## Section: Scientific Foundations

### Diagnostics

A further monitoring step, often called *fault isolation*,
consists in determining which (subsets of) components
of the parameter vector $\theta $ have been affected by the change.
Solutions for that are now described.
How this relates to diagnostics is addressed afterwards.

#### Isolation.

The question: *which (subsets of) components of $\theta $ have changed ?*,
can be addressed
using either nuisance parameters elimination methods or a multiple hypotheses testing
approach [27] . Here we only sketch two intuitively simple
statistical nuisance elimination techniques, which proceed by projection and rejection,
respectively.

The fault vector $\eta $ is partitioned into an informative part and a nuisance part, and the sensitivity matrix $\mathcal{J}$, the Fisher information matrix $\mathbf{F}={\mathcal{J}}^{T}\phantom{\rule{0.166667em}{0ex}}{\Sigma}^{-1}\phantom{\rule{0.166667em}{0ex}}\mathcal{J}$ and the normalized residual $\overline{\zeta}={\mathcal{J}}^{T}\phantom{\rule{0.166667em}{0ex}}{\Sigma}^{-1}\phantom{\rule{0.166667em}{0ex}}{\zeta}_{N}$ are partitioned accordingly

A rather intuitive statistical solution to the isolation problem,
which can be called *sensitivity* approach, consists in projecting
the deviations in $\eta $ onto the subspace generated by the components ${\eta}_{a}$
to be isolated, and deciding between ${\eta}_{a}={\eta}_{b}=0$
and ${\eta}_{a}\ne 0$, ${\eta}_{b}=0$.
This results in the following test statistics :

${t}_{a}={\overline{\zeta}}_{a}^{\phantom{\rule{0.166667em}{0ex}}T}\phantom{\rule{0.277778em}{0ex}}{\mathbf{F}}_{aa}^{-1}\phantom{\rule{0.277778em}{0ex}}{\overline{\zeta}}_{a}\phantom{\rule{4pt}{0ex}},$ | (6) |

where ${\overline{\zeta}}_{a}$ is the partial residual (score). If ${t}_{a}\ge {t}_{b}$, the component responsible for the fault is considered to be $a$ rather than $b$.

Another statistical solution to the problem of isolating ${\eta}_{a}$ consists
in viewing parameter ${\eta}_{b}$ as a nuisance, and using an existing method
for inferring part of the parameters while ignoring and being robust to
the complementary part. This method is called *min-max approach*.
It consists in replacing the nuisance parameter component ${\eta}_{b}$
by its least favorable value, for deciding between
${\eta}_{a}=0$ and ${\eta}_{a}\ne 0$, with ${\eta}_{b}$ unknown.
This results in the following test statistics :

${t}_{a}^{*}={\overline{\zeta}}_{a}^{\phantom{\rule{0.166667em}{0ex}}*\phantom{\rule{0.166667em}{0ex}}T}\phantom{\rule{0.277778em}{0ex}}{\mathbf{F}}_{a}^{*-1}\phantom{\rule{0.277778em}{0ex}}{\overline{\zeta}}_{a}^{\phantom{\rule{0.166667em}{0ex}}*}\phantom{\rule{4pt}{0ex}},$ | (7) |

where ${\overline{\zeta}}_{a}^{\phantom{\rule{0.166667em}{0ex}}*}\stackrel{\Delta}{=}{\overline{\zeta}}_{a}-{\mathbf{F}}_{ab}\phantom{\rule{0.277778em}{0ex}}{\mathbf{F}}_{bb}^{-1}\phantom{\rule{0.277778em}{0ex}}{\overline{\zeta}}_{b}$ is the effective residual (score) resulting from the regression of the informative partial score ${\overline{\zeta}}_{a}$ over the nuisance partial score ${\overline{\zeta}}_{b}$, and where the Schur complement ${\mathbf{F}}_{a}^{*}={\mathbf{F}}_{aa}-{\mathbf{F}}_{ab}\phantom{\rule{0.277778em}{0ex}}{\mathbf{F}}_{bb}^{-1}\phantom{\rule{0.277778em}{0ex}}{\mathbf{F}}_{ba}$ is the associated Fisher information matrix. If ${t}_{a}^{*}\ge {t}_{b}^{*}$, the component responsible for the fault is considered to be $a$ rather than $b$.

The properties and relationships of these two types of tests are investigated in [26] .

#### Diagnostics.

In most SHM applications, a complex physical system, characterized by a generally non identifiable parameter vector $\Phi $ has to be monitored using a simple (black-box) model characterized by an identifiable parameter vector $\theta $. A typical example is the vibration monitoring problem in module 4.2 , for which complex finite elements models are often available but not identifiable, whereas the small number of existing sensors calls for identifying only simplified input-output (black-box) representations. In such a situation, two different diagnosis problems may arise, namely diagnosis in terms of the black-box parameter $\theta $ and diagnosis in terms of the parameter vector $\Phi $ of the underlying physical model.

The isolation methods sketched above are possible solutions to the former. Our approach to the latter diagnosis problem is basically a detection approach again, and not a (generally ill-posed) inverse problem estimation approach [3] . The basic idea is to note that the physical sensitivity matrix writes $\mathcal{J}\phantom{\rule{0.166667em}{0ex}}{\mathcal{J}}_{\Phi \theta}$, where ${\mathcal{J}}_{\Phi \theta}$ is the Jacobian matrix at ${\Phi}_{0}$ of the application $\Phi \mapsto \theta \left(\Phi \right)$, and to use the sensitivity test (6 ) for the components of the parameter vector $\Phi $. Typically this results in the following type of directional test :

${\chi}_{\Phi}^{2}={\zeta}^{T}\phantom{\rule{0.166667em}{0ex}}{\Sigma}^{-1}\phantom{\rule{0.166667em}{0ex}}\mathcal{J}\phantom{\rule{0.166667em}{0ex}}{\mathcal{J}}_{\Phi \theta}\phantom{\rule{0.166667em}{0ex}}{\left({\mathcal{J}}_{\Phi \theta}^{T}\phantom{\rule{0.166667em}{0ex}}{\mathcal{J}}^{T}\phantom{\rule{0.166667em}{0ex}}{\Sigma}^{-1}\phantom{\rule{0.166667em}{0ex}}\mathcal{J}\phantom{\rule{0.166667em}{0ex}}{\mathcal{J}}_{\Phi \theta}\right)}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathcal{J}}_{\Phi \theta}^{T}\phantom{\rule{0.166667em}{0ex}}{\mathcal{J}}^{T}\phantom{\rule{0.166667em}{0ex}}{\Sigma}^{-1}\phantom{\rule{0.166667em}{0ex}}\zeta \gtrless \lambda \phantom{\rule{4pt}{0ex}}.$ | (8) |

It should be clear that the selection of a particular parameterization $\Phi $ for the physical model may have a non negligible influence on such type of tests, according to the numerical conditioning of the Jacobian matrices ${\mathcal{J}}_{\Phi \theta}$.

As a summary, the machinery in
modules
3.2 ,
3.3
and
3.4
provides us with a generic framework for designing monitoring algorithms for
continuous structures, machines and processes.
This approach assumes that a model of the monitored system is available.
This is a reasonable assumption within the field of applications described
in module
4.2 ,
since most mechanical processes rely on physical principles
which write in terms of equations, providing us with models.
These important *modeling* and *parameterization*
issues are among the questions we intend to investigate within
our research program.

The key issue to be addressed within each parametric model class is
the residual generation, or equivalently the choice of the
*parameter estimating function*.