the eigensystem realization algorithm (era)...decomposition. matrix realization. eigenvalue problem...

The Eigensystem Realization Algorithm(ERA)

Chair of Structural Mechanics & Monitoring, ETH Zürich

Workflow overview

Data assembly

Decomposition

Matrix Realization

Eigenvalue problem solving

Extract system properties

Assemble the selected data sets into a Hankel

Matrix and a Shifted Hankel Matrix


Workflow overview

Data assembly

Decomposition

Matrix Realization



Decompose the Hankel Matrix using

Singular Value Decomposition


Workflow overview

Data assembly

Decomposition

Matrix Realization



Extract the new controllability and

observability matrix;Calculate the system

realization matrix


Workflow overview

Data assembly

Decomposition

Matrix Realization



Solve the eigenvalue problem for the

system realization matrix


Workflow overview

Data assembly

Decomposition

Matrix Realization



Calculate natural frequencies and

damping factors using the obtained eigenvalues


Data acquisition

PCAccelerationsignals

Note: The ERA is implemented for the case of free response data. Therefore Impact (Hammer, drop-weight) tests would be generally suitable.


Preprocessing

Selection


Preprocessing

Selection(keep the part that corresponds to free response)


Preprocessing

Selection


Data assembly

y1 y2

y3

y4

y5

yn-1 yn

Hankel Matrix:

Shifted Hankel Matrix:

Channel 1Channel 2Channel 3Channel 4Channel i

The ERA works by exploiting the relationship of the series of outputs from different points (channels) of the structure to fundamental system properties (Markov Parameters)

1 ny y


Decomposition

Assume the state – space representation of a dynamic system


Decomposition


Assume an impulse force, at t = 0, and 0 Initial Conditions

Chair of Structural Mechanics & Monitoring, ETH ZürichCh i f St t l M h i & M it i ETH Zü i h

Decomposition


Assume an impulse force, at t = 0, and 0 Initial Conditions

Decomposition

By iterating system in time



Decomposition

Iterate the system in time starting from I.C.

Markov Parameters

These constant parameters are termed & are system characteristics:

Decomposition

By constructing the Hankel matrix of the Markov Parameters yi :


Decomposition


Which is equivalent to the matrix product:


Decomposition

Controllability matrix

Observabilitymatrix



Decomposition

Controllability matrix

Observabilitymatrix

In order to obtain these matrices we perform Singular Value Decomposition for H1:


Matrix Realization

Product of Singular Value Decomposition :

TIP:

New controllability

matrix

New observability

matrix


System «Realization» – What does it mean?

Note: The Decomposition 1H PQ= is not unique!

In fact by using a different number of time shifts k, and totalmeasurements n, different alternatives can occur.

This is due to the fact that the set of system matrices (A, B, C) areonly one (out of many) realizations of the system that connects theinputs u (loads) to the output y (measurements).

Therefore, if (A,B,C) are one realization, leading to the followingsystem:


1ì i i

i i i

x Ax Buy Cx Du+ = += +


Note: The Decomposition 1H PQ= is not unique!

the set of matrices, 1 1, TB, CTTAT − − is also a further, equivalentRealization of the system that links inputs u to outputs y, with:

11 1

1

ˆ ˆˆ ˆ

ì i i ì i i

i i ii i i

x TAT x TBu x Ax Buy Cx Duy CT x Du

−+ +

−

= + = +⇒= += +

Then, under the transform 1x Tx x T x−= ⇒ =

Therefore the internal state of the ERA identified system is not necessarily the one that corresponds to the structural dofs but some transformation of it.

x



Note: The ERA-identified system is NOT the original physical coordinates system!

The matrices (Ad, Bd, Cd) that are defined via discretization of theoriginal differential equation of the structural dynamics problem:

are only one (out of many) realizations of the system that connectsthe inputs u (loads) to the output y (measurements).The standard implementation of the ERA will NOT return thesephysical coordinate system matrices, but rather a transformation ofthem using matrix T.


( ) ( )

1

1

0with A ,c c

ì d i d ic c

i i i

tA t Ad d c d

x A x B ux A x B uMu u Ku f

y Cx Duy Cx Du

e B e d B A A I Bτ τ

+

∆∆ −

= += + + Γ + = ⇒ ⇒ = += +

= = = −∫

Matrix Realization

By using the new observabilityand the new controllability

matrices:

Realization of A

replaced by

replaced by

Then, using the Shifted Hankel Matrix :


How many singular values do we keep? 2 × the expected modes. This defines the dimension of A

Matrix Realization

Realization of C“Output matrix”

Realization of B“Control matrix”



Eigenvalues

By solving the eigenvalueproblem

Eigenvectors


of other realizations



How are the eigenvalues of A linked to the eigenvalues of other realizations1ˆ ?A TAT −=

The above linear transformation preserves the eigenvalues, i.e.,

( ) ( ) ( )1ˆ ˆeig A eig TAT eig A−= = = Λ

Additionally, the mode shapes in terms of the measured variable y(which is physical) will be preserved:

( ) ( )ˆ ˆ ˆeigenvectors CA eigenvectors CA CV= =


Conversion for discrete time to continuous time representation

For obtaining the mode shapes:


Furthermore, we know the relationship between the eigenvalues λ of the discrete and the original continuous system, λc :

Original continuous system

C cx A x B uy Cx Du= += +


Damping factors;

Natural frequencies;


For the continuous state-space system, describing the equation of motion

1 10 with C c

C

x A x B u IAM K M Cy Cx Du − −

= + = − −= +

( ) ( )2( ) 1i c i i ic i eig A iλ ζ ω ζ ω= = − ± −

it holds that:

*the eigenvectors of A, ΦΑ are also linked to the structural system’s mode shapes Φ, as

( )Re AΦ = Φ

Therefore:

Extension for Random Input

It has already been mentioned that the ERA operates using outputmeasurements of impulse response data. However, it possible toappropriately extend the method so as to account for response to ameasured input loading.

The ERA as an input-output Id method

Assuming measurements of the input f(t) and output of the systemx(t) are available from m measurement locations.

The Frequency Response Function (FRF) may be extracted as:


( ) ( )( )

, 1xfi

ff

S jH j i m

S jω

ωω

= =

Extension for Random Input

The ERA as an input-output Id method

The Frequency Response Function (FRF) may be extracted as1:

( ) ( )( )

, 1xfi

ff

S jH j i m

S jω

ωω

= =

Then by applying the Inverse Fourier Transfom, the ImpulseResponse Functions (IRF) per measurement channel (usually thisimplies per dof) are obtained.

The ERA method, as described previously can then be implementedon the IRFs which essentially simulate the system’s response toimpulse.

Chair of Structural Mechanics & Monitoring, ETH Zürich 1See the Appendix for the proof

Extension for White Noise (Ambient Data)

For the case of ambient (operational) loads, it is assumed that theexcitation and responses are stationary random processes.Assuming that the structural parameter matrices are deterministic,post-multiplying the Eq. of motion by a reference scalar responseprocess and taking the expected value of each side yields:

The Natural Excitation Technique (NExT)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2, , , ,i ii i

i i i i

XX FXXX XX

ME X t X t CE X t X t KE X t X t E F t X t

MR t t CR t t KR t t R t t

+ + = ⇒ + + =

where denote the displacement and excitation stochastic vector process respectively. Additionally, for weakly (or strongly) stationary processes, we know that:

( ) ( ), X t F t

( )2iX t

( ) ( ) ( ) ( ) th2 1, , where denotes the m derivative.m

mABA B

R R t t mτ τ τ= = −


Extension for White Noise (Ambient Data)

The Natural Excitation Technique (NExT)

Recognizing that the responses of the system are uncorrelated to the disturbance for t>0, and assuming that the random vector processes

are weakly stationary, we can eliminate the term on theright-hand side:

, ,X X X

Thus, the vector of displacement cross-correlation functions (wrt toa reference Xi), satisfies the homogeneous differential equation ofmotion.Using a similar approach it can be shown that the accelerationcorrelation functions also satisfy this equation (Beck et al. 1994).

We can therefore employ the ERA for the correlation signals!

( ) ( ) ( ) 0i i iXX XX XXMR CR KRτ τ τ+ + =


Practical Guildeines to Using ERA, ERA-NEXT

ERAThe ERA has three main parameters to be determined:the number of rows and columns of the Hankel matrix (m and n), and thenumber of poles (which is 2×modes) to identify.

NEXTThe number of points of the FFT, and the reference channel used tocalculate the cross-correlation function are the two main parameters forNExT.

*The length and sampling frequency of the time domain records affectthe modal identification procedure.

Usually, this is an iterative procedure where the analyst performs theidentification with a set of parameters, analyzes the output of thisprocess and changes the parameters depending on the results obtained.

Main Hyperparameters – as defined by Caicedo 2011


https://onlinelibrary.wiley.com/doi/full/10.1111/j.1747-1567.2010.00643.x


• Before starting the modal identification procedure, it is useful tocreate a few cross-spectral density functions and determine if cleanpeaks are shown within the spectrum. These peaks indicate possiblenatural frequencies.

• NEXT assumes white noise as the driving excitation. This implies thatthe record to be used should be stationary, and that therefore themean and variance should be constant over time.

• Longer records are preferred for noisy data, while relatively shortrecords can be used for relatively clean data.

• The record length should be selected based on the expectedfrequency of the structure. Lower frequency structures, such ascable-stayed bridges, require longer records to capture the samenumber of cycles than higher frequency structures.

Preprocessing – as defined by Caicedo 2011




• The sample frequency should be at least twice the higherfrequency of interest to comply with the Nyquist criterion but it isrecommended to select a sampling frequency not much higher thanthat.

• Anti-aliasing filters should be used when resampling the data.However, it is important to consider that anti-aliasing filters couldappear as poles on the signals due to filter’s dynamic characteristics.

• NEXT requires a reference channel to calculate the cross-correlationfunction. One option is to choose a reference channel with highamplitude, low noise to-signal ratio, and far from a node of vibrationof the modes of interest. Using

Preprocessing – as defined by Caicedo 2011




• The number of rows and columns is chosen based on the number ofexpected natural frequencies. A first approximation to the number ofmodes can be obtained by counting the peaks of the cross-spectraldensity function.

• A rule of thumb for the number of columns of the Hankel matrixis to use four times the number of expected modes (twice thenumber of expected poles. A low-rank Hankel matrix could lead tomissing some physical modes of vibration but this is usually a goodstarting point.

• The number of rows is a set based on the number of points available in the cross-spectral density function. The goal is to use as much data from the spectral density function as possible without including noisy signals found at the end of the cross-correlation function.

Hankel Matrix Dimension – as defined by Caicedo 2011




Specifying the correct number of poles, or model order, is probably themost important step in the modal identification procedure:

• If the model order is too high, fictitious modes of vibration occur.• If the model order is too small, some of the modal parameters might

not be identified.

Stabilization diagrams are an effective tool to determine the correctnumber of poles. The idea behind stabilization diagrams is to repeat theidentification process with a different number of poles (or modes) eachtime.• Stable poles should remain constant for all or most of the iterations.

Stabilization Diagram



Example: stabilization diagram for a 3 degree of freedom system:

Stabilization Diagram


0 10 20 30 40 50 60

Frequency (Hz)

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

Num

ber o

f Mod

esStabilization Chart

Appendix

As mentioned in Lecture 1, the system’s response to a random input canbe obtained via discrete convolution with the IRF:

Derivation of the FRF formula:

[ ] [ ] [ ]xft

R x t f tτ τ∞

=−∞

= −∑

[ ] [ ] [ ]0x t h t f

ττ τ∞

== −∑

On the other hand, the cross-correlation of two discrete time signals isdefined as:

(1)

(2)

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

( ) ( ) ( )

0

xf fft t

xf xx

x t f t h t f f t R R h

S j S j H j

τ

τ τ τ τ τ τ τ

ω ω ω

∞ ∞ ∞

=−∞ =−∞ =

− = − − ⇒ = ∗

=

∑ ∑ ∑However, convolution in the time domain is multiplication in the frequency domain. Thus, by taking the Fourier Transform we obtain:


the eigensystem realization algorithm (era)...decomposition. matrix realization. eigenvalue problem...

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