The Eigensystem Realization Algorithm (ERA)

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

Eigenvalue problem solving

Extract system properties

Decompose the Hankel Matrix using Singular Value Decomposition

Workflow overview

Data assembly

Decomposition

Matrix Realization

Eigenvalue problem solving

Extract system properties

Extract the new controllability and

observability matrix; Calculate the system

realization matrix

Workflow overview

Data assembly

Decomposition

Matrix Realization

Eigenvalue problem solving

Extract system properties

Solve the eigenvalue problem for the system

realization matrix

Workflow overview

Data assembly

Decomposition

Matrix Realization

Eigenvalue problem solving

Extract system properties

Calculate natural frequencies and

damping factors using the obtained eigenvalues

Data acquisition

PC Acceleration signals

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 1

Channel 2

Channel 3

Channel 4 Channel 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 the state – space representation of a dynamic system

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

Decomposition

Assume the state – space representation of a dynamic system

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

Decomposition

By iterating system in time

Assume the state – space representation of a dynamic system

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

By constructing the Hankel matrix of the Markov Parameters yi :

Which is equivalent to the matrix product:

Decomposition

Controllability matrix

Observability matrix

By constructing the Hankel matrix of the Markov Parameters yi :

Decomposition

Controllability matrix

Observability matrix

By constructing the Hankel matrix of the Markov Parameters yi :

Decomposition

Controllability matrix

Observability matrix

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

Matrix Realization

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

In fact by using a different number of shifts k, and total measurements n, different alternatives can occur. And this is due to the fact that if matrices (A, B, C) are a realization of the system:

Then matrices, 1 1, TB, CTTAT − − are also a realization through the system: 1

11

ì i i

i i i

x TAT x TBuy CT x Du

−+

−

= +

= +

1ì i i

i i i

x Ax Buy Cx Du+ = += +

Under the transformation: x Tx=

Therefore the state that occurs from the ERA is not necessarily the that corresponds to the structural dofs but some transformation of it.

x

x

Matrix Realization

By using the new observability and the new controllability matrices:

Realization of A

replaced by

replaced by

Then, using the Shifted Hankel Matrix :

Matrix Realization

Realization of C

“Output matrix”

Realization of B

“Control matrix”

Eigenvalue problem solving

Eigenvalues

By solving the eigenvalue problem

Eigenvectors

Extract system properties

Damping factors;

Conversion for discrete time to continuous time

representation

Natural frequencies;

For obtaining the mode shapes:

Extension for Random Input

It has already been mentioned that the ERA operates using output measurements of impulse response data. However, it possible to appropriately extend the method so as to account for response to a measured input loading.

The ERA as an input-output Id method

Assuming measurements of the input f(t) and output of the system x(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

= =

Then by applying the Inverse Fourier Transfom, the Impulse Response Functions (IRF) per measurement channel (usually this implied per dof) are obtained. The ERA method, as described previously can then be implemented on the IRFs which essentially simulate the system’s response to impulse.

Extension for Random Input

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

Proof of the FRF extraction 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 is defined 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:

Extension for White Noise (Ambient Data)

For the case of ambient (operational) loads, it may be assumed that the excitation and responses are each stationary random processes. Assuming that the structural parameter matrices are deterministic, postmultiplying the Eq. of motion by a reference scalar response process 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

( )1 2X t

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

mABA B

R R t t mτ τ τ= = −

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 write: , ,X X X

Thus, the vector of displacement process correlation functions, satisfies the homogeneous differential equation of motion. Using a similar approach it can be shown that the acceleration process correlation functions also satisfy this equation (Beck et al. 1994). We can therefore employ the ERA for the correlation signals!

( ) ( ) ( )1 2, 0i i iXX XX XXMR t t CR KRτ τ τ+ + =