Generalized Eigenvalue Decomposition in Time Domain Modal Parameter Identification

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1 Wenliang Zhou 1 Lang Mekra Nth America, LLC, 101 Tillessen Bloulevard, Ridgeway, SC zhouwl@gmail.com David Chelidze Department of Mechanical Engineering & Applied Mechanics, University of Rhode Isl, Kingston, RI chelidze@egr.uri.edu Generalized Eigenvalue Decomposition in Time Domain Modal Parameter Identification This paper is intended to point out the relationship among current time domain modal analysis methods by employing generalized eigenvalue decomposition. Ibrahim time domain (ITD), least-squares complex exponential (LSCE) eigensystem realization algithm (ERA) methods are reviewed chosen to do the comparison. Refmulation to their iginal fms shows these three methods can all be attributed to a generalized eigenvalue problem with different matrix pairs. With this general fmat, we can see that single-input multioutput (SIMO) methods can easily be extended to multi-input multioutput (MIMO) cases by taking advantage of a generalized Hankel matrix a generalized Toeplitz matrix. DOI: / Cresponding auth. Contributed by the Technical Committee on Vibration Sound of ASME f publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received January, 007; final manuscript received July 8, 007; published online November 1, 007. Review conducted by Jean Zu. Paper presented at the 006 ASME International Mechanical Engineering Congress IMECE006, November 5 10, 006, Chicago, IL. 1 Introduction A fundamental problem in experimental modal analysis is to extract modal parameters of dynamical structures. These parameters include natural frequencies, damping ratios, mode shapes. With this infmation, we can validate update theetical finite element models gain a better understing of cresponding structures. Another widely used application of these modal parameters is to monit the health of a structure. During past decades, a vast range of modal parameter identification methods has been developed. These can be divided into two main categies: frequency domain time domain methods. Frequency domain methods tend to provide me accurate results when the der of modes are relatively low, but they suffer specific problems associated with the fast Fourier transfm FFT analysis, such as the leakage. In addition, frequency domain methods often need detailed input infmation to calculate frequency response functions. Time domain methods manipulate signals in time domain usually need only output responses. This property makes time domain methods advantageous in practical tests. In addition, time domain methods are able to differentiate two modes that are close to each other. This paper is concerned with the time domain methods is intended to provide a unifm fmulation based on generalized eigenvalue decomposition. Based on the number of input output channels used in tests, modal analysis algithms can be classified as single-input singleoutput SISO, e.g., the complex exponential CE method 1, single-input multioutput SIMO methods, e.g., the least-squares complex exponential LSCE method, Ibrahim time domain ITD method 3 6, the multi-input multioutput MIMO methods e.g., polyreference complex exponential PRCE method 7, eigensystem realization algithm ERA 8,9, stochastic subspace-based modal analysis methods CE, LSCE, PRCE are all based on the Prony s they fm a natural extension from SISO to SIMO then to MIMO. The CE method 1 uses a square output matrix to estimate the coefficients of the Prony s polynomial is sensitive to noise. LSCE was introduced by Brown et al. in Its fmulation is similar to the CE except that the displacement matrix is constructed by a multichannel output signal, pseudoinverse technique is employed to estimate the coefficients of the Prony s polynomials. F both methods, a pri singular value decomposition can significantly reduce the effect of noise. In 198, Vold et al.7 further extended LSCE to a MIMO case. The PRCE 7 method overcomes the problem associated with the SIMO method, when one of the modes may not be present in the output responses. Ibrahim has published a series of papers since 1973 developed the so-called ITD method. The iginal paper started with the state space description used the output displacement to reconstruct the governing equations of structures 3. Later, a me general fm was given, which was based on the explicit mathematical fm of the output response 4. In this new algithm, the output signal can be displacement, velocity, acceleration. The modal confidence fact 5 was also introduced to identify the real modes from the spurious computational modes. A detailed examination on the perfmance of ITD was given in Ref. 6. Two me sophisticated time domain modal parameter identification algithms are the eigensystem realization algithm ERA 8 stochastic subspace-based modal analysis methods Both were developed from control they. Using a dynamical system in its state space fm, ERA provides a minimum der realization of the system matrix by using free decay responses. ERA method also provides modal amplitude coherence modal phase collinearity to quantitatively identify system noisy modes. A detailed evaluation of the effect of noise on ERA was given in Ref. 9. The stochastic subspace-based modal analysis uses the they of stochastic subspace realization with an assumption that inputs are white noise. A good review of the popular modal parameter identification methods can be found in Ref. 1. A unified matrix polynomial fmulation of the current modal identification methods is given in Ref. 13. In what follows, three popular time domain modal parameter identification methods LSCE, ITD, ERA are chosen to explicitly show the underlying relationship among these methods. Original derivations of these methods are presented briefly followed by refmulation based on generalized eigenvalue decomposition. This refmulation can be readily used to extend SISO modal analysis algithms to SIMO MIMO cases. The discussion conclusion are given at the end. Time Domain Modal Analysis Methods.1 Ibrahim Time Domain (ITD) Method. The ITD method is examined first to provide a unifm fmat by generalized eigenvalue decomposition. ITD was introduced in In that iginal paper, the auths used state variables with a length of time sequence, where n is the degree of freedom of a system, to Journal of Vibration Acoustics Copyright 008 by ASME FEBRUARY 008, Vol. 130 /

2 identify the system matrix modal parameters. This method relies on the exact fmat of system s governing equation, the time signal should be clean displacement. In a later paper 4, a me general algithm independent of the state space construction was proposed. The algithm allows the analytical signals to be displacement, velocity, acceleration measurements. The success of this updated algithm relies on the fact that output signals have an exponentially decaying fm. The governing equations of motion f an n-degree-of-freedom free vibration system can be written as Mẍ + Cẋ + Kx = 0 1 where M, C, K are nn mass, damping, stiffness matrices, ẍ, ẋ, x represent n-dimensional acceleration, velocity, displacement vects. The solution to these equations can be expressed in the fm of x i t = ik e s k t, i =1,...,n x i t = i e st where i = i1, i,..., i, s=s 1,s,...,s T, t represents scalar time. i will constitute the desired mode shape matrix. F a lightly damped system, s k = k nk + j dk provides the natural frequency the damping ratio. nk, dk, k represent kth undamped, damped natural frequencies, damping ratio. j is the imaginary unit. Suppose time responses from n senss are sampled over a time period equal to t, where t is the sampling time interval. Then from Eq., we have x1t0 x1t1 x1t 1 x t 0 x t 1 x t 1 x n t 0 x n t 1 x n t =11 n1 n nes1t0 es1t1 es1t 1 e s t 0 e s t 1 e s t e s t 0 e s t 1 e s 1 t X = e st 3 where XR n is the trajecty matrix composed by n-channel output signals, = 1,,..., n T, t=t 0,t 1,...,t 1. Then we consider the same time responses shifted by one sampling time interval here one sampling time is chosen just f convenience x1t1 x1t x1t x t 1 x t x t x n t 1 x n t x n t =11 n1 n nes1t0 es1t1 es1t 1 e s t 0 e s t 1 e s t e s t 0 e s t 1 e s 1 t X = e st e, where =es1t t s t e s X R n is the one sampling time shifted trajecty matrix. R is a diagonal matrix with the entries equal to e skt. Similarly, another time-shifted response matrix with respect to X can be fmed X = e st Equations 3 5 can be combined as follows: X X = e st X = e st 6 5 X X = e st Xˆ = e st 7 Equations 6 7 can be manipulated to eliminating e st get Ā = Ā = Xˆ X 1 8 where ĀR is called the system matrix, containing the infmation of the dynamical system. R is the complex mode shape matrix. Solving the eigenvalue problem of Eq. 8 provides us with all modal parameters. The above are the maj steps in ITD analysis. F the time periods longer than, Eq. 8 can be rewritten as Xˆ X T X X T 1 = Me generally, we consider the joint diagnolization of matrix R 1 R R 1 = X X T = e st e st T T R = X Xˆ T = e st e st T T 10 Then, Eq. 9 can be rewritten using generalized eigenvalue decomposition as R 1 T = R T 1 11 From Eq. 11, we can see the inverse transpose of the generalized eigenvect matrix from matrix pair R 1,R provides the complex mode shapes the inverse of the generalized eigenvalues contains the natural frequencies damping ratios. In summary, the procedures f perfming ITD by generalized eigenvalue decomposition are as follows: construct X Xˆ as befe assemble two matrices R 1 =X X T R =X Xˆ T perfm generalized eigenvalue decomposition to the matrix pair R 1,R extract modal parameters from the generalized eigenvalues eigenvects It is noted that the smooth thogonal decomposition SOD based modal analysis method 14 can be looked as another version of ITD, where the differentiated matrix instead the time shifted matrix is used. However, the SOD-based method can wk directly with an n-dimensional data matrix.. Least-Squares Complex Exponential Method. Now ITD will be used as the baseline, efft will be made to refmulate LSCE to a similar fm. LSCE PRCE 7 constitute an extension of CE from SISO to SIMO then the MIMO case. All of them are based on the Prony s they. Here, LSCE is chosen to compare to ITD because both belong to the SIMO case. To better present the relationship, CE is discussed first. CE, LSCE, PRCE start with the impulse response function IRF, which has the same fm as Eq.. F CE, assuming the time responses are sampled at +1 stances, one gets / Vol. 130, FEBRUARY 008 Transactions of the ASME

3 x i t 0 = ik ; x i t 1 = ik e s k t ; ;x i t = ik e s k t just x i t 0 = ik ; x i t 1 = ik V k ; ;x i t = ik V k 1 13 with V k =e s kt. To solve the unknowns V k ik, we use the Prony s method, which states if V k appear in complex conjugate pairs then there exists a polynomial in V k of der l here l= with real coefficients, such that the following equation holds: V k + V k + + V k =0 14 Here, the solutions to V k provide the modal parameters in complex conjugate pairs. Then we multiply both sides of Eq. 13 with 0,..., sum them together j x i t j = j ik V k j = ik j V k j 15 Substituting Eq. 15 into Eq. 14, weget j x i t j =0 16 To solve Eq. 16, we put a single time response x i into the matrix fm as follows: xit0 xit1 xit x i t 1 x i t x i t +1 x i t 1 x i t x i t 4n 1 0 Setting =1, Eq. 17 transfms to xit0 xit1 xit 1 x i t 1 x i t x i t x i t 1 x i t x i t 4n = 1 1= xit x i t +1 x i t 4n X i = xˇ i 19 X i R, xˇ ir the subscript i means the matrices are constructed from a single output response. Equation 18 Eq. 19 can be used to solve f. Then substituting into Eq. 14 can lead to the roots of the polynomial, therefe, the natural frequencies damping ratios. The modal residues ik are solvable by rewriting Eq. 13 as V 1 V V i x i t i1 V 1 V V i=xit0 0 x i t 1 The mode shapes can be obtained by combining the modal residues calculated from different sampling senss. The above are the basic ideas of CE. The extension from CE to LSCE is quite straightfward. Instead of using a single output response, LSCE estimates the coefficients in Eq. 14 by using several output time histies, the mode shape matrix is obtainable in one step. F each output time histy, we have equations similar to Eq. 18 Eq. 19, then we assemble p output responses into one matrix as X1 1 X xˇ = xˇ X p p 1 xˇ is then obtained by using pseudoinverse technique. The mode shapes calculation follows the similar idea as CE, but using several output responses, V 1 V V 1 V 1 V V V 1 V V = x1t0 xt0 xt0 x 1 t 1 x t 1 x t 1 x 1 t 1 x t 1 x t 1 in matrix fmat where = 1 V V T = X 1 1 V 1 V V V 1 V V 1 1 V 1 V 1 V 1 1 =11 T 1 1 = x1t0 xt0 xt0 x 1 t 1 x t 1 x t 1 X x 1 t 1 x t 1 x t 1 3 Here, provides the complex mode shape matrix. Equations 1 3 are the fundamental steps of LSCE. CE LSCE use the Prony s technique to extract the modal parameters, whereas ITD achieves this by an eigenvalue decomposition a generalized eigenvalue decomposition. At a first glance, they look different. However, further transfm of CE LSCE shows both methods are essentially equal to ITD. First, let us consider the CE case. Equation 18 is used to calculate the coefficients, which is used f solving the polynomial roots of Eq. 14. If we add me time instants to the righth side of Eq. 18 as xit xit xit x i t +1 x i t x i t 3 x i t +1 x i t 4n 1 xit1 4 x i t x i t +1 x i t 4n 1 then Eq. 18 becomes: Journal of Vibration Acoustics FEBRUARY 008, Vol. 130 /

4 where X i ß=X i ß T X i = X i, ß= = xit0 xit1 xit 1 x i t 1 x i t x i t X i x i t 1 x i t x i t 4n xit xit x i t x i t 3 x i t +1 X i =xit1 x i t x i t +1 x i t 4n 1 Here, ß ß T is called the companion matrix of the polynomial pu pu = u + u u 1 + u 7 where u is the polynomial variable. The roots of pu can be found by calculating the eigenvalue decomposition of its companion matrix, it is usually superi to the conventional root searching methods. With companion matrix ß, the computation of Eq. 5 then solving V k can be accomplished in a single step by doing an eigenvalue decomposition ß T i = i X i X i 1 i = i 8 where is a diagonal matrix with its entries as e s it. The subscript i in i is used to differentiate i from mode shape matrix since this is a single time output case. As can be seen, Eq. 8 is quite similar to Eq. 8. Again f general output signals, Eq. 8 is expressed as X i X i T X i X i T 1 i = i 9 Written in a generalized eigenvalue decomposition fm, Eq. 9 becomes X i X T i T i = X i X T i T i 1 30 Equations 8 9 look quite similar to Eq. 9 except here matrix X i, X i, i are composed of single output response. To further examine the meaning of matrix i T, we consider Eq. 0 add me time instants to the right-h side. Equation 0 becomes We also have 1 V1 V1 1 1 V =1 T V V 1 V V i1v1 iv iv i1 V 1 i V 3 3 i1 V 1 i V i1 V 1 i V i V 3 i V 1 = Xi VT i V iv T V 0 0 = X i where =V V Equations can be combined as X i X i 1 i = i 35 Examining Eqs. 35 8, we can see i= i. From Eq. 3, we have V 1 V V 0 i 0 i 1i1 0 0 i= V 1 1 V 1 V 36 Therefe, i is constructed by the multiplication of model residues with cresponding poles. Now CE has been fmulated into a generalized eigenvalue decomposition fm, which is quite similar to ITD. However, ITD was developed initially f SIMO case it is me appropriate to relate LSCE with ITD. In fact, when multioutput time histies are used, the relationship between LSCE ITD is even clearer. Starting with the Eq. 1, which aims at estimating the matrix ß T by using several output responses, here f an n-degree-of-freedom system, we assume we have n output channels. Instead of shifting a single time output 1 times as CE, we can construct a matrix by assembling the iginal n-channel output histy with its one sampling time shifted output histy. Now, Eq. 5 becomes X 1 ß T X 3 ß T X = Xˆ 37 Here we use different row der from the iginal LSCE since it will not affect the estimation of ß T. From Eq. 37, we can estimate ß T calculate its eigenvalue decomposition to obtain modal parameters. This can be written in a generalized eigenvalue decomposition fm X = X where iv T = X i = i1 i i i1 V 1 i V i V i i1 V 1 i V i V 1 1 i1 V 1 i V i V X X T = X Xˆ T T 1 which looks exactly the same as ITD Eigensystem Realization Algithm. ERA was developed by Juang Pappa in 1985 from system realization viewpoint 8. It is a MIMO algithm that can be used f modal parameter identification model reduction of dynamical systems. The first step of this algithm is to fmulate a generalized Hankel matrix, which contains the Markov parameters. Then the realization matrices, which can reproduce system s input-output relationship, are derived. Modal parameters are extracted from the realized system matrices. The detailed derivation of ERA can be / Vol. 130, FEBRUARY 008 Transactions of the ASME

5 found in Ref. 8. Here, only maj steps of the algithm are listed used to construct a relationship with ITD. ERA starts with the state-variable description of Eq. 1 xj +1 = Axj + Bfj yj = Cxj 39a 39b where xr is a state variable vect assuming an n-degree-of-freedom system, yr p is an observation vect, A R is a state transition matrix characterizing the dynamics of the system, BR m is an input matrix, CR p is an output matrix, fr m is a control input vect, j is the sampling index. It is assumed that the system is observed by n senss y has a dimension of p=n. The objective of ERA is to recover the three matrices A,B,C, such that y is reproducible from Eq. 39. F the free impulse response, the Markov parameters Yj are expressed from control they as Yj = CA j 1 B 40 where Yj has a dimension of nl l is the dimension of input channels. In practice, it is constructed by columnwise concatenation of observation vects resulting from l different inputs Yj = y 1 j,y j,...,y l j 41 The ERA procedure can be summarized as follows: Fm a generalized Hankel matrix of dimension rnsl, = Yj Yj +1 Yj + s Yj +1 Yj + Yj + s +1 Hj 1 4 Yj + r Yj + r +1 Yj + r + s where r s are some kind of numbers, which need to be defined optimally, Calculate singular value decomposition of the generalized Hankel matrix H0 H0 = PDJ T 43 Examine the nonzero flo of the singular values usually it is just the dimension of the phase space: in this case truncate the matrices J P by keeping only their first columns. Construct nrn E n lsl E l matrices E n T = I n E l T = I l where I l R ll I n R nn are identity matrices. Estimate the realization matrices as A = D 1/ J T H1JD 1/, B = D 1/ J T E l, C = E n T PD 1/ Calculate all modal parameters using the estimated system matrices A C. As a result of the above procedure, the eigenvalues of A provide natural frequencies damping ratios. Assume the eigenvect matrix of A is, then the mode shape matrix is obtained using following transfmation: = C = E n T PD 1/ 46 Equation 45 constitutes a minimum realization f the given system. After examining the singular values of H0 truncating P J matrices, A=D 1/ P T H1JD 1/ has a rank of. From control they, a nonsingular linear transfmation T of the state vect xj leads to a transfmed realization triple, such as TAT 1,TB,CT 1, which is also a realization of the given dynamical system. TAT 1 is called similarity transfmation, which will not change the eigenvalues of the matrix A. To relate ERA to ITD, matrices H0 H1 are considered after the singular value decomposition, which means both have the number of rows equal to the dimension of the system here. Using a nonsingular matrix T=PD 1/, the similarity transfmation of A is given by TAT 1 = PD 1/ D 1/ P T H1JD 1/ D 1/ P 1 = PP T H1JD 1 P 1 = H1H0 47 where denotes a pseudoinverse, P J are unitary matrices from singular value decomposition as mentioned above. Thus, P 1 =P T J =J T. Natural frequencies damping ratios can be extracted from the eigenvalue decomposition of the transfmed matrix TAT 1 H1H0. To identify mode shapes, we first look at the matrix C multiplied by T 1 : CT 1 = E n T PD 1/ D 1/ P T = E n T 48 Therefe, from Eqs , we know the eigenvect matrix of TAT 1 H1H0 directly provides mode shapes. In summary, ERA in the SIMO case can be fmulated as the following eigenvalue problem: H1H0 =, in a generalized eigenvalue decomposition fm, H0H0 T T = H0H1 T T Equation 50 shows that ERA can also be realized by generalized eigenvalue decomposition, which is quite similar to the ones used f ITD LSCE. 3 Discussion Conclusion In addition to ERA, stochastic subspace-based modal parameter extraction methods 10 1 are also developed from a control they viewpoint. Similar to ERA, stochastic subspace methods aim to identify state space model Eq. 39 modal parameters using only the output response. These methods assume the input is a stochastic process white noise, while ERA is based on free impulse responses. Although the generalized Hankel matrix is used in ERA, stochastic subspace-based methods start with the generalized Toeplitz matrix. Various stochastic subspace-based modal parameter extraction algithms exist, such as a datadriven based algithm 11, covariance-based algithm 10, etc. The other steps in the stochastic subspace modal parameter extraction procedures are quite similar to the ERA, described in detail in Sec..3. Thus, if in the ERA derivation the Hankel matrix is replaced by the Toeplitz matrix, the stochastic subspace methods can also be refmulated into a generalized eigenvalue problem. In summary, this paper examines current popular time domain modal analysis methods explicitly refmulates them by use of generalized eigenvalue decomposition. As shown in the above discussion, ITD, LSCE, ERA are essentially similar if fmulated by eigenvalue decomposition, me generally, generalized eigenvalue decomposition. With this unifm fm, we can see the matrix X i in CE can be replaced by X as in ITD, therefe generalizing CE to the SIMO case. Also in ITD, X can be replaced by H as in ERA, this generalizes ITD from SIMO to the MIMO case. In addition, the generalized eigenvalue decomposition fmalism makes it clear that the crucial point f extracting modal parameters in time domain analysis is to simultaneously diagonalize two matrices, one of which is the crelation matrix of the output signal itself, the other is the cross-crelation matrix with its time-shifted output signal its differentiated matrix as Ref. 14. Me imptantly, this simultaneous diagonalization Journal of Vibration Acoustics FEBRUARY 008, Vol. 130 /

6 makes the time domain modal analysis methods similar to some algithms in BSS 15,16 since both employ this simultaneous diagonalization procedure 17. Acknowledgment This paper is based on a wk suppted by the NSF Grant No. CMS References 1 Mendes, M., Silva, J. M., 1997, Theetical Experimental Modal Analysis, Research Studies Press, Cambridge, UK. Brown, D. L., Allemang, R. J., Zimmerman, R., Mergeay, M., 1979, Parameter Estimation Techniques f Modal Analysis, SAE Technical Paper Series, 7901, pp Ibrahim, S. R., Mikulcik, E. C., 1973, A Time Domain Modal Vibration Test Technique, Shock Vib. Bull., 434, pp Ibrahim, S. R., Mikulcik, E. C., 1977, A Method f the Direct Identification of Vibration Parameters From the Free Response, Shock Vib. Bull., 474, pp Ibrahim, S. R., 1978, Modal Confidence Fact in Vibration Testing, Shock Vib. Bull., 481, pp Pappa, S. R., Ibrahim, S. R., 1981, A Parameter Study of the Ibrahim Time Domain Identification Algithm, Shock Vib. Bull., 513, pp Vold, H., Kundrat, J., Rocklin, G. T., Russel, R., 198, A Multi-Input Modal Estimation Algithm f Mini-Computers, SAE Technical Paper Series, 80194, pp Juang, J. N., Pappa, R. S., 1985, An Eigensystem Realization Algithm f Modal Parameter Identification Model Reduction, J. Guid. Control Dyn., 85, pp Juang, J. N., Pappa, R. S., 1986, Effects of Noise on Modal Parameters Identified by the Eigensystem Realization Algithm, J. Guid. Control Dyn., 93, pp Peeters, B., Roeck, G. D., 1999, Reference-Based Stochastic Subspace Identification f Output-Only Modal Analysis, Mech. Syst. Signal Process., 136, pp Overschee, P. V., Mo, B. D., 1993, Subspace Algithm f the Stochastic Identification Problem, Automatica, 93, pp Benveniste, A., Fuchs, J. J., 1985, Single Sample Modal Identification of a Nonstationary Stochastic Process, IEEE Trans. Autom. Control, 301, pp Allemang, R. J., Brown, D. L., 1998, A Unified Matrix Polynomial Approach to Modal Identification, J. Sound Vib., 113, pp Chelidze, D., Zhou, W., 006, Smooth Orthogonal Decomposition Based Vibration Mode Identification, J. Sound Vib., 93-5, pp Tong, L., Liu, R. W., Soon, V. C., Huang, Y. F., 1991, Indeterminacy Identifiability of Blind Identification, IEEE Trans. Circuits Syst., 385, pp Belouchrani, A., Abed-Meraim, K., Cardoso, J. F., Moulines, E., 1997, A Blind Source Separation Technique Using Second-Order Statistics, IEEE Trans. Signal Process., 45, pp Zhou, W., Chelidze, D., 007, Blind Source Separation Based Vibration Mode Identification, Mech. Syst. Signal Process., doi: / j.ymssp to be published / Vol. 130, FEBRUARY 008 Transactions of the ASME