## Section: New Results

### damage detection for mechanical structures

#### Damage detection and localisation

Participants : Michael Doehler, Luciano Gallegos, Laurent Mevel.

The Stochastic Dynamic Damage Locating Vector approach is a vibration-based damage localization method based on a finite element model of a structure and output-only measurements in both reference and damaged states. A stress field is computed for loads in the null space of a surrogate of the change in the transfer matrix at the sensor positions for some values in the Laplace domain. Then, the damage location is related to positions where the stress is close to zero. Robustness of the localization information can be achieved by aggregating results at different values in the Laplace domain. So far, this approach and in particular the aggregation is deterministic and does not take the uncertainty in the stress estimates into account. In this paper, the damage localization method is extended with a statistical framework. The uncertainty in the output-only measurements is propagated to the stress estimates at different values of the Laplace variable and these estimates are aggregated based on statistical principles. The performance of the new statistical approach is demonstrated both in a numerical application and a lab experiment, showing a significant improvement of the robustness of the method due to the statistical evaluation of the localization information [24] , [39] .

#### An Innovations Approach to Fault Diagnosis in Linear Time-Varying Descriptor Systems

Participant : Qinghua Zhang.

This work has been carried out in collaboration with Abdouramane Moussa-Ali (LSIS/CNRS/Université de Toulon).

Many modern engineering systems can be modeled by explicit ordinary differential equations (ODE) in state-space form.
Such state-space equations have a long-term mathematical history, and a large number of analytical and numerical tools have been developed for their study.
Nevertheless, some systems cannot be described by such explicit state-space models, but described by *implicit* differential equations, known as differential-algebraic equations (DAE).
After linearization along a trajectory and discretization in time, a nonlinear DAE system is approximately described by *implicit* discrete time state-space equations, known as *descriptor system equations*.
In this work, fault diagnosis is studied for time varying descriptor systems. The Kalman filter for descriptor systems is first revisited by completing existing results
about its properties that are essential for the purpose of fault diagnosis. Based on the analysis of the effects of the considered
actuator and sensor faults on the innovation of the descriptor system Kalman filter, it is shown that the considered fault diagnosis problem in time varying
descriptor systems is equivalent to a classical linear regression problem formulated by appropriately filtering the input-output data.
Following this result, algorithms for fault diagnosis through maximum likelihood estimation are then developed.
The results of this study have been presented at [44] .

#### Statistical detection and isolation of additive faults in linear time-varying systems

Participant : Qinghua Zhang.

This work has been carried out in collaboration with Michèle Basseville (IRISA/CNRS).

Model-based approaches to fault detection and isolation (FDI) have been mostly studied in the literature for linear time invariant (LTI) systems. In practice, quite often time-varying and/or nonlinear properties of the monitored system cannot be neglected. One of the possible approachs to dealing with nonlinear systems is based on the linearization along the actual or nominal trajectory of the monitored system. Such a linearization generally leads to linear time-varying (LTV) systems, whereas the more basic LTI approximation is usually related to the linearization around a single working point. It is thus clear that methods for FDI in LTV systems are much more powerful than their LTI counterparts. In the present work, we address the FDI problem for LTV systems subject to parametric additive faults. The proposed approach is statistical, by combining a generalized likelihood ratio (GLR) test with the Kalman filter that cancels out the dynamics of the faults effects in the considered LTV systems. With this approach, it is possible to perform fault isolation when the number of sensors is smaller than the number of assumed faults, under an appropriate assumption about the excitation of the system. The results of this study have been published in [31] .

#### Robust subspace damage detection

Participants : Michael Doehler, Laurent Mevel.

In the last ten years, monitoring the integrity of the civil infrastructure has been an active research topic, including in connected areas as automatic control. It is common practice to perform damage detection by detecting changes in the modal parameters between a reference state and the current (possibly damaged) state from measured vibration data. Subspace methods enjoy some popularity in structural engineering, where large model orders have to be considered. In the context of detecting changes in the structural properties and the modal parameters linked to them, a subspace-based fault detection residual has been recently proposed and applied successfully, where the estimation of the modal parameters in the possibly damaged state is avoided. However, most works assume that the unmeasured ambient excitation properties during measurements of the structure in the reference and possibly damaged condition stay constant, which is hardly satisfied by any application. This paper addresses the problem of robustness of such fault detection methods. It is explained why current algorithms from literature fail when the excitation covariance changes and how they can be modified. Then, an efficient and fast subspace-based damage detection test is derived that is robust to changes in the excitation covariance but also to numerical instabilities that can arise easily in the computations. Three numerical applications show the efficiency of the new approach to better detect and separate different levels of damage even using a relatively low sample length [20] , [37] , [19] .

#### Sensor placement

Participant : Michael Doehler.

Deciding on the position of sensors by optimizing the utility of the monitoring system over a structure lifetime is typically forbidden by computational cost. Sensor placement strategies are, instead, usually formulated for a pre-selected number of sensors and are based on cost functions that can be evaluated for any arrangement without the need for simulations. This paper examines the performance of two such schemes, the first one is derived directly from a technique that detects damage from the shift of a chi-square distribution from central to non-central and takes the optimal arrangement as the one that maximizes the sensitivity of the non-centrality to all parameter changes of equal norm. The second scheme selects the sensor arrangement as that which maximizes a weighted version of the norm of the sensitivity of the covariance of the output to all feasible changes in system parameters. The performance of the two schemes is tested in simulations [34] .

#### Reflectometry for external post-tensioned cable monitoring

Participant : Qinghua Zhang.

This work has been carried out in collaboration with IFSTTAR, EDF, ENS Cachan and Andra.

Nowadays a considerable number of bridges is reaching an age when repairs become necessary. In some bridges, external post-tension cables are placed in ducts within which the residual internal space is imperfectly filled with a fluid cement grout. Detecting the defaults of filling is visually impossible from the outside. Among non-destructive detection techniques proposed for cable health monitoring, reflectometry techniques offer remarkable advantages in that they can monitor cables in concrete deviator (embedded in concrete) and they do not require human intervention inside the bridge. In this work, the application of reflectometry techniques to cable health monitoring has been investigated via numerical simulations and laboratory experiments. The results of this study have been presented at [55] .

#### Efficient Computation of Minmax Tests for Fault Isolation and Their Application to Structural Damage Localization

Participants : Michael Doehler, Laurent Mevel.

Fault detection and isolation can be handled by many different approaches. This paper builds upon a hypothesis test that checks whether the mean of a Gaussian random vector has become non-zero in the faulty state, based on a chi2 test. For fault isolation, it has to be decided which components in the parameter set of the Gaussian vector have changed, which is done by variants of the chi2 hypothesis test using the so-called sensitivity and minmax approaches. While only the sensitivity of the tested parameter component is taken into account in the sensitivity approach, the sensitivities of all parameters are used in the minmax approach, leading to better statistical properties at the expense of an increased computational burden. The computation of the respective test variable in the minmax test is cumbersome and may be ill-conditioned especially for large parameter sets, asking hence for a careful numerical evaluation. Furthermore, the fault isolation procedure requires the repetitive calculation of the test variable for each of the parameter components that are tested for a change, which may be a significant computational burden. In this paper, dealing with the minmax problem, we propose a new efficient computation for the test variables, which is based on a simultaneous QR decomposition for all parameters. Based on this scheme, we propose an efficient test computation for a large parameter set, leading to a decrease in the numerical complexity by one order of magnitude in the total number of parameters. Finally, we show how the minmax test is useful for structural damage localization, where an asymptotically Gaussian residual vector is computed from output-only vibration data of a mechanical or a civil structure [41] .

#### Inverse problems in damage detection

Participant : Dominique Siegert.

Reinforced concrete beams are widely employed in civil engineering structures. To reduce the maintenance financial cost, structure damages have to be detected early. To this end, one needs robust monitoring techniques. The paper deals with the identification of mechanical parameters, useful for Structural Health Monitoring, in a 2D beam using inverse modeling technique. The optimal control theory is employed. As an example, we aim to identify a reduction of the steel bar cross-section and a decrease of the concrete Young modulus in damaged areas. In our strategy, the beam is instrumented with strain sensors, and a known dynamic load is applied. In the inverse technique, two space discretizations are considered: a fine dicretization to solve the structural dynamic problem and a coarse discretization for the beam parameter identification. To get the beam parameters, we minimize a classical data misfit functional using a gradient-like algorithm. A low-cost computation of the functional gradient is performed using the adjoint equation. The inverse problem is solved in a general way using engineer numerical tools: Python scripts and the free finite element software Code Aster. First results show that a local reduction of the steel bar cross-section and a local decrease of concrete Young modulus can be detected using this inverse technique [27] .

#### NDT by active thermography coupled with infrared shearography

Participant : Jean Dumoulin.

As infrastructures are aging, the evaluation of their health is becoming crucial. To do so, numerous Non Destructive Testing (NDT) methods are available. Among them, thermal shearography and active infrared thermography represent two full field and contactless methods for surface inspection. The synchronized use of both methods presents multiples advantages. Most importantly, both NDT are based on different material properties. Thermography depend on the thermal properties and shearography on the mechanical properties. The cross-correlation of both methods result in a more accurate and exact detection of the defects. For real site application, the simultaneous use of both methods is simplified due to the fact that the excitation method (thermal) is the same. Active infrared thermography is the measure of the temperature by an infrared camera of a surface subjected to heat flux. Observation of the variation of temperature in function of time reveal the presence of defects. On the other hand, shearography is a measure of out-of-plane surface displacement. This displacement is caused by the application of a strain on the surface which (in our case) take the form of a temperature gradient inducing a thermal stress [58] , [49] , [50] .