Particle Filtering for State and Parameter Estimation in Gas Turbine

Engine Fault Diagnostics *

Najmeh Daroogheh1, Nader Meskin2 and Khashayar Khorasani1

Abstract— In this paper, a novel method for a time-varyingparameter estimation technique using particle filters isproposed based on the concept of Recursive Prediction Error(RPE). According to the proposed method, a parallel structurefor both state and parameter estimation in a nonlinearnon-Gaussian system is developed. The performance of thedeveloped framework is evaluated in an application to the gasturbine engine state and health parameters estimation by usingdifferent scenarios. The developed method is identified to beapplicable for fault diagnosis of an engine system while it issubjected to concurrent and simultaneous loss of effectivenessfaults in the system components.

I. INTRODUCTION

One of the main concerns in model-based fault diagnosis

and failure prognosis methods is the dependency of their

efficiency and accuracy on the dynamic models that are used

in order to describe the behavior of the system. Linear and

Gaussian models that are developed to achieve a satisfactory

performance in diagnosis and prognosis applications utilize

less complicated representations of the actual nonlinear dy-

namic models. Although these models may have acceptable

performance in short-term applications, as the prediction

horizon extends to include long-term behavior of the system,

such as the case in prognosis application, the linear models

would not necessarily lead to an accurate solution. Therefore,

development of model-based algorithms must be investigated

that can satisfy the prediction requirements of both short-

term and long-term behaviors of the system [1].

A general solution to the problem of state estimation in

nonlinear state space systems is described by the Bayesian

recursive methods. Particle Filter (PF) is one of the most

popular recursive nonlinear state estimation methods [2],

[3] which solves the Bayesian recursive relations using

Sequential Monte Carlo (SMC) methods. SMC methods are a

set of simulation-based techniques that provide an interesting

approach to compute the posterior distributions of states,

so that the statistical estimates can be easily computed [2].

In this work, nonlinear Bayesian and Sequential Monte-

Carlo (SMC) methods are applied to develop a consistent

framework for both state and parameter estimation in fault

*This publication was made possible by NPRP grant No. 4-195-2-065from the Qatar National Research Fund (a member of Qatar Foundation).The statements made herein are solely the responsibility of the authors.

1N. Daroogheh and K. Khorasani are with the Department ofElectrical and Computer Engineering, Concordia University, Mon-treal, Quebec, Canada n daroog@encs.concordia.ca andkash@ece.concordia.ca

2N. Meskin is with the Department of Electrical Engineering, QatarUniversity, Doha, Qatar nader.meskin@qu.edu.qa

diagnosis problem of a gas turbine engine system. The main

contribution of this work is development of an algorithm

that utilizes the Recursive Prediction Error (RPE) methods

[4] to estimate the nonlinear system parameters with particle

filtering approach. These parameters are due to changes be-

cause of faults in the system components. The RPE methods

refer to a set of recursive techniques that are used for off-line

and on-line identification based on a quadratically convergent

scheme [4]. The application of the RPE techniques for on-

line state and parameter estimation in the Hidden Markov

Models (HMM) is shown to be more memory efficient in

the sense of computational complexity [5].

A general method that is capable of simultaneously esti-

mating the fixed parameters and the time-varying states of a

system is addressed in [6]. This method is based on the Se-

quential Monte Carlo method in which an artificial evolution

dynamic model is considered for the parameters. In order to

overcome the degeneracy problem due to particle filtering,

kernel smoothing technique [7], as a method of smoothing

for the approximation of the parameter conditional density is

applied. The estimation algorithm is improved by reinterpre-

tation of the artificial evolution algorithm according to the

shrinkage scaling concept. This notion is based on shrinkage

kernel locations [7].

In cases where the static system parameter is unknown

and needs to be estimated from different online or off-line

approaches, two main classes of estimation methods based on

numerical particle methods and maximum likelihood param-

eter estimation are proposed [8]. These methods have been

developed to only estimate static parameters of a model. The

combination of particle filtering and gradient algorithm based

on the stochastic approximation technique was utilized in [9],

[10] to implement an adaptive fixed parameter estimation

method. The efficiency of this method is assessed to be

more reliable than the specific particle filtering methods for

parameter estimation.

The application of SMC methods for fault diagnosis and

failure prognosis in a gas turbine engine has been proposed

in [1]. In their work, the crack propagation process is

considered as a fault feature, and its propagation is predicted

in time to obtain the Remaining Useful Life (RUL) of the

turbine blade within the prognostics framework.

In this work, by using the prediction error to correct the

changes in the system parameters, a novel method is pro-

posed for the parameter estimation in nonlinear state-space

models based on the PF. This method is capable of capturing

the changes in the parameters due to faults and differentiate

them from the changes caused by other factors, e.g. due to

2013 American Control Conference (ACC)Washington, DC, USA, June 17-19, 2013

978-1-4799-0178-4/$31.00 ©2013 AACC 4343

changes in the system operational conditions. The amplitude

of faults can also be determined. In the implementation of

the algorithm, a parallel structure is proposed for both state

and parameter estimation in the PF framework. The hidden

states, and the variations in the system health parameters are

estimated through two parallel filters. This structure is used

for component fault diagnostics in a jet engine application.

II. PROBLEM STATEMENT

In this work, it is desired to develop a dual algorithm for

state/parameter estimation for addressing the problem of fault

detection, isolation and identification of a nonlinear dynamic

state-space model. The dynamics of the system is governed

by a known discrete-time nonlinear stochastic model given

by,

xk = fk(xk−1, θTk−1λ(xk−1), uk−1, wk), (1)

yk = hk(xk, θTk λ(xk), vk), (2)

where fk : Rnx×Rnθ×R

nu×Rnw −→ R

nx is the nonlinear

function defining the state at the next time step k+1, k ∈ N,

θk ∈ Rnθ is the unknown fault parameter vector at time k

and in the steady state for a healthy system it is equal to 1,

λk : Rnx −→ Rnθ is a differentiable function in terms of

system states and it determines the relationship between the

system states and health parameters, and uk is the known

input. The function hk : Rnx × Rnθ × R

nv −→ Rny is the

nonlinear function representing the relationship between the

state, parameters and the measurements at time k, and wk, ζkand vk are the uncorrelated white noise sequences with zero-

mean and covariance matrices Qk, Lk and Rk, respectively.

The assumed fault that affects the system parameters

is multiplicative. This fault causes changes in the state

transition function and also the measurement function at time

k.

III. THE BACKGROUND

A. Nonlinear Bayesian Filtering Problem

In the framework of Bayesian filtering formulation for

a dynamic system state estimation, the main goal is to

construct the posterior probability density function (pdf) for

the state based on the measurements received at each time

instant. A recursive algorithm for updating the estimation

results has been proposed in [1] for this purpose.

Consider the unobserved process (states), denoted by

X = {xk, k ∈ N}, to be an Rnx valued Markov process

with initial distribution p(x0) and the transition probability

p(xk|xk−1). This probability is defined by the system state

equation (3) with the noise sequence of {wk}k> 0. The

parameters of the model are considered to be constant, that

is

xk = fk(xk−1, θTλ(xk−1), uk−1, wk) ≈ p(xk|xk−1). (3)

On the other hand, the process of noisy observations,

denoted by Y = {yk, k ∈ N}, is assumed to be conditionally

independent given the process, and the marginal distribution

p(yk|xk) is defined as

yk = hk(xk, θTλ(xk), vk) ≈ p(yk|xk). (4)

Consider the signal and observations (measurements) up

to time k are denoted by x0:k = {x0, ..., xk} and y1:k ={y1, ..., yk}, respectively. The main goal is to recursively

estimate the posterior distribution, p(x0:k|y1:k}, and the

marginal distribution, p(xk|y1:k), which is also known as

the filtering distribution.

B. Sequential Monte Carlo Methods and Particle Filtering

Sequential Monte Carlo (SMC) methods , that are also re-

ferred to as Particle Filtering methods, are a set of algorithms

that are used to approximate the optimal filtering by repre-

senting the probability density function with a population of

particles. The main objective is to approximate the integrals

that are used to take expectations for generating a priori

state pdf estimate for the next time step, when a large number

of samples are drawn from the posterior distributions.

In particle filters, the probability density function is repre-

sented by N random samples, known as particles {x(i)0:k, i =

1, ..., N} and their associated normalized importance weights

w(i), (∑N

i=1 w(i)k = 1). This method can be interpreted as a

sampling algorithm where at time k, the posterior distribution

p(x0:k|y1:k) is approximated as [11],

p(x0:k|y1:k) =N∑i=1

w(i)k δ(x0:k − x

(i)0:k), (5)

where δ(.) is the Dirac delta function. The estimate of the

target state at time k is as follows xk ≈∑N

i=1 w(i)k x

(i)k ,

where the new samples x(i)k are discrete samples drawn from

a chosen importance density function, q(xk|xk−1, y1:k). This

function is a kernel function used to extend the current

paths of x(i)0:k. The special case occurs when the prior

distribution is selected as the importance distribution [11],

q(x0:k|y1:k) = p(x0:k) = p(x0)∏k

j=1 p(xj |xj−1). In this

case the importance weights satisfy w(i)k = w

(i)k−1p(yk|x

(i)k ).

The degeneracy phenomenon in particle population is one

of the major problems in the implementation of the sequential

importance sampling PF. It may cause poor approximation

of the target distribution. The degeneracy problem is directly

related to the variance of the importance weights [1], [12].

In order to solve the degeneracy problem, a resampling

algorithm is performed to remove the particles with small

weights [13], [14]. Hence, a new particle population is

generated (which is denoted by {x(i)0:k}i=1,...,N ), by n times

sampling from equation (5). After resampling, the new

particle population is an i.i.d sample from the empirical

distribution (6). Therefore, the weights are reset to w(j)k =

1N

, and we have

pNk (x0:k) =1

N

N∑i=1

δ(x0:k − x(i)0:k), (6)

where pNk denotes the approximated posterior distribution.

Standard PF methods are extended by including an aux-

iliary variable that is designated by µik+1, that helps the PF

to be adapted more effectively [15]. This auxiliary variable

is chosen as an estimate of xk+1 for each time step. For

example, the mean or mode of the distribution p(xk+1|xk)

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can be chosen to form µik+1. The resampled states are then

sampled from p(xk+1|xk), with the probability proportional

to g(i)k+1 that is defined as g

(i)k+1 ∝ w

(i)k p(yk+1|µ

(i)k+1). The

associated weight update law at time k+1 becomes w(i)k+1 =

p(yk+1|x(i)k+1)

p(yk+1|µ(i)k+1)

.

Another problem in discrete approximation with particle

filters is the loss of diversity among the particles. The

Regularized Particle Filters (RPF) [16] have been proposed

as a method to overcome this problem. The main idea

in RPF is to change the discrete approximation from the

importance function into a continuous one, such that the

resampling step is actually changed into simulations from a

continuous distribution. The new particle population would

have N different locations, therefore the loss of diversity

among particles can be completely avoided. In this work

RPF with the rescaled kernel structure is mainly used for

state estimation in a nonlinear system model.

IV. FILTERING FRAMEWORK

In the parallel state/parameter estimation model proposed

in this paper, two separate filters are implemented for state

and parameter estimation. At each time step for the state

(parameter) estimator filter, the parameters (states) at that

time step are considered to be known and fed into the

filter from the parameter (state) estimator filter. The filtering

structure for states and parameters is constructed as follows.

A. Filtering for States

Consider the dynamical model (3), in which at time k the

state vector is xk and the assumed fixed parameter vector

θ is denoted as p(xk|xk−1, θ). By applying the observation

equation (4), the observation density becomes p(yk|xk, θ). It

must be noted that each yk is considered to be conditionally

independent from the past states and observations, given the

current state xk and the parameter θ. This is the case in many

models and practical applications [6].

In order to approximate the posterior distribution

p(xk, θ|y1:k) by applying the Monte Carlo approach, suppose

at time k we have a sample of current states x(1)k , ..., x

(N)k

and their associated weights w(1)k , ..., w

(N)k , that together

represent the Monte Carlo importance sample approximation

to the desired posterior. As time evolves from k to k+1 and

the new observation yk+1 is received, the posterior density

of xk+1 is calculated as,

p(xk+1, θ|Dk+1) ∝ p(yk+1|xk+1, θ)p(xk+1, θ|Dk)

∝ p(yk+1|xk+1, θ)p(xk+1|θ,Dk)p(θ|Dk),(7)

where Dk+1 = {Dk, yk+1}, Dk = {y1:k}, and p(θ|Dk) is

considered to be 1, since it is assumed that the parameter

value is known in the state estimator filter. Therefore, the

known θ from the conditioning statements can be dropped.

The term p(yk+1|xk+1) is the likelihood function. The

prior for the state at time k + 1 is implemented as:

p(xk+1|Dk) =

∫p(xk+1|xk)p(xk|Dk)dxk. (8)

By substituting the integral in equation (8) with its Monte

Carlo approximation (weighted summation over the particles

x(i)k using the RPF structure), the update rule becomes,

p(xk+1|Dk) ≈ p(yk+1|xk+1)N∑i=1

w(i)k p(xk+1|x

(i)k )

≈N∑i=1

w(i)k

|A−1|

hnxK(

1

hA−1(xk − x

(i)k+1|k)),

(9)

where w(i)k = w

(i)k p(yk+1|xk+1), i = 1, ..., N , xk is

obtained from the continuous Gaussian distribution, and K is

the regularization kernel that is considered to be a symmetric

density function on Rnx [16]. The matrix A is chosen in

order to achieve unit covariance in the new x(i)k population

and AAT = S where S is the empirical covariance matrix

of the resampled samples. The constant h is the optimal

bandwidth for the kernel [16].

B. Filtering for Parameters

In the parameter estimation step of the algorithm it is

assumed that the parameter is not known and is time-

varying. Hence, the term p(θ|Dk) in equation (7) would

not necessarily be equal to one. It is essential to consider

a dynamical model for the parameter evolution in order to

estimate the density function p(θk|Dk).The most common dynamical model considered for the

parameter propagation (in case of fixed model parameter)

is the artificial evolution law. In this model, small random

disturbances are added to the parameter [6] as follows:

θk+1 = θk + ζk+1, ζk+1 ∼ N(0,Wk+1), (10)

where Wk+1 is the variance matrix, and θk and ζk+1 are

conditionally independent given Dk.

In artificial evolution dynamic model for the system with

constant parameters, it is assumed that the constant parameter

is time-varying. This may cause loss of information between

time steps, and hence the precision of the parameter estima-

tion algorithm would become limited. However, this model

is not capable of dealing with estimation of the parameter

when it is changing due to faults in the system components,

which is the main concern in this work.

In this paper, a new approach is proposed in order to

model the parameter dynamics. This new structure is called

modified artificial evolution and is suitable for dealing with

the changes in the fixed parameters due to faults. The

problem of information loss between time steps in case when

the parameter is fixed is also addressed by utilizing a kernel

smoothing method [6]. The proposed method is described in

more detail in the following subsections.

1) Modified Artificial Parameter Evolution: The tradi-

tional artificial evolution update law for a parameter is now

modified to include the output prediction error as an extra

term. This modification enables one to deal with the varia-

tions in the parameters that can affect the system outputs. In

order to obtain this dynamical model, we consider a function

of the prediction error (ǫ(k, θ) = yk−yk) as the cost function

that must be minimized with respect to the parameter θ. We

4345

consider the special case where the cost function, J(θ) is

only a function of ǫ(k, θ), i.e.

J(θ) = E(l(ǫ(k, θ))), (11)

where, l(ǫ(k, θ)) = 12ǫ(k, θ)ǫ(k, θ)

T. The minimization of

J(θ) with respect to θ yields,

d

dθl(ǫ(k, θ))T =

dǫ

dθǫ(k, θ) = −

dyk

dθǫ(k, θ), (12)

and by applying the gradient descent method the new pa-

rameter update law for the next time step becomes,

θk+1 = θk + γkψkǫ(k, θ) + ζk+1, (13)

where ψk = dyk

dθ=

dhk(xk,θTk λ(xk))

dθand γk is a time-varying

adaptive step size. Since the variation of each parameter

may cause change in several elements of the output, after

fitting the prediction errors at each time step to a Gaussian

distribution (using the maximum likelihood estimator), the

variance of this estimation is a suitable measure for the level

of change in the evolution law.

2) Kernel Smoothing of a Parameter: The developed

algorithm must also be able to track a parameter when it is

considered to be fixed, and the problem of information loss

should also be addressed. Hence, for achieving this objective,

the correlations between θk and the random noise ζk+1 are

introduced [6]. It is assumed that in the steady state condition

when the parameter is fixed, the gradient of the predicted

output (ψk) is constant which is denoted by Ψ.

Considering the modified artificial evolution equation (13),

the relationship between the variances of both sides of

equation (13) when the covariance matrix is considered to

be non-singular and also the prediction error at time k is

uncorrelated from the parameter value at that time instant, is

as follows

V (θk+1|Dk) = V (θk|Dk) + 2γ2kΨVyΨT +Wk+1

+2C(θk, ζk+1|Dk) + 2C(γkΨǫk, ζk+1|Dk),(14)

where V denotes the variance and C denotes the covariance

and Vy is the variance of the observation noise and,

V (γkΨǫk|Dk) = (E[γk|Dk])2V (Ψǫk|Dk)

+ (E[Ψǫk|Dk])2V (γk|Dk)

+ V (γk|Dk)V (Ψǫk) = 2γ2kΨVyΨT.

(15)

In order to have no information loss in case when θ is

constant, we must have,

V (θk+1|Dk) = V (θk|Dk) = Vθk , (16)

which implies that,

2C(θk, ζk+1|Dk) + 2C(γkΨǫk, ζk+1|Dk)

= −1

2Wk+1 − γ2kΨVyΨ

T.(17)

Therefore, negative correlations are needed to remove

the unwanted information loss. In order to approximate the

parameter evolution with a normal distribution [6], by apply-

ing the shrinkage of kernel locations concept, a conditional

normal distribution is obtained,

p(θk+1|θk) ∼

N(θk+1|Ak+1θk + (I −Ak+1)θk, (I −A2k+1)Vθk),

(18)

where the mean value of this Gaussian distribution at each

time step θk+1 is found from equation (13), when we substi-

tute θk by its modified version according to the shrinkage

kernel method. The shrinkage matrix Ak+1, is calculated

from the minimum eigenvalue of the desired correlation

matrix that is multiplied by the inverse of the parameter

variance at time k as,

Ak+1 = min(eig(1

2Wk+1+γ

2kΨVyΨ

T )V −1θ ))I = aI. (19)

The step size γk is not necessarily constant (also in

the case of fixed parameter due to the estimation error in

parameters), but it is bounded. Therefore, the value of a

can be calculated in general by using the bounds of γkinstead of its actual value. The smoothing factor of the

normal distribution variance is calculated from the shrinkage

factor as (I−A2k+1). In the implementation of the parameter

estimation filter the structure of the Auxiliary Particle Filter

(APF) is used [6], [15]. More detail on the APF was

presented in Section III.

V. FAULT DIAGNOSIS OF A GAS TURBINE ENGINE

A. FDI Formulation

In the implementation of the fault detection and isolation

(FDI) using the proposed state/parameter estimation method,

the parameter estimate is considered as the feature to detect

and isolate the faults in the system components. The residuals

are generated from the parameter estimates in the healthy and

faulty operational modes of the system as,

rk = θ0 − θk,

θ0 = argmax(−log(p(θ0|y1:T)),(20)

where θ0 denotes the estimated parameter with the probabil-

ity density of p(θ0|y0:T ) (conditioned to the observations up

to time T ), obtained from the collected estimated values in

the healthy mode of the system when they are fitted to a nor-

mal distribution. The time window T is chosen according to

the convergence time of the parameter estimation algorithm.

The second standard deviation of the estimation distribution

of the parameter is considered as the confidence interval and

bound.

B. Model Overview

The application of the proposed PF method for state and

parameter estimation in a gas turbine engine is presented in

this section. This approach can be used in fault diagnosis

of the engine system through the fault detection, isolation

and identification that are all accomplished by utilizing this

algorithm. The performance of the proposed algorithm for

state/parameter estimation is tested under two main scenarios

for deficiencies in the system health parameters due to

simultaneous and concurrent faults.

4346

The model used in this paper for performing the proposed

state/parameter estimation method is a single spool gas

turbine jet engine model [17]. The system states are the

combustion chamber pressure and the temperature, PCC and

TCC, the spool speed N and the nozzle outlet pressure PNLT.

The continuous-time system state equations are as follows,

TCC =1

cvmcc[(cpTCmC + ηCCHumf

− cpTCCmT)− cvTCC(mC +mf −mT)],

N =ηmechmTcp(TCC − TT)−mCcp(TC − Td)

JN( π30 )

2,

PCC =PCC

TCCTCC +

γRTCC

VCC(mc +mf −mT),

PNLT =TM

VM(mT +

β

β + 1mC −mNozzle).

(21)

The system health parameters are the compressor and

turbine efficiency and the mass flow capacity. A fault vector

is considered to affect the system health parameters as

denoted by θ = [θηC , θmC , θηT , θmT ]T. Each parameter

variation is due to the variation of this fault vector which

is considered as a multiplicative fault type. The proposed

dynamical equations for fault parameters in the estimation

filter according to the modified artificial evolution are given

by,

˙θ(t) = γΨ(y − y) + ζ(t), (22)

where γ is the adaptive step size, y ∈ R5 is the output vector,

Ψ ∈ R4×5 is the Jacobian matrix of the output equations in

terms of the fault parameters that is found from the sensitivity

analysis of the change in each output due to the unit change

in each of the parameters in the cruise steady state operation

mode of the engine. The y ∈ R5 is the vector of predicted

output from the particle filter. Also, ζ ∈ R5 is the noise added

to the parameters and is drawn from a normal distribution

with the variance of Wk.

The gas turbine measured outputs are considered to be the

output pressure and temperature of the compressor and the

turbine and the spool speed, namely

y1(t) = TC = Tdiffuser[1 +1

ηC[(

PCC

Pdiffuzer)

γ−1γ − 1]],

y2(t) = PCC, y3(t) = N, y4(t) = PNLT,

y5(t) = TCC[1− ηT(1− (PNLT

PCC)

γ−1γ ].

(23)

In order to discretize the system equations for implementa-

tion with a particle filter, the simple Euler Backward method

is applied. More detail on the variables in equations (21) and

(23) are presented in [17].

C. Simulation Scenarios

In this section, two main scenarios are applied to the sys-

tem (21) in order to evaluate the effectiveness of the proposed

state/parameter estimation algorithm for fault diagnosis of

simultaneous and concurrent faults in the components (com-

pressor and turbine). All the simulations are conducted in

the cruise steady-state flight condition mode in which the

system parameters are considered to be constant or slowly

time-varying.

In order to show the effectiveness of the proposed algo-

rithm, the records of applying the traditional kernel smooth-

ing method (without using the RPE term in the parameter

estimation algorithm) and the method with the step size

considered as a decreasing series of time [10] are also

presented. In our work, an adaptive step size is proposed

based on the variance of the maximum likelihood estimation

of the 5 outputs of the system at each time step. The residuals

corresponding to the parameter estimates are obtained for

each scenario. Based on the simulation results, a convergence

time of 2 seconds is obtained for the proposed algorithm. In

order to illustrate the performance of the proposed algorithm

Tables I, II, III, and IV provide the Mean Absolute Error

(MAE) of the state and parameter estimates for the last data

set after convergence (the last 2 seconds of simulations).

To choose the number of particles for implementation

of the state and parameter estimation filters, a quantitative

study have been conducted. Hence, based on the MAE in

the steady state condition and considering the computational

time for the algorithm implementation, the particle number of

N = 50 is chosen for this application as it shows acceptable

performance and convergence time. The shrinkage coefficient

is evaluated to be around 0.93. The initial distributions (mean

and covariance matrices) of the states and parameters are

selected based on the cruise flight operational condition.

1) Concurrent Faults in the Compressor and Turbine

Health Parameters: In this scenario, the effects of concurrent

faults in both compressor and turbine are studied by applying

a sequential pattern for faults affecting the system compo-

nents. First, at time t = 5 seconds the compressor efficiency

is dropped by 5% (fault severity), then at t = 10 seconds the

same fault affects the compressor mass flow capacity, and at

t = 15 seconds the turbine efficiency is affected, and finally

at t = 20 seconds the same scenario has occurred to the

turbine mass flow capacity.

The results of applying three different methods for esti-

mating the system health parameters are presented in Figure

1. It follows from Figure 1 that traditional kernel smoothing

method with the artificial evolution parameter dynamics is

not capable of following the parameter changes and is only

suitable for constant parameter estimation problem. The

figure shows that the developed modified artificial evolution

dynamics can follow the parameter changes. However, in the

case with an adaptive step size, the post fault convergence

time and its accuracy is better as compared to the case with

the decreasing step size (γk = 0.1(k+2)0.63 ) [10].

For the FDI task and generation of residuals, the developed

method with an adaptive step size is considered while the

residuals related to all the four parameters in the filter are

shown in Figure 2. The red dotted lines show the confidence

bounds for the residuals and are calculated from equation

(20).

4347

0 5 10 15 20 25 300.74

0.75

0.76

0.77

0.78

0.79

0.8

0.81

0.82

0.83

Time (seconds)

Eff

icie

ncy (

pe

rce

nt)

Compressor Efficiency

0 5 10 15 20 25 3018.5

19

19.5

20

20.5

21

21.5

22

22.5

Time (seconds)

Ma

ss F

low

(kg

/s)

Compressor Mass Flow Capacity

0 5 10 15 20 25 300.8

0.82

0.84

0.86

0.88

0.9

0.92

Time (seconds)

Eff

icie

ncy (

pe

rce

nt)

Turbine Efficiency

0 5 10 15 20 25 30

4.9

5

5.1

5.2

5.3

5.4

5.5

5.6

Time (seconds)

Ma

ss F

low

(kg

/s)

Turbine Mass Flow Capacity

variable step size

decreasing step size

zero step size

real parameter

Fig. 1. Estimated health parameters for concurrent faults in the turbineand compressor parameters.

0 10 20 30−0.08

−0.06

−0.04

−0.02

0

0.02

0.04Residual for Compressor Efficiency Fault

0 10 20 30−0.08

−0.06

−0.04

−0.02

0

0.02Residual for Compressor Mass Flow Fault

0 10 20 30−0.06

−0.04

−0.02

0

0.02Residual for Turbine Efficiency Fault

0 10 20 30−0.1

−0.05

0

0.05

0.1Residual for Turbine Mass Flow Fault

Fig. 2. Residuals related to the concurrent faults scenario in the turbineand compressor parameters.

From analyzing the residuals, the time of fault occurrence

in each component and its severity can be distinguished. In

order to have a quantitative interpretation on the precision of

the proposed algorithm for state and parameter estimations,

the results related to the 5% fault severity in terms of the

MAE of the estimates for the last 2 seconds of simulations

(following the algorithm convergence) after each change are

summarized in Table I. The presented results show that the

maximum MAE for the states is between 0.2 − 0.7 percent

of their nominal values. In the case of parameters for ηC and

mC the maximum MAE is located around 0.5% and for ηTand mT is around 1% of their nominal values. The accuracy

of the parameter estimation algorithm for different levels of

severity is also presented in Table II.

2) Simultaneous Faults in the Compressor and Turbine

Health Parameters: In the second scenario, the input fuel

flow to the system is changed at t = 2 seconds by decreasing

TABLE I

STATE/PARAMETER ESTIMATION MAES IN THE CASE OF CONCURRENT

FAULTS SCENARIO

state 1st Fault 2nd Fault 3rd Fault 4th Fault 5th Fault

PCC 0.0414 0.0544 0.0569 0.0566 0.0511

N 8.7438 16.4753 9.7447 18.0585 10.6354

TCC 1.5429 3.2456 2.2386 4.3557 2.7713

PNLT 0.0138 0.0244 0.0216 0.0355 0.0242

ηC 0.0027 0.0036 0.0040 0.0040 0.0032

mC 0.0730 0.1236 0.1075 0.1775 0.1180

ηT 0.0019 0.0027 0.0022 0.0124 0.0132

mT 0.0452 0.0574 0.0618 0.0754 0.0595

TABLE II

PARAMETER ESTIMATION MAXIMUM MAES IN THE CASE OF

CONCURRENT FAULTS SCENARIO FOR DIFFERENT FAULT SEVERITY

parameter 1% 3% 5% 10%

ηC 0.0032 0.0040 0.0033 0.0037

mC 0.1058 0.1775 0.1178 0.1116

ηT 0.0044 0.0132 0.0221 0.0440

mT 0.0608 0.0754 0.0602 0.0652

it by 2%. Also, a simultaneous fault in all the 4 health

parameters of the system is applied at time t = 10 seconds.This fault again causes a 5% loss of effectiveness in the

turbine and compressor efficiencies and mass flow capacities.

The related parameter estimation results for applying the

previously mentioned three estimation methods are presented

in Figure 3. Again in this scenario the effectiveness of our

proposed method as compared to the other applied methods

is highlighted .

Considering decreasing the step size yields a better com-

promise for our proposed method (with the adaptive step

size), however we are interested in an algorithm that is ca-

pable of tracking the parameter changes under all conditions.

Therefore, we consider the adaptive step size for implemen-

tation of the FDI task. The residuals in Figure 4 show that

in case of changes in the system input (t = 2 seconds) the

selected residuals do not exceed their confidence bound.

0 5 10 15 200.7

0.75

0.8

0.85

0.9

Time (seconds)

Eff

icie

ncy (

pe

rce

nt)

Compressor Efficiency

0 5 10 15 2018

18.5

19

19.5

20

20.5

21

21.5

22

Time (seconds)

Ma

ss F

low

(kg

/s)

Compressor Mass Flow Capacity

0 5 10 15 200.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

Time (seconds)

Eff

icie

ncy (

pe

rce

nt)

Turbine Efficiency

0 5 10 15 203.5

4

4.5

5

5.5

Time (seconds)

Ma

ss F

low

(kg

/s)

Turbine Mass Flow Capacity

variable step size

decreasing step size

zero step size

real parameter

Fig. 3. Estimated health parameters for simultaneous faults in the turbineand compressor parameters

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0 5 10 15 20−0.15

−0.1

−0.05

0

0.05Residual for Compressor Efficiency Fault

0 5 10 15 20−0.06

−0.04

−0.02

0

0.02Residual for Compressor Mass Flow Fault

0 5 10 15 20−0.15

−0.1

−0.05

0

0.05Residual for Turbine Efficiency Fault

0 5 10 15 20−0.15

−0.1

−0.05

0

0.05

Time (seconds)

Residual for Turbine Mass Flow Fault

Fig. 4. Residuals related to the simultaneous faults scenario in the turbineand compressor parameters.

TABLE III

STATE/PARAMETER ESTIMATION MAES IN THE CASE OF

SIMULTANEOUS FAULTS SCENARIO

state Before Fault After Fault

PCC 0.0541 0.0449

N 8.4156 11.8833

TCC 1.7027 2.0001

PNLT 0.0185 0.0179

ηC 0.0033 0.0031

mC 0.0839 0.0840

ηT 0.0023 0.0143

mT 0.0572 0.0570

The accuracy of the state/parameter algorithm can also be

justified in this case using the results that are summarized

in Table III, where the MAEs before and after the fault

occurrence are reported. The performance of the parameter

estimation algorithm for different severity levels is justified

in Table IV. The results in Tables III and IV show that

the maximum MAE of the estimation of both states and

parameters in the case of simultaneous faults in the system

components is between 0.1 − 0.6% of their nominal values

but in the worst case the post fault MAE for the estimate of

ηT is 1% of its nominal value. It is concluded that in the

case of simultaneous faults in the system components, the

faults with the minimum severity of 6% in the compressor

and 1% in the turbine components can be estimated.

The developed algorithm is capable of detecting, isolating

and estimating the component faults (in the compressor and

the turbine) in the engine application with an accuracy of

0.5% for the compressor faults and 1% for the turbine faults.

TABLE IV

PARAMETER ESTIMATION MAXIMUM MAES IN CASE OF SIMULTANEOUS

FAULTS SCENARIO FOR DIFFERENT FAULT SEVERITY

parameter 1% 3% 5% 10%

ηC 0.0031 0.0032 0.0033 0.0028

mC 0.1014 0.1307 0.0840 0.0891

ηT 0.0043 0.0132 0.0143 0.0442

mT 0.0664 0.0608 0.0572 0.0569

VI. CONCLUSION

In this paper, the particle filtering algorithm is developed in

order to estimate the gas turbine engine states and variations

in its health parameters. A parallel structure is proposed

for simultaneous state and parameter estimation problem.

The state estimator is implemented by using the Regularized

Particle Filter while the parameter estimator is implemented

by using the Auxiliary Particle Filter through considering

a kernel smoothing. A novel method is also proposed for

estimating time-varying parameters in the particle filtering

framework. The results of applying our method for parameter

estimation under healthy and also faulty cases are presented

that show the acceptable performance of the algorithm for a

challenging fault diagnostic application.

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