Team i4s

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Section: Scientific Foundations


See module  6.5 .

Our approach to on-board detection is based on the so-called asymptotic statistical local approach, which we have extended and adapted [5] , [4] , [2] . It is worth noticing that these investigations of ours have been initially motivated by a vibration monitoring application example. It should also be stressed that, as opposite to many monitoring approaches, our method does not require repeated identification for each newly collected data sample.

For achieving the early detection of small deviations with respect to the normal behavior, our approach generates, on the basis of the reference parameter vector $ \theta$0 and a new data record, indicators which automatically perform :

These indicators are computationally cheap, and thus can be embedded. This is of particular interest in some applications, such as flutter monitoring, as explained in module  4.4 .

As in most fault detection approaches, the key issue is to design a residual , which is ideally close to zero under normal operation, and has low sensitivity to noises and other nuisance perturbations, but high sensitivity to small deviations, before they develop into events to be avoided (damages, faults, ...). The originality of our approach is to :

This is actually a strong result, which transforms any detection problem concerning a parameterized stochastic process into the problem of monitoring the mean of a Gaussian vector .


The behavior of the monitored system is again assumed to be described by a parametric model Im1 ${{\#119823 _\#952 ~,~\#952 \#8712 \#920 }}$ , and the safe behavior of the process is assumed to correspond to the parameter value $ \theta$0 . This parameter often results from a preliminary identification based on reference data, as in module  3.2 .

Given a new N -size sample of sensors data, the following question is addressed : Does the new sample still correspond to the nominal model Im13 $\#119823 _\#952 _0$  ? One manner to address this generally difficult question is the following. The asymptotic local approach consists in deciding between the nominal hypothesis and a close alternative hypothesis, namely :

Im14 ${\mtext (Safe)~\#119815 _0:~\#952 =\#952 _0~~~~\mtext and~~~~\mtext (Damaged)~\#119815 _1:~\#952 =\#952 _0+\#951 /\sqrt N}$(1)

where $ \eta$ is an unknown but fixed change vector. A residual is generated under the form :

Im15 ${\#950 _N=1/\sqrt N~\munderover \#8721 {k=0}NK{(\#952 _0,Z_k)}=\sqrt N~\#119974 _N{(\#952 _0)}~.}$(2)

If the matrix Im16 ${\#119973 _N=-~\#119812 _\#952 _0{[{\#119974 }_N^'{(\#952 _0)}]}}$ converges towards a limit Im17 $\#119973 $ , then the central limit theorem shows  [35] that the residual is asymptotically Gaussian :

Im18 ${\#950 _N\mfrac {}{~N\#8594 \#8734 }~~~~\#8594 \mfenced o={  ~\mtable{...}}$(3)

where the asymptotic covariance matrix $ \upper_sigma$ can be estimated, and manifests the deviation in the parameter vector by a change in its own mean value. Then, deciding between $ \eta$ = 0 and $ \eta$$ \ne$0 amounts to compute the following $ \chi$2 -test, provided that Im17 $\#119973 $ is full rank and $ \upper_sigma$ is invertible :

Im19 ${\#967 ^2=\mover \#950 ¯^{~T}~{\#119813 }^{-1}~\mover \#950 ¯\#8823 \#955 ~.}$(4)


Im20 ${\mover \#950 ¯\mover =\#916 {\#119973 }^T~\#931 ^{-1}~\#950 _N~~\mtext and~~\#119813 \mover =\#916 {\#119973 }^T~\#931 ^{-1}~\#119973 }$(5)

With this approach, it is possible to decide, with a quantifiable error level, if a residual value is significantly different from zero, for assessing whether a fault/damage has occurred. It should be stressed that the residual and the sensitivity and covariance matrices Im17 $\#119973 $ and $ \upper_sigma$ can be evaluated (or estimated) for the nominal model. In particular, it is not necessary to re-identify the model, and the sensitivity and covariance matrices can be pre-computed off-line.


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