An Examination of the ARX as a Residuals
Generator for Damage Detection
Dionisio Bernal1, Daniele Zonta2, Matteo Pozzi2
1Northeastern
University, Civil and Environmental Engineering Department, Center for Digital
Signal Processing, Boston, 02115
2University
of Trento, DIMS, via Mesiano 77, 38050 Trento, Italy
Model Based Damage Detection Schemes
Data
Model-A
Interrogate changes in the models
New Data
Model-B
measurement
+
-
residual
Reference signal computed using model A
This Paper
y
res  y  yˆ
ŷ
res has contributions from damage
and from non-damage related
causes.
One would like, of course, to make choices that accentuate the contribution
of damage over that of the non-damage sources.
pdf
pdf
poor
g ( res )
healthy
good
damaged
g ( res )
The objective here IS NOT to propose a particular residual
based metric or to propose a specific damage detection
approach but to gain some insight into the implication of
using an open loop or a closed-loop residual generator.
Open Loop Residual Generator
(t)
u(t)
(t)
system
+
+
y(t) +
-
u(t)
Open Loop Residual (OL)
I-O
model
Data in a reference condition
In Equation Form:
measurements from actual system = y(k)
+
OL - residuals
k
 h(k  j)u( j)  yˆ (k )
j 0
n

j 1
-
model
n
j
yˆ (k  j )    j u (k  j )  yˆ (k )
j 0
model
Equivalently
OL and ARX residuals:
measurements from actual system = y(k)
+
OL - residuals
n
 n

n
n
y(k )    j yˆˆ(k  j )    j u (k  j )    OL (k )
ˆ
 j y(k  j )   j u (k j j1)  y(k )
j 0

j 1
j 0


measurements from actual system = y(k)
+
n

j 1
n
j
y (k  j )  
j 0
n
 n
ARX- residuals
-  j u (k  j )    ARX (k )
y (k )    j y (k  j )  
 j u (k j j1)  yˆ (k ) j 0

How is the resolution of the damage detection affected by selection of the
OL or ARX structures (if any)?
n
 n

y(k )    j y(k  j )    j u (k  j )    ARX (k )
j 0
 j 1

n
 n

y(k )    j yˆ (k  j )    j u (k  j )    OL (k )
j 0
 j 1

n
  u(k  j)  
j
j 0
n
OL
(k )  y(k )   j yˆ (k  j )
j 1
n
n
j 1
j 1
 j y(k  j )   OL (k )   j yˆ (k  j )   ARX (k )
n
n
j 1
j 1
 j y(k  j )   OL (k )   j yˆ (k  j )   ARX (k )
y(k  j )  yˆ (k  j )   OL (k  j )
Taking 0 = -I
n
 
j 0
j OL
(k  j )   ARX (k )
the ARX residuals are given by the convolution of the OL ones with
the AR part of the reference model.
OLARX Transfer Matrix:
n
 j OL (k  j)   ARX (k )
j 0
The OL ARX transfer matrix is then
Recall that the Laplace – Z
connection is:
z  e st
X ( s )  X ( z )t
n
 OL ( z )   j z  j   ARX ( z )
j 0
n
T ( z )    j z  j
j 0
negative of the z-transform of
the AR part of the model
There are no finite poles in the s-plane
(but there are zeros)
t11 ( z ) . t1m ( z )    OL ,1 ( z )    ARX ,1 ( z ) 

 

 .

.
.  .  
.


 .
. tmm ( z )   OL ,m ( z )   ARX ,m ( z ) 
The roots of the polynomials that define the entries of T(z)
are zeros.
Here we focus not on the individual zeros of the transfer
functions but on the Transmission zeros of the OL ARX
transfer matrix as whole
t11 ( z ) . t1m ( z )    OL ,1 ( z )    ARX ,1 ( z ) 

 

 .

.
.  .  
.


 .
. tmm ( z )   OL ,m ( z )   ARX ,m ( z ) 
Transmission Zeros (TZ):
The values of z for which the matrix on the left side looses
rank.
T ( z )  0
Transmission Zeros of T
The transmission zeros determine at which frequencies the ARX structure
is likely1 to lead to large attenuation of the OL residuals.
The poles of the system in the
undamaged state (i.e., the values
where y(z) goes to infinity) are
the zeros of the OL ARX
transfer matrix.
n
T ( z )    j z  j
j 0
n

j 0
n
j
y (k  j )    j u (k  j )  0
j 0
 n
j 
y( z )    j z 
 j 0

1
(z-transform and rearranging)
n
j

z
 j u( z)
j 0
evaluation on the unit circle is what matters –
damping can play an important role.
1-
Large attenuation from OL to ARX residual in an un-damped
system is at the frequencies of the reference model.
Output measurement
Input
k
1.5k
m
1.2k
m
m
0.1% classical damping
Damage simulated as a reduction from k to 0.2 k (very large to separate
frequencies and make the effect clear)
Undamaged System Frequencies
(Hz)
Damaged System Frequencies
(Hz)
0.79
2.25
3.04
0.46
2.06
2.53
undamaged frequencies
damaged frequencies
2
Amplitude of Residual Fourier
10
1
10
0
10
-1
OL
10
-2
10
ARX
-3
10
-4
10
0
0.5
1
1.5
2
2.5
3
3.5
frequency (Hz)
4
Damping can Play an Important Role:
Z-plane
S-plane
1
The OL ARX relation in time is determined by the behavior on the unit
circle in the Z-plane or on the imaginary in the s-plane. Damping moves
the zeros away from the unit circle and this makes inference more difficult.
Same Scalar Example:
1
10
T
5%
0
10
2%
-1
10
undamped
-2
10
-3
10
-4
10
0
1
2
3
4
5
6
7
8
9
10
freq (Hz)
Note that the zero connected with the second pole is not evident on
the imaginary even at 2% damping.
Zero Directions:
T ( z )  0
Are the vectors 
The Zero directions are of interest because in the multidimensional case what gets
annihilated at the TZ is the projection of the OL residual in this direction.
n
T ( z )    j z  j
j 0

j 
y( z )    j z 
 j 0

n
1
n
 z
j 0
j
j
u( z)
Inspection shows that the zero
directions coincide with the complex
eigenvectors of the system (at the
sensor locations).
“Direction” of the Damaged OL Residual
Pseudo-force perspective (undamaged
reference)
M q  Cd q  ( K  K )q  u (t )
q  qu   q
M ( q )  Cd ( q )  K ( q )  K  q
M ( q )  Cd ( q )  K ( q )  K  q
1
 q(s)    M  s  Cd  s  K  K  q(s)
2
2*dof
 q( s )   
j 1
Pre-multiplying by the matrix
that selects the measured
coordinates
 j Tj
K  q(s)
s  j
 Tj  K  q(s)
 OL (s)     1, j
s  j
j 1
2*dof
The OL residual vector near the jth undamaged pole has a shape that is
dominated by the jth mode
Since the TZ direction coincides with the eigenvector of the undamaged
system the OLARX attenuates the full OL residual vector at the
undamaged poles.
Example: 5DOF system
Absolute Magnitude of OLARX Transfer for a 5DOF
system with a single output sensor (over specified order)
Undamped
2
T
1.6
1.2
0.8
0.4
0
0
5
10
15
f(Hz)
System natural frequencies
20
25
OLARX as a function of the order of the ARX
model - SDOF
Undamped
4
T
n=2
3
n=4
2
n=20
n=10
1
0
0
1
2
3
4
5
6
7

0
8
9
As the order increases the transfer seems to approach an all pass filter with a notch
at the natural frequency.
The frequency composition of the contribution
of damage and spurious sources to the OL
residual vector is the piece that completes the
discussion.
Contributions to the OL Residual:
u
u


+
v
u

+
+
(a)
 OL
-
+
-
u
+
+
(b)
+


-

v
+

+
+
+
(c)
d
+
(d)
+
+
 OL   d     
damage dependent
v
 OL   d     
Typically broad band.
Narrow band but not as much as the damage.
Very narrow band – because the system is narrow band and the “damage
equivalent” excitation (the pseudo-forces) is rich in the system frequencies.
Some Numerical Results:
Monte Carlo Examination:
Case1- One output at coordinate #4
Case2- Output sensors at {1,3,5}
200 simulations
Stiffness proportional damping -1% in the first mode
10% stiffness loss
Deterministic input at coordinate#3
Unmeasured disturbances at all coordinates (2-5% of the RMS of the
deterministic input)
Measurement noise (5% RMS of the associated
measurement)
CASE 1
Metric is the 2-Norm of the residual normalized by the 2norm of the associated measurement
CASE 2
Metric is the 2-Norm of the residual normalized by the 2norm of the associated measurement
Sum of all 3 outputs
Summary and Closing Remarks
•The ARX residuals are the sum filtered version of the OL
ones.
•The essential feature of the filtering is attenuation near
the resonant frequencies of the undamaged system.
• The damage contribution to the OL residual is the most
narrow band of all, and it is concentrated near the poles this suggests that the OL is likely to offer better contrast in
many cases
• numerical results, although limited in scope, appear to
support the previous statement.