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روش های شناسایی در سیستم های سازه ای
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
Identification Methods for Structural Systems
by: Dr B. Moaveni
3
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
Identification Methods for Structural Systems
by: Dr B. Moaveni
4
ERA- Data Acquisition
Identification Methods for Structural Systems
by: Dr B. Moaveni
5
ERA-Preprocessing
Identification Methods for Structural Systems
by: Dr B. Moaveni
6
ERA-Preprocessing
Identification Methods for Structural Systems
by: Dr B. Moaveni
7
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)
Identification Methods for Structural Systems
by: Dr B. Moaveni
8
Decomposition
Assume the state space representation of a dynamic system
Assume an impulse force, at t = 0, and ZERO Initial Conditions
Identification Methods for Structural Systems
by: Dr B. Moaveni
9
Assume the state space representation of a dynamic system
Markov Parameters
Decomposition
Identification Methods for Structural Systems
by: Dr B. Moaveni
10
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
Identification Methods for Structural Systems
by: Dr B. Moaveni
11
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
Identification Methods for Structural Systems
by: Dr B. Moaveni
12
Matrix Realization
1H PQ
Identification Methods for Structural Systems
by: Dr B. Moaveni
13
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
Identification Methods for Structural Systems
by: Dr B. Moaveni
14
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
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سازه دو طبقه: مثال
Identification Methods for Structural Systems
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.:
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سازه دو طبقه: مثال
Identification Methods for Structural Systems
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);
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سازه دو طبقه: مثال
Identification Methods for Structural Systems
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
18
سازه دو طبقه: مثال
Identification Methods for Structural Systems
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
19
سازه دو طبقه: مثال
Identification Methods for Structural Systems
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
20
سازه دو طبقه: مثال
Identification Methods for Structural Systems
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
سازه دو طبقه: مثال
Identification Methods for Structural Systems
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
Identification Methods for Structural Systems
by: Dr B. Moaveni
22
Extracting M, C and K
Identification Methods for Structural Systems
by: Dr B. Moaveni
23
Extracting M, C and K
Identification Methods for Structural Systems
by: Dr B. Moaveni
24
Extracting M, C and K
Identification Methods for Structural Systems
by: Dr B. Moaveni
25
Extracting M, C and K
Identification Methods for Structural Systems
by: Dr B. Moaveni
26
Extracting M, C and K
Identification Methods for Structural Systems
by: Dr B. Moaveni
27
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.