A Prognostic and Health Management Framework for Aero-Engines Based on a Dynamic Probability Model and LSTM Network

Abstract

In this study, a prognostics and health management (PHM) framework is proposed for aero-engines, which combines a dynamic probability (DP) model and a long short-term memory neural network (LSTM). A DP model based on Gaussian mixture model-adaptive density peaks clustering algorithm, which has the advantages of an extremely short training time and high enough precision, is employed for modelling engine fault development from the beginning of engine service, and principal component analysis is introduced to convert complex high-dimensional raw data into low-dimensional data. The model can be updated from time to time according to the accumulation of engine data to capture the occurrence and evolution process of engine faults. In order to address the problems with the commonly used data driven methods, the DP + LSTM model is employed to estimate the remaining useful life (RUL) of the engine. Finally, the proposed PHM framework is validated experimentally using NASA’s commercial modular aero-propulsion system simulation dataset, and the results indicate that the DP model has higher stability than the classical artificial neural network method in fault diagnosis, whereas the DP + LSTM model has higher accuracy in RUL estimation than other classical deep learning methods.

Full text

Citation: Huang, Y.; Tao, J.; Sun, G.;
Zhang, H.; Hu, Y. A Prognostic and
Health Management Framework for
Aero-Engines Based on a Dynamic
Probability Model and LSTM
Network. Aerospace 2022,9, 316.
https://doi.org/10.3390/
aerospace9060316
Academic Editor: Ernesto Benini
Received: 16 April 2022
Accepted: 8 June 2022
Published: 10 June 2022
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aerospace
Article
A Prognostic and Health Management Framework for
Aero-Engines Based on a Dynamic Probability Model and
LSTM Network
Yufeng Huang 1, Jun Tao 1, *, Gang Sun 1, Hao Zhang 1and Yan Hu 2
1Department of Aeronautics & Astronautics, Fudan University, Shanghai 200433, China;
yfhuang19@fudan.edu.cn (Y.H.); gang_sun@fudan.edu.cn (G.S.); haozhang20@fudan.edu.cn (H.Z.)
2AECC Commercial Aircraft Engine Co., Ltd., Shanghai 200241, China; icegull007@163.com
*Correspondence: juntao@fudan.edu.cn
Abstract:
In this study, a prognostics and health management (PHM) framework is proposed for
aero-engines, which combines a dynamic probability (DP) model and a long short-term memory
neural network (LSTM). A DP model based on Gaussian mixture model-adaptive density peaks
clustering algorithm, which has the advantages of an extremely short training time and high enough
precision, is employed for modelling engine fault development from the beginning of engine service,
and principal component analysis is introduced to convert complex high-dimensional raw data into
low-dimensional data. The model can be updated from time to time according to the accumulation
of engine data to capture the occurrence and evolution process of engine faults. In order to address
the problems with the commonly used data driven methods, the DP + LSTM model is employed
to estimate the remaining useful life (RUL) of the engine. Finally, the proposed PHM framework
is validated experimentally using NASA’s commercial modular aero-propulsion system simulation
dataset, and the results indicate that the DP model has higher stability than the classical artificial
neural network method in fault diagnosis, whereas the DP + LSTM model has higher accuracy in
RUL estimation than other classical deep learning methods.
Keywords: dynamic probability (DP); prognostics and health management (PHM); long short-term
memory (LSTM); remaining useful life (RUL)
1. Introduction
Aero-engines are core machinery systems with complex structures, high levels of
integration and poor working conditions, of which the reliable and efficient operations
are crucial to the flight safety of aircraft. Prognostics and health management (PHM) is an
effective maintenance technique to achieve safe and reliable operations of machines and
systems, and plays a significant role in the operations of aero-engines [
1
–
3
]. Incomplete
statistics showed that failures of gas path components account for more than 90% of all
engine failures, 60% of the aero-engine maintenance costs are spent on gas-path compo-
nents [
4
]. However, due to the unique manufacturing technologies and special materials
of the aero-engine, it is difficult to maintain and replace the components of engines fre-
quently [
5
]. The PHM system is capable of determining whether a gas-path component has
failed and deciding whether it needs to be repaired or replaced, thus can reduce routine
maintenance costs and time. Fault diagnosis and remaining useful life (RUL) estimation
are research emphases of PHM.
In general, PHM approaches can be categorized into model-based methods [
6
,
7
] and
data-driven methods [
8
,
9
]. Model-based methods include physical models, structural
analysis, contact analysis cumulative damage models, cyclic fatigue, and crack propaga-
tion models, etc., [
10
]. Obviously, they need a detailed mathematical model of the aero-
engine [
11
]. In addition, their reliability decreases as the system nonlinearities, complexity,
Aerospace 2022,9, 316. https://doi.org/10.3390/aerospace9060316 https://www.mdpi.com/journal/aerospace

Aerospace 2022,9, 316 2 of 21
and modeling uncertainties increase. Data-driven methods can be roughly divided into
two categories, namely machine learning algorithms and probability models, and they do
not require deep knowledge of the engine mechanism, and mostly depend on real-time or
collected historical data from the engine sensors and measurements, so they have attracted
considerable attention and have been developed rapidly. Commonly used data-driven
methods include artificial neural network (ANN), support vector machine (SVM), k-means
clustering algorithm, Bayesian method, Markov model, Gaussian distribution, etc. [
12
–
18
].
The operation and external conditions of aero-engine change over time and, therefore,
the time-varying problems, have become the main challenge. Machine learning can be
used for fault diagnosis, but it is not flexible enough to deal with time-varying problems
and is difficult to update as data accumulates. The traditional ANN is also known as the
black box model. Its construction process does not reflect the actual operation law of the
engine. In addition, it has the limitations of weak generalization ability and difficulty in
dealing with time-varying problems, etc. Chen et al. [
1
] proposed a new deep learning
method called deep belief network (DBN) for engine fault diagnosis. Compared with the
traditional back propagation (BP) model, it has been greatly improved, but its essence
is still ANN, which has the above-mentioned drawbacks. Compared with ANN, the
probability model has unique advantages in dealing with time-varying problems due
to its solid mathematical background. The Gaussian mixture model (GMM) is a typical
probability model, which can fit the fault monitoring features (FMFs) of random distribution
by a combination of a finite number of Gaussian components (GCs) [
19
]. Avendaño
Valencia et al. [
20
] proposed a stochastic framework based on the Gaussian mixture random
coefficient model for structural health state monitoring under time-varying conditions,
and their results showed that GMM has great flexibility in dealing with time-varying and
uncertain problems. Qiu et al. [
21
] proposed an enhanced dynamic Gaussian mixture
model-based damage monitoring method for aircraft structural health monitoring (SHM).
Fang et al. [
22
] proposed a probability modeling-based aircraft structural health monitoring
framework under time-varying conditions.
However, the probability model is rarely used in aero-engine PHM systems, especially
the GMM model. The difficulty of applying the probability model to aero-engines lies in
the data of aero-engine contain noise, which is much more complex than those of other
objects such as aircraft structural analysis [
23
]. In addition, the biggest disadvantage of
the traditional GMM model is that the initial values have a great influence on the result,
and manual selection is required. At present, the most common improvement is to use the
k-means clustering algorithm [
24
], but it is still unable to achieve complete self-adaptation.
A new method called adaptive density peaks clustering algorithm (ADPC) can solve these
problems and realize adaptive initial clustering. Another difficulty in aero-engine fault
diagnosis lies in the difficulty of obtaining a large amount of failure data for an engine.
Most of the engine’s life cycle is in the non-failure state, and it is a gradual process for an
engine from health to failure. Therefore, it is necessary to design a dynamic model that
makes full use of the normal data and can be updated as the data accumulates.
RUL estimation is another focus in the PHM framework. Data-driven approaches
are typical algorithms for RUL estimation. Soualhi et al. [
25
] developed a data-driven
approach for bearing RUL prediction using the Hilbert–Huang transform (HHT) and the
SVM. Li et al. [
26
] proposed a smooth transition auto-regression model combined with
the Bayesian model to estimate the RUL. Listou et al. [
27
] proposed a semi-supervised
learning method for RUL prediction, which reduced the amount of marker training data.
However, these methods also suffer from some deficiencies. For instance, the imperfection
of expert knowledge may cause the handcrafted feature to fail to effectively reflect the
engine degradation, and these methods do not propose a good solution mechanism for the
utilization of historical data and current data. In addition, the prediction accuracy of these
methods is not optimal.
In the construction of the PHM framework, Che et al. [
1
] proposed a framework
combining DBN and long short-term memory neural network (LSTM) methods. In his

Aerospace 2022,9, 316 3 of 21
framework, fault diagnosis and RUL estimation are not deeply linked, and health indicators
are not fully utilized in RUL estimation, which makes it necessary to mine information from
engine sensor data again before RUL estimation, which is not the most efficient. Li et al. [
28
]
proposed a framework for deriving system requirements for PHM system development
to provide a solution for predicting RUL. Similarly, the framework does not consider a
technical route that combines fault diagnosis with RUL estimation.
Given the above, an aero-engine PHM framework based on GMM-ADPC algorithm
and LSTM network is proposed in this study. In this study, a new GMM-ADPC algorithm
is proposed to construct probability distribution space of engine data. Based on the GMM-
ADPC algorithm, a dynamic probability (DP) model is proposed for modeling engine fault
development. This model has a solid mathematical foundation and can make full use of
engine life cycle data. And principal component analysis (PCA) is used to convert complex
high-dimensional raw data into low-dimensional data. For the purpose of addressing
the problems with the commonly used data-driven methods, the DP + LSTM model
is introduced for RUL estimation. Here, the engine fault probability distribution data
constructed by the DP model is used as the input of the LSTM network, which realizes the
information transmission between the two modules, avoids sensor noise interference to a
certain extent, and improves the stability and accuracy of the PHM framework.
The rest of this paper proceeds as follows. Section 2introduces the DP model and
LSTM algorithms. Section 3details the architecture and the realization of the framework.
Section 4provides the validation results of the framework in NASA’s dataset. Finally, the
conclusion of this work is given in Section 5.
2. PHM Basic Theory
2.1. Probability Modeling
The core algorithm of the probability model is the GMM-ADPC algorithm, and the
probability difference measuring method is used to quantify the difference between two
probability models so as to generate fault detection indexes.
2.1.1. GMM-ADPC Algorithm
GMM is an extension of single Gaussian probability density function. It is a weighted
sum of a finite number of GCs. Assume
X=[x1,x2,···,xi,···,xN]
denote a feature sample
set composed by NFMFs, i= 1, 2,
. . .
,N, where
xi=[x1,x2,···,xD]
represents a FMF with
D dimensionality. Equation (1) expresses the probability density function of GMM and
Equation (2) expresses the GC.
Φ(xi|ζ)=
K
∑
k=1
wkϕk(xi|µk,Σk)(1)
ϕk(xi|µk,Σk)=1
(2π)D
2|Σk|1
2
e−1
2(xi−µk)TΣk−1(xi−µk)(2)
where
ζ={(w1,µ1,Σ1),···,(wk,µk,Σk),···,(wK,µK,ΣK)}
is the most important parame-
ter set of GMM. The number of GCs is Kand
k=
1, 2,
···
,
K
. The parameter
ζ
,
wk
,
µk
and
Σk
denote the mixture weight, mean, and covariance matrix of the k-th GC, respectively,
|·|is the determinant value, and Tis the transpose.
Usually, the Expectation-Maximization (EM) algorithm is used to construct GMM [
29
].
However, the drawback of the EM algorithm is that the initial values of
ζ
will greatly
affect the result, which results in reduced stability of GMM. Some methods, such as the
Bayesian non-parametric clustering approach and enhanced dynamic GMM method, have
been proposed to determine
ζ
[
21
,
30
]. However, the ideal approach is adaptive and not
computationally intensive. In addition, since each sample set belongs to a different FMF,
the selected method is required to have good generalization performance. In fact, GMM
represents the probability distribution of FMFs, with each GC representing a cluster. ADPC
is an improved clustering algorithm based on probability density distribution [
31
,
32
]. The

Aerospace 2022,9, 316 4 of 21
main advantage of the ADPC algorithm is that it could effectively identify clustering centers
and cut-off distances with low-dimensions or arbitrary data sets. The ADPC algorithm
contains the following two main steps:
Step 1: Automatic identification of the cut-off distance.
First, define a variable Hto represent the uncertainty of the system expressed as
Equations (3)–(5). If the values of Hare smaller, the uncertainty of the system will be
smaller, which is in favor of clustering.
H=−
n
∑
i=1 δi∑
j
e−(dij
dc)
2!log δi∑
j
e−(dij
dc)
2!(3)
δi=min
ρi<ρjdij (4)
ρi=∑
j
e−(dij
dc)
2
(5)
where
dij
is the distance between FMF
xi
and FMF
xj
,
ρi
is the local density of FMF
xi
. Make
the
dc
gradually increase from 0 until Hhas the minimum value, in which case
dc
is the
most appropriate cut-off distance.
For some samples, it is difficult to find the cut-off distance that meets the above
requirements. In this case,
dc
can be set as the top 1% to 2% of the distance between all data
points [31].
Step 2: Automatically identify clustering centers.
Clustering centers should have both large
ρi
and
δi
values. Define a variable
γ
ex-
pressed as Equation (6). Sample points with larger
γ
values are more suitable for cluster-
ing centers.
γi=ρiδi(6)
In addition, the number of cluster centers needs to be determined. Firstly, calculate
the
γ
value of each FMF and sort them. Let
tendi
be a criterion for determining the number
of cluster centers, and tendiexpressed as Equation (7).
tendi=(i−1)γi−1−γi
γi−γi+1
(7)
Then, select the nFMFs with the largest
γ
value, and calculate the
tendi
value for each
FMF. If the
tendi
value of the m-th FMF is the largest, then the former m
−
1 FMFs are taken
as the clustering center.
After the initial clustering is completed, the mean, covariance, and weight of FMFs
belonging to each cluster can be obtained and can be used as the initial value
ζ
of the
EM algorithm.
2.1.2. Probability Difference Measuring Method
In this paper, two probability models are constructed, as detailed in the following sec-
tions. Appropriate rules for quantifying the difference between the two models need to be
determined. Some methods such as Renyi divergence and Kullback-Leibler divergence [
33
]
have been proposed to measure the difference. However, these methods are not symmetric
and normalized. Qiu et al. [
21
] used the Monte Carlo simulation method in probability simi-
larity measuring and achieved good results [
34
]. Firstly, let
XMC ={x1,x2,···,xR}
denote a
large number of random samples that are generated by Monte Carlo sampling. Secondly, the
posterior probability of
XMC
, is denoted as
P(XMC |ζ)={Φ(x1|ζ),Φ(x2|ζ),···,Φ(xR|ζ)}T
,
which can be calculated by Equations (1) and (2). Finally, the difference between the two

Aerospace 2022,9, 316 5 of 21
probability models can be calculated by Equation (8). In this paper,
Di f f (ζ1,ζ2)
actually
denotes the fault detection indexes.
Di f f (ζ1,ζ2)=1−P(XMC|ζ1)TP(XMC |ζ2)T
kP(XMC |ζ1)k · kP(XMC |ζ2)k(8)
2.2. Long Short-Term Memory Networks
LSTM model based recurrent neural network (RNN) can adaptively learn the rep-
resentative information through multiple non-linear transformations [
35
–
37
]. Compared
with the traditional ANN, LSTM can remember all the historical information entered and is
suitable for dealing with time-varying problems. Compared with RNN, LSTM has been im-
proved in two main aspects. First, in order to solve the limitation of information forgetting,
the cell state is split into the short-term state
ht
and the long-term state
ct
. Second, the cell
states are regulated by three control gates, the forget gate, the input gate, and the output
gate [38]. The architecture of LSTM can be described in Figure 1.
Aerospace 2022, 9, x FOR PEER REVIEW 5 of 21
to be determined. Some methods such as Renyi divergence and Kullback-Leibler diver-
gence [33] have been proposed to measure the difference. However, these methods are
not symmetric and normalized. Qiu et al. [21] used the Monte Carlo simulation method
in probability similarity measuring and achieved good results [34]. Firstly, let
 
12
, , ,
MC R
= Xx x x
denote a large number of random samples that are generated by
Monte Carlo sampling. Secondly, the posterior probability of
MC
X
, is denoted as
( ) ( ) ( ) ( )
 
12
, , , T
MC R
P
   
=    Xx x x
, which can be calculated by Equations (1)
and (2). Finally, the difference between the two probability models can be calculated by
Equation (8). In this paper,
( )
12
,Diff

actually denotes the fault detection indexes.
( )
( ) ( )
( ) ( )
12
12
12
,1
TT
MC MC
MC MC
PP
Diff PP

 
=− 
XX
XX
(8)
2.2. Long Short-Term Memory Networks
LSTM model based recurrent neural network (RNN) can adaptively learn the repre-
sentative information through multiple non-linear transformations [35–37]. Compared
with the traditional ANN, LSTM can remember all the historical information entered and
is suitable for dealing with time-varying problems. Compared with RNN, LSTM has been
improved in two main aspects. First, in order to solve the limitation of information forget-
ting, the cell state is split into the short-term state
t
h
and the long-term state
t
c
. Second,
the cell states are regulated by three control gates, the forget gate, the input gate, and the
output gate [38]. The architecture of LSTM can be described in Figure 1.
Figure 1. Building block of long short-term memory (LSTM) network [1].
A typical LSTM is illustrated in Figure 1, and the hidden layer contains three gates:
forget gate, input gate, and output gate. The functions of these three gates are: information
forgetting, long-term state updating, and short-term state updating.
1. Information forgetting. The states removed from the previous long-term state
1t−
c
are controlled by the forget gate
t
f
. The
t
f
can be described by Equation (9).
Figure 1. Building block of long short-term memory (LSTM) network [1].
A typical LSTM is illustrated in Figure 1, and the hidden layer contains three gates:
forget gate, input gate, and output gate. The functions of these three gates are: information
forgetting, long-term state updating, and short-term state updating.
1.
Information forgetting. The states removed from the previous long-term state
ct−1
are
controlled by the forget gate ft. The ftcan be described by Equation (9).
ft=σwf·[ht−1,xt]+bf(9)
where
σ
is the sigmoid function,
xt
is the previous current time,
wf
is the weight
vectors, bfis the bias term of the forget gate, and “·” means matrix multiplication.
2.
Long-term state updating. The input gate layer determines what values will be
updated. The input gate
it
and candidate value vector
c0
t
are expressed by Equations
(10) and (11).
it=σ(wi·[ht−1,xt]+bi)(10)
c0
t=Tanh(wc·[ht−1,xt]+bc)(11)
where (wi,wc)are the weight vectors, and (bi,bc)are bias terms.

Aerospace 2022,9, 316 6 of 21
Then, the new long-term cell state ctcan be obtained by Equation (12).
ct=ft⊗ct−1⊕it⊗c0
t(12)
where (⊗,⊕)are element-wise multiplication and addition.
3.
Short-term state updating. The function of the output gate is to change the long-term
state to the short-term state. Equation (13) describes the output gate ot.
ot=σ(wo·[ht−1,xt]+bo)(13)
Finally, the short-term state of the cell unit at time tcan be described as Equation (14).
ht=ot⊗Tanh(ct)(14)
In this study, LSTM implements the prediction of sequential to point, the dimension of
the input sequence is 10. The loss function is mean squared error (MSE), and the optimizer
is the ADAM algorithm, which is an extension of the gradient descent algorithm.
3. Design of the Aero-Engine PHM Framework
3.1. DP Model for Fault Monitoring
Data collected by aero-engine sensors vary over time and contain noise [
39
]. Further-
more, aero-engines are designed based on failure-tolerance, which means that the engine
will keep a healthy state in the early stage of the engine, and the influence of fault is minimal
and even far lower than the time-varying influence. As time goes on, the influence of the
fault becomes greater and greater until the engine is unable to function properly. Therefore,
it is necessary to find a method that can not only eliminate the influence of noise but also
capture the accumulation of engine fault. Generally, the operating state of the engine cannot
be directly reflected by sensor data at a certain moment. Deriving the engine state from the
physical meaning of the data itself is difficult and complex. Therefore, the core idea of the
proposed framework is to construct the probability distribution of engine life cycle data,
which is dynamically updated. Different health states necessarily correspond to different
probability distributions. Double probability models are constructed to represent the engine
health state and the health monitoring state, respectively, and the monitoring probability
model must be updatable so as to reveal the progressive variation trend. Once the double
probability models are constructed, the engine fault can be quantified by comparing the
difference between the two probability models. It should be noted that this method is
proposed on the assumption that the time-varying influences of the two states are the same.
In addition, the DP model is designed as standardized architecture. In the PHM
field, some model-driven methods such as Kalman Filtering, particle filter [
40
], and so
on are all aimed at fixed objects. When the engine model is different, the PHM model
needs to be modified. Some data-driven methods such as ANN, SVM, and so on also have
limited generalization capability [
14
,
15
]. In contrast to these methods, the DP model can be
designed as standardized architecture that is suitable for different engine models, because
the DP model is updated dynamically with the accumulation of engine data, so it does not
require much prior knowledge or complex model parameter adjustment, and considering
the inevitability of data transfer in the framework, the proposed PHM framework in this
paper will use a generalized data interface between the parts. In addition, the input and
output data of the framework are normalized. Another significant advantage of the DP
model is that it does not need training as ANN does, so this method is more efficient
than ANN.
3.2. Combining the DP Model and LSTM for the PHM Framework
The modular hierarchical structure is a prominent feature of the proposed PHM
framework, and the framework contains four blocks, as shown in Figure 2. The first step
of the framework is to obtain sensors data for the entire life cycle and RUL information

Aerospace 2022,9, 316 7 of 21
of the engine, which is large and contains noise. Therefore, in this step, it is necessary to
clean these sensors’ data and reduce their dimensions through the PCA technique [
41
].
Then, the data after dimension reduction should be standardized. The preprocessed data is
divided into baseline data and monitoring data, which are passed into Block 1 and Block 2,
respectively. These two blocks constitute the double probability models. Block 1 constructs
the baseline probability model based on the baseline data; that is, the data under the engine
health state. For the same engine, the baseline probability model remains unchanged and
is updated for different engines. Block 2 constructs the monitoring probability model
based on real-time monitoring data, which needs to be updated in real-time. After the
construction of the double probability models, the difference (
Di f f (ζB,ζO)
, where
ζB
and
ζO
are the parameters of baseline and monitoring probability models, respectively) between
the two models can be used to evaluate the degree of engine failure, and that is what Block 3
does. In this paper, the normalized
Di f f (ζB,ζO)
are used as fault detection indexes. Block
4 is the second part of the PHM framework-RUL estimation. The large amount of fault
detection index data generated by the fault diagnosis module is taken as the training sample
of the LSTM network. In this way, the interference of sensor data noise can be avoided.
In addition, since the probability model contains the information of the entire data set,
it is difficult for the fault detection indexes to be disturbed by a very small number of
abnormal data. Therefore, the framework combining the two models has better stability.
RUL prediction can be started from any time of different engines. A threshold value can be
selected to conduct RUL evaluation according to the fault detection index curve.
Aerospace 2022, 9, x FOR PEER REVIEW 8 of 21
Figure 2. DP-LSTM modeling-based aero-engine PHM Framework.
4. Results and Discussions
In order to further evaluate the PHM model, a turbofan engine performance degra-
dation dataset, which is generated by commercial modular aero-propulsion system simu-
lation (C-MAPSS) [42], is utilized. Each example within the turbofan dataset is a time se-
ries signal of various sensor data and operating conditions data which is measured peri-
odically over the life-cycle of the turbofans [43].
4.1. Data Sets Characterization
As shown in Figure 3, a turbofan engine normally includes a fan, low pressure com-
pressor (LPC), low pressure turbine (LPT), high pressure compressor (HPC), high pres-
sure turbine (HPT), combustor, and a nozzle. The C-MAPSS data sets are multiple multi-
variate time series. Each dataset has been partitioned into training and test sample sets.
Each dataset (i.e., a 24-element vector) includes 21 characteristic sensors for engine health
data recording. With the preprocessing method, 14 sensors that are currently available
Figure 2. DP-LSTM modeling-based aero-engine PHM Framework.

Aerospace 2022,9, 316 8 of 21
4. Results and Discussions
In order to further evaluate the PHM model, a turbofan engine performance degrada-
tion dataset, which is generated by commercial modular aero-propulsion system simulation
(C-MAPSS) [
42
], is utilized. Each example within the turbofan dataset is a time series signal
of various sensor data and operating conditions data which is measured periodically over
the life-cycle of the turbofans [43].
4.1. Data Sets Characterization
As shown in Figure 3, a turbofan engine normally includes a fan, low pressure com-
pressor (LPC), low pressure turbine (LPT), high pressure compressor (HPC), high pressure
turbine (HPT), combustor, and a nozzle. The C-MAPSS data sets are multiple multivariate
time series. Each dataset has been partitioned into training and test sample sets. Each
dataset (i.e., a 24-element vector) includes 21 characteristic sensors for engine health data
recording. With the preprocessing method, 14 sensors that are currently available onboard
for many commercial turbofan engines are selected for PHM in this study [
44
]. Table 1
shows the description of selected sensors.
Aerospace 2022, 9, x FOR PEER REVIEW 9 of 21
onboard for many commercial turbofan engines are selected for PHM in this study [44].
Table 1 shows the description of selected sensors.
Figure 3. Diagram of engine in C-MAPSS.
Table 1. Fourteen selected sensors in C-MAPSS.
No.
Sensor Abbreviation
Description
Units
1
T24
Total temperature at low pressure compressor outlet
R
2
T30
Total temperature at high pressure compressor outlet
R
3
T50
Total temperature at low pressure turbine outlet
R
4
P30
Total pressure at high pressure compressor outlet
psia
5
Nf
Physical fan speed
rpm
6
Nc
Physical core speed
rpm
7
Ps30
Static pressure at high pressure compressor outlet (Ps30)
psia
8
Phi
Ratio of fuel flow to Ps30
pps/psi
9
NRf
Corrected fan speed
rpm
10
NRc
Corrected core speed
rpm
11
BPR
Bypass ratio
-
12
Ht Bleed
Burner fuel–air ratio
-
13
W31
High pressure turbine coolant bleed
lbm/s
14
W32
Low pressure turbine coolant bleed
lbm/s
After the raw data is selected, the Z-score method is used to standardize the 14 sensor
parameters. The Z-score actually reflects the relative standard distance from an element
to the mean. It can be calculated as:
( )
/zx

=−
(15)
where z is the z-score, x is the value of the element, μ is the population mean, and σ is the
standard deviation.
In this paper, four datasets (Engine #1-#4) are selected to validate the DP model, and
80 datasets (60 datasets as training samples and 20 datasets as testing samples) are selected
to validate the LSTM model.
4.2. Fault Diagnosis
This section corresponds to Block 1, Block2, and Block3 in the PHM framework dia-
gram. In Section 4.1, a high dimensional dataset containing 14 sensor parameters was ob-
tained. Because of the limitation of DP model in processing high-dimensional data, PCA
Figure 3. Diagram of engine in C-MAPSS.
Table 1. Fourteen selected sensors in C-MAPSS.
No. Sensor Abbreviation Description Units
1 T24 Total temperature at low pressure compressor outlet ◦R
2 T30 Total temperature at high pressure compressor outlet ◦R
3 T50 Total temperature at low pressure turbine outlet ◦R
4 P30 Total pressure at high pressure compressor outlet psia
5 Nf Physical fan speed rpm
6 Nc Physical core speed rpm
7 Ps30 Static pressure at high pressure compressor outlet (Ps30) psia
8 Phi Ratio of fuel flow to Ps30
pps/psi
9 NRf Corrected fan speed rpm
10 NRc Corrected core speed rpm
11 BPR Bypass ratio -
12 Ht Bleed Burner fuel–air ratio -
13 W31 High pressure turbine coolant bleed lbm/s
14 W32 Low pressure turbine coolant bleed lbm/s

Aerospace 2022,9, 316 9 of 21
After the raw data is selected, the Z-score method is used to standardize the 14 sensor
parameters. The Z-score actually reflects the relative standard distance from an element to
the mean. It can be calculated as:
z=(x−µ)/σ(15)
where zis the z-score, xis the value of the element,
µ
is the population mean, and
σ
is the
standard deviation.
In this paper, four datasets (Engine #1–#4) are selected to validate the DP model, and
80 datasets (60 datasets as training samples and 20 datasets as testing samples) are selected
to validate the LSTM model.
4.2. Fault Diagnosis
This section corresponds to Block 1, Block2, and Block3 in the PHM framework
diagram. In Section 4.1, a high dimensional dataset containing 14 sensor parameters was
obtained. Because of the limitation of DP model in processing high-dimensional data, PCA
is used to construct a two-dimensional FMF. The data of the four engines processed by PCA
is shown in Figure 4.
Aerospace 2022, 9, x FOR PEER REVIEW 10 of 21
is used to construct a two-dimensional FMF. The data of the four engines processed by
PCA is shown in Figure 4.
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
-4
-2
0
2
4
6
8
10
12 Top 25% cycles
After 75% cycles
PCA-1
PCA-2
(a) Engine #1
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
-4
-2
0
2
4
6
8
10
12 Top 25% cycles
After 75% cycles
PCA-1
PCA-2
(b) Engine #2
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
-4
-2
0
2
4
6
8
10
12 Top 25% cycles
After 75% cycles
PCA-1
PCA-2
(c) Engine #3
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
-4
-2
0
2
4
6
8
10
12 Top 25% cycles
After 75% cycles
PCA-1
PCA-2
(d) Engine #4
Figure 4. Two-dimensional PCA plot of the four engines.
As can be seen in the figure, the red dots represent the data of the top 25% cycles,
which are very concentrated, and there is little difference in the first principal component
among the data points. Based on experience, we can assume that the engine is in a healthy
state for the top 25% cycles, and the data of top 25% cycles is considered to be baseline
data. Here, 25% is a conservative estimate and does not mean that the engine will fail after
25%. Figure 4 also tells us that the data for the after 75% cycles of the engine is heavily
dispersed, which means that the operating data of the engine during this period has grad-
ually deviated from the data of the health state.
After the preprocessing of the original data is completed, the initial classification of
these data can be achieved by the ADPC algorithm, so as to obtain the initial values of

required by the GMM. Although the ADPC algorithm can adaptively identify clustering
centers and cut-off distance, the research in this paper finds that the method has limita-
tions when dealing the sample sets with small sizes. Therefore, in the early stage of engine
operation, the sample size is still small, and a limiter is added to the ADPC algorithm to
keep the number of clustering centers and cut-off distance unchanged. Therefore, a fixed
number of cluster centers and cut-off distance are used for the top 50% of the engine full
life cycles. In addition, the number of cluster centers is set between the interval [2,6], and
Figure 4. Two-dimensional PCA plot of the four engines.

Aerospace 2022,9, 316 10 of 21
As can be seen in the figure, the red dots represent the data of the top 25% cycles,
which are very concentrated, and there is little difference in the first principal component
among the data points. Based on experience, we can assume that the engine is in a healthy
state for the top 25% cycles, and the data of top 25% cycles is considered to be baseline
data. Here, 25% is a conservative estimate and does not mean that the engine will fail
after 25%. Figure 4also tells us that the data for the after 75% cycles of the engine is
heavily dispersed, which means that the operating data of the engine during this period
has gradually deviated from the data of the health state.
After the preprocessing of the original data is completed, the initial classification of
these data can be achieved by the ADPC algorithm, so as to obtain the initial values of
ζ
required by the GMM. Although the ADPC algorithm can adaptively identify clustering
centers and cut-off distance, the research in this paper finds that the method has limitations
when dealing the sample sets with small sizes. Therefore, in the early stage of engine
operation, the sample size is still small, and a limiter is added to the ADPC algorithm to
keep the number of clustering centers and cut-off distance unchanged. Therefore, a fixed
number of cluster centers and cut-off distance are used for the top 50% of the engine full
life cycles. In addition, the number of cluster centers is set between the interval [
2
,
6
], and
the difference between the number of cluster centers of two adjacent samples cannot be
more than two. The idea is to prevent violent oscillations in rare cases. The above measures
can ensure the accuracy and stability of the established model. The variation of the number
of clustering centers in the full life cycles is shown in Figure 5. This figure reflects that the
number of GCs recognized by the ADPC is changing adaptively to the changing monitoring
feature space along with the engine life cycle.
Aerospace 2022, 9, x FOR PEER REVIEW 11 of 21
the difference between the number of cluster centers of two adjacent samples cannot be
more than two. The idea is to prevent violent oscillations in rare cases. The above
measures can ensure the accuracy and stability of the established model. The variation of
the number of clustering centers in the full life cycles is shown in Figure 5. This figure
reflects that the number of GCs recognized by the ADPC is changing adaptively to the
changing monitoring feature space along with the engine life cycle.
0 2 4 6 8 100 150 200 250 30
0
2
3
4
5
6
Monitoring
Probability Model
The number of GCs
C
y
cles
Baseline
Probability Model
0 2 4 6 8 100 150 200 250 30
0
2
3
4
5
6
Monitoring
Probability Model
Baseline
Probability Model
The number of GCs
C
y
cles
(a) Engine #1 (b) Engine #2
02468 50 100 150 200 250 30
0
2
3
4
5
6
Monitoring
Probability Model
Baseline
Probability Model
The number of GCs
C
y
cles
02468 100 150 200 250 30
0
2
3
4
5
6
Monitoring
Probability Model
Baseline
Probability Model
The number of GCs
C
y
lces
(c) Engine #3 (d) Engine #4
Figure 5. The number of GCs along with the engine life cycle.
After the initial clustering of the original data using the ADPC algorithm is com-
pleted. The initial values
ζ
can be determined, and the EM algorithm is used to build
the GMM. The implementation process is shown in Figure 6.
The GMM model for the health state and monitoring state need to be constructed.
This kind of DP model is also called the dynamic double probability model. Among them,
data from the top 25% of the engine life cycles is used to construct the baseline probability
model, and this model remains unchanged in the process of engine fault diagnosis. Data
from after 75% of the engine life cycles is used to construct the monitoring probability
model, which is continuously updated with the increase of the engine life cycles. In the
fault diagnosis stage, the most important thing is to get the engine fault detection indexes,
as is shown in Figure 7. In the probability difference measuring method, the number of
Monte Carlo samples is R = 10,000. Table 2 shows the relevant parameters of the four en-
gines and the fault detection indexes in case of engine failure. It can be seen from the figure
that the fault detection indexes of the top 25% cycles are zero. This is because the engine
is in a healthy state at this stage and failure monitoring is not carried out. In the fault
monitoring stage, the fault detection index’s variation trend of the four engines is basically
Figure 5. The number of GCs along with the engine life cycle.

Aerospace 2022,9, 316 11 of 21
After the initial clustering of the original data using the ADPC algorithm is completed.
The initial values
ζ
can be determined, and the EM algorithm is used to build the GMM.
The implementation process is shown in Figure 6.
Aerospace 2022, 9, x FOR PEER REVIEW 12 of 21
the same. Since the initial value and total life cycles of each engine are slightly different
(this is a characteristic of the C-MAPSS data set itself), the four curves do not completely
coincide in the early stage, but they tend to coincide very well in the later stage. And all
four engines have almost the same fault detection index at the end of the cycle. These
results are quite consistent with the real failure evolution law of engine.
Figure 6. Steps of E-M algorithm.
050 100 150 200 250 300
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Fault detection index
Cycles
Engine #1
Engine #2
Engine #3
Engine #4
Figure 7. Failure monitoring results in C-MAPSS data.
Table 2. Parameters related to the four engines.
Engine No.
Full Life Cycle
Fault Detection Index at the End of the Cycle
Engine #1
287
0.5768
Engine #2
269
0.5814
Engine #3
276
0.5885
Engine #4
283
0.5747
Figure 6. Steps of E-M algorithm.
The GMM model for the health state and monitoring state need to be constructed.
This kind of DP model is also called the dynamic double probability model. Among them,
data from the top 25% of the engine life cycles is used to construct the baseline probability
model, and this model remains unchanged in the process of engine fault diagnosis. Data
from after 75% of the engine life cycles is used to construct the monitoring probability
model, which is continuously updated with the increase of the engine life cycles. In the
fault diagnosis stage, the most important thing is to get the engine fault detection indexes,
as is shown in Figure 7. In the probability difference measuring method, the number of
Monte Carlo samples is R= 10,000. Table 2shows the relevant parameters of the four
engines and the fault detection indexes in case of engine failure. It can be seen from the
figure that the fault detection indexes of the top 25% cycles are zero. This is because the
engine is in a healthy state at this stage and failure monitoring is not carried out. In the
fault monitoring stage, the fault detection index’s variation trend of the four engines is
basically the same. Since the initial value and total life cycles of each engine are slightly
different (this is a characteristic of the C-MAPSS data set itself), the four curves do not
completely coincide in the early stage, but they tend to coincide very well in the later stage.
And all four engines have almost the same fault detection index at the end of the cycle.
These results are quite consistent with the real failure evolution law of engine.
Table 2. Parameters related to the four engines.
Engine No. Full Life Cycle Fault Detection Index at the End of the Cycle
Engine #1 287 0.5768
Engine #2 269 0.5814
Engine #3 276 0.5885
Engine #4 283 0.5747
In order to verify the superiority of the proposed model, BP and DBN models are used
as comparison, among which the BP model is a classic algorithm, whereas the DBN model
is a new and effective method used for engine fault diagnosis in recent years. Figure 8
indicates the analysis results of five samples, which are also from the C-MAPSS data (The
relevant data of BP and DBN models are from reference [
1
]). The results show that the fault
detection indexes obtained by BP or DBN models oscillate violently.

Aerospace 2022,9, 316 12 of 21
Aerospace 2022, 9, x FOR PEER REVIEW 12 of 21
the same. Since the initial value and total life cycles of each engine are slightly different
(this is a characteristic of the C-MAPSS data set itself), the four curves do not completely
coincide in the early stage, but they tend to coincide very well in the later stage. And all
four engines have almost the same fault detection index at the end of the cycle. These
results are quite consistent with the real failure evolution law of engine.
Figure 6. Steps of E-M algorithm.
050 100 150 200 250 300
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Fault detection index
Cycles
Engine #1
Engine #2
Engine #3
Engine #4
Figure 7. Failure monitoring results in C-MAPSS data.
Table 2. Parameters related to the four engines.
Engine No.
Full Life Cycle
Fault Detection Index at the End of the Cycle
Engine #1
287
0.5768
Engine #2
269
0.5814
Engine #3
276
0.5885
Engine #4
283
0.5747
Figure 7. Failure monitoring results in C-MAPSS data.
Aerospace 2022, 9, x FOR PEER REVIEW 13 of 21
In order to verify the superiority of the proposed model, BP and DBN models are
used as comparison, among which the BP model is a classic algorithm, whereas the DBN
model is a new and effective method used for engine fault diagnosis in recent years. Fig-
ure 8 indicates the analysis results of five samples, which are also from the C-MAPSS data
(The relevant data of BP and DBN models are from reference [1]). The results show that
the fault detection indexes obtained by BP or DBN models oscillate violently.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Sample 5
Sample 4
Sample 3
Sample 2
Fault detection index
BP
DBN
Sample 1
Figure 8. Comparison of BP and DBN for fault diagnosis of several samples [1].
In order to compare the effect of the model more specifically, the first-order differ-
ence of the predicted fault detection indexes and the corresponding variance value are
obtained, as shown in Figure 9. The variance of the proposed DP model is 0.015, whereas
the variance of the BP and DBN models are 0.035 and 0.024, respectively, as shown in
Table 3. Obviously, the proposed DP model has lower difference variances and better fault
diagnosis results compared with the BP model and DBN model. Unlike the DBN and other
ANN methods, the key to the DP model is to construct the probability distribution of en-
gine data set in a specific space, which is the statistical result of a large number of data.
Therefore, the DP model has the ability to integrate historical data and current data, and
its stability is bound to be better.
050 100 150 200
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
First difference value
Cycle
(a) DBN
050 100 150 200
-0.04
-0.02
0.00
0.02
0.04
First difference value
Cycle
(b) DP
Figure 9. First difference value of the fault detection indexes.
Figure 8. Comparison of BP and DBN for fault diagnosis of several samples [1].
In order to compare the effect of the model more specifically, the first-order difference
of the predicted fault detection indexes and the corresponding variance value are obtained,
as shown in Figure 9. The variance of the proposed DP model is 0.015, whereas the variance
of the BP and DBN models are 0.035 and 0.024, respectively, as shown in Table 3. Obviously,
the proposed DP model has lower difference variances and better fault diagnosis results
compared with the BP model and DBN model. Unlike the DBN and other ANN methods,
the key to the DP model is to construct the probability distribution of engine data set in a
specific space, which is the statistical result of a large number of data. Therefore, the DP

Aerospace 2022,9, 316 13 of 21
model has the ability to integrate historical data and current data, and its stability is bound
to be better.
Aerospace 2022, 9, x FOR PEER REVIEW 13 of 21
In order to verify the superiority of the proposed model, BP and DBN models are
used as comparison, among which the BP model is a classic algorithm, whereas the DBN
model is a new and effective method used for engine fault diagnosis in recent years. Fig-
ure 8 indicates the analysis results of five samples, which are also from the C-MAPSS data
(The relevant data of BP and DBN models are from reference [1]). The results show that
the fault detection indexes obtained by BP or DBN models oscillate violently.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Sample 5
Sample 4
Sample 3
Sample 2
Fault detection index
BP
DBN
Sample 1
Figure 8. Comparison of BP and DBN for fault diagnosis of several samples [1].
In order to compare the effect of the model more specifically, the first-order differ-
ence of the predicted fault detection indexes and the corresponding variance value are
obtained, as shown in Figure 9. The variance of the proposed DP model is 0.015, whereas
the variance of the BP and DBN models are 0.035 and 0.024, respectively, as shown in
Table 3. Obviously, the proposed DP model has lower difference variances and better fault
diagnosis results compared with the BP model and DBN model. Unlike the DBN and other
ANN methods, the key to the DP model is to construct the probability distribution of en-
gine data set in a specific space, which is the statistical result of a large number of data.
Therefore, the DP model has the ability to integrate historical data and current data, and
its stability is bound to be better.
050 100 150 200
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
First difference value
Cycle
(a) DBN
050 100 150 200
-0.04
-0.02
0.00
0.02
0.04
First difference value
Cycle
(b) DP
Figure 9. First difference value of the fault detection indexes.
Figure 9. First difference value of the fault detection indexes.
Table 3. Comparison results of variance values of three models.
Model Difference Variance (10−2)
BP 3.5
DBN 2.4
DP 1.5
It is expected that the dynamic double probability model is able to capture the manifold
of the healthy state and map differences between degradation trajectories into different
regions of 2D FMF space. This visualization is given in Figure 10 using the first two
principal components combined with the fault detection indexes. As can be seen from the
figure, blue data points representing engine health status are mainly concentrated around
PCA-1 =
−
3. As PCA-1 increases, the value of fault detection indexes also increases. The
fault monitoring index reaches the maximum at about PCA-1 = 10 for all four engines,
which means engine failure. It is clear that the DP model can well identify the evolution
process of engine failure.
4.3. RUL Estimation
The DP + LSTM model is applied for RUL estimation. It is necessary to select appropri-
ate parameters for LSTM models to avoid local optimum and fitting errors. As a matter of
experience, the optimal parameter combinations of the LSTM model are shown in Table 4.
The 80 representative engines in the C-MAPSS dataset are used to verify the reliability of
the LSTM model, in which the training and test subsets are divided into a ratio of 3:1. The
training data of LSTM is the fault detection indexes for each engine.
Table 4. Designs based for LSTM networks.
Model Parameters Value
Layer 3
Hidden units [128, 64, 64]
Dropout [0.3, 0.3, 0]
Batch size 100
Epoch 100
Input shape [10, 1]
Output shape [1, 1]

Aerospace 2022,9, 316 14 of 21
Aerospace 2022, 9, x FOR PEER REVIEW 14 of 21
Table 3. Comparison results of variance values of three models.
Model
Difference Variance (10−2)
BP
3.5
DBN
2.4
DP
1.5
It is expected that the dynamic double probability model is able to capture the man-
ifold of the healthy state and map differences between degradation trajectories into dif-
ferent regions of 2D FMF space. This visualization is given in Figure 10 using the first two
principal components combined with the fault detection indexes. As can be seen from the
figure, blue data points representing engine health status are mainly concentrated around
PCA-1 = −3. As PCA-1 increases, the value of fault detection indexes also increases. The
fault monitoring index reaches the maximum at about PCA-1 = 10 for all four engines,
which means engine failure. It is clear that the DP model can well identify the evolution
process of engine failure.
(a) Engine #1
(b) Engine #2
(c) Engine #3
(d) Engine #4
Figure 10. Two-dimensional PCA plot of the fault detection index.
4.3. RUL Estimation
The DP + LSTM model is applied for RUL estimation. It is necessary to select appro-
priate parameters for LSTM models to avoid local optimum and fitting errors. As a matter
of experience, the optimal parameter combinations of the LSTM model are shown in Table
Figure 10. Two-dimensional PCA plot of the fault detection index.
To verify the superiority of the proposed method, the RNN and gated recurrent unit
(GRU) network, which is a variant of LSTM, are implemented as comparisons [
1
], Mean
absolute error (MAE) is used as a training loss function, and the MAE values of the three
models are shown in Figure 11. The results show that the training loss of these three models
decreases gradually with the increasing epoch. When the epoch reaches 100, the training
loss of LSTM is lower than that of RNN but higher than that of GRU. During the last
20 epochs, the mean loss is 0.028.
Figure 12 plots the prediction result of four testing sets from 60% and 70% of the
monitoring cycles. As can be seen from the figure, the predicted results are in good
agreement with the actual results. Especially near the cycle of engine failure, the actual
value is highly coincident with the predicted value. High precision prediction can be
achieved whether the prediction starts from 60% or 70% of the monitoring period. LSTM is
a time series prediction model, and the prediction ability it has learned does not include
the prediction after engine failure. Therefore, when the prediction curve tends to be stable,
it means that the engine is about to fail. In addition, the prediction curve flattens out after
the failure point and shows little growth. These prove the reliability and accuracy of the
DP model and LSTM model proposed in this paper. The threshold needs to be set for RUL
estimation since the initial state of each engine in the C-MAPSS data set is different, and
the threshold value will vary slightly. The threshold value of the four engines selected in
Figure 12 can be set to about 0.55.

Aerospace 2022,9, 316 15 of 21
Aerospace 2022, 9, x FOR PEER REVIEW 15 of 21
4. The 80 representative engines in the C-MAPSS dataset are used to verify the reliability
of the LSTM model, in which the training and test subsets are divided into a ratio of 3:1.
The training data of LSTM is the fault detection indexes for each engine.
Table 4. Designs based for LSTM networks.
Model Parameters
Value
Layer
3
Hidden units
[128, 64, 64]
Dropout
[0.3, 0.3, 0]
Batch size
100
Epoch
100
Input shape
[10, 1]
Output shape
[1, 1]
To verify the superiority of the proposed method, the RNN and gated recurrent unit
(GRU) network, which is a variant of LSTM, are implemented as comparisons [1], Mean
absolute error (MAE) is used as a training loss function, and the MAE values of the three
models are shown in Figure 11. The results show that the training loss of these three mod-
els decreases gradually with the increasing epoch. When the epoch reaches 100, the train-
ing loss of LSTM is lower than that of RNN but higher than that of GRU. During the last
20 epochs, the mean loss is 0.028.
020 40 60 80 100
0.020
0.025
0.030
0.035
0.040
0.090
0.095
0.100
MAE
Epoch
LSTM
RNN
GUR
Figure 11. The loss contrast among LSTM, GRU, and RNN.
Figure 12 plots the prediction result of four testing sets from 60% and 70% of the
monitoring cycles. As can be seen from the figure, the predicted results are in good agree-
ment with the actual results. Especially near the cycle of engine failure, the actual value is
highly coincident with the predicted value. High precision prediction can be achieved
whether the prediction starts from 60% or 70% of the monitoring period. LSTM is a time
series prediction model, and the prediction ability it has learned does not include the pre-
diction after engine failure. Therefore, when the prediction curve tends to be stable, it
Figure 11. The loss contrast among LSTM, GRU, and RNN.
Aerospace 2022, 9, x FOR PEER REVIEW 16 of 21
means that the engine is about to fail. In addition, the prediction curve flattens out after
the failure point and shows little growth. These prove the reliability and accuracy of the
DP model and LSTM model proposed in this paper. The threshold needs to be set for RUL
estimation since the initial state of each engine in the C-MAPSS data set is different, and
the threshold value will vary slightly. The threshold value of the four engines selected in
Figure 12 can be set to about 0.55.
050 100 150 200
0.0
0.1
0.2
0.3
0.4
0.5
0.6 Threshold
Fault detection index
Cycle (Monitoring state)
Target index
Predicted index ( 60%)
Predicted index ( 70%)
Point 1
Point 2
(a) Test #1
050 100 150 200
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Point 2
Point 1
Fault detection index
Cycle (Monitoring state)
Target index
Predicted index ( 60%)
Predicted index ( 70%)
Threshold
(b) Test #2
050 100 150 200
0.0
0.1
0.2
0.3
0.4
0.5
0.6 Threshold
Point 2
Point 1
Fault detection index
Cycle (Monitoring state)
Target index
Predicted index ( 60%)
Predicted index ( 70%)
(c) Test #3
020 40 60 80 100 120 140 160 180
0.0
0.1
0.2
0.3
0.4
0.5
0.6 Threshold
Point 2
Point 1
Fault detection index
Cycle (Monitoring state)
Target index
Predicted index ( 60%)
Predicted index ( 70%)
(d) Test #4
Figure 12. RUL prediction of 4 engines.
The engine cycles corresponding to the threshold can be determined according to the
prediction curve; that is, the cycle when the failure is predicted. To get a more detailed
understanding of the model’s accuracy, we calculated the relative error of prediction for
20 testing sets, as shown in Figures 13 and 14. When predicted from 60% cycles, the mean
relative error of the testing is 0.024%. When predicted from 70% cycles, the mean relative
error of the testing is 0.019%. Obviously, the prediction accuracy is slightly higher when
starting from 70% cycles, because time series prediction models generally have a certain
degree of cumulative error. In general, the relative errors of both of them remain below
6%, which proves the high accuracy of the proposed model.
Figure 12. RUL prediction of 4 engines.

Aerospace 2022,9, 316 16 of 21
The engine cycles corresponding to the threshold can be determined according to the
prediction curve; that is, the cycle when the failure is predicted. To get a more detailed
understanding of the model’s accuracy, we calculated the relative error of prediction for
20 testing sets, as shown in Figures 13 and 14. When predicted from 60% cycles, the mean
relative error of the testing is 0.024%. When predicted from 70% cycles, the mean relative
error of the testing is 0.019%. Obviously, the prediction accuracy is slightly higher when
starting from 70% cycles, because time series prediction models generally have a certain
degree of cumulative error. In general, the relative errors of both of them remain below 6%,
which proves the high accuracy of the proposed model.
Aerospace 2022, 9, x FOR PEER REVIEW 17 of 21
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
140
160
180
200
220
240
260
280
300
Cycles
Test set
Actual value
Predicted value
(a) Prediction from 60% cycles
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
150
200
250
300
Cycles
Test set
Actual value
Predicted value
(b) Prediction from 70% cycles
Figure 13. The residuals of the actual RUL and estimated RUL on 20 testing sets.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Relative Error
Test set
Predict from 60% cycles
Predict from 70% cycles
Figure 14. The relative error of RUL prediction on 20 testing sets.
Several classical RUL estimation methods are compared to verify the superiority of
the proposed method, and the RUL prediction errors of the five models are listed in Table
5 (relevant data of the model used for comparison come from the reference [1]). Compared
with DBN + LSTM, LSTM, RNN, and GRU, the average RUL estimation error of DP +
LSTM model is 4.4, which decreases by 21%, 41%, 51%, and 48% (the data of these five
models are all from the C-MAPSS dataset). The result shows that proposed DP + LSTM
model has higher accuracy than those classical time series prediction models. In fact, sev-
eral other methods belong to the ANN model, which can also be called a black box model.
In essence, they achieve prediction by learning the inherent laws of a large amount of data.
These methods are sensitive to data, and the hyper-parameters have a great impact on the
model effect, and the adjustment of hyper-parameters is a complex process. The DP +
LSTM method proposed in this study is the combination of probability model and ANN
Figure 13. The residuals of the actual RUL and estimated RUL on 20 testing sets.
Aerospace 2022, 9, x FOR PEER REVIEW 17 of 21
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
140
160
180
200
220
240
260
280
300
Cycles
Test set
Actual value
Predicted value
(a) Prediction from 60% cycles
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
150
200
250
300
Cycles
Test set
Actual value
Predicted value
(b) Prediction from 70% cycles
Figure 13. The residuals of the actual RUL and estimated RUL on 20 testing sets.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Relative Error
Test set
Predict from 60% cycles
Predict from 70% cycles
Figure 14. The relative error of RUL prediction on 20 testing sets.
Several classical RUL estimation methods are compared to verify the superiority of
the proposed method, and the RUL prediction errors of the five models are listed in Table
5 (relevant data of the model used for comparison come from the reference [1]). Compared
with DBN + LSTM, LSTM, RNN, and GRU, the average RUL estimation error of DP +
LSTM model is 4.4, which decreases by 21%, 41%, 51%, and 48% (the data of these five
models are all from the C-MAPSS dataset). The result shows that proposed DP + LSTM
model has higher accuracy than those classical time series prediction models. In fact, sev-
eral other methods belong to the ANN model, which can also be called a black box model.
In essence, they achieve prediction by learning the inherent laws of a large amount of data.
These methods are sensitive to data, and the hyper-parameters have a great impact on the
model effect, and the adjustment of hyper-parameters is a complex process. The DP +
LSTM method proposed in this study is the combination of probability model and ANN
Figure 14. The relative error of RUL prediction on 20 testing sets.

Aerospace 2022,9, 316 17 of 21
Several classical RUL estimation methods are compared to verify the superiority of
the proposed method, and the RUL prediction errors of the five models are listed in Table 5
(relevant data of the model used for comparison come from the reference [
1
]). Compared
with DBN + LSTM, LSTM, RNN, and GRU, the average RUL estimation error of DP + LSTM
model is 4.4, which decreases by 21%, 41%, 51%, and 48% (the data of these five models
are all from the C-MAPSS dataset). The result shows that proposed DP + LSTM model has
higher accuracy than those classical time series prediction models. In fact, several other
methods belong to the ANN model, which can also be called a black box model. In essence,
they achieve prediction by learning the inherent laws of a large amount of data. These
methods are sensitive to data, and the hyper-parameters have a great impact on the model
effect, and the adjustment of hyper-parameters is a complex process. The DP + LSTM
method proposed in this study is the combination of probability model and ANN model.
Solid mathematical basis is the advantage of probability model, which is an important
factor for the DP + LSTM model to be more superior.
Table 5. RUL estimation error of different models.
Model Point 1 (Cycles) Point 2 (Cycles) Average (Cycles)
DP + LSTM 5.1 3.7 4.4
DBN + LSTM 6.9 4.4 5.6
LSTM 8.2 6.8 7.5
RNN 10.2 7.9 9.0
GRU 10.0 7.0 8.5
4.4. PHM Application Example
Standardizing the data processing flow of the PHM framework is one of the aims of
this study. Algorithm 1 summarizes the function realization process of the PHM framework.
Algorithm 1. PHM framework process.
Input: Aero-engine raw sensor data.
Process 1: Data preprocessing
(1) Data collation and standardization (z-score).
(2) Data dimension reduction based on PCA method.
Process 2: DP model construction
(1) The preprocessed data are fed to the double probability models.
(2) Construct the baseline probability model.
(3) Construct the dynamic monitoring probability model.
(4) Difference measures for double probability models.
(5) Output fault detection indexes.
Process 3: RUL Estimation
(1) Training LSTM network based on fault detection indexes.
(2)
The prediction of engine RUL at the current time is realized from any cycle point in
the engine life cycle.
(3) Output engine RUL.
Output: Fault detection indexes and RUL.
An engine data set in the C-MAPSS data set is selected to show the processing results
of the proposed PHM framework, as shown in Figure 15a, which shows 7 of the 14 sets
of raw sensor data for the engine. It can be seen that noise greatly interferes with sensor
data, and the change trends of sensors are inconsistent in the whole life cycle of the engine.
Figure 15b shows the data after dimension reduction. Figure 15c,d, respectively, show
the results of fault diagnosis and RUL estimation respectively. The proposed framework
realizes data analysis and mining from the original data of the engine to monitor engine
health and realize the estimation of RUL.

Aerospace 2022,9, 316 18 of 21
Aerospace 2022, 9, x FOR PEER REVIEW 19 of 21
Figure 15. Example of PHM framework data processing flow.
5. Conclusions
In this study, a PHM framework combining the DP model and LSTM model is pro-
posed for fault diagnosis and RUL estimation of aero-engine. Firstly, the DP model con-
sisting of a baseline probability model and a monitoring probability model is constructed,
in which the baseline probability model reflects the operating characteristics of the en-
gine’s healthy state, and the monitoring probability model reflects the failure occurrence
and evolution process of the engine. A GMM-ADPC algorithm is employed for modeling
engine fault development, and the PCA method is adopted to reduce the dimension of the
input data. Secondly, the probability difference measuring method is used to quantify the
difference between the two probability models so as to obtain the fault detection indexes.
Thirdly, the DP + LSTM model is introduced for a time series prediction of fault detection
indexes, so as to estimate the RUL of the engine. Finally, the PHM framework is estab-
lished by integrating the aforementioned models. The experimental results on the degra-
dation datasets obtained by the C-MAPSS indicated that the proposed DP model can cap-
ture the process of engine failure well, and the DP + LSTM model can perform RUL esti-
mation well. By comparing the results of the proposed method with some classical meth-
ods, it is shown that the proposed method has better stability and accuracy.
To sum up, the PHM framework proposed in this study can adequately realize the
functions of fault diagnosis and RUL estimation.
Author Contributions: Conceptualization, Y.H. (Yufeng Huang), G.S. and J.T.; methodology, Y.H.
(Yufeng Huang); software, H.Z.; validation, Y.H. (Yufeng Huang); formal analysis, J.T.; investiga-
tion, Y.H. (Yan Hu); resources, G.S.; data curation, Y.H. (Yufeng Huang); writing—original draft
preparation, Y.H. (Yufeng Huang); writing—review and editing, G.S. and J.T.; visualization, Y.H.
(Yufeng Huang); supervision, J.T.; project administration, G.S.; funding acquisition, Y.H. (Yan Hu).
All authors have read and agreed to the published version of the manuscript.
Figure 15. Example of PHM framework data processing flow.
5. Conclusions
In this study, a PHM framework combining the DP model and LSTM model is pro-
posed for fault diagnosis and RUL estimation of aero-engine. Firstly, the DP model consist-
ing of a baseline probability model and a monitoring probability model is constructed, in
which the baseline probability model reflects the operating characteristics of the engine’s
healthy state, and the monitoring probability model reflects the failure occurrence and
evolution process of the engine. A GMM-ADPC algorithm is employed for modeling
engine fault development, and the PCA method is adopted to reduce the dimension of the
input data. Secondly, the probability difference measuring method is used to quantify the
difference between the two probability models so as to obtain the fault detection indexes.
Thirdly, the DP + LSTM model is introduced for a time series prediction of fault detection
indexes, so as to estimate the RUL of the engine. Finally, the PHM framework is established
by integrating the aforementioned models. The experimental results on the degradation
datasets obtained by the C-MAPSS indicated that the proposed DP model can capture the
process of engine failure well, and the DP + LSTM model can perform RUL estimation well.
By comparing the results of the proposed method with some classical methods, it is shown
that the proposed method has better stability and accuracy.
To sum up, the PHM framework proposed in this study can adequately realize the
functions of fault diagnosis and RUL estimation.

Aerospace 2022,9, 316 19 of 21
Author Contributions:
Conceptualization, Y.H. (Yufeng Huang), G.S. and J.T.; methodology, Y.H.
(Yufeng Huang); software, H.Z.; validation, Y.H. (Yufeng Huang); formal analysis, J.T.; investiga-
tion, Y.H. (Yan Hu); resources, G.S.; data curation, Y.H. (Yufeng Huang); writing—original draft
preparation, Y.H. (Yufeng Huang); writing—review and editing, G.S. and J.T.; visualization, Y.H.
(Yufeng Huang); supervision, J.T.; project administration, G.S.; funding acquisition, Y.H. (Yan Hu).
All authors have read and agreed to the published version of the manuscript.
Funding:
This research was co-funded by the Shanghai Pujiang Program (No. 20PJ1402000) and the
AECC Commercial Aircraft Engine Co., Ltd. (No. AR0973.00RW.001).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data are available by contacting the corresponding author.
Conflicts of Interest: The authors declare no conflict of interest.
Nomenclature
ADPC adaptive density peaks clustering HPT high pressure turbine
ANN artificial neural network LPC low pressure compressor
BP back propagation LPT low pressure turbine
C-MAPSS commercial modular aero-propulsion LSTM long short-term memory
system simulation neural network
DBN deep belief network MAE mean absolute error
DP dynamic probability MSE mean squared error
EM expectation-maximization PHM prognostics and health management
FMF fault monitoring feature PCA principal component analysis
GRU gated recurrent unit RNN recurrent neural network
GC Gaussian component RUL remaining useful life
GMM Gaussian mixture model SHM structural health monitoring
HHT Hilbert-Huang transform SVM support vector machine
HPC high pressure compressor
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