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Lecture 8

باسمه تعالی

Eigensystem Realization Algorithm(ERA)

Identification Methods for Structural Systems

by: Dr B. Moaveni

2

• The Eigensystem realization algorithm (ERA) is a system

identification technique popular in civil engineering, in particular in

structural health monitoring.

• ERA can be used as a modal analysis technique and generates a

system realization using the time domain response (multi-)input and

(multi-)output data.

• The ERA was proposed by Juang and Pappa [2] and has been used for

system identification of aerospace structures such as the Galileo

spacecraft, turbines, civil structures and many other type of systems.

[Ref] Wikipedia, last modified 26 September 2014 at 21:06.

مقدمه

http://en.wikipedia.org/wiki/System_identification

http://en.wikipedia.org/wiki/Civil_engineering

http://en.wikipedia.org/wiki/Structural_health_monitoring

http://en.wikipedia.org/wiki/Modal_analysis

http://en.wikipedia.org/wiki/System_realization

http://en.wikipedia.org/wiki/Galileo_spacecraft


by: Dr B. Moaveni

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Workflow overview

Data assemblyAssemble the selecteddata sets into a Hankel Matrix and a Shifted Hankel Matrix

Decomposition

Decompose the Hankel Matrix using Singular Value Decomposition

Matrix RealizationExtract the new Controllability and Observabilitymatrix; Calculate the system realization matrix

Eigenvalue problem solving

Solve the eigenvalue problem for the system realization matrix

Extract system properties

Calculate natural frequencies and damping factorsusing the obtained eigenvalues


by: Dr B. Moaveni

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ERA- Data Acquisition


by: Dr B. Moaveni

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ERA-Preprocessing


by: Dr B. Moaveni

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ERA-Preprocessing


by: Dr B. Moaveni

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DATA Assembly

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)


by: Dr B. Moaveni

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Decomposition

Assume the state space representation of a dynamic system

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


by: Dr B. Moaveni

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Assume the state space representation of a dynamic system

Markov Parameters

Decomposition


by: Dr B. Moaveni

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By constructing the Hankel matrix of the Markov Parameters :

Decomposition

1

1 2

22 3 1

1

1 2 21 1

11

1

:

n

n

nn

n n nn n n n

no c

n

o

CB CAB CA By y y

y y y CAB CA B CA BH

y y y CA B CA B CA B

C

CAH B AB A B

CA

Obser

: c

vability Matrix

Controllability Matrix


by: Dr B. Moaveni

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In order to obtain these two matrices (Controllability and Observability) we

perform Singular Value Decomposition (SVD) for H1:

Decomposition & Matrix Realization

11

1

21

no c

n

T T

T

C

CAH B AB A B

CA

P UH U V U V PQ

Q V

Note: The decomposition is NOT unique.1H PQ


by: Dr B. Moaveni

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Matrix Realization

1H PQ


by: Dr B. Moaveni

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Matrix Realization

Using the shifted Hankel matrix

1 12 2

1 12

1

ˆo c o cH A A H

P H Q

H PQ

A

1

1

ˆ ˆ

ˆˆ ˆ

ˆ

ˆ ˆ

ˆ ˆ

n

n

C

CAP

CA

Q AB A BB


by: Dr B. Moaveni

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Extract System Properties

Using the shifted Hankel matrix

1 1 2

1 2

ˆ ˆ

ˆ

i i Av v n

ni i

x Ax Bu

V v v vy Cx

21 2

ln

Re

i ci

cii

c

d c in ci

s

s

i

T

T sample time

1 2For obtaining the mode shapes: nV v v v

15

سازه دو طبقه: مثال


by: Dr B. Moaveni

1 1 1 1 2 1 2 1 1 2 1 2

2 2 2 2 1 2 2 1

1 1 1 2 1 1 1 2 1 1 1 1

2 2 2 2 2 2 2 2

1

( ) ( ) ( ) ( ) 0

( ) ( ) 0

0

0 0 0

0

0

m x c x y c x x k x y k x x

m x c x x k x x

m x c c c x k k k x c ky y

m x c c x k k x

m

1 1 2 1 1 1 2 1 1 1 2 1 1 2 1

2 2 2 2 2 2 2 2 2 2 2 2

1 1 1 2 1 1

2 2 2 2 2

1 1

1 1

0

0

x c c c x k k k x c c c k k ky y

m x c c x k k x c c k k

m x c c c x

m x c c x

1 2 1 1 1 2 1 1 2 1

2 2 2 2 2 2 2

k k k x c c c k k ky y

k k x c c k ky y

2

2

0 0Iy

I

-1 -1

-1 -1

X XM C M K

Y M C M K X

Absolute Motion Eqs.:

16



by: Dr B. Moaveni

% system matrices

m1 = 100; % Mgr

k1 = 200000; % kN/m

m2 = 80; % Mgr

k2 = 200000; % kN/m

Mass = diag([m1 m2]);

Stiffness = [k1+k2 -k2;-k2 k2];

% add 5% modal damping

K_ = Mass^(-1/2)*Stiffness*Mass^(-1/2);

wn = sqrt(eig(K_));

A = [ones(length(wn),1) wn.^2];

b = 2*wn.*[0.05;0.05];

x = A\b;

Damping = x(1)*Mass + x(2)*Stiffness;

Ass = [zeros(2) eye(2);-inv(Mass)*Stiffness

-inv(Mass)*Damping];

Bss = [zeros(2,1);-ones(2,1)];

Css = [-inv(Mass)*Stiffness -

inv(Mass)*Damping];

Dss = -ones(2,1);

sys = ss(Ass,Bss,Css,Dss);

17



by: Dr B. Moaveni

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.5

0

0.5

1

time(sec)

y1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.5

0

0.5

1

1.5

y2

time(sec)

x0 =

1.0e-03 *

-0.2435

-0.6747

-0.4520

-0.9619

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by: Dr B. Moaveni

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-10

-5

0

5

10

time(sec)

y1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-10

-5

0

5

10

y2

time(sec)

x0 =

0.0012

-0.0017

-0.0011

-0.0008

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by: Dr B. Moaveni

H1 =

1.1303 0.6971 -0.1857 -1.0084

-1.6423 -1.2349 -0.4090 0.4139

0.6971 -0.1857 -1.0084 -1.3210

-1.2349 -0.4090 0.4139 0.8564

-0.1857 -1.0084 -1.3210 -0.9639

-0.4090 0.4139 0.8564 0.7726

-1.0084 -1.3210 -0.9639 -0.1324

0.4139 0.8564 0.7726 0.3016

H2 =

0.6971 -0.1857 -1.0084 -1.3210

-1.2349 -0.4090 0.4139 0.8564

-0.1857 -1.0084 -1.3210 -0.9639

-0.4090 0.4139 0.8564 0.7726

-1.0084 -1.3210 -0.9639 -0.1324

0.4139 0.8564 0.7726 0.3016

-1.3210 -0.9639 -0.1324 0.7464

0.8564 0.7726 0.3016 -0.2303

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by: Dr B. Moaveni

P =

-0.6814 0.5790 0.3366 -0.0245

1.1082 -0.1948 0.2498 -0.0455

-0.1080 0.9883 0.1986 0.0131

0.5723 -0.6513 0.2318 -0.0002

0.5665 0.8822 0.0368 0.0444

-0.0851 -0.7021 0.2025 0.0471

0.9655 0.3278 -0.0929 0.0670

-0.5498 -0.3890 0.1320 0.0920

Q =

-1.2920 -1.2270 -0.6003 0.2429

0.6354 -0.3691 -1.1129 -1.2352

-0.3525 0.2281 0.1403 -0.3759

-0.0300 0.0878 -0.0966 0.0454

21



by: Dr B. Moaveni

>> eig_Ahat=log(eig(Ahat))/Ts

eig_Ahat =

-3.7445 +74.7962i

-3.7445 -74.7962i

-1.4929 +29.8207i

-1.4929 -29.8207i

>> eig(Ass) =

-3.7445 +74.7962i

-3.7445 -74.7962i

-1.4929 +29.8207i

-1.4929 -29.8207i

wn =

74.8899

74.8899

29.8581

29.8581

zeta =

0.0500

0.0500

0.0500

0.0500


by: Dr B. Moaveni

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Extracting M, C and K


by: Dr B. Moaveni

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by: Dr B. Moaveni

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مراجع

1. E. Chatzi, Identification Methods for Structural Systems, 2013.

2. J. H. Suk, Investigation and Solution of Problems for Applying Identification

Methods to Real Systems, PhD Thesis, University of Washington, 2009.

3. N. N. Nielsen, Dynamic Response of Multistory Buildings, PhD Thesis, 1964.

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