### 4.2.4 Residual covariance

It is assumed that the testing data under parameter $P$ can be described by the interpolated state-space model (4.11). To evaluate changes in the corresponding residual (4.18) at parameter $P$ with the statistical test in (4.19) the residual covariance at external parameter $P$ is required. This is derived in accordance to (3.30) from the covariance of the underlying Hankel matrix $\Sigma_{\mathcal{H}(P)}$ at parameter $P$. It is either re-computed from the current data set $\mathcal{Y}_N(P)$ in the testing state, or it is related to the covariances of the reference data under parameters $\{P_1, P_2, \dots, P_u\}$ as follows.

Suppose that under each reference parameter $P_j$ a data set $\mathcal{Y}_{j,M}$ of length $M$ is available in the reference state of the structure. The Hankel matrix estimate $\hat{\mathcal{H}}_j$ obtained from $\mathcal{Y}_{j,M}$ is asymptotically Gaussian distributed with (cf. (3.5))

$$ \sqrt{M} \mathrm{vec}(\hat{\mathcal{H}}_j - \mathcal{H}_j) \rightarrow \mathcal{N}(0, \Sigma_{\mathcal{H}_j}), \qquad (4.20) $$

where $\Sigma_{\mathcal{H}_j}$ is the asymptotic Hankel matrix covariance that can be easily estimated by means of the sample covariance as in (2.36). Using (4.13) the covariance of the Hankel matrix under $P$ can be approximated as a weighted mean of the Hankel matrix covariances under $P_j$, $j = 1, \dots, u$, as

$$ \underline{\Sigma}_{\mathcal{H}(P)} = \left( \sum_{j=1}^{u} \rho_j^4(P) \right)^{-1} \sum_{j=1}^{u} \rho_j^4(P) \Sigma_{\mathcal{H}_j}, \qquad (4.21) $$

where the factor in front of the sum ensures that the sum of the weights is one.

With this result, the asymptotic covariance of residual (4.18) can be developed. With an uncertainty afflicted reference null space, the covariance related to the null space estimate as well as the covariance related to the testing Hankel matrix estimate contribute to the residual covariance, denoted by $\underline{\Sigma}_1(P)$ and $\underline{\Sigma}_2(P)$, respectively. Following (3.30), the residual covariance yields

$$ \tilde{\underline{\Sigma}}_{\zeta}(P) = \frac{N}{M} \underline{\Sigma}_1(P) + \underline{\Sigma}_2(P), \qquad (4.22) $$

where

$$ \underline{\Sigma}_1(P) = (\underline{\mathcal{H}}(P)^T \otimes I_s) \mathcal{P}_{t,s} \underline{\Sigma}_{S(P)} \mathcal{P}_{t,s}^T (\underline{\mathcal{H}}(P) \otimes I_s), $$

$$ \underline{\Sigma}_2(P) = (I_{qr} \otimes \underline{S}(P)^T) \underline{\Sigma}_{\mathcal{H}(P)} (I_{qr} \otimes \underline{S}(P)). $$
$P_{t,s}$ is the permutation matrix in (2.42), $t$ and $s$ are the number of rows and columns of $\underline{S}(P)$. The asymptotic covariance matrix $\underline{\Sigma}_{S(P)}$ of the estimated reference null space is derived from the covariance of the underlying Hankel matrix and depends on the strategy that is adopted for the left reference null space computation in section 4.2.2.

**Residual covariance for strategy 1** When the left reference null space is obtained from the interpolated Hankel matrix $\underline{\mathcal{H}}(P)$, $\underline{\Sigma}_{S(P)}$ is directly related to $\underline{\Sigma}_{\mathcal{H}(P)}$ with the sensitivity in (3.24). It writes as

$$ \underline{\Sigma}_{S(P)} = \mathcal{J}_{\underline{S}(P), \underline{\mathcal{H}}(P)} \underline{\Sigma}_{\mathcal{H}(P)} \mathcal{J}_{\underline{S}(P), \underline{\mathcal{H}}(P)}^{T}, \quad (4.23) $$

with $\mathcal{J}_{\underline{S}(P), \underline{\mathcal{H}}(P)} = -\mathcal{P}_{s,t}(U_1 D_1^{-1} V_1^T \otimes U_1^T)$, where $U_1$, $D_1$, and $V_1$ are the column space, the matrix with the non-zero singular values, and the row space that are computed from $\underline{\mathcal{H}}(P)$ in the reference state.

**Residual covariance for strategy 2** When the left reference null space is obtained from an observability matrix $\underline{\mathcal{O}}(P)$ computed from interpolated modal parameters, the covariance of the estimates of the modal parameter vectors $\theta_j^*$, $j = 1, \dots, u$ is first related to their interpolation $\theta^*(P)$, then to the observability matrix $\underline{\mathcal{O}}(P)$, and finally to its left null space to obtain $\underline{\Sigma}_{S(P)}$ as follows: In the first step, the covariance $\underline{\Sigma}_{\theta_j^*}$ of the modal parameter vectors under each reference parameter $P_j$ is obtained with respect to the used system identification method, e.g. for stochastic system identification (SSI) with the sensitivities from (2.53), (2.56), and (2.58) to (2.60) leading to

$$ \underline{\Sigma}_{\theta_j^*} = \mathcal{J}_{\theta_j^*, \mathcal{H}_j} \underline{\Sigma}_{\mathcal{H}_j} \mathcal{J}_{\theta_j^*, \mathcal{H}_j}^{T}. \quad (4.24) $$

Then, the covariance of the interpolated modal parameters follows similarly to (4.21) as

$$ \underline{\Sigma}_{\theta^*(P)} = \left( \sum_{j=1}^{m} \rho_j^2(P) \right)^{-1} \sum_{j=1}^{m} \rho_j^2(P) \underline{\Sigma}_{\theta_j^*}. \quad (4.25) $$

The covariance of the observability matrix follows as $\underline{\Sigma}_{\underline{\mathcal{O}}(P)} = \mathcal{J}_{\underline{\mathcal{O}}(P), \underline{\theta}^*(P)} \underline{\Sigma}_{\underline{\theta}^*(P)} \mathcal{J}_{\underline{\mathcal{O}}(P), \underline{\theta}^*(P)}^T$, where the sensitivity $\mathcal{J}_{\underline{\mathcal{O}}(P), \underline{\theta}^*(P)}$ of the observability matrix towards the modal parameters is detailed in (2.64). Finally, the covariance of its left null space $\underline{\Sigma}_{S(P)}$ is directly related to $\underline{\Sigma}_{\underline{\mathcal{O}}(P)}$ with (3.24), where $U_1$, $D_1$, and $V_1$ are the column space, the matrix with the
non-zero singular values, and the row space that are computed from $\mathcal{O}(P)$ in the reference state

**Residual covariance for strategy 3** In the third strategy the left reference null space is similarly obtained from the interpolated observability matrix $\mathcal{O}(P)$, but where $\theta^*(P)$ is computed on a regression analysis instead of interpolation. At this point it is assumed that the covariance computation is only slightly affected by the exact modal parameter uncertainties related to the estimate of $\theta^*(P)$, and therefore in this approach the same covariance computation for $\Sigma_{S(P)}$ is carried out as described for strategy 2. The weights are not very much different, since both consider a linear behavior between two neighboring reference temperature.

### 4.2.5 Discussion and comparison to previous approaches

The developed strategies account for environmental variation in subspace-based damage detection by creating an appropriate reference under the current parameter $P$ of the testing state, based on reference measurements at values {$P_1, P_2, \dots, P_u$} that are in general different from $P$. Using the knowledge of the external parameter in addition to the vibration measurements (e.g., the temperature needs to be measured), these strategies have the potential to define a more precise reference for the current environmental condition than other approaches that mix data blindly from different but unknown environmental conditions for the definition of a reference. In particular, in the context of subspace-based damage detection under environmental variation, a previously developed approach is the *average approach* from [Bal+08a], where the left reference null space is computed on the mean of the Hankel matrices $\frac{1}{u}\sum_{j=1}^{u}\mathcal{H}_j$ that are obtained from measurements in the reference state under different environmental conditions. While this approach provides some robustness in practice when the external parameter is unknown, it is not optimal particularly when the current external parameter is quite far from the mean. Besides, a previous approach based on weighted reference Hankel matrices has been presented in [FKB13], which is similar to the developed *strategy 1*, whereas this work provides in addition a theoretical justification for the Hankel matrix interpolation and derives the adapted covariance computation.

Within the developed temperature rejection strategies, *strategy 1* based on the Hankel matrix interpolation in the reference state is the simplest and the most straight-forward to implement, as it is purely data-driven and modal analysis is not required. *Strategy 2*
requires in addition modal parameter estimation, but only once from the reference mea-
surements. Modal analysis in the testing state is never required. Using their interpolation
to construct an appropriate left reference null space, strategy 2 considers more directly
the well-known effect of environmental variation on the modal parameters than on the
related Hankel matrices in strategy 1. Moreover, with this approach a reference null space
can be computed robustly even if the excitation covariance between the reference data
sets changes, since the eigenstructure is independent from the excitation covariance. This
is also guaranteed with *Strategy 3* that is based on individual regressions of the reference
modal parameters. It is again more complex, and offers in addition an interesting possi-
bility to extrapolate for cases where *P* is outside the interval of the reference parameters,
thanks to the definition of the regression functions. In either case, if structure-specific
behavior in dependence of the environmental parameter is known a priori, it can be taken
into account in the definition of the respective interpolation or regression function.

**4.3 Proof of concept and discussion**

In the previous section, the theory for a SSDD approach robust to changes in ex-
ternal parameters such as temperature was derived. In this section these developments
are validated on numerical simulation data and the results are discussed. The method is
evaluated regarding its damage detection performance and compared to the existing tem-
perature rejection approach using average reference matrices from [Bal+08a] as outlined
in section 4.2.5. The test performance is measured by means of probability of detection
(POD) and the receiver operating characteristic (ROC) curves, that deliver information
on the ratio between POD and probability of false alarms (PFA) for thresholds computed
for different PFA limits, as described in section 2.4.4. The residual is formulated such that
the uncertainty of the reference null space due to its estimation from data is taken into
account, accordingly to the developments in chapter 3.

After introducing the study setup used for the generation of the numerical data, in
section 4.3.2 the test performance is evaluated when the reference null space is computed
with the three proposed robust strategies from the previous section. The test results
are compared to results when no effects from the external parameter are considered in
the reference setup and when the reference null space is computed with the average
method. In this context the impact of the uncertainty consideration of the reference null
space is studied. In section 4.3.3 it is shown how the covariance computation affects the
<figure><img src="image_29.png" /><figcaption>Figure 4.2 – Mass-spring chain and temperature dependent stiffness.</figcaption></figure>

test procedure and its performance, i.e. if the covariance is interpolated or re-computed
in the testing state. In section 4.3.4 the test performance is analyzed with respect to
different reference setups, i.e. for different combinations of reference parameter states and
for different choices of the interpolation function. Finally, it will be shown in section
4.3.5 that the new approach can be applied even when the excitation covariance changes
between the reference states.

4.3.1 Study setup

The numerical data for validation of the theoretical developments from section 4.2 is
obtained from an eight mass-spring-damper as illustrated in Fig. 4.2 (left). It is set up
by masses m₁ = m₃ = m₅ = m₇ = 1 and m₂ = m₄ = m₆ = m₈ = 2 and stiffness
k₁ = k₃ = k₅ = k₇ = 200 and k₂ = k₄ = k₆ = k₈ = 100. Classical damping is defined
such that the damping ratio of all modes is 2%. The structure is excited by white noise at
elements 1, 3, 5 and 7, and velocities are recorded at the same elements with a sampling
frequency of 8 Hz. Measurement noise with 5% standard deviation of the signals is added.

The external parameter is considered to be the temperature. Its effect is simulated by
changing the stiffness of all elements in accordance to a non-linear function as depicted in
Fig. 4.2 (right). This stiffness-temperature relation is much stronger than in real applica-
tions such that the limits of the developed methods can be shown. Damage is simulated
by reducing the temperature dependent stiffness of element 3 by 1.5%.

The test performance of the different strategies that account for the environmental
variation is evaluated in a Monte Carlo simulation with vibration data generated at tem-
peratures between 0°C and 20°C, using 10 data sets with length N = 200,000 at each
temperature step. The respective damage detection methods are set up in the reference
state at selected reference temperatures, as detailed in the following. At these reference
temperatures, 10 additional Monte Carlo data sets are generated for the determination of
the threshold for damage detection, which is set up for 0.1% probability of false alarms.
<figure><img src="image_30.png" /><figcaption>Figure 4.3 – Test values at different testing temperatures computed from a single reference data set at 10°C for the whole temperature range (left) and with zoom on the temperatures close to the reference temperature (right).</figcaption></figure>

4.3.2 Damage detection under changing temperatures

When the conventional SSDD from (3.2) is applied to a dynamical system that depends
on an external parameter, changes in this parameter in the testing state with respect to the
reference state lead to a significant number of false alarms. Fig. 4.3 shows the test values
from the Monte Carlo simulation at different testing temperatures when the reference null
space and the residual covariance are computed from data at only one single reference
temperature T₁ = 10°C, and the associated empirical threshold as described before. At
the reference temperature T₁ = 10°C the test performs well, as can be seen in the zoom
on Fig. 4.3 (right): the test values from data of the reference state are below the threshold
(cyan dots), i.e. no false alarms occur, and the test values from data of the damaged state
are above the threshold (black diamonds), so damage is detected. However, for any testing
temperature different from the reference temperature at 10°C, the test values computed
on data from the undamaged state also react strongly leading to false alarms (red dots)
at all temperatures except the reference temperature. In fact, the test is not able to
distinguish between changes due to damage and changes due to temperature effects. For
temperatures below 10°C, the test values in the damaged state are even lower than the
values in the undamaged state, as the effects of decreasing stiffness caused by damage
seem to be weaker than increasing stiffness from temperature effects. It becomes clear
that a proper setup of the reference and a meaningful threshold is of vital importance for
a reliable and robust damage detection method, especially to avoid false alarms.

In [Bal+08a] a more robust *average approach* for SSDD was derived that has been
discussed in section 4.2.5. The approach uses the averaged Hankel matrices from different
<figure><img src="image_31.png" /><figcaption>Figure 4.4 – Test values at different testing temperatures computed with the *average approach* [Bal+08a].</figcaption></figure>

reference temperature states of the structure for the definition of the left reference null space and the computation of the covariance matrix for the computation of (3.2) and (3.11). In Fig. 4.4 the numerical data is evaluated by means of these averaged matrices considering reference data from the system at five reference temperatures $T_1 = 0°C$, $T_2 = 5°C$, $T_3 = 10°C$, $T_4 = 15°C$, and $T_5 = 20°C$. It shows the test values in the undamaged and the damaged state for testing temperatures between 0°C and 20°C. Compared to the conventional algorithm, the test values for the undamaged and damaged states of the system are better separated for each of the testing temperatures and false alarms can be avoided. However, it is impossible to separate the test values from the undamaged and damaged states with a common threshold for all temperatures. The threshold that is obtained based on the test values from the reference temperature states does only permit the detection of damages at temperatures above 19°C, while the damages at all other temperatures remain undetected. In this respect, the damage sensitivity of the test is poor.

The new approach derived in the previous section uses a temperature adapted reference null space $\mathcal{S}(P)$ in the residual (4.18) providing information on the current testing temperature linked to the temperature depending system behavior to improve the test performance. In the following the robustness of this approach to environmental variation will be evaluated considering the three strategies: *strategy 1* with the left reference null space $\mathcal{S}(P)$ computed on the interpolated Hankel matrix is denoted by “S(H) interpolated” in the figures, *strategy 2* based on the interpolation of modal parameters to retrieve the left reference null space from the parametric observability matrix is denoted by “S(O) interpolated”, and *strategy 3* based on the regression of the modal parameters
<figure><img src="image_32.png" /><figcaption>Figure 4.5 – Test values at different testing temperatures computed with the three strategies of the new method.</figcaption></figure>

is denoted by “S(O) regression”. The considered reference temperatures are $T_1 = 0°C$, $T_2 = 5°C$, $T_3 = 10°C$, $T_4 = 15°C$, and $T_5 = 20°C$, same as for the average approach in Fig. 4.4.

The weights for the interpolation are generated for each testing temperature, considering a Gaussian function with $\sigma^2 = 11$ centered at the testing temperature. For the interpolation close to the borders of the interval of the reference temperatures, where the interpolation function would overlap strongly outside of the interval, the weights of the reference points are re-evaluated for the interpolation assuming a (virtual) linear extrapolation outside the interval.

The statistical test in (4.19) is performed with an interpolated residual covariance from (4.22) considering the Gaussian weighting function used for the interpolation of the reference null space.

Fig. 4.5 shows the test values with the three strategies in the undamaged and the damaged states for all testing temperatures. In the undamaged state the test values at the reference temperatures (cyan dots) are quite stable for each of the strategies. At the
<figure><img src="image_33.png" /><figcaption>Figure 4.6 – Test performance of the average method and the three strategies of the new method. Probability of detection (left) and ROC curves for 0.8 % stiffness reduction (right).</figcaption></figure>

testing temperatures in between the reference temperatures in the undamaged state the test values increase in particular for strategy 1 (“S(H) interpolated”) but remain overall quite stable. This shows that the left reference null space $S(P)$ and the covariance can be computed well for any testing temperature based on the interpolation or regression strategies. From test values at the reference temperatures empirical thresholds can be set up, which are valid globally over all temperatures. In the S(H) strategy about 25% false alarms occur, while in both S(O) strategies the rate is around 1 to 2%. It seems indeed that the temperature dependency of the modal parameters is easier to cover by an interpolation or regression rule, than it is directly for the Hankel matrix. Nevertheless, in all test versions the test values in the damaged state are separated clearly from the reference values. Damage can be detected even for the very small damage of 1.5% stiffness loss at one element.

The test performances for damages in element 3 of increasing extent are shown by the POD in Fig. 4.6 (left) for the average approach and the three developed strategies for robustness to environmental variation. All three developed strategies show good performance for small damages, with the S(O) approaches outperforming the S(H) approach, whereas the previous average approach can detect bigger damages only. This is confirmed by the ROC curves of the three strategies and the average approach in the right plot in Fig. 4.6, where the POD is plotted against the PFA. The POD and PFA values are computed for stiffness reduction of 0.8 % at element 3 varying the allowed false positive rate for the computation of the threshold. The average approach is very close to the line of no performance, that is when the POD equals the PFA. The performances of the new
<figure><img src="image_34.png" /><figcaption>Figure 4.7 – Test values at different testing temperatures when the uncertainties of the reference null space are neglected (left) and ROC curves for 0.8 % stiffness reduction computed with S(H) strategy.</figcaption></figure>

algorithms and especially the S(O) strategies approach the best performance point, which
is where no false alarms occur and the damage of 0.8 % can be detected in any case.

**Neglecting uncertainties in the reference null space** When in the robust damage detection procedure the uncertainties of the reference null space are neglected, i.e. Σ₁ = 0 which corresponds to the origin residual from [BAB00] recalled in (2.24), the damage detection test using the “S(H) interpolation” strategy yields the test values in Fig. 4.7 (left). Compare to Fig. 4.5 (top) the test values increase significantly at any testing temperature and show bigger variance, also regarding the reference temperatures, which leads to a bigger threshold. As a consequence, the test that takes the uncertainties into account shows a much better test performance as illustrated by the ROC curves in Fig. 4.7 (right) computed for stiffness reduction of 0.8 % at element 3, varying the allowed false positive rate for the computation of the threshold.

**4.3.3 Covariance computation from test data**

In statistical damage detection such as in (4.19) the residual covariance $\tilde{\Sigma}_{\zeta}$ is an
important part of the test, since it carries the information of the natural variance of the
residual values stemming from the uncertainties in the underlying data in contrary to
changes due to damage. So far interpolated residual covariances as described in section
4.2.4 have been used in the damage detection test (4.19). Another possibility is to re-
compute the covariance in the testing state from testing data. This leads to increasing
computational effort and Fig. 4.8 shows that there is no significant benefit from this
<figure><img src="image_35.png" /><figcaption>Figure 4.8 – Test values at different testing temperatures when the covariance is computed from the testing data (left) and ROC curves for 0.8 % stiffness reduction computed with S(H) strategy.</figcaption></figure>

procedure. On the left side the test values of the S(H) strategy is shown for a stiffness
reduction of 0.8 %, which behave similarly as in Fig. 4.5 (top). The test performance
measured by the ROC curves on the right side increases only slightly, when the covariance
is re-computed in the testing state. The interpolated covariance can be applied reliably
also overcoming problems to estimate the covariance when only short data is available in
the testing state.

4.3.4 Considerations for reference setup

In the previous a reference setup has been used considering reference data at five
reference temperatures with distance of Δ*T* = 5°C. In the following results from two
more reference setups are analyzed regarding the robustness of the developed method and
considerations for the design of the reference setup. Tbl. 4.1 summarizes the settings of the
evaluated reference setups, with setup "A" denoting the previous reference configuration.
In setup "B" the reference data at the borders is not taken into account. This provides
information on the capability of the algorithms to yield an extrapolated temperature
adapted reference null space outside the reference temperature interval. In reference setup
"C" the distance between the reference temperatures is bigger leading to conclusion for
the choice of weighting function with respect to the available reference data. Setup "C2"
considers the same reference data, but uses a wider Gaussian weighting function (W2
with σ² = 25 instead of W1 with σ² = 11) for the interpolation. These modifications are
made to optimize the computation of the temperature adapted reference null space with
respect to the bigger distance between the reference temperatures.
<table><caption>Table 4.1 – Reference setups</caption><thead><tr><th>setup name</th><th>reference T °C</th><th>weighting function</th></tr></thead><tbody><tr><td>A</td><td>[0 5 10 15 20]</td><td>W1</td></tr><tr><td>B</td><td>[5 10 15]</td><td>W1</td></tr><tr><td>C</td><td>[0 10 20]</td><td>W1</td></tr><tr><td>C2</td><td>[0 10 20]</td><td>W2</td></tr></tbody></table>

The respective test results at testing temperatures between 0 and 20°C are shown for the new approaches and the average method in Fig. 4.9. At the top the results form reference setup "A" are plotted. In the second line the test values of reference setup "B" are shown. This is followed by setup "C" and "C2" at the bottom.

When in the reference setup "B" no reference data is taken into account at the borders, i.e. at 0°C and 20°C, the test values for temperatures below 5°C and above 15°C are computed with extrapolated system matrices. The extrapolated system matrices lead to increasing reference test values for all methods, since the reference null spaces are less good fitting than in setup "A", where additional information was available from the reference data at the borders. In the undamaged state in all strategies the false alarm rate increases, especially the average method and the S(H) strategy are not able to distinguish the undamaged from the damaged state reliably. However, the S(O) strategies are robust for testing temperatures that are not too far from the reference temperatures. In the damaged state, all new approaches are able to detect damage reliably, while the average method fails to distinguish between damage and undamaged case.

When the distance between the reference temperatures increases as in setup "C" the test values computed with the temperature adapted reference matrices are much smaller at the reference temperatures than at the testing temperatures in between. Computing the threshold from these test values leads to a significant number of false alarms. This is due to the weighting function that was optimized for the temperature model in Fig. 4.2 combined with the reference setup "A" with temperature distance of 5°C. In the present setup the distance is 10°C and the used weighting function is too sharp, i.e. when the testing temperature is one of the reference temperatures this reference point has a strong effect, while the other reference points are neglected. When the testing temperature is different from the reference temperatures, all reference points gets the same weight.

With these findings the interpolation function is modified in setup "C2", i.e. a wider weighting function is chosen. For the S(H) strategy the effect is small, but in the S(O) strategies the false alarm rate decreases. With all new approaches damages is detected
<figure><img src="image_36.png" /><figcaption>Figure 4.9 – Test values at different testing temperatures for different reference setups.</figcaption></figure>

reliably, which is not the case with the average method.

### 4.3.5 Effects of changes in the excitation covariance

The previous results apply when the excitation covariance is constant for all reference data sets. In real applications in the structural engineering field this may not hold, and changes e.g. in the traffic or wind conditions may lead to changes in the covariance of the excitation, i.e. the variance in $w_{j,k}$ in (2.66). This perturbs the interpolation of the Hankel matrix in (4.13).

Fig. 4.10 (left) shows the test values computed with strategy S(H) in the undamaged
<figure><img src="image_37.png" /><figcaption>Figure 4.10 – Test values when reference and testing data sets have different excitation covariance, computed with S(H) interpolation (left) and S(O) interpolation (right) strategies.</figcaption></figure>

and the damaged state when the stiffness at element 3 is decreased by 4%. The standard deviation of the excitation in the reference state is changed from 1 to 0.2 and the standard deviation of the excitation in the test state is uniformly randomly varied between 1 and 10. Many damages cannot be detected and false alarms occur.

This problem can be overcome by means of the strategies, which uses the interpolated eigenstructure of the system (4.16) to compute the observability matrix $O(P)$, since the modal parameters are independent from the excitation covariance. Fig. 4.10 (right) shows the test values computed with the "S(O) interpolation" strategy with the above defined changes in the excitation covariance. Those changes are reflected in the variance of the test values in the unchanged and in the damaged system but the different system states are separated clearly and damage detection is successful.

## 4.4 Summary

In this chapter the theory for a damage detection procedure has been derived that is robust to changes in the environmental conditions, and the results have been validated in a numerical study.

The robust approach is based on a model interpolation leading to a temperature adapted reference null space. Three strategies for computing the reference null space were considered. Firstly, the reference null space is computed from an interpolated Hankel matrix. This is convenient as the system identification step is avoided, but gives less good results in the interpolation step of the Hankel matrix and requires sufficient reference points. Secondly, interpolated modal parameters lead to an interpolated observability ma-
trix, from which the reference null space can be computed. With this procedure good
results could be reached also for few reference points. Both approaches need some extra
effort, if the testing temperature is at the border or outside the range of the reference
temperatures, e.g. higher than the highest reference point or lower than the lowest refer-
ence point, by introducing virtually extrapolated reference points. Thirdly, a null space
is computed from an observability matrix based on modal parameters obtained from a
regression analyis. In this method the regression function has to be optimized carefully,
especially when only few reference points are available.

The strategies that are based on the interpolated observability matrix can also be
applied robustly when changes in the excitation covariance between the reference data
sets occur, since the modal parameters are independent from the excitation covariance,
whereas the Hankel matrix interpolation fails under these circumstances.

The new approach outperforms the existing average approach from [Bal+08a], which
is not able to detect damages when the temperature effect is higher than the effect due to
damage, while the newly developed method is still sensitive. With a temperature adapted
reference null space false alarm rate can be decreased significantly when a well designed
weighting or regression function is used taking at least the two neighbored reference points
into account for interpolation.

4.5 Dissemination

Parts of this chapter have been published in

[Vie+19] E. Viefhues, M. Döhler, Q. Zhang, F. Hille, and L. Mevel. « Subspace-based damage detection with rejection of the temperature effect and uncertainty in the reference ». In: *Proceedings of IOMAC 2019 - 8th International Operational Modal Analysis Conference*. Copenhagen, Denmark, 2019

[Vie+20] E. Viefhues, M. Döhler, F. Hille, and L. Mevel. « Fault detection for linear parameter varying systems under changes in the process noise covariance ». In: *IFAC-PapersOnLine* 53.2 (2020). 21th IFAC World Congress, pp. 13668–13673

[Vie+21] E. Viefhues, M. Döhler, P. Simon, R. Herrmann, F. Hille, and L. Mevel. « Stochastic subspace-based damage detection of a temperature affected beam structure ».
In: *Proceedings of SHMII-10 2021 - 10th International Conference on Structural Health Monitoring of Intelligent Infrastructure*. Porto, Portugal, 2021, pp. 1-6

A journal paper based on this chapter is in preparation for submission to "Mechanical Structures and Signal Processing".
<figure><img src="image_38.png" /></figure>
PART III

Applications
<figure><img src="image_39.png" /></figure>
# DAMAGE DETECTION OF A STEEL FRAME
CONSIDERING UNCERTAINTIES IN THE
REFERENCE MATRIX

## 5.1 Introduction

In chapter 3 the theory for a subspace-based damage detection with estimated reference null space was derived. The developed damage detection test has been validated on simulation data, where its behavior was shown to be coherent with the theoretically expected properties thanks to the consideration of the uncertainty related to the reference data. Moreover, it was shown to be robust for different data lengths with an increased performance especially for short reference data lengths compared to the conventional test. In this chapter, the tests are applied to a laboratory steel frame structure for an evaluation of the test performance on experimental data.

The experiments were performed within the scope of work from [Hil18] at the Federal Institute of Materials Research and Testing (BAM), where damage detection of off-shore wind turbine jacket structures with subspace-based methods was evaluated. A laboratory steel structure has been designed, such that reversible damage can be introduced to the structure. The eigenstructure of the down-scaled test specimen is comparable to the modes of real jacket structures in the lower frequencies within the limits of minimum material thickness.

In the following, the test setup and the dynamical properties of the laboratory tests are summarized. Then damage detection with the algorithms from chapter 3 is performed using the experimental data, and the effect of data length is discussed.
## 5.2 Experimental setup and dynamical properties

The laboratory steel frame structure represents a 1 : 10 scaled two-dimensional section of a jacket-type support structure of an off-shore wind turbine, as it is shown in Fig. 5.1 in the lab of BAM. The structure is of 2.36 m height, 1.92 m width at the bottom, and 1.62 m width at the top. It is made of steel pipe components with an I-sectional steel beam on the top. At the bottom the structure is bolted to the floor. At the top the structure is fixed perpendicular to the in-plane-direction.

<figure><img src="image_40.png" /><figcaption>Figure 5.1 – Laboratory testing structure (left) and sensor positions (right).</figcaption></figure>

The structural parts are mainly welded, except for the area where a reversible damage can be introduced. This damage area is defined by a K-node, which is fixed to the structure by means of bolted end plates as demonstrated in Fig. 5.2. The damage is designed such that the global behavior is affected. By loosening the bolts at the lower brace successively, the progress of a fatigue crack-like damage can be introduced. A high number of bolts is used to provide a high resolution of the induced stiffness loss. At the columns 26 and at the diagonals 18 M6 screws are installed, from which seven screws are used to simulate damage, while the remaining are needed for structural safety. In the reference state all bolts are screwed tight. For increasing damage, 1, 2, 3, 5, or 7 bolts are unscrewed. Loosening 1 and 2 bolts results in less than 1% reduction of the moment of inertia. When 3, 5, and 7 bolts are loosened, this represents a fatigue crack, leading to reduction of the moment of inertia of about 3%, 10%, and 30%, respectively [Hil18].
<figure><img src="image_41.png" /><figcaption>Figure 5.2 – Reversible damage detail (left) and screw configuration (right).</figcaption></figure>

An electrodynamic shaker of type “LDS V406“ (Co. Brüel & Kjaer) equipped with an extra mass of 5 kg is used to excite the structure with a random white noise signal in the range of 10 to 1000 Hz. The excitation direction is approximately 30° rotated out of the in-plane-direction. Accelerations are measured with a sampling rate of 2500 Hz at nine locations in accordance to Fig. 5.1 (right). The sensors have a resonance frequency of more than 10 Hz and were of type “KD41V“ (Co. MMF Radebeul, 8 pieces) and of type “7251A“ (Co. Endevko, 1 piece).

The dynamical properties of the steel frame are evaluated in [Hil18] with the covariance
driven SSI. 50 robust modes can be identified under different test scenarios. Also the higher
modes are identified, since it is assumed that they are affected by damage. The table 5.1
recalls the frequencies in the undamaged case.

Further details of the test setup and dynamical properties can be found in [Hil18]. In
the following application, data is used from tests without changes in the environmental
and excitation conditions.
<table><caption>Table 5.1 – Frequencies of the steel frame in Hz from [Hil18]</caption><thead><tr><th></th><th>f1</th><th>f2</th><th>f3</th><th>f4</th><th>f5</th><th>f6</th><th>f7</th><th>f8</th><th>f9</th><th>f10</th></tr></thead><tbody><tr><th>f1</th><td>31.4</td><td>48.2</td><td>65.1</td><td>78.7</td><td>81.1</td><td>93.9</td><td>108.0</td><td>149.6</td><td>150.4</td><td>164.5</td></tr><tr><th>f11</th><td>168.9</td><td>179.5</td><td>182.5</td><td>192.9</td><td>201.7</td><td>208.2</td><td>211.4</td><td>232.9</td><td>238.6</td><td>280.4</td></tr><tr><th>f21</th><td>283.1</td><td>292.8</td><td>304.5</td><td>311.4</td><td>337.1</td><td>360.1</td><td>393.2</td><td>415.1</td><td>431.1</td><td>441.7</td></tr><tr><th>f31</th><td>452.9</td><td>486.9</td><td>500.8</td><td>520.2</td><td>524.2</td><td>535.3</td><td>544.2</td><td>562.8</td><td>574.0</td><td>587.6</td></tr><tr><th>f41</th><td>595.6</td><td>603.4</td><td>623.7</td><td>633.7</td><td>685.5</td><td>717.7</td><td>750.5</td><td>785.6</td><td>812.3</td><td>856.8</td></tr></tbody></table>

## 5.3 Damage detection with uncertain reference null space

For damage detection, the reference system is defined when all bolts are tightened, and five damaged states are considered with 1, 2, 3, 5, and 7 loosened bolts. For each system state, eight data sets are prepared. One data set in the undamaged state is used to compute the reference matrix $\hat{S}$, as well as the residual covariance $\hat{\Sigma}$ and $\hat{\Sigma}$ for the conventional and the new test, respectively.

Since in the system identification more than 50 modes are found, parametrization with respect to the damage sensitive modal parameters is not an easy task. This motivates for the application of the non-parametric test (3.33), which is the most convenient test setup with a high practicability, since the system identification step is also avoided in the reference state. It corresponds to a direct detection of changes in the residual. The number of degrees of freedom of the test is related to the rank of the converged covariance matrices $\Sigma$ or $\hat{\Sigma}$ of the tests, which however cannot be evaluated with limited data. Nevertheless, an upper bound is given by their size, which is limited by the number of data blocks in their estimation [DMH14]. In this application, $n_b = 500$ data blocks have been used for the computation of $\hat{\Sigma}$ and $\hat{\Sigma}$, hence the degrees of freedom of the $\chi^2$ test distributions are limited by 500, where the threshold allowing for 0.1% false positive alarms is at 603.

Fig. 5.3 shows the test values in the undamaged reference state of the conventional test (left) and when the uncertainty of the reference is considered in the new test procedure (right) for different reference data lengths $M$. The testing data length is about 8 seconds with $N = 20,480$. In accordance with the numerical evaluation in section 3.3
<figure><img src="image_42.png" /><figcaption>Figure 5.3 – Test values of the conventional (left) and the new test (right) in the undamaged reference state computed with different reference data lengths $M$.</figcaption></figure>

the conventional test leads to test values clearly above the threshold in Fig. 5.3 (left). Moreover, the test values depend on the reference data length $M$ used for the computation of the reference null space and the covariance matrix. They tend to decrease with increasing $M$ as it could similarly be seen in the numerical simulation in Fig. 3.5. With the new testing method only the test values for the very short data length $M = 5,000$ are above the threshold in Fig. 5.3 (right), and indeed it can also be observed in Fig. 3.5 that the test value mean has not converged yet for very small $M$. For the other used data lengths, the test values are consistently below the theoretical threshold and do not show any dependence on $M$.

The test values in different system states are presented in Fig. 5.4 for both tests, where the reference data length $M = 15,000$ was chosen and $N = 20,480$. Both tests are able to indicate changes in the system by an increase in the test values, even for damages below

<figure><img src="image_43.png" /><figcaption>Figure 5.4 – Test values of the conventional (left) and the new test (right) in the reference and at different damaged states.</figcaption></figure>
1% stiffness reduction. Two different thresholds are marked: the theoretical threshold from above and an empirical threshold, that is computed from the values in the reference state allowing for 0.1% false alarms. Notice that a threshold computed from such a small sample size is not very reliable. Still, with the new test the empirical threshold coincides with the theoretical one, thanks to very stable test values. The undamaged and the damaged state can be distinguished reliably. With the conventional test method, however, the theoretical and the empirical threshold are significantly different. The method fails to recognize the healthy state when using the theoretical threshold, while with the empirical threshold the test sensitivity is small. In particular, the smallest damage of one loose bolt cannot be detected with the conventional test, even when using the empirical threshold, while this damage can already be detected with the new test.

The effect of testing data length *N* is evaluated in Fig. 5.5. Test values, which are computed with three different testing lengths *N*, and the empirical and the theoretical threshold are shown. The reference matrices are estimated from a data set of length *M* = 15,000, the thresholds are defined such that the maximum number of false alarms is 0.1%. When the uncertainties of the reference null space are taken into account with the new test (right), the reference values are mostly independent from *N* and the theoretical and empirical threshold are similar. The damaged state is better separated from the reference state, when the testing data length *N* increases. Thus, damage can be detected even with very small damages, which is in accordance to the formulation of the

<figure><img src="image_44.png" /><figcaption>Figure 5.5 – Test values of the conventional (left) and the new test (right) in the reference and at different damaged states for different *N*.</figcaption></figure>
close hypothesis (2.26). In contrary, the conventional test method shows an unpredictable
damage detection behavior, when the data length *N* increases, leading to larger test val-
ues in all states. This was pointed out in the context of Fig. 3.4. Since both tests show
some variance of the test values in the reference state, the small sample size perturbs the
setup of a reliable empirical threshold, making the big advantage of a theoretical based
threshold obvious. Furthermore, the new test shows a better damage sensitivity, where
smaller damages can already be detected compared to the conventional test.

**5.4 Summary**

In this chapter the damage detection method for a residual with uncertain reference (chapter 3) has been applied to experimental data from a laboratory steel structure. Since over 50 modes were found for the structure, a non-parametric test procedure was used, which avoids the identification of the damage sensitive modes.

The results correspond to the findings in the numerical study. The new test shows good accordance to theoretically expected values, while the conventional test leads to higher test values due to the bias that is introduced by neglecting the uncertainties in the reference state. It was confirmed that the new test is very robust to changes in data lengths *M* and *N* and thus allows for computing the threshold on a theoretical basis. Such theory-based a priori thresholds offer important benefits, especially when only few reference data is available and bad reference estimate or small sample size perturb the setup of an empirical threshold. Moreover, the new test had a better damage sensitivity than the conventional test in this case study. When the uncertainty of the reference null space is neglected in the conventional test method, the test values and the test performance are not predictable and consequently require comprehensive study of the reference state based on data to e.g. derive reliable thresholds.

**5.5 Dissemination**

Parts of this chapter have been published in

[Vie+22] E. Viefhues, M. Döhler, F. Hille, and L. Mevel. « Statistical subspace-based damage detection with estimated reference ». In: *Mechanical Systems and Signal Processing* 164 (2022)
<figure><img src="image_45.png" /></figure>
# DAMAGE DETECTION OF A TEMPERATURE AFFECTED LABORATORY BEAM STRUCTURE

## 6.1 Introduction

Three strategies for a stochastic subspace-based damage detection (SSDD) approach that is robust to changes in the environmental conditions such as temperature have been presented in chapter 4 and validated on numerical data. The approaches are based on or motivated by the model interpolation approach for linear parameter varying (LPV) systems from [Zha18] leading to a reference null space that is adapted for the current parameter state, e.g. for the temperature conditions in the testing state.

Up to now, the interpolation-based temperature rejection approach was validated theoretically on numerical data of a mass-spring-damper. A very simple temperature model in section 4.3 was used that simulates the effect of temperature on the dynamical behavior of the structure by decreasing stiffness of the springs with increasing temperature. In this section the simplest of the developed algorithms, namely the one based on the interpolated Hankel matrix, is applied to data from a reinforced concrete beam tested in a climate chamber in the testing facilities of the Federal Institute for Materials Research and Testing (BAM) at defined temperatures and at three different system states. The data is obtained by experiments that have been performed by Simon et al. [Sim+20]. It is shown that the new test behaves robustly towards temperature effects and still provides a reliable sensitivity towards damages.

In the following the test setup and the dynamical properties of the test specimen are presented. Then, the damage detection approach with the interpolated Hankel matrix from (4.13) is applied to the experimental data and the robustness towards temperature effects and the test sensitivity are discussed and compared to previous approaches.
6.2 Experimental setup and dynamical properties

A reinforced concrete beam is tested in a climate chamber at defined temperatures and at different system states. The test setup is shown in Fig. 6.1. The beam of 2.72 m span is supported at two points. The cross section is reinforced with two bars of 16 mm diameter at the bottom and no additional stirrups.

Three geophones measure vertical vibration velocities on the top of the beam. The structure is excited by an external shaker with a white noise signal in the range of 30 Hz to 2000 Hz, to simulate ambient excitation as it will occur similarly in e.g. bridges under operational conditions. The shaker is installed upside down (Fig. 6.2) and isolated against the changing temperature conditions. The data is recorded with 5 kHz.

Three different system states are considered: the undamaged reference state, damage
level 1 after the beam was loaded in the midspan with 20 kN, and damage level 2 after the
structure was loaded up to 28 kN. In the second damage level obvious cracks have been
occurred in the middle of the beam. The load is removed after introducing the damage.

The experimental temperatures are chosen to be -25°C, -5°C, 5°C, 25°C, and 40°C.
The temperatures are measured at several points inside the chamber and with two em-
bedded sensors in the middle of the beam. In each system state and at all five testing
temperatures the velocities are recorded, after reaching a constant temperature level in
the beam. Further details on this experiment are described in [Sim+20].

With a SSI-algorithm the eigenstructure of the system is identified. Due to high noise

<figure><img src="image_46.png" /><figcaption>Figure 6.1 – Test setup of the concrete beam with locations of geophones, shaker, and loading for the damage introduction.</figcaption></figure>
<figure><img src="image_47.png" /><figcaption>Figure 6.2 – Concrete beam in the climate chamber.</figcaption></figure>

impact above 500 Hz, the data is sampled down to 400 Hz. Fig. 6.3 shows the first four eigenfrequencies in dependence of the temperature, which can be clearly identified in every system state and at every temperature. The first and the second eigenfrequency are quite close. This is due to the construction of the shaker installation, which works as a tuned mass damper to the beam structure. Simulation results from a model of the corresponding structure with a mass damper have shown a good accordance to these first four eigenfrequencies.

Each frequency is plotted within its 5%-range to give an idea of the quantity of the temperature and the damage effect. The black full line marks the frequencies in the undamaged state. Effects due to temperature become more obvious for the second and the third frequency. While the damage level 2 leads to significant decrease of the eigenfrequencies of about 5%, the effect of damage level 1 does not become obvious in the first and the second eigenfrequency. The third and fourth frequencies increase by about 3% due to damage. This was not expected and cannot be explained from the information gained from the experiments. However, this fact has no impact on the usability of the data for the application study.

## 6.3 Damage detection with temperature effects

For the damage detection the data is sampled down to 400 Hz, and data sets with 350,000 data points are prepared at each temperature state. For the setup of the reference matrices the first half of the respective data set is used, such that $M = 175,000$, which is
<figure><img src="image_48.png" /><figcaption>Figure 6.3 – 1st to 4th eigenfrequencies at different system states and at different temperatures. The y-axis is scaled to about 5 % of each frequency.</figcaption></figure>

about seven minutes. For the damage detection test the data sets are split into eight data samples each of length *N* = 40,000 or 100 seconds. In this first application to real data a fully data-driven non-parametric test setup (4.19) is used where the residual is computed from an interpolated Hankel matrix (4.13). The residual covariance is derived with (4.22) from the interpolated Hankel matrix covariance (4.21) and without consideration of the uncertainty in the reference null space, i.e. Σ₁ = 0 in accordance to the conventional test in (2.37). Other test variants are further investigated in the subsequent chapter 7.

To illustrate the effect of the changing temperature to damage detection of the concrete beam, the interpolation based method is compared to 1/ taking a fixed reference at one of the temperatures (without consideration of the temperature effect), and 2/ the averaging approach from [Bal+08a] that defines a fixed reference by mixing data from different reference temperatures.

**Fixed reference** The tests are done firstly for the case where only data from one fixed reference temperature is considered in the reference state. This corresponds to the conventional residual in (2.24). The null space and the residual covariance in (2.37) are computed from a data set at 5°C. Fig. 6.4 shows the test values of the hypothesis test
(2.33) at different testing temperatures in the undamaged state and at the two damage levels. For the setup of the testing Hankel matrices $\tilde{H}$ data samples of 100 seconds are used. From the test values in the reference state for the undamaged structure at 5°C (grey bars) a threshold is defined, which allows for false positive alarms of 1% maximum.

It becomes obvious that changes in the temperature in the undamaged state (green bars) affect the test values in a similar way as the different damages do. This results in a high number of false alarms in the undamaged state, namely for all cases where the test temperature is different from the reference temperature. While all test values in the damaged cases also exceed the threshold, they cannot be distinguished from the test values in the reference state due to the temperature effect.

This is in accordance to the findings with the numerical simulation data, where in Fig. 4.3 all test values at temperatures different from the reference temperature are above the threshold, no matter if they classify the damaged or the undamaged case.

**Averaging approach from [Bal+08a]** The temperature rejection approach in [Bal+08a] that was discussed in section 4.2.5 uses data from different temperatures in the reference state together in one merged data set. From this merged data set the averaged reference Hankel matrix and its left null space are computed with, as well as the covariance of the resulting residual. In Fig. 6.5 the test values from this method are presented, where reference data samples from three reference temperatures, 25°C, 5°C, and 40°C, are merged in one reference data set. The testing Hankel matrices $\tilde{H}$ are computed from a 100 second data sample. A threshold is defined from data in the reference state, thus of the undamaged structure at the three reference temperatures (grey bars), allowing for 1% false alarms.

The structure is correctly classified as undamaged if the testing temperature corresponds to one of the used reference temperatures. Information about all reference states is contained in the merged reference data set and thus considered in the reference null space and the covariance matrix.

If the temperature differs from the reference temperatures, the changes in the structure due to temperature lead to higher test values even in the healthy states exceeding the threshold. The structure is detected to be damaged. These changes due to new testing temperatures have a quantitatively similar effect to the test values as damages do. While damages can be detected at all temperature levels, they cannot be distinguished from the healthy state at testing temperatures that are not contained in the reference data.
<figure><img src="image_49.png" /><figcaption>Figure 6.4 – Fixed reference null space and covariance matrix. Test values at different testing temperatures in the undamaged and damaged states. Reference data at 5°C.</figcaption></figure>

<figure><img src="image_50.png" /><figcaption>Figure 6.5 – Averaged reference null space and covariance matrix. Test values at different testing temperatures in the undamaged and damage states. Reference data at -25°C, 5°C, and 40°C.</figcaption></figure>

<figure><img src="image_51.png" /><figcaption>Figure 6.6 – Interpolated reference null space and covariance matrix. Test values at different testing temperatures in the undamaged and damage states. Reference data at -25°C, 5°C, and 40°C.</figcaption></figure>
**Approach with interpolated Hankel matrix** With the robust SSDD approach from chapter 4 using the interpolated Hankel matrix the information about the temperature depending behavior of the structure at testing temperatures different from the reference temperatures can qualitatively be carried by an adequate weighting function within the interpolation step. In Fig. 6.6 test values of the experimental data are presented that are computed with null space derived from an interpolated Hankel matrix (4.13). The residual covariance is computed with (4.22) from the interpolated Hankel matrix covariance (4.21). The uncertainties in the reference null space are neglected, i.e. $\Sigma_1 = 0$ which corresponds to the conventional test with (2.37).

In the reference state three reference temperatures are considered 25°C, 5°C, and 40°C as in the previous paragraph. For the weighting of the respective Hankel matrices $H_j$ in (4.13) a Gaussian function with a variance of 30, centered at the testing temperature, is applied. A threshold is computed in the reference state of the undamaged structure at the three reference temperatures (grey bars), allowing for 1% false alarms.

With this approach the undamaged state can be distinguished clearly from the damaged state for any testing temperature. There is no difference in the test values, whether data from the testing temperature was available for the reference set up or not. An adequate null space can be found in any case, and false alarms in the undamaged state are avoided. The damages can be detected reliably.

In general, the test values of the interpolation method are higher than in the other computation methods. This can be explained from the choice of the weighting of the local reference models. For any testing temperature, the Gaussian function with variance of 30 leads to a weighted mixture between all of the used reference models. This means that even at the reference temperatures, where data for a more precise reference null space exists, an interpolated reference null space is used, which is affected by the other reference null spaces. This leads to some bias and as a result to higher test values of the hypothesis test. However, in the damaged case this effect persists, and damages still can be detected reliably.

It could be confirmed that the usage of bell-shaped functions centered at the testing temperature is adequate, if the temperature effects can be assumed to be approximately linear for each section. However, the choice of the variance of these weighting functions is quite important as Fig. 6.7 shows. If a sharp weighting function is used ($\sigma^2 = 10$) in Fig. 6.7 (top) the test values at the reference temperatures are much lower than at the temperatures in between, since in the latter case two local models are considered
for the interpolation, while at the reference temperatures only the local model from the
reference temperature itself is taken into account. This holds for the test values in the
undamaged as well as in the damaged case. When a wide interpolation function is chosen
(bottom), this leads to results that are similar to the averaging approach in Fig. 6.5. The
undamaged structure at temperatures different from the reference temperatures is falsely
classified as damaged. In the damaged case the test values increase. In the present case
the interpolation worked well with a function with σ² = 30 variance (Fig. 6.6) and it is
recommended not to use weighting functions that neglect nearby local models, even if a
good local model is known from existing reference data. In this case the test values for
testing temperatures different from the reference ones would be higher and lead to more
false alarms.

<figure><img src="image_52.png" /><figcaption>Figure 6.7 – Effect of weighting function in interpolated reference null space and covariance matrix. Sharp weighting function (top) and wide weighting function (bottom). Test values at different testing temperatures in the undamaged and damaged states. Reference data at -25°C, 5°C, and 40°C.</figcaption></figure>
## 6.4 Summary

In this chapter a first test of the robust methods from chapter 4 on real data has been performed. Experimental data from a concrete beam tested in a climate chamber at five different temperatures and three different system states has been used. With the goal to provide a data-driven damage detection procedure strategy 1 from chapter 4 based on direct Hankel matrix interpolation (4.13) has been applied together with a non-parametric test procedure. The residual covariance was derived once in the reference state from the interpolated covariance from (4.22) with $\Sigma_1 = 0$.

The results correspond to the findings in the numerical study in section 4.3. It could be shown that the interpolation approach is able to distinguish between damaged and undamaged states reliably, and false alarms can be avoided, while previous methods fail to classify the undamaged state correctly when the testing temperature is different from the temperatures for which reference data is available. Some suggestions were made regarding the choice of an adequate weighting function.

## 6.5 Dissemination

Parts of this chapter have been published in

[Vie+21] E. Viefhues, M. Döhler, P. Simon, R. Herrmann, F. Hille, and L. Mevel. « Stochastic subspace-based damage detection of a temperature affected beam structure ». In: *Proceedings of SHMII-10 2021 - 10th International Conference on Structural Health Monitoring of Intelligent Infrastructure*. Porto, Portugal, 2021, pp. 1–6
<figure><img src="image_53.png" /></figure>
# DAMAGE DETECTION OF A TESTING BRIDGE EXPOSED TO AMBIENT TEMPERATURE VARIATIONS

## 7.1 Introduction

An approach for the stochastic subspace-based damaged detection that is robust to changes in environmental conditions of a mechanical structure was derived in chapter 4. The environmental effects are considered as external parameters, e.g. temperature, affecting the dynamical behavior of the structure. In the method a reference null space is derived that is adapted for the parameter in the testing state. In a numerical study the developed damage detection tests were proven to be robust to such parameter changes and led to meaningful global thresholds allowing for sensitive damage detection.

A first application to real data is described in chapter 6, where the algorithms of strategy 1 from chapter 4 are used with data of a laboratory structure tested in a climate chamber under defined conditions. In this chapter the robust approaches are applied to data from a testing bridge located on an outdoor testing field that is exposed to real environmental conditions. All strategies from chapter 4 are used, and it will be shown that the test with a temperature adapted reference null space also performs well for data under more realistic and less controllable conditions.

In the following, the experimental setup and the dynamical properties of the testing bridge are presented. Then, the three strategies for damage detection robust to temperature variations from chapter 4 are applied to the experimental data. The test robustness is evaluated similarly to section 4.3 in several reference setups and compared to previous temperature rejection approaches.
## 7.2 Experimental setup and dynamical properties of the structure

Within this thesis and related to an interdisciplinary project in the Federal Institute of Materials Research and Testing Germany (BAM) a multi-sensor equipped testing bridge with variable pre-stressing is designed and constructed. The 24 m long concrete structure, displayed in Fig. 7.1, is built on an outside testing field of BAM. Its purpose is to provide data for this work and for future research on developments and applications of SHM strategies and new testing hardware [Ung+17; LMK18; ZU18; Pir+19; Lia+21]. Data can be generated under near-realistic environmental conditions, and the system state can be varied well defined and reversibly.

<figure><img src="image_54.png" /><figcaption>Figure 7.1 – Testing bridge.</figcaption></figure>

The bridge is designed as a two-span girder with 12 m span each supported with floating bearings at two points in each of the three axis as detailed in Fig. 7.2. The U-shaped cross-section is 90 cm wide and 30 cm high with 15 cm top thickness. The ratio of static height to length is chosen such that eigenfrequencies are obtained similar to those from realistic civil engineering structures and the resulting structural mass and damping allow for artificial and environmental excitation. Two external tendons are located between the flanges of the cross-section allowing for variations of the pre-stressing between 0 and 1200 kN. The pre-stressing load is applied at the beginning and at the end of the bridge (Fig. 7.3, left).
<figure><img src="image_55.png" /><figcaption>Figure 7.2 – Static system of the testing bridge.</figcaption></figure>

<figure><img src="image_56.png" /><figcaption>Figure 7.3 – Pre-stressing facility (left) and encased shaker (right).</figcaption></figure>

For the evaluation of the temperature rejection approach vibration data is recorded for 14 days while the structure is excited by an 80 kg shaker with white noise signal between 2 and 80 Hz for 10 minutes every half hour. The shaker is encased to be protected from direct sun and water (Fig. 7.3, right). It is located at 7.8 m from the beginning of the bridge with an eccentric position as shown in Fig. 7.4. The data is recorded with 5 geophones located at an eccentric axis (y1 in Fig. 7.4) and one additional geophone at axis y2 to cover torsional effects sampled at a frequency of 1200 Hz.

Three embedded and one external temperature sensors enable the monitoring of the temperature conditions. The positions of the embedded sensors are seen in Fig. 7.4. They are located in the middle span of the bridge. A linear temperature variation over the cross section is observed. However, the structural temperature during this experiment is derived with respect to the central sensor without detailed consideration of positive or negative temperature gradients along the cross-section. Structural temperatures between 8°C and
<figure><img src="image_57.png" /><figcaption>Figure 7.4 – Test setup testing bridge with geophones and shaker for excitation.</figcaption></figure>

24°C are measured.

Before the experiment started, the test specimen has been pre-loaded with a four tons weighting shifted over the bridge from on side to the other, such that in the case of no pre-stressing the section is cracked significantly. During the experiment the pre-stressing was set to 450 kN from day 1 to day 7 and then lowered to 250 kN.

The dynamical properties of the bridge are identified with a covariance-driven SSI algorithm with data sampled down to 40 Hz and filtered between 1 and 20 Hz. Fig. 7.5 (left) exemplarily shows the stabilization diagram of one data set at 15°C with pre-stressing of 450 kN. The first three modes can be identified clearly from each considered data set. The identification of the fourth mode is afflicted with more uncertainties as marked by the horizontal bars at the identified modes in the stabilization diagram. The frequencies at 15°C are summarized in Tab. 7.1 together with the damping ratios. The respective modeshapes at 15°C are presented in Fig. 7.6. They are in accordance to the expected first four modeshapes of a two-span beam including some torsional effects in the first mode (blue dot).

Decreasing the pre-stressing load to 250 kN leads to changes in the frequencies of about 12%, which increases the effect of the temperature, significantly. A smaller effect due to pre-stressing release to the dynamical properties was expected from results of previous tests. Those were performed before the pre-stressing of the structure has been
<figure><img src="image_58.png" /><figcaption>Figure 7.5 – Stabilization diagram at 15°C.</figcaption></figure>

<table><caption>Table 7.1 – Frequencies and damping ratios at 15°C.</caption><thead><tr><td>frequency</td><td>damping ratio</td></tr></thead><tbody><tr><td>3.54 Hz</td><td>1.39 %</td></tr><tr><td>5.67 Hz</td><td>1.04 %</td></tr><tr><td>14.88 Hz</td><td>0.90 %</td></tr><tr><td>17.73 Hz</td><td>2.39 %</td></tr></tbody></table>

<figure><img src="image_59.png" /><figcaption>Figure 7.6 – First four modeshapes of the testing bridge.</figcaption></figure>

<figure><img src="image_60.png" /></figure>

fully released to 0 kN and the load of four tons has been applied, while the present experiments are made after this full release. This must be kept in mind when performing the damage detection and evaluating the test regarding its sensitivity.

The temperature effect to the dynamical properties of the structure can be derived from Fig. 7.7, where the fourth mode shows no clear dependence on the temperature.

# 7.3 Damage detection with temperature effects

## 7.3.1 General settings and temperature effect

The damage detection with the roust methods from chapter 4 is evaluated with the experimental data sampled down to 40 Hz and filtered between 1 Hz and 20 Hz. Only data is used that is recorded with activated shaker yielding data sets of 10 minutes length at several temperatures. The first half of these data sets is considered for the reference
<figure><img src="image_61.png" /><figcaption>Figure 7.7 – Temperature effect to the first four eigenfrequencies of the testing bridge.</figcaption></figure>

data, the second part is used for testing.

From the reference data the reference matrix $S$ and the residual covariance from (4.22) are computed. The damage detection is processed with five data sets of 50 seconds length at $T = [8, 9, ..., 24]^{\circ}\text{C}$ and additionally at $10.5^{\circ}\text{C}$ and $19.5^{\circ}\text{C}$.

For the interpolation method weights are computed from a Gaussian function with variance of $\sigma^2 = 64$ centered at the testing temperature. To improve the results at the borders, virtual reference points are used that are extrapolated from the two nearest reference points assuming a linear relationship. When the reference matrix $S$ is computed from the observability matrix, the first four modes as identified in the previous section are used as system parameter $\theta_j^*$ in (4.16) or (4.17).

Two system states are considered: One system with pre-stressing load of 450 kN, referred to as undamaged and a damaged system with the pre-stressing load decreased to 250 kN. A threshold for damage detection is computed from the data in the undamaged state at the reference temperatures allowing for 0.1% false alarms.

When the standard damage detection test setup e.g. from (3.2) is applied that uses reference data from only one fix reference temperature at $T = 15^{\circ}\text{C}$ for the computation of the reference matrix and the covariance the test values in Fig. 7.8 are obtained at different testing temperatures. When the testing temperature is different from $15^{\circ}\text{C}$ the test values are significantly larger than those at the reference temperature. Hence, setting up an empirical threshold with the test values at the reference temperature leads to false alarms at any other testing temperature for the undamaged structure. This is in accordance to Fig. 4.3 for the numerical simulation data. Due to the high impact of the changes in the pre-stressing on the dynamical properties the test values in the damaged state are very large and from the experimental data no conclusions can be drawn for the sensitivity
<figure><img src="image_62.png" /><figcaption>Figure 7.8 – Test values with reference data from one fixed reference temperature.</figcaption></figure>

of the robust damage detection. In the following the robust damage detection test from
chapter 4 is applied and compared to results from the average approach from [Bal+08a].

**7.3.2 Robust damage detection**

First, a reference setup denoted by “A“ is designed that is based on five data sets at the reference temperatures T₁ = 8°C, T₂ = 10.5°C, T₃ = 15°C, T₄ = 19.5°C, and T₅ = 24°C. Fig. 7.9 presents the test values in the undamaged and the damaged state when the three strategies robust to temperature changes and an averaged reference matrix S are used. The test values at the reference temperatures in the reference state are bold plotted. The horizontal dashed line denotes the empirical threshold for damage detection leading to a red dotted false alarm if it is exceeded.

In the undamaged state at the reference temperatures, the test values of the three
interpolation versions follow a flat curve. This is in accordance to the numerical results
in Fig. 4.5. While again the test values of the S(H)-version show some variation due to
inaccuracies in the interpolation of the Hankel matrix, the test values computed from a
observability matrix based null space are very stable. The thresholds, which are based on
values at the reference temperatures, are at a similar range for each testing procedure.
Very few false alarms occur. Thus, compared to the numerical study, the more realistic
temperature and noise effects, confirm or even increase the robustness of the interpolation
approach towards false alarms in the undamaged state.

In contrary, the values computed with the average method are affected by temperature,
even when testing at the reference temperatures. This effect was already observed (more
clear) with the numerical data (Fig. 4.4). At testing temperatures between the reference
points the test values show high variance and despite the high threshold, many false
<figure><img src="image_63.png" /><figcaption>Figure 7.9 – Test values of the 4 algorithms for different testing temperatures in the undamaged and the damaged case, computed for reference setup A.</figcaption></figure>

<table><caption>Table 7.2 – Reference setups for testing bridge</caption><thead><tr><th>setup name</th><th>reference T</th></tr></thead><tbody><tr><td>A</td><td>[8.0 10.5 15.0 19.5 24.0]</td></tr><tr><td>B</td><td>[10.5 15.0 19.5]</td></tr><tr><td>C</td><td>[8.0 15.0 24.0]</td></tr><tr><td>D</td><td>[10.5 19.5]</td></tr><tr><td>E</td><td>[8.0 24.0]</td></tr></tbody></table>

alarms occur.

With all algorithms damage can be detected clearly at any testing temperature and the ratios of threshold to test value in the damaged state are in a similar range. However, due to the high effect of the pre-stressing change to the dynamical system no more conclusion can be drawn regarding the sensitivity of the tests. In the following, the test performance is evaluated in the reference state and through the probability of false alarms (PFA).

### 7.3.3 Effect of reference setup

The previous results revealed the capability of the interpolation method being very robust towards false alarms, due to the computation of a very well fitting reference null space and covariance for the testing temperature. This motivates for deeper analysis of the effect of different reference setups within the temperature rejection methods.

Four additional reference setups are evaluated with the three algorithms robust to temperature changes and the average approach considering different combinations of reference temperatures as summarized in table 7.2. Fig. 7.10 shows the test values for the reference setups “B” to “E”.

Comparing scenario “A” and “B”, the robustness of the new temperature rejection approach is not affected obviously. In both setups the distance between the reference
<figure><img src="image_64.png" /><figcaption>Figure 7.10 – Test values of the 4 algorithms for different testing temperatures in the undamaged case, computed for reference setup B, C, D and E from top to bottom.</figcaption></figure>

temperatures is 4.5°C, but in “B“ only the central reference temperatures are taken into account in the reference state although testing temperatures outside the reference temperature range occur. With the interpolation approach this is an extrapolation task, which is handled by the weightings for the considered reference points, similar to the interpolation between two reference points. Especially the algorithms using the observability matrix computed from the eigenstructure show very stable test values, due to a good interpola-
<figure><img src="image_65.png" /><figcaption>Figure 7.11 – Weights for different reference setups with weighting function W1 (left) and with optimized weighting functions (right).</figcaption></figure>

tion of the reference null space. With the average method the variation in the test values
increases, in particular at testing temperatures differing from the reference temperatures.

In the setups “C“, “D“ and “E“ the PFA of the new algorithms with temperature
adapted reference null space increases. This is due to the change of the distance between
the reference temperatures in these scenarios. In setup “A“ and “B“ the reference null
space and the covariance are very well interpolated with the chosen weights. In both
cases the distance between the reference temperatures is about 4.5°C. In the remaining
variants the ΔT is about 9°C and 16°C, while the weighting functions are not modified
with respect to this change. Consequently, the chosen weights do not cover the qualitative
temperature depending behavior of the structure in these cases. However, the false alarms
with the average method exceed by far the PFA of the interpolation method.

In an optimization step for the reference setups “C“ to “E“ modified weighting func-
tions are used that are shown in Fig. 7.11. In the left plot the weights are shown that
are generated with the initial weighting function. With respect to the bigger distance ΔT
between the reference temperatures in these setups a wider weighting function is designed
with an increased σ² = 196 (right). This ensures that not mainly one reference point but
several points are considered for the computation of the reference matrix and the residual
covariance, and that the nearby reference points are weighted similarly. Notice that if in
the limit an extremely wide weighting function is used, the results with the interpolated
Hankel matrix will become similar to the results from the average approach.

The effect of the optimized weights can be seen from the PFA in Fig. 7.12. For scenario
“C“ the temperature adapted reference matrices can be optimized in each strategy by the
choice of a proper weighting function reflecting the temperature behavior of the structure
<figure><img src="image_66.png" /><figcaption>Figure 7.12 – Effect of optimized weighting function to false positives for the interpolation approach in reference setups C, D, and E.</figcaption></figure>

with respect to the reference systems. Notice that the improvement of the algorithm using the observability matrix with regression analysis for the eigenstructure is due to a different weighting of the covariance matrices for the interpolated covariance. In setup “D” significant improvement of the interpolation of the observability matrix can be reached by a wider weighting function, while for the interpolation of the Hankel matrix information from only two reference points is too low. In scenario “E” false alarms cannot be avoided by changing the weighting function, as the distance between the reference points is very far and only two reference points are available.

7.4 Summary

In this chapter the temperature rejection approaches from chapter 4 for subspace-
based damage detection were applied to experimental data from a testing bridge that is
exposed to environmental conditions. Experiments have been carried out recording data
for 14 days providing measurements at temperatures between 8 and 24°C.

The application showed that the new method with the temperature adapted reference
matrices provide robust algorithms with significant reduced false alarm rate when com-
pared to other approaches. In particular when reference data is available at a temperature
range and the structure is tested at a temperature outside of this range this can be solved
properly with the proposed method with extrapolation from the reference points.

The effect of the weighting function design has been analyzed and it could be shown
that the test performance can be improved by choosing a weighting function that is
adapted to the reference setup and the temperature depending behavior.

Compared to results in the numerical study in section 4.3 with the more realistic data
providing a more realistic effect of the temperatures to the dynamics of the structure
the new algorithms perform better and an optimization of the weighting function for the
interpolation succeeds.

**7.5 Dissemination**

Parts of this chapter are in preparation for submission to "Mechanical System and Signal Processing".
## Considered aspects

In vibration-based damage detection in engineering structures automated, data-driven methods are particularly of interest. In the context of the present thesis developments were made that aim at higher robustness and sensitivity of such methods for application on engineering structures.

The developments are based on the covariance-driven stochastic subspace-based damage detection (SSDD) method. In this method the damage indicator is derived from a residual related to output data covariances. The statistical properties of the damage indicator can be described precisely for the reference and the testing system state when the statistical properties of the residual are known. Previously, uncertainties of the testing data are considered in the SSDD, but it was assumed that the reference is perfectly known without uncertainties. This is true in ideal cases but does not hold in real world applications, since the reference usually is estimated from data and thus contains considerable uncertainties. The resulting test behaves unpredictable. Furthermore, environmental variability perturbs the damage detection with SSDD, similar to many other approaches, since the residual is sensitive to those changes, too. This has motivated a robust approach for the method.

The developments within the thesis accounted for the following aspects of robust vibration-based damage detection with the SSDD method:

* consideration of all uncertainties of the residual in statistical evaluation for damage detection,

* robustness to environmental variability without explicit temperature model.

## Summary, improvements and limits

(1) Damage detection with uncertain reference Damage detection methods that derive reference models of a system from data are in particular appealing for automated applications. This is the case in the SSDD, but so far the uncertainties related to those
estimated reference models have not been considered in the tests leading to inherent bias and unpredictable test behavior.

In the context of this thesis this gap was closed. The covariance evaluation of the residual was carried out considering uncertainties related to the testing data and the estimated reference. Its distribution properties were evaluated on the basis of the local approach [BBM87]. A general formulation was derived that also applies when the data length in the reference state is different from the length in the testing state. The associated hypothesis test has been formulated in a parametric and a non-parametric version and its statistical properties have been derived from the distribution properties of the residual. Considering the test statistics as damage indicator, the impact of the new developed residual is not only academic. The crucial part of defining a threshold between healthy and damaged state now can be done apriori based on theoretical considerations without extensive empirical studies on reference data.

The test behavior and thereby the distribution properties of the new residual have been validated on numerical simulation data of a simple structure. The test showed very good accordance to the theoretical predicted behavior and the results were independent from data lengths in the reference and the testing state, system parametrization, and reference computation method. It was also demonstrated that neglecting the uncertainty of the reference null space leads to an inherent bias in the residual, which is eliminated with the new formulation resulting in a damage detection test with significant higher sensitivity.

The new test has been applied to experimental data of a steel frame structure. The results correspond to those evaluated on numerical data, and it could be shown that an upper bound for the threshold can be found apriori from a theoretical value. Furthermore, the covariance computation related to the reference data is also part of the second contribution on robustness to changing external parameters with further application.

From these developments the practical applications could be improved as follows:

* The behavior of the new test is predictable and independent from data length and null space computation procedure.

* A damage detection threshold can be defined apriori.

* The approach is also suitable when only relatively few data is available in the reference state.

* The damage detection is more sensitive due to avoided bias in the residual.

Some limitations can be defined as follows:
* Careful estimation of the data covariance matrix is required, since its quality affects the uncertainty quantification of the residual.

* When the non-parametric test is applied only an upper bound for the threshold can be computed, since a theoretical rank of the (estimated) covariance matrix cannot be determined.

**(2) Damage detection under changing environmental parameters** Damage detection of structures exposed to environmental variability often fails, since the environmental effects lead to changes of the dynamical system similar to damage effects.

In this thesis, an approach has been derived for robustness of the SSDD to environmental changes. The procedure is inspired by a model interpolation approach formulated for a state-space representation from [Zha18] postulating an engineering structure under environmental changes as a linear parameter varying system with linear time-invariant behavior at fixed parameter points. Assuming reference data at a few reference parameter points, a parameter adapted reference null space has been derived using three different strategies: two interpolation methods for either the covariance Hankel matrix or the modal parameters and an approach using regression analysis of the modal parameters. The parameter adapted null space is computed in the testing state for the particular testing parameter value and then used in the residual definition of the SSDD. The uncertainties of the interpolated reference null space have been quantified and it was shown how to compute them directly from the interpolated covariance matrices of either the Hankel matrix or the modal parameters, depending on the chosen strategy.

The developed strategies have been validated on numerical data of a simple structure with a non-linear parameter depending behavior and compared to an existing approach from [Bal+08b] using an average procedure, where, however, the actual temperature is not needed to be known. It could be shown that the new approach is much more sensitive to damage, while false alarms due to changes in the environmental conditions could be mostly avoided. Considerations of different reference setups, i.e. different number and distances of reference parameter values, illustrated the strengths and weaknesses of the developed strategies. Robust results could be achieved even when the testing data is recorded under a parameter outside of the reference parameter interval or when changes in the excitation covariance are supposed. Again, considering the uncertainties of the reference improves the test results significantly.

The developed approaches were applied to experimental data from a concrete beam
in a climate chamber, as well as to data from a testing bridge. This testing bridge was constructed within this thesis and related to an interdisciplinary project at BAM. It is subjected to natural daily temperature variations. The bridge is equipped with several sensor types and provides the possibility to activate and deactivate two external pre-stressing tendons for the simulation of different structural states. The experimental results from the developed approach showed similar robustness to temperature variations as the numerical study.

From these contribution the following improvements for practical applications can be concluded:

* The robust approach is widely insensitive to changing external parameters without requiring explicit parameter model formulation.

* The approach works with reference data sets from only few parameter working points and even succeeds under parameters outside of the reference parameter interval.

* The rate of false alarms under changing external parameters is reduced compared to previous approaches.

* The approach is more sensitive to damage under changing external parameters than previous approaches.

* The approaches are validated under near-realistic conditions.

The following weakness of the approach can be stated:

* The definition of the interpolation function in accordance to the available reference setup may be a critical task. However, an individual reference design for the investigated structure and its qualitative parameter depending behavior can optimize the damage detection performance.

## Outlook

It was in the scope of this thesis to contribute to automated damage detection methods with regard to the special requirements of engineering structures. The methods were derived theoretically and were validated on numerical and experimental data. While the developed methods have improved robustness for real-world applications, they also arise further research questions.

For practical applications it would be important to investigate in more detail the estimation of the residual covariance matrix. In the present applications to experimental
data this was always a critical task, since the covariance estimation from relatively few
data blocks in comparison to the residual dimension may be challenging and can lead to
numerical problems in its inversion for the statistical test. This is also important when
the threshold of the non-parametric test should be derived from the (theoretical) rank
of the residual covariance matrix. Another subject that should be addressed is the fur-
ther investigation of the effect of the weighting functions for robustness to environmental
variation and their optimal choice. In this context also applications for multivariate de-
pendencies should be considered. This is a structure-depending task and thus data from
various, more complex structures would be desirable. Moreover, the validation on data
that was recorded under realistic conditions, i.e. traffic and wind, long term monitoring
etc., should be performed in the next steps.

From a technical point of view the residual covariance related to the parameter adapted
null space corresponding to the regression approach should be derived and evaluated with
respect to the interpolation approaches. Furthermore, the developed strategies for damage
detection under environmental variation should be extended to damage localization and
quantification [DMZ16], as well as to rigorous evaluation of the methods’ sensitivity to
damage [MDV21], to provide solutions for the further steps in the damage diagnosis. This
requires the derivation of a parametric version of the robust approach with the need of
corresponding, parameter dependent sensitivities.
<figure><img src="image_67.png" /></figure>
# Introduction

## Contexte

La recherche sur les méthodes automatisées de diagnostic d'endommagements basées sur les mesures vibratoires pour la surveillance de l'intégrité des ouvrages d'art (Structural Health Monitoring, SHM) a pris de l'ampleur au cours des dernières décennies, car de nombreuses structures ont atteint la fin de leur cycle de vie et se détériorent. Ces méthodes sont basées sur le fait que le comportement dynamique d'une structure peut être modifié par un ou des endommagements. Elles comparent les données mesurées dans un état de référence (sain) à celles de l'état actuel.

Des conditions particulières doivent être prises en compte pour l'application des méthodes de SHM aux ouvrages d'art : Comme l'excitation de la structure n'est généralement pas mesurée, les méthodes basées uniquement sur l'excitation ambiante sont particulièrement intéressantes. De plus, les caractéristiques calculées à partir de données mesurées ont une variabilité statistique intrinsèque, qui doit être prise en compte avant de décider si un changement dans une feature ou un paramètre est significatif, indiquant un endommagement. Lorsque des changements significatifs sont détectés, il faut distinguer entre les changements dus à des endommagements et ceux dus à la variabilité des conditions environnementales afin d'éviter les fausses alertes. Ces problématiques pour le traitement robuste des données sont au coeur de cette thèse, basée sur la méthodologie suivante.

## Méthodologie

La méthode de détection stochastique des endommagements basée sur le sous-espace (SSDD) présentée dans [BAB00] s'applique aux mesures de sortie d'un système dynamique, représenté par un modèle d'espace d'état. Dans un résidu, le noyau gauche d'une matrice de Hankel construite à partir des covariances de mesures dans l'état de référence est confrontée à la matrice de Hankel calculée à partir de données du système actuel. Dans un test d'hypothèse le résidu est évalué statistiquement en tenant compte des
incertitudes des données dans l'état actuel dues aux bruits (excitations inconnues, bruits de mesure) par le biais de la matrice de covariance du résidu. Ainsi, les propriétés statistiques du test de détection d'endommagements peuvent être caractérisées précisément dans les conditions saines et endommagées d'une structure.

## Contributions

La méthode SSDD existante jusqu'à présent ne répond pas à certaines exigences fondamentales concernant le traitement des incertitudes liées aux données de référence, ainsi que la prise en compte des changements dans les variables environnementales qui peuvent perturber la détection des endommagements. La contribution de cette thèse à ces problématiques peut être résumée comme suit :

(1) **Détection d'endommagements avec incertitudes dans la référence** Les algorithmes existants considèrent seulement les incertitudes du résidu qui sont liées aux données de test dans l'état actuel, et il est supposé que la référence est parfaitement connue. C'est vrai dans des cas académiques idéalisés, mais dans les applications réelles la référence est généralement estimée à partir de données et peut donc contenir des incertitudes considérables. Par conséquent, avec l'approche existante, les propriétés statistiques du résidu ne correspondent pas aux propriétés théoriques attendues. Le comportement du test devient imprévisible et les seuils pour la décision doivent être fixés empiriquement à partir d'études sur les données de référence. Pour lever ce verrou, les propriétés statistiques théoriques du résidu sont dérivées, conduisant au calcul de la covariance du résidu qui tient compte à la fois des incertitudes liées aux données de référence et de test. Le test d'hypothèse associé montre une très bonne conformité avec le comportement théorique attendu. Avec la nouvelle évaluation du résidu, un seuil peut maintenant être établi à priori basé sur la théorie, sans études empiriques approfondies sur les données de référence. De plus, le nouveau calcul améliore la sensibilité de test aux petits endommagements.

(2) **Détection d'endommagements sous changements de paramètres environnementaux** De nombreuses méthodes automatisées de détection d'endommagements basées sur les vibrations échouent en raison de la variabilité des conditions environnementales, car elles peuvent affecter le comportement dynamique des structures d'une manière similaire aux endommagements [Soh07]. Pour la SSDD, seuls quelques travaux ont été réalisés sur la robustesse aux changements des paramètres environnementaux tels que
la température. Toutes ces approches sont basées sur des études empiriques et ne sont pas optimales lorsque la température dans l'état actuel est différente de la température moyenne des ensembles de données de référence, ou bien elles nécessitent des ensembles de données de référence pour autant de scénarios de température que possible. Dans cette thèse, une approche robuste est dérivée qui s'inspire d'une approche d'interpolation de modèle. Des stratégies d'interpolation ou d'analyse de régression sont proposées pour calculer un noyau de référence adapté à la valeur du paramètre environnemental actuel, et ceci à partir de données dans l'état non endommagé à seulement quelques valeurs de paramètres environnementaux de référence. Le test de détection d'endommagements est développé en tenant compte des incertitudes liées au noyau de référence qui est basé sur l'interpolation ou la régression. Le test résultant s'avère être robuste et toujours sensible aux dommages. Pour la validation expérimentale, un pont d'essai équipé de plusieurs capteurs est construit dans le cadre de cette thèse sur un champ d'essai extérieur de l'Institut fédéral allemand de recherche et d'essai des matériaux (BAM).

# Chapitre 1 – État de l'art

Habituellement, la détection d'endommagements est considérée comme un proces-
sus triple, comprenant l'extraction d'un résidu sensible aux endommagements à partir
des données, la comparaison du résidu dans l'état de référence et l'état actuel, et enfin
l'évaluation (statistique) de cette comparaison. Les concepts existants pour l'extraction
des résidus utilisent des informations provenant des différentes étapes de la procédure
d'identification du système et peuvent être classés en approches basées sur les paramètres
modaux (par exemple, les fréquences et les déformées modales) et les paramètres associés,
sur les paramètres des modèles de séries temporelles ou les caractéristiques dans le do-
maine fréquentiel, ou sur le signal ou les fonctions de traitement du signal. Les stratégies
d'évaluation existantes vont des méthodes graphiques, telles que les cartes de contrôle,
aux méthodes statistiques, telles que l'analyse des outliers ou les tests d'hypothèse, ces
dernières permettant de prendre en compte mathématiquement les incertitudes intrin-
sèques des résidus (voir Fig. 1.2).

Les méthodes automatisées de détection d'endommagements par vibration qui sont ro-
bustes aux changements des conditions environnementales prennent en compte les paramètres
environnementaux mesurés, par exemple en utilisant des approches de régression ou
d'interpolation ou des méthodes de classification ; soit elles omettent la prise en compte
explicite des paramètres environnementaux en utilisant des résidus et des indicateurs d'endommagements robustes, ou des modèles de conditions normales de grande taille, représentant l'influence des paramètres sur les résidus sensibles aux endommagements.

# Chapitre 2 – Contexte des méthodes

Les développements de cette thèse sont basés sur la détection stochastique des endommagements basée sur le sous-espace (SSDD) [BAB00; DMH14]. Cette méthode est basées sur les propriétés des l'identification sous-espace stochastique (SSI) basée sur les covariance des données, à l'issue d'un modèle d'espace d'état représentant une structure mécanique linéaire invariante dans le temps (LTI). Le résidu utilise des matrices de Hankel contenant des covariances des mesures et est défini comme le produit du noyau gauche de la matrice de Hankel dans l'état de référence S et de la matrice de Hankel estimée à partir d'un ensemble de données de mesure actuelles de longueur N dans l'état actuel $\hat{\mathcal{H}}$ par

$$ \zeta \stackrel{\text{def}}{=} \sqrt{N} \text{vec}(S^T \hat{\mathcal{H}}). $$

Sa distribution asymptotique est Gaussienne, avec une moyenne nulle dans l'état sain et une moyenne différente de zéro dans l'état endommagé. Son évaluation statistique dans un test d'hypothèse donne une statistique de test de détection des endommagements, dont la distribution asymptotique est $\chi^2$, prenant en compte les incertitudes du résidu liées aux données de test par la covariance du résidu $\Sigma_{\zeta}$. Le teste s'écrit

$$ t = \zeta^T \Sigma_{\zeta}^{-1} \mathcal{J}_{\zeta} (\mathcal{J}_{\zeta}^T \Sigma_{\zeta}^{-1} \mathcal{J}_{\zeta})^{-1} \mathcal{J}_{\zeta}^T \Sigma_{\zeta}^{-1} \zeta. $$

$\mathcal{J}_{\zeta}$ est la sensibilité asymptotique du résidu par rapport aux paramètres du système dans l'état de référence. La covariance du résidu a un impact significatif sur la sensibilité du test. Son calcul est basé sur la covariance empirique de la covariance des sorties du système.
Chapitre 3 – Détection d'endommagements par méthodes sous-espaces tenant compte des incertitudes dans le noyau de référence

Jusqu’à présent, la covariance du résidu de la SSDD est dérivée à partir de l’incertitude de la matrice de Hankel, où le noyau gauche de de référence est supposé être parfaitement connu sans aucune contribution à l’incertitude du résidu. Ceci est vrai dans des cas idéaux, alors que dans les applications du monde réel, la référence S est généralement estimée à partir de données. Négliger ses incertitudes dans la covariance du résidu a pour conséquence que le test ne suit plus sa distribution théoriquement attendue de $\chi^2$. Cela devient particulièrement pertinent pour l’application en pratique lorsque peu de données du système de référence sont disponible, ne permettant pas le calcul d’un seuil empirique, ou donc un seuil théorique est désirable.

Dans ce chapitre, la théorie pour le résidu de la SSDD sous incertitudes du noyau de référence est dérivée. Le résidu est nouvellement évalué en le considérant comme une fonction des deux variables aléatoires $\hat{S}$ et $\hat{H}$. Les perturbations de ces deux variables sont propagées au résidu, conduisant au calcul de sa covariance asymptotique et au test d’hypothèse associé. L’application du nouveau test aux données numériques montre un comportement statistique correct et prévisible, permettant également le calcul d’un seuil théorique a priori comme illustré dans la Fig. 1 (droite), alors qu’au contraire, le test conventionnel qui néglige les incertitudes de référence s’écarte significativement des valeurs attendues (gauche). Dans des études ultérieures, le nouveau test montre une plus grande sensibilité aux petits endommagements (Fig. 2, à gauche), son indépendance vis-à-vis de la longueur des données (Fig. 2, à droite) et d’autres paramètres de la procédure de calcul.

Chapitre 4 – Détection d'endommagements par méthodes sous-espaces sous conditions environnementales variées

L’application du diagnostic automatisé des endommagements basé sur les vibrations à la surveillance de l’intégrité des ouvrages d’art (SHM) échoue souvent en raison de l’évolution des conditions environnementales, car le comportement dynamique des struc-
<figure><img src="image_68.png" /><figcaption>Figure 1 – Histogrammes du test dans les états de référence et endommagé, pour le test conventionnel (gauche) et le nouveau test (droite).</figcaption></figure>

<figure><img src="image_69.png" /><figcaption>Figure 2 – Probabilité de détection (gauche) et dépendance du test de la longueur de données *M* dans l'état de référence (droite).</figcaption></figure>

tures est affecté [Soh07]. Le résidu de la SSDD est également sensible à de tels changements. Nous développons la théorie d'une procédure SSDD robuste à la variabilité environnementale, inspirée de l'interpolation du modèle de [Zha18]. Les données enregistrées à quelques températures de référence seulement sont utilisées pour créer un noyau gauche de référence adapté à la température actuelle. Trois stratégies de calcul du noyau de référence sont proposées. La première stratégie utilise un noyau de référence calculé à partir d'une matrice de Hankel interpolée. La deuxième stratégie interpole les paramètres modaux conduisant à une matrice d'observabilité interpolée, à partir de laquelle le noyau de référence est calculé. Dans la troisième stratégie, un noyau gauche est calculé à partir d'une matrice d'observabilité qui est basée sur les paramètres modaux obtenus à partir d'une analyse de régression. Conformément au chapitre précédent, la théorie pour le calcul de la covariance du résidu où le noyau est adapté à la température est dérivée. La méthode est validée dans une étude numérique. Par rapport à une approche existante de
[Bal+08a] qui échoue lorsque l'effet de la température est plus élevé que l'effet dû aux endommagements, la nouvelle approche est robuste et toujours sensible aux endommagements comme le montre la Fig. 3. En utilisant le noyau gauche de référence adapté à la température, le taux de fausses alarmes peut être réduit de manière significative. Les stratégies basées sur la matrice d'observabilité sont plus performantes que l'interpolation directe de la matrice de Hankel.

<figure><img src="image_70.png" /><figcaption>Figure 3 – Valeurs du test pour toutes les strategies développées, ainsi que pour l’approche existante [Bal+08a] aux températures de test variées et les références prises à 0, 5, 10, 15 and 20°C.</figcaption></figure>

Chapitre 5 – Détection d'endommagements dans un
cadre en acier en tenant compte des incertitudes dans
la référence

La méthode de détection d'endommagements pour le résidu avec référence incer-
taine est validée sur des données expérimentales provenant d'un modèle de laboratoire
d'une structure de fondation d'éolienne off-shore permettant des changements structurels
réversibles. Les données ont été obtenues dans le cadre des travaux de Hille [Hil18] à
l'Institut fédéral de recherche et d'essais des matériaux (BAM). Une procédure de test
non paramétrique est utilisée pour éviter l'identification des modes sensibles aux endom-
magements.

Les résultats correspondent aux conclusions de l'étude numérique. Dans l'état de
référence, le test conventionnel conduit à des valeurs d'essai plus élevées en raison d'un
biais qui est introduit en négligeant les incertitudes dans l'état de référence (Fig. 4,
à gauche). Le nouveau test, au contraire, montre une bonne conformité aux valeurs
théoriquement attendues (droite), ce qui permet d'utiliser un seuil calculé sur une base
théorique. De plus, le nouveau test a une meilleure sensibilité aux endommagements que le test conventionnel, et des études plus poussées ont confirmé que le nouveau test est très robuste aux changements de longueur des données. Avec le test conventionnel, les valeurs et les performances du test ne sont pas prévisibles et nécessitent donc une étude approfondie de l'état de référence basée sur des données suffisamment longues pour, par exemple, dériver des seuils fiables.

<figure><img src="image_71.png" /><figcaption>Figure 4 – Valeurs du test conventionnel (gauche) et du nouveau test (droite) dans l'état de référence et plusieurs états endommagés.</figcaption></figure>

# Chapitre 6 – Détection d'endommagements d'une poutre en laboratoire affectée par la température

Une première application des méthodes robustes du chapitre 4 sur les données réelles d'une poutre en béton testée dans une chambre climatique est effectuée. La poutre est soumise à cinq températures différentes et à trois états de système différents dans le cadre des travaux de Simon et al. [Sim+20] dans le hall d'essai de l'Institut fédéral de recherche et d'essai des matériaux (BAM). La procédure de détection d'endommagements non paramétrique et entièrement basée sur les données est appliquée en utilisant l'interpolation directe de la matrice de Hankel pour calculer le noyau de référence adapté à la température.

Les résultats correspondent aux conclusions de l'étude numérique. L'approche d'interpolation est capable de distinguer de manière fiable les états sains et endommagés, en évitant les fausses alarmes comme le montre la Fig. 5 (en haut), alors que les méthodes précédentes ne parviennent pas à classer correctement l'état sain lorsque la température dans l'état de test
est différente des températures pour lesquelles des données de référence sont disponibles,
comme l'illustre l'image du bas.

<figure><img src="image_72.png" /><figcaption>Figure 5 – Valeurs du test avec la stratégie d’interpolation des matrices Hankel (en haut), et avec l’approche existante [Bal+08a] (en bas). Les données de référence ont été utilisées à -25°C, 5°C, et 40°C.</figcaption></figure>

Chapitre 7 – Détection d'endommagements sur un
pont d'essai exposé à des variations de température
ambiente

Les approches robustes aux changements de température pour la SSDD sont appliquées
aux données expérimentales d'un pont d'essai (Fig. 6) situé sur un terrain d'essai extérieur
de l'Institut fédéral de recherche et d'essai des matériaux (BAM) exposé à des conditions
environnementales réelles. Les expériences incluant la conceptualisation du pont ont été
réalisées dans le cadre de cette thèse.

Les nouvelles méthodes avec le noyau de référence adapté à la température permet une
détection fiable des endommagements avec une réduction significative du taux de fausses
alarmes par rapport à d'autres approches, comme l'illustre la Fig. 7. En particulier, lorsque
des données de référence sont disponibles dans une plage de température et que la structure
est testée à une température en dehors de cette plage, la méthode proposée permet de
résoudre correctement ce problème par extrapolation à partir des points de référence.
Dans des études ultérieures, l'effet de la conception de la fonction de pondération est analysé, ce qui permet d'améliorer les performances du test en choisissant des fonctions de pondération adaptées au comportement dépendant de la température.

<figure><img src="image_73.png" /><figcaption>Figure 6 – Testing bridge.</figcaption></figure>

<figure><img src="image_74.png" /><figcaption>Figure 7 – Test values of the 4 algorithms for different testing temperatures in the undamaged and the damaged case, computed for reference setup A.</figcaption></figure>

## Conclusions

Dans cette thèse, des développements ont été réalisés dans le but d'améliorer la robustesse et la sensibilité de la SSDD pour une application sur les ouvrages d'art.

Jusqu'à présent, dans la procédure SSDD existante, les incertitudes liées aux modèles de référence estimés n'ont pas été prises en compte, ce qui entraînait un biais inhérent et un comportement imprévisible du test. Cette lacune a été comblée et l'évaluation de la covariance du résidu a été effectuée en tenant compte des incertitudes liées aux données d'essai et à la référence estimée. La partie cruciale de la définition d'un seuil entre l'état sain et l'état endommagé peut maintenant être faite a priori sur la base de considérations théoriques sans études empiriques approfondies sur les données de référence. La méthode
développée a été validée sur des données numériques et expérimentales montrant une
bonne conformité avec le comportement théorique attendu et conduisant à une sensibilité
de test plus élevée. Cependant, une estimation minutieuse de la matrice de covariance des
données est nécessaire, car sa qualité affecte la quantification de l'incertitude du résidu.

Deuxièmement, une approche pour la SSDD a été dérivée qui est robuste aux change-
ments environnementaux. La méthode s'inspire d'une approche d'interpolation de modèle
conduisant à un noyau gauche de référence adapté aux paramètres (de température) sans
connaissance explicite de la dépendance de la structure du paramètre. Trois stratégies
différentes ont été proposées, basées sur des approches d'interpolation ou de régression de
la matrice de Hankel ou des paramètres modaux. Les incertitudes du noyau de référence
adapté aux paramètres ont été quantifiées, et il a été montré comment les calculer en fonc-
tion de la stratégie choisie. La validation sur des données numériques et expérimentales a
montré une sensibilité de test significativement plus élevée que les approches précédentes,
et les fausses alarmes dues à des changements dans les conditions environnementales ont
pu être évitées en grande partie. L'approche nécessite une conception minutieuse de la
fonction d'interpolation.

L'estimation de la matrice de covariance du résidu devrait faire l'objet d'une étude plus approfondie dans le futur, car dans une application pratique, l'estimation de la covariance à partir d'un nombre relativement faible de blocs de données par rapport à la dimension du résidu peut être difficile et entraîner des problèmes numériques. Un autre sujet qui devrait être abordé, bien qu'il s'agisse d'une tâche dépendant de la structure, est l'évaluation de l'effet des fonctions de pondération pour la robustesse aux variations environnementales et leur choix optimal. En outre, une version paramétrique de l'approche robuste est nécessaire pour l'extension à la localisation et à la quantification des endommagements [DMZ16] et l'évaluation rigoureuse de la sensibilité des méthodes aux endommagements [MDV21].
<figure><img src="image_75.png" /></figure>
[AB82] R.J. Allemang and D.L. Brown. « A correlation coefficient for modal vector analysis ». In: *Proceedings of the 1st International Modal Analysis Conference*. Orlando, Florida, USA, 1982, pp. 110–116.

[Ad99] M.M. Abdel Wahab and G. de Roeck. « Damage detection in bridges using modal curvatures: application to a real damage scenario ». In: *Journal of Sound and Vibration* 226.2 (1999), pp. 217–235.

[Ada+78] R.D. Adams, P. Cawley, C.J. Pye, and B.J. Stone. « A vibration technique for non-destructively assessing the integrity of structures ». In: *Journal of Mechanical Engineering Science* 20.2 (1978), pp. 93–100.

[All+19] S. Allahdadian, M. Döhler, C. Ventura, and L. Mevel. « Towards robust statistical damage localization via model-based sensitivity clustering ». In: *Mechanical Systems and Signal Processing* 134 (2019), p. 106341.

[All03] R.J. Allemang. « The modal assurance criterion – Twenty years of use and abuse ». In: *Sound and Vibration* 37 (2003), pp. 14–23.

[Avc+21] O. Avci, O. Abdeljaber, S. Kiranyaz, M. Hussein, M. Gabbouj, and D.J. Inman. « A review of vibration-based damage detection in civil structures: From traditional methods to Machine Learning and Deep Learning applications ». In: *Mechanical Systems and Signal Processing* 147 (2021), p. 107077.

[BAB00] M. Basseville, M. Abdelghani, and A. Benveniste. « Subspace-based fault detection algorithms for vibration monitoring ». In: *Automatica* 36.1 (2000), pp. 101–109.

[Bal+08a] É. Balmès, M. Basseville, F. Bourquin, L. Mevel, H. Nasser, and F. Treyssède. « Merging sensor data from multiple temperature scenarios for vibration-based monitoring of civil structures ». In: *Structural Health Monitoring* 7.2 (2008), pp. 129–142.
[Bal+08b] É. Balmès, M. Basseville, L. Mevel, H. Nasser, and W. Zhou. « Statistical model-based damage localization: a combined subspace-based and substruc- turing approach ». In: *Structural Control and Health Monitoring* 15.6 (2008), pp. 857–875.

[Bal+09] É. Balmès, M. Basseville, L. Mevel, and H. Nasser. « Handling the tem- perature effect in vibration-based monitoring of civil structures: a combined subspace-based and nuisance rejection approach ». In: *Control Engineering Practice* 17.1 (2009), pp. 80–87.

[Bas+10] M. Basseville, F. Bourquin, L. Mevel, H. Nasser, and F. Treyssède. « Han- dling the temperature effect in vibration monitoring: two subspace-based analytical approaches ». In: *Journal of Engineering Mechanics* 136.3 (2010), pp. 367–378.

[BB15] L. Balsamo and R. Betti. « Data-based structural health monitoring using small training data sets ». In: *Structural Control and Health Monitoring* 22.10 (2015), pp. 1240–1264.

[BBM87] A. Benveniste, M. Basseville, and G.V. Moustakides. « The asymptotic local approach to change detection and model validation ». In: *IEEE Transactions on Automatic Control* 32.7 (1987), pp. 583–592.

[Ber02] D. Bernal. « Load vectors for damage localization ». In: *Journal of Engineer- ing Mechanics* 128.1 (2002), pp. 7–14.

[Ber10] D. Bernal. « Load vectors for damage location in systems identified from op- erational loads ». In: *Journal of Engineering Mechanics* 136.1 (2010), pp. 31– 39.

[Ber13] D. Bernal. « Kalman filter damage detection in the presence of changing pro- cess and measurement noise ». In: *Mechanical Systems and Signal Processing* 39.1-2 (2013), pp. 361–371.

[BF85] A. Benveniste and J.-J. Fuchs. « Single sample modal identification of a non- stationary stochastic process ». In: *IEEE Transactions on Automatic Control* 30.1 (1985), pp. 66–74.

[BHS16] M. Baeßler, F. Hille, and S. Said. « Bauwerksüberwachung von Brücken im Kontext saisonaler Einflüsse ». In: *Proceedings of Messen Im Bauwesen*. Berlin, Germany, 2016.
[Bhu+19] M.D.H. Bhuyan, G. Gautier, N. Le Touz, M. Döhler, F. Hille, J. Dumoulin, and L. Mevel. « Vibration-based damage localization with load vectors under temperature changes ». In: *Structural Control and Health Monitoring* 26.11 (2019), e2439.

[BM07] A. Benveniste and L. Mevel. « Nonstationary Consistency of Subspace Methods ». In: *IEEE Transactions on Automatic Control* 52.6 (2007), pp. 974–984.

[BMG04] M. Basseville, L. Mevel, and M. Goursat. « Statistical model-based damage detection and localization: subspace-based residuals and damage-to-noise sensitivity ratios ». In: *Journal of Sound and Vibration* 275.3 (2004), pp. 769–794.

[Bro+01] J.M.W. Brownjohn, P.-Q. Xia, H. Hao, and Y. Xia. « Civil structure condition assessment by FE model updating:: methodology and case studies ». In: *Finite Elements in Analysis and Design* 37.10 (2001), pp. 761–775.

[Bro+18] J.M.W. Brownjohn, S.-K. Au, Y. Zhu, Z. Sun, B. Li, J. Bassitt, E. Hudson, and H. Sun. « Bayesian operational modal analysis of Jiangyin Yangtze River Bridge ». In: *Mechanical Systems and Signal Processing* 110 (2018), pp. 210–230.

[CB02] G. Casella and R.L. Berger. *Statistical inference*. Pacific Grove, CA, USA: Duxbury Press, 2002.

[CB08] E.P. Carden and J.M.W. Brownjohn. « ARMA modelled time-series classification for structural health monitoring of civil infrastructure ». In: *Mechanical Systems and Signal Processing* 22.2 (2008), pp. 295–314.

[CF04] E.P. Carden and P. Fanning. « Vibration based condition monitoring: a review ». In: *Structural Health Monitoring* 3.4 (2004), pp. 355–377.

[CM11] E.P. Carden and A. Mita. « Challenges in developing confidence intervals on modal parameters estimated for large civil infrastructure with stochastic subspace identification ». In: *Structural Control and Health Monitoring* 18.1 (2011), pp. 53–78.
[Cro+12] E.J. Cross, G. Manson, K. Worden, and S.G. Pierce. « Features for damage detection with insensitivity to environmental and operational variations ». In: *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences* 468.2148 (2012), pp. 4098–4122.

[Der+08] A. Deraemaeker, E. Reynders, G. De Roeck, and Kullaa J. « Vibration-based structural health monitoring using output-only measurements under changing environment ». In: *Mechanical Systems and Signal Processing* 22.1 (2008), pp. 34–56.

[Der+15] N. Dervilis, I. Antoniadou, R.J. Barthorpe, E.J. Cross, and K. Worden. « Robust methods for outlier detection and regression for SHM applications ». In: *International Journal of Sustainable Materials and Structural Systems* 2.1-2 (2015), pp. 3–26.

[DFP98] S.W. Doebling, C.R. Farrar, and M.B. Prime. « A summary review of vibration-based damage identification methods ». In: *The Shock and Vibration Digest* 30.2 (1998), pp. 91–105.

[DM12] M. Döhler and L. Mevel. « Fast multi-order computation of system matrices in subspace-based system identification ». In: *Control Engineering Practice* 20.9 (2012), pp. 882–894.

[DM13a] M. Döhler and L. Mevel. « Efficient multi-order uncertainty computation for stochastic subspace identification ». In: *Mechanical Systems and Signal Processing* 38.2 (2013), pp. 346–366.

[DM13b] M. Döhler and L. Mevel. « Subspace-based fault detection robust to changes in the noise covariances ». In: *Automatica* 49.9 (2013), pp. 2734–2743.

[DMH14] M. Döhler, L. Mevel, and F. Hille. « Subspace-based damage detection under changes in the ambient excitation statistics ». In: *Mechanical Systems and Signal Processing* 45.1 (2014), pp. 207–224.

[DMZ16] M. Döhler, L. Mevel, and Q. Zhang. « Fault detection, isolation and quantification from Gaussian residuals with application to structural damage diagnosis ». In: *Annual Reviews in Control* 42 (2016), pp. 244–256.

[Döh+14] M. Döhler, F. Hille, L. Mevel, and W. Rücker. « Structural health monitoring with statistical methods during progressive damage test of S101 Bridge ». In: *Engineering Structures* 69 (2014), pp. 183–193.
[Döh+15] M. Döhler, L. Marin, L. Mevel, and D. Bernal. « Operational modal analysis with uncertainty quantification for SDDLV-based damage localization ». In: *MATEC Web of Conferences* 20 (2015).

[DVG12] J. Dong, M. Verhaegen, and F. Gustafsson. « Robust fault detection with statistical uncertainty in identified parameters ». In: *IEEE Transactions on Signal Processing* 60.10 (2012), pp. 5064–5076.

[Far+00] C.R. Farrar, P.J. Cornwell, S.W. Doebling, and M.B. Prime. *Structural health monitoring studies of the Alamosa Canyon and I-40 Bridges*. Tech. rep. Los Alamos National Lab. (LANL), Los Alamos, NM (United States), 2000.

[Far+94] C.R. Farrar, W.E. Baker, T.M. Bell, K.M. Cone, T.W. Darling, T.A. Duffey, A. Eklund, and A. Migliori. *Dynamic characterization and damage detection in the I-40 bridge over the Rio Grande*. Tech. rep. Los Alamos National Lab. (LANL), Los Alamos, NM (United States), 1994.

[FC03] P.J. Fanning and E.P. Carden. « Damage Detection based on Single-Input-Single-Output Measurements ». In: *Journal of Engineering Mechanics* 129.2 (2003), pp. 202–209.

[FDN01] C.R. Farrar, S.W. Doebling, and D.A. Nix. « Vibration-based structural damage identification ». In: *Philosophical Transactions: Mathematical, Physical and Engineering Sciences* 359.1778 (2001), pp. 131–149.

[FJ97] C.R. Farrar and G.H. James III. « System identification from ambient vibration measurements on a bridge ». In: *Journal of Sound and Vibration* 205.1 (1997), pp. 1–18.

[FK13] S.D. Fassois and F.P. Kopsaftopoulos. « Statistical time series methods for vibration based structural health monitoring ». In: *New Trends in Structural Health Monitoring*. Ed. by Wiesław Ostachowicz and J. Alfredo Güemes. Vienna: Springer Vienna, 2013, pp. 209–264.

[FKB13] C.P. Fritzen, P. Kraemer, and I. Buethe. « Vibration-based damage detection under changing environmental and operational conditions ». In: *Advances in Science and Technology* 83 (2013), pp. 95–104.
[FKK13] C.P. Fritzen, M. Klinkov, and P. Kraemer. « Vibration-Based Damage Diagnosis and Monitoring of External Loads ». In: *New Trends in Structural Health Monitoring*. Ed. by Wiesław Ostachowicz and J. Alfredo Güemes. Vienna: Springer, Vienna, 2013, pp 149–208. ISBN: 978-3-7091-1389-9.

[FM95] M. Friswell and J.E. Mottershead. *Finite element model updating in structural dynamics*. Springer Netherlands, 1995. ISBN: 978-90-481-4535-5.

[FMG03] C. P. Fritzen, G. Mengelkamp, and A. Güemes. « Elimination of temperature effects on damage detection within a smart structure concept. » In: *Proceedings of 4th International Workshop on Structural Health Monitoring*. Stanford, CA, USA, 2003, pp. 1530–1538.

[Fri07] M. Friswell. « Damage identification using inverse methods ». In: *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences* 365.1851 (2007), pp. 393–410.

[FSF01] M.L. Fugate, H. Sohn, and C.R. Farrar. « Vibration-based damage detection using statistical process control ». In: *Mechanical systems and signal processing* 15.4 (2001), pp. 707–721.

[FW07] C.R. Farrar and K. Worden. « An introduction to structural health monitoring ». In: *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences* 365.1851 (2007), pp. 303–315.

[FW12] C.R. Farrar and K. Worden. *Structural health monitoring: A machine learning perspective*. Chichester, West Sussex, U.K: Wiley, 2012. ISBN: 978-1-119-99433-6.

[Gau+17] G. Gautier, L. Mevel, J.-M. Mencik, R. Serra, and M. Döhler. « Variance analysis for model updating with a finite element based subspace fitting approach ». In: *Mechanical Systems and Signal Processing* 91 (2017), pp. 142–156.

[GDM20] S. Greś, M. Döhler, and L. Mevel. « Variance computation for system matrices and transfer function from input/output subspace system identification ». In: *IFAC-PapersOnLine* 53.2 (2020). 21st IFAC World Congress, pp. 933–938.
[GDM21] S. Greś, M. Döhler, and L. Mevel. « Uncertainty quantification of the modal assurance criterion in operational modal analysis ». In: *Mechanical Systems and Signal Processing* 152 (2021), p. 107457.

[GGW17] J. Gu, M. Gul, and X. Wu. « Damage detection under varying temperature using artificial neural networks ». In: *Structural Control and Health Monitoring* 24.11 (2017), e1998.

[GK17] Y. Goi and C.-W. Kim. « Damage detection of a truss bridge utilizing a damage indicator from multivariate autoregressive model ». In: *Journal of Civil Structural Health Monitoring* 7 (2017), pp. 153–162.

[GN09] M. Gul and F. Necati Catbas. « Statistical pattern recognition for Structural Health Monitoring using time series modeling: Theory and experimental verifications ». In: *Mechanical Systems and Signal Processing* 23.7 (2009), pp. 2192–2204.

[Gre+17] S. Greś, M.D. Ulriksen, M. Döhler, R.J. Johansen, P. Andersen, L. Damkilde, and S.A. Nielsen. « Statistical methods for damage detection applied to civil structures ». In: *Procedia Engineering* 199 (2017). X International Conference on Structural Dynamics, EURODYN 2017, pp. 1919–1924.

[Gre+21a] S. Greś, M. Döhler, P. Andersen, and L. Mevel. « Kalman filter-based subspace identification for operational modal analysis under unmeasured periodic excitation ». In: *Mechanical Systems and Signal Processing* 146 (2021), p. 106996.

[Gre+21b] S. Greś, M. Döhler, P. Andersen, and L. Mevel. « Subspace-based Mahalanobis damage detection robust to changes in excitation covariance ». In: *Structural Control and Health Monitoring* 28.8 (2021), e2760.

[Gre+21c] S. Greś, M. Döhler, P. Andersen, and L. Mevel. « Uncertainty quantification for the modal phase collinearity of complex mode shapes ». In: *Mechanical Systems and Signal Processing* 152 (2021), p. 107436.

[Han70] E.J. Hannan. *Multiple time series*. New York: Wiley, 1970.

[HF09] J. Hios and S. Fassois. « Stochastic identification of temperature effects on the dynamics of a smart composite beam: Assessment of multi-model and global approaches ». In: *Smart Materials and Structures* 18 (2009), p. 035011.
[Hil18] F. Hille. « Unterraumbasierte Detektion von Strukturschäden an Jacket-Gründungen von Offshore-Windenergieanlagen ». Doctoral Thesis. Berlin: Technische Universität Berlin, 2018.

[HO14] S. Hoell and P. Omenzetter. « Damage detection in a wind turbine blade based on time series methods ». In: *7th European Workshop on Structural Health Monitoring*. Nantes, France, 2014, pp. 237–244.

[Hu+15] W.-H. Hu, S. Said, R. Rohrmann, W. Rücker, A. Cunha, and A. Rogge. « Vibration-based structural health monitoring of a highway bridge based on continuous dynamic measurements during 14 years ». In: *Proceedings of SHMII-7 - 7th International conference on structural health monitoring of intelligent infrastructure*. Turin, Italy, 2015.

[Ike14] K. Ikeda. « Error analysis of state approaches in subspace identification methods ». In: *2014 Proceedings of the SICE Annual Conference (SICE)*. Sapporo, Japan, 2014, pp. 1685–1690.

[JC16] I.T. Jolliffe and J. Cadima. « Principal component analysis: a review and recent developments ». In: *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences* 374.2065 (2016), p. 20150202.

[JMM12] A. Jhinaoui, L. Mevel, and J. Morlier. « Subspace instability monitoring for linear periodically time-varying systems ». In: *16th IFAC Symposium on System Identification*. Brussels, Belgium, 2012, pp. 380–385.

[Jua94] J.-N. Juang. *Applied system identification*. Englewood Cliffs, NJ, USA: Prentice Hall, 1994.

[Kál63] R. Kálmán. « Mathematical description of linear dynamical systems ». In: *Journal of The Society for Industrial and Applied Mathematics, Series A: Control* 1 (1963), pp. 152–192.

[KBD13] S.M. Kojidi, D. Bernal, and M. Döhler. « Change detection under environmental variability ». In: *5th International Operational Modal Analysis Conference, IOMAC 2013* (2013), pp. 1–8.
[KF16] P. Kraemer and H. Friedmann. « Experimental validation of stochastic sub-space algorithms for structural health monitoring of offshore wind turbine towers and foundations ». In: *8th European Conference in Structural Health Monitoring*. Bilbao, Spain, 2016.

[Kra91] M.A. Kramer. « Nonlinear principal component analysis using autoassociative neural networks ». In: *AIChE Journal* 37.2 (1991), pp. 233–243.

[Kul03] J. Kullaa. « Damage detection of the Z24 bridge using control charts ». In: *Mechanical Systems and Signal Processing* 17.1 (2003), pp. 163–170.

[Kul14] J. Kullaa. « Structural health monitoring under non-linear environmental or operational influences ». In: *Shock and Vibration* 2014 (2014).

[KWA19] T. Kernicky, M. Whelan, and E. Al-Shaer. « Vibration-based damage detection with uncertainty quantification by structural identification using non-linear constraint satisfaction with interval arithmetic ». In: *Structural Health Monitoring* 18.5-6 (2019), pp. 1569–1589.

[KXY18] H. Kordestani, Y.-Q. Xiang, and X.-W. Ye. « Output-only damage detection of steel beam using moving average filter ». In: *Shock and Vibration* 2018 (2018).

[LD07] C. Y. Liu and J. T. DeWolf. « Effect of temperature on modal variability of a curved concrete bridge under ambient loads ». In: *Journal of Structural Engineering* 133.12 (2007), pp. 1742–1751.

[Le 56] L. Le Cam. « On the asymptotic theory of estimation and testing hypotheses ». In: *Berkeley Symposium on Mathematical Statistics and Probability*. 1956.

[Leo06] Cornelius T. Leondes, ed. *Mems/Nems, (1) Handbook Techniques and Applications Design Methods*. Springer, US, 2006.

[Lia+21] C.M. Liao, F. Hille, D. Fontoura Barosso, and E. Niederleithinger. « Monitoring of a prestressed bridge model by ultrasonic measurement and vibration recordings ». In: *Proceedings of 8th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering COMPDYN 2021*. Athens, Greece, 2021.
[Lim+21] M.P. Limongelli, E. Manoach, S. Quqa, P.F. Giordano, B. Bhowmik, V. Pakrashi, and A. Cigada. « Vibration Response-Based Damage Detection ». In: *Structural Health Monitoring Damage Detection Systems for Aerospace*. Ed. by Jasiūnienė E. Sause M.G.R. Springer Aerospace Technology, 2021.

[Lim10] M.P. Limongelli. « Frequency response function interpolation for damage detection under changing environment ». In: *Mechanical Systems and Signal Processing* 24 (2010), pp. 2898–2913.

[Lim19] M.P. Limongelli. « Vibration-based structural health monitoring: Challenges and opportunities ». In: *Advances in Engineering Materials, Structures and Systems: Innovations, Mechanics and Applications*. 2019, pp. 1999–2004.

[LLM08] J. Liu, X. Liu, and X. Ma. « First-order perturbation analysis of singular vectors in singular value decomposition ». In: *IEEE Transactions on Signal Processing* 56.7 (2008), pp. 3044–3049.

[LM11] X.-B. Lam and L. Mevel. « Uncertainty quantification for eigensystem-realization-algorithm, a class of subspace system identification ». In: *IFAC Proceedings Volumes* 44.1 (2011). 18th IFAC World Congress, pp. 6529–6534.

[LMK18] S. Liehr, S. Münzenberger, and K. Krebber. « Wavelength-scanning coherent OTDR for dynamic high strain resolution sensing ». In: *Optic Express* 26.8 (2018), pp. 10573–10588.

[Man02] G. Manson. « Identifying damage sensitive, environment insensitive features for damage detection ». In: *Proceedings of the International Conference on Identification in Engineering Systems*. Swansea, Wales, 2002, pp. 187–197.

[Mao12] Z. Mao. « Uncertainty quantification in vibration-based structural health monitoring for enhanced decision-making capability ». PhD thesis. University of California, San Diego, Jan. 2012.

[MCC12] F. Magalhães, A. Cunha, and E. Caetano. « Vibration based structural health monitoring of an arch bridge: From automated OMA to damage detection ». In: *Mechanical Systems and Signal Processing* 28 (2012), pp. 212–228.

[MDM16] P. Mellinger, M. Döhler, and L. Mevel. « Variance estimation of modal parameters from output-only and input/output subspace-based system identification ». In: *Journal of Sound and Vibration* 379 (2016), pp. 1–27.
[MDV21] A. Mendler, M. Döhler, and C. Ventura. « A reliability-based approach to determine the minimum detectable damage for statistical damage detection ». In: *Mechanical Systems and Signal Processing* 154 (2021), p. 107561.

[Men+20] A. Mendler, M. Döhler, C. Ventura, and L. Mevel. « Clustering of redundant parameters for fault isolation with Gaussian residuals ». In: *IFAC-PapersOnLine* 53 (2020). 21st IFAC World Congress, pp. 13727–13732.

[Men20] A. Mendler. « Minimum diagnosable damage and optimal sensor placement for structural health monitoring ». PhD thesis. University of British Columbia, 2020.

[Mig+12] L.F.F. Miguel, L.F.F. Miguel, J. Kaminski, and J.D. Riera. « Damage detection under ambient vibration by harmony search algorithm ». In: *Expert Systems with Applications* 39.10 (2012), pp. 9704–9714.

[NK07] K.K. Nair and A. Kiremidjian. « Time series based structural damage detection algorithm using Gaussian mixtures modeling ». In: *Journal of Dynamic Systems Measurement and Control* 129.3 (May 2007), pp. 285–293.

[Oli+18] G. Oliveira, F. Magalhães, Á. Cunha, and E. Caetano. « Vibration-based damage detection in a wind turbine using 1 year of data ». In: *Structural Control and Health Monitoring* 25.11 (2018), e2238.

[PB94] A.K. Pandey and M. Biswas. « Damage detection in structures using changes in flexibility ». In: *Journal of Sound and Vibration* 169.1 (1994), pp. 3–17.

[PBS91] A.K. Pandey, M. Biswas, and M.M. Samman. « Damage detection from changes in curvature mode shapes ». In: *Journal of Sound and Vibration* 145.2 (1991), pp. 321–332.

[PD00] B. Peeters and G. De Roeck. « One year monitoring of the Z24-bridge: Environmental influences versus damage events ». In: *Proceedings of SPIE - The International Society for Optical Engineering* 2 (2000).

[PD99] B. Peeters and G. De Roeck. « Reference-based stochastic subspace identification for output-only modal analysis ». In: *Mechanical Systems and Signal Processing* 13.6 (1999), pp. 855–878.

[Pee+04] B. Peeters, H. Van der Auweraer, P. Guillaume, and J. Leuridan. « The PolyMAX frequency-domain method: A new standard for modal parameter estimation? » In: *Shock and Vibration* 11 (2004), pp. 395–409.
[PGS07] R. Pintelon, P. Guillaume, and J. Schoukens. « Uncertainty calculation in (operational) modal analysis ». In: *Mechanical Systems and Signal Processing* 21.6 (2007), pp. 2359–2373.

[Pir+19] S. Pirskawetz, K.P. Gründer, D. Kadoke, and J. Unger. « Überwachung von Stahlbetonbrücken - Ein Modellprojekt ». In: *Proceedings of 22. Kolloquium Schallemission und 3. Anwenderseminar Zustandsüberwachung mit geführten Wellen*. Karlsruhe, Germany, 2019.

[PMD01] B. Peeters, J. Maeck, and G. De Roeck. « Vibration-based damage detection in civil engineering: excitation sources and temperature effects ». In: *Smart Materials and Structures* 10.3 (2001), pp. 518–527.

[Ram+10] L.F. Ramos, L. Marques, P.B. Lourenço, G. De Roeck, A. Campos-Costa, and J. Roque. « Monitoring historical masonry structures with operational modal analysis: Two case studies ». In: *Mechanical Systems and Signal Processing* 24.5 (2010), pp. 1291–1305.

[RF10] C. Rainieri and G. Fabbrocino. « Automated output-only dynamic identification of civil engineering structures ». In: *Mechanical Systems and Signal Processing* 24.3 (2010), pp. 678–695.

[Roh+00a] R. Rohrmann, M. Baeßler, S. Said, W. Schmid, and W. Rücker. « Structural causes of temperature affected modal data of civil structures obtained by long time monitoring, No.141 ». In: *Proceedings of SPIE - The International Society for Optical Engineering* 1 (2000).

[Roh+00b] R. Rohrmann, S. Said, W. Schmid, W. Rücker, and M. Baeßler. « Zustandsuntersuchung und Dauerüberwachung von Bauwerken mit Hilfe modaler Parameter ». In: *Vortragsband der VDI-Schwingungstagung Experimentelle und rechnerische Modalanalyse sowie Identifikation dynamischer Systeme*. Kassel, Germany, 2000, pp. 327–350.

[RPD08] E. Reynders, R. Pintelon, and G. De Roeck. « Uncertainty bounds on modal parameters obtained from stochastic subspace identification ». In: *Mechanical Systems and Signal Processing* 22.4 (2008). Special Issue: Crack Effects in Rotordynamics, pp. 948–969.
[RWD14] E. Reynders, G. Wursten, and G. De Roeck. « Output-only structural health monitoring in changing environmental conditions by means of nonlinear system identification ». In: *Structural Health Monitoring* 13.1 (2014), pp. 82–93.

[Ryt93] A. Rytter. « Vibrational based inspection of civil engineering structures ». English. PhD thesis. Aalborg, Denmark: Aalborg University, 1993.

[SCF00] H. Sohn, J. Czarnecki, and C.R. Farrar. « Structural health monitoring using statistical process control ». In: *Journal of Structural Engineering* 126 (11 2000), pp. 1356–1363.

[SCN09] F. Schoefs, A. Clément, and A. Nouy. « Assessment of ROC curves for inspection of random fields ». In: *Structural Safety* 31.5 (2009), pp. 409–419.

[SCS16] M.D. Spiridonakos, E.N. Chatzi, and B. Sudret. « Polynomial Chaos Expansion Models for the Monitoring of Structures under Operational Variability ». In: *ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering* 2.3 (2016), B4016003.

[SDL15] E. Simoen, G. De Roeck, and G. Lombaert. « Dealing with uncertainty in model updating for damage assessment: A review ». In: *Mechanical Systems and Signal Processing* 56-57 (2015), pp. 123–149.

[SF01] H. Sohn and C.R. Farrar. « Damage diagnosis using time series analysis of vibration signals ». In: *Smart Materials and Structures* 10.3 (2001), pp. 446–451.

[Sim+20] P. Simon, R. Schneider, E. Viefhues, S. Said, R. Herrmann, and M. Baeßler. « Vibration-based structural health monitoring of a reinforced concrete beam subject to varying ambient temperatures using Bayesian methods ». In: *Proceedings of XI International Conference on Structural Dynamics EURODYN 2020*. Athens, Greece, 2020.

[SM13] S. Sankararaman and S. Mahadevan. « Bayesian methodology for diagnosis uncertainty quantification and health monitoring ». In: *Structural Control and Health Monitoring* 20.1 (2013), pp. 88–106.

[SMS99] R.P.C. Sampaio, N.M.M. Maia, and J.M.M. Silva. « Damage detection using the frequency-response-function curvature method ». In: *Journal of Sound and Vibration* 226.5 (1999), pp. 1029–1042.
[Soh+98] H. Sohn, M. Dzwonczyk, E.G. Straser, K.H. Law, T.H.-Y. Meng, and A.S. Kiremidjian. « Adaptive modeling of environmental effects in modal parameters for damage detection in civil structures ». In: *Smart Structures and Materials 1998: Smart Systems for Bridges, Structures, and Highways*. International Society for Optics and Photonics. 1998, pp. 127–138.

[Soh07] H. Sohn. « Effects of environmental and operational variability on structural health monitoring ». In: *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences* 365(1851) (2007), pp. 539–560.

[SSS10] M. Sun, W.J. Staszewski, and R.N. Swamy. « Smart sensing technologies for structural health monitoring of civil engineering structures ». In: *Advances in Civil Engineering* (2010).

[SW10] C. Surace and K. Worden. « Novelty detection in a changing environment: A negative selection approach ». In: *Mechanical Systems and Signal Processing* 24.4 (2010), pp. 1114–1128.

[SWF02] H. Sohn, K. Worden, and C.R. Farrar. « Statistical damage classification under changing environmental and operational conditions ». In: *Journal of Intelligent Material Systems and Structures* 13.9 (2002), pp. 561–574.

[TA18] A. Tributsch and C. Adam. « An enhanced energy vibration-based approach for damage detection and localization ». In: *Structural Control and Health Monitoring* 25.1 (2018), e2047.

[Tan+19] Z. Tang, Z. Chen, Y. Bao, and H. Li. « Convolutional neural network-based data anomaly detection method using multiple information for structural health monitoring ». In: *Structural Control and Health Monitoring* 26.1 (2019), e2296.

[Tan05] J. Tang. « Frequency response based damage detection using principal component analysis ». In: *2005 IEEE International Conference on Information Acquisition*. Hong Kong, China, 2005.

[Ube+17] F. Ubertini, G. Comanducci, N. Cavalagli, A. Laura Pisello, A. Luigi Materazzi, and F. Cotana. « Environmental effects on natural frequencies of the San Pietro bell tower in Perugia, Italy, and their removal for structural per-
formance assessment». In: *Mechanical Systems and Signal Processing* 82 (Jan. 2017), pp. 307–322.

[Ulr+16] M.D. Ulriksen, D. Tcherniak, P.H. Kirkegaard, and L. Damkilde. « Opera- tional modal analysis and wavelet transformation for damage identification in wind turbine blades ». In: *Structural Health Monitoring* 15.4 (2016), pp. 381– 388.

[Ung+17] J. Unger, K.P. Gründer, F. Hille, S. Liehr, S. Maack, E. Niederleithinger, S. Piskawetz, A. Rogge, S. Said, E. Viefhues, and S. Zorn. « Innovative Messsys- teme zum Brückenmonitoring am Beispiel einer Versuchsbrücke – vom Sensor bis zur Zustandsprognose ». In: *Proceedings of Messen im Bauwesen 2017*. Berlin, Germany, 2017.

[Van+05] S. Vanlanduit, E. Parloo, B. Cauberghe, P. Guillaume, and P. Verboven. « A robust singular value decomposition for damage detection under changing operating conditions and structural uncertainties ». In: *Journal of Sound and Vibration* 284.3 (2005), pp. 1033–1050.

[VD96] P. Van Overschee and B. De Moor. *Subspace identification for linear systems: theory, implementation, applications*. Dordrecht, The Netherlands: Kluwer, 1996.

[Ven+14] C. Ventura, P. Andersen, L. Mevel, and M. Döhler. « Structural health mon- itoring of the Pitt River Bridge in British Columbia, Canada ». In: *Proceed- ings of WCSCM - 6th World Conference on Structural Control and Moni- toring*. Barcelona, Spain, July 2014.

[Vie+17] E. Viefhues, M. Döhler, F. Hille, and L. Mevel. « Stochastic subspace-based damage detection with uncertainty in the reference null space ». In: *IWSHM - 11th International Workshop on Structural Health Monitoring*. Stanford, United States, 2017.

[Vie+18] E. Viefhues, M. Döhler, F. Hille, and L. Mevel. « Asymptotic analysis of subspace-based data-driven residual for fault detection with uncertain ref- erence ». In: *IFAC-PapersOnLine* 51.24 (2018). 10th IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes SAFEPRO- CESS 2018, pp. 414–419.
[Vie+19] E. Viefhues, M. Döhler, Q. Zhang, F. Hille, and L. Mevel. « Subspace-based damage detection with rejection of the temperature effect and uncertainty in the reference ». In: *Proceedings of IOMAC 2019 - 8th International Operational Modal Analysis Conference*. Copenhagen, Denmark, 2019.

[Vie+20] E. Viefhues, M. Döhler, F. Hille, and L. Mevel. « Fault detection for linear parameter varying systems under changes in the process noise covariance ». In: *IFAC-PapersOnLine* 53.2 (2020). 21th IFAC World Congress, pp. 13668–13673.

[Vie+21] E. Viefhues, M. Döhler, P. Simon, R. Herrmann, F. Hille, and L. Mevel. « Stochastic subspace-based damage detection of a temperature affected beam structure ». In: *Proceedings of SHMII-10 2021 - 10th International Conference on Structural Health Monitoring of Intelligent Infrastructure*. Porto, Portugal, 2021, pp. 1–6.

[Vie+22] E. Viefhues, M. Döhler, F. Hille, and L. Mevel. « Statistical subspace-based damage detection with estimated reference ». In: *Mechanical Systems and Signal Processing* 164 (2022).

[VSF14] K. Vamvoudakis-Stefanou, J. Sakellariou, and S. Fassois. « Output-only statistical time series methods for structural health monitoring: A comparative study ». In: *Proceedings of 7th European Workshop on Structural Health Monitoring, Nantes, France*. July 2014.

[Wae+16] J. Waeytens, B. Rosić, P.-E. Charbonnel, E. Merliot, D. Siegert, X. Chapeleau, R. Vidal, V. le Corvec, and L.-M. Cottineau. « Model updating techniques for damage detection in concrete beam using optical fiber strain measurement device ». In: *Engineering Structures* 129 (2016). SI: Structures Rehabilitation, pp. 2–10.

[WMF00] K. Worden, G. Manson, and N.R.J. Fieller. « Damage detection using outlier analysis ». In: *Journal of Sound and Vibration* 229.3 (2000), pp. 647–667.

[Wor+07] K. Worden, C.R. Farrar, G. Manson, and G. Park. « The fundamental axioms of structural health monitoring ». In: *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science* 463.2082 (2007), pp. 1639–1664.
[WSF02] K. Worden, H. Sohn, and C.R. Farrar. « Novelty detection in a changing environment: Regression and interpolation approaches ». In: *Journal of Sound and Vibration* 258.4 (2002), pp. 741–761.

[Yan+05a] A.-M. Yan, G. Kerschen, P. De Boe, and J.-C. Golinval. « Structural damage diagnosis under varying environmental conditions—Part I: A linear analysis ». In: *Mechanical Systems and Signal Processing* 19.4 (2005), pp. 847–864.

[Yan+05b] A.-M. Yan, G. Kerschen, P. De Boe, and J.-C. Golinval. « Structural damage diagnosis under varying environmental conditions—part II: local PCA for non-linear cases ». In: *Mechanical Systems and Signal Processing* 19.4 (2005), pp. 865–880.

[YDG04] A.-M. Yan, P. De Boe, and J.-C. Golinval. « Structural damage diagnosis by Kalman model based on stochastic subspace identification ». In: *Structural Health Monitoring* 3.2 (2004), pp. 103–119.

[YG06] A.-M. Yan and J.-C. Golinval. « Null subspace-based damage detection of structures using vibration measurements ». In: *Mechanical Systems and Signal Processing* 20.3 (2006), pp. 611–626.

[YK19] W.-J. Yan and L.S. Katafygiotis. « An analytical investigation into the propagation properties of uncertainty in a two-stage fast Bayesian spectral density approach for ambient modal analysis ». In: *Mechanical Systems and Signal Processing* 118 (2019), pp. 503–533.

[Zha18] Q. Zhang. « LPV system local model interpolation based on combined model reduction ». In: *IFAC-PapersOnLine* 51.15 (2018). 18th IFAC Symposium on System Identification SYSID 2018, pp. 1104–1109.

[ZR09] V. Zabel and W. Rücker. « Detection of a fatigue crack by vibration tests ». In: *Proceedings of 27th Int national Modal Analysis Conference - IMAC-XXVII* (2009).

[ZU18] S. Zorn and J.F. Unger. « Probabilistic traffic load identification for concrete bridges ». In: *Proceeding of the 6th International Conference on Integrity-Reliability-Failure*. 2018.
<figure><img src="image_76.png" /></figure>
<figure><img src="image_77.png" /></figure>
**Titre :** Détection statistique d'endommagements pour les ouvrages d'art par méthodes sous-espaces sous conditions environnementales

**Mot clés :** Détection d'endommagements, Méthodes sous-espaces, Vibrations, Quantification d'incertitude, Variation de température, Ouvrages d'art

**Résumé :** La détection automatisée d'endommagements basée sur les mesures vibratoires est pertinente pour la surveillance de l'intégrité des ouvrages d'art. Dans ce contexte, la détection basée sur les techniques sous-espace (SSDD) compare statiquement les mesures à un modèle de référence. Dans cette thèse, une nouvelle approche est proposée pour améliorer la robustesse de la SSDD pour un usage applicatif réaliste. Tout d'abord, un test statistique est formulé tenant compte des incertitudes statistiques liées aux erreurs sur le modèle obtenu sur des données de référence. Cela conduit à

une description précise des propriétés statistiques du test et des seuils d'alarme. Deuxièmement, une approche a été développée pour tenir compte des effets environnementaux sur la SSDD. A partir de mesures de référence dans plusieurs conditions environnementales différentes, un test est proposé détectant un défaut par rapport à une référence obtenue par interpolation. Les méthodes développées sont validées par des simulations numériques et appliquées à des données expérimentales obtenues en laboratoire et sur structures installées en extérieur.

**Title:** Subspace-based damage detection in engineering structures considering reference uncertainties and temperature effects

**Keywords:** Damage detection, Subspace methods, Vibrations, Uncertainty quantification, Environmental effects, Civil structures

**Abstract:** Automated vibration-based damage detection is of increasing interest for structural health monitoring of engineering structures. In this context, stochastic subspace-based damage detection (SSDD) compares measurements from a testing state to a data-driven reference model in a statistical framework. In this thesis theoretical developments have been proposed to improve the robustness of SSDD for realistic applications conditions. First, a statistical test has been proposed considering the statistical uncertainties about the model obtained from the refer-

ence data. This leads to a precise description of the test's distribution properties and damage detection thresholds. Second, an approach has been developed to account for environmental effects in SSDD. Based on reference measurements at few different environmental conditions, a test is derived with respect to an adequate interpolated reference. The proposed methods are validated in numerical simulations and applied to experimental data from the laboratory and outdoor structures.