7.2 Damage detection residual based on robust normalization
7.2.5 Application and computation of the damage detection tests
tnpglobalandtglobalis addressed. Subsequently, both tests are validated on a numerical example.
7.2.5.1 Covariance of the residualΣζ
First, the covariance of the residual (7.24) is developed. A general expression for the first order perturbation of ˆζwrites
∆ ˆζθ ≈√ Nh
JHˆθ∗
ref
JHˆθtest
i
"
vec(∆ ˆHθref∗ ) vec(∆ ˆHθtest)
#
. (7.29)
A final expression for the covariance of the residual is given in the following Lemma.
Lemma 7.6 The covariance of the residual yields Σζ= lim E
(∆ ˆζθ)∆( ˆζθ)T
(7.30)
=JHref(θ∗)ΣHref(θ∗)JHTref(θ∗)+JHtest(θ∗)ΣHtest(θ∗)JHTtest(θ∗) ,
whereJHref(θ∗) =I(p+1)r⊗UsUsT−Iqr0⊗I(p+1)r andJHtest(θ∗) = (Ztest† Zref)T⊗ UkerUkerT .
7.2 Damage detection residual based on robust normalization 113
Proof: For derivation of the necessary Jacobians, see Appendix E.2.
An estimate of the asymptotic covariance of the residual ˆΣζ can be computed with consistent estimates of matrices used in Lemma 7.6. A consistent estimate of the asymptotic covariance of Hankel matrices can be found e.g. in [D¨oh11].
7.2.5.2 Example of non-parametric χ2 test
This section presents the results from the non-parametric test, introduced in Proposi-tion 7.27, evaluated on the numerical simulaProposi-tions of the chain-like system introduced in Section 7.1.1.
reference healthy 5% damage10% damage
101 102 103 104
Q = I Q = randn(ndof,ndof)2 Q = I Q = randn(ndof,ndof)2
reference healthy 5% damage10% damage
104 105 106 107
Figure 7.5: Damage detection with non-parametric residual based on robust normalization.
Three randomly chosen excitation levels in the tested data sets (left). Random excitation in every data set (right).
Figure 7.5 (left) displays the evolution of the test over time for three different states of the system, namely one healthy and two damage states. As it can be seen, the test is able to separate the different damages despite being computed for different values of the excitation. Figure 7.5 (right) displays the same type of information forQ changing for each simulated data set. That shows that the damage can be identified without any prior knowledge of the excitation properties.
To illustrate the distribution of the test statistics a numerical Monte Carlo simula-tions are conducted for the considered excitation cases.
102 103
2 test values 10-5
10-4 10-3 10-2
probability
Q = randn(ndof,ndof)2 Q = I Q = randn(ndof,ndof)2 mean value
102 103 104
2 test values 10-5
10-4 10-3 10-2
probability
histogram tested data sets mean value
Figure 7.6: Distribution of non-parametric residual based on robust normalization. Three randomly chosen excitation levels in the tested data sets (left). Random excitation in every data set (right).
The Monte Carlo simulations exhibit the distribution of the test for different excitation matricesQ, first for three selectedQ=Q1, Q2, Q3 illustrated on Figure
7.6 (left). There, in the healthy case, all three distributions are superposed, showing the robustness of the test to the excitation changes. As for the damaged cases, slight changes in the mean of the distribution can be seen between the histograms of the pair Q2,Q3andQ1, but still different damage levels are visibly separated. Figure 7.6 (right) shows the capability of the test whenQis changing all the time for each simulated data set and is in principal unknown. It can be observed that the distributions of the test for safe and damage states are well separated and different damage levels can be distinguished.
Notice that the mean value of the test distribution when the structure is safe is high and unpredictable. This is due to the lack of Jacobian in the computation of the tnpglobaltest, see Proposition 7.27. The Jacobian takes into account the sensitivity of the test with respect to each of the system parameters. Therefore, when using such Jacobian the mean value of the test can be predicted, and a threshold for damage can be theoretically established.
In the next section, an analytical derivation of the JacobianJθζ∗ is given. This will yield the full expression oftglobaland its evaluation on the former numerical example.
7.2.5.3 Residual sensitivity with respect to system parameterJθζ∗ This section is devoted to the derivation of Jacobian Jθζ∗. Let θ0 andθ00 be given variables in the parameter space. Defineh
Href(θ0) Htest(θ00)
i∈R(p+1)r×2qr0 as a matrix constructed from juxtaposed exact Hankel matrices, both describing state of the system under respective parametersθ0andθ00. Its SVD writes as
h
Href(θ0) Htest(θ00) i
(7.31)
=Us(θ0, θ00)Ds(θ0, θ00)h
Vs,refT (θ0, θ00) Vs,testT (θ0, θ00) i +Uker(θ0, θ00)Dker(θ0, θ00)h
Vker,refT (θ0, θ00) Vker,testT (θ0, θ00) i
. Now introduce parametrized normalization matrices such that
Zref(θ0, θ00) =Ds(θ0, θ00)Vs,refT (θ0, θ00), Ztest(θ0, θ00) =Ds(θ0, θ00)Vs,testT (θ0, θ00).
Then the theoretical residual can be written as a function of two variablesθ0andθ00as ζ(θ0, θ00) = vec
Htest(θ00)Ztest† (θ0, θ00)Zref(θ0, θ00)− Href(θ0)
. (7.32)
Since the parametrized residualζ(θ0, θ00) is derived with respect to the system parame-ter the notation ofZref(θ0, θ00) andZtest(θ0, θ00) is kept for the clarity of the derivative.
The expression ofζ(θ0, θ00) yields
∀θ0=θ00=θ∗−→ζ(θ0, θ00) = 0.
The derivative ofζ(θ0, θ00) with respect to the system parametersθ0 andθ00evaluated at the point (θ∗, θ∗) coincide with the Jacobian matrixJθζ∗.
Lemma 7.7 JacobianJθζ∗ can be expressed as Jθζ∗=
(Zref† Zref)T⊗UkerUkerT
JθH∗ref , (7.33)
7.2 Damage detection residual based on robust normalization 115
where JθH∗ref = vec (∂Href/∂θ (θ∗)), Zref =Zref(θ∗, θ∗), Ztest =Ztest(θ∗, θ∗) and Uker=Uker(θ∗, θ∗). Consistent estimates of these matrices can be used in the estima-tion ofJθζ∗.
Proof: See Appendix E.3.
Computation of the JacobianJθH∗ref can vary based on the chosen parametrization of the structural system, which can be done e.g. with the modal parameters identified through the system identification, modal parameters of an analytic model of the system or any other parameter with a functional relation between, in this case, the parametric Hankel matrix and the system model. A review of different parametrization schemes can be found in [All17]. In this chapter the chain system is parametrized with its analytical stiffness, which is connected to the parametric Hankel matrixHref(θ∗) by the following chained sensitivities
JθH∗ref =JΓHrefJλ,φΓ Jµ,ψλ,φJθµ,ψ∗ ,
whereJΓHref is the sensitivity of the Hankel matrix with respect to the parametric observability matrix,Jλ,φΓ is the sensitivity of the parametric observability with respect to the continuous poles of the system,Jµ,ψλ,φis the Jacobian of the eigenvalues of the continuous system with respect to its discrete eigenvalues andJθµ,ψ∗ is the Jacobian of the discrete eigenvalues of the system with respect to the selected system parameter.
Computation of the following Jacobians is not elaborated in this chapter. For a detailed computation schemes see [All17, D¨oh11, BBM+08].
7.2.5.4 Example of parametricχ2 test
Here the capabilities of the parametric test are investigated on the numerical example.
The protocol is very similar to the one in Section 7.2.5.2 and illustrates the test formulated in (7.28).
reference healthy 5% damage10% damage
100 101 102 103 104
Q = I Q = randn(ndof,ndof)2 Q = I Q = randn(ndof,ndof)2
reference healthy 5% damage 10% damage
100 101 102 103 104
Figure 7.7: Damage detection with parametric residual based on the robust normalization.
Three randomly chosen excitation levels in the tested data sets (left). Random excitation in every data set (right).
The results illustrated on Figure 7.7 (left) are also quite similar to the ones depicted on Figure 7.6, where values of the damage detection test for all three different Q are well separated and the damage is well detected. As expected, no false alarms are present in the reference state. The mean value of the test corresponds to the rank of the Jacobian matrixJθζ∗ which yields the number of the independent system parameters (in this case equal to 6). The test values for completely unknown and
changingQare plotted on Figure 7.7 (right) and show a good capability for the test to be robust to false alarms in the reference (safe) state and to detect damages in both damage states. For a final depiction of the robustness of the proposed test the Monte Carlo simulations are conducted.
100 101 102 103
2 test values 10-5
10-4 10-3 10-2 10-1
probability
Q = randn(ndof,ndof)2 Q = I Q = randn(ndof,ndof)2 expected value mean value
100 101 102 103 104
2 test values 10-5
10-4 10-3 10-2 10-1
probability
histogram tested data sets mean value expected value
Figure 7.8: Distribution of parametric residual based on robust normalization. Three randomly chosen excitation levels in the tested data sets (left). Random excitation in every data set (right).
Figure 7.8 (left) shows the histograms of the test values obtained from the Monte Carlo simulations. It can be observed that the parametric test is very robust to the changes in the excitation covariance and that the test values from the damaged structure are very well separated from the ones corresponding to the safe structure.
Similar information is shown on Figure 7.8 (right), where a good separation between test values computed on the safe structure and the test values due to different level of damages can be observed.
Contrary to the non parametric test, the mean of the distribution of thetglobal
test computed on the safe structure is both stable and very close to its theoretical value equal to 6. In addition to that, a threshold for assessing that damage occurred in the structure can be determined a priori based on the theoretical distribution of the residual under H0 and its quantiles.
Notice also that the mean of the test is still showing some fluctuations due toQin the damage states. That is because the covariance of the residual Σζ depends onQ.
Albeit not preventing to detect damages, in this case, these fluctuations might affect the ability of the test to separate between different damage levels.