# The Fisher Information Matrix as a Relevant Tool for  Sensor Selection in Engine Health Monitoring

Authors: S. Borguet; O. Léonard
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

Engine health monitoring has been an area of intensive research for many years. Numerous methods
have been developed with the goal of determining a faithful picture of the engine condition. On the
other hand, the issue of sensor selection allowing an efficient diagnosis has received less attention
from the community. The present contribution revisits the problem of sensor selection for engine
performance monitoring within the scope of information theory. To this end, a metric that integrates
the essential elements of the sensor selection problem is defined from the Fisher information matrix. An example application consisting in a commercial turbofan engine illustrates the enhancement that can be expected from a wise selection of the sensor set.


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## Body

## 1. Introduction

In the last years, condition-based maintenance has been widely promoted in the jet engine
community. A maintenance schedule adapted to the level of deterioration of the
engine leads to many advantages such as improved operability and safety or
reduced life cycle costs. In this framework, generating a reliable information
about the health condition of the engine is a requisite.In this
contribution, module performance analysis, also known asgas path analysis, is considered. Its
purpose is to assess the changes in engine module performance on the basis of
measurements collected along the gas path of the engine [1]. This approach allows to
track the evolution of a particular engine relative to some baseline
performance which may be engine specific or fleet averaged. The gas path
analysis approach is illustrated in Figure 1 and is governed by the following
principles:Figure 1
The gas path analysis approach to jet engine diagnostics.(i)
the condition of the components (e.g., fan,
lpc, hpc, hpt, lpt, nozzle) can be represented by a set of performance
indicators, so-calledhealth
parametersw,
which are typically correcting factors on their efficiency and flow capacity;(ii)
a degradation (progressive or accidental)
affecting the engine induces a modification of its performance, which results
in a drift of somemeasurementsy collected along the engine flow path (e.g.,
temperatures, pressures, mass flows, shaft speeds);(iii)
aphysical or mathematical model, of the
engine that relates the variation of the health parameters w to the drift of the gas-path measurements y can be derived.
As
depicted in Figure 1, the diagnosis problem (or health parameter estimation
problem) can be considered as the inverse problem of performance simulation.
Mathematically speaking, the health parameters are estimated by minimising a
cost function ρ(⋅) of the residuals defined as the difference
between the measured value y and the model prediction y^w:(1)w^=argminw{ρ(y−y^w)}.Next to the
definition of the cost function and the choice of an appropriate method to
solve the inverse problem, which has been an area of intensive research (see
[2] for a detailed
review), the selection of a relevant configuration of the sensors is an
important issue. Current sensors remain intrusive, their integration is limited
by many constraints such as cost, weight, or harsh environment. As a
consequence, the number of sensors is usually kept to a minimum and hence
outweighed by the number of health parameters. This makes the estimation
problem ill posed. Moreover, the measurements are corrupted with noise. The
efficiency of the diagnosis algorithm depends on the sensor configuration which
should therefore be selected so as to optimise the amount of information that
is carried about the engine condition.Optimal sensor
selection has been investigated only sparsely in the engine health monitoring
community, few contributions indeed address this issue. Most of the work so far
is based on linear approaches. In [3], a metric based on the sensitivity of the sensors with
respect to the parameters is defined. The use of the condition number of the
influence coefficient matrix that relates measurements and parameters is also
investigated in order to perform the sensor selection in the case of a square
problem (i.e., as many sensors as parameters). In [4], a figure of merit based on
the steady-state error covariance matrix of a Kalman filter is used to perform
the optimisation of the sensor set, again in the case of a square problem. This
study also takes into account the cost of the different sensors. In [5], the strong link between sensor selection and
observability is underlined. A sensor configuration is considered as
informative if it avoids both redundancy between measurements and correlation
between fault directions. The solution of the sensor selection problem relies
on a method for observability analysis developed in [6].The same topic
has received much more attention in other fields such as chemical or structural
engineering. As a general trend (see for instance [7–10]),
the sensor selection problem is addressed in the framework of information
theory. Metrics based on the observability Grammian or on the Fisher
information matrix are defined in order to optimise the configuration of the
experiment.In the light of
these considerations, the present contribution revisits the problem of sensor
selection for turbine engine performance monitoring within the scope of
information theory. The application consists in a generic turbofan model
developed in the frame of the European OBIDICOTE project. A metric that handles
the aforementioned particularities is proposed. With the research being focused
on the metric definition itself, the optimal configuration is obtained through
a brute-force technique. The quality of the resulting sensor set is assessed on
the basis of fault cases that can be expected on a turbofan engine, and an
observability analysis is performed to get more knowledge about the sensor
configuration.


## 2. Statement of the Problem

The scope of
this section is to present the theoretical foundation of the methodology
developed for sensor selection. First, the model relating the observations to
the parameters is described. Elements of information theory are then introduced
in the scope of sensor selection, with a particular focus on the Fisher
information matrix (FIM). Metrics based on the FIM are subsequently proposed in
order to optimise the measurement configuration for a given set of health
parameters. Finally, guidelines to perform an observability analysis of a given
configuration are proposed.


### 2.1. Simulation Model

One of the
master pieces of the gas path analysis approach is a simulation model of the
engine. Considering steady-state operation of the engine, these simulation tools
are generally nonlinear aerothermodynamic models based on mass, energy, and
momentum conservation laws applied to the engine flow path. Equation (2)
represents such an engine model where u are the variables defining the operating point
of the engine (e.g., fuel flow, altitude, Mach number), w are the aforementioned health parameters, and y are the gas path measurements:(2)y=𝒢(u,w).In the frame of
turbine engine diagnosis, the statistical model is rarely used in the previous
form stated by (2). Indeed, the quantity of interest is the difference between
the actual engine health condition and a reference one represented by baseline
values wbl.
Assuming a linear relationship between the measurements and the health
parameters, as well as fixed operating conditions, the model is re-formulated
according to(3)y=ybl+G(w−wbl)⟺δy=Gδw,where(4)ybl=𝒢(u,wbl),G=∂∂w𝒢(u,w)|w=wbl are, respectively, the prediction of the measurements
and the Jacobian matrix of the engine model around the baseline wbl.A random
variableϵ∈𝒩(0,Ry) which accounts for sensor inaccuracies and
modelling errors is added to the deterministic linearised model (3) in order to
reconcile the observed measurements and the model predictions. Equation (5) is
therefore termed the statistical model:(5)δy=Gδw+ϵ.The statistical
model (5) can further be scaled to a linear system with a noise distribution ϵ˜∈𝒩(0,I) provided that the covariance matrix Ry is positive definite. The scaled model is
given by G˜=(Ry)−1G,
where the scaling factor takes into account the relative accuracy of each
sensor.(6)ϵ˜=δy˜−G˜δw,The scaled statistical model is actually the
probability density function of the residuals ϵ˜ parameterised by the health parameters δw:(7)p(ϵ˜∣δw)=1(2π)mexp(−12ϵ˜Tϵ˜).


### 2.2. Fisher Information Matrix

The idea behind
sensor selection is to optimise the amount of information conveyed by the
measurements about the parameters to be estimated. Optimal information can be
defined in various ways, such as maximum response of the measurements to a
change in the health condition, minimum uncertainty in the estimated
parameters, or maximum orthogonality between the measurements to name a few.
This question will be addressed in more details in the next section. It is
desirable to base the optimisation on a quantity that captures these properties
and that allows easy comparison between different configurations.For the kind of
static systems described by (6), the Fisher information matrix (FIM) is a
mathematical entity that possesses the aforementioned features. Indeed, the FIM
quantifies the amount of information that an observation carries about an
unknown parameter. Mathematically, the FIM is defined as the variance of the
score function associated to the estimation problem [11]. Loosely speaking, it is
intimately linked to the sensitivity of the measurements with respect to the
parameters.Considering the
joint probability distribution of the residuals and the parametersp(ϵ˜,δw),
which characterises the estimation problem, the general term of the FIM is
defined as(8)FIMi,j=defE{∂∂δwilogp(ϵ˜,δw)⋅∂∂δwjlogp(ϵ˜,δw)},where E{⋅} is the mathematical expectation operator and log,
the natural logarithm. The existence of the FIM is conditioned on some
regularity assumptions of the probability density. In the following, two cases
widely used in practice are studied.Consider first
the estimation of the health parameters in a maximum likelihood (ML) framework.
The health parameters are seen in this case as deterministic variables that are
assessed from the available measurements. Consequently, the joint probability
density function is equal to the probability density function of the residuals
conditioned on the health parameters:(9)pML(ϵ˜,δw)=p(ϵ˜∣δw).Clearly, the Gaussian distribution used to model p(ϵ˜∣δw) is such that its natural logarithm as well as
its first derivative with respect to the parameters can be computed for any
values of these parameters. Substituting expression (7) in relation (8), the
FIM writes down:(10)FIMML=G˜TG˜.Consider now
the Bayesian approach to estimation. Both the measurements and the health
parameters are seen as random variables. The joint probability density function
is here equal to the product of the probability density function of the
residuals conditioned on the health parameters and the a priori probability
density function of the health parameters:(11)pBayes(ϵ˜,δw)=p(ϵ˜∣δw)⋅p(δw).The prior
distribution of the parameters is chosen as δw=𝒩(0,Q),
where Q is the prior covariance matrix of the
parameters. Moreover, the parameters and the measurements are assumed to be statistically independent. Under these
hypotheses, the FIM takes the form(12)FIMBayes=G˜TG˜+Q−1.Interestingly,
the Cramer-Rao inequality [12] states that the diagonal terms of the inverse of the
FIM (provided that it exists) are asymptotic lower bounds on the variance of
any unbiased estimator of δw.
This underlines the strong coupling between information, observability, and
estimation.Engine
performance monitoring is characterised in practice by negative redundancy,
which means that the number of parameters exceeds the number of sensors (n>m). In this case, the ML formulation leads to
an underdetermined estimation problem that has no unique solution. Bayesian
estimation has to be applied as a palliative. As already mentioned, sensor
selection relies here on the FIM. Comparing (10) and (12), it can be seen that
the FIM in the Bayesian case is equal to the one of the ML case, which
effectively depends on the sensor configuration, plus the contribution of the a
priori knowledge. This latter term can be interpreted as an additional information
that makes the mathematical problem well posed. As a conclusion, the sensor
selection problem will be solved by using the FIM of the ML framework, even in
the case m<n.


### 2.3. Figures of Merit for Sensor Selection

In the previous
section, it has been shown that the FIM is a relevant mathematical entity on
which the sensor selection can rely. Practically, two main objectives govern
the design of a diagnosis tool. On the one hand, high sensitivity is desirable
in order to provide an early detection of a fault. On the other hand, the
user's confidence in the system is conditioned by a minimum false alarm rate.
These objectives can be achieved by determining the configuration that
maximises both the orthogonality between the sensors and the orthogonality
between the parameters. The former means that two sensors should not react in
the same way to any engine fault. The latter means that two parameters should
have a distinct signature on the observations. Various scalar figures of merit
based on the FIM can describe these objectives:(1)
thecondition numberCN of the FIM is defined as the ratio of the
largest to the smallest singular value. The condition number is linked to the
rank of a matrix and to the difficulty in performing its inversion. In the case m<n,
the rank of the FIM is at best equal to m which can be ensured by minimising the
condition number. In that way, no sensor is redundant with another;(2)
thetraceTr of the FIM is defined as the sum of the
singular values. In the case m<n,
the sum is limited to the first m singular values. The trace is a measure of the
global sensitivity of the sensors with respect to the parameters and hence has
to be maximised. It is linked to some extent to the sensitivity measure defined
in [3];(3)
thedeterminantDet of the FIM is defined as the product of the
singular values. Again, the product is restricted to the first m singular values. Indeed, as stated in point 1,
at most m singular values are nonzero. This quantity has
to be maximised too as the inverse of the determinant is a measure of the
overall uncertainty on the estimated parameters. Furthermore, this criterion
tends to standardise the singular values and hence the contribution of each
sensor. This criterion is similar to the one used in [5].
Relations (13) give the
mathematical form of the figures of merit, where σi,i=1,…,m are the singular values, see next section for
a definition of singular value, of the FIM sorted in descending
order:(13)CN=σ1σm,Tr=∑i=1mσi,Det=∏i=1mσi.Each of those
metrics puts emphasis on a particular aspect of the problem at hand. To perform
the sensor selection, it is thus proposed to combine them in an aggregated
figure of merit:(14)FOM=−W1log(CN)+W2log(Tr)+W3log(Det),where Wi are factors that allow both normalisation of
the magnitude of the various components and variable weighting between the
objectives. For the problem considered in this study, FOM has to be maximised with respect to the sensor
configuration under the constraint of a fixed number of sensors.


### 2.4. Metrics for Observability Analysis

The purpose of
the observability analysis as considered in the present study is twofold:
firstly assessing the contribution of each sensor to the estimation problem and
secondly quantifying the observability level of each health factor. In simple
words, the observability level measures the possibility to estimate correctly a
given parameter.As pointed by
Brown in [13], the
classical test for observability, that is, checking the invertibility (or full
rank) of the system matrix—G˜ here—gives merely a boolean answer. In the
case of negative redundancy, the system is always classified as nonobservable.
Practically, it is however partially observable, and it is worth determining the observability level of the different
parameters as well as the unobservable fault directions. Inspired by the work
of Provost [6], a
method for generating this information is proposed in the following.The intent is
to derive metrics for both the sensors and the parameters, the computation is
therefore based on the scaled Jacobian matrix rather than on the FIM. Singular
value decomposition (SVD) (see [14]) of G˜ serves as the mathematical tool to carry out the
observability analysis:(15)G˜=UΣVT=U[Σ1Σ2][V1V2]T,where
(i)
U∈ℛm×m is an orthogonal matrix whose columns define
an orthonormal basis for the output (measurement) space;(ii)
V∈ℛn×n is an orthogonal matrix whose columns define
an orthonormal basis for the input (parameter) space;(iii)
Σ∈ℛm×n is a rectangular matrix whose main diagonal
contains the singular values σ,
which can be seen as stretching factors along the directions defined in U and V.
A more
practical interpretation of the SVD is proposed hereafter. The scaled Jacobian
matrix G˜ can be seen as an operator that maps the space
of the health parameters (input space) to the space of the measurements (output
space). In this representation, each health parameter (resp., measurement) is
one orthogonal base vector of the input (resp., output) space. Application of
the SVD to the matrix consists essentially in a change of reference frame in
both the input and output spaces. The position of any health condition of the
engine is now measured according to the columns of matrix V,
and the corresponding measurement coordinates are expressed in terms of the
columns of matrix U.
The major difference between the two representations (original and SVD) lies in
the mapping from the input space to the output space. In the SVD
representation, the mapping is given by the diagonal matrix Σ which simplifies the description of the
input-output relation.Given the meaning
of the columns ofU and the diagonal terms of Σ,
a sensitivity index SI is defined for each sensor as a weighted norm
of each row of U: (16)SIi=UiΣTΣUiT,i=1,…,m.This
sensitivity index is an image of the overall contribution of a given sensor to
the estimation process. The higher the SI,
the more important the influence of the associated sensor in the estimation of
the health parameters.An
observability index for the parameters, and to a larger extent for any fault
(i.e., combination of several parameters), could be defined in a similar way as
the sensitivity index is. Nonetheless, in the case of negative redundancy, a
loss of information is introduced due to the fact that the number of parameters
outweighs the number of sensors. From linear algebra theory, it can be stated
that the rank of the Jacobian matrixG˜ is at best equal to m.
Hence, n−m singular values are equal to zero. As
highlighted in the second part of (15), matrix Σ is partitioned in two where Σ1T gathers the m nonzero singular values and Σ2T is a submatrix containing the n−m null singular values. Matrix V is accordingly split column-wise into the
observable subspace V1 and its unobservable counterpart V2,
also known as the null space, which does not carry any information about the
state of the system [15].The definition
of the observability index proposed in the present study quantifies this loss
of information. For some faultf,
the observability index OI is computed according to(17)OIf=∥V1Tf∥2∥VTf∥2,where the numerator is a measure of the intensity of
fault f in the observable subspace, and the
denominator represents the total intensity of fault f.Following this
definition, the observability index is bounded by zero and one. An
observability index equal to unity means that the considered fault has no
component located in the null space and can therefore be estimated with a great
accuracy. On the contrary, a small observability index characterises a large
loss of information. The estimation of the associated fault is hence less
accurate.The fault
directions defined by the columns of matrixV2 are unobservable in the sense that they
produce no shift in the measurements. Any fault direction that is a linear
combination of these vectors has obviously the same detrimental property. The
examination of the unobservable fault directions gives clues about possible
confusions between weakly observable faults.


## 2.1. Simulation Model

One of the
master pieces of the gas path analysis approach is a simulation model of the
engine. Considering steady-state operation of the engine, these simulation tools
are generally nonlinear aerothermodynamic models based on mass, energy, and
momentum conservation laws applied to the engine flow path. Equation (2)
represents such an engine model where u are the variables defining the operating point
of the engine (e.g., fuel flow, altitude, Mach number), w are the aforementioned health parameters, and y are the gas path measurements:(2)y=𝒢(u,w).In the frame of
turbine engine diagnosis, the statistical model is rarely used in the previous
form stated by (2). Indeed, the quantity of interest is the difference between
the actual engine health condition and a reference one represented by baseline
values wbl.
Assuming a linear relationship between the measurements and the health
parameters, as well as fixed operating conditions, the model is re-formulated
according to(3)y=ybl+G(w−wbl)⟺δy=Gδw,where(4)ybl=𝒢(u,wbl),G=∂∂w𝒢(u,w)|w=wbl are, respectively, the prediction of the measurements
and the Jacobian matrix of the engine model around the baseline wbl.A random
variableϵ∈𝒩(0,Ry) which accounts for sensor inaccuracies and
modelling errors is added to the deterministic linearised model (3) in order to
reconcile the observed measurements and the model predictions. Equation (5) is
therefore termed the statistical model:(5)δy=Gδw+ϵ.The statistical
model (5) can further be scaled to a linear system with a noise distribution ϵ˜∈𝒩(0,I) provided that the covariance matrix Ry is positive definite. The scaled model is
given by G˜=(Ry)−1G,
where the scaling factor takes into account the relative accuracy of each
sensor.(6)ϵ˜=δy˜−G˜δw,The scaled statistical model is actually the
probability density function of the residuals ϵ˜ parameterised by the health parameters δw:(7)p(ϵ˜∣δw)=1(2π)mexp(−12ϵ˜Tϵ˜).


## 2.2. Fisher Information Matrix

The idea behind
sensor selection is to optimise the amount of information conveyed by the
measurements about the parameters to be estimated. Optimal information can be
defined in various ways, such as maximum response of the measurements to a
change in the health condition, minimum uncertainty in the estimated
parameters, or maximum orthogonality between the measurements to name a few.
This question will be addressed in more details in the next section. It is
desirable to base the optimisation on a quantity that captures these properties
and that allows easy comparison between different configurations.For the kind of
static systems described by (6), the Fisher information matrix (FIM) is a
mathematical entity that possesses the aforementioned features. Indeed, the FIM
quantifies the amount of information that an observation carries about an
unknown parameter. Mathematically, the FIM is defined as the variance of the
score function associated to the estimation problem [11]. Loosely speaking, it is
intimately linked to the sensitivity of the measurements with respect to the
parameters.Considering the
joint probability distribution of the residuals and the parametersp(ϵ˜,δw),
which characterises the estimation problem, the general term of the FIM is
defined as(8)FIMi,j=defE{∂∂δwilogp(ϵ˜,δw)⋅∂∂δwjlogp(ϵ˜,δw)},where E{⋅} is the mathematical expectation operator and log,
the natural logarithm. The existence of the FIM is conditioned on some
regularity assumptions of the probability density. In the following, two cases
widely used in practice are studied.Consider first
the estimation of the health parameters in a maximum likelihood (ML) framework.
The health parameters are seen in this case as deterministic variables that are
assessed from the available measurements. Consequently, the joint probability
density function is equal to the probability density function of the residuals
conditioned on the health parameters:(9)pML(ϵ˜,δw)=p(ϵ˜∣δw).Clearly, the Gaussian distribution used to model p(ϵ˜∣δw) is such that its natural logarithm as well as
its first derivative with respect to the parameters can be computed for any
values of these parameters. Substituting expression (7) in relation (8), the
FIM writes down:(10)FIMML=G˜TG˜.Consider now
the Bayesian approach to estimation. Both the measurements and the health
parameters are seen as random variables. The joint probability density function
is here equal to the product of the probability density function of the
residuals conditioned on the health parameters and the a priori probability
density function of the health parameters:(11)pBayes(ϵ˜,δw)=p(ϵ˜∣δw)⋅p(δw).The prior
distribution of the parameters is chosen as δw=𝒩(0,Q),
where Q is the prior covariance matrix of the
parameters. Moreover, the parameters and the measurements are assumed to be statistically independent. Under these
hypotheses, the FIM takes the form(12)FIMBayes=G˜TG˜+Q−1.Interestingly,
the Cramer-Rao inequality [12] states that the diagonal terms of the inverse of the
FIM (provided that it exists) are asymptotic lower bounds on the variance of
any unbiased estimator of δw.
This underlines the strong coupling between information, observability, and
estimation.Engine
performance monitoring is characterised in practice by negative redundancy,
which means that the number of parameters exceeds the number of sensors (n>m). In this case, the ML formulation leads to
an underdetermined estimation problem that has no unique solution. Bayesian
estimation has to be applied as a palliative. As already mentioned, sensor
selection relies here on the FIM. Comparing (10) and (12), it can be seen that
the FIM in the Bayesian case is equal to the one of the ML case, which
effectively depends on the sensor configuration, plus the contribution of the a
priori knowledge. This latter term can be interpreted as an additional information
that makes the mathematical problem well posed. As a conclusion, the sensor
selection problem will be solved by using the FIM of the ML framework, even in
the case m<n.


## 2.3. Figures of Merit for Sensor Selection

In the previous
section, it has been shown that the FIM is a relevant mathematical entity on
which the sensor selection can rely. Practically, two main objectives govern
the design of a diagnosis tool. On the one hand, high sensitivity is desirable
in order to provide an early detection of a fault. On the other hand, the
user's confidence in the system is conditioned by a minimum false alarm rate.
These objectives can be achieved by determining the configuration that
maximises both the orthogonality between the sensors and the orthogonality
between the parameters. The former means that two sensors should not react in
the same way to any engine fault. The latter means that two parameters should
have a distinct signature on the observations. Various scalar figures of merit
based on the FIM can describe these objectives:(1)
thecondition numberCN of the FIM is defined as the ratio of the
largest to the smallest singular value. The condition number is linked to the
rank of a matrix and to the difficulty in performing its inversion. In the case m<n,
the rank of the FIM is at best equal to m which can be ensured by minimising the
condition number. In that way, no sensor is redundant with another;(2)
thetraceTr of the FIM is defined as the sum of the
singular values. In the case m<n,
the sum is limited to the first m singular values. The trace is a measure of the
global sensitivity of the sensors with respect to the parameters and hence has
to be maximised. It is linked to some extent to the sensitivity measure defined
in [3];(3)
thedeterminantDet of the FIM is defined as the product of the
singular values. Again, the product is restricted to the first m singular values. Indeed, as stated in point 1,
at most m singular values are nonzero. This quantity has
to be maximised too as the inverse of the determinant is a measure of the
overall uncertainty on the estimated parameters. Furthermore, this criterion
tends to standardise the singular values and hence the contribution of each
sensor. This criterion is similar to the one used in [5].
Relations (13) give the
mathematical form of the figures of merit, where σi,i=1,…,m are the singular values, see next section for
a definition of singular value, of the FIM sorted in descending
order:(13)CN=σ1σm,Tr=∑i=1mσi,Det=∏i=1mσi.Each of those
metrics puts emphasis on a particular aspect of the problem at hand. To perform
the sensor selection, it is thus proposed to combine them in an aggregated
figure of merit:(14)FOM=−W1log(CN)+W2log(Tr)+W3log(Det),where Wi are factors that allow both normalisation of
the magnitude of the various components and variable weighting between the
objectives. For the problem considered in this study, FOM has to be maximised with respect to the sensor
configuration under the constraint of a fixed number of sensors.


## 2.4. Metrics for Observability Analysis

The purpose of
the observability analysis as considered in the present study is twofold:
firstly assessing the contribution of each sensor to the estimation problem and
secondly quantifying the observability level of each health factor. In simple
words, the observability level measures the possibility to estimate correctly a
given parameter.As pointed by
Brown in [13], the
classical test for observability, that is, checking the invertibility (or full
rank) of the system matrix—G˜ here—gives merely a boolean answer. In the
case of negative redundancy, the system is always classified as nonobservable.
Practically, it is however partially observable, and it is worth determining the observability level of the different
parameters as well as the unobservable fault directions. Inspired by the work
of Provost [6], a
method for generating this information is proposed in the following.The intent is
to derive metrics for both the sensors and the parameters, the computation is
therefore based on the scaled Jacobian matrix rather than on the FIM. Singular
value decomposition (SVD) (see [14]) of G˜ serves as the mathematical tool to carry out the
observability analysis:(15)G˜=UΣVT=U[Σ1Σ2][V1V2]T,where
(i)
U∈ℛm×m is an orthogonal matrix whose columns define
an orthonormal basis for the output (measurement) space;(ii)
V∈ℛn×n is an orthogonal matrix whose columns define
an orthonormal basis for the input (parameter) space;(iii)
Σ∈ℛm×n is a rectangular matrix whose main diagonal
contains the singular values σ,
which can be seen as stretching factors along the directions defined in U and V.
A more
practical interpretation of the SVD is proposed hereafter. The scaled Jacobian
matrix G˜ can be seen as an operator that maps the space
of the health parameters (input space) to the space of the measurements (output
space). In this representation, each health parameter (resp., measurement) is
one orthogonal base vector of the input (resp., output) space. Application of
the SVD to the matrix consists essentially in a change of reference frame in
both the input and output spaces. The position of any health condition of the
engine is now measured according to the columns of matrix V,
and the corresponding measurement coordinates are expressed in terms of the
columns of matrix U.
The major difference between the two representations (original and SVD) lies in
the mapping from the input space to the output space. In the SVD
representation, the mapping is given by the diagonal matrix Σ which simplifies the description of the
input-output relation.Given the meaning
of the columns ofU and the diagonal terms of Σ,
a sensitivity index SI is defined for each sensor as a weighted norm
of each row of U: (16)SIi=UiΣTΣUiT,i=1,…,m.This
sensitivity index is an image of the overall contribution of a given sensor to
the estimation process. The higher the SI,
the more important the influence of the associated sensor in the estimation of
the health parameters.An
observability index for the parameters, and to a larger extent for any fault
(i.e., combination of several parameters), could be defined in a similar way as
the sensitivity index is. Nonetheless, in the case of negative redundancy, a
loss of information is introduced due to the fact that the number of parameters
outweighs the number of sensors. From linear algebra theory, it can be stated
that the rank of the Jacobian matrixG˜ is at best equal to m.
Hence, n−m singular values are equal to zero. As
highlighted in the second part of (15), matrix Σ is partitioned in two where Σ1T gathers the m nonzero singular values and Σ2T is a submatrix containing the n−m null singular values. Matrix V is accordingly split column-wise into the
observable subspace V1 and its unobservable counterpart V2,
also known as the null space, which does not carry any information about the
state of the system [15].The definition
of the observability index proposed in the present study quantifies this loss
of information. For some faultf,
the observability index OI is computed according to(17)OIf=∥V1Tf∥2∥VTf∥2,where the numerator is a measure of the intensity of
fault f in the observable subspace, and the
denominator represents the total intensity of fault f.Following this
definition, the observability index is bounded by zero and one. An
observability index equal to unity means that the considered fault has no
component located in the null space and can therefore be estimated with a great
accuracy. On the contrary, a small observability index characterises a large
loss of information. The estimation of the associated fault is hence less
accurate.The fault
directions defined by the columns of matrixV2 are unobservable in the sense that they
produce no shift in the measurements. Any fault direction that is a linear
combination of these vectors has obviously the same detrimental property. The
examination of the unobservable fault directions gives clues about possible
confusions between weakly observable faults.


## 3. Application of the Method

### 3.1. Engine Layout

The application
considered as a test case is a large bypass ratio, mixed-flow turbofan. The
engine performance model was developed in the frame of the OBIDICOTE project, a
Brite-Euram project for onboard identification, diagnosis, and control of
turbofan engine, and is detailed in [16]. A schematic of the engine is sketched in Figure 2,
where the location of the eleven health parameters and the station numbering
are also indicated. One control variable is considered in the following,
namely, fuel flow rate Wfuel fed in the combustor.Figure 2
Turbofan layout with health parameters location.The sensor
selection problem is formulated for the case of an onboard engine performance
monitoring tool. As steady-state operation of the engine is achieved nearly
exclusively during the cruise phase, the corresponding operating point is
selected for this study. Cruise conditions are defined in Table1.Table 1
Cruise point definition.
LabelWfuelAltitudeMachΔTISAValue0.350 kg/s10668 m0.800.0 KSensors that
may be fitted on the engine are listed with their associated uncertainty (noise
level) in Table2. They mostly consist in total temperature and pressure at
various intercomponent planes as well as both spool speeds.Table 2
Available sensors (uncertainty is three times
the standard deviation).
LabelP13P26NLPP3P49P5Uncertainty±100 Pa± 500 Pa±6 rpm±5000 Pa±500 Pa±300 PaLabelT13T26NHPT3T49T5Uncertainty±2 K±2 K±12 rpm±2 K±2 K±2 K


### 3.2. Optimisation of the Sensor Configuration

The
optimisation problem is to determine them (among twelve) sensors that maximise the
figure of merit of (14) for the set of n=11 health parameters depicted in Figure 2 at
cruise conditions. The only parameter that still needs to be set is the number
of sensors that are installed on the engine. According to the information
reported in [17], a
number of m=7 sensors seem a reasonable choice for
contemporary commercial turbofans.This
optimisation problem is of combinatorial nature. Although algorithms are
especially dedicated to this kind of problem, the optimal configuration is
found here by means of a brute force technique. The figure of merit is computed
for every of the 792 (12 choose 7 binomial coefficient) possible combinations.
Unit weights have been applied to each component of theFOM (Wi=1.0,i=1,2,3).Table3 reports
the first ten optimal configurations based on the figure of merit defined in
(14). The last column on the right gives the value of the figure of merit for
each setup. At first sight, the values may appear quite close to each other,
but the reader should recall that the figure of merit is defined on a
logarithmic scale. The bottom line in the table gives the frequency of each
sensor in the first ten best sensor sets. To complete the picture, the first
five optimal configurations according to each component of the FOM are given in Tables 4, 5, and 6. The
last column in each of these tables is also expressed in a logarithmic scale.Table 3
First ten optimal configurations.
P13T13P26T26NLPNHPP3T3P49T49P5T5FOM1√√√√√√√−1.42542√√√√√√√−1.59683√√√√√√√−1.70144√√√√√√√−1.73155√√√√√√√−1.89366√√√√√√√−2.04927√√√√√√√−2.17098√√√√√√√−2.25409√√√√√√√−2.257610√√√√√√√−2.28199090988810702Table 4
First five optimal configurations based on the
condition number of the FIM.
P13T13P26T26NLPNHPP3T3P49T49P5T5CN1√√√√√√√−1.98822√√√√√√√−2.06343√√√√√√√−2.07134√√√√√√√−2.15335√√√√√√√−2.1851505055335202Table 5
First five optimal configurations based on the
trace of the FIM.
P13T13P26T26NLPNHPP3T3P49T49P5T5Tr1√√√√√√√2.01222√√√√√√√2.01173√√√√√√√2.00304√√√√√√√2.00255√√√√√√√2.0003404055315503Table 6
First five optimal configurations based on the
determinant of the FIM.
P13T13P26T26NLPNHPP3T3P49T49P5T5Det1√√√√√√√−1.36452√√√√√√√−1.45573√√√√√√√−1.62694√√√√√√√−1.63685√√√√√√√−1.6608504055345301It can be seen
that the optimal configuration according to the fullFOM (i.e., first line of Table 3) is ranked,
respectively, fourth, first, and second according to the condition number, the
trace, and the determinant of the FIM. Hence, the combined metric elegantly
merges the different objectives of the optimisation problem. Looking at the
last line of each table, it can be deduced that T13, T26, and P5 are not good candidate to the optimal
instrumentation whatever the metric considered. On the contrary, P13,P26,P49, and the spool speeds appear to be mandatory in
the optimised configuration. This point will be discussed in more details
subsequently.As highlighted
in [18], installation
of an interturbine pressure sensor becomes more difficult on modern engines.
Consequently, the search has been carried out a second time by removing P49 from the list of available sensors, leaving a
mere 330 configurations to be examined.The three
sensor sets selected for further investigations are presented in Table7. Set A is the standard instrumentation defined
in the frame of the OBIDICOTE project and serves as the baseline for
comparison, set B is the optimal
one when considering P49 as available, and set C is the optimal one when discarding P49 from the list. Note that the notion of
optimality is relative to the metric defined by (14).Table 7
Selected sensors sets for further
investigations.
SetAP13T13NLPNHPP3T3T5SetBP13P26NLPNHPP3P49T49SetCP13P26NLPNHPP3T3T49Examining Table7, one first notices that P13,
both spool speeds, and P3 are common to the three sensor sets. P13 is indeed required to correctly assess fan
performance. Concerning the spool speeds and P3,
these sensors are mandatory for control purpose. It is thus a good point that
they appear in the optimal sets. Furthermore, spool speeds are generally very
accurately measured, which is one of the reasons of their presence in the
optimal configurations. Set B is
characterised by the presence of P49.
This sensor is in fact the only one in the list that allows full
differentiation of lpt and hpt faults.It is also of
interest to point out that setsB and C contain, respectively, 4
and 3 pressure sensors out of 7. This is explained by the fact that pressure
sensors are more informative about the engine condition than temperature
sensors. It may appear strange that lpt exhaust pressure P5 does not appear in the optimal selection. For
a mixed-flow engine, this quantity is nonetheless highly correlated with fan
outlet pressure P13.
So selecting P13 discards P5 and vice versa. It can also be seen that
exhaust gas temperature (defined as T49 or T5,
depending on the manufacturer) is part of the optimal sensor sets. An EGT sensor
is already fitted on every engine as this quantity serves as a global indicator
of engine health for the pilot (so-called EGT margin).


### 3.3. Evaluation of the Sensor Sets

In order to
evaluate the benefits brought by optimal instrumentation, a series of fault
cases, taken from [19]
and summarised in Table 8, are considered. They are intended to represent
degradations that can be expected on modern turbofans for all individual
components. Considered faults involve single as well as multiple health
parameters. In the present application, faults have a constant magnitude.
Snapshot data are simulated by means of the engine performance model. Gaussian
noise, whose magnitude is specified in Table 2, is added to the generated
measurements to make them closer to test data.Table 8
Considered fault cases.
a−1% onSW12R−0.5% onSE12fan, lpc−0.7% onSW2R−0.4% onSE2b−1% onSE12c−1% onSW26R−0.7% onSE26hpcd−1% onSE26e−1% onSW26Rf+1% onSW42Rhptg−1% onSW42R−1% onSE42h−1% onSE42i−1% onSE49lptj−1% onSW49R−0.4% onSE49k−1% onSW49Rl+1% onSW49R−0.6% onSE49m+1% onA8IMPnozzlen−1% onA8IMPA diagnosis
tool based on a Kalman filter (see [20] for further details) processes the data. Given its
stochastic nature, each test case is run twenty times with different
realisation of measurement noise. Table 9 reports the averaged results in a
very synthetic way. Identification of the fault is declared successful if the
estimation error lies within its bounds (see [21] for more details) and is
indicated by a check mark (√) in the table.Table 9
Fault
identification results with the 3 sensor sets.
Fault caseabcdefghijklmnSetA√√√√√√√√√×××√√SetB√√√√√√√√√√√√×√SetC√√√√√√√√√×√√×√The standard
instrumentation, setA, performs
already quite well. It can however be seen that it fails in diagnosing
correctly most of the lpt faults (cases j, k, and l). Sensor set B, featuring the interturbine pressure
sensor, resolves this issue, as previously mentioned. However, fault case m is now misidentified. The increase in
exhaust nozzle area is confounded with a fan fault. Finally, set C achieves some improvement with respect
to set A too. Most of the lpt
faults but case j are correctly
assessed. As for set B, the
diagnosis tool is misled by fault case m.


### 3.4. Observability Analysis of the Sensor Sets

The study of
the optimal sensor sets is concluded by an observability analysis as per the
concepts defined in a previous section. Figure3 depicts the observability
indices of the health parameters for the three sensors sets. Concerning set A, the observability of most of the
parameters is quite good, except for the booster flow factor, SW2R,
and the lpt flow factor SW49R.
The optimal set B improves the
observability of those parameters, on the other hand, it degrades the
observability of SE12 and SE26 due to the loss, respectively, of T13 and T3.
The nozzle area is also affected. Finally, optimal set C offers an intermediate solution. With
respect to set A, the
observability of the lpc is improved as well as the one of the core components.
As for set B, one can notice a
decrease in observability for SE12 and A8IMP.Figure 3
Comparison of
the observability indicesOI.In Figure4,
the sensitivity indices are compared for each sensor set. From the left
picture, it can be seen that both temperature sensors in the cold section,
namely, T13 and T3,
contribute to a lesser extent in set A. The center picture, associated to set B, illustrates the effect of the
optimisation. The distribution of sensitivity indices is more uniform, and
values are higher. Finally, the right graph shows the sensitivity indices for
set C. Again, the general trend
reveals an increase in the average level. T3 has the lowest contribution among all sensors.Comparison of
the sensitivity indicesSI.


## 3.1. Engine Layout

The application
considered as a test case is a large bypass ratio, mixed-flow turbofan. The
engine performance model was developed in the frame of the OBIDICOTE project, a
Brite-Euram project for onboard identification, diagnosis, and control of
turbofan engine, and is detailed in [16]. A schematic of the engine is sketched in Figure 2,
where the location of the eleven health parameters and the station numbering
are also indicated. One control variable is considered in the following,
namely, fuel flow rate Wfuel fed in the combustor.Figure 2
Turbofan layout with health parameters location.The sensor
selection problem is formulated for the case of an onboard engine performance
monitoring tool. As steady-state operation of the engine is achieved nearly
exclusively during the cruise phase, the corresponding operating point is
selected for this study. Cruise conditions are defined in Table1.Table 1
Cruise point definition.
LabelWfuelAltitudeMachΔTISAValue0.350 kg/s10668 m0.800.0 KSensors that
may be fitted on the engine are listed with their associated uncertainty (noise
level) in Table2. They mostly consist in total temperature and pressure at
various intercomponent planes as well as both spool speeds.Table 2
Available sensors (uncertainty is three times
the standard deviation).
LabelP13P26NLPP3P49P5Uncertainty±100 Pa± 500 Pa±6 rpm±5000 Pa±500 Pa±300 PaLabelT13T26NHPT3T49T5Uncertainty±2 K±2 K±12 rpm±2 K±2 K±2 K


## 3.2. Optimisation of the Sensor Configuration

The
optimisation problem is to determine them (among twelve) sensors that maximise the
figure of merit of (14) for the set of n=11 health parameters depicted in Figure 2 at
cruise conditions. The only parameter that still needs to be set is the number
of sensors that are installed on the engine. According to the information
reported in [17], a
number of m=7 sensors seem a reasonable choice for
contemporary commercial turbofans.This
optimisation problem is of combinatorial nature. Although algorithms are
especially dedicated to this kind of problem, the optimal configuration is
found here by means of a brute force technique. The figure of merit is computed
for every of the 792 (12 choose 7 binomial coefficient) possible combinations.
Unit weights have been applied to each component of theFOM (Wi=1.0,i=1,2,3).Table3 reports
the first ten optimal configurations based on the figure of merit defined in
(14). The last column on the right gives the value of the figure of merit for
each setup. At first sight, the values may appear quite close to each other,
but the reader should recall that the figure of merit is defined on a
logarithmic scale. The bottom line in the table gives the frequency of each
sensor in the first ten best sensor sets. To complete the picture, the first
five optimal configurations according to each component of the FOM are given in Tables 4, 5, and 6. The
last column in each of these tables is also expressed in a logarithmic scale.Table 3
First ten optimal configurations.
P13T13P26T26NLPNHPP3T3P49T49P5T5FOM1√√√√√√√−1.42542√√√√√√√−1.59683√√√√√√√−1.70144√√√√√√√−1.73155√√√√√√√−1.89366√√√√√√√−2.04927√√√√√√√−2.17098√√√√√√√−2.25409√√√√√√√−2.257610√√√√√√√−2.28199090988810702Table 4
First five optimal configurations based on the
condition number of the FIM.
P13T13P26T26NLPNHPP3T3P49T49P5T5CN1√√√√√√√−1.98822√√√√√√√−2.06343√√√√√√√−2.07134√√√√√√√−2.15335√√√√√√√−2.1851505055335202Table 5
First five optimal configurations based on the
trace of the FIM.
P13T13P26T26NLPNHPP3T3P49T49P5T5Tr1√√√√√√√2.01222√√√√√√√2.01173√√√√√√√2.00304√√√√√√√2.00255√√√√√√√2.0003404055315503Table 6
First five optimal configurations based on the
determinant of the FIM.
P13T13P26T26NLPNHPP3T3P49T49P5T5Det1√√√√√√√−1.36452√√√√√√√−1.45573√√√√√√√−1.62694√√√√√√√−1.63685√√√√√√√−1.6608504055345301It can be seen
that the optimal configuration according to the fullFOM (i.e., first line of Table 3) is ranked,
respectively, fourth, first, and second according to the condition number, the
trace, and the determinant of the FIM. Hence, the combined metric elegantly
merges the different objectives of the optimisation problem. Looking at the
last line of each table, it can be deduced that T13, T26, and P5 are not good candidate to the optimal
instrumentation whatever the metric considered. On the contrary, P13,P26,P49, and the spool speeds appear to be mandatory in
the optimised configuration. This point will be discussed in more details
subsequently.As highlighted
in [18], installation
of an interturbine pressure sensor becomes more difficult on modern engines.
Consequently, the search has been carried out a second time by removing P49 from the list of available sensors, leaving a
mere 330 configurations to be examined.The three
sensor sets selected for further investigations are presented in Table7. Set A is the standard instrumentation defined
in the frame of the OBIDICOTE project and serves as the baseline for
comparison, set B is the optimal
one when considering P49 as available, and set C is the optimal one when discarding P49 from the list. Note that the notion of
optimality is relative to the metric defined by (14).Table 7
Selected sensors sets for further
investigations.
SetAP13T13NLPNHPP3T3T5SetBP13P26NLPNHPP3P49T49SetCP13P26NLPNHPP3T3T49Examining Table7, one first notices that P13,
both spool speeds, and P3 are common to the three sensor sets. P13 is indeed required to correctly assess fan
performance. Concerning the spool speeds and P3,
these sensors are mandatory for control purpose. It is thus a good point that
they appear in the optimal sets. Furthermore, spool speeds are generally very
accurately measured, which is one of the reasons of their presence in the
optimal configurations. Set B is
characterised by the presence of P49.
This sensor is in fact the only one in the list that allows full
differentiation of lpt and hpt faults.It is also of
interest to point out that setsB and C contain, respectively, 4
and 3 pressure sensors out of 7. This is explained by the fact that pressure
sensors are more informative about the engine condition than temperature
sensors. It may appear strange that lpt exhaust pressure P5 does not appear in the optimal selection. For
a mixed-flow engine, this quantity is nonetheless highly correlated with fan
outlet pressure P13.
So selecting P13 discards P5 and vice versa. It can also be seen that
exhaust gas temperature (defined as T49 or T5,
depending on the manufacturer) is part of the optimal sensor sets. An EGT sensor
is already fitted on every engine as this quantity serves as a global indicator
of engine health for the pilot (so-called EGT margin).


## 3.3. Evaluation of the Sensor Sets

In order to
evaluate the benefits brought by optimal instrumentation, a series of fault
cases, taken from [19]
and summarised in Table 8, are considered. They are intended to represent
degradations that can be expected on modern turbofans for all individual
components. Considered faults involve single as well as multiple health
parameters. In the present application, faults have a constant magnitude.
Snapshot data are simulated by means of the engine performance model. Gaussian
noise, whose magnitude is specified in Table 2, is added to the generated
measurements to make them closer to test data.Table 8
Considered fault cases.
a−1% onSW12R−0.5% onSE12fan, lpc−0.7% onSW2R−0.4% onSE2b−1% onSE12c−1% onSW26R−0.7% onSE26hpcd−1% onSE26e−1% onSW26Rf+1% onSW42Rhptg−1% onSW42R−1% onSE42h−1% onSE42i−1% onSE49lptj−1% onSW49R−0.4% onSE49k−1% onSW49Rl+1% onSW49R−0.6% onSE49m+1% onA8IMPnozzlen−1% onA8IMPA diagnosis
tool based on a Kalman filter (see [20] for further details) processes the data. Given its
stochastic nature, each test case is run twenty times with different
realisation of measurement noise. Table 9 reports the averaged results in a
very synthetic way. Identification of the fault is declared successful if the
estimation error lies within its bounds (see [21] for more details) and is
indicated by a check mark (√) in the table.Table 9
Fault
identification results with the 3 sensor sets.
Fault caseabcdefghijklmnSetA√√√√√√√√√×××√√SetB√√√√√√√√√√√√×√SetC√√√√√√√√√×√√×√The standard
instrumentation, setA, performs
already quite well. It can however be seen that it fails in diagnosing
correctly most of the lpt faults (cases j, k, and l). Sensor set B, featuring the interturbine pressure
sensor, resolves this issue, as previously mentioned. However, fault case m is now misidentified. The increase in
exhaust nozzle area is confounded with a fan fault. Finally, set C achieves some improvement with respect
to set A too. Most of the lpt
faults but case j are correctly
assessed. As for set B, the
diagnosis tool is misled by fault case m.


## 3.4. Observability Analysis of the Sensor Sets

The study of
the optimal sensor sets is concluded by an observability analysis as per the
concepts defined in a previous section. Figure3 depicts the observability
indices of the health parameters for the three sensors sets. Concerning set A, the observability of most of the
parameters is quite good, except for the booster flow factor, SW2R,
and the lpt flow factor SW49R.
The optimal set B improves the
observability of those parameters, on the other hand, it degrades the
observability of SE12 and SE26 due to the loss, respectively, of T13 and T3.
The nozzle area is also affected. Finally, optimal set C offers an intermediate solution. With
respect to set A, the
observability of the lpc is improved as well as the one of the core components.
As for set B, one can notice a
decrease in observability for SE12 and A8IMP.Figure 3
Comparison of
the observability indicesOI.In Figure4,
the sensitivity indices are compared for each sensor set. From the left
picture, it can be seen that both temperature sensors in the cold section,
namely, T13 and T3,
contribute to a lesser extent in set A. The center picture, associated to set B, illustrates the effect of the
optimisation. The distribution of sensitivity indices is more uniform, and
values are higher. Finally, the right graph shows the sensitivity indices for
set C. Again, the general trend
reveals an increase in the average level. T3 has the lowest contribution among all sensors.Comparison of
the sensitivity indicesSI.


## 4. Discussion

To complete the
analysis of the results, some issues that may lead to further developments of
the proposed methodology are discussed in the following.The first
question is related to the assumptions on which the proposed approach relies.
The metric defined for sensor selection has indeed been derived for a
linearised system, around a given operating point and reference health
conditions. Yet, the behaviour of the engine is essentially nonlinear. It would
be of interest to evaluate the effects of the nonlinearities on the selection
process. On the other hand, the method should be extended to multipoint
estimation. A solution could consist in computing the metric from a weighted
sum of Fisher information matrices derived for various conditions (both
operating and health). Another concern is that the current methodology does not
take into account preferential directions for some faults (e.g., the
efficiencies are not expected to improve over time). This might impact the
sensor selection and should hence be integrated within the figure of merit.The present
contribution is dedicated to the selection of the optimal sensor configuration
for a given set of health parameters. Variations of this problem could make up
another interesting field of research. A first variant consists in the
selection of the best sensors to add to an existing set (for instance imposed
by control system requirements). A second one is the selection of the subset of
the most observable health parameters given a sensor configuration in order to
have a square problem (m=n). This problem has already been investigated
within the scope of thrust estimation in [22]. The reduction is however carried out in a
transformed space, so that the health parameters become actually tuners which
turn out to be of little interest for diagnostics purpose.With a
redefined metric, problems such as the minimisation of the number of sensors
that allow a satisfactory estimation of a given set of parameters could be
investigated. A valuable complementary study could consist in the assessment of
the modification in the observability properties of the system in case a sensor
is removed either to simplify the configuration or because it is faulty.Finally, the
search for the optimal sensor configuration is achieved here by means of a brute
force technique. The primary focus of the present paper is indeed the
definition of relevant metrics that describe the problem under consideration.
An optimisation algorithm fitted to the combinatorial nature of the problem
could supersede the brute force technique which rapidly turns prohibitive as
far as computational burden is concerned. In this framework, multiobjective
optimisation could be considered. This would leave the designer to select the
optimal configuration by trading off the metrics a posteriori rather than a
priori through the definition of an aggregated figure of merit. Additional
features such as cost and reliability of the sensors should be taken into
account in the definition of the metric as they are important factors from an
industrial standpoint.


## 5. Conclusion

In this
contribution, the problem of optimal selection of the sensor configuration for
diagnostics has been revisited from the viewpoint of information theory. From
sound mathematical arguments, the Fisher information matrix has appeared to be
a relevant quantity for the problem of sensor selection. A figure of merit
addressing various issues such as sensor noise, negative redundancy, or
orthogonality has been defined based on the Fisher information matrix.The selection of
the optimal sensor configuration with respect to the defined metric has been
performed by a naive brute force technique. The enhancement brought by the use
of an optimal instrumentation has been underlined with the estimation of a
series of simulated but still realistic fault cases that may occur on a
contemporary turbofan engine.



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*Source: 784749-2008-10-22.xml*