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proceedings
Proceedings
Efficient Use of the Output Information to Improve
Modal Parameter Estimation †
Oscar Olarte 1,*, Mahmoud El-Kafafy 1,2 and Patrick Guillaume 1
1Mechanical Engineering Department, Vrije Universiteit Brussel, Acoustics and Vibration Research Group,
B-1050 Brussels, Belgium; Mahmoud.El-Kafafy@vub.be (M.E.-K.); Patrick.Guillaume@vub.be (P.G.)
2Mechanical Design Engineering Department, Helwan University, Cairo 11790, Egypt
*Correspondence: oolarter@vub.ac.be
† Presented at the 18th International Conference on Experimental Mechanics (ICEM18), Brussels, Belgium,
1–5 July 2018.
Published: 26 June 2018
Abstract:
In modal identification, the value of the model parameters and the associated uncertainty
depends on the quality of the measurements. The maximum likelihood estimator (
mle
) is a consistent
and efficient estimator. This means that the value of the parameters trends asymptotically close to
the true value, while the variance of such parameters is the lowest possible with the associated data.
The
mle
implementation and application can be complex and generally need strong computational
requirements. In applications where the number of inputs and outputs are elevated (as in modal
analysis) is common to reduce the covariance matrix to a diagonal one where only the variances are
considered. This implementation is still consistent but not efficient. However, it generates acceptable
results. The current work shows that using efficiently the output information as complement to the
input–output relations, it is possible to improve the model identification reaching similar levels than
the mle, while reducing the execution time and the computational load.
Keywords: modal parameters estimation; maximum likelihood estimator; output–output relations
1. Introduction
The input-output (transfer function relation) frequency domain approach requires the direct
measurement of the input–output signals. In some applications such measurements, specially the
inputs of the system, are difficult to assess, and/or in addition to the applied excitation other/others
unknown sources can go into the system generating correlations in the variables. The previous scenario
is common in modal analysis with the additional factor that the number of outputs can easily reach
hundreds, even thousands.
An estimator that has the desired properties of consistency and efficiency is the maximum
likelihood (mle) estimator. This is, the value of the parameters trends asymptotically close to the
true parameter value (consistency), while the variance of such parameters is the lower possible
with the associated data (efficiency) [
1
,
2
]. The implementation of this estimator could be complex
since it requires the estimation of the covariance matrix. This is an important issue since when the
number of variables is high (as in modal analysis) this requires the development of high amount of
experiments. In many cases, in order to reduce the number of experiments and the computational
processing time, the covariance matrix is reduced to a diagonal one where only the variances are
present. This implementation (called here diagonal implementation of the mle (
Dmle
)) still generates a
consistent estimation but the efficiency lost (it is not possible to reach the lower variance in the model
and parameters associated with the used data), however the result are quite acceptable [2,3].
Proceedings 2018,2, 519; doi:10.3390/ICEM18-05391 www.mdpi.com/journal/proceedings
Proceedings 2018,2, 519 2 of 7
In the present work, in a similar way as in the
Dmle
where the covariance matrix is reduced
to a diagonal one, two estimators are introduced. But in addition to the input–output relations,
the output–output relations are used as complement to the input–output information. Similar to
a traditional input–output transfer functions the output–output functions are defined as the
frequency-domain ratio between two outputs, and describes the relative admittance between two
measurements. The output–output relations are interpreted as the response data normalized by a
reference response instead of by the input excitation [
4
,
5
]. Unlike the transfer function, these functions
are independent of the input information and its associated noise [3].
The article starts describing the
mle
estimator, subsequently introduces the
Dmle
followed by the
approaches considering the only output relations. The comparison between the different estimators is
presented by mean of a Monte-Carlo simulation examples. The different estimators are assessed based
on the global uncertainty reached by the estimators and the computational execution time. Finally the
conclusions are presented.
2. Maximum Likelihood Estimator (m le)
The parametric transfer function considered here is scalar matrix description or common
denominator transfer function described as
Y(ωf) = N(s)
d(s)X(ωf)(1)
but can be extended to left and right matrix fraction description models. In (1)
N(s)
is a q-output by
p-input polynomial matrix.
Y(ωf)
is q-output by one vector and
X(ωf)
is p-input by one vector.
The
nij (s)
locations of the
N(s)
matrix are given by
nij (s) = ai j
0+aij
1s+
...
+aij
αij sαi j
while the
denominator is given by
d(s) = b0+b1s+
...
+bβsβ
. The index
αij
and
β
refer to the order of
the polynomials nij (s)and d(s)respectively.
Assuming that the noise on the real and imaginary part of the spectra of the input and output
signals are independent, zero mean, Gaussian random variables, it can be proven [
1
] that the
mle
of
the parameters is found by minimizing the cost function
v=1
2
F
∑
k=1
ξH(k)C−1
ξ(k)ξ(k)(2)
with Hthe complex conjugate transport operation and
ξ(k) = hN(k),−d(k)Ii"X(k)
Y(k)#(3)
The matrix C−1
ξ(k)is the inverse of the covariance matrix
Cξ(k) = hN(k),−d(k)IiCov ("X(k)
Y(k)#)hN(k),−d(k)IiH(4)
with
Cov {·}
the covariance matrices of the perturbation noise of the inputs and outputs at the
frequency lines k. The perturbation noise can be determined from a priori measurements [6].
Diagonal Implementation of the mle Estimator (Dmle)
The mle estimator is optimum since it follows the properties of consistency and efficiency and
converge to the noiseless solution [
2
]. However, the procedure requires to estimate the covariance
matrix and its inverse (4). which, carries a significant experimental time consumption and computation
load. To reduce the computational load it is possible to assume independence between the inputs and
Proceedings 2018,2, 519 3 of 7
outputs. In this sense the covariance matrix (4) becomes a diagonal matrix and from here a fast and
practical implementation is achieved [7].
3. Extension of Diagonal-m le Using Output-Output Relations (Doml e)
The equation error (3) considers the input-output relation and from here its dependency of
the input signal and related noise. However, along the output–output relations this dependency is
set aside.
Output–output relations are obtained by the ratio of two response spectra. This is, the ratio
between the output qand ris Tq,r(ω) = Yq(ω)
Yr(ω). From (1) the output-output equations are reduced to:
Tq,r(ω) = Nq,p(ω)
Nr,p(ω)(5)
with
Nq,p(ω)
and
Nr,p(ω)
the numerator polynomials occurring in the transfer function
Gq,p=Nq,p(ω)
d(ω)
and
Gr,p=Nr,p(ω)
d(ω)
. Observe that the input excitation signal is eliminated by taking the
ratio between the response-output spectra.
The equation error is extended to the output-output equations as:
ξ(k) = "N(k)−d(k)I
0T(k)#"X(k)
Y(k)#(6)
and the covariance matrix became
Cξ(k) = "N(k)−d(k)I
0T(k)#Cov ("X(k)
Y(k)#)"N(k)−d(k)I
0T(k)#H
(7)
with Tthe matrix of the numerator relations.
The
mle
solution of (2) requires the inverse of (7) which is a singular matrix since there exist
linear dependency between the rows. However, as in the
Dmle
, it is assumed that covariance matrix is
diagonal, or in other words it is assumed that the noise is uncorrelated between the inputs and outputs,
and that the contribution of the non-diagonal terms are irrelevant. The covariance matrix (7) becomes
diagonal and includes the input–output and output–output relations and allow a fast implementation.
This implementation will provide consistent estimators although non efficient. However, the estimation
using the output–output relation will provide better results in terms of efficiency than the common
Dmle implementation.
4. Diagonal Implementation of the m le Using Orthogonal Equations (D⊥ml e)
The dimension of the output–output equations is function of the number of outputs. In fact it
is the combination without repetition of q-outputs taken 2 at a time. This combination can easily
reach considerable values for few number of variables. For example: for a system with 10 outputs the
number of output–output equations is already 45, for 100 the number of equations reach 4950 and for
more of 200 outputs the number exceed the twenty thousand!
A generalization to the output–output equations that keep the number of relation in similar
proportion to the number of input-output equations is achieved by taken the orthogonal value of the
numerator such that the effect of the input (and associated noise) is eliminated from the equations.
This is, from the input-output relation (1) by left multiplication at both sides of the equation by the
orthogonal value of the numerator (N⊥(s)) we have
N⊥(s)Y(ωf) = 0 (8)
observe that the effect of the input and associated noise has been removed from the equation.
Proceedings 2018,2, 519 4 of 7
Similar to the previous section the equation error and associated covariance have the shape of (6)
and (7) respectively, but the explicit outpu–output relations
T(k)
is substituted by the general form
N⊥(k)
. Here again assuming that the variables are independent taking only the information of the
diagonal, the cost function (2) can be implemented.
5. Experiment
Four different estimators
mle
,
Dmle
,
Domle
and
D⊥mle
are compared in a system under two
scenarios: The first scenario presents uncorrelated independent noise added at the inputs and outputs
of the system with levels that reach signal to noise ratios (SNR) of 70, 50, 40, 30, 20, 15 and 10 dB.
The second scenario covers the conditions of the first one, but includes the presence of an unknown
excitation. The unknown excitation reach attenuations (regarding the known excitation) of
−
50 and
−
10 dB. Table 1summarizes the employed levels while the Figure 1illustrates the experimented
scenarios. The different estimators will be assessed by mean of the general variance (GV) of the
parameters of the system (the GV can be seen as a scalar measure of the overall multidimensional
scatter of the analyzed data. Lower the GV value is, lower variance in the parameters) and the
computational execution time.
Table 1. SNR and attenuation levels used for the input, output and unknown excitation.
Title 1 SNR
SNR level at the input 10, 15, 20, 30, 40, 50 70
SNR level at the output 10, 15, 20, 30, 40, 50 70
Unknown excitation attenuation −50, −10
Figure 1.
Experimental scenarios. (
a
) This scenario only includes the presence of uncorrelated
independent noise at the inputs and outputs; (
b
) This scenario includes the presence of uncorrelated
independent noise at the inputs and outputs and presence of unknown forces acting on the system.
6. Results
1st Scenario: Independent and Uncorrelated Noise.
In this scenario only independent and
uncorrelated noise are present at the inputs and outputs of the system. The Figure 2shows the general
variance on the parameters for the different estimators (Figure 2a) and the execution time (Figure 2b).
For low levels of noise at the input the estimators generate similar results. However, when the SNR of
the input becomes lower (high presence of noise) and the SNR at the outputs grow, the estimators using
complementary output equations (
Domle
and
D⊥mle
) show better results than the
Dmle
estimator.
In fact the results of the Domle and D⊥mle are superimposed on the mle results.
The mean execution time for the different estimators is shown in the Figure 2b. According with
the results, the
Dmle
estimator requires less time while the
mle
estimator expend at least one order of
magnitude more. The
Domle
and the
D⊥mle
estimators present execution times lower than the half
required for the mle being lower for the estimator where the output–output relations are explicit.
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Figure 2.
Results of the
mle
,
Dmle
,
Domle
and
D⊥mle
estimators when only independent and
uncorrelated noise is present at the input and output of the system. (
a
) General variance; (
b
) mean
execution time.
2nd Scenario: Independent Uncorrelated and Correlated Noise.
Similar than the first scenario
uncorrelated noise has been added into the inputs and outputs of the system, but in addition, one
unknown excitation is set, which will generate correlated noise in the system. The result shows, as in the
previous case, that at high levels of noise at the input, the outputs relations improve the identification
of the system reducing the associated variance when is compared with the
Dmle
implementation.
The unknown excitation generate add correlation in the variables so that the
Domle
and the
D⊥mle
move away of the
mle
results. But in any case these estimators still present better performance,
in general variance terms, than the
Dmle
. See Figure 3. The execution time show similar results as in
the first scenario: The
Dmle
implementation shows the lowest time consumption, the
mle
estimator the
highest and the proposed methods,
Domle
and
D⊥mle
, show intermedian execution times. See Figure 4.
Figure 3.
General Variance of the
mle
,
Dmle
,
Domle
and
D⊥mle
estimators when an unknown
excitation is present. The unknown attenuation excitation regarding the known excitation is:
(a)−50 dB
;
(b)−10 dB.
Proceedings 2018,2, 519 6 of 7
Figure 4.
Mean execution time of the
mle
,
Dmle
,
Domle
and
D⊥mle
estimators when an unknown
excitation is present. The unknown attenuation excitation regarding the known excitation is:
(a)−50 dB
;
(b)−10 dB.
7. Conclusions
The properties of consistency and efficiency of the
mle
estimator makes it the best choice in the
identification of a system. However, the practical implementation of this estimator is complex since
requires the estimation and inverse calculation of the covariance matrix. The diagonal implementation
of the
mle
estimator (
Dmle
) reduces the number of required experiments as well as the computation
requirements and execution time. The price to pay for this simplified and faster implementation
is the lost of the estimator’s efficiency. In order to improve the results of the
Dmle
estimator, the
article propose to use conveniently the output information. The
Domle
estimator uses explicitly all the
possible combinations of output–output relations, while the
D⊥mle
uses the orthogonal projection of
the numerator. The proposed methods improve the
Dmle
results since the covariance of the model
become closest to the
mle
results, specially in the case where the input signal is highly contaminated
by noise.
Author Contributions:
P.G. and O.O. conceived and design the proposed estimators, wrote the paper, develop
the softwatre implementation of the algorithms, proposse de experiments and analyzed the data.M.E.-K. orientate
the data analysis and orientate the software implementation.
Acknowledgments: This work is financially supported by SRP-OPTIMech Vrije Universiteit Brussels (VUB).
Conflicts of Interest:
The authors declare no conflict of interest. The founding sponsors had no role in the design
of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the
decision to publish the results.
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2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
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