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

New Data Model-B

Interrogate changes in the models

Reference signal computed using model A

measurement

+

-residual This Paper

y

yˆres y 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.

poor good

pdf pdfdamagedhealthy

( )g res( )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.

y(t)u(t)

(t)

(t)

++

u(t)

-

+Open Loop Residual (OL)

I-O model

system

Open Loop Residual Generator

Data in a reference condition

0

ˆ( ) ( ) ( )k

j

h k j u j y k

model

measurements from actual system = y(k)+

-OL - residuals

1 0

ˆ ˆ( ) ( ) ( )n n

j jj j

y k j u k j y k

model

Equivalently

In Equation Form:

measurements from actual system = y(k)+

-OL - residuals

1 0

ˆ ˆ( ) ( ) ( )n n

j jj j

y k j u k j y k

measurements from actual system = y(k)+

-

ARX- residuals

1 0

ˆ( ) ( ) ( )n n

j jj j

y k j u k j y k

How is the resolution of the damage detection affected by selection of the OL or ARX structures (if any)?

OL and ARX residuals:

1 0

( ) ( ) ( ) ( )n n

j j ARXj j

y k y k j u k j k

1 0

ˆ( ) ( ) ( ) ( )n n

j j OLj j

y k y k j u k j k

1 0

( ) ( ) ( ) ( )n n

j j ARXj j

y k y k j u k j k

1 0

ˆ( ) ( ) ( ) ( )n n

j j OLj j

y k y k j u k j k

0 1

ˆ( ) ( ) ( ) ( )n n

j OL jj j

u k j k y k y k j

1 1

ˆ( ) ( ) ( ) ( )n n

j OL j ARXj j

y k j k y k j k

1 1

ˆ( ) ( ) ( ) ( )n n

j OL j ARXj j

y k j k y k j k

ˆ( ) ( ) ( )OLy k j y k j k j

0

( ) ( )n

j OL ARXj

k j k

Taking 0 = -I

the ARX residuals are given by the convolution of the OL ones with the AR part of the reference model.

0

( ) ( )n

j OL ARXj

k j k

0

( ) ( )n

jOL j ARX

j

z z z

The OL ARX transfer matrix is then

0

( )n

jj

j

T z z

There are no finite poles in the s-plane (but there are zeros)

s tz e

Recall that the Laplace – Z connection is:

( ) ( )X s X z t

negative of the z-transform of the AR part of the model

OLARX Transfer Matrix:

11 1 ,1 ,1

, ,

( ) . ( ) ( ) ( )

. . . . .

. . ( ) ( ) ( )

m OL ARX

mm OL m ARX m

t z t z z z

t z z 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

Transmission Zeros (TZ):

The values of z for which the matrix on the left side looses rank.

11 1 ,1 ,1

, ,

( ) . ( ) ( ) ( )

. . . . .

. . ( ) ( ) ( )

m OL ARX

mm OL m ARX m

t z t z z z

t z z z

( ) 0T z

0

( )n

jj

j

T z z

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.

0 0

( ) ( ) 0n n

j jj j

y k j u k j

(z-transform and rearranging)1

0 0

( ) ( )n n

j jj j

j j

y z z z u z

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.

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.

0.1% classical damping

m m m

1.5k k 1.2k

Output measurement

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)

0.792.253.04

0.462.062.53

Input

0 0.5 1 1.5 2 2.5 3 3.5 410

-4

10-3

10-2

10-1

100

101

102

frequency (Hz)

Am

plitu

de o

f Res

idua

l Fou

rier

damaged frequencies

undamaged frequencies

ARX

OL

1

Z-plane S-plane

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.

Damping can Play an Important Role:

0 1 2 3 4 5 6 7 8 9 1010

-4

10-3

10-2

10-1

100

101

T

freq (Hz)

5%

2%

undamped

Note that the zero connected with the second pole is not evident on the imaginary even at 2% damping.

Same Scalar Example:

Zero Directions:

( ) 0T z 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.

1

0 0

( ) ( )n n

j jj j

j j

y z z z u z

0

( )n

jj

j

T z 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

( ) ( )dMq C q K K q u t

uq q q

( ) ( ) ( )dM q C q K q K q

Pseudo-force perspective (undamaged reference)

12( ) ( )dq s M s C s K K q s

( ) ( ) ( )dM q C q K q K q

2*

1

( ) ( )Tdof

j j

j j

q s K q ss

2*

1,1

( )( )

Tdofj

OL jj j

K q ss

s

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

f(Hz)

T

0 5 10 15 20 250

0.4

0.8

1.2

1.6

2

Absolute Magnitude of OLARX Transfer for a 5DOF system with a single output sensor (over specified order)

System natural frequencies

Undamped

0 1 2 3 4 5 6 7 8 90

1

2

3

4

0

n=2

n=4

n=10n=20

T

OLARX as a function of the order of the ARX model - SDOF

As the order increases the transfer seems to approach an all pass filter with a notch at the natural frequency.

Undamped

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

(b)

+

-

++

+

v

u

u

OL

(a)

+

-

++

u

u

d+

-

+

+

(c)

-

+

++

+

v

v

(d)

OL d

Contributions to the OL Residual:

damage dependent

OL d

Very narrow band – because the system is narrow band and the “damage equivalent” excitation (the pseudo-forces) is rich in the system frequencies.

Narrow band but not as much as the damage.

Typically broad band.

Some Numerical Results:

Monte Carlo Examination:

10% stiffness loss

Deterministic input at coordinate#3Unmeasured disturbances at all coordinates (2-5% of the RMS of the deterministic input)

Measurement noise (5% RMS of the associated measurement)

200 simulations

Case1- One output at coordinate #4

Case2- Output sensors at {1,3,5}

Stiffness proportional damping -1% in the first mode

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

Metric is the 2-Norm of the residual normalized by the 2-norm of the associated measurement

CASE 2

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.