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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.

Proceedings 2018,2, 519 5 of 7

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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