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 This Paper - Reference signal computed using model A
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
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 y(t) u(t) (t) (t) + - Open Loop Residual (OL) I-O model system Data in a reference condition
In Equation Form: OL - residuals Equivalently measurements from actual system = y(k) + OL - residuals - model Equivalently model
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)?
Taking 0 = -I the ARX residuals are given by the convolution of the OL ones with the AR part of the reference model.
OLARX 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)
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
Transmission Zeros (TZ): The values of z for which the matrix on the left side looses rank.
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.
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
OL ARX undamaged frequencies damaged frequencies frequency (Hz) Amplitude of Residual Fourier OL ARX
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: 5% 2% undamped freq (Hz) Note that the zero connected with the second pole is not evident on the imaginary even at 2% damping.
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).
“Direction” of the Damaged OL Residual Pseudo-force perspective (undamaged reference)
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 OLARX attenuates the full OL residual vector at the undamaged poles.
Example: 5DOF system
Absolute Magnitude of OLARX 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
OLARX 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.
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: + - + v u (a) u + + - u + + (b) - + + v (d) - + + (c) damage dependent
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 2-norm of the associated measurement
CASE 2 Metric is the 2-Norm of the residual normalized by the 2-norm 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.