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4.2 Damage Detection

Damage detection, and thence the core part of SHM, consists in the detection of some change in one or more damage-sensitive properties of a system over time. The damage detection problem is therefore a change detection problem that requires the comparison between two system states before and after the occurrence of change. The state of the system before the change is commonly referred to as baseline, and any property sample collected while the system is in its baseline state is called a baseline sample. Note that in its baseline state the system may be undamaged or may present some pre-existing damage. It must be stressed that each gathered property sample y will be corrupted by some random noise ε with prob- ability density function p (ε). If µ represents the true underlying value of a damage-sensitive property when its sample y was gathered, then y can be expressed as

y = µ + ε (4.1)

Because the noise ε is a random variable, and µ is a deterministic if unknown variable, it follows that the damage-sensitive property y will also be a random variable with probability density function p y
= p µ + ε
. Therefore, changes in a damage-sensitive property of a system will be reflected by changes in the probability density function p y
of the gathered property samples y . Assume that before an unknown change time tc the probability density

function of y is given by p0 y
, and that after tcthe probability density function of y is given

by p1 y
. A change detection algorithm is a procedure to detect the onset of change in the probability density function of y and to estimate tc. Formally, there are two possible change

detection problems:

• Simple Change Detection. The first and simplest scenario is based on the principle of simple hypothesis testing and assumes that both the baseline probability density function p0 y
and the damaged probability density function p1 y
are known. The task therefore consists solely in estimating tc.

• Composite Change Detection. The second, more complex scenario is based on the principle of composite hypothesis testing and assumes that the baseline probability density function p0 y
is known and that the damaged probability density function

p1 y
= p y | θ1
is a differently parameterised version of the baseline probability density function p0 y
= p y | θ0
. The task therefore consists in estimating tc and

the parameter change ν = θ1− θ0. Note that such assumption is generally trivial since the probability density function of the noise ε is almost invariably assumed to be in-

dependent of the true underlying value µ of the damage-sensitive property and con- sequently independent of the occurrence of the change. Therefore, before the change one has p0 y
= µ0+ p (ε) while after the change one has p1 y
= µ1+ p (ε), and it fol- lows that the probability density function of y is identical before and after the change up to the change in the mean ν = µ1−µ0of the true underlying value µ of the damage- sensitive property that one intends to detect.

Chapter 5 will thoroughly discuss change detection and will introduce the Cumulative Sum (CUSUM) algorithm for the solution of simple change detection problems and Generalised Likelihood Ratio (GLR) algorithm for the solution of composite change detection problems. It will be shown that these algorithms can optimally solve change detection problems so that the required number of samples of y is minimised, the probability of detecting a change is maximised and the probability of false-calling a change is minimised.

Besides detecting the occurrence of change, the biggest damage detection challenge consists in establishing the value of the damage-sensitive properties while a system is in its baseline state. There are two main approaches for modelling the baseline state of a system [2].

If a high-fidelity model of the damage-sensitive properties of a system while it remains in its baseline state can be precisely computed in advance, then the damage detection problem is a composite change detection problem that seeks to determine whether a newly gathered property sample is consistent or not with the model. In one case the system is still in its baseline state, while in the other case damage has grown. Collected samples are also typ- ically utilised to actively update the model. After an initial baseline sample is gathered to finely tune the model to a specific system, any further collected sample deemed to be rep- resentative of damage growth can be fed to the model to locate the damage, diagnose it and formulate a prognosis, as well as to update the model to a new baseline state. This approach has been utilised extensively and successfully, perhaps most notably in the case of the finite element updating methodology [2, 104–107].

Conversely, if a high-fidelity model of the damage-sensitive properties of a system while it re- mains in its baseline state cannot be precisely computed in advance, as in the case of the sig- nal from a permanently installed guided wave sensor monitoring a petrochemical pipeline, then damage detection will have to be based on some form of machine learning [108, 109]. Machine learning is concerned with constructing, i.e. learning, computational relationships between variables given an observed set of noisy samples known as the learning set. These computational relationships should model the underlying mechanism that generated the

4.2. Damage Detection

samples, and once constructed they can be utilised to predict the probability density func- tion of samples that may or may not belong to the learning set. Machine learning for damage detection problems can be fundamentally divided into unsupervised and supervised learn- ing.

Supervised learning is applied when there is availability of samples of the damage-sensitive properties of a system in its undamaged and damaged states. In this case, a supervised learn- ing algorithm is utilised to create a statistical model of the system in all its states, and the damage detection problem then becomes a simple change detection problem seeking to de- termine whether a newly gathered sample of the damage-sensitive properties of a system is consistent with any of the undamaged or damaged states. A typical example of the utilisa- tion of supervised learning is in the detection of damage in rotating machinery [2], in which case algorithms can rely on large databases of vibration data compiled by running nominally identical pieces of machinery to some threshold damage condition or to failure. By utilising a supervised learning algorithm to compare the vibrations of a monitored piece of opera- tional machinery to those contained in a database, it becomes possible to detect damage growth, diagnose it and produce a prognosis.

Conversely, unsupervised learning is applied when there is availability only of samples of the damage-sensitive properties of a system while it is in its baseline state. These baseline samples are utilised to derive a statistical model of the damage-sensitive properties while the system is in its baseline state, and the model is then utilised to predict the likely future evolution of the damage-sensitive properties were the system to remain in its baseline state. The damage detection problem therefore becomes a composite change detection problem which seeks to determine whether a newly gathered properties sample is consistent or not with the obtained statistical model. In one case the system is still in its baseline state, while in the other case damage has grown. Damage detection based on unsupervised learning can typically only detect and locate the growth of damage, but cannot in general formulate a diagnosis or a prognosis. However, if information exists about the changes in the measured damage-sensitive properties that a particular damage mode is typically likely to produce, then in principle it could be possible to diagnose the damage and formulate a prognosis by comparing the difference between the newly gathered properties sample and the statistical model to the change in damage-sensitive properties that a particular damage configuration would have likely produced had it occurred.

It is worth noting that there exists a number of approaches that claim to be baseline-less and not to require baseline samples or a comparison between two system states to ascertain

the growth of damage. However, upon close analysis it can be concluded that these claims are based on a terminology misunderstanding [2]. For example, strain energy methods [110] rely on the assumption that the monitored structure in its baseline, undamaged state be- haves as an Euler-Bernoulli beam, while time-reversal acoustics [111] assumes that when it is in its baseline, undamaged state the monitored structure is an ideal linear elastic solid ex- hibiting the time-reversal property, even though this assumption may not be experimentally verifiable [2]. Similarly, other non-linear acoustic approaches [112–114] also assume that the monitored structure behaves like an ideal linear elastic solids unless damage grows. One of the major issues with approaches that rely on very specific assumptions to model a sys- tem’s undamaged state is that they can only be utilised to detect a change in the system state from undamaged to damaged but cannot be utilised to monitor damage growth, since their underlying assumptions often break down when the system is partly but not yet critically damaged.

In principle, a true baseline-less approach would be one capable of distinguishing between different system states without any information about the composition of the sample set under analysis except that the samples have been gathered in a given temporal sequence. Therefore, up to some statistical confidence and subject to a minimum damage size that is of interest to detect, a true baseline-less approach should be capable of partitioning the sample set into two or possibly more subsets representative of the baseline condition and of the various stages of damage growth and development.

Importantly, the effectiveness of damage detection, and therefore of the core part of SHM, can be hindered by changes in Environmental and Operational Conditions (EOC). As dis- cussed, damage detection consists in the detection of some change in the damage-sensitive system properties over time, and is therefore a change detection problem that requires com- parison between two system states before and after the occurrence of change, i.e. of dam- age growth. However, if the measured damage-sensitive system properties were to change not just following damage growth but also following EOC variations, then damage growth and changes in EOC could be mistaken for each other. The effects of EOC variations on the measured damage-sensitive system properties represent arguably the major issue hindering practical large-scale SHM deployment [1, 2, 102, 115].