
Additionally, the transient component can be obtained in a similar way. Let us consider the whole measurement, including the
non-steady state blocks as well, one obtains the measurement coherence function estimate:
𝛾
=𝑆..[..∗]..[..∗]
𝑆..[..∗]..[..∗]
𝑆..[..∗]..[..∗]𝑆..[..∗]..[..∗]
(25)
where 𝑃∗ refers to all measured (transient and steady-state) periods.
The transient component of the multiple coherence function estimate is then given by:
𝛾
=1−(𝛾
−𝛾
) (26)
It is important to stress that the (multiple) coherence function estimate do not consider the model (e.g., H1 or BLA FRF) used
during the estimation process, it is purely a quantity of the signals.
A further disadvantage with respect to the variance technique elaborated in the previous section is that multiple coherence
function and its components are the identical for all input channels (for a given output). Furthermore, the components of the
multiple coherence function give an absolute number between zero and one, therefore it is difficult to judge how much is the
real (noticeable) effect on the FRF estimate, whereas the variance level estimates answers that directly.
5 ILLUSTRATION
5.1 Description of the measurement
To illustrate the capabilities of the proposed framework, multiple input measurements (GVT) of an F-16 aircraft are discussed
in this section. For an illustration of the measurement, see Figure 5. The aircraft is equipped with two dummy missiles with
similar mass and inertia properties as the live payloads.
The two wings are simultaneously excited by 2 shakers. At the driving (excitation) point there are 2 force cells. The responses
of the structure are measured with acceleration sensors. The shakers are fed with orthogonally shifted full random phase
multisines. The sampling frequency is 200 Hz with a period length is 4096. The excitation frequency range is between 4.541 Hz
and 25 Hz with a frequency resolution of 0.0489 Hz.
In the excitation frequency band, the aircraft has about 12 resonance modes. The first few modes below 5 Hz correspond to
rigid body motions of the structure. The first flexible mode around 5.2 Hz corresponds to wing bending deformations. The
dummy payloads are mounted at the wing tips with a T-shaped connection interfaces. These connection interfaces are the main
source of nonlinearities - as it can be seen later in the data analysis. The mode involving the most substantial nonlinear
distortions is the wing torsion mode located around 7.3 Hz.
A similar single input measurement campaign and its benchmark data are openly accessible [36]. As previously turned out, the
F-16 GVT has been proved to be highly challenging. To illustrate an even more challenging, higher complexity problem,
however, we consider a MIMO setup with a broader frequency range of excitation.
For the sake of simplicity, only high and low levels of experiments are shown at the driving points. In each experiment, there
are 7 orthogonally shifted multisine realizations (i.e., total 14 realizations). In each random realization there are 3 periods
measured.
Figure 5: F-16 ground vibration testing measurement.
5.2 Estimating the transient
Before carrying out any study of the measurement, it is crucial to ensure that the measured signals considered are in a steady
state, so a simple transient analysis is performed. The first realization of the measured time-domain input-output is shown in
Figure 6. The input signal is the measured force, the output signal is the measured acceleration at the left and right wings. In
order to simplify the analysis, only the high excitation level is considered. It is because the exponentially decaying transient
has the highest impact on this level. To estimate the length of the transient, the last period – that is assumed to be in steady-
state – is subtracted from every preceding block. The differences are shown in logarithmic scale. This allows us to easily
identify the transient: the exponential decay results in a linear decay on a logarithmic scale. It can clearly be seen that that the
input has minimal transient effect (as expected due to the fast dynamic of the shaker). The main transient effect comes from
the output measurements. Observe that the only the first block distorted by the transient. In the pre-processing steps, every first
block (period) is discarded in all (phase-shifted) realization.
Figure 6: The first realization of the input force (left) and output acceleration (right) are shown in time domain. Figures in the
first and third column show the periods in the excitation (z) direction. Figures in the second and fourth column show the
differences between the (corresponding) last block minus every preceding block in logarithmic scale. The first row shows the
signals at right wing. The second row shows the signals at the left wing. The greyed area refers to the transient block.
5.3 Analysis of the measured signals
This section concerns the analysis of the input (force) and output (acceleration) signals. The input-output measurements at the
driving points are shown in Figure 7.
The left and right measured forces (i.e., inputs) have almost identical, flat signal spectra, however, their noise estimates are
different. For example, at low level of excitation, the right wing’s error estimate has a slight trend (it is not a straight line), and
it looks ‘noisier’ compared to the left wing estimates. Under normal circumstances, the measurement noise (estimate) should
be a (nearly) flat line (due to the whiteness assumptions applied in this work, see Section 2.2). Deviation from this indicates
the presence of (weak) nonlinear distortions. In both, low and high level excitation cases, the SNR of the right wing is
approximately 5 dB better than the left wing.
In case of the acceleration signals, it can be observed that the left-right signal spectra are different. It turned out to be due to
the asymmetric configuration of the plane. It can be observed that at high level of excitation, the resonance frequencies have
been shifted. This type of behaviour is due to the (strong) nonlinearities.
A further interesting thing to point out that the (input-output) estimated noise levels move together with the signal estimates.
Increasing the excitation level results in decreasing the SNR. This type of behavior is usually due to (weak) nonlinearities, like
friction. A detailed analysis on the input-output data can be found in [23] [24] [37].
10 20 30 40 50 60
time [sec]
-100
-50
0
50
100
150
A
m
p
l
i
t
u
d
e
[
N
]
Input RWNG:100A:+Z
1 2 3
10 20 30 40 50 60
time [sec]
-60
-40
-20
0
20
db(u last -u 1,p)
Differences between last period and all periods
3-1 3-2 3-3
10 20 30 40 50 60
time [sec]
-100
-50
0
50
100
150
A
m
p
l
i
t
u
d
e
[
N
]
Input LWNG:000A:+Z
1 2 3
10 20 30 40 50 60
time [sec]
-60
-40
-20
0
20
40
db(u last -u2, p)
Differences between last period and all periods
3-1 3-2 3-3
10 20 30 40 50 60
time [sec]
-1
0
1
Amplitude [m/s
2
]
Output RWNG:100A:-Z
1 2 3
10 20 30 40 50 60
time [sec]
-60
-40
-20
0
db(y last -y1,p )
Differences between last period and all periods
3-1 3-2 3-3
10 20 30 40 50 60
time [sec]
-2
-1
0
1
2
Amplitude [m/s
2
]
Output LWNG:000A:-Z
1 2 3
10 20 30 40 50 60
time [sec]
-60
-40
-20
0
db(y last -y2,p )
Differences between last period and all periods
3-1 3-2 3-3
Figure 7: The input and output signals and their noise estimates measured at the right and left wings in frequency domain at
the low and high amplitude levels.
5.4 FRF analysis
The next step in the proposed analysis framework is the assessment of the frequency response functions. Figure 8 shows the
BLA FRF estimates at the driving points for low and high level of excitation. Observe that the FRM elements differ a lot from
each other. Due to the asymmetric configuration of the aircraft, that the transfer from left to right side does not equal to the
transfer from the right to the left side. Observe, that the level-wise comparison of the FRFs also shows differences (as it was
the case with the output measurements). The resonance frequencies have been shifted even though the difference between the
low and high level of excitation is only 15 dB.
Figure 8: FRFs at low and high excitation levels.
5.4.1 FRF analysis using variances
The use of periodic random realizations of multisines allow us to estimate the noise and nonlinear distortions as it demonstrated
in Figure 9 for the high level excitation case. Using this information, it is fairly simple to tell that in a certain frequency range
what the source of the domination error is. For example, around 3 Hz (the first resonance that correspond to a rigid body
motion) the main source of the distortion is due to noise, because there is a signal-to-nonlinearity ratio (SNLR) of 40 dB, and
an SNR of 35 dB. At the most dominant resonance around 7.3 Hz (that correspond to the torsional mode of the wing), it can be
clearly observed that the domination error source is due to nonlinearities (the SNLR is approximately 30 dB whereas the SNR
is around 60 dB). Using the classical H1 framework (that shows only the coherence function or the indirect total variance
estimate) this kind of extra information would have been possible to be obtained.
The noise and nonlinearity estimates are shown in Figure 9 for the entire measurements. The repetition of blocks improves the
SNR, the application of random realizations improves the SNLR. So, this kind of SNR, SNLR information describes the
goodness of the whole measurement. Using the details of the multidimensional averaging and variance estimation section of
this paper (Section 4.2-4.3), it is possible to tell how much is the (expected) SNR and SNLR for one randomly chosen period.
The conversion from measurement-wise to the period-wise form is simple: the noise variance estimates must be multiplied by
𝑃 ∙ 𝑀, and the nonlinear variance estimates by 𝑀.
The period-wise distortion estimates are shown in Figure 10. Observe that the noise estimate remains similar, but there is a
significant elevation of the nonlinear distortion estimates. This type of view (period-wise normalization) is very important,
-100
-80
-60
-40
-20
RWNG:100A:-Z /RWNG:100A:+Z RWNG:100A:-Z /LWNG:000A:+Z
5 10 15 20 25
-100
-80
-60
-40
-20
Magnitude [m/s2/N] (dB)
LWNG:000A:-Z /RWNG:100A:+Z
5 10 15 20 25
Frequency (Hz)
LWNG:000A:-Z /LWNG:000A:+Z
FRF low FRF high
when someone wants to build an accurate model that mimics the true behaviour of the measurement. Further, this period-wise
view provides more details that can be used in case of control, or other target applications. This can also help the novice user
to decide if a linear or nonlinear data-driven modelling framework is needed. The relative error of properly chosen linear model
can – under best circumstances – be in the vicinity of the SNR. In case of a properly chosen nonlinear modelling framework,
this relative error can be in the vicinity of SNLR. In the F16 aircraft situation, around 30 dB can be gained by using adequate
nonlinear modelling tools. This information has a high significance because the cost of nonlinear modeling is very high.
Figure 9: FRFs, noise and nonlinear distortions estimated at driving points at low and high excitation levels.
Figure 10: FRFs, noise and nonlinear distortions estimated with respect to one period at driving points at high excitation levels.
5.4.2 FRF analysis using coherence components
The use of distortion variances is simple, but it requires some hands-on experience. Many industrial users prefer to use the
multiple coherence function instead because its simplicity to assess the entire measurement quality. When the multiple
coherence function estimate is 1, it indicates that there is 100% linear correlation between the output and input signals. Lower
multiple coherence estimates indicate the presence of significant distortions. Such distortions can be due to noise, transient,
frequency leakage, nonlinearities. Because the multisines are periodic, frequency leakage is automatically excluded (provided
that the assumptions in section 2.2 are fulfilled).
Using the classical (multiple) coherence function, there is no possibility to separate difference types of distortions. The
advantage of the proposed estimation framework that it allows us to directly split up the multiple coherence function estimate
into noise, nonlinearity and transient (multiple coherence) components, as it is detailed in Section 4.4. This exciting feature
will allow practitioners to have a detailed view at the distortions with the ‘classical’ tools they know.
The coherence components of the aircraft measurements are shown in Figure 11. The classical multiple coherence estimate
(represented by the pink lines) shows how much ‘imperfections’ are present in the experiment. However, using only this
information, one cannot be certain what the origin of the distortions are. When the multiple coherence function estimate is split
into components, one can see that the main source of the distortions comes from the transient (the dashed lines). When an
appropriate number of delay blocks are cut out from the measurement (as it is shown in Section 5.2), this component is virtually
eliminated. The second largest source of distortions can be found in the nonlinear coherence components (see red lines). The
Magnitude [m/s
2
/N] (dB)
Magnitude [m/s2/N] (dB)
last remaining coherence component is the noise coherence components (thin black lines). Looking at these components, it
confirms that the measurement was (indeed) of high quality.
The main source of nonlinear distortions can easily be identified around 7.3 Hz when looking at Figure 10 and Figure 11. This
observation is completely in-line with a previous (independent) modal analysis work [38] of the F16 SISO GVT benchmark
data. It was pointed out that the variable stiffness of the bolted T joints (between the payload and the wing) is the main source
of nonlinearity as it acts as a cubic spring this is (or in other works, it acts as hardening springs).
The main advantage of the proposed BLA distortions variance view that different normalization modes can be used, and relative
information is shown. This allows the users to directly estimate the SNR/SNLR levels. Furthermore, for each pair of FRF (i.e.,
input-output channels) there are different variance levels, whereas with the coherence component, there is only one fixed
estimate per output channel. This means that with the coherence components we can get the averaged behavior per output using
all input – instead of per output per input information.
Figure 11: The BLA FRF estimations and their coherences components. The pink line shows the classical multiple coherence
estimate. The dashed line refers to the transient coherence component. The red line refers to the nonlinear coherence
component. The thin black line refers to the noise coherence component.
6 CONCLUSION
In this work a novel, user-friendly framework was developed that aims at enhancing the model assessment capabilities using
the efficient, highly tailorable state-of-the-art multisines. The proposed MIMO framework uses a novel orthogonalization
technique that offers many attractive features, such as optimal excitation of multiple input structures. Due to the pseudo-
randomness and the orthogonality, the multisines are very rich, persistently exciting MIMO optimized signals.
The statistical features of the excitation signal allowed us to tailor the Best Linear Approximation framework such that
advanced distortion information can be obtained. The proposed methodology was successfully illustrated on the F-16 GVT
data, and turned out to be useful because:
The orthogonal excitation technique that
o optimally excites structures with multiple inputs,
o improves the signal-to-noise ratio of the measurements,
o allows advanced characterization of system under tests,
o avoids classical signal processing issues such as spectral leakage.
The tailed multisine based BLA process is simple but robust method that provides at each level of excitation:
o an advanced FRM estimation,
o detailed distortion information: total, noise and nonlinear distortion estimates,
o the multiple coherence function is slit up into transient, noise and nonlinearity level components.
The framework is generic enough to be used for (virtually) almost any vibro-acoustic structures (where external user-defined
excitation is possible). The technique will allow industrial practitioners to obtain more advanced, detailed information in
assessing the measurement with already existing metrics such as variances and multiple coherence functions. This means that
that an (inexperienced) user can easily:
decide if a linear framework is still accurate and safe enough to be used, and
tell how much improvement can be gained using a nonlinear framework.
ACKNOWLEDGEMENTS
This work was funded by the Strategic Research Program SRP60 of the Vrije Universiteit Brussel.
0.2
0.4
0.6
0.8
1
Coherence
RWNG:100A:-Z /LWNG:000A:+Z
5 10 15 20 25
-100
-80
-60
-40
-20
Magnitude [m/s2/N] (dB)
LWNG:000A:-Z /RWNG:100A:+Z
5 10 15 20 25
Frequency (Hz)
0.6
0.7
0.8
0.9
1
LWNG:000A:-Z /LWNG:000A:+Z
FRF high Coherence - total measurement high Coherence - Transient high
Coherence - NL high Coherence - noise high
-100
-80
-60
-40
-20
RWNG:100A:-Z /RWNG:100A:+Z
STATEMENTS
Conflict of Interest: The author declares that they have no conflict of interest.
The datasets generated during and/or analyzed during the current study are available from the corresponding author on
reasonable request.
7 REFERENCES
[1]
G. Kerschen, K. Worden, A.F. Vakakis, J.-
C. Golinval, "Past, present and future of nonlinear system identification in
structural dynamics,," Mechanical Systems and Signal Processing, vol. 20, no. 3, pp. 505-592, 2006.
[2]
K. Worden, G.R. Tomlinson, Nonlinearity in Structural Dynamics: Detection, Identification and Modelling,, Bristol :
Institute of Physics Publishing, 2001.
[3]
J. Schoukens, K. Godfrey, M. Schoukens, "Nonparametric data-
driven modeling of linear systems: estimating the
frequency response and impulse response function," IEEE Control Systems Magazine, vol. 38, no. 4, pp. 49-88, 2019.
[4]
J. Schoukens, L. Ljung, "Nonlinear System Identification: A User-Oriented Road Map,"
IEEE Control Systems
Magazine, vol. 39, no. 6, pp. 28-99, 2019.
[5]
P. Fargette, U. Füllekrug, G. Gloth, B. Levadoux, P. Lubrina, H. Schaak, M. Sinapius, "Tasks for improvement in Ground
Vibration Testing of large aircraft. In Proc. IFASD 2001 Int. Forum on Aeroelasticity and Struct. Dyn., Madrid, Jun
2001.," in IFASD 2001 Int. Forum on Aeroelasticity and Struct. Dyn., Madrid, June 2001.
[6]
P. Lubrina, "Ground vibration experiments on large civil aircraft for engine imbalance purpose," in
IFASD 2003 Int.
Forum on Aeroelasticity and Struct. Dyn., Amsterdam, June 2003.
[7]
H. Climent, "Aeroelastic and Structural Dynamics Tests at EADS CASA," in
the 18th Annual Symposium of the Society
of Flight Test Engineers SFTE, Madrid, June 2007.
[8]
C.R. Pickrel, G.C. Foss, A. Phillips, R.J. Allemang, D.L. Brown, "New Concepts GVT," in
IMAC 24 Int. Modal Analysis
Conf.,, St. Louis (MO), , Jan-Feb 2006..
[9]
H. Van Der Auweraer, D. Otte, J. Leuridan, W. Bakkers, "Modal testing with multiple sinusoidal excitation," in
IMAC 9
Int. Modal Analysis Conf., Firenze, 1991.
[10]
D. Otte, H. Van Der Auweraer, J. Debille, J. Leuridan, "Enhanced force vector appropriation methods for normal mode
testingm Los Angeles," in February 1993, Los Angeles, IMAC 11 Int. Modal Analysis Conf..
[11]
J. Schoukens, R. Pintelon, Y. Rolain, Mastering System Identification in 100 exercises, New Jersey: John Wiley & Sons,
ISBN: 978047093698, 2012.
[12]
L. Lauwers, J. Schoukens, R. Pintelon and M. Enqvist, "Nonlinear Structure Analysis Using the Best Linear," in
Proceedings of International Conference on Noise and Vibration Engineering, Leuven, 2006.
[13]
A. Esfahani, J. Schoukens and L. Vanbeylen, "Using the Best Linear Approximation With Varying Excitation Signals
for Nonlinear System Characterization," IEEE Transaction on Instrumentation and Measurement, vol. 65, pp. 1271-
1280, 2016.
[14]
H. K. Wong, J. Schoukens and K. Godfrey, "Analysis of Best Linear Approximation of a Wiener–
Hammerstein System
for Arbitrary Amplitude Distributions," IEEE Transactions on Instrumentation and Measurement, vol. 61, no. 3, pp. 645-
654, 2012.
[15]
J. Schoukens and R. Pintelon, "Study of the Variance of Parametric Estimates of the Best Linear Approximation of
Nonlinear Systems," IEEE Transactions on Instrumentation and Measurement, vol. 59, no. 12, pp. 3156-3167, 2010.
[16]
Schoukens, J.; Vaes, M.; Pintelon, R., "Linear System Identification in a Nonlinear Setting. Nonparametric Analysis of
the Nonlinear Distortions and Their Impact on The Best Linear Approximation.," IEEE Control Systems Magazine,
vol.
36, pp. 38-69, 2016.
[17]
T. Dobrowiecki, J. Schoukens, P. Guillaume, "Optimized excitation signals for MIMO frequency function
measurements," IEEE Transactions on Instrumentation and Measurements,, vol. 55, pp. 2072-2079, 2006.
[18]
T. Dobrowiecki and J. Schoukens, "Linear approximation of weakly nonlinear MIMO systems,"
IEEE International
Instrumentation and Measurement Technology Conference (I2MTC), pp. 1607-1612, 2004.
[19]
J. Schoukens, R. Pintelon, G. Vandersteen, Y. Rolain, "Design of excitation signals for nonparametric measurements on
MIMO-
systems in the presence of nonlinear distortions, IEEE International Instrumentation and Measurement
Technology Conference," in IEEE International Instrumentation and Measurement Technology Conference
, Austin,
2010.
[20]
R. Pintelon, J. Schoukens, System Identification: A Frequency Domain Approach, 2nd ed., New Jersey: Wiley-
IEEE
Press, ISBN: 978-0470640371, 2012.
[21]
P. Z. Csurcsia, B. Peeters, J. Schoukens, "User-friendly nonlinear nonparametric estimation framework for vibro-
acoustic
industrial measurements with multiple inputs," Mechanical Systems and Signal Processing, vol. 145, 2020.
[22]
Gómez González, A., J. Rodríguez, X. Sagartzazu, A. Schumacher, and I. Isasa., "Multiple Coherence Method in Time
Domain for the Analysis of the Transmission Paths of Noise and Vibrations with Non-
Stationary Signals," in
International Vibration and Noise Conference, Leuven, Belgium, 2010.
[23]
P. Z. Csurcsia, B. Peeters, J. Schoukens, T. De Troyer, "Simplified Analysis for Multiple Input Systems: A Toolbox
Study Illustrated on F-16 Measurements," Vibration, vol. 3, no. 2, pp. 70-84, 2020.
[24]
P. Csurcsia and T. de Troyer, "An empirical study on decoupling PNLSS models illustrated on an airplane," IFAC-
PapersOnLine, vol. 7, no. 673-678, p. 54, 2021.
[25]
P. Csurcsia, B. Peeters and J. Schoukens, "The best linear approximation of MIMO systems: Simplified nonlinearity
assessment using a toolbox," in Proceedings of ISMA 2020 -
International Conference on Noise and Vibration
Engineering and USD 2020 - International Conference on Uncertainty in Structural Dynamics
, Leuven, Belgium, 2020.
[26]
P. Z. Csurcsia and J. Lataire, "Nonparametric Estimation of Time-variant Systems Using 2D Regularization,"
IEEE
Transactions on Instrumentation & Measurement, vol. 65, no. 5, pp. 1259-1270, 2016.
[27]
P. Z. Csurcsia, J. Schoukens, I. Kollár, "Identification of time-varying systems using a two-dimensional B-
spline
algorithm," in 2012 IEEE International Instrumentation and Measurement Technology Conference
, Graz, Austria, 2012.
[28]
P. Z. Csurcsia, J. Schoukens, I. Kollár, "A first study of using B-splines in nonparametric system identification," in
IEEE
8th International Symposium on Intelligent Signal Processing, Funchal, Portugal, 2013.
[29]
P. Z. Csurcsia, "Static nonlinearity handling using best linear approximation: An introduction," Pollack Periodica ,
vol.
8, no. 1, 2013.
[30]
W. Heylen, P. Sas, Modal Analysis Theory and Testing, Leuven: Lirias, 2005.
[31]
S. M. Kay, Modern Spectral Estimation: Theory and Application, Prentice Hall Signal Processing Series, 1988.
[32]
B. Peeters, M. El-
Kafafy, P. Guillaume and H. Van der Auweraer, "Uncertainty propagation in Experimental Modal
Analysis," in Conference Proceedings of the Society for Experimental Mechanics Series, 2014.
[33]
P. Z. Csurcsia, B. Peeters, J. Schoukens, "The Best Linear Approximation of MIMO Systems: First Results on Simplified
Nonlinearity Assessment," in IMAC International Modal Analysis Conference, Orlando, 2019.
[34]
B. Cornelis, A. Toso, W. Verpoest, B. Peeters, "Improved MIMO FRF estimation and model updating for robust Time
Waveform Replication on durability test rigs," in International Conference on Noise and Vibration Engineering
, Leuven,
2014.
[35]
J. Schoukens, R. Pintelon and Y. Rolain, Mastering System Identification in 100 exercises, New Jersey: John Wiley &
Sons, 2012.
[36]
J.P. Noël and M. Schoukens, "F-16 aircraft benchmark based on ground vibration test data," in
2017 Workshop on
Nonlinear System Identification Benchmarks, Brussels, Belgium, 2017.
[37]
P. Z. Csurcsia, J. Decuyper and T. D. Troyer, "Nonlinear Modelling of an F16 Benchmark Measurement," in
Conference
Proceedings of the Society for Experimental Mechanics Series, Florida, U.S., 2022.
[38]
T. Dossogne, J. Noël, C. Grappasonni, G. Kerschen, B. Peeters, J. Debille, M. Vaes, J. Schoukens, "Nonlinear Ground
Vibration Identification of an F-16 Aircraft -
Part II Understanding Nonlinear Behaviour in Aerospace Structures Using
Sine-sweep Testing," in Conference: International Forum on Aeroelasticity and Structural Dynamics
, Saint Petersburg,
Russia, 2015.
