Section: Scientific Foundations
Detection
Our approach to onboard detection is based on the socalled 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 :

The early detection of a slight mismatch between the model and the data;

A preliminary diagnostics and localization of the deviation(s);

The tradeoff between the magnitude of the detected changes and the uncertainty resulting from the estimation error in the reference model and the measurement noise level.
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 :

Design the residual basically as a parameter estimating function,

Evaluate the residual thanks to a kind of central limit theorem, stating that the residual is asymptotically Gaussian and reflects the presence of a deviation in the parameter vector through a change in its own mean vector, which switches from zero in the reference situation to a nonzero value.
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 $\{{\mathbf{P}}_{\theta}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}\theta \in \Theta \}$, 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 ${\mathbf{P}}_{{\theta}_{0}}$ ? 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 :
$\text{(Safe)}\phantom{\rule{3.33333pt}{0ex}}{\mathbf{H}}_{0}:\phantom{\rule{3.33333pt}{0ex}}\theta ={\theta}_{0}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\text{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\text{(Damaged)}\phantom{\rule{3.33333pt}{0ex}}{\mathbf{H}}_{1}:\phantom{\rule{3.33333pt}{0ex}}\theta ={\theta}_{0}+\eta /\sqrt{N}$  (1) 
where $\eta $ is an unknown but fixed change vector. A residual is generated under the form :
${\zeta}_{N}=1/\sqrt{N}\phantom{\rule{0.277778em}{0ex}}\sum _{k=0}^{N}K({\theta}_{0},{Z}_{k})=\sqrt{N}\phantom{\rule{0.277778em}{0ex}}{\mathcal{K}}_{N}\left({\theta}_{0}\right)\phantom{\rule{4pt}{0ex}}.$  (2) 
If the matrix ${\mathcal{J}}_{N}=\phantom{\rule{0.166667em}{0ex}}{\mathbf{E}}_{{\theta}_{0}}\left[{\mathcal{K}}_{N}^{\text{'}}\left({\theta}_{0}\right)\right]$ converges towards a limit $\mathcal{J}$, then the central limit theorem shows [29] that the residual is asymptotically Gaussian :
${\zeta}_{N}\frac{}{\phantom{\rule{3.33333pt}{0ex}}N\to \infty}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\to \left\{\phantom{\rule{1.em}{0ex}}\begin{array}{cc}\mathcal{N}(0,\Sigma )\hfill & \text{under}\phantom{\rule{4.pt}{0ex}}{\mathbf{P}}_{{\theta}_{0}}\phantom{\rule{4pt}{0ex}},\hfill \\ \\ \mathcal{N}(\mathcal{J}\phantom{\rule{0.166667em}{0ex}}\eta ,\Sigma )\hfill & \text{under}\phantom{\rule{4.pt}{0ex}}{\mathbf{P}}_{{\theta}_{0}+\eta /\sqrt{N}}\phantom{\rule{4pt}{0ex}},\hfill \end{array}\right.$  (3) 
where the asymptotic covariance matrix $\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 $\mathcal{J}$ is full rank and $\Sigma $ is invertible :
${\chi}^{2}={\overline{\zeta}}^{\phantom{\rule{0.166667em}{0ex}}T}\phantom{\rule{0.166667em}{0ex}}{\mathbf{F}}^{1}\phantom{\rule{0.166667em}{0ex}}\overline{\zeta}\gtrless \lambda \phantom{\rule{4pt}{0ex}}.$  (4) 
where
$\overline{\zeta}\stackrel{\Delta}{=}{\mathcal{J}}^{T}\phantom{\rule{0.166667em}{0ex}}{\Sigma}^{1}\phantom{\rule{0.166667em}{0ex}}{\zeta}_{N}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\text{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathbf{F}\stackrel{\Delta}{=}{\mathcal{J}}^{T}\phantom{\rule{0.166667em}{0ex}}{\Sigma}^{1}\phantom{\rule{0.166667em}{0ex}}\mathcal{J}$  (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 $\mathcal{J}$ and $\Sigma $ can be evaluated (or estimated) for the nominal model. In particular, it is not necessary to reidentify the model, and the sensitivity and covariance matrices can be precomputed offline.