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.

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 as

The gas path analysis approach to jet engine diagnostics.

the condition of the components (e.g., fan,
lpc, hpc, hpt, lpt, nozzle) can be represented by a set of performance
indicators, so-called

a degradation (progressive or accidental)
affecting the engine induces a modification of its performance, which results
in a drift of some

a

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 [

The same topic
has received much more attention in other fields such as chemical or structural
engineering. As a general trend (see for instance [

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.

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.

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 (

A random
variable

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 (

Considering the
joint probability distribution of the residuals and the parameters

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:

Engine
performance monitoring is characterised in practice by negative redundancy,
which means that the number of parameters exceeds the number of sensors (

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:

the

the

the

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 [

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 [

Given the meaning
of the columns of

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 matrix

The definition
of the observability index proposed in the present study quantifies this loss
of information. For some fault

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 matrix

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 [

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 Table

Cruise point definition.

Label | Altitude | Mach | ||

Value | 0.350 kg/s | 10668 m | 0.80 | 0.0 K |

Sensors that
may be fitted on the engine are listed with their associated uncertainty (noise
level) in Table

Available sensors (uncertainty is three times the standard deviation).

Label | ||||||

Uncertainty | ||||||

Label | ||||||

Uncertainty |

The
optimisation problem is to determine the

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 the

Table

First ten optimal configurations.

−1.4254 | |||||||||||||

2 | −1.5968 | ||||||||||||

3 | −1.7014 | ||||||||||||

4 | −1.7315 | ||||||||||||

5 | −1.8936 | ||||||||||||

6 | −2.0492 | ||||||||||||

7 | −2.1709 | ||||||||||||

8 | −2.2540 | ||||||||||||

9 | −2.2576 | ||||||||||||

10 | −2.2819 | ||||||||||||

9 | 0 | 9 | 0 | 9 | 8 | 8 | 8 | 10 | 7 | 0 | 2 |

First five optimal configurations based on the condition number of the FIM.

1 | −1.9882 | ||||||||||||

2 | −2.0634 | ||||||||||||

3 | −2.0713 | ||||||||||||

−2.1533 | |||||||||||||

5 | −2.1851 | ||||||||||||

5 | 0 | 5 | 0 | 5 | 5 | 3 | 3 | 5 | 2 | 0 | 2 |

First five optimal configurations based on the trace of the FIM.

2.0122 | |||||||||||||

2 | 2.0117 | ||||||||||||

3 | 2.0030 | ||||||||||||

4 | 2.0025 | ||||||||||||

5 | 2.0003 | ||||||||||||

4 | 0 | 4 | 0 | 5 | 5 | 3 | 1 | 5 | 5 | 0 | 3 |

First five optimal configurations based on the determinant of the FIM.

1 | −1.3645 | ||||||||||||

−1.4557 | |||||||||||||

3 | −1.6269 | ||||||||||||

4 | −1.6368 | ||||||||||||

5 | −1.6608 | ||||||||||||

5 | 0 | 4 | 0 | 5 | 5 | 3 | 4 | 5 | 3 | 0 | 1 |

It can be seen
that the optimal configuration according to the full

As highlighted
in [

The three
sensor sets selected for further investigations are presented in Table

Selected sensors sets for further investigations.

Set | |||||||

Set | |||||||

Set |

Examining Table

It is also of
interest to point out that sets

In order to
evaluate the benefits brought by optimal instrumentation, a series of fault
cases, taken from [

Considered fault cases.

−1% on | −0.5% on | fan, lpc | |

−0.7% on | −0.4% on | ||

−1% on | |||

−1% on | −0.7% on | hpc | |

−1% on | |||

−1% on | |||

+1% on | hpt | ||

−1% on | −1% on | ||

−1% on | |||

−1% on | lpt | ||

−1% on | −0.4% on | ||

−1% on | |||

+1% on | −0.6% on | ||

+1% on | nozzle | ||

−1% on |

A diagnosis
tool based on a Kalman filter (see [

Fault identification results with the 3 sensor sets.

Fault case | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Set | ||||||||||||||

Set | ||||||||||||||

Set |

The standard
instrumentation, set

The study of
the optimal sensor sets is concluded by an observability analysis as per the
concepts defined in a previous section. Figure

Comparison of
the observability indices

In Figure

Comparison of
the sensitivity indices

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 (

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.

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.

Estimated value

Scaled value

Nozzle exit area (baseline value:

Baseline conditions

Fisher information matrix

High-pressure compressor

High-pressure turbine

Low-pressure compressor

Low-pressure turbine

Number of gas path measurements

Number of health parameters

Rotational speed

Total pressure at station

Efficiency scaler at station

Flow capacity scaler at station

Total temperature at station

Operating point parameters

Health parameters

Observed measurements

Measurement noise vector

Singular value

A Gaussian probability distribution with
mean