Mechanical Systems and Signal Processing (2002) 16(5), 873–883

doi:10.1006/mssp.2001.1430, available online at http://www.idealibrary.com on

INDIRECT INPUT IDENTIFICATION INMULTI-SOURCEENVIRONMENTS BY PRINCIPALCOMPONENTANALYSIS

M.-S. Cho and K.-J. Kim

Department of Mechanical Engineering, Center for Noise and Vibration Control, KAIST,Science Town, Taejon 305-701, South Korea

(Received 18 October 1999, accepted 26 May 2001)

The paper deals with problems in the indirect input identification when the number ofsimultaneously usable measurement and processing channels is smaller than the totalnumber of response points. In such a case, the output spectral matrix essential to theindirect input identification can be obtained by processing the responses with respect to areference point in a sequential manner, which is the so-called transmissibility functionapproach. This conventional transmissibility function approach is applicable only to a casewhere the number of independent excitation sources is just one regardless of the number ofinput points at which the input forces are to be estimated. In this paper, the technique isextended to other cases where the number of independent sources is greater than one basedupon the principal component analysis. A method to identify the number of independentexcitation sources is also presented. The validity of the proposed method is demonstratedby a numerical example.

# 2002 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

The accurate identification of operational excitation forces in vibrating mechanicalsystems is an important issue from the aspects of design, control and diagnosis. Sincedirect measurement of the input forces is often very difficult or almost impossible, inputsare practically identified indirectly, i.e. from response measurements under operationalconditions and system characteristics [1–4].

When the number of simultaneously usable measurement channels is smaller than thatof the actual output response points, the whole output spectral matrix can be built in asequential way using a single reference signal, which is the conventional transmissibilityfunction approach [5]. This approach yields correct estimations of the whole outputspectral matrix only if the number of independent excitation sources is just one regardlessof the number of points at which the input forces are going to be estimated. Otherwise,estimations of the response spectral matrix vary with the location of the reference pointand consequently, input estimations become distorted. Therefore, when the number ofindependent excitation sources are greater than one, it is necessary first of all todecompose the responses so that each of the decomposed responses become independentof the others and then apply the conventional approach to each decomposed response. Thedecomposition is done in this paper by the principal component analysis (PCA).

The PCA is a technique to extract compact information from a matrix by investigatingits dimensionality, which was introduced in multivariate statistics for data reduction [6]and applied to the determination of operational deflection shapes in multi-sourceenvironments [7–8]. The basic idea of the PCA approach in the multi-source environmentsis to decompose the outputs into several principal components which are independent of

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M.-S. CHO AND K.-J. KIM874

each other. The number of principal components of the output spectral matrix is equal tothe number of non-zero singular values of the matrix. However, it seldom happens thatsingular values of the spectral matrix become perfectly zeroes due to effects of manyunknown factors such as the measurement noise. In such a case, the number of theprincipal components can be determined by ignoring singular values smaller than a givenlevel. In this paper, a technique to determine the number of principal components basedon the noise power level is suggested and a technique to estimate the noise power levelfrom the measured frequency response functions (FRF) and output spectral matrix undersome assumptions is presented in Appendix A. The proposed approach is applied to anumerical example to show its validity.

The paper is organised as follows. First, the basic formulation of the indirect inputidentification is reviewed and the conventional transmissibility function approach toobtain the output spectral matrix is explained for a single excitation source environment.Then, the new transmissibility function approach based upon the PCA is proposed and thecriterion for determining the number of principal components is presented. Finally, thevalidity of the proposed method is demonstrated using a simulation study.

2. BASIC FORMULATIONS FOR INDIRECT INPUT IDENTIFICATION

For a linear mechanical system where input excitations are measured at m points andoutput responses at n points, the output spectral matrix SYY of dimension n� n can beexpressed by [9]

SYY ¼ HHSFFH ð1Þ

where H is the m� n FRF matrix between the inputs and outputs, SFF the m�m inputspectral matrix, superscript H the Hermitian transpose of a complex matrix, andfrequency dependence of each matrix is omitted for the simplicity in expression.

The input spectral matrix SFF in equation (1), which is of interest in this paper, can beobtained from the output spectral matrix by taking pseudo-inverse of the FRF matrixwhen the number of output responses is greater than or equal to the one of input forces.That is, the minimum norm solution for the input force is given by

#SSFF ¼ ðHHÞþSYY ðHÞþ ð2Þ

where + and ^ denote, respectively, the pseudo-inverse of a matrix and the estimation of avariable. It is noted that, in order for equation (1) to be successful in practice, the systemFRF matrix as well as the output spectral matrix must be correctly provided. In this paper,the focus in on the construction of the latter under the assumption that the former isavailable.

3. CONSTRUCTION OF OUTPUT SPECTRAL MATRIX WITH LIMITED NUMBEROF MEASUREMENT CHANNELS

The output spectral matrix, SYY , can be obtained without any difficulty if the wholeresponses can be measured simultaneously. In reality, however, the number ofsimultaneously measurable responses is often limited by the number of channels of agiven data acquisition and processing device, although the cost per channel in such adevice has been greatly reduced recently. The output spectral matrix can then be estimatedby measuring and processing the responses in a sequential manner, i.e. by fixing a givenchannel onto one reference output point and roving the rest of the available channels overthe other output points. This is the conventional transmissibility approach and will be

INDIRECT FORCE IDENTIFICATION VIA PCA 875

reviewed in the next section and a critical problem inherent in this approach will bediscussed as well. A solution to this problem will be proposed based upon the PCA inSection 3.2, which is called a new transmissibility approach.

3.1. CONVENTIONAL TRANSMISSIBILITY APPROACH

When the output spectral matrix in equation (2) cannot be obtained from simultaneousmeasurements, it can be estimated in a sequential manner by using the following equation:

Sij ¼Sir

SrrS�jr: ð3Þ

That is, one channel of the data acquisition and analysis system is fixed onto anarbitrarily chosen reference point and the remaining, at least one, channels are roved toprocess the responses from the other points of interest in association with the responsefrom the reference point in a sequential manner. The key assumption of this approach isthat the choice of the reference point should not be influential on the estimation results.This assumption is, however, guaranteed only if the number of independent excitationsources equal to the rank of the input spectral matrix, SFF , is just one regardless of thenumber of input measurement points.

Let us go back to equation (1) to look into ranks of matrices on both sides. It can beassumed that excitation forces are generated at frequencies other than the systemresonance frequencies. Then, the rank of the left-hand side is determined by the rank of theinput spectral matrix on the right-hand side because the rank of the system matrix onthe right-hand side is full at such non-resonance excitation frequencies. If, therefore, thenumber of independent excitation sources is just one regardless of the number of inputmeasurement points, the rank of the output spectral matrix becomes one, meaning thatevery column or row is proportional to the others. Therefore, a crucial drawback of theconventional transmissibility method is that estimations of the output spectral matrix maybe inconsistent depending on the choice of the reference point if the number ofindependent excitation sources are greater than one.

3.2. A NEW TRANSMISSIBILITY FUNCTION APPROACH IN MULTI-SOURCE ENVIRONMENT

3.2.1. A new approach based upon principal component analysis

When a mechanical system is subjected to more than one independent excitationsources, columns of the output spectral matrix SYY are not proportional to each otherand, in fact, form a vector space, the dimension of which is determined by the rank of SYY .Hence, in order to obtain the output spectral matrix correctly by the conventionaltransmissibility function approach, it is necessary to decompose the output spectral matrixin such a way that the rank of each decomposed matrix may be one. The decompositioncan be carried out using the PCA as explained below.

Let the number of independent excitation sources be p; then construct the spectralmatrix SRR from the responses at r reference points, where r should be greater than orequal to p. That is, simultaneous measurements should be obtained for at least r points.Then, carry out the singular value decomposition on SRR as follows:

SRR ¼ USxxUH ð4Þ

where Sxx is a p� p diagonal matrix containing singular values of SRR in descending orderand U is an r� p unitary matrix. The relative importance of the singular values orprincipal components can be determined by comparing the ratios of singular values,li=liþ1, i ¼ 1 to (r� 1). It can also be done by comparing with unity the ratio ofsummation of the first l singular values to the summation of all singular values, e.g.

M.-S. CHO AND K.-J. KIM876

Pli¼1 li=

Prj¼1 lj, l ¼ 1 or r [6]. In this paper, an objective criterion using noise power level

is proposed, which will be presented in the next section.The p principal component vector x can be obtained from the relation with the r

reference response vector R and the unitary matrix U as follows:

Rr�q ¼ Ur�pnp�q or x ¼ UTR ð5Þ

where q is the number of data sets for ensemble averaging. The principal componentsderived from a completely mathematical point of view are mutually independent and,hence, can be considered as either responses from optimal reference response locations in amultidimensional transmissibility measurement [8] or independent virtual inputs. Once theprincipal components or independent virtual inputs are available, the output spectralmatrix SYY can be decomposed as follows:

SYY ¼ Sx1

YY þ Sx2

YY þ � � � þ SxrYY ð6Þ

where Sx1

YY ;Sx2

YY ; . . . ; SxrYY are spectral matrices corresponding to the principal compo-

nents. Since the rank of each decomposed matrix is now one, the conventionaltransmissibility approach can be applied as follows:

Sxkij ¼

SixkSxkxk

S�jxk: ð7Þ

Cross-spectrums between the principal components and the responses in equation (7) canalso be obtained from cross-spectrums between the whole responses and referenceresponses and unitary matrix as follows:

SYx ¼1

qY�nT ¼

1

qY�ðRTUÞ ¼ SYRU: ð8Þ

A schematic diagram of the proposed approach is presented in Fig. 1. The determinationof p or how many principal components to take into consideration in the decompositiongiven by equation (6) in practice is another problem as mentioned above.

3.2.2. Determination of the number of principal components

The number of principal components of the reference output spectral matrix istheoretically equal to the number of non-zero singular values. Since it seldom happens,however, that singular values become perfectly zeroes in real situations, it is not a matterof course to determine the rank of the matrix from singular value calculations. One of themajor reasons for the non-zero singular values in the analysis of the output spectral matrixis suspected to be the measurement and computational noise. Therefore, the number of theprincipal components can be determined by ignoring singular values smaller than a givenvalue related to the noise level. In this section, issues of determination of the criterion toobtain the number of principal components are dealt with by assuming that themeasurement noise in the output signals raises such problems.

Let *SSYY ¼ SYY þ SNN be the output spectral matrix contaminated by a noise spectralmatrix SNN . The measurement noise at the responses is assumed to be mutuallyuncorrelated with each other and, further, to have the same auto-power spectral density.That is, the noise spectral matrix is assumed to be the identity matrix I multiplied by thepower of the noise s2

N as expressed below

SNN ¼ s2N � I: ð9Þ

Carry out the PCA on the noise-contaminated output spectral matrix *SSYY as shown inequation (10a) and construct a new output spectral matrix *SSYY by taking only the first p

Figure 1. Schematic diagram of the proposed method.

INDIRECT FORCE IDENTIFICATION VIA PCA 877

singular values as shown in equation (10b):

*SSYY ¼ USxxðUÞH

¼ Up Uðn�pÞ��� � S

pxx 0

0 Sðn�pÞxx

" #Up Uðn�pÞ��� �H

ð10aÞ

*SSYY ¼ UpSpxxðU

pÞH ð10bÞ

where Up and Uðn�pÞ denote, respectively, the first p and the remaining (n� p) columns ofthe unitary matrix U and S

pxx a p� p diagonal matrix composed of the first p singular

values. Then, an error matrix EðpÞ, which is a function of p, can be defined by projectingthe difference between the noise-contaminated but reduced output matrix *SSYY and the

M.-S. CHO AND K.-J. KIM878

noiseless output matrix SYY onto the base matrix U as follows [10]:

EðpÞ ¼ U11ð *SSYY � SYY Þ ¼ðUpÞH *SSYY

0

" #�UHSYY ¼

ðUpÞHSNN

�ðUn�pÞHSYY

" #: ð11Þ

In equation (11), the zero submatrix, 0, results from the singular value truncation of theoutput spectral matrix and ðUPÞH *SSYY is equal to ðUPÞH *SSYY due to the orthogonality ofunitary singular vector. Taking the Frobenius norm of the error matrix EðpÞ yields

EðpÞj jj jF ¼Xpi¼1

UHi SNN � SHNNUi þ

Xni¼pþ1

UHi SYY � SHYYUi

!1=2

ð12aÞ

¼Xpi¼1

UHi SNN � SHNNUi �

Xpi¼1

UHi SYY � SHYYUi þ

Xni¼1

UHi SYY � SHYYUi

!1=2

ð12bÞ

where Ui is the ith column of matrix U. Since the last term in the parentheses of equation(12b) is independent of p, a necessary condition for minimising the Frobenius norm is thatp be chosen such that the second term in equation (12b) may be greater than the first.Although this requirement need not be a necessary and sufficient condition for minimisingthe Frobenius norm, it can be at least a guide for determining the rank of the outputspectral matrix. By substituting equation (9) into the requirement, it can be stated by thefollowing equation that

s4NU

Hi Uj5UH

i SYYSHYYUi ði ¼ 1; . . . ; pÞ: ð13Þ

The ith singular value of the noise-contaminated output spectral matrix can be representedby

li ¼ UHi*SSYYUi ð14aÞ

and its square can be represented as follows:

l2i ¼ UH

i*SSYY *SS

H

YYUi ¼ UHt SYYS

HYY þ ðSYY þ SHYY Þ � s

2N þ s4

N � I �

Ut

¼ UHi SYYS

HYY þ ð *SSYY þ *SS

H

YY Þ � s2N � s4

N � I�

Ui

¼ UHi SYYS

HYYUj þ 2li � s2

N � s4N �UH

i Ui: ð14bÞ

Then, by substituting equation (13) into equation (14b) the requirement can be rewrittenas

l2i > 2lis2

N ð15aÞ

or

li > 2s2N : ð15bÞ

That is, the number of principal components p can be obtained in such a way that thesingular values smaller than twice the noise power level may be discarded. In order toapply this technique the noise power level should be provided, which is another difficultproblem to solve in general because it is not known in practice. In Appendix A, a methodto estimate the noise power level under some assumptions is presented, which will beemployed in this study.

INDIRECT FORCE IDENTIFICATION VIA PCA 879

4. A SIMULATION STUDY

A simulation study was made to show the validity of the proposed technique by taking apinned–pinned Euler beam as a sample structure whose physical characteristics anddynamic properties are shown in Fig. 2 and Table 1. The responses were assumed to bemeasurable from 19 equally spaced points and the frequency range of interest for the forceestimation was chosen as 0–30Hz where the first four modes were included.

The beam was excited at three points 6, 10, 14 with two independent random sources u1,and u2; f6 ¼ u1, f10 ¼ 0:4u1 þ 0:6u2, f14 ¼ u2 and the responses were generated at 19 pointsby superimposing the first ten modes. That is, the responses were obtained using the(19� 3) FRF matrix as follows:

Hðf Þ ffiX10r¼1

Ur&UUT

r

f 2r � f 2 þ j2zrfrfð16Þ

where Ur ¼ Fr120

�Fr

220

�� � �Fr

1920

�� �Tand &UUr ¼ Fr

620

�Fr

1020

�Fr

1420

�� �T; respectively, de-

note the mode shape vector at the output points and the input points calculated from the

mode shape function FrðxÞ ¼ ð1=ffiffiffiffiffiffi2p

pÞ sinðrpx=LÞ. Then, contamination of the output

power spectrum was done by adding white noise onto the signals as follows:

*SSYY ¼ SYY þ SYYj jmin�u � a � I ð17Þ

where u represents a random variable with uniform distribution between 0 and 1 and a anattenuation factor which was selected as 0.05 for the illustration here. The dimension ofthe identity matrix I is the same as the one of SYY and SYYj jmin denotes the minimum of theresponse auto-spectrums.

The relative error in the force identification is defined using Frobenius norm of the errormatrix as below

e ¼SFF � #SSFF

��� ������ ���F

SFFj jj jF� 100ð%Þ ð18Þ

Figure 2. Test structure: pinned–pinned uniform Euler beam with proportional viscous damping. Massdensity rA ¼ 1; bending stiffness EIðxÞ ¼ 1; modal damping ratio zr ¼ 0:005; beam length L ¼ 1; undampednatural frequency fr ¼ pr2=2; mode shape function FrðxÞ ¼ ð1=

ffiffiffiffiffiffi2p

pÞ sinðrpx=LÞ.

Table 1

Natural frequencies of pinned–pinned Euler beam in simulation study

Mode 1 2 3 4 5 6 7 8 9 10

Nat. freq. (Hz) 1.57 6.28 14.1 25.1 39.3 56.5 77.0 100.5 127.2 157.1Freq. range of interest 0–30Hz

Figure 3. Singular values of reference spectral matrix and number of principal components: (a) Singular valuesfor reference signals without noise; (b) singular values for reference signals with 5% noise and (c) number ofprincipal components obtained by the criterion in Section 3.2.2.

M.-S. CHO AND K.-J. KIM880

where SFF is the true input spectral matrix and #SSFF its estimation from the responses. Thenumber of measurement channels which can be used simultaneously was chosen as eightand both the conventional transmissibility and the new transmissibility approach based

Figure 4. Comparison of identified force error by transmissibility and PCA approach: }, trans.; ——, PCA.

INDIRECT FORCE IDENTIFICATION VIA PCA 881

upon PCA were applied to the 19 output responses. In the conventional approach,the results obtained by choosing point 3 as the reference point are presented herebecause this point yielded minimum error in the estimation of the input spectral matrix.In the PCA approach, four points 3, 8, 10, 17 were chosen as the reference pointsbecause the magnitudes of the FRFs in the frequency range of interests werelargest.

Singular values obtained from the four reference signals when there is no noise at theoutput measurements are shown in Fig. 3(a), where it can be seen that the number ofprincipal components can be determined easily as two by visual observation over most ofthe frequency of interest. Singular values for noise-contaminated spectral matrix areshown in Figure 3(b), where it can be seen that it is not so easy to determine the number ofprincipal components just by visual observation. Results for the number of the principalcomponents determined by the proposed technique are shown in Figure 3(c), where thenumber of principal components is two over most of the frequency range of interest exceptnear resonant frequencies. The reason for the malestimation near the resonant frequenciesis that only one mode is predominant although the number of independent sources is two.

Errors in the identification of input forces by the conventional transmissibility functionapproach and the PCA approach are shown in Fig. 4, where it can be seen that theconventional technique is very unsuccessful while the proposed PCA technique issuccessful except near the resonance frequencies.

5. CONCLUSION

The objective of this study was to propose a method to reduce the estimation errors inthe indirect force identification in multi-source environments by applying the PCAapproach and verify the proposed method by a numerical example. The conclusionsderived therefrom can be summarised as follows.

The conventional transmissibility approach is not applicable to cases where the numberof independent excitation sources is greater than one. A new approach based upon thePCA can resolve the problems caused by the multiple source environments and, hence, itcan be usefully applied to cases where the number of simultaneously usable channels islimited. The proposed technique is believed to be as cost-effective as the conventional

M.-S. CHO AND K.-J. KIM882

technique in an environment where the independent sources are multiple becauseresponses from many points can be dealt with by a limited number of measurementchannels.

REFERENCES

1. F. D. Bartlett and W. G. Flannelly 1979 Journal of American Helicopter Society 24, 10–18.Model verification of force determination for measuring vibration loads.

2. N. Giansante, R. Jones and N. J. Calapodas 1982 Journal of American Helicopter Society 27,58–64. Determination of in-flight helicopter loads.

3. M. Mauri 1983 Proceedings of the International Modal Analysis Conference, 586–590. True iceforce by deconvolution.

4. N. Okubo, S. Tanabe and T. Tatsuno 1985 Proceedings of the International Modal AnalysisConference, 685–690. Identification of forces generated by a machine under operating condition.

5. O. Dossing 1988 Sound and Vibration, 18–26. Structural stroboscopy}measurement ofoperational deflection shapes.

6. K. V. Mardia, J. T. Kent and J. M. Bibby 1979 Multivariate Analysis, pp. 213–246. London,New York: Academic Press.

7. J. Leuridan, P. Van de Ponseele and D. Otte 1990 Proceedings of the International ModalAnalysis Conference, 413–421. Operational deflection shapes in multisource environments.

8. S. D. Tucker and H. Vold 1992 Proceedings of the International Modal Analysis Conference,1217–1220. A practical way of using principal response analysis to generate operating shapes.

9. J. S. Bendat and Piersol adsa Random Random Data.10. J. N. Juang and R. S. Pappa 1986 Journal of Guidance, Control and Dynamics 9, 294–303.

Effects of noise on modal parameters identified by the eigensystem realization algorithm.

APPENDIX A: ESTIMATION OF OUTPUT MEASUREMENTNOISE LEVEL

In actual measurements, it is not at all easy to differentiate the noise from the signal.Yet, the technique proposed in the paper requires the noise power level at the responses inorder to set up a criterion for the singular value cut-off. In this appendix, a procedure toestimate the output noise power level under some assumptions from the measured FRFsand the contaminated output spectral matrix is presented.

The measurement noise is assumed to be uncorrelated with each other as well as thesignals and have the same auto-power spectra1 density. Then, the output spectral matrixcan be written as follows:

*SSYY ¼ HHSFFHþ SNN ¼ HHSFFHþ s2N � I: ðA1Þ

Equation (A1) can be rewritten as

SNN ¼ *SSYY � SYY ¼ *SSYY �HHSFFH

¼ *SSYY �HH #SSFFHþHHð #SSFF � SFF ÞH ðA2Þ

where #SSFF is the input spectral matrix estimated from the contaminated output spectralmatrix *SSYY as described by

#SSFF ¼ ðHHÞþ *SSYY ðHÞþ: ðA3Þ

By substituting equation (A1) into equation (A3) as below

#SSFF ¼ ðHHÞþ HHSFFHþ s2N � I

�ðHÞþ

¼ SFF þ s2NðH

HÞþðHÞþ ðA4Þ

INDIRECT FORCE IDENTIFICATION VIA PCA 883

the error in the force identification can be obtained as

#SSFF � SFF ¼ s2NðH

HÞþðHÞþ: ðA5Þ

Substituting equation (A5) into equation (A2) and using the SVD of the FRF matrix H asgiven below

Hm�n ¼ Vm�mStt�nWHn�n ðA6Þ

the following relationship can be derived:

In � ðWmÞðWmÞH �

� s2N � I ¼ *SSYY �HH #SSFFH: ðA7Þ

In equation (A7), Vm is a unitary matrix composed of the first m columns of the unitarymatrix V satisfying the following relation:

WmðWmÞH þWn�mðWn�mÞH ¼ In ðA8Þ

where In is the n� n identity matrix. Therefore, equation (A7) can be rewritten by usingequation (A8) as follows:

Wn�mðWn�mÞH � s2N � I ¼ *SSYY �HH #SSFFH: ðA9Þ

Taking the Frobenius norm of each side of equation (A9), the measurement noise level canbe finally estimated as below

s2Nðf Þ ¼

1ffiffiffiffiffiffiffiffiffiffiffiffin�m

p *SSyy �HH #SSffH��� ������ ���

FðA10Þ

which shows that one can estimate the measurement noise level from the noisy outputspectral matrix and the input spectral matrix is estimated from it.