# Spatial Filtering for Structural Health Monitoring

**DOI:**https://doi.org/10.1007/978-3-642-36197-5_95-1

## Synonyms

## Introduction

Assessing the integrity of the structures in real time is a very important topic for which many methods have been developed in the last decades. Today, structural health monitoring (SHM) is gaining more and more attention: in the case of bridges, the maximum loads tend to increase (increase of the vehicle weights), while most of the structures are coming to the end of their theoretical lifetime. In addition, exceptional events such as collisions or earthquakes can cause more severe and fast deteriorations. Optimal maintenance calls for an early detection of small damage in structures, as it is well known that limited and frequent repairs are much less costly than major repairs or total rebuilding after collapse. Current monitoring practice consists in scheduled maintenances including visual inspections, ultrasounds, eddy current, magnetic field, or radiography techniques (Hellier 2003). All these experimental methods require however that the vicinity of the flaw is known and that the proximity to be inspected is accessible. Moreover, these local inspections are labor intensive and therefore very expensive. A major problem is that traditional monitoring is noncontinuous which means that if a critical damage occurs between two inspections, it might lead to catastrophic structural failure. One of the most relevant examples is the I-35W Mississippi River Bridge case (Rofidal 2007): this bridge collapsed in August 2007 killing 13 people and injuring 145, despite annual inspection.

A general trend for new structures and bridges is a lighter and more slender design, which tends to increase the levels of vibrations under ambient excitation. While these levels of vibrations need to be controlled as they could be detrimental to the lifetime of the structure, they can also be used for the continuous monitoring of the structure without disruption or decrease of functionality. The basic idea is that the occurrence of damage alters the structural parameters which in turn affect the vibration characteristics. Vibration SHM of civil engineering structures relies on ambient vibrations, as artificial excitation of such large structures is often unpractical.

Based on this basic concept, many vibration-based SHM techniques have been developed in the last decades using mainly eigenfrequencies, damping ratios, or mode shapes (Doebling et al. 1998). The reason of this popularity is the ease of measuring modal parameters or frequency responses on real structures thanks to recent advances in sensors and sensing systems and in the development of efficient operational modal analysis techniques (Reynders and De Roeck 2008; Reynders et al. 2012). Such advances are so important that more and more very large bridges are instrumented with larger and larger sensor networks. China has been the driving force in this direction with the massive instrumentation of bridges in the Hong Kong area, the largest one being the Stonecutters Bridge with more than 1,500 sensors among which are 58 accelerometers and 853 dynamic and static strain sensors (Ni et al. 2012). In Europe, the Messina bridge project, designed to be the largest cable-stayed bridge in the world would include a very large monitoring system with more than 3,000 sensors (De Neumann et al. 2011).

These technological advances have opened the way to real-time automated SHM of bridges. A major problem related to the use of such very large sensor networks is to find adequate techniques to post-process the data: intelligent methods are needed in order to take advantage of the enormous amount of information provided by these large networks. In fact, operational modal analysis is not yet fit for automated modal analysis using very large sensor networks.

With that perspective, rather than identifying online the full set of modal properties of the structure, an alternative is to condense the measured data while keeping the information about the potential damage occurring in the structure. The technology presented in this entry is spatial filtering, which consists in using a linear combiner to condense the information from very large networks of sensors into one or just a few “virtual sensors.” Such virtual sensors can be designed in order to react strongly to damage while being insensitive to environmental changes or even to react to a damage occurring in a specific location along the bridge.

Rytter (Rytter 1993) has proposed a hierarchical decomposition of the SHM process in four levels, which has been widely accepted in the SHM literature: detection (level 1), localization (level 2), quantification (level 3) of the damage, and prediction of the remaining service life of the damaged structure (level 4). As the level of SHM increases, the knowledge about the damage increases and, usually, the complexity of the method increases as well. The method based on spatial filtering can deal with levels 1 and 2 by comparing data measured in the current unknown state with data measured from the structure assumed to be undamaged.

The first section of this entry deals with measurement of raw time-domain data and details the spatial filtering technique for data reduction. The second section is devoted to feature extraction, and the third section deals with the diagnostics, both for damage detection and localization. The last section presents an experimental illustration of the SHM technique on a 3.78 m long steel I-beam tested in the laboratory.

## Measurement of Raw Time-Domain Data

The first building block of the SHM system is the sensor network. As stated earlier, a current trend is to implement very large sensor networks on large civil engineering structures. In such cases, it is often necessary to perform data reduction in order to decrease the power consumption and the bandwidth needed to transmit the data and to facilitate the data storage and post-processing. For SHM applications, an optimal reduction is one that significantly reduces the amount of data while keeping most of the information about the damage. The technique presented in this entry for data reduction is spatial filtering.

### Spatial and Modal Filtering

*f*(

*t*) equipped with a network of

*n*sensors whose time-domain output is denoted by

*y*

_{ k }(

*t*) as shown in Fig. 2. The dynamic time-domain response at each sensor can be decomposed into a sum of contributions of the

*N*mode shapes excited by the ambient force:

*a*

_{ i }(

*t*) is the modal amplitude of mode

*i*and

*ϕ*

_{ ki }is the projection of mode shape

*i*on sensor

*k*. The application of spatial filtering with coefficients

*α*

_{ k }leads to a single sensor output

*g*(

*t*):

The general scheme of spatial filtering can be used for different purposes. A first idea is to condense the information into the modal coordinates of the undamaged structure, which allows to reduce the data *y* _{ k }(*t*) from a very large network of n sensors to a limited set of *a* _{ i }(*t*) time series from *N* modal sensors. This idea is motivated by the fact that the vibration of structures typically involves only a few mode shapes which are excited in a given frequency band of interest.

*α*

_{ k }must be orthogonal to all the modes of the structure in a frequency band of interest, except mode

*l*:

*α*} = {

*α*

_{1}⋯

*α*

_{ n }}

^{ T }and

*e*

_{ l }= {0 0 ⋯ 1 ⋯ 0 0}

^{ T }(all components set to 0 except the

*l*th component) and [

*C*] is a matrix whose columns correspond to the

*N*mode shapes projected on the

*n*sensors of the array. Equation 5 can be solved only if there are at least as many sensors as there are mode shapes (

*n*≥

*N*). In this case, matrix [

*C*] is rectangular and the system of equations is overdetermined (several solutions exist which satisfy Eq. 5). The minimum norm solution is usually adopted by computing the pseudo-inverse (regularized with singular value decomposition) of [

*C*]

^{ T }(Deraemaeker et al. 2008).

*N*mode shapes in the frequency band of interest. In the frequency domain, the FRF of the modal filter tuned on mode

*l*is given by

*b*

_{ l }depends on the excitation level and position. It corresponds to the FRF of a single degree of freedom system which presents a single peak at frequency

*ω*

_{ l }.

Using modal filtering, the amount of data from a large network of *n* sensors can be significantly reduced to just a few modal filters. Such modal filters are virtual sensors which measure the amplitude of vibration of each mode separately. Typically, for a bridge excited by the ambiance, only a few (up to ten) mode shapes are relevant, and the information can be drastically reduced. The computation of the linear combiner coefficients *α* _{ k } is based on Eq. 5 which requires the knowledge of matrix [*C*]. This matrix should be built using experimentally identified mode shapes in order to avoid the need for a numerical model of the structure to be monitored. This can be achieved thanks to operational modal analysis, using, for instance, stochastic subspace-based methods (Reynders and De Roeck 2008).

### Effect of Damage on Modal Filters

*C*]

^{ T }so that Eq. 5 will now be violated. In other terms, the damage will alter the mode shapes, and the coefficients of the linear combiner will not be tuned anymore. This will result in the reappearance of the filtered peaks, as illustrated in Fig. 3. This is the central idea of vibration-based SHM based on modal filters, as detailed in Deraemaeker et al. (2008).

## From Data to Information

*S*

_{ gg }(

*ω*) of

*g*(

*t*) should be computed. This quantity is directly related to the amplitude of the FRF

*G*(

*ω*) as follows (Ewins 1984):

*G*(

*ω*), they will be filtered in

*S*

_{ gg }(

*ω*) as well. When damage occurs, the reappearance of spurious peaks should be monitored based on the power spectral density of the output of the modal filter

*S*

_{ gg }(

*ω*).

### Feature Extraction

*s*(

*ω*) be the frequency dependent amplitude in the frequency range (

*ω*

_{ a },

*ω*

_{ b }) (Fig. 4). For ambient vibrations,

*s*(

*ω*) is the PSD

*S*

_{ gg }(

*ω*). The frequency band is typically defined by

*ω*

_{ a }= 0.95

*ω*

_{ i }and

*ω*

_{ b }= 1.05

*ω*

_{ i }, where

*ω*

_{ i }is the angular eigenfrequency. A peak indicator is then computed in this frequency interval:

*RV F*is the root variance frequency defined by

*FC*is the frequency center defined by

Theoretically, *I* _{Peak} is equal to 1 if *s*(*ω*) is constant and decreases when the peak grows. Figure 4 gives an example of *I* _{Peak} values computed between *ω* _{ a } and *ω* _{ b } when a spurious peak grows around *ω* _{ i }. The advantage of that feature is that it is very sensitive to the peak growth but not to the level of the excitation force, which is particularly needed for ambient vibrations. More details on the peak indicator computation can be found in Deraemaeker and Worden (2010).

## Diagnostics

The last building block of the SHM system is the diagnostics. It consists in assessing, based on the monitored features (here the peak indicators) whether the structure is healthy or damaged, and possibly in giving indication on the location of damage.

### Statistical Analysis of the Features

In phase I, a set of samples are collected and analyzed to infer statistical characteristics of the process when it is assumed to be in control (i.e., when the structure is undamaged). The aim of this step is to compute the control limits (upper control limit *UCL* and/or lower control limits *LCL*) between which the feature should be included if the process stays in control. Those limits are governed by the statistical distribution *f*(*x*) of the quality characteristic and the probability 1 − *γ* that any in-control sample will fall inside the control limits. There are control limits that can be computed to detect a shift of the mean value of the process or a shift of the variance of the process.

Once a set of reliable control charts has been established (phase I), the process is under monitoring (phase II). The process state is unknown (it might be in or out of control), and if a sample falls outside the control limits previously computed, it is considered as an abnormal value, and a warning is triggered. Phase I fixes the probability of type I (false alarms) and type II (missing alarms) errors. Because the control limit values are based on the number of samples in the in-control set of data, the statistical distribution *f*(*x*), and the *γ* value, the statistical analysis must be done very carefully.

Typically, one can find two families of control charts in the literature: the univariate control charts and the multivariate control charts. The first family will be used if there is only one feature to be monitored, while the second one is used when several features are monitored at the same time. In the present application, this means that if one checks the appearance of only one spurious peak around one given natural frequency, the univariate control chart will be applied on that feature, while a multivariate control chart will be used if one checks the spurious peaks around several eigenfrequencies in each modal filter. Finally, there are two categories of control charts in each family: the Shewhart control chart and the time-weighted control charts. The first category monitors each sample independently while the second category considers the previous samples to monitor the current sample, which allows to detect smaller shifts. It has been found that the best results are obtained when the Shewart control charts are applied, because the time weighted control charts increased too much the number of type I errors. For this reason, only the univariate and the Hotelling T^{2} control charts are presented.

**Individual control chart**Consider that only one feature

*x*following a normal distribution is monitored (e.g., one

*I*

_{ Peak }value in each modal filter). The individual control chart will monitor that individual feature

*x*. The control limits are

*n*, then

*d*

_{2}= 1.128. In fact, \( \frac{\overline{ MR}}{d_2} \) is an estimate of the standard deviation σ of

*x*assumed to follow a normal distribution in phase I. Equation 11 is therefore based on a choice of

*γ*= 0.027. The individual control chart is designed to detect a shift of \( \overline{x} \).

**Hotelling**

**T**^{ 2 }

**control chart**If two or more features are monitored at the same time, monitoring these two quantities independently by applying two or more univariate control charts can be very misleading, especially if those features are correlated. On the opposite, the Hotelling

*T*

^{2}control chart is designed for the monitoring of several features simultaneously. Consider

*p*features following a

*p*-normal distribution. The Hotelling

*T*

^{2}control chart monitors the Mahalanobis distance

*T*

^{2}:

*p*×

*p*estimated covariance matrix of features,

*x*is the current

*p*× 1 feature vector, and \( \overline{x} \) is the

*p*× 1 vector of estimated mean values of

*x*vectors (only the undamaged samples are considered to obtain Σ and \( \overline{x} \)). Since the Mahalanobis distance is always positive, only the upper control limit

*UCL*based on an

*F*distribution is considered:

*p*is the number of variables,

*m*is the number of samples in the set of data in phase I, and

*γ*is such that there is a probability of 1 −

*γ*that any in-control sample will fall between the control limits. Like the individual control chart, the Hotelling

*T*

^{2}control chart detects a change of \( \overline{x} \).

### Damage Detection and Localization

*n*sensors installed on the structure are grouped in several smaller sensor networks, each consisting of

*m*sensors. Modal filters can be built for each of these local sensor networks resulting in independent

*local modal filters*(Fig. 6).

*I*contains sensors

*y*

_{ 1,I }, …,

*y*

_{ m,I }, the output of its modal filter tuned to mode

*l*is given by

*α*

_{ k,I }coefficients are computed in order to satisfy the following condition:

*ϕ*

_{(k,I)i }is the

*k*th (

*k*= 1, …,

*m*) component of the

*i*th mode shape projected on the

*I*th local sensor network. If a damage occurs under spatial filter

*I*and if the sensor responses are locally sensitive to damage, the mode shape will only be altered in that spatial filter. As a result, only the spatial filter

*I*will have spurious peaks, indicating the location of the damage.

The efficiency of the approach relies therefore on a very strong assumption: damage in a local filter will cause a local change of the mode shape which is limited to the very close vicinity of the damage location. The fulfillment of this requirement depends on the type of measured quantity which is considered. Two different approaches coexist in the literature: in Mendrok and Uhl (2010), accelerometers are used, while in Tondreau and Deraemaeker (2013), dynamic strain sensors are used. These two approaches have been compared in Tondreau and Deraemaeker (2011), showing that the method based on strain sensors has a higher sensitivity to damage and better localization capabilities. This highlights the importance of the choice of the type of sensor which is part of the first building block of the SHM system. Applications of spatial filtering techniques have been limited so far to accelerometers or dynamic strain sensors.

## Illustrative Example

### Description of the Case Study

*i*) [1:5], (

*ii*) [4:8], (

*iii*) [8:12], (

*iv*) [12:16], and (

*v*) [16:20]. The damage scenarios are described in Table 1. For the undamaged case, the measurement is performed 350 times in order to infer the statistical properties of the peak indicators. For each damaged case, 50 measurements are performed. Each measurement is referred to as a statistical

*sample*.

Damage scenarios

Case | Samples | Location of damage (sensor) | Location of damage (local filter) |
---|---|---|---|

1 | 1–350 | None | None |

2 | 351–400 | 1 | [1:5] |

3 | 401–450 | 2 | [1:5] |

4 | 451–500 | 3 | [1:5] |

5 | 501–550 | 4 | [1:5] and [4:8] |

6 | 551–600 | 5 | [1:5] and [4:8] |

The modal filters tuned on the two first bending mode shapes of the beam at 64 and 230 Hz are applied for each of the five local filters separately. The feature vector therefore consists in two peak indicators in each local filter (appearance of peak at 64 Hz for the modal filter tuned to 230 Hz, and appearance of peak at 230 Hz for the modal filter tuned to 64 Hz). Note that the peak indicator used in this example is slightly different from the peak indicator presented in section “Spatial and Modal Filtering” but shares similar properties. As the feature vector is multivariate, the Hotelling *T* ^{2} control chart has been applied to automate the damage localization in each local filter. The first 200 undamaged samples have been considered to estimate the covariance matrix, as well as to compute the control limit (*γ* is fixed to 0.25 %).

*T*

^{2}control chart. There is only one missing alarm in local filter [1:5] for a damage at sensor four. However, that missing alarm is compensated thanks to the overlapping of the local filters. Indeed, the damage at sensors four and five is correctly located in local filter [4:8]. The results show that the method has successfully, and automatically, localized all the damage cases.

## Summary

There is a strong incentive for the development of online automated SHM techniques for large civil infrastructures. The objective of such systems is to be able to assess the structural integrity of safety critical civil infrastructure in real time. This is particularly important to detect the onset of damage due to aging or more severe damage due to accidental event such as an earthquake or a collision.

With the deployment of very large sensor networks on structures, alternatives to modal identification techniques can be interesting when the focus lies in fast and efficient damage detection and localization. This is the aim of the method presented in this chapter. The three important ingredients are (i) the use of a linear combiner to perform data reduction in the time domain through modal filtering, (ii) the transformation of the time-domain output of modal filters to the frequency domain and the subsequent feature extraction to detect the appearance of spurious peaks, and (iii) the use of control charts to automate the damage detection and localization process.

The fully integrated and automated methodology allows to process efficiently data from large sensor networks and to condense it into very limited information for diagnostics in the form of control charts. The efficiency of this technology has been illustrated on a laboratory experiment of a 3.78 m long steel I-beam.

## Cross-References

Long Term Monitoring for Large Scale Civil Structures

Operational Modal Analysis in Civil Engineering: An Overview

Smart Sensing and Smart Material Applications

Stochastic Structural Identification from Vibrational and Environmental Data

System and Damage Identification of Civil Structures

Vibration-Based Damage Identification: The Z24 Bridge Benchmark

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