A Modal Identification Algorithm Combining Blind Source Separation and State Space Realization

Transcription

1 Journal of Signal and Information Proessing, 03, 4, Published Online May 03 ( 73 A Modal Identifiation Algorithm Combining Blind Soure Separation and State Spae Realization Sot MNeill Stress Engineering Servies, In., Houston, USA. sot.mneill@stress.om Reeived January 3 rd, 03; revised February 8 th, 03; aepted Marh 9 th, 03 Copyright 03 Sot MNeill. This is an open aess artile distributed under the Creative Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. ABSTRACT A modal identifiation algorithm is developed, ombining tehniques from Seond Order Blind Soure Separation (SOBSS) and State Spae Realization (SSR) theory. In this hybrid algorithm, a set of orrelation matries is generated using time-shifted, analyti data and assembled into several Hankel matries. Dissimilar left and right matries are found, whih diagonalize the set of nonhermetian Hankel matries. The omplex-valued modal matrix is obtained from this deomposition. The modal responses, modal auto-orrelation funtions and disrete-time plant matrix (in state spae modal form) are subsequently identified. System eigenvalues are omputed from the plant matrix to obtain the natural frequenies and modal frations of ritial damping. Joint Approximate Diagonalization (JAD) of the Hankel matries enables the under determined (more modes than sensors) problem to be effetively treated without restritions on the number of sensors required. Beause the analyti signal is used, the redundant omplex onjugate pairs are eliminated, reduing the system order (number of modes) to be identified half. This enables smaller Hankel matrix sizes and redued omputational effort. The modal auto-orrelation funtions provide an expedient means of sreening out spurious omputational modes or modes orresponding to noise soures, eliminating the need for a onsisteny diagram. In addition, the redution in the number of modes enables the modal responses to be identified when there are at least as many sensors as independent (not inluding onjugate pairs) modes. A further benefit of the algorithm is that identifiation of dissimilar left and right diagonalizers prelude the need for windowing of the analyti data. The effetiveness of the new modal identifiation method is demonstrated using vibration data from a 6 DOF simulation, 4-story building simulation and the Heritage ourt tower building. Keywords: Modal Identifiation; Blind Soure Separation; State Spae Realization; Analyti Signal; Complex Modes. Introdution An introdution to blind soure separation applied to modal identifiation is provided in this setion, along with a review of previous developments in the literature... Blind Soure Separation Applied to Modal Identifiation In the last several years researh has evolved the appliation of Blind Soure Separation (BSS) tehniques to solve the Modal ID entifiation (MID) problem. BSS J attempts to find soure omponents, s j t C, with presribed properties embedded in measured data I x i t C. The tehniques appliable for modal identifiation assume that the measured data is a linear mixture (as opposed to onvolutive) of the omponents. Suppose that there are I hannels of measured data and J omponents. The relation between the omponents and the measured data an be written as I I t J J t x A s () where A is the (onstant) mixing matrix and the dimensions have been plaed parenthetially in the supersript. Note that all quantities are generally omplex-valued. The objetive of BSS is to simultaneously estimate the mixing matrix, A, and the vetor of omponents, s t, from the observed data, x t. Due to the number of variables involved, this task requires a haraterization of t the soure omponents, s. Many BSS tehniques use seond order statistial information (e.g. orrelation struture) to desribe the omponents. It is appropriate to onsider the inverse relationship of (), J JI t I t. s W x ()

2 74 A Modal Identifiation Algorithm Combining Blind Soure Separation and State Spae Realization The de-mixing matrix, W, is the (generalized, if neessary) inverse of the mixing matrix, A. The BSS task is to estimate A and s t. Assume that the rank of A is given by rank A min I, J. This an easily be satisfied by judiious hoie of sensor loations. If I J, then () is a fully determined problem for the soures, s. t If I J, then () is an over determined problem. Similarly, if I J, then () is an under determined problem. Most BSS algorithms only treat the fully determined or over determined ases. The potential appliation of BSS on vibration data is fairly obvious. In linear vibration, the physial responses, x t, are onstruted from the modal responses, q t, by multipliation of the modal matrix, Φ, x t Φq. t One might onsider using BSS to estimate both the (inverse) modal matrix and the modal responses, q t Φ x t. (4) The onnetion with Equations () and () is immediate. The deomposition, one ahieved, allows for estimation of natural frequeny and damping ratios using simple SDOF methods applied to qj t. Note that the modal matrix and modal responses are real valued if the topology of the damping matrix is restrited (e.g. proportional damping) and are otherwise omplex-valued... Disussion of Previous Developments Early attempts at using BSS for MID are simply applied BSS methods, suh as the Algorithm for Multiple Unknown Signal Extration (AMUSE) [] and Seond Order Blind Identifiation (SOBI) [] algorithms, diretly to measured vibration data [3-5]. However, little insight was provided regarding the relationship between the soure omponents identified by suh algorithms to modal responses. In addition, the methods suggested were not apable of identifying omplex-valued modes. This presents a problem for appliation to measured data as most physial strutures and mahinery exhibit omplexvalued modes. Reently, a framework for Blind Modal Identifiation (BMID) was developed to perform MID using BSS methods [6-9]. It was first shown that the SOBI algorithm is suitable for the modal identifiation task for systems with real-valued modes (diagonally damped systems). The SOBI algorithm finds soure omponents in measured physial response data that are unorrelated irrespetive of time shifts relative to one another. Speifially, the SOBI algorithm finds omponents that approximately produe diagonal time shifted orrelation matries. A omplex-valued version of SOBI is now disussed, though the algorithm was originally pre- (3) sented for the real-valued mixing model. Consider the time shifted orrelation matrix of observed data x t, H Rxx τ E x t x t. (5) Here τ is a time shift, H, represents the Hermetian (omplex onjugate) transpose and x t has been entralized by removing the mean from eah hannel. SOBI employs Joint Approximate Diagonalization (JAD) to find the demixing matrix, W A, that approximately diagonalizes several time shifted orrelation matries with time shift, k,,, K, k xx τ H diagonal, WR W k. (6) k This problem an be solved by minimizing the off di- H agonal terms of WRx kw for the K orrelation matries, K k H min off WR W. (7) W Joint approximate diagonalization was first solved in two steps: ) prewhitening the data by transforming with a unitary matrix found from Prinipal Component Analysis (PCA); ) applying the final rotation matrix obtained by applying Givens rotations on the data orrelation matrix [0]. The mixing matrix was then onstruted as the produt of two unitary matries. Several single-step algorithms were later developed (e.g., [,] and the referenes therein). By appealing to the orrelation struture of the modal t, the soure omponents, s t responses, q, where shown to orrespond to the modal responses and the mixing matrix, A, was shown to orrespond to the system mode shapes,. Namely, the modal responses were stated to have nearly diagonal time-shifted orrelation matries. Damping and finite data length was shown to result in nonzero off-diagonals, ausing the orre- xx q t and t spondene between s to be approximate, resulting in error in the mode shapes and modal responses. This error was mitigated by introduing a windowing tehnique. A Gaussian window is applied to the data set before applying SOBI. Windowing the data auses the windowed modal responses to have diagonal orrelation matries, improving the quality of the estimated mixing matrix [6,7]. A further objetive of BMID is to extend SOBI to systems with omplex-valued modes (non-diagonally damped systems), allowing for estimation of omplex-valued mode shapes, natural frequenies and modal damping. The omplex-valued mode shapes an be expressed as, k i, (8) r where i is the imaginary unit, and the subsripts r and i indiate the real and imaginary parts, respetively. This i

3 A Modal Identifiation Algorithm Combining Blind Soure Separation and State Spae Realization 75 was aomplished simply by applying SOBI on an expanded data set. The measured data set is augmented with its Hilbert transform to yield the double-sized mixing problem, xr t r iqr t ˆ ˆ. r t (9) x i r qr t In (9) the ˆ symbol indiates the Hilbert transform. Note that though Equation (9) ontains the imaginary part of the omplex modes, the imaginary unit is not present. Therefore all the terms are real-valued and (9) may be solved using a real-valued implementation of SOBI. Consider the analyti signals assoiated with the measured data and the modal responses, x t xr t ixi t xr t iˆ xr t, (0) q t q t iq t q t iqˆ t. r i r r In [9] it was shown that the omplex mixing problem, x t q t, () x t ixˆ t i q t iqˆ t, r r r i r r is equivalent to the double-sized problem (9). It was also noted that solving () instead of (9) using a omplexvalued implementation of SOBI avoids a post-proessing step, known as the pairing routine, needed to math the modal responses to their Hilbert transform pairs. In addition, it was suggested in referene [9] that a single-step JAD routine an be used to solve (), and thus avoid the whitening step ommon in many of the early JAD algorithms. A omplex-valued version of the high-performane Weighted Exhaustive Diagonalization with Gauss iterations (WEDGE) algorithm [-3], was used to solve for the omplex mode shapes and modal responses. Many of the obstales involved in applying BSS for MID were overome by the proposed BMID framework. However one of the remaining limitations of the method is that the under determined ase, where the number of sensors is less than the number of modes present in the measured data I J, annot be solved. In pratie, this issue is often addressed by band-pass filtering the measured data into two or more bandwidths where I J and applying BMID on eah of the bands, separately. It is desirable from a time and effort standpoint to minimize the amount of bands neessary. In addition allowane for identifiation of more soures (modal responses) than sensors an allow for better quality modal parameters, as more noise modes an be identified and separated from the strutural modes. Most of the methods thus far proposed to extend BSS to the underdetermined ase exploited a tehnique from multilinear algebra, known as Parallel Fator analysis (PARAFAC) [4-6]. The PARAFAC algorithm em- ployed in these methods was that of [7], due to its onvergene properties. In [4,5], PARAFAC was applied to the measured data diretly for estimation of real and omplex modes, respetively. The Blind Modal Identifiation for under determined data (BMIDUD) algorithm was developed by the author [6]. The algorithm applies PARAFAC to the analyti data set, omposed of the measured data and orresponding Hilbert transform. After applying BMIDUD to two sets of simulated data, performane of the BMIDUD algorithm was observed to be fairly good in the underdetermined ase. However, two shortomings an be noted. First, the number of modes identified is limited by, J J I I. () The maximum number of uniquely identifiable soures for several values of I is listed in aording to Table. Seond, some residual mixing was evident in the modal auto-orrelation funtions. The first issue an be quite restritive when the number of sensors is small. The seond issue may result in error in modal parameter estimates. Another approah to the underdetermined ase is to assemble a large Hankel matrix from the data orrelation matries, as first suggested in [8]. This method does not suffer from the identifiability ondition of Equation (). In addition, the modal auto-orrelation funtions are easily obtained in a post-proessing step. The method was extended in [9], where methods from state-spae realization were ombined with SOBI for improved modal identifiation. It was also shown that when dissimilar left and right diagonalizers are applied to the Hankel matrix, the orret mode shapes an be obtained without the need for windowing the data. In a related result, it was proven that the off diagonals of the modal response orrelation are nonzero unless the system is onservative (i.e., does not have damping), as surmised in [6,7]. In the remainder of this paper, the Blind Modal Identifiation using Hankel Matries (BMIDHM) algorithm is developed. JAD is applied on a set of Hankel orrelation matries, generated from analyti data to estimate the state-spae system and modal parameters. The algorithm is applied to two sets of simulated output-only data and one set of measured ambient response data. The first data set was obtained from a simulated 6 DOF massspring-dashpot system with non-diagonal damping, exited by random noise. The seond and third data sets represent ambient building vibration from simulation and real-world measurement, respetively. In eah ase, a Table. Maximum number of soures. I J max

4 76 A Modal Identifiation Algorithm Combining Blind Soure Separation and State Spae Realization minimal number of sensors is used to identify as many modes as possible. The method is similar to that of [9], with the exeption that the analyti signal is exploited. This leads to a redution of the system order (to be identified) by a fator of two, reduing the required Hankel matrix size.. State Spae System Preliminaries Theoretial bakground on seond-order and state-spae systems is provided in this setion. Consider the familiar seond order Equation of Motion (EOM) for a linear system, Mx t Cx t Kx t B f t (3) J Here x t R is the displaement vetor for all L DOF, f t R is the fore vetor, B is the fore influene matrix mapping the fore vetor to physial oordinates, M is the positive definite mass matrix, C is the positive semi-definite damping matrix and K is the positive semi-definite stiffness matrix. Matries M, C and K are real valued and symmetri. Beause K is positive semi-definite in general, the struture will have rigid body modes. However the rigid body modes and responses are generally not of interest in the modal identifiation problem. Therefore we will only onsider osillatory responses. The seond order EOM an be reast into a ontinuous-time state-spae model, z t Az t B f t (4) y t C z t D f t ν t, T T T J where t t t R I t t z x x is the state vetor, I y R is the output vetor and ν R represents J J measurement noise. The plant, A R, input influ- J L ene, B R I J output influene, C R,and di- I L ret feedthrough, D R, matries are given by: 0 I 0 A,, B M K M C M B C Hd 0Hv 0A Ha 0 A,or (5) C Hd HM a K Hv HM a C, D H M B. a The permutation matries H d, H v and H a, ontain rows with a single unit element with all other elements null, suh that they pik-off the measured DOF. As an example, for aeleration sensing at all DOF, Hd Hv0, and H I. a The set of differential equations an be deoupled us- J J ing the transformation, z t t, where, C is the matrix of eigenvetors of the plant matrix and q q T H T J t t t C is the modal state vetor. The matrix takes the form,, (6) J J where C is the matrix of omplex-valued sys- J J tem mode shapes, C is the diagonal matrix of independent (not inluding onjugate pairs) plant matrix eigen values, and the symbol represents the omplex onjugate. The diagonal system is given by: t t B f t t t t y C D f 0 0.,, where (7) The ontinuous-time state-spae system an be transformed into a disrete-time system with sampling interval Δt by: Ad exp At, Cd C, Dd D, t (8) B exp A B d A I A B. d 0 d The transformation results in the disrete-time system, k Ad kbd f k, (9) y k C k D f k ν k. d d The set of differene equations an be deoupled through the transformation z k k, yielding, k k Bd f k, ykcd kdd f k, where (0) expt 0. 0 exp t It was shown in [8,9] that for τ suffiiently large (ompared to the deay rate of the fore and noise orrelation funtions), the orrelation matries take the form, H Ryy L R, () R C, LC R 0, d d as terms involving the random input fore and measurement noise dissipate quikly with inreasing time shift, τ. The matrix R 0 is the orrelation of the state-spae modal responses (at zero time shift). Referene [9] provided the proof that R 0 is diagonal if and only if the system is onservative, i.e., there is no system damping. The relative magnitude of the off diagonal terms an be signifiant when damping is present. It an also be seen from () that nondiagonality of R 0 auses R yy to be nonhermetian for 0. Equation () an be ompared diretly to the JAD problem solved in SOBI (6). In SOBI, it is assumed that

5 A Modal Identifiation Algorithm Combining Blind Soure Separation and State Spae Realization 77 the data orrelation matries, R yy, are Hermetian for all τ and, therefore, the right diagonalizing matrix fator is the Hermetian transpose of the left. In fat the nonhermetian part of the orrelation matries is disarded before appliation of Hermetian JAD. This leads to errors in the estimated mode shapes and modal responses when SOBI is applied. Windowing the data prior to appliation of SOBI alleviates this, as the orrelation matries of the windowed data are nearly Hermetian and the windowed modal orrelation matrix is nearly diagonal, [6]. Diagonalization of the nonwindowed orrelation matries, R yy, is preferred but requires separate left and right matrix fators. 3. Additional Preliminaries In this setion, additional developments are provided that set the stage for the BMIDHM algorithm. 3.. Uniqueness of the Nonhermetian Diagonalization Diagonalization of nonhermetian orrelation matries with separate left and right matrix fators, as in (), is essential to estimate aurate modal parameters, while avoiding windowing the data set. However, it is known that suh a diagonalization of a single orrelation matrix is not unique. Consider, for example, that a single orrelation matrix an be diagonalized by left and right singular vetors or left and right eigenvetors. Further to this point, an alternate set of left and right matrix fators an be derived that diagonalize R yy to the set of arbitrary diagonal matries, H τ L LN, R RM,where N M. () Here M is an arbitrary invertible matrix and the diagonalization holds when N an be found satisfying the presribed relationship between N and M. If we onsider a single value of, =, N an easily be obtained. However for multiple values of τ, a matrix N annot generally be found that satisfies the above equation for all τ. Therefore the deomposition is unique for all pratial purposes. 3.. Analyti State-Spae Form In this setion the analyti state-spae form will be derived onsidering aeleration sensing. Forms for displaement and veloity sensing are similar. The output equation in (0) with aeleration sensing an be written as, y k k k q H a H M B f kν q k a. (3) If the redundant onjugate pairs are removed the first term beomes, H a qk. Realling that the modal responses an be written as analyti signals, [9], and onstruting the analyti signals of output, fore and measurement noise, the analyti state-spae output equation is arrived at, k k k y H a q M B f ν k. (4) In (4), a tilde has been plaed over the modal responses, q k, to remind the reader that they are analyti signals. It should be noted that though the modal responses are analyti signals for free response and impulse response ases, they are not stritly analyti signals in the general fored response ase. In the ase of random exitation however, modal responses are analyti up to a random additive term, whih does not affet the ross-orrelation funtions. The true modal responses an then be substituted by their orresponding analyti signals for modal identifiation purposes. The state equation an also be written with redundant onjugate pairs removed, yielding the analyti state equation, q k q k Bd f k,where, JJ C : J,: Jexp t,and, JJ C : J,: J,i.e., (5) : J,: J,and, J J J :, :. In Equation (5), Matlab notation is used for row and olumn partitions of a matrix. It an be seen from (5) that the analyti state-spae system order is J, while representing the same dynamis as the original state-spae system of (0) with order J. This proves advantageous for modal identifiation, as the desired system order is redued by half, requiring smaller Hankel matrix size to identify the desired number of modes. The analyti, disrete-time, state-spae form an be summarized as, q k q k Bd f k, ykcanqk HaM B f kνk,where, Can Hd displaement sensing, (6) Can Hv veloity sensing, C H aeleration sensing. an a Note that in all ases, Can onsists of row partitions of saled mode shapes. (Reall from properties of eigenvetors that mode shapes saled by arbitrary omplexvalued onstant are still mode shapes.) The orrelation funtions for suffiiently large τ an be written as,

6 78 A Modal Identifiation Algorithm Combining Blind Soure Separation and State Spae Realization H Ryy L R,where, RC, LC R 0. an an qq (7) In (7), the right and left matrix fators, R and L respetively, have been redefined. 4. The BMIDHM Algorithm The BMIDHM algorithm is derived in this setion. The analyti signal of measured responses is first onstruted by adding the Hilbert transform as the imaginary part, y t y t iˆ y t. (8) r IPIP A set of K Hankel matries, yy C, is assembled, whereby the blok elements are the orrelation matries of the analyti response data, yy Ryy Ryy Ryy P P Ryy Ryy Ryy. Ryy P Ryy P Ryy P (9) Here, τ is the time shift operator and P is the number of blok rows and olumns in the Hankel matrix. Nonhermetian JAD is applied to the set of Hankel matries, yielding the deomposition, L yy L P L H L R,. yy r P R R R (30) The fast-onverging nonhermetian JAD algorithm of [0] was hosen for the deomposition. The algorithm results in the matries L and R diretly, as opposed to the algorithm suggested in [9], whih results in their generalized inverses, neessitating an additional inversion step. Note that dimensions of L, and R are IP J, J J, and IP J, respetively, where J is the system order, J IP. The mode shapes (C an matrix) are obtained as the first J rows of R. In pratie, the system order may be asertained by omparing the magnitudes of the diagonal elements of. It was shown that by deomposing the Hankel matries, up to IP omplex modes an be identified. Beause the analyti state-spae form is used, the redundant omplex onjugate modes are not inluded in the dynamis and are not identified in the deomposition. Therefore, all IP modes represent independent system modes (no onjugate pairs), as opposed to the method of [9], where they represent the independent modes and the omplex-onjugate pairs. Summarizing, a neessary ondition for identifying all J modes in the proposed methodology is IP J, whereas the ondition of [9] is IP J. The approximately diagonal plant matries an be reovered for any τ by inverting (30), H L R (3) yy. A set of T samples of the modal auto-orrelation funtions an then be taken from the diagonal elements of, Rqq Rqq 0, τ,,, T. (3) Sine only the diagonal elements of R qq are de- sired, and is diagonal, the matrix R qq 0 an be omitted without loss of generality. This is due to the fat that the diagonal elements of Rqq are simply equal to the diagonal elements of saled by the diagonal elements of R qq 0. Therefore the modal auto-orrelation funtions an be obtained by, r diag R diag,,,,. (33) q qq T Samples of the modal responses an be estimated by, q k R Y, (34) T T T T where Y y k y k y kp. The estimates will ontain random noise due to negleting the random fore (if sensors are near input loations) and measurement noise terms in (34). If P is large enough, however, random fore and noise terms average out. A least squares estimate of the plant matrix,, an be obtained by, P: T J,: J PJ : TJ,: J,where, (35) H P τ J : J,: J L τ R. yy Note that the estimated disrete-time plant and output influene matries will approximately be in state-spae modal oordinates, i.e., is approximately diagonal. Disrete time eigenvalues an be obtained as the eigenvalues of, and, an be onverted to ontinuous time eigen-values by, diag t ln eig. (36) Here, eig is the operator that omputes the eigen-values of a matrix. Natural frequenies, ω n, and modal damping ratios, ζ j, an be reovered through the relation,, j j j nj i nj j. Alternatively, the natural frequenies and modal damp-

7 A Modal Identifiation Algorithm Combining Blind Soure Separation and State Spae Realization 79 ing ratios an be omputed from the modal orrelation funtions using simple SDOF methods. 5. Appliation to Simulation and Measured Data In this setion, results from appliation of the BMIDHM algorithm to simulated and measured data are presented. In eah ase the number of sensors utilized in the data set is kept low to evaluate the apabilities of the method when applied to the under determined problem. 5.. Simulated 6 DOF System with Two Sensors In order to investigate the effetiveness of the BMIDHM method, a six DOF mass-spring-dashpot system with non-diagonal damping was onstruted. Three of the six DOF were hosen as sensor loations I 3. The system was exited by unorrelated, Gaussian-random, white noise applied to eah of the six DOF. White noise was added to the response data with RMS equal to 0% of the maximum RMS among the response DOF. Figure illustrates the displaement response along with the Fast Fourier Transform (FFT) of the displaement at eah of the sensor loations. The BMIDHM algorithm was applied to the measured data, seleting J 6. The modal parameters (natural frequenies, modal damping ratios and mode shapes) of the six identified modes are ompared to modal parameters of the six analytial modes in Table. Mode shapes are ompared using a orrelation oeffiient known as the Modal Assurane Criterion (MAC), []. The MAC value between two modes and is given by, MAC, H. (37) A value near indiates a good shape orrelation. A value near 0 indiates poor orrelation. It an be seen that natural frequeny and mode shape estimates ompare extremely well, while the estimated modal damping ratios exhibit slightly more error. The modal auto-orrelation funtions and their FFTs, whih are the power spetral densities, are shown in Figures and 3. The shape observed in both the time domain and frequeny domain indiates that the modes are ompletely isolated. 5.. Simulated 4-Story Building Vibration A model of a four story building was reated by the University of British Columbia (UBC) as a benhmark for damage detetion. In this setion, modal identifiation of the undamaged struture with asymmetri floor mass is onsidered. The analytial model of a four story building mokup was modeled with Euler frame elements. The Figure. 6 DOF simulation-measured responses. Table. 6 DOF simulation-modal parameters. Analytial BMIDHM Mode Freq. Damp. Freq. Damp. (Hz) (%) (Hz) (%) MAC Figure. 6 DOF simulation-estimated modal response ovariane funtions - 3. FEM model with node numbering is shown in Figure 4. The floors in the mokup are fairly rigid. Notie that

8 80 A Modal Identifiation Algorithm Combining Blind Soure Separation and State Spae Realization Figure 3. 6 DOF simulation-estimated modal response ovariane funtions 4-6. Z Y Figure 4. UBC building-fem model. the FEM model does not inlude expliit models for eah floor. Instead, a rigidizing onstraint was applied to the stiffness matrix as an ad-ho step before simulation. For simulation, % diagonal-damping is applied in the modal domain. The resulting model ontained 0 DOF. Note that the modeling and simulation was done in a MAT- LAB pakage provided by the UBC. The struture was exited in the transverse diretion by a simulated diagonally oriented shaker at the top enter of the building. Gaussian random exitation, filtered between 0-00 Hz was applied for 50 seonds. The time interval for integration was 0.00 seonds. Aelerations were measured at 6 loations. A pair of planar aelerometers was simulated on eah floor, adjaent to X the enter olumns by outputting x and y aelerations at nearby nodes. Measurement noise was simulated by adding Gaussian random noise to eah measurement. The RMS of the noise was seleted as 0% of the RMS of the measurement with the highest response level. A plot of the Complex Mode Indiator Funtion (CMIF) is shown in Figure 5. The influene of modes on the response shape an be seen as well as the additive measurement noise. Note that the first two modes are losely spaed modes (~9 Hz) representing the first sway modes in the x and y diretions. The poorly exited third mode (~4 Hz) is the first torsional mode. The influene of many frequenies on the response shape an be seen as well as the additive measurement noise. Due to these harateristis, this data set presents a hallenge to the BMIDHM algorithm. Data from five of the 6 available sensors were hosen as the data set I 5. Modal identifiation was performed on the response data (ignoring the input fore data) using BMIDHM, seleting J 3. Two noise modes were separated out based on the poor spetral and temporal shape of the identified modal responses and low ontribution to the vibration response. The modal parameters of the remaining modes are ompared to modal parameters of the analytial modes in Table 3. Note that the frequeny CMIF Frequeny [Hz] Figure 5. UBC building simulation-first CMIF. Table 3. UBC building simulation-modal parameters. Analytial BMIDHM Mode Freq. Damp. Freq. Damp. (Hz) (%) (Hz) (%) MAC

9 A Modal Identifiation Algorithm Combining Blind Soure Separation and State Spae Realization 8 and damping results for mode 3 were obtained by SDOF fitting methods using data around the peaks of the modal spetra. It an be seen that natural frequeny and mode shape estimates ompare well, but estimated modal damping ratios tend to have more error. Modal response auto-orrelation funtions and their FFTs are shown in Figures 6-8. The shape observed in both the time domain and frequeny domain indiates that the modes are ompletely isolated for all modes exept for the third. A peak in the spetrum is visible for the poorly-exited third mode at the orret frequeny of 4.35 Hz, however another large peak is present as well as a high noise level above 30 Hz. Bandpass filtering the data from 0-0 Hz results in improved spetral and temporal shape (Figure 9) Heritage Court Tower Building The struture analyzed in this setion is the Heritage Court Tower (HCT) building. The desription of the building, provided in [] is summarized here. The HCT is loated in downtown Vanouver, British Columbia, Canada. It is a 5-story reinfored onrete shear ore building. The building is essentially retangular in shape with only small projetions and setbaks. The building tower is short and stout, having a height to width aspet ratio of approximately.7 in the east-west diretion and Figure 7. UBC building simulation-estimated modal response ovariane funtions 5-8. Figure 8. UBC building simulation-estimated modal response ovariane funtions 9 -. Figure 6. UBC building simulation-estimated modal response ovariane funtions in the north-south diretion. An overview of the building is presented in Figure 0. Typial floor dimensions of the upper floors are about 5 meters by 3 me-

10 8 A Modal Identifiation Algorithm Combining Blind Soure Separation and State Spae Realization Figure 9. UBC building simulation-estimated modal response ovariane funtions 3 (Filtered). Figure 0. Photograph of Heritage Court Tower (HCT) building. ters, while the dimensions of the lower three levels are about 36 meters by 30 meters. The footprint of the building below ground level is about 4 meters by 36 meters. The height of the first story is 4.7 meters, while most of the other levels have heights of.7 meters. Elevator and stairs orridors are onentrated at the enter ore of the building and form the main lateral resisting elements against potential wind and seismi lateral and torsional fores. The tower struture sits on top of four levels of reinfored onrete underground parking. A parking struture extends approximately 4 meters beyond the tower in the south diretion forming an L-shape with the tower. The parking struture and first floors of the tower are fairly flush on the remaining three sides. Beause the building sits to the north side of the underground parking struture, oupling of the torsional and lateral modes of vibration was expeted primarily in the EW diretion. Four sets of ambient response data were olleted with eight aelerometers. Six of the aelerometers were roving transduers that hanged loation for eah data set. Two referene aelerometers remained in a fixed loation. This enables the mode shapes omputed from the four data sets to be ombined into one modal matrix. The data sets ontained around 35 seonds of data, sampled at 40 Hz. The third data set was analyzed using BMIDHM as well as the state-spae realization method, known as ovariane-driven Stohasti Subspae Identifiation (SSI). Covariane-driven SSI is essentially an appliation of ERA on ross-orrelation funtions instead of impulse response funtions. More details of the SSI algorithm an be found in [3]. Channels, 4, 7 and 8 were hosen as referene hannels for SSI. The BMIDHM and SSI methods were applied on the ambient responses diretly. The first two Complex Mode Indiator Funtions (CMIFs) are shown in Figure. From the CMIF plot, it an be seen that the first three modes are losely spaed. In fat the first two modes (~.5 Hz) are so losely spaed that they are nearly the same frequeny. As a result, the first CMIF does not exhibit two distint peaks. However the presene of the seond mode at ~.5 Hz is indiated by the peak in the seond CMIF. There is a sharp peak in the first CMIF around 8.7 Hz. This is most likely harmoni noise due to a mehanial devie suh as a motor, fan or pump. Also notie that above 0 Hz, there are no distint peaks visible. This data set presents a hallenge to the BMID method due to the small number of sensors, presene of losely spaed modes and the presene of noise. In order to evaluate the apabilities of BMIDHM for the underdetermined problem, the frequeny band from CMIF Frequeny [Hz] Figure. HCT building The first two CMIFs.

11 A Modal Identifiation Algorithm Combining Blind Soure Separation and State Spae Realization Hz is foused on, only three of the eight available sensors are hosen I 3. All eight sensors are used in the SSI algorithm so that results from BMIDHM an be ompared to an aurate set of modal parameters. Modal identifiation was performed on the ambient response data using BMIDHM, seleting J 0. Three noise modes were disarded based on the poor spetral and temporal shape of the identified modal responses and low ontribution to the vibration response. The SSI method was performed with several different Toeplitz matrix sizes, resulting in several realizations. Consisteny diagrams were formed for eah filter band, and the best realization was hosen. The onsisteny diagrams an be found in [8]. Modal parameters from the two methods are presented in Table 4. Modal parameters from BMIDHM applied using three sensors ompare quite will to those obtained using SSI applied on all eight sensors, though damping error is signifiant for some modes. Modal auto-orrelation funtions from the BMIDHM method are shown in Figures and 3. Note that the time sale is different in the two figures. The FFTs of the random modal responses omputed using (34) are provided in Figure 4. The modal response autoorrelation funtions exhibit lean, monotone deay and the spetral peaks are seen to be quite well-separated in the modal responses and modal auto-orrelations. Figure. ht building-estimated modal response ovariane funtions Conlusions Methods employed in state-spae realization theory and seond order blind soure separation were ombined, formulating an new modal identifiation algorithm. The algorithm is applied on measured vibration (output-only) data and results in estimates of the system: omplex-valued modal matrix; modal responses and auto-orrelation funtions (even in the underdetermined ase); state-spae disrete-time plant, output-influene and observability matries in modal form; natural frequenies and frations of modal damping. Table 4. HCT building simulation-modal parameters. SSI BMIDHM Mode Freq. Damp. Freq. Damp. (Hz) (%) (Hz) (%) MAC Figure 3. HCT building-estimated modal response ovariane funtions 4-7. Compared to the first pratial adaptation of BSS to modal identifiation, [6], several improvements an be ited: The whitening, windowing and pairing routines are

12 84 A Modal Identifiation Algorithm Combining Blind Soure Separation and State Spae Realization Figure 4. HCT building-fft of estimated modal responses. preluded. Appliation to the underdetermined problem (more independent modes than sensors) is effetively handled, resulting in high-quality modal parameters, modal auto-orrelation funtions and modal responses with few sensors. The state-spae plant matrix and orresponding system eigenvalues are readily available. REFERENCES [] L. Tong, V. C. Soon, Y. Huang and R. Liu, AMUSE: A New Blind Identifiation Algorithm, Proeedings of IEEE International Symposium on Ciruits and Systems, New Orleans, -3 May 990, pp doi:0.09/iscas [] A. Belouhrani, K. Abed-Meraim, J.-F. Cardoso and E. Moulines, A Blind Soure Separation Tehnique Using Seond-Order Statistis, IEEE Transations on Signal Proessing, Vol. 45, No., 997, pp doi:0.09/ [3] S. Chauhan, R. J. Allemang, R. Martell and D. L. Brown, Appliation of Independent Component Analysis and Blind Soure Separation Tehniques to Operational Modal Analysis, Proeedings of the 5th International Modal Analysis Conferene, Orlando, 9- February 007. [4] F. Ponelet, G. Kershen, J.-C. Golinval and D. Verhelst, Output-Only Modal Analysis Using Blind Soure Separation Tehniques, Mehanial Systems and Signal Proessing, Vol., No. 6, 007, pp doi:0.06/j.ymssp [5] W. Zhou and D. Chelidze, Blind Soure Separation Based Vibration Mode Identifiation, Proeedings of the 5th International Modal Analysis Conferene, Orlando, 9- February 007, pp [6] S. I. MNeill and D. C. Zimmerman, A Framework for Blind Modal Identifiation Using Joint Approximate Diagonalization, Mehanial Systems and Signal Proessing, Vol., No. 7, 007, pp doi:0.06/j.ymssp [7] S. I. MNeill, Modal Identifiation Using Blind Soure Separation Tehniques, Ph.D. Dissertation, University of Houston, Houston, 007. [8] S. I. MNeill and D. C. Zimmerman, Blind Modal Identifiation Applied to Output-Only Building Vibration, Proeedings of the 8th International Modal Analysis Conferene, Jaksonville, 00. [9] S. I. MNeill, An Analyti Formulation for Blind Modal Identifiation, Journal of Vibration and Control, Vol. 8, No. 4, 0, pp. -. doi:0.77/ [0] A. Belouhrani, K. Abed-Meraim, J.-F. Cardoso and E. Moulines, A Blind Soure Separation Tehnique Using Seond-Order Statistis, IEEE Transations on Signal Proessing, Vol. 45, No., pp doi:0.09/ [] P. Tihavsky and A. Yeredor, Fast Approximate Joint Diagonalization Inorporating Weight Matries, IEEE Transations of Signal Proessing, Vol. 57. No. 3, 009, pp doi:0.09/tsp [] P. Tihavsky, A. Yeredor and J. Nielsen, A Fast Approximate Joint Diagonalization Algorithm Using a Criterion with a Blok Diagonal Weight Matrix, Proeedings of the 008 International Conferene on Aoustis, Speeh and Signal Proessing, Las Vegas, 3 Marh-4 April 008, pp [3] P. Tihavsky, UWEDGE_C Complex Version (Version for Complex-Valued Matries and Complex-Valued Mixing) of the Algorithm UWEDGE, [4] F. Abazarsa, S. Ghahari, F. Nateghi and E. Tairoglu, Response-Only Modal Identifiation of Strutures Using Limited Sensors, Strutural Control and Health Monitoring, Vol. 0, No. 6, 03, pp doi:0.00/st.53 [5] F. Abazarsa, S. Ghahari, F. Nateghi and E. Tairoglu, Blind Modal Identifiation of Non-Classially Damped Systems from Free or ambient Vibration Reords, Earthquake Spetra, in Press. [6] S. I. MNeill, Extending Blind Modal Identifiation to the Underdetermined Case for Ambient Vibration, Proeedings of the ASME 0 International Mehanial Engineering Congress & Exposition, Houston, 9-5 November 0. [7] L. De Lathauwer and J. Castaing, Blind Identifiation of Underdetermined Mixtures by Simultaneous Matrix Diagonalization, IEEE Transations on Signal Proessing, Vol. 56, No. 3, 008, pp doi:0.09/tsp [8] J. Antoni and S. Chauhan, Seond Order Blind Soure Separation Tehniques (SO-BSS) and Their Relation to Stohasti Subspae Identifiation (SSI) Algorithm, Proeedings of the 8th International Modal Analysis Conferene (IMAC), 00. [9] J. Antoni and S. Chauhan, A Study and Extension of Seond-Order Blind Soure Separation to Operational Modal Analysis, Journal of Sound and Vibration, Vol. 33, No. 4, 03, pp doi:0.06/j.jsv

13 A Modal Identifiation Algorithm Combining Blind Soure Separation and State Spae Realization 85 [0] A.-J. van der Veen, Joint Diagonalization via Subspae Fitting Tehniques, International Conferene on Aoustis, Speeh, and Signal Proessing ICASSP, Salt Lake City, 7- May 00, pp [] R. J. Allemang and D. L. Brown, A Correlation Coeffiient for Modal Vetor Analysis, Proeedings of the st International Modal Analysis Conferene, Orlando, 8-0 November 98, pp [] C. Ventura, R. Brinker, E. Dasotte and P. Anderson FEM Updating of the Heritage Court Building Struture, Proeedings of the 9th International Modal Analysis Conferene, Kissimmee, 5-8 February, 00, pp [3] B. Peeters, System Identifiation and Damage Detetion in Civil Engineering, Ph.D. Dissertation, Katholike Universite Leuven, Belgium, 000.