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Multiple-Model Based Sensor Fault Diagnosis Using Hybrid Kalman
Filter Approach for Nonlinear Gas Turbine Engines *
Bahareh Pourbabaee1, Nader Meskin2and Khashayar Khorasani1
Abstract—In this paper, an efficient sensor fault detection and
isolation (FDI) strategy is proposed based on multiple-model
(MM) approach. The scheme is composed of hybrid kalman
filters (HKF) by integrating a nonlinear gas turbine engine model
that represents the operational engine model with a number
of piecewise linear (PWL) models to estimate sensor outputs.
The proposed FDI scheme is capable of detecting and isolating
permanent sensor bias faults during the entire operational regime
of the engine by interpolating the PWL models using a Bayesian
approach. Another important aspect of our proposed FDI strat-
egy is its effectiveness within the engine life cycle by periodically
updating the model to the degraded health parameters, that one
estimated by means of an off-line trend monitoring system that
is based on post flight data. The simulation results demonstrate
the effectiveness of our proposed online sensor fault diagnosis
scheme as well as the robustness of our technique with respect
to the engine health parameters degradations.
I. INTRODUCTION
An effective fault diagnosis, prognosis and health monitor-
ing (DPHM) solution can play a critical role in improving the
system availability, safety and reliability as well as reducing
the maintenance cost and risks of catastrophic failures. Over
the past few years, many researchers have focused on propos-
ing sophisticated fault detection and isolation (FDI) schemes
constituting as a significant component of an DPHM system
[1]–[3].
The efficiency of approaches that are used to perform dif-
ferent types of fault diagnosis can be measured by evaluating
various factors such as promptness of the fault detection,
sensitivity to an incipient fault, low false alarms and missed
fault detections rates, and accuracy of the fault severity estima-
tion. The compromises among these criteria require extensive
knowledge of the monitored system as well as the operating
system conditions [4]. The above factors can also be affected
by the accuracy of the system model used and fault diagnosis
method, the severity of the injected fault and also the current
health condition of the operational system.
Since early research on FDI, gas turbines have been one of
the challenging application areas which have received much
attention. The main theme of research in gas turbines FDI is
based on the Gas Path Analysis (GPA) which enables the ac-
tuators, sensors and components fault diagnosis by observing
*This publication was made possible by NPRP grant No. 4 - 195 - 2 - 065
from the Qatar National Research Fund (a member of Qatar Foundation). The
statements made herein are solely the responsibility of the authors.
1B. Pourbabaee and K. Khorasani are with the Department of Elec-
trical and Computer Engineering, Concordia University, Montreal, Canada.
b pourba@ece.concordia.ca and kash@ece.concordia.ca
2N. Meskin is with the Department of Electrical Engineering, Qatar
University, Doha, Qatar. nader.meskin@qu.edu.qa
the engine’s parameters such as the rotor speed, temperature
and pressure at each stage and the fuel flow rates [5], [6].
Different fault diagnosis techniques have been developed in
the literature for gas turbines based on the GPA ranging from
Kalman filters [7]–[9], neural networks [10], fuzzy logic [11],
genetic algorithms [12], and hybrid diagnosis [13]. Another
popular FDI approach is the multiple-model (MM) based
filtering method that consists of a bank of estimators - one for
each of the hypothesized fault modes and one for the healthy
mode. This method has a simple decision logic as well as it
directly provides fault identification information [14]–[17].
In this paper, the MM-based approach is used for sensor
fault detection and isolation in a single spool jet engine. The
proposed estimator for the MM-based scheme is an extension
of the hybrid kalman filter [9] that consists of a single
nonlinear on-board engine model (OBEM) with piecewise
linear (PWL) models to cover the entire engine operating
regime. The system works robustly with respect to health
parameters degradation in a wide range of operating conditions
by decomposing the regime into certain simpler sub-regions
each represented by a PWL model. In this paper, we have
applied a Bayesian approach to generate a general combined
model based on the PWL models’ normalized weights as this
provides us with a soft transition or interpolation among the
PWL models.
One of the significant non-fault related factors leading to
false alarms is the aging effect of the engine’s components.
This phenomenon is associated with the gradual degradation
of the health parameters from their original health baselines
which has an effect on the performance of the fault diagnosis
method and may lead to false alarms. Usually, the engine
health parameters which are the efficiencies and the mass flow
rates of the turbine and the compressor need to be estimated
after a certain number of flights by using health parameter
estimation methods [18]. The applied method can be used off-
line, since the health parameters are degraded more slowly
as compared to an abrupt fault. Consequently, the proposed
fault diagnosis framework should be capable of periodically
being updated by the last estimated health parameters to switch
to the updated health baseline and to keep its efficiency.
This method is useful in preventing false alarm occurrences
due to the health parameters degradations. Therefore, another
contribution of this work is to propose an update on the health
condition of the engine components for sensor FDI algorithm
and to analyze the robustness of the proposed method with
respect to the estimation errors of health degradations through
performing a number of simulations.
II. HYBRID KALMAN FILTER (HKF) DESIGN
The design procedure for an HKF [9] is similar to a general
Kalman filter. We start from a discrete-time model of a gas
turbine engine as represented by:
X(k+ 1) = F(X(k), H(k), u(k), e(k)),
Y(k) = G(X(k), H(k), u(k), e(k)) + v(k),(1)
where X∈Rn,H∈Rr,Y∈Rq,u∈Rp,v∈Rqand
e∈R2denote the state variables, health parameters, sensor
measurements, input signal, Gaussian zero-mean measurement
noise and environmental parameters including the altitude and
the Mach number, respectively. The above model is linearized
at different operating points corresponding to environmental
conditions to generate a linear state-space model as follows:
X(k+ 1) =A(X(k)−Xss ) + B(u(k)−uss)
+L(H(k)−Href ) + E(e(k)−ess),
Y(k)−Yss =C(X(k)−Xss) + M(H(k)−Hss )
+F(e(k)−ess) + v(k),(2)
where A,B,C,L,M,Eand Fdenote the state-space
matrices and Xss,uss ,Yss and ess denote the steady-state
values defining the operating point corresponding to each
linearized model. The effects of the reference health baseline
is also incorporated by using Href in the linearized model.
For each linearized model, a linear Kalman filter is designed
to estimate both the state variables and the sensor outputs as
follows: ˆ
X(k+ 1) =A(ˆ
X(k)−Xss) + B(u(k)−uss )
+K(k)(Y(k)−ˆ
Y(k)),
ˆ
Y(k) =C(ˆ
X(k)−Xss),(3)
where Kdenotes the Kalman filter gain matrix. According to
(3), the linear Kalman filter does not take into account the
effects of health parameters degradations from their nominal
values. Also, it does not have the required level of robustness
to handle all the health degradations that occur during the
entire life cycle operation of the engine. Therefore, it is
essential to update the health status of the engine for the
Kalman filters to maintain the accuracy of the state estimation
and the efficiency of the FDI algorithm through the entire
engine life cycle operation. For this purpose, Kobayashi in [9]
has proposed a hybrid Kalman filter (HKF) structure to take
into account not only the influence of the health degradations
in the output estimation process, but also to ensure this
procedure is less restrictive.
The HKF consists of two main blocks that include the
nonlinear on-board engine model (OBEM) and piecewise
linear (PWL) models which are extracted by linearizing the
OBEM at different operating points without considering health
degradations. The OBEM works in parallel with the opera-
tional engine by using the nonlinear mathematical model
XOBEM(k+ 1) = FM(XOBEM(k),ˆ
Href (k), u(k), W (k)),
YOBEM(k) = GM(XOBEM (k),ˆ
Href (k), u(k), W (k)),(4)
where b
Href and Wdenote the estimated health baseline
and the measured variables including the ambient temperature
and pressure associated with the operational condition of the
engine and FMand GMdenote the nonlinear mathematical
models of the engine. The health parameters are periodically
updated to their recently estimated values in order to generate
the state variables and sensor measurements in the vicinity
of the operational engine condition. In the next step, a linear
Kalman filter is designed by using (3) for each PWL model and
the corresponding steady-state value of the Kalman gain matrix
is saved into a look-up table in addition to the PWL models’
state-space matrices. Subsequently, the HKF formulation is
developed by modifying (3) through replacing the steady-state
variables by the OBEM state variables and outputs and also
by using the saved fixed Kalman gain as follows:
ˆ
X(k+ 1) = A(ˆ
X(k)−XOBEM(k)) + K(Y(k)−ˆ
Y(k)),
ˆ
Y(k) = C(ˆ
X(k)−XOBEM(k)) + YOBEM (k).(5)
where the effects of the input signal and also the Bmatrix
have been removed from the HKF formulation since they have
already been accounted for by the OBEM [9]. After updating
the OBEM health parameters, there is no longer a need to
redesign the A,Cand Kmatrices that have previously been
computed at the nominal engine condition. In a real application
and implementation, the OBEM represents a nonlinear engine
model which does not capture all the dynamical behavior
of the operational engine. Modeling errors always exist in
addition to the health parameters estimation errors in the HKF
algorithm, which lead to system output mismatch between the
operational engine and the OBEM. An acceptable range of the
engine health parameters estimation errors for the HKF will
be obtained in Section V.
In order to apply the HKF framework as an FDI strategy,
the PWL models are generated for each fault hypothesis at
different operating points. Moreover, the PWL models need
to be interpolated to cover the entire operational regime.
The detail description of this process is provided in the next
section.
III. PIECEWISE LINEAR MOD EL S INTE RP OLATI ON
Instead of using crisp boundaries between the PWL models,
it is smoother to have a soft interpolation among them and
to generate a combined general model for the entire flight
envelope. Therefore, using the engine input the operational
regime is divided into certain sub-regions for which a PWL
model is derived for each region. These sub-regions are related
to different flight modes such as the take-off, climbing, cruise
and landing modes. The number of the corresponding PWL
models for each mode influences the HKF estimation accuracy
and the FDI scheme efficiency. In this paper, the number
of PWL models is determined in order to avoid false alarm
occurrences in the range of the applied health degradation.
One of the most important advantages of the HKF is in
the small number of required PWL models to cover a wide
operating range during the entire operational regime of the
engine. The number for the HKF is less than that of a standard
piecewise linear Kalman filter due to the substitution of the
steady-state variables in (5) by the OBEM which is a valid
representation of the operational engine [9].
The PWL models are obtained by linearizing the OBEM
and taking into account the effects of the injected sensor bias
faults at each operating point as follows:
X(k+ 1) = FM(X(k), H(k), u(k), W (k)),(6)
Y(k) =GM(X(k), H(k), u(k), W (k))
+bjajδ(k−tf) + v(k), j = 1,...,(M+ 1)
where Mis the number of sensors, bjrepresents the jth sensor
bias fault magnitude and ajrepresents the fault location vector
that has a unit value for the jth sensor and the other elements
are zero, and δ(k−tf)denotes a unit step function that occurs
at tfcorresponding to the fault occurrence time. The engine
health parameters degradations have not been considered in
the above linearization process for all (M+ 1) healthy and
faulty sensor modes. In the next step, the HKF is designed for
the jth sensor mode in the ith operating region, as follows:
ˆ
X(i,j)(k+ 1) =Ai(ˆ
X(i,j)(k)−XOBEM (k))
+K(i,j)(Y(k)−ˆ
Y(i,j)(k)),
ˆ
Y(i,j)(k) =Ci(ˆ
X(i,j)(k)−XOBEM (k))
+YOBEM(k) + bjajδ(k).(7)
Each linear model is valid around the associated operat-
ing point. Although the operating range has been increased
through the use of the OBEM instead of the steady-state
variables, still none of the PWL models are valid over the
entire range of input variations. Therefore, each PWL model
could have a validity function based on its normalized weight
that is obtained by means of the Bayes formula. There are
various approaches including [19], [20] for defining such
weighting functions, however in this paper a Bayesian ap-
proach is selected to compute the weights. For this purpose, the
residual vector and the covariance matrix which are generated
by the HKF are used to compute the likelihood function for
the jth sensor mode in the ith operating region as follows:
γ(i,j)(k) = Y(k)−ˆ
Y(i,j)(k), S(i,j)(k) = cov(γ(i,j)(k)),
f(i,j)(γ(i,j)(k)) = 1
(2π)q/2qS(i,j)(k)×
exp[−1
2(γ(i,j)(k))T(S(i,j)(k))(−1)(γ(i,j )(k))],(8)
for i= 1, . . . , L and j= 1,...,(M+ 1), where Ldenotes
the number of operating points and Mdenotes the number
of engine sensors. The innovation sequence that is generated
by the hybrid Kalman filter, γ(k), is a Gaussian white noise
process with zero mean and covariance matrix S(k)which
is calculated numerically. The normalized weight for the jth
sensor is updated recursively using the Bayes formula as
follows:
w(i,j)(k) = f(i,j)[γ(i,j)(k)]w(i,j)(k−1)
PL
i=1 f(i,j)[γ(i,j)(k)]w(i,j)(k−1) .(9)
The weights computed above should also remain bounded
in order to avoid them from becoming close to zero as follows:
if w(i,j)(k)> α then w(i,j)(k) = w(i,j)(k),
if w(i,j)(k)≤αthen w(i,j)(k) = α, (10)
where the magnitude of αas a design parameter has an
effect on the speed of the model switching. Large values of
αlead to faster switchings while smaller values will cause
a slower transition between the PWL models. Following the
computation of the normalized weights, the general combined
model remains valid for the entire operational regime which
is obtained with respect to the weights of all the PWL models
designed for each sensor. The set of equations (11) below show
how the general combined model’s innovation vector and the
covariance matrix are designed by using the PWL models and
their associated normalized weights for each sensor, that is
γj
c(k) =
L
X
i=1
w(i,j)(k)γ(i,j)(k),
Sj
c(k) =
L
X
i=1
[w(i,j)(k)]2S(i,j)(k).(11)
The above procedure is known as the soft switching or
interpolation between the PWL models, where γcand Sc
denote the combined innovation vector and covariance matrix
for the healthy and faulty sensor modes. Consequently, (M+1)
linear time-varying combined models operate together through
the entire operational regime to form a multiple HKF structure
for the FDI algorithm.
IV. MULTIPLE-MODEL-BASED FDI SCHEME
In this section, the overall structure of the MM-based fault
detection and isolation scheme is presented. It is assumed
that the fault location vector ajcan take on only one of
(M+ 1) representative values. Therefore, at each operating
point, there are (M+ 1) PWL models, one for the healthy
sensors and Mfor different faulty sensors that have been
designed and integrated with only one OBEM to generate
a multiple HKF (MHKF) structure. The innovation vectors
and the covariance matrices are also calculated by the use
of the MHKF and interpolated to construct the combined
model matrices as given by (11). Finally, there are (M+ 1)
combined models for different engine’s healthy and faulty
scenarios to be applied for the MM-based FDI algorithm for
the entire operational regime. In the MM-based approach [21]–
[23], the hypothesis conditional probability Pj(k)is defined as
the probability that aas a fault parameter assumes the value
of aj(for j= 1, . . . , M + 1) conditioned on the observed
measurement history up to the kth sample, that is: Pj(k) =
Pr[a=aj|Y(k) = Yk],where Y(k)is the measurement
history random variable with Y(1), Y (2), . . . , Y (k)partitions
displaying the available measurements up to the kth sample
time. Similarly, Ykis the measurement history vector that
has the partitions of Y1, Y2, . . . , Yk. Therefore, the conditional
probability can be computed recursively as follows:
Pj(k) = fy(k)|a,Y (k−1)(yj|aj, Yk−1)Pj(k−1)
PM
h=1 fy(k)|a,Y (k−1)(yh|ah, Yk−1)Ph(k−1) ,
(12)
where fy(k)|a,Y (k−1)(yj|aj, Yk−1)is the Gaussian density
function for the current measurement as given by:
fy(k)|a,Y (k−1)(yj|aj, Yk−1) = 1
(2π)q/2rSj
c(k)
×
exp[−1
2(γj
c(k))T(Sj
c(k))(−1)(γj
c(k))] (13)
for j= 1, . . . , M + 1, where γj
c(k)and Sj
c(k)denote the
innovation vectors and the covariance matrices of a bank
of (M+ 1) combined models that are associated with the
fault parameters. If the fault parameter vector has the value
aj, the probability of the jth model will be larger than the
others since its corresponding residual vector and also the
determinant of the residual covariance matrix will be much
smaller than those that are predicted by the other filters and
which are mismatched with the assumed fault scenario. Hence,
the condition of the system and the location of a faulty sensor
can be detected and isolated based on evaluating the Pj(k)
and finding its maximum value. Consequently, the above MM-
based approach is capable of detecting and isolating various
kinds of faults. Fig. 1 shows the structure of the MM-based
FDI algorithm that uses the MHKF for the corresponding
multiple operating points.
V. SIMULATION RESULTS
In this section, simulation results and performance evalua-
tion of the proposed sensor fault diagnosis scheme correspond-
ing to different fault scenarios are presented for a nonlinear
mathematical model of a commercial single spool jet engine,
previously developed in [21]. This model is generated based
on the rotor and volume dynamics behavior and is validated by
the GSP 10 software [24]. For our study, this model is extended
to the entire flight profile and is simulated in SIMULINK to be
used as both the real engine and the OBEM. The real engine
operates at a given health condition, while the OBEM health
parameters are periodically updated to their recently estimated
values which are assumed to be already estimated by an off-
line trend monitoring solution. The set of nonlinear state-space
equations that are used for the engine are as follows (refer to
the Nomenclature section of [21] for physical meaning and
definition of the variables):
˙
TCC =1
cvmCC
[(cpTC˙mC+ηCCHu˙mf−cpTCC ˙mT)
−cvTCC( ˙mC+ ˙mf−˙mT)],
˙
N=ηmech ˙mTcp(TCC −TT)−˙mCcp(TC−Td)
JN (π
30 )2,
˙
PT=RTM
VM
( ˙mT+β
β+ 1 ˙mC−˙mn),
˙
PCC =PCC
TCC
˙
TCC +γRTCC
VCC
( ˙mC+ ˙mf−˙mT),(14)
where TCC,N,PTand PCC denote the combustion
chamber temperature, the shaft rotational speed, the turbine
pressure and the combustion chamber pressure, respectively.
There is also one actuator that supplies the fuel flow ( ˙mf)
as well as five sensors measuring [TC, PC, N, TT, PT],
where TC=Tdh1 + 1
ηC[(PCC
Pd)γ−1
γ−1]i,TT=
TCC h1−γT[1 −(PT
PCC )γ−1
γ]iand PC, denoting the
compressor temperature, turbine temperature and pressure,
respectively. Also [γC, γT,˙mC,˙mT]is the health parameter
vector. The flight condition is defined by two environmental
variables, namely the altitude and the Mach number. The
ambient temperature and pressure can be computed according
to the following equations:
Tamb =Ts−6.5h
1000, Pamb = 0.8 + Psexp(−gM h
288R),(15)
where Ts= 288 ◦Kand Ps= 1.01325 atm. are set to the
standard condition, R= 8.31447 is a constant and gis the
gravitational acceleration, Mand hindicate the altitude and
the Mach number, respectively. The ambient parameters can be
measured by the associated sensors and applied to the OBEM.
Also, the same control input is applied to both the actual
engine and the OBEM. The system is simulated for 520 (sec)
with the sampling rate of 0.01 sec. The profiles of the altitude,
Mach number and the fuel flow rate are shown in Fig. 2.
In this paper, the FDI scheme is implemented for the entire
flight profile including the take-off or climbing, cruise and
the landing. Therefore, according to the overall structure of
our proposed FDI algorithm in Fig. 1, six PWL models are
generated at each operating point for the five faulty sensor
modes as well as one healthy mode when the injected bias
fault is 3% of its nominal value. Consequently, the state
variables and the sensor outputs are estimated by means of the
MHKF. Finally, six general combined matrices are calculated
through the use of (11) and are applied in the MM-based
structure to detect and isolate sensor bias faults that occur
at different points of the flight profile. For our simulations,
there are five operating points that are assumed for handling
the entire flight profile. This is the minimum number of PWL
models for each faulty mode which prevents the occurrence of
false alarms in the range of the applied health degradations.
TABLE I shows the corresponding fuel flow rates as well as
the flight conditions for all the applied PWL models that are
designed for each sensor mode. In order to efficiently track
the variations of the system input during the climbing and
landing situations, two PWL models are derived for each of
these two conditions; while only one PWL model is derived
for the cruise condition corresponding to the applied constant
input.
During fault detection and isolation processes, a mode
probability is generated for each combined model using (12).
Normally, when the sensors are healthy, the first mode proba-
bility related to the healthy mode is at the maximum. Once a
fault has occurred, the healthy mode probability is decreased
and the mode probability for the corresponding occurred fault
is increased until it becomes the maximum among all the mode
probabilities.
According to Section I, the parameter degradations due to
the engine aging is one of the factors that should be considered
for an efficient FDI strategy. Therefore, in all the simulation
results the performance of the FDI strategy is evaluated in
OBEM
+
++
+
+-+
-
()
()
(,)()
(,)(+)
,()
.()
(,)
Offline PWL Kalman Filter
,(),,()
,(),,()
,(),,()
Bayesian Weight Estimation
+
+
++
Combination of
L linear Models
for jth Sensor
Mode
(,())
=,()
Conditional
Hypothesis
Probability
Evaluator
MAX
Fault
Detection
&
Isolation
Ai
W(k)
(),()
(),()
(+)(),(+)()
(),()
PWL(i , j)
Model from
Look - up Table by the
dimension of (L , M+1)
,(),,()
+
u(k)
Y(k)
(,)
(,)
,
=,()
(,)
(,)
,
()
Fig. 1: MM-based FDI scheme that uses the HKF corresponding to multi-operating regime.
(a) (b) (c)
Fig. 2: Profiles of the fuel flow rate (a), altitude (b) and the Mach number (c) during a flight mission.
TABLE I: The operating point specifications corresponding to
the designed PWL models.
Flight Condition Fuel Flow Rate Mach Num. Altitude
(Kg/m2) (ft)
Model 1 0.38 0.2109 4070.538
Model 2 0.38 0.6585 12708.33
Model 3 0.25 0.85 16404.2
Model 4 0.3 0.5402 10424.87
Model 5 0.3 0.1203 2322.835
presence of the compressor’s or the turbine’s health parameters
degradations. Normally, there is a difference between the real
and the estimated health degradations due to the parameter
estimation approach error. The robustness of the FDI algorithm
and the HKF structure will be evaluated subsequently in this
section with respect to the percentage of the health parameters
estimation errors.
A. Case 1: False Alarm Evaluation
Many factors such as dynamic mismatch between the
OBEM and the real engine or a large difference between
the OBEM and the real engine’s percentage of parameters
degradations may lead to false alarm flags. In order to evaluate
the efficiency of the FDI scheme in terms of preventing false
alarms, our proposed algorithm is implemented over the entire
flight profile as shown in Fig. 2, while the compressor’s and
the turbine’s health parameters degradations are updated for
the OBEM. The robustness of the FDI scheme in terms of the
maximum percentage of parameters degradations estimation
errors has also been considered. The estimation error is due to
the difference between the percentage of the health parameter
degradations for the real engine and the OBEM. The FDI
algorithm is robust to the maximum value of 3% for the
compressor’s health parameters estimation error while this
value is 2% for the turbine’s parameters estimation error. The
algorithm has declared no false alarms in the range of the
above estimation errors, however the chance of false alarms
will be increased if the estimation errors are further increased.
This implies that, if for example, the compressor’s mass flow
rate and the efficiency are degraded by 4% in the real engine,
the same parameters should be degraded by at least 1% for
the OBEM; otherwise, a false alarm will be declared.
In this scenario, the mode probability of the healthy sen-
sors is approximately one whereas the other probabilities
corresponding to the faulty sensor scenarios are all almost
zero during the entire flight profile. Fig. 3 depicts the mode
probabilities for the healthy sensors in presence of health
degradations.
Fig. 3: Mode probabilities for the healthy sensors in presence
of 3% or 2% of health degradation estimation errors for the
compressor and the turbine parameters, respectively.
B. Case 2: 3% Sensor Bias Fault Diagnosis
In this section, the performance of the sensor FDI scheme
with respect to the fault detection time and the robustness
of the algorithm to the percentage of parameter degradation
estimation errors are evaluated. This evaluation is conducted
during the entire flight profile which lasts for 520 seconds,
when a preset bias fault with the severity of 3% from its nom-
inal value has occurred for a single sensor. TABLE II shows
the fault detection time for each single sensor fault scenario
at different stages of the flight profile including the climbing,
cruise and landing modes as well as the maximum limit of
the estimation errors of the health parameters degradations for
which the system is still robust corresponding to less than
8 seconds of fault detection time. Furthermore, Fig. 4 depicts
the mode probabilities for three selected faulty scenarios when
the bias fault occurs at different instants of the flight profile.
The fault is 3% of the sensors’ nominal values, while the
compressor’s parameter degradation estimation errors are set
at the maximum level as indicated in TABLE II.
As discussed earlier, the estimation error is actually the
difference between the percentages of the health parameters
degradations of the actual engine and the OBEM. By increas-
ing the estimation errors a larger mismatch between the actual
engine and the OBEM is obtained, the detection time will be
increased, and there will also be a chance of a false alarm flags.
Moreover, the fault detection time could be highly decreased
by assuming a lower percentage of the health degradations or
a more accurate off-line trend monitoring system.
By comparing the results in TABLE II, it can also be
concluded that the sensor fault detection time during the cruise
mode is much less than that of the other flight modes since
there is less variation of thrust and flight ambient conditions. In
spite of a large input and ambient condition variations during
the climbing and the landing modes, it is still possible to detect
a sensor fault by applying the HKF scheme as well as the PWL
models interpolation or soft switching. In order to show this
capability, the sensor faults occur at tf= 50 (sec) during the
climbing mode, at tf= 250 (sec) during the cruise mode and
also at tf= 450 (sec) during the landing mode. The fault
detection times are also indicated in TABLE II for each faulty
scenario.
C. Case 3: Sensor Fault Diagnosis for Different Severities
In real applications there is no guarantee that the sensor
fault severity always matches the predefined level of 3%.
Therefore, it is essential to investigate the performance of the
MM-based FDI scheme for an injected sensor bias fault having
different severities starting from the minimum detectable bias
values and not matched with the predefined level of fault
severity with the designed PWL models. Fig. 5 shows the
fault detection times as a function of fault severity for each
of the five faulty modes. In this figure all the sensor faults
occur at tf= 250 (sec) during the cruise mode. TABLE III
also shows the average detection times for all the faulty modes
as a function of the fault severities, while the injected faults
occur at different stages of the flight profile. Corresponding
to all the above simulations, the estimation errors for the
health parameters degradations are at the maximum level of
the compressor health degradation as indicated in TABLE II.
It can be observed from TABLE III that the higher the
TABLE III: The average sensor fault detection times for all
fault modes as a function of the fault severity at different
stages of flight profile, considering the maximum level of the
compressor’s parameters degradation estimation errors.
Fault Occurrence Time 2% 3% 4% 5% 6%
tf= 50 sec 38.58 5.68 4.52 4.12 3.94
tf= 250 sec 5.12 3.64 3.62 3.68 3.66
tf= 450 sec 18.76 4.86 4.64 4.48 4.38
Fig. 5: Fault detection times as a function of the fault severity
for the five sensor faulty modes considering the maximum
level of compressor parameters degradation estimation errors.
fault severity the earlier the detection time. Also, the minimum
detectable sensor bias fault is 2% and requires more time to be
detected as compared to the higher fault severities, especially
during the climbing and the landing flight modes due to the
rapid variations of the engine thrust and operational conditions.
VI. CONCLUSIONS
In this paper, a sensor fault detection and isolation algorithm
for the gas turbine engine is proposed based on the application
of the hybrid kalman filter (HKF) approach as well as the
multiple-model based framework. Despite the use of linear
Kalman filters, the HKF is capable of capturing the nonlinear-
ities of the system by integrating a nonlinear on-board engine
model (OBEM) with the piecewise linear (PWL) models to
cover the entire operating range of the engine. Compared
to the PWL Kalman filters, our proposed approach requires
less number of PWL models with a larger feasible operating
range due to the replacement of the steady-state values by the
OBEM state variables and outputs. Moreover, the HKF has
similar numerical stability and robustness as the PWL Kalman
filters, since the OBEM does not receive any feedback from
the estimation process which may lead to instability due to
presence of large estimation errors.
Another important contribution of this work is in the inclu-
sion of the effects of health parameters degradations in our
proposed sensor FDI scheme through updating them offline to
their estimated values. This enables one to prevent a false
alarm flags due to the engine health parameters deviations
from their nominal values which may lead to large mismatches
between the OBEM and the actual engine. Therefore, unlike
most of the previous works in the literature which have not
considered the influence of the engine deterioration on the FDI
performance as well as the risk of a fault occurrence during the
0100 200 300 400 500 600
0
0.5
1
1.2
Time (Sec)
Mode Probability
M#1 M#2 M#3 M#4 M#5 M#6
0100 200 300 400 500 600
0
0.5
1
1.2
Time (Sec)
Mode Probability
M#1 M#2 M#3 M#4 M#5 M#6
0100 200 300 400 500 600
0
0.5
1
1.2
Time (Sec)
Mode Probability
M#1 M#2 M#3 M#4 M#5 M#6
(a) (b) (c)
Fig. 4: Mode probabilities for 3% bias fault injected at tf= 50 (sec) to the TCsensor (a), at tf= 250 (sec) to the PCsensor
(b) and at tf= 450 (sec) to the Nsensor (c) in presence of the maximum level of compressor’s health parameters degradation
estimation errors.
TABLE II: Sensor fault detection time (FDT) corresponding to the maximum acceptable limit on health parameters degradation
estimation errors (HPDEE) at different stages of the flight profile.
Faulty Scenario HPDEE% Sensor FDT (Sec)
tf= 50 tf= 250 tf= 450
Fault on TCSensor Compressor ∆ ˙mC= 2.5 ∆ηC= 2.5 3.7 4.1 6.3
Turbine ∆ ˙mT= 2.5 ∆ηT= 1.5 3.5 5.1 5.9
Fault on PCSensor Compressor ∆ ˙mC= 1 ∆ηC= 1 3.4 5.9 8
Turbine ∆ ˙mT= 1 ∆ηT= 0.3 3.4 2.7 2.5
Fault on NSensor Compressor ∆ ˙mC= 1.2 ∆ηC= 2 6.5 2.6 3.5
Turbine ∆ ˙mT= 1 ∆ηT= 0.8 6.5 2.5 2.9
Fault on TTSensor Compressor ∆ ˙mC= 0.2 ∆ηC= 0.2 7.8 3 3.1
Turbine ∆ ˙mT= 0.1 ∆ηT= 0.1 6 2.2 2.3
Fault on PTSensor Compressor ∆ ˙mC= 0.3 ∆ηC= 0.3 7 2.6 3.4
Turbine ∆ ˙mT= 0.3 ∆ηT= 0.1 8 2.2 2.2
climbing and landing modes, our proposed sensor FDI scheme
is sufficiently efficient to be used for the entire flight mission
profile during the engine life cycle through interpolating and
combining PWL models.
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