1. 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, Trento, Italy

2. Model Based Damage Detection Schemes
Data
Model-A
Interrogate changes in the models
New Data
Model-B
measurement +
residual
This Paper -
Reference signal computed using model A

3. res has contributions from damage and from non-damage related causes.y
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
healthy
damaged
poor
good

4. 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.

5. Open Loop Residual Generator
y(t)
u(t)
(t)
(t) + -
Open Loop Residual (OL)
I-O model
system
Data in a reference condition

6. In Equation Form: OL - residuals Equivalently
measurements from actual system = y(k) +
OL - residuals -
model
Equivalently
model

7. OL and ARX residuals: measurements from actual system = y(k) +
OL - residuals -
measurements from actual system = y(k) +
ARX- residuals -
How is the resolution of the damage detection affected by selection of the OL or ARX structures (if any)?

9. Taking 0 = -I
the ARX residuals are given by the convolution of the OL ones with the AR part of the reference model.

10. OLARX Transfer Matrix:
The OL ARX transfer matrix is then
Recall that the Laplace – Z connection is:
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)

11. 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

12. Transmission Zeros (TZ):
The values of z for which the matrix on the left side looses rank.

13. 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.
(z-transform and rearranging)
1- evaluation on the unit circle is what matters – damping can play an important role.

14. Large attenuation from OL to ARX residual in an un-damped system is at the frequencies of the reference model.
Output measurement
Input
1.5k k
1.2k m 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

15. OL ARX undamaged frequencies damaged frequencies
frequency (Hz)
Amplitude of Residual Fourier OL
ARX

16. 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.

17. Same Scalar Example: 5% 2%
undamped
freq (Hz)
Note that the zero connected with the second pole is not evident on the imaginary even at 2% damping.

18. Zero Directions: 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.
Inspection shows that the zero directions coincide with the complex eigenvectors of the system (at the sensor locations).

19. “Direction” of the Damaged OL Residual
Pseudo-force perspective (undamaged reference)

20. Pre-multiplying by the matrix that selects the measured coordinates
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.

21. Example: 5DOF system

22. Absolute Magnitude of OLARX Transfer for a 5DOF system with a single output sensor (over specified order)
Undamped
f(Hz) 5 10 15 20 25
0.4
0.8
1.2
1.6 2
System natural frequencies

23. OLARX as a function of the order of the ARX model - SDOF
Undamped 1 2 3 4 5 6 7 8 9  0
n=2
n=4
n=10
n=20
As the order increases the transfer seems to approach an all pass filter with a notch at the natural frequency.

24. The frequency composition of the contribution of damage and spurious sources to the OL residual vector is the piece that completes the discussion.

25. Contributions to the OL Residual:+ - 
+ v u 
(a) u  + + - u
+ +
(b) - + 
+ v
(d)  - + 
+
(c)
damage dependent

26. 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.

27. Some Numerical Results:

28. 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)

29. CASE 1
Metric is the 2-Norm of the residual normalized by the 2-norm of the associated measurement

30. CASE 2
Metric is the 2-Norm of the residual normalized by the 2-norm of the associated measurement
Sum of all 3 outputs

31. 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.