Uncertainty Quantification forEigensystem-Realization-Algorithm, A

Class of Subspace System Identification ?

Xuan-Binh Lam ∗ Laurent Mevel ∗∗

∗ INRIA, Centre Rennes - Bretagne Atlantique, 35042 Rennes, France(e-mail: xuan.lam@inria.fr).

∗∗ INRIA, Centre Rennes - Bretagne Atlantique, 35042 Rennes, France(e-mail: laurent.mevel@inria.fr)

Abstract: In Operational Modal Analysis, the modal parameters (natural frequencies, dampingratios and mode shapes), obtained from Stochastic Subspace Identification of structures, aresubject to statistical uncertainty from ambient vibration measurements. It is hence neccessaryto evaluate the confidence intervals of these obtained results. This paper will propose analgorithm that can efficiently estimate the uncertainty on modal parameters obtained from theEigensystem-Realization-Algorithm (ERA). The algorithm is validated on a relevant industrialexample.

Keywords: Operational Modal Analysis, Stochastic Subspace Identification, ErrorQuantification, Mechanical and Aerospace

1. INTRODUCTION

The design and maintenance of mechanical structuressubject to noise and vibrations is an important topic inmechanical engineering. It is an important component ofcomfort (cars and buildings) and contributes signicantlyto the safety related aspects of design and maintenance(aircrafts, aerospace vehicles and payloads, civil struc-tures). Requirements from these application areas arenumerous and demanding. Laboratory and in-operationtests are performed on the prototype structure, in orderto get so-called modal models, i.e., to extract the modesand damping factors (these correspond to system poles),the mode shapes (corresponding eigenvectors), and loads.These results are used for updating the design model for abetter fit to data, and sometimes for certification purposes(e.g., in flight domain opening for new aircrafts).

The estimation of modal parameters of structures caneasily be carried out by using Stochastic Subspace Iden-tification methods on sensor measurements. Benvenisteand Fuchs (1985) proved that the Instrumental Variablemethod and what was called the Balanced Realizationmethod for linear eigenstructure identification are consis-tent in a nonstationary context. From that on, the familyof subspace algorithms has been extensively studied (see inLarimore (1983); Van Overschee and De Moor (1996)) andhas expanded rapidly. There are a number of convergencestudies on subspace methods in the literature (see Deistleret al. (1995); Bauer and Jansson (2000); Bauer et al.(1999); Chiuso and Picci (2004)) to mention just a few ofthem. These papers provide deep and technically difficultresults including convergence rates. Our objective is toderive simple formula for such sensitivities. Sensitivities? This work was supported by the European project FP7-NMP CP-IP 213968-2 IRIS.

for the algorithms considered in this paper, ERA, are notaddressed by those papers.

The uncertainty on modal parameters appears for manyreasons, e.g. finite number of data samples, undefinedmeasurement noises, nonstationary excitations, nonlinearstructure, model order reduction,..., then the system iden-tification algorithms do not yield the exact system matri-ces. Practically, the statistical uncertainty of the obtainedmodal parameters at a chosen system order can be com-puted from the uncertainty of the system matrices, whichdepends on the covariance of the corresponding subspacematrix. Not knowing the model order yields to use empiri-cal multi-order procedure such as the stabilization diagram(Peeters and De Roeck (1999)), where modes of the systemare assumed to stabilize when the model order increases.

In Reynders et al. (2008), it has been shown how confi-dence intervals of modal parameters can be determinedfrom the covariances of the system matrices and the co-variances of subspace matrices. The current paper willexpand on this and compare sensitivities for two output-only system identification methods, namely output-onlyStochastic Subspace Identification (SSI) and Eigensystem-Realization-Algorithm (ERA, see in Juang and Pappa(1985)) System Identification. Subspace identification isbased on the computation of one subspace matrix from thecorrelation tail. Unlike subspace algorithm, ERA computesthe system matrices using the information of both (k)- and(k+1)-lag of shifted correlation tails.

In this paper, following the lines of (Reynders et al.(2008)), an algorithm will be developed for estimatingthe confidence intervals in ERA system identification. Theuncertainty on state transition matrix is derived, based onthe uncertainties of (k)- and (k+1)-lag subspace matrices.

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A relevant industrial example is applied to ERA estimates.The efficiency of these algorithms and lag effect are alsotaken into account. Comparison with subspace algorithmestimates is also performed.

2. STOCHASTIC SUBSPACE IDENTIFICATION

2.1 The General SSI Algorithm

The discrete time model in state-space form is:{Xk+1 = AXk + Vk+1

Yk = CXk(1)

with the state X ∈ Rn, the output Y ∈ Rr, the statetransition matrix A ∈ Rn×n and the observation matrixC ∈ Rr×n. The state noise V is unmeasured and assumedto be Gaussian, zero-mean, white.

Let r be the number of sensors, p and q be chosenparameters with (p + 1)r ≥ qr ≥ n. From the outputdata, a matrix Hp+1,q ∈ R(p+1)r×qr is built accordingto a chosen SSI algorithm, see e.g. Benveniste and Mevel(2007) for an overview. The matrix Hp+1,q will be called“subspace matrix” in the following, and the SSI algorithmis chosen such that the corresponding subspace matrixenjoys (asymptotically for a large number of samples) thefactorization property

Hp+1,q = Op+1 Zq (2)into the matrix of observability

Op+1def=

CCA

...CAp

(3)

and a matrix Zq depending on the selected SSI algorithm.

Let N be the number of available samples and Yk ∈ Rr,{k ∈ 1, . . . , N} the vector containing the sensor data.Then, the “forward” and “backward” data matrices

Y+p+1 =

1√N − p− q

Yq+1 Yq+2 . . . YN−pYq+2 Yq+3 . . . YN−p+1

......

. . ....

Yq+p+1 Yq+p+2 . . . YN

,

Y−q =1√

N − p− q

Yq Yq+1 . . . YN−p−1

Yq−1 Yq . . . YN−p−2

......

. . ....

Y1 Y2 . . . YN−p−q

(4)

are built. For the covariance-driven SSI (see also Ben-veniste and Fuchs (1985), Peeters and De Roeck (1999)),the subspace matrix H(cov)

p+1,q = Y+p+1Y−q

T is built, whichenjoys the factorization property (2), where Zq is thecontrollability matrix.

For simplicity, let p and q be given, skip the subscripts ofHp+1,q, Op+1 and Zq. The eigenstructure of the system(1) is retrieved from a given matrix H.

The observability matrix O is obtained from a thin Singu-lar Value Decomposition (SVD) of the matrix H and itstruncation at the desired model order n:

H=UΣV T

= [U1 U0 ][

Σ1 00 Σ0

]V T , (5)

O=U1Σ1/21 . (6)

Note that the singular values in Σ1 ∈ Rd×d must be non-zero and hence O is of full column rank. The observationmatrix C is then found in the first block-row of theobservability matrix O. The state transition matrix Ais obtained from the shifting invariance property of O,namely as the least squares solution of

O↑A = O↓, where O↑ def=

CCA

...CAp−1

, O↓ def=

CACA2

...CAp

.(7)

The eigenstructure (λ, ϕλ) results fromdet(A− λI) = 0, Aφλ = λφλ, ϕλ = Cφλ, (8)

where λ ranges over the set of eigenvalues of A. From λ,the natural frequency and damping ratio are obtained, andϕλ is the corresponding mode shape.

There are many papers on the used identification tech-niques. A complete description can be found in Ben-veniste and Fuchs (1985), Van Overschee and De Moor(1996), Peeters and De Roeck (1999), Benveniste andMevel (2007), and the related references. A proof of non-stationary consistency of these subspace methods can befound in Benveniste and Mevel (2007).

2.2 ERA (Eigensystem-Realization-Algorithm)

Another variant of realization algorithm based on thecomputation of the subspace matrices is called ERA(Eigensystem-Realization-Algorithm) (see in Juang andPappa (1985)). It is based on the general remark that onecan compute the subspace matrix H not using the firstlags of the correlation tail. Defining H(k) as

H(k) =

Rk+1 Rk+2 . . . Rk+qRk+2 Rk+3 . . . Rk+q+1

......

. . ....

Rk+p+1 . . . . . . Rk+p+q

, (9)

in which the correlations are related to the factorizationRj

def= E(Yl+j Y

Tl

)= CAjG (10)

with the cross-covariance between the state and the ob-served outputs G = E [Xl Y

Tl ].

Then, a Singular Value Decomposition is performed onH(k) as

H(k) = [U1 U0 ][

Σ1 00 Σ0

] [V T1V T0

](11)

The state transition matrix will be defined as

A=(O†1)H(k+1)

(Z†1), (12)

where † means Moore-Penrose pseudo-inverse, and

O†1 = (Σ1)−12 UT1 , Z

†1 = V1 (Σ1)−

12 . (13)

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If the correlations are computed from cross spectra, themethod is called NEXT-ERA; but without loss of general-ity, it is just assumed that the correlations are computedfrom time samples. The dimensions of A relates to thedimensions of U1,Σ1, V1. And as such, a stabilization di-agram is obtained by performing the computation of Afor multiple model orders and keeping as stable poles themodes which repeat over multiple model orders.

3. CONFIDENCE INTERVALS

3.1 Descriptions of SSI Confidence Intervals algorithm

The statistical uncertainty of the obtained modal param-eters at a chosen system order can be computed fromthe uncertainty of the system matrices, which depends onthe covariance of the corresponding subspace matrix H.The latter can be evaluated by cutting the sensor datainto blocks on which instances of the subspace matrixare computed. So, this offers a possibility to compute theconfidence intervals of the modal parameters at a certainsystem order without repeating the system identification.In Reynders et al. (2008), this algorithm was described indetail for the covariance-driven SSI. The uncertainty ∆Aand ∆C of the system matrices A and C are connected tothe uncertainty of the subspace matrix through a Jacobianmatrix [

vec∆Avec∆C

]=[JAJC

]vec∆H

= JA,C vec∆H, (14)where vec is the vectorization operator, Jacobian JA canbe defined as vec∆A = JA vec∆H, Jacobian JC can bedefined as vec∆C = JC vec∆H.

Then, the uncertainty of the modal parameters (naturalfrequency f , damping ratio d and mode shape φ) is derivedfrom

∆fµ = Jfµ

[vec∆Avec∆C

], ∆dµ = Jdµ

[vec∆Avec∆C

], (15)

and

∆φµ = Jφµ

[vec∆Avec∆C

]. (16)

The Jacobians Jfµ , Jdµ and Jφµ are computed for eachmode µ. Finally, the covariances of the modal parametersare obtained as

cov(fµ) = JfµJA,C cov(vec H) JTA,C JTfµ

cov(dµ) = JdµJA,C cov(vec H) JTA,C JTdµ

cov(φµ) = Jφµ JA,C cov(vec H) JTA,C JTφµ(17)

where cov(vecH) is the covariance of the vectorized sub-space matrix. After retrieving the uncertainties on the sys-tem matrices A and C, the calculation of the uncertaintieson the frequency and damping is straightforward. How-ever, for the mode shape, there is an issue of normalizationas each one is defined up to an unknown constant. Thiswas addressed in Dohler et al. (2010).

3.2 Derivation of ERA Confidence Intervals

In this section, for ERA, it is investigated how the covari-ances of modal parameters can be derived from the covari-ance of subspace matrices taking care of the uncertaintiesof observability, controllability and system matrices.

Firstly, the uncertainty on the system matrix A is afunction of the sensitivities of H(k+1), O†1 and Z†1 :

∆A= ∆[ (O†1)H(k+1)

(Z†1) ]

=[∆(O†1) ]H(k+1)

(Z†1)

+(O†1) [

∆H(k+1)] (Z†1)

+(O†1)H(k+1)

[∆(Z†1) ]. (18)

The uncertainty on the vectorized system matrix A isrewritten as

vec∆A=((Z†1

TH(k+1)T

)⊗ Id

)vec

(∆(O†1))

+(Z†1

T⊗O†1

)vec∆H(k+1)

+(Id ⊗

(O†1H(k+1)

))vec

(∆(Z†1))

, (19)

where Id is identity matrix with dimension d. ⊗ is theKronecker product. The uncertainty of H(k+1) can simplybe estimated by cutting the signals.

The uncertainty on the pseudo-inverse of observabilityO1 can be defined directly from the singular values andsingular vectors by

∆(O†1)

= ∆(

Σ−12

1 UT1

)=[∆(

Σ−12

1

) ]UT1 + Σ−

12

1 ∆(UT1)

=−12

Σ−32

1 (∆Σ1)UT1 + Σ−12

1 ∆(UT1)

(20)

The uncertainty of O†1 is now vectorized as

vec(

∆(O†1))

=(U1 ⊗

(−1

2Σ−

32

1

))vec∆Σ1

+(I(p+1)r ⊗ Σ−

12

1

)vec

(∆(UT1))

=(U1 ⊗

(−1

2Σ−

32

1

))vec∆Σ1

+(I(p+1)r ⊗ Σ−

12

1

)PU1 vec∆U1 (21)

where PU1 is a matrix that can permutate vec∆U1 tovec

(∆(UT1))

.

Similarly, the uncertainty on the pseudo-inverse of control-lability Z1 can be descibed as

∆(Z†1)

= ∆(V1Σ−

12

1

)= (∆V1) Σ−

12

1 + V1∆(

Σ−12

1

)= (∆V1) Σ−

12

1 + V1

(−1

2

)Σ−

32

1 ∆Σ1 (22)

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and reconstructing it in vectorized form leads to

vec(

∆(Z†1))

=(

Σ−12

1 ⊗ Iqr)vec∆V1

+(Id ⊗

(−1

2V1Σ−

32

1

))vec∆Σ1.(23)

The sensitivity of the left singular vectors can be related tothe uncertainty of subspace matrix H(k) (see in Pintelonet al. (2007))

vec∆U1 =L1d

B†1C1

...B†dCd

vec∆H(k) (24)

with a selection matrix defined by

L1d = Id ⊗[I(p+1)r O(p+1)r×qr

](25)

and

Bj =

I(p+1)r −H(k)

σj

− (H(k))T

σjIqr

, (26)

Cj =1σj

[vTj ⊗ (I(p+1)r − ujuTj )(uTj ⊗ (Iqr − vjvTj ))P

], (27)

P =(p+1)r∑k1=1

qr∑k2=1

E(p+1)r×qrk1k2

⊗ Eqr×(p+1)rk2k1

, (28)

where σj is the eigenvalue at system order j {j ∈ 1, . . . , d},uj (resp. vj) is column number j of U (resp. V ). E(p+1)r×qr

k1k2is a (p + 1)r × qr matrix whose element is 1 at position(k1, k2) and zero elsewhere.

The sensitivity of eigenvalues is addressed as (see inPintelon et al. (2007))

vec(∆Σ1) = S3d

(v1 ⊗ u1)T...

(vd ⊗ ud)T

vec∆H(k), (29)

in which S3d is a selection matrix

S3d =d∑s=1

Ed2×d

(s−1)d+s,s (30)

The sensitivity of right eigenvectors (see in Pintelon et al.(2007)) is then specified by

vec∆V1 =L2d

B†1C1

...B†dCd

vec∆H(k) (31)

with selection matrix

L2d = Id ⊗[Oqr×(p+1)r Iqr

]. (32)

Especially, vec∆H(k) can be simplified by making use of ablock-storing matrix M (k)

vec∆H(k) = S(k)4 vec∆M (k) (33)

where

M (k) =

R1

R2

...Rp+q+k

(34)

S(k)4 =

Ir ⊗ S(k)

5,1

Ir ⊗ S(k)5,2

...Ir ⊗ S(k)

5,q

(35)

S(k)5,t =

[O(p+1)r×(t−1+k)r I(p+1)r O(p+1)r×(q−t)r

](36)

Finally, the uncertainty of system matrix A can be shownin vectorization form

vec∆A= JA vec∆M (k), (37)

in which JA is a Jacobian matrix

JA =N1

(v1 ⊗ u1)T...

(vd ⊗ ud)T

S(k)4

+N2

B†1C1

...B†dCd

S(k)4

+(Z†1

T⊗O†1

)S

(k+1)4 (38)

with the matrices

N1 =((Z†1

TH(k+1)T

)⊗ Id

)(U1 ⊗

(−1

2Σ−

32

1

))S3d

+(Id ⊗

(O†1H(k+1)

))(Id ⊗

(−1

2V1Σ−

32

1

))S3d,

(39)

N2 =((Z†1

TH(k+1)T

)⊗ Id

)(I(p+1)r ⊗ Σ−

12

1

)PU1 L1d

+(Id ⊗

(O†1H(k+1)

))(Σ−

12

1 ⊗ Iqr)L2d.

(40)

Likewise, the uncertainty of the vectorized system matrixC is

vec∆C = JC vec∆M (k) (41)

with Jacobian matrix

JC = (Id ⊗ SC) (Bd + Cd)S(k)4 , (42)

where

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SC = [ Ir Or×pr ] , (43)

Bd =(Id ⊗

(12U1Σ−

12

1

))S3d

(v1 ⊗ u1)T...

(vd ⊗ ud)T

(44)

Cd =(

Σ121 ⊗ I(p+1)r

)L1d

B†1C1

...B†dCd

. (45)

Finally, the uncertainty of system matrices can be joinedtogether [

vec∆Avec∆C

]=[JAJC

]vec∆M (k)

= JA,C vec∆M (k) (46)

Then, the covariances of the modal parameters are ob-tained as

cov(fµ) = Jfµ JA,C cov(vec M (k)) JTA,C JTfµ

cov(dµ) = Jdµ JA,C cov(vec M (k)) JTA,C JTdµ

cov(φµ) = JφµJA,C cov(vec M (k)) JTA,C JTφµ

(47)

4. NUMERICAL EXAMPLES

The S101 bridge (Siringoringo et al. (2010)) connectedSalzburg - Vienna carriage way in Austria. That is apost-tensioned concrete bridge with main span of 32 m,side spans of 12 m, and the width of 6.6 m. The deck iscontinuous over the piers. This bridge, contructed in 1960,has been a typical overpass bridge in Austria nationalhighway. In the current paper, the ambient vibration datais collected on 15 sensors. The original sampling frequencyis 500 Hz with 165000 time samples available. The data isdecimated to 35.7 Hz and only five modes are taken intoaccount.

Fig. 1. S101 bridge in Reibersdorf, Austria

4.1 Modal analysis

For the output-only modal analysis of the ambient vibra-tion data of the S101 bridge, similar parameters for bothsubspace algorithm and ERA are employed. 64 correlations

(p + 1 = q = 32) are used, leading to Hankel matriceswith 32 block rows and columns. The resulting multi-orderdiagrams are presented in Figure 2 and Figure 3.

Fig. 2. Stabilization diagram with subspace algorithm(natural frequency vs. model order)

Fig. 3. Stabilization diagram with ERA (natural frequencyvs. model order)

The summary of the frequencies and damping ratios ofthe five identified modes is given in Table 1 and Table2, for both subspace identification and ERA. ERA-fl isthe ERA which uses first-lag and second-lag subspacematrices, ERA-sl is the ERA which utilizes second-lagand third-lag subspace matrices, ERA-tl is the ERA whichemploys third-lag and fourth-lag subspace matrices.

The differences in the obtained frequencies between sub-space identification and ERA are small, less than 0.5%, forall five modes. In the case of damping ratios, the differencesare bigger because of higher uncertainty in the estimationof damping ratios.

Table 1. Identified frequencies with subspacealgorithm and ERA

Mode Frequency f (Hz)Subspace ERA-fl ERA-sl ERA-tl

1 4.039 4.038 4.037 4.037

2 6.282 6.283 6.284 6.283

3 9.682 9.684 9.683 9.684

4 13.284 13.283 13.290 13.307

5 15.721 15.720 15.763 15.630

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Table 2. Identified damping ratios with sub-space algorithm and ERA

Mode Damping Ratio d (%)Subspace ERA-fl ERA-sl ERA-tl

1 0.759 0.754 0.762 0.756

2 0.617 0.608 0.588 0.573

3 1.193 1.189 1.185 1.228

4 1.436 1.424 1.309 1.135

5 1.638 1.966 2.360 2.475

4.2 Confidence Intervals

For the computation of confidence intervals on modalparameters, 24 time lags, leading to p+1 = q = 12, and 40model orders are utilized due to the limitation in computermemory.

Table 3. Frequency confidence intervals withsubspace algorithm and ERA

Mode Frequency confidence intervals (%)Subspace ERA-fl ERA-sl ERA-tl

1 0.115 0.110 0.122 0.100

2 0.092 0.093 0.092 0.089

3 0.133 0.134 0.156 0.159

4 0.471 0.300 0.247 0.573

5 1.148 1.825 2.485 1.442

Table 4. Damping-ratio confidence intervalswith subspace algorithm and ERA

Mode Damping-ratio confidence intervals (%)Subspace ERA-fl ERA-sl ERA-tl

1 17.791 17.819 17.757 16.352

2 19.383 19.239 20.401 19.900

3 16.845 13.302 11.332 11.960

4 28.919 15.012 56.777 65.019

5 60.522 70.800 161.749 130.691

In Table 3 and Table 4, the confidence intervals (standarddeviation) of the natural frequencies and damping ratiosof the five modes are presented, respectively. Confidenceintervals of the frequencies are much smaller than thoseof damping ratios. This is coherent with statistical theory,since the lower bound of the covariance given by Fisherinformation matrix is smaller for the frequencies than forthe damping ratios. Besides, for this application, confi-dence bounds on modal parameters of ERA are relativelysimilar as those obtained with the subspace algorithm.While shifting the lags of ERA, the confidence intervalfluctuations seem to be stable, and the ERA-fl may supplythe most comparable results to subspace identification.

5. CONCLUSIONS

In this paper, the output-only system identification andconfidence intervals on modal parameters are derived andimplemented for both subspace algorithm and ERA. Allthe methods were successfully applied and tested on theambient vibration data of the S101 overpass bridge.

The subspace algorithm and ERA give comparable results.The quality of stabilization diagrams as well as frequenciesof subspace algorithm and ERA are almost similar. The

damping ratios are slightly different due to an expectedlyhigher uncertainty on estimation.

The confidence intervals on modal parameters are alsocomputed. It is observed that the uncertainty for ERAis relatively similar with that associated to subspace algo-rithm. When taking into account the lag effect for ERA,ERA-fl seems to be the most reasonable selection for thedesigners dealing with ERA.

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