An Adaptive Estimation Algorithmfor Aircraft Engine Performance Monitoring

Olivier Leonard, Sebastien Borguet and Pierre Dewallef

University of Liege - Turbomachinery Group

Chemin des chevreuils 1 - 4000 Liege - Belgium

Abstract

In the frame of turbine engine performance monitoring, system identification procedures areoften used to adapt a simulation model of the engine to some observed data through a set of so-called health parameters. Doing so, the values of these health parameters are intended to representa faithful image of the actual health condition of the engine. The Kalman filter has been widelyused to achieve the identification procedure in real time, on-board, applications. However, in orderto achieve a proper filtering of the measurement noise, the health parameters must be supposedto vary in time relatively slowly preventing any abrupt accidental events to be tracked effectively.This contribution presents a procedure called adaptive filtering and based on a covariance match-ing method, intended to automatically release the health parameters once an accidental event isdetected. This enables the Kalman filter to deal with both continuous and abrupt fault conditions.

INTRODUCTION

In the last years, improving the availability of aircraft turbine engines as well as mini-

mizing their maintenance costs turned out to be key issues for airline companies. In this

context, engine manufacturers are more and more involved in research programs intended

to dynamically adapt the maintenance planning based on the actual engine health condition

rather than using fixed schedule based on the number of running hours.

Basically, the monitoring of the engine health condition can be carried out by observing

the drifts from some reference values of a set of measurements performed on-board, while

the engine is in service. However, as a modification in the engine health condition generally

involves a drift on nearly all of the measurements at the same time, the identification of

the underlying component fault turns out to be uneasy. Moreover, the measurement drifts

are corrupted by random errors whose amplitude is comparable to the measurement drifts

induced by the component faults of interest. To improve the significance of the measurement

drifts, it is appropriate to translate them into a set of numerical parameters intended to

1

represent the health condition of the main engine components.

The link between the measurement drifts and the health parameter deviations is done by

an aero-thermodynamical simulation model. Such models are developed for design, perfor-

mance analysis or control purposes. They are based on the resolution of a set of equations

expressing mass, momentum and energy balances inside the engine. Besides their capability

to predict most of the measurements taken along the gas path of the engine in its whole

operating range, these models may also be parameterized. The nature of these parameters

may vary from one manufacturer to another but they usually consist in efficiencies and flow

capacities related to the main engine components. The simulation model is therefore able to

predict the value of the measurements corresponding to specific ambient conditions (e.g. the

ambient temperature and pressure, the humidity, . . . ), control conditions (e.g. the power

setting) and health conditions through a given set of health parameters. Doing so, the

performance monitoring task is achieved by the resolution of an inverse problem where the

health parameter deviations are determined based on the observation of the measurement

drifts.

Within this framework, system identification techniques bring an answer to the rather

poor physical meaning of the raw measurement drifts. Indeed, these techniques are intended

to adapt a simulation model of the engine to the observed data through the set of health

parameters. Accordingly, the set of adapted health parameters represents a faithful image of

the engine health condition. Moreover, system identification techniques provide the user with

a large set of methodologies intended to filter the random measurement noise. In the turbine

engine literature, the use of system identification techniques together with measurements

performed along the engine gas path is usually referred to as Gas Path Analysis (GPA). The

interested reader is merely referred to [1] for a simple but yet complete introduction of the

GPA approach to turbine engine health condition monitoring.

Among the large amount of available system identification methods, the Kalman filter is

considered herein as an appropriate approach. The Kalman filter provides an optimal update

of the health parameters (in the least-squares sense) each time new data are available, at

the cost of a very low computational load. However, the correct filtering of the random

measurement errors requires proper constraints to be set on the variation rate of the health

parameters. In the frame of engine health monitoring, the transition model of the health

parameters is based on the assumption of a relatively slow variation. Consequently, the

2

Kalman filter tracks engine wear with a good accuracy. Engine wear is the primary cause of

deterioration ; it affects all components at the same time and evolves smoothly with time.

On the other hand abrupt faults, due for instance to foreign object damage, may happen

too. These events generally impact one (at most two) component at a time. The Kalman

filter is less capable of following short-time-scale variations and consequently the estimated

fault is spread on several components. This behavior is referred to as a “smearing effect” in

the literature [3].

Different strategies have already been considered in the turbine engine literature to over-

come this issue. One of the most advanced consists in determining whether or not the mea-

surement drifts are related to an abrupt component fault before using them in the Kalman

filter. When an abrupt component fault is detected, a bank of Kalman filters is used, each

of which considering a specific accidental event. Provided that at least one of these specific

filters correctly fits the situation at hand, the abrupt fault can be isolated and accomodated

by the simulation model and the generic Kalman filter can be used again. Even if succesful

applications of this procedure are found in the literature (see for example [1]), the setup of

the fault partition and of the corresponding bank of specific Kalman filters, as well as the

selection of the most relevant one, turn out to require a significant experience from the user.

The present contribution investigates a different approach in which the constraints on

the health parameter variation rate are adapted on-line such that they best fit the actual

situation. Therefore, both abrupt and continuous faults can be processed by a single Kalman

filter.

THE SIMULATION MODEL

In the framework of diagnosis, the purpose of the engine simulation model is to reproduce

the value of some variables measured along the engine gas path. Among the measurable

quantities, some are independent variables, such as the uncontrollable ambient conditions

(temperature and pressure, humidity, . . . ) or the command variables (e.g. the power set-

ting). In the present document, no distinction is made between the command variables and

the uncontrollable inputs. All operating conditions are aggregated into a column vector

denoted uk where k is a discrete time index. The other measurable quantities are dependent

variables and are stored in a m-dimensional vector yk simply called the measurements.

3

Besides their dependency upon the operating conditions uk, the measurement yk are also

a function of the actual engine health condition1.

To take advantage of this dependency, the simulation model is parameterized by a set of

so-called health parameters denoted by a p-dimensional vector wk. Doing so, the simulation

model is a vector function noted G(uk,wk). In this contribution, the simulation model is

restricted to the steady-state prediction of gas path measurements (temperatures, pressures,

thrust, air mass flow rates, . . . ) and rotational spool speeds. Due to the nature of the model,

the health parameters are restricted to the efficiency, the flow capacity or the effective cross-

section of the main components of the engine.

As a deterministic model is a simplification of the real-world process, it is unlikely to

perfectly match the random errors corrupting the observed measurements. To establish the

equality, a random additive error term εk is defined according to:

yk = G(uk,wk) + εk (1)

As the value of εk cannot be known beforehand, only its probability to fall in certain

ranges can be described analytically. A common assumption consists in modeling the be-

havior of εk by a zero-mean (E(εk) = 0), white and Gaussian random variable with covariance

matrix defined by:

E(εkεTl ) =

Rr if k = l

0 if k 6= l(2)

With some abuse of notations, the operator E(·) represents the mathematical expectation

of a random variable2 and the superscript T is the vector-matrix transpose. The specification

of the equation (1) together with the covariance matrix Rr is called the statistical system

model because it combines the deterministic system model G(·) to the stochastic description

of the random measurement errors.

In the frame of turbine engine diagnosis the quantity of interest is the difference between

the actual engine health condition and a reference one. In this contribution, this reference

1 Strictly speaking, the measurement yk are also a function of the transient effects taking place inside theengine (i.e. mechanical inertia, heat transfers and gas dynamic) but, as only steady-state behaviors areconsidered herein, these effects are not taken into account in the present document.

2 See [4] for a more proper statement of the concept of mathematical expectation of a random variable.

4

value is represented by a so-called prior value which designates a value of the health pa-

rameters w−k which is available to the user prior to the knowledge of the measurements yk.

To take this consideration into account, a residual is defined as the difference between the

measurements yk and those predicted by the model using the prior value of the parameters

w−k :

rk , yk − G(uk, w−k ) (3)

In order to reduce the computational burden a first order linearization of the simulation

model around the prior health parameter values is used (see [7] for a discussion about the

use of linear models in turbine engine diagnosis):

Gk ,∂G(uk,wk)

∂wk

∣∣∣∣wk=bw−k (4)

The multiplying factor Gk is a m× p influence matrix which indicates how the measure-

ments are drifting from the assumed values when each health parameter moves away from

its prior value. According to (3) and (4) the model equation (1) becomes:

εk = rk − Gk(wk − w−k ) (5)

As the influence matrix Gk depends upon the operating conditions, its estimation requires

p additional runs of the simulation model at each time step if it is calculated by forward or

backward differences (2p for central differences). From a computational point of view, this

situation is not favorable since the processors currently available onboard of aircraft engines

are not yet able to achieve such tasks in real-time. However, for the application detailed

further in the document, the matrix Gk does not change significantly for the considered range

of operating conditions and the influence matrix is computed beforehand for nominal health

parameters and cruise flight conditions. Yet, when large ranges of operating conditions are

considered, it may be advantageous to exploit the nonlinearities of the simulation model

with respect to the operating conditions by processing important ranges of flight conditions,

as outlined in [8]. The influence matrix can be interpolated between a bank of matrices

computed beforehand for a set of control conditions (usually the engine pressure ratio or

the non-dimensional low pressure spool speed) and corrected for the actual inlet conditions

(typically the total pressure and temperature) which provides a high accuracy together with

5

a low computational load.

THE KALMAN FILTER

The most objective way to solve the estimation problem would consist in a maximum

likelihood approach in which the health parameters are considered as unknown quantities

that can take any possible value. Within the present Gaussian framework, this comes down

to determine the value of wk minimizing the inner-product εTk R−1

r εk. However, due to the

fact that the number of health parameters to be determined is higher than the number of

available measurements yk for each data sample, the maximum likelihood approach leads to

a non-unique solution.

To overcome this issue while keeping a sample by sample data processing, the health pa-

rameter deviations are considered, symmetrically to the random errors, as Gaussian random

variables with mean and covariance matrix respectively defined by w−k = E(wk|yk−1) and

P−w,k = E[(wk − w−k )(wk − w−k )T |yk−1

]. Again with some abuse of notations, the operator

E(·|yk−1) designates the conditional mathematical expectation of a random variable before

the data sample at time step k is used to update the health parameters [4].

Once the current data sample yk is observed, the application of the Bayes’ rule allows to

build the posterior probability density function describing the probability of occurrence of

the health parameters at time step k. The appealing property of using Gaussian random

variables together with the first order Taylor expansion of equation (4) is that the posterior

probability distribution is still Gaussian, which enables the estimation problem to be solved

analytically.

Mathematically, the estimation problem comes down to the determination of the posterior

mean and covariance matrix of the health parameters respectively defined by wk = E(wk|yk)

and Pw,k = E[(wk − wk)(wk − wk)

T |yk

]. The use of the posterior conditional mathematical

expectation E(·|yk) underlines the fact that the latter quantities are obtained once the current

measurement sample yk is used.

As for Gaussian random variables the mathematical expectation is equivalent to the

maximum of the posterior probability density function (MAP estimator), the posterior mean

6

wk is computed by the resolution of a maximization problem3. After some algebraic steps,

the maximization problem yields the following quadratic minimization problem (see [6] for

more details):

J (wk) = (wk − w−k )T (P−w,k)−1(wk − w−k ) + εT

k R−1r εk (6)

The first term on the right hand side in the previous expression simply expresses that

the values of wk must remain in a neighborhood of the prior values w−k (the shape of the

neighborhood being specified by the covariance matrix P−w,k).

The resolution of the quadratic minimization problem only involves basic linear algebra

operations. Indeed, it comes down to find the value of wk which cancels the derivatives of

the quadratic objective function with respect to wk which yields:

wk = w−k + Kkrk with Kk = P−w,kGTk

(GkP

−w,kG

Tk + Rr

)−1(7)

From the definition of the prior and posterior covariance matrices and exploiting the

previous update rule (7), it yields:

Pw,k = (I−KkGk)P−w,k−1 (8)

The use of the previous relations poses no serious problems provided that proper values for

w−k and P−w,k are available. A very interesting feature of the Kalman filter is that it is able to

exploit a so-called transition model for the health parameters. This enables the computation

of proper values for w−k and P−w,k from the results obtained at the previous time step and

enables the update rules (7) and (8) to be used recursively.

As very few information is usually known beforehand about the way the health parameters

will vary in time, the health parameter transition equation is often restricted to a quasi-

stationary process described by:

wk = wk−1 + ωk (9)

where ωk is called the process noise and is modeled by a zero-mean (E(ωk) = 0), white and

3 It has to be noted that the present approach is not the most generic methodology used to deduce theKalman filter. Yet, the interested reader is referred to [2,5] for a more generic approach.

7

FIG. 1: Performance monitoring tool based on an Kalman filter for the case of a turbofan engine

Gaussian random variable with covariance matrix defined by:

E(ωkωTl ) =

Rw,k if k = l

0 if k 6= l(10)

From the definition of w−k and ωk and noting that E(wk−1|yk−1) = wk−1, it is relatively

straightforward to deduce that w−k = wk−1 and that P−w,k = Pw,k−1 + Rw,k.

In order to provide the reader with an overall picture of the whole procedure, the update

mechanism of the health parameters is shortly depicted in FIG. 1 and more precisely in

algorithm 1 where it is referred to as the Kalman filter. At the current time step k, the

previously estimated health parameters are used by the simulation model to generate a

prediction of the measured values. The comparison between the model outputs and the

observed measurements gives a residual rk which is an image of the distance between the

health parameters and the actual engine health condition. The Kalman filter determines the

new estimate wk through relation (7) and updates the health parameter covariance matrix

(not represented in the figure).

8

Algorithm 1 The Kalman filter

Require: w0, Pw,0, Rr, Rw,k

1: for k > 0 do2: w−k = wk−1 and P−w,k = Pw,k−1 + Rw,k

3: acquire uk and yk

4: rk = yk − G(uk, wbl)− Gk(w−k − wbl)

5: compute the influence matrix through (4)

6: K = P−w,kGTk

(GkP

−w,kG

Tk + Rr

)−1

7: wk = w−k + Krk and Pw,k = (I−KGk)8: end for

PERFORMANCE MONITORING

In the present application it is assumed that the health parameters vary independently

such that the covariance matrix Rw,k is strictly diagonal. Therefore, the covariance matrix is

completely determined by the p diagonal terms called herein the variance of the process noise.

Even if the resulting transition model (9) appears very simple, the covariance matrix Rw,k

enables to control the stochastic character of the time series formed by the health parameters

wk: low variances of the process noise mean slow variations while high variances suppose

fast variations. In normal operation, the health parameters are varying in a relatively

predictable manner and the process noise variance can be set according to the maximum

expectable degradation rate of the health parameters. Of course, the more important the

data acquisition rate is, the lower the process noise variance can be which allows the random

measurement noise to be appropriately filtered without penalizing the faithfulness of the

performance tracking.

However, optimizing the performance tracking for gradual degradations also strongly

deteriorates its behavior when accidental events are encountered (e.g. a foreign object

ingestion). Indeed, the latter events involve large deviations on a small subset of the health

parameters in a very short period of time (typically less than a second). As the health

parameter variations necessary to adapt the simulation model to the actual engine health

condition exceed the range specified by the health parameter variance, the Kalman filter

spreads the fault on all of the health parameters. This behavior, providing the user with a

poor diagnosis, is sometimes referred to as a smearing effect in the turbine engine literature

[3]. Practically, a value for the health parameter variance giving a good balance between

fast tracking abilities and efficient random noise filtering is very difficult to find and the

9

resulting algorithm is more likely to behave poorly in both situations.

ADAPTIVE ESTIMATION

To improve the tracking abilities of the filter without sacrificing the reliability of the

estimation, the present contribution investigates a so-called adaptive estimation procedure.

By adaptive estimation, it is meant that the estimation algorithm is able to automatically

adapt the health parameter variance to both abrupt and continuous engine degradations.

The methodology used in this contribution is strongly inspired from the one developed by

Jazwinski in [9] and is intended to ensure consistency between the predicted residuals rk

and their statistics. The present methodology is developed for the specific kind of transition

model specified by relation (9). The interested reader will find the developments for a more

general family of transition models in [9].

To do so, the update of the health parameters is delayed of N time steps. Therefore,

a buffer of N + 1 residuals, denoted {rk−N , . . . , rk}, is formed by comparing the effective

measurements yk to those obtained through the simulation model when the prior value

w−k−N is used. This gives the following definition for the residuals:

rk−l , yk−l − G(uk, w−k−N) = Gk−l(wk−N−1 − wk−N−1) + Gk−l

N∑i=l

ωk−i + εk−l (11)

Since, by definition, E(wk−N−1|yk−N−1) = wk−N−1, E(εk) = 0 ∀ k and E(ωk) = 0 ∀ k,

it yields that E(rk−l|yk−N−1) = 0. This conditional mathematical expectation indicates

that the residuals are computed using the most recent health parameter estimate (wk−N−1).

Similarly, it can be proved, after some algebraic steps, that:

E(rk−lrTk−m|yk−N−1) = Gk−lPw,k−N−1G

Tk−m+Rrδm,l+(N+1−max (m, l))Gk−lRw,kG

Tk−m (12)

where max (m, l) is equal to m if m ≥ l and l otherwise and δm,l is equal to 1 if m = l and 0

otherwise. Practically, the off-diagonal terms of the matrix rk−lrTk−m are extremely sensitive

to the measurement noise. To make sure that the data are statistically representative,

it is chosen to restrict the present method to the diagonal terms of the previous matrix

equality. Therefore, ensuring the consistency of the residuals with their statistics is done by

10

determining Rw,k such that:

diag(rk−lr

Tk−m

)= diag

(E(rk+lr

Tk+m|yk−N−1)

)(13)

where the operator diag (·) designates the vector made of the diagonal values of a square

matrix. In a previous publication by the authors [6], the delay N is set to zero in order to

avoid a delayed health parameter estimation. In this way, the matrix Rw,k is determined

such as the squared residuals r2k match the diagonal terms of the right hand side of relation

(13) when N = m = n = 0. However, such an approach neglects the fact that, in tur-

bine engine performance monitoring, the level of measurement noise is relatively high and

Rr & GkRw,kGTk . As a consequence, the estimation of Rw,k is mainly driven by the random

measurement noise and the resulting adaptive procedure turns out to be too sensitive to

the random errors. In [6], this effect is overcome with the help of a second Kalman filter

estimating the diagonal terms of Rw,k. Even though it effectively stabilizes the estimation

procedure, the setup of the complete algorithm involves many parameters to be tuned which,

in the end, turns out to be uneasy and does not fulfill the simplicity requirement stated in

the introduction section.

In our search to simplify the complete estimation procedure, and according to the ap-

proach devised in [9], it appeared that the best approach consists in averaging the residuals

rk−l over the N + 1 samples of the buffer according to:

rk =1

N + 1

N∑l=0

rk−l (14)

Doing so, the health parameter estimation is delayed of N time steps but the mean rk is

also expected to be less sensitive to the measurement noise. Indeed, it can be shown that

the mean rk is a white and Gaussian random variable with zero-mean (E(rk|yk−N−1) = 0)

and covariance matrix given by (after some algebraic steps using equations (12) and (14)):

E(rkrTk |yk−N−1) = Gk,NPw,k−NG

T

k,N + Rr +N∑

l=0

Gk,lRw,kGT

k,l (15)

11

where the m× p matrix Gk,l and the m×m matrix Rr are defined as:

Gk,l =1

N + 1

l∑i=0

Gk−i and Rr =1

N + 1Rr (16)

Applying equation (15) to the covariance matching (13) leads to verify the following

equality:

dk , r2k − diag

(Gk,NPw,k−NG

T

k,N + Rr

)= Bkfk (17)

where, for sake of simplicity, the matrix Rw,k is reduced to a vector containing the diagonal

values according to:

fk = diag (Rw,k) and Bk =N∑

l=0

G2

k,l (18)

Since in turbine engine diagnosis there are often more health parameters than available

measurements, the matrix Bk is rectangular (m < p) and the previous equation has no

unique solution. Consequently, the resolution of the previous equation is best done by a

maximum a posteriori approach where the prior value of fk is simply set to fmin = diag (Rminw )

with a covariance matrix Pf . Therefore, an adaptive estimation procedure is obtained by

setting the diagonal terms of the covariance matrix Rw,k to the vector fk given by:

fk = fmin + max[0 ,

(P−1

f + BTk Bk

)−1BT

k dk

](19)

The covariance matrix Pf may be easily built by setting its diagonal terms as fmax/3

where fmax are the maximum allowed values for fk. Loosely speaking, fmax reflects the

squared maximum amplitude of the expected abrupt events. Additionally to the diagonal

terms, it is often found very helpful to set some correlations into the covariance matrix Pf

in order to decrease the sensitivity of the estimation to the measurement noise. Indeed, it

may happen that the health parameter variations generated by abrupt component faults are

not totally independent. For example, the flow capacity and the efficiency of the fan usually

vary together when a fault occurs. Such correlations are set into the covariance matrix by

following the method detailed in [1]. It consists in setting the off-diagonal terms of the

matrix Pf according to:

Pf (i, j) = Pf (j, i) = ρi↔j

√Pf (i, i) · Pf (j, j) (20)

12

FIG. 2: Structure of the adaptive estimation algorithm

where 0 ≤ ρi↔j ≤ 1 is the correlation factor. Theoretically, nothing prevents from specifying

negative values for the correlation factor, but this makes no sense in this application since the

values of fk always rise together whatever the direction of the health parameter deviations.

There is no rule of thumb for setting the correlation factors and they strongly depend upon

the application at hand. However, a value of 0.5 is often found to be a good compromise if

the only available information is that some health parameters “may usually vary together”.

The estimation algorithm described in FIG. 1 with the Kalman filter of algorithm 1 can

be now updated to achieve adaptive estimation. This newly developed procedure is sketched

in FIG. 2. Basically, the residuals are stored in a buffer of size N + 1 whose purpose is to

provide the residual rk−N and to compute the mean residual rk. The latter is used in turn by

the adaptive estimation, namely the relation (19), to determine a proper covariance matrix

Rw,k. Finally, the updated covariance matrix Rw,k and the residual rk−N are used by the

Kalman filter of algorithm 1 to update the estimated health parameters.

Even if the procedure is relatively straightforward, the data buffering, represented by

a simple box in FIG. 2, needs to be commented. Indeed, as the estimation is delayed

of N samples, the residuals rk−l given by the buffer would be related to the prior health

parameters w−k−l−N for l = 0, . . . , N , and not to the most recent prior estimate (namely

w−k−N) as required by both the adaptive estimation and the Kalman filter. To correct this

13

effect, without re-estimating the whole buffer at each time step, the residuals are updated

by the following rule:

rk−l = rk−l − Gk−l(wk−N − w−k−N) ∀ l = 0, ..., N (21)

STABILIZATION OF THE ALGORITHM

Since the square of a Gaussian random variable is no longer Gaussian, the residuals dk

defined in (17) are not Gaussian. As a consequence, the use of the estimation formula (19)

may result in very noisy values for fk with small values of N (i.e. N ' 10). This generates

spurious covariance rise which decreases the stability of the estimation with respect to the

measurement noise. Of course, the estimated values fk can be filtered on L time steps, as

advised in [9], but the corresponding average does not completely correct the problem.

A better approach consists in testing whether or not it is purposeful to apply the value

of fk. If not, a covariance matrix is used which corresponds to normal degradations. This

matrix is denoted Rminw . The test used herein is the well-known χ2 (read chi-squared)

test (see [4]). It is based on the assumption according which, if no abrupt fault occurs,

rk must be an m-dimensional Gaussian random variable with zero mean and covariance

matrix E(rkrTk |yk−N−1) given by relation (13) where Rw,k = Rmin

w . If the latter covariance

matrix is denoted Pr,k, the Mahalanobis distance defined by qk = rTk (Pr,k)

−1 rk follows a

χ2m probability density function with m degrees of freedom. The test compares the scalar

qk with the integral Xα of the χ2 distribution; if qk is equal to the quantile Xα of the χ2m

distribution defined by α =∫ Xα

0χ2

m (x)dx then the probability of obtaining qk or a larger

value in the “null hypothesis” (i.e. no abrupt component fault is present) is given by 1−α.

In the present application, Xα is computed beforehand by setting 1− α to a low value (e.g.

10−6). The use of the statistical test comes down to set the diagonal terms of the covariance

matrix Rw,k according to relation (19) if qk ≥ Xα and to Rminw otherwise.

APPLICATION ON A TURBOFAN ENGINE

The aforementioned methodology is applied to a turbofan layout representative of cur-

rently available commercial aircraft propulsors. It is a two-spool, mixed flow turbofan whose

14

fan

lpc hpc combustorhpt lpt

inlet

SW12RSE12

SW2RSE2

SW26RSE26

SW41RSE41

SW49RSE49 A8IMP

nozzle

2 26 3

13

41 49 6 8

12

1

Wfuel

FIG. 3: The turbofan layout used as an application test case. The symbols hpc and lpc referrespectively to the high and low pressure compressors while hpt and lpt refer to the high and lowpressure turbines.

general structure is sketched in FIG. 3. This turbine engine is a virtual, but still realistic,

engine used in the frame of the OBIDICOTE project4. The simulation model used further

in the application is basically an aero-thermodynamical model parameterized by a set of 11

health parameters represented in the boxes in FIG. 3. These health parameters are the flow

coefficients (SWiR) and the efficiency drops (SEi) of the main components of the engine,

namely the fan, the high and low pressure compressors (hpc and lpc) and the high and low

pressure turbines (hpt and lpt), together with the nozzle area (A8IMP). The simulation

model is described in [10] to which the interested reader is referred for more details.

Additionally to the simulation model, a set of 14 typical accidental component faults is

also supplied. The fault cases, extracted from [11] and detailed in TABLE I, are intended

to be representative of some of the most likely component faults that can be expected on a

current turbofan engine. Since they have been used by many authors in the turbine engine

literature, these test cases can be considered as benchmarks.

To identify the component faults, a set of 7 measurements is used. The positions and

the accuracies of these measurements are also extracted from [11] and are listed in TABLE

II. These measurements are intended to represent those available on-board of a commercial

4 A Brite/Euram project concerned with on-board identification and control of turbofan engines.

15

a -0.7% on SW2R -0.4% on SE2 fan, lpc-1% on SW12R -0.5% on SE12

b -1% on SE12c -1% on SW26R -0.7% on SE26 hpcd -1% on SE26e -1% on SW26Rf +1% on SW41R hptg -1% on SW41R -1% on SE41h -1% on SE41i -1% on SE49 lptj -1% on SW49R -0.4% on SE49k -1% on SW49Rl +1% on SW49R -0.6% on SE49m +1% on A8IMP Nozzlen -1% on A8IMP

TABLE I: Accidental fault cases of a turbofan engine.

index name Description Accuracyu(1) T1 Inlet total temperature 2Ku(2) P1 Inlet total pressure 100Pau(3) Pamb ambient pressure 100Pau(4) WFE Fuel mass flow rate 2g/sy(1) T13 Afterfan total temperature 2Ky(2) P13 Afterfan total pressure 100Pay(3) T3 HPC outlet total temperature 2Ky(4) P3 HPC outlet total pressure 5000Pay(5) Nlp Low pressure spool speed 4rpmy(6) Nhp High pressure spool speed 12rpmy(7) T6 LPT outlet total pressure 2K

TABLE II: Operating conditions uk and measurements yk available on-board of the turbofan layoutconsidered in the present application. The accuracies are 3 times the standard deviation.

turbofan.

The measurements used further in the applications are fictive measurements generated

by the simulation model for constant cruise flight conditions (altitude=10800m, Mach=0.8,

∆Tiso=0K). Random measurement errors are added to all of the simulated measurements

and to all of the operating conditions by using a zero-mean Gaussian random variable with

standard deviations set to the third of the accuracies mentioned in TABLE II. The sequence

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is 5000 seconds long with a data acquisition rate of 2 samples per second.

In order to represent a slow degradation due to normal operation, a reference fault case is

considered where nearly all of the health parameters are varying simultaneously. This fault

case, referred to as the 9p case further in the document, has already been investigated in [12]

and consists of the following degradations occurring progressively in 5000 seconds: -1.5%

on SW12R, -1.2% on SE12, -1% on SW2R, -1% on SE2, -2.3% on SW26R, -1.4% on SE26,

+0.88% on SW41R, -1.6% on SE41, -1.3% on SE49. To underline the benefits of the newly

developed method, the accidental fault cases listed in TABLE I are successively added, one

at a time, to the reference slow degradation mentioned above. Each abrupt component fault

is added to the slowly drifting fault at time t=2500 s. Doing so, the behavior of the diagnosis

tool is illustrated for both slow and fast drifting faults.

APPLICATION RESULTS

A good indicator of the quality of the performance tracking is the maximum RMS (root

mean square) estimation error (the average is done on the whole flight sequence) expressed in

percent of deviation from the nominal values. If only slowly drifting faults have to be tracked,

the algorithm 1 can be used effectively with a constant covariance matrix Rw,k = Rminw . Such

a situation is represented in FIG. 4. The actual values of the health parameters are drawn in

plain lines with the corresponding health parameter name on the right axes. FIG. 4 shows

clearly that identified values, plotted with the symbols, agree with their actual values. The

maximum RMS error erms remains below 0,15% which hints at the good tracking ability of

the Kalman filter for normal degradations.

However, if an abrupt component fault occurs, such a procedure fails to track and isolate

the fault. This is well illustrated by adding the fault case ‘a’ from TABLE I to the previous

slowly drifting faults. The results given by the previous diagnosis tool are summarized in

FIG. 5. The plain lines representing the actual values of the health parameters exhibit a

brutal drift of 4 health parameters, namely the efficiency and the flow capacity of both

the fan and the low pressure compressor at t=2500 seconds (i.e. SW12R, SE12, SW2R,

SE2). In this case, the Kalman filter is unable to follow the abrupt deviation of the 4 health

parameters. As a result, the maximum RMS estimation error rises above 0.4% which is

visualized in FIG. 5 by the spreading of the fault on several other components such as the

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FIG. 4: Identification results obtained on a slowly drifting fault on nearly all of the health param-eters and without adaptive filtering

high pressure compressor (SW26R) and the nozzle (A8IMP). The present test case is a good

illustration of the Kalman filter divergence in the presence of a rapid measurement drift.

Even at the end of the sequence, the fault is still not isolated which provides the user with

a wrong diagnosis.

To illustrate the advantage of using an adaptive estimation algorithm, the same fault case

is used but now enabling the update of the process noise covariance matrix Rw,k depicted in

FIG. 2 with a delay of 50 samples (i.e. N = 50). To improve the results, some correlation

factors are set in the covariance matrix Pf according to (20). A correlation of 0.7 is set

between SW12R, SE12, SW2R and SE2, an other of 0.7 between SW26R and SE26, one

of 0.5 between SW41R and SE41 and one of 0.1 between SW49R and SE49. The results

are presented in FIG. 6. The first thing to notice is that the use of an adaptive estimation

algorithm does not deteriorates the accuracy when only slowly drifting faults are present.

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FIG. 5: Identification results given without adaptive filtering when both slowly drifting faults andan abrupt component fault (fault case ’a’) are present

However, when an abrupt fault occurs, the diagonal terms of the covariance matrix Rw,k

related to the fan and the low pressure compressor are increased. Consequently, the Kalman

filter puts more emphasis on more recent data and the health parameters can be quickly

adapted. As a result, the identified values remain close to their actual ones and no smearing

effect is observed. Accordingly, the maximum RMS error computed for the present test case

remains below 0.15% for the whole sequence.

To have a more general idea of the efficiency of the newly developed diagnosis tool, the

set of 15 test cases is run with and without adaptive filtering. These results are summarized

in TABLE III in terms of maximum RMS error. The values mentioned in the table are

averaged over 10 independent runs in order to ensure their reliability. Since the standard

deviations of the estimated health parameters (i.e. the square root of the diagonal terms

of the covariance matrix Pw,k) are approximately 0.1%, a test case whose maximum RMS

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FIG. 6: Identification results given when both slowly drifting faults and an abrupt component fault(fault case ’a’) are present and using adaptive filtering

error remains below 0.25% is considered as successful. For sake of simplicity, this is denoted

by a checkmark in TABLE III. The first line in TABLE III refers to the situation where

only the continuous fault is present. As it can be expected, both algorithms give the same

results. However, when an abrupt component fault is introduced at t=2500 seconds, the

results given by the adaptive filtering are far better than those obtained without adaptive

filtering. Indeed, without adaptive filtering only 4 test cases (namely cases ’d’, ’f’, ’i’ and

’m’) are solved satisfactorily while with adaptive estimation, all of the test cases (but one)

are successful.

CONCLUSIONS

In the present contribution, an adaptive estimation algorithm has been developed to

achieve the performance monitoring of aircraft turbine engines for both normal degradations

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Withoutadaptivefiltering

With adaptivefiltering

9p 0.17% X 0,18% X9p + a 0.41% - 0.18% X9p + b 0.27% - 0,17% X9p + c 0.39% - 0,24% X9p + d 0,16% X 0.17% X9p + e 0,35% - 0.18% X9p + f 0,21% X 0.16% X9p + g 0,31% - 0.17% X9p + h 0,25% X 0.16% X9p + i 0,23% X 0.17% X9p + j 0,52% - 0.57% -9p + k 0,46% - 0.23% X9p + l 0,44% - 0.23% X9p + m 0,16% X 0.18% X9p + n 0.34% - 0.18% X

TABLE III: Comparison of the results obtained with and without adaptive filtering. The resultsare the maximum RMS errors averaged on 10 successive runs.

and abrupt accidental events. The improvements of the new methodology with respect to

a more generic tool, based on a Kalman filter, have been illustrated on a typical turbofan

application. The accurate performance estimation achieved by the adaptive estimation al-

gorithm allows an efficient performance monitoring and a better component fault isolation.

Moreover, the implementation of the estimation algorithm is relatively straightforward and

requires only basic matrix operations. The computational burden is also very limited which

is compatible with the requirements of an on-board application.

The only drawback of the method is that the estimation is delayed of N time steps. High

values for N gives a more important delay but a more accurate abrupt fault detection while

lower delays give faster but less accurate results. In practice, values of N from 10 to 100

are usually sufficient to achieve a very accurate performance tracking. As a consequence,

the improvement in terms of convergence speed of the performance tracking is usually more

important than the time delay induced by the data buffering.

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ACKNOWLEDGEMENTS

This work is funded by the Walloon Region of Belgium in the frame of its Program FIRST

and is supported by Techspace Aero, Safran Group.

BIBLIOGRAPHY

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[10] A. Stamatis, K. Mathioudakis, J. Ruiz, and B. Curnock. Real-time engine model implemen-tation for adaptive control and performance monitoring of large civil turbofans. ASME papernumber GT2001-362, 2001.

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