EACS 2012 – 5th European Conference on Structural Control

Genoa, Italy – 18-20 June 2012 Paper No. # 036

Data Fusion for Frequency Domain Stochastic Subspace Identification Philippe MELLINGER*,1,2, Michael DÖHLER2, Laurent MEVEL2, Gabriel BROUX1 1

Dassault Aviation Quai Marcel Dassault, Saint Cloud, France [email protected], [email protected] Inria Rennes – Bretagne Atlantique Campus de Beaulieu, 35042 Rennes, France [email protected], [email protected] 2

ABSTRACT In Operational Modal Analysis (OMA) of large structures it is often needed to process output-only sensor data from multiple non-simultaneously recorded measurement setups, where some reference sensors stay fixed, while the others are moved between the setups. A standard approach to process the data together for global system identification is to transfer the data into frequency domain and merge it there. However, this only works if the unmeasured ambient excitation remains stationary throughout all setups. As the ambient excitation can be different from setup to setup, the amplitude of the measured data can be different as well and the data has to be normalized. Recently, a method has been developed for covariance- and data-driven Stochastic Subspace Identification (SSI) to automatically normalize and merge the data from multiple setups in order to obtain the global modal parameters (natural frequencies, damping ratios, mode shapes), instead of doing the SSI for each setup separately. In this paper, we adapt this approach to multi-setup SSI in frequency domain, where we use spectra data instead of time series data. We demonstrate the advantages of the new merging approach in the frequency domain and apply it to a relevant industrial large scale example, where we compare the estimation results of the modal parameters between the time and frequency domain approaches. Keywords: Stochastic subspace identification, modal analysis, frequency domain, multiple sensor setups.

1 INTRODUCTION Subspace-based system identification methods have been proven efficient for the identification of the eigenstructure of linear multi-output systems [1]-[3]. An important application of these methods is Operational Modal Analysis, where the modal parameters (frequencies, damping ratios and mode shapes) are identified for mechanical, civil or aeronautical structures subject to natural, unmeasured and nonstationary excitation [3]-[5]. To obtain vibration measurements at many coordinates of a structure with only few sensors, it is common practice to use multiple sensor setups for the measurements. For these multi-setup measurements, some of the sensors, the so-called reference sensors, stay fixed throughout all the setups, while the other sensors are moved from setup to setup. By merging in some way the corresponding data while taking into account possible different ambient excitations between the 1 *

Corresponding author

measurements, this allows to perform modal identification as if there was a large number of sensors. A global merging approach was proposed in [6],[7], where the data from the different setups is normalized and merged first, followed by a global system identification step. Recently, this approach was generalized to a large range of stochastic subspace algorithms in [8]. In this paper, this global merging approach is extended to subspace identification in the frequency domain [9]. We show the benefits of the use of frequency domain data and successfully apply the new algorithm to vibration data from Z24 Bridge which is an European benchmark described in [10]. 2 STOCHASTIC SUBSPACE IDENTIFICATION 2.1 Time Domain 2.1.1

Modeling and modal identification

The behavior of a mechanical system is assumed to be described by a stationary linear dynamical system {

̈( ) ( )

̇( ) ̈( )

( ) ̇( )

( ) ( )

(1)

where denotes continuous time, vector ( ) contains the displacements of the structure at its degrees of freedom under unknown forces ( ) , which are assumed to be white noise. are respectively mass, damping and stiffness matrices of the system. Matrices , , define acceleration, velocity and displacement sensor positions, collecting accelerations ̈ ( ), velocities ̇ ( ) and displacements ( ) at selected positions on the structure. These measurements are gathered into the output vector ( ) . The eigenstructure { obtained by solving, {

(

} of the continuous stationary linear dynamical system (1) is

) (

Using the state variable

(2)

) ( )

[

( ) ] and sampling system (1) at sampling frequency ̇( )

⁄ yields the discrete time model {

(3)

( ) and where ( ). Let be the model order. respectively the state transition matrix and the observation matrix, and input and output noise.

and

and are are respectively

The eigenstructure { } found by solving (2) can be equivalently obtained by the eigenstructure { } of the discrete model (3) which is obtained by solving

2

{

(

)

(4)

as the discrete and continuous modes and mode shapes are related by | |

Then, frequencies and damping ratios of the system are obtained by 2.1.2

and and

. ( ) . | |

Stochastic subspace identification algorithm

The generic subspace identification algorithm is based on the so-called subspace matrix , which is built from the data according to the selected subspace algorithm. is chosen such that the factorization property (5) is satisfied, where

[

]

(6)

is the pth-order observability matrix of dimension and is a matrix depending on the selected subspace algorithm. The observation matrix is extracted from the first rows of the observability matrix . The state space transition matrix is obtained by the following least squares resolution (7)

where

[

denotes the pseudo-inverse and

To obtain the state space transition matrix ( ) .

],

[

of rank , the parameter

] has to be chosen such that

In the following, the covariance-driven SSI approach [1], [3] is used, where

, [

(

)

(8)

]

and denotes the expectation operator. Then, matrix in decomposition (5) is the stochastic [ ] ( ). controllability , where Assuming centered, stationary and ergodic signals containing N samples, an unbiased estimate ̂ of is obtained by filling matrix (8) with the empirical covariances ̂ ̂



.

(9) 3

Then, an estimate of the observability matrix is obtained from the left part of the singular values decomposition (SVD) of ̂ truncated at the desired model order : ̂

[

̂

][

][

]

(10)

,

(11)

where contains the first singular values and the first columns of . The eigenstructure with the modal parameters is finally obtained from both the system matrices and as outlined above. Note that the assumption of stationary white noise excitation can be relaxed and in fact eigenstructure identification using subspace algorithm is still consistent under a non stationary excitation [1],[5],[7]. 2.2 Frequency domain 2.2.1

Modeling

Assume the number of samples large enough to describe the whole spectrum. The states, the inputs and the outputs are considered to be N-periodic to perform a discrete Fourier transformation (DFT) of the linear time-invariant model (3), resulting in the following description in frequency domain (12) (13) where

,

,

and

are respectively the DFT of the sequences {

{ } and spectral lines has no influence on matrices and . 2.2.2

}, {

}, {

} and {

} at the

. Note that model (3) is linear time-invariant and the DFT

Subspace matrix

In (8), contains the covariances of the time-domain output data. In frequency domain, these covariances have to be expressed by the respective frequency data. To use the DFT, the assumption has to be made that the signals are N-periodic, i.e. . Then, the covariances can be defined by ̂



,

(14)

as ∑

(15)

and, for the shifted signal, ∑

.

(16)

4

2.2.3

Frequency band reduction

An advantage of using the SSI algorithm in frequency domain is the ease of selecting frequency bands without the need of band-pass filters. Each spectral line l is related to its frequency value through the frequency resolution by (

)

,

(17)

where is the sampling frequency. Reducing the frequency band is equivalent to considering the spectral lines , instead of . In this way, only the elements of the spectrum are considered. This truncation has no influence on the frequency resolution . Then, the minimal and maximal frequencies are no more and but, ( ) and ( ). The reduced frequency band referring to (17), ] and its width is given by is defined by [ (

)

(

)

,

(18)

( ) is the number of selected spectral lines in the frequency band. To where implement this band-pass filter in the estimation of the subspace matrix ̂ , the covariances corresponding to (14) are defined as ̂



(

)

(19)

where is the discrete Fourier coefficient at spectral line in the reduced frequency band. The more the frequency band is reduced, the more the calculation time is sped up. However, has to be set large enough for a good estimation of the covariances. 2.2.4

Modal identification after a frequency band reduction

Because the frequency band reduction changes the discrete Fourier coefficients, the link between the discrete and the continuous domain explained in Section 2.1.1 is no longer correct. Due to the band reduction, the sampling frequency has to be replaced by the reduced sampling frequency . The obtained eigenvalues of the discrete state transition matrix are given by ⁄

.

(20)

Moreover, the spectral line on . This rotation is given by

has to be defined as first line. For doing this, a rotation is applied

, where finally obtained by

(21) ⁄ (| |)

. Continuous eigenvalues (

( )

)

5

within the reduced frequency band are

(22)

The modal frequencies and damping are then obtained by the classical formula described in Section 2.1.1. 3 MULTI-SETUP STOCHASTIC SUBSPACE IDENTIFICATION 3.1 Time and frequency domain description Instead of a single record for the output ( ) of the system (3), Ns records ( ⏟

(

)

(

)

)( ⏟

(

)

(

)

)

( ⏟

(

)

(

)

)

(23)

(

)

are now available collected successively. Each record s contains data from a fixed reference ( ) sensor pool containing ( ) sensors, and data from a moving sensor pool containing sensors. Each record s = 1, …, Ns corresponds to a state-space realization in the form [6]-[8] ( ) (

{

( )

( )

(

) ( )

)

(

)

(24)

) ( )

(

with a single state transition matrix . Note that the unmeasured excitation ( ) can be different for each measurement s as the environmental conditions can slightly change between the measurements. Note also that the observation matrix ( ) is independent of the specific measurement setup if the reference sensors are the same throughout all setups s = 1, …, Ns. Analogously to Section 2.2.1, if spectra from multi-setup measurements ( ⏟

(

)

(

)

)( ⏟

(

)

(

)

)

( ⏟

(

)

(

)

)

(25)

are available, the corresponding state-space system in frequency domain writes as ( ) (

{ (

( ) ) )

(

( ) ( )

)

(

)

(26)

( )

3.2 Merging algorithm In [6], [7] a method was described to normalize and merge data from multiple setups to obtain global modal parameters (natural frequencies, damping ratios, mode shapes). The normalization is important because the natural excitation may differ between setups. As the normalization and merging step is done first, only one system identification of the global system is necessary, instead of performing system identification of each setup separately and then merging the results. This global merging approach, which is valid for the covariance-driven SSI, was generalized in [8] to a large range of subspace algorithms. It applies to all subspace algorithms with a factorization property 6

, which is satisfied by both methods in Sections 2.1.2 and 2.2.2. Thus, it can be applied to both timeand frequency domain data from multi-setup measurements. It consists in the following steps: a) From the data of each setup s, build the matrix ( b) SVD of ( ) as in (10) to get observability matrix ( )

c) Separate into reference sensors (

( (

)

(

) ( )

)

and , where the former contains the information w.r.t. the ) ) ) and the latter w.r.t. moving sensors ( ( ).

( d) Compute the “normalized” observability matrix part ̃

where denotes the pseudoinverse ( ) ( e) Interleave the matrices and ̃ (

)

)

(

)

(

(

)

)

(

)

,

, j = 1,…,Ns, to a global observability matrix

)

) ( ) ( ) , where ( ( ) ( ) f) Do global system identification of system ({(3),(4)} or {(12),(13)}) with SSI from Sections ( ) 2.1.2 or 2.2.2 starting at Equation (8) using

Figure 1: Multi-setup system identification with merging scheme from [8]

4 NUMERICAL RESULTS 4.1 Z24 Bridge As it can be seen in Figure 2, the Z24 bridge spanned the A1 Bern-Zurich motorway in Switzerland. It consists of 9 setups with each 33 sensors, except setup 5 having 27 sensors, among which five reference sensors were fixed throughout all the measurements. The data has been sampled at with a measurement time of 655 seconds, resulting in 65,516 samples per channel. For a detailed description refer to [10]. The spectra of the first patch are displayed in Figure 3.

7

Figure 2: Z24 Bridge

Figure 3: Spectra of the first patch at [0 Hz ; 30 Hz] 4.2 Time-domain and frequency-domain modal analysis For the output-only modal analysis of the ambient vibration data of the Z24 bridge we use covariance driven subspace identification approach in both time and frequency domain to compute the subspace matrix . The time domain-based and frequency domain-based SSI are respectively called “TDSSI” and “FDSSI”. First we compare results in time and frequency domain using the same parameters . The modal parameters are obtained through stabilization diagrams, where the models are identified at orders n = 1,…,150. In Figure 4, the respective stabilization diagrams are presented, giving quite similar results for most of the identified modes.

Figure 4: Stabilization diagrams of SSI in time domain (left) and frequency domain (right). 8

From the stabilization diagrams, the stable system modes are selected. In Table 1, the results are summarized. Also, the MAC value between mode shapes obtained by FDSSI and TDSSI are stated. Table 1 : Comparison of identified modal parameters by FDSSI and TDSSI Modal parameters Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 Mode 9 Mode 10

Frequencies (Hz) FDSSI TDSSI 3.854 3.854 4.890 4.889 9.786 9.785 10.30 10.30 12.31 12.30 13.40 13.41 17.25 17.33 19.18 19.18 19.64 19.68 26.92 26.91

Damping ratios (%) FDSSI TDSSI 0.71 0.71 1.35 1.35 1.28 1.28 1.70 1.70 3.54 3.51 3.43 3.42 6.59 6.56 3.24 3.22 4.58 4.36 4.75 4.77

MAC FDSSI-TDSSI 1.00 1.00 1.00 1.00 1.00 1.00 0.92 0.99 0.99 1.00

The results of both methods are clearly similar, except for some of the higher order mode shapes. This might be due to the weak signal noise ratio that can be observed in Figure 3. The identified mode shapes from FDSSI are presented in Figure 5 and a comparison of their MAC values is done in Figure 6. All the obtained mode shapes are of high quality except the apparently low excited mode 7 [11]. Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

Mode 9

Mode 10

Figure 5: Mode shapes obtained by FDSSI.

Figure 6: MAC values between identified mode shapes by FDSSI and TDSSI. As in Table 1, one can observe in Figure 6 that the MAC diagonal reaches 1 for all the modes, while the off-diagonal values are close to 0. 9

As explained in Section 2.2, one benefit of using the system identification in frequency domain is to use some band frequency selections. This also speeds up the calculation of by allowing the reduction of both its size and the number of elements in each computed covariance, as the number of frequency samples is reduced within the frequency band [ ⁄ ]. Moreover, the selection of a specific frequency band reduces the model order, as only modes within this band are identified. Then, the parameters p and q of matrix can be reduced in the same way. The obtained results by using this reduced frequency band are referred to as Selected FDSSI. Figure 7 shows the stabilization diagram obtained by considering the frequency band [ ] with the procedure described in Sections 2.2.3-2.2.4. The system order is set as n = 1,…, 13 and p = q = 25. To observe the influence of the band reduction on the identification, one can compare this stabilization diagram with stabilization diagrams obtained by both FDSSI and ] in Figure 8. In Table 2, a comparison is made TDSSI in Figure 4, which are on [ between the values of frequencies, damping ratios and mode shapes of the first four modes that are found in this frequency band. MAC values are evaluated between the identified mode shapes by Selected FDSSI and FDSSI in Figure 9.

Figure 7: Stabilization diagram from Selected FDSSI with frequency band reduction to [3 Hz ; 11 Hz].

Figure 8: Zoom in full stabilization diagrams from SSI in frequency domain (left) and time domain (right). In these stabilization diagrams, it can be observed that the first four modes are equivalently identified in terms of frequencies by Selected FDSSI, FDSSI and TDSSI.

10

Table 2: Comparison of identified modal parameters by Selected FDSSI, FDSSI and TDSSI Modal parameters Mode 1 Mode 2 Mode 3 Mode 4

Selected FDSSI 3.864 4.879 9.790 10.34

Frequencies (Hz) Zoomed FDSSI 3.854 4.890 9.786 10.30

Zoomed TDSSI 3.854 4.889 9.785 10.30

Selected FDSSI 0.71 1.53 1.37 1.57

Damping ratios (%) Zoomed Zoomed FDSSI TDSSI 0.71 0.71 1.35 1.35 1.28 1.28 1.70 1.70

MAC Selected FSSI – Zoomed FDSSI 1.00 1.00 1.00 0.98

Figure 9: MAC values between identified mode shapes by Selected and full FDSSI. The results obtained from the reduction of the frequency band show the strong compliance with the results from FDSSI and TDSSI methods. Attention has to be paid on the risk of a sharp frequency band reduction. Typically, timedomain covariance computation requires a sufficient number of samples to be valid. This consideration must also be taken into account in the frequency domain, where it implies that frequency band of interest should be large enough with respect to the frequency resolution.

5 CONCLUSION In this paper, a time domain multi-setup subspace method was extended to frequency domain. We have shown that subspace methods in frequency domain are interesting for the reduction of the frequency band. Without any deterioration of the results, the frequency band reduction allows focusing on desired modes and reducing the system order and the size of the involved matrices. This significantly reduces the computation time of the identification. Further work includes handling different frequency information such as FRF or cross spectra and using uncertainties quantification to remove spurious modes as it is done in time domain [12]. ACKNOWLEDGEMENTS The authors of this work were partially supported by the European projects FP7-PEOPLE2009-IAPP 251515 ISMS, FP7-NMP CP-IP 213968-2 IRIS as well as the ANRT French Funding Agency. Data were collected during the European project SIMCES and provided by the SAMCO association.

11

REFERENCES [1] Benveniste, A. & Fuchs, J.J. 1985. Single sample modal identification of a non-stationary stochastic process. IEEE Transactions on Automatic Control, AC-30(1):101-109. [2] Van Overschee, P. & De Moor, B. 1996. Subspace Identification for Linear Systems, Kluwer. [3] Peeters, B. & De Roeck, G. 1999. Reference-based stochastic subspace identification for output-only modal analysis. Mechanical Systems and Signal Processing, 13(6):855-878. [4] Juang, J.-N. 1994. Applied System Identification, Prentice-Hall. [5] Benveniste, A. & Mevel, L. 2007. Nonstationary consistency of subspace methods. IEEE Transactions on Automatic Control, AC-52(6) 974-984. [6] Mevel, L., Basseville, M., Benveniste, A. & Goursat, M. 2002. Merging sensor data from multiple measurement setups for nonstationary subspace-based modal analysis. Journal of Sound and Vibration, 249(4) 719-741. [7] Mevel, L., Benveniste, A., Basseville, M. & Goursat, M. 2002. Blind subspace-based eigenstructure identification under nonstationary excitation using moving sensors. IEEE Transactions on Signal Processing, SP-50(1) 41-48. [8] Döhler, M. & Mevel, L. 2012. Modular subspace-based system identification from multi-setup measurements. IEEE Transactions on Automatic Control, under revision. [9] Cauberghe, B. 2004. Applied frequency-domain system identification in the field of experimental and operational modal analysis, Ph.D. Thesis, Vrije Universiteit Brussel. [10] Maeck, J. & De Roeck, G. 2002. Description of the Z24 Benchmark. Mechanical Systems and Signal Processing, 17(1), 127–131. [11] Döhler, M., Lam X.-B. & Mevel, L. 2010. Crystal clear data fusion in subspace system identification and damage detection. In Proceedings of the The Fifth International Conference on Bridge Maintenance, Safety and Management (IABMAS), Philadelphia, USA. [12] Döhler, M., Lam, X.-B. & Mevel, L. 2011. Uncertainty Quantification for Stochastic Subspace Identification on Multi-Setup Measurements. Proc. of 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, USA.

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