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Abstract

In this article, an improved particle filter with electromagnetism-like mechanism algorithm is proposed for aircraft engine gas-path component abrupt fault diagnosis. In order to avoid the particle degeneracy and sample impoverishment of normal particle filter, the electromagnetism-like mechanism optimization algorithm is introduced into resampling procedure, which adjusts the position of the particles through simulating attraction–repulsion mechanism between charged particles of the electromagnetism theory. The improved particle filter can solve the particle degradation problem and ensure the diversity of the particle set. Meanwhile, it enhances the ability of tracking abrupt fault due to considering the latest measurement information. Comparison of the proposed method with three different filter algorithms is carried out on a univariate nonstationary growth model. Simulations on a turbofan engine model indicate that compared to the normal particle filter, the improved particle filter can ensure the completion of the fault diagnosis within less sampling period and the root mean square error of parameters estimation is reduced.

Keywords

Aircraft engine fault diagnosis particle filter electromagnetism-like mechanism non-Gaussian noise

Introduction

The performance and reliability of aircraft engine are the significant guarantee of the aircraft’s performance and flight safety. However, the complex structure and harsh operating environment make the aircraft engine to be a fault-prone system. Of all the engine failures, gas-path component fault accounts for more than 90%. In order to reduce maintenance costs and improve flight safety, engine performance estimation and fault diagnosis timely and accurately must be taken into account.^1,2

Gas-path failures result in the changes in the performance parameters, such as thermodynamic efficiency and flow capacity, which will further result in changes in the rotational speed, temperature, pressure, and other measured parameters.³ Therefore, aircraft engine gas-path fault diagnosis is mainly achieved through estimating health parameters according to the changes in measurement parameters, after which the health condition of the gas-path components is analyzed and finally the inspection and maintenance are carried out.

Currently, model-based approach is most widely used in gas-path fault diagnosis of aircraft engine, such as Kalman filter (KF), extended Kalman filter (EKF), and unscented Kalman filter (UKF). The KF algorithm is simple, but it is only applicable to linear systems.⁴ The EKF algorithm utilizes the linear filtering theory to solve the nonlinear filtering problem by linearizing the nonlinear model. If the system is strongly nonlinear, linearization errors are non-negligible, which decreases the filtering accuracy and can even cause filtering divergence.⁵ The UKF algorithm can better approximate the nonlinear characteristics of state equation and obtain higher estimation accuracy than EKF without linearization of nonlinear model.⁶ However, the EKF and UKF methods are not applicable for non-Gaussian noise systems, whereas in practice, the process and measurement noise of engine systems are complex, which may not meet the Gaussian distribution assumption.

Particle filtering is a class of filtering methods arising in recent years that is suitable for nonlinear and non-Gaussian system. It is a methodology, based on Bayesian filtering and sequential Monte Carlo, whose basic idea is approximately recursive computation of posterior probability distributions using a set of weighted particles. Breaking through the theory frame of KF, theoretically, the particle filter (PF) is a strongly applicable approach without any limitations to statistical properties of the process noise and the measurement noise.^7,8 However, there is a particle degeneracy phenomenon because the normal PF method, without taking the latest measurement information into account, regards the prior probability as a suboptimal importance function. To overcome the particle degeneracy, resampling technique is used to remove the smaller weighted particles and to copy the larger weighted particles, but it will cause particle impoverishment.^9,10 In view of the main problems of particle filtering, numerous improved methods have been put forward, basically focusing on different choices of the important density functions and resampling methods, such as extended Kalman particle filter (EKPF)¹¹ and Markov chain Monte Carlo (MCMC)-based particle filter¹². Recently, the PF based on intelligent optimization has been developed. Fang et al.¹³ introduce a particle swarm optimization algorithm into particle filter (PSOPF) so that importance sampling distribution moves to higher posterior probability region, which increases the state estimation accuracy. However, the algorithm needs several known empirical parameters which will affect the filter accuracy directly. Zhang et al.¹⁴ propose a method that uses artificial immune particle filter (AIPF) to select the required particles after repeatedly calculating and mutating operations. While the diversity of the particles is enhanced, the algorithm makes lower convergence speed and much higher level of computational complexity because every step the initial antibodies (particles) and the clonal antibodies are mutated and the weights need to be recalculated. Hence, the AIPF is not applicable to parameter estimation for high-dimensional and multi-parameter systems such as aircraft engine model. Electromagnetism-like mechanism (EM) is introduced into PF in Chen et al.¹⁵ The algorithm utilizes the electromagnetism law to improve the distribution of the particles, and it increases the estimated accuracy. However, the algorithm lacks the Local Search procedure which is very important in EM,¹⁶ and the EM is just iterated once every simulation step. These will affect the filter accuracy. In addition, the constant movement coefficient of the EM in this article is only suitable for a specific model.

For the reasons described above, an improved EM-based particle filtering (EMPF) algorithm is proposed in this article. The algorithm, simulating interaction force between charged particles, drives the particles to move toward the high-likelihood region and ensures the particles’ diversity and thus relieves the particle degeneracy and sample impoverishment. At the same time, with the consideration of the latest measurement information in the optimization process, this article attempts to enhance the ability of tracking abrupt fault. Finally, the effectiveness of the particle filtering method for engine gas-path abrupt fault diagnosis is simulated and validated on a turbofan engine.

Particle filtering

Particle filtering was proposed in 1993 by NJ Gordon et al.⁷ The basic idea is to draw random samples (particles) from the importance distribution and then adjust the weights and position of the particles constantly according to the measurement information to approximate the true state posterior distribution. Finally, the weighted sum of particles replaces the integration item so as to obtain the minimum variance estimation of the state. Consider the discrete nonlinear system represented by the following state-space model

x_{k} = f (x_{k - 1}, u_{k - 1}) + ω_{k - 1} z_{k} = h (x_{k}, u_{k}) + υ_{k}

(1)

where $x_{k}$ , $z_{k}$ , and $u_{k}$ are the state vector, the measurement vector, and the known input at time k, respectively. $f (\cdot)$ is the system transition function, $h (\cdot)$ is the measurement function, ${ω_{k}}$ is the system noise sequence, and ${υ_{k}}$ is the measurement noise sequence.

Assume the state prior distribution $p (x_{0} | z_{0}) = p (x_{0})$ is known. Then, the filtering problem is to estimate the posterior probability density function (PDF) $p (x_{k} | z_{1 : k})$ based on the given measurement data sequence ${z_{1 : k}}$ . Assuming at time $k - 1$ the posterior PDF $p (x_{k - 1} | z_{1 : k - 1})$ known, state prediction equation can be expressed via the Chapman–Kolmogorov equation as

p (x_{k} | z_{1 : k - 1}) = \int p (x_{k} | x_{k - 1}) p (x_{k - 1} | z_{1 : k - 1}) d x_{k - 1}

(2)

State update equation can be expressed via Bayes’ rule as

p (x_{k} | z_{1 : k}) = \frac{p (z_{k} | x_{k}) p (x_{k} | z_{1 : k - 1})}{p (z_{k} | z_{1 : k - 1})} = \frac{p (z_{k} | x_{k}) p (x_{k} | z_{1 : k - 1})}{\int p (z_{k} | x_{k}) p (x_{k} | z_{1 : k - 1}) d x_{k}}

(3)

Analytic solutions for the integrals in recurrence relations (2) and (3) just exist in a restrictive set of dynamic systems. Therefore, the PF adopts sequential Monte Carlo sampling method to approximate the optimal Bayesian solution. Under the assumption that N samples ${x_{k}^{i}, i = 0, \dots, N}$ can be drawn independently from the posterior PDF $p (x_{k} | z_{1 : k})$ , the posterior PDF can be approximated as

p (x_{k} | z_{1 : k}) \approx \frac{1}{N} \sum_{i = 1}^{N} δ (x_{k} - x_{k}^{i})

(4)

where $δ (\cdot)$ is the Dirac delta function.

In general, sampling directly from the posterior PDF is intractable, so a set of weighted samples ${x_{k}^{i}, w_{k}^{i}}_{i = 1}^{N}$ can be generated from an easy sampled importance density function $q (x_{k} | x_{k - 1}, z_{k})$ by setting the values of the weights equal to

w_{k}^{i} = w_{k - 1}^{i} \frac{p (z_{k} | x_{k}^{i}) p (x_{k}^{i} | x_{k - 1}^{i})}{q (x_{k}^{i} | x_{k - 1}^{i}, z_{k})}

(5)

In the normal PF, it is convenient to select the prior distribution as the importance distribution, that is

q (x_{k}^{i} | x_{k - 1}^{i}, z_{k}) = p (x_{k}^{i} | x_{k - 1}^{i})

(6)

Thus, equation (5) is reduced to

w_{k}^{i} = w_{k - 1}^{i} p (z_{k} | x_{k}^{i})

(7)

Normalize the weights and the posterior distribution can be expressed as

p (x_{k} | z_{1 : k}) \approx \sum_{i = 1}^{N} w_{k}^{i} δ (x_{k} - x_{k}^{i})

(8)

The law of large numbers can guarantee the equation approximate the true posterior probability $p (x_{k} | z_{1 : k})$ when $N \to \infty$ .

Improved particle filtering algorithm

Because the PF selects the prior PDF as the important density function without considering the latest measurement information, samples of the important density have a large deviation from samples of the true posterior density. As the algorithm evolves continuously, the weight variances increase, and all but few particles have negligible weights. As a result, significant computational resources are wasted on updating particles with minimum relevance to the posterior density, and it leads to the particle degeneracy phenomenon in turn. Although the resampling step reduces the effects of the phenomenon, it loses the particles’ diversity and results in sample impoverishment. To solve these problems, in this article, each sampling particle is assumed as a charged particle, the position of which is adjusted according to the attraction–repulsion mechanism of the electromagnetism theory.

EM algorithm

The EM, proposed by Birbil and Fang¹⁶ who were motivated by the attraction–repulsion mechanism of the electromagnetism law, is an innovative population-based meta-heuristic algorithm for global optimization methods. By adopting the mechanism, each solution is regarded as a charged particle attracted or repulsed by others, moving to the optimal solution under certain rules in EM algorithm, and the convergence properties have been proved.¹⁷ Calculating the maximum value of the objective function $g (x)$ as an example, the basic procedures of EM algorithm are provided as follows:

Initialization. The procedure is used to sample N particles ${x_{1}, x_{2}, \dots, x_{N}}$ randomly from the feasible domain as the initial particles, which is n dimensional, where $x_{i} = {x_{i}^{1}, x_{i}^{2}, \dots, x_{i}^{n}}$ . After that, the objective function value $g (x_{i})$ for each particle is calculated and the particle that currently has the best (maximum) function value is stored in $x_{best}$ .

Local search. The procedure is used to gather the local information for each sample particle, in which random linear search algorithm is used in a certain area to find better particle in order to achieve local update. Experiments show that when applying local search to the current best particle only, it can balance the number of evaluations and the accuracy of the results. After the procedure, the current best particle $x_{best}$ is updated.

Calculation of force vector. Total force calculation is the most important step in EM algorithm that combines the local information and the global information effectively. Imitating the superposition principle of electromagnetic theory, EM algorithm provides information for the next search through computing the total force.

The charge of each particle i determines its power of attraction or repulsion. This charge $q_{i}$ is evaluated as

q_{i} = \exp (- n \frac{g (x_{i}) - g (x_{best})}{\sum_{k = 1}^{N} (g (x_{k}) - g (x_{best}))}), \forall i

(9)

In this way, the particles with better function values possess higher charges. It is worth noticing that unlike electrical charges, no signs are attached to the charge of an individual particle; instead, the direction of a particular force between two particles is decided after comparing their objective function values. Hence, the total force $F_{i}$ exerted on particle i is computed by the following equation

F_{i} = \sum_{j \neq i}^{N} {\begin{matrix} (x_{j} - x_{i}) \frac{q_{i} q_{j}}{{‖ x_{j} - x_{i} ‖}^{2}} if g (x_{j}) > g (x_{i}) \\ (x_{i} - x_{j}) \frac{q_{i} q_{j}}{{‖ x_{j} - x_{i} ‖}^{2}} if g (x_{j}) \leq g (x_{i}) \end{matrix}, \forall i

(10)

As seen in equation (10), between each of the two particles, the particle with a larger objective function value attracts the other one. Contrarily, the particle with smaller objective function value repels the other one as shown in Figure 1. Since the current best particle $x_{best}$ has the maximum objective function value, it acts as a particle with absolute attraction that attracts all other particles in the population.

Figure 1.

The principle of attraction–repulsion mechanism.

4. Movement. When the total force $F_{i}$ is calculated, the corresponding particle is moved in the direction of the force by a random step length as follows

x_{i} = x_{i} + λ \frac{F_{i}}{‖ F_{i} ‖} (RNG), i = 1, 2, \dots, N

(11)

where $λ \in (0, 1)$ is the random step length. In the movement process, RNG is a vector whose components denote the allowed feasible movement step toward the upper bound $U_{k}$ or the lower bound $L_{k}$ .

After the above procedures, an iteration of EM algorithm is completed and the position of each particle is updated.

Improved EMPF algorithm

The performance of particle filtering is improved by utilizing the EM in the sampling process which moves the particles toward a better particle set and keeps a certain distance. Two improvements are proposed during EM algorithm applied in the PF:

Because of the complex computations and large calculations for the nonlinear dynamic model operation in practical application, the estimation time will multiply due to particle optimization using the EM algorithm. So, for the sake of filtering accuracy and efficiency, only the particles whose weights are less than the mean weight ${\bar{w}}_{k} = 1 / N \sum_{i = 1}^{N} (w_{k}^{i})$ will be optimized by EM algorithm and the rest remain stationary. On the other hand, the termination criterion is that the current best particle is changed less than $ε$ or the number of iterations equal to 10.

It can be seen from equation (10) that if two particles are very close, the denominator part will tend to be infinitely small. As a result, $(| | x_{j} - x_{i} | |^{2})^{- 1}$ may be infinite and causes an overflow error. In addition, when the charge is constant, the force between the close particles is greater than the particles with far distance. Moreover, if a particle with larger magnitude of charge is relatively far from the current particle, the force between the particles will be smaller than the force between the particles that have smaller magnitude of charge but are closer, which is not conducive for the particle to move toward the optimal particle. Therefore, the distance factor in the force formula is weakened or even eliminated to accelerate the calculation speed and avoid falling into local optimum.¹⁸ In this article, the denominator part in equation (10), which causes overflow error and introduces the distance factor, is removed. The improved particle force calculation formula is as follows

F_{i} = \sum_{j \neq i}^{N} {\begin{matrix} (x_{j} - x_{i}) q_{i} q_{j} if g (x_{j}) > g (x_{i}) \\ (x_{i} - x_{j}) q_{i} q_{j} if g (x_{j}) \leq g (x_{i}) \end{matrix}, \forall i

(12)

The flow chart of EMPF algorithm is shown in Figure 2. The specific implementation steps are as follows:

Parameter initialization. At time $k = 0$ , generate initial particles $x_{0}^{i} ~ p (x_{0})$ with $w_{0}^{i} = 1 / N$ , $i = 1, 2, \dots, N$ . Set the upper bound $U_{k}$ and lower bound $L_{k}$ for each state parameter.

Importance sampling and weights evaluating. Draw new particles from importance sampling distribution, that is, state transition PDF: $x_{k}^{i} ~ p (x_{k} | x_{k - 1}^{i})$ . Calculate the importance weights $w_{k}^{i}$ by equation (7) once new measurement is available and calculate the mean weight ${\bar{w}}_{k}$ .

Particles adjustment by EM. Set the particle weight $w_{k}^{i}$ as the objective function value for the optimization process and store the particle with maximum weight in $x_{best}$ . Apply random linear search on the current optimal particle to generate a better one. Then, calculate the magnitude of charge of each particle according to equation (9). The moving particle set S is formed by the particles whose weight is less than the mean weight ${\bar{w}}_{k}$ . The total force exerted on each particle is computed by the improved formula (12). Finally, by equation (11), particles are moved along the direction of the force. After the optimization process of EM, particles are encouraged to converge to the true posterior probability distribution. Recalculate the weights of the particles in set S and normalize the weights of all particles

w_{k}^{i} = w_{k}^{i} {(\sum_{i = 1}^{N} (w_{k}^{i}))}^{- 1}

(13)

State estimation

{\hat{x}}_{k} = \sum_{i = 1}^{N} w_{k}^{i} x_{k}^{i}

(14)

Resampling. Calculate effective sample size $N_{eff} = 1 / \sum_{i = 1}^{N} {(w_{k}^{i})}^{2}$ , if $N_{eff} < N_{t}$ , where $N_{t}$ is a fixed threshold, resample particles to be a new sample set with equal weight ${x_{0 : k}^{i}, N^{- 1}}_{i = 1}^{N}$ . Set $k = k + 1$ and go back to step 2.

Figure 2.

The procedure of EMPF algorithm.

Comparison of filters for univariate nonstationary growth model

In this section, four filter methods are applied to the univariate nonstationary growth model (UNGM), which is popular in econometrics and has been used previously in Gordon et al.⁷ and Kitagawa.¹⁹ This model is highly nonlinear and is bimodal in nature. The state-space equations for this model can be written as

\begin{matrix} x_{k} = 0.5 x_{k - 1} + \frac{25 x_{k - 1}}{1 + x_{k - 1}^{2}} + 8 \cos (1.2 (k - 1)) + ω_{k} \\ y_{k} = \frac{x_{k}^{2}}{20} + υ_{k}, k = 1, \dots, M \end{matrix}

(15)

where $ω_{k} ~ N (0, σ_{ω})$ and $υ_{k} ~ N (0, σ_{υ})$ are zero-mean Gaussian white noise. The likelihood $p (y_{k} | x_{k})$ has bimodal nature when $y_{k} > 0$ , and when $y_{k} < 0$ , it is unimodal. The bimodality makes the problem more difficult to address using conventional methods.

The estimation performance of the EKF, PF, PSOPF, and EMPF is assessed in terms of the root mean square error (RMSE): $RMSE = [1 / M \sum_{k = 1}^{M} {(x_{k} - {\hat{x}}_{k})}^{2}]^{1 / 2}$ . The variance of measurement noise $σ_{υ} = 1$ and the variance of process noise (1) $σ_{ω} = 10$ and (2) $σ_{ω} = 20$ . The number of simulation steps $M = 50$ . All the simulations are performed on a personal computer with i3-3220 (3.30 and 3.29 GHz) processor, 2 GB memory, and Windows XP operation system in a MATLAB R2012a environment.

In total, 100 independent simulations were performed to compare the four filters. Figure 3 shows the true states and the estimates obtained using the EKF and PF, PSOPF, and EMPF with $N = 100$ particles for a single simulation when $σ_{ω} = 10$ . Figure 4 gives the estimation error at every step. Note the tendency of the EKF to track the opposite mode of the bimodality, especially when $(x_{k}^{2} / 20) / υ_{k}$ is small. In addition, the values of error for the EKF are much higher, pointing to the occurrence of divergence of the filter. Clearly, the PF, PSOPF, and EMPF outperform the EKF significantly for this highly nonlinear model. But the PF also tracks the opposite mode at few points, for example, $k = 15, 18, 37$ . It is because the values of process noise are large at these points that several particles are symmetrical to the true states but the likelihood is high. Since the intelligent optimization algorithms are introduced to optimize the particles in a larger region, the PSOPF and EMPF estimate the states well.

Figure 3.

Estimation results for UNGM.

Figure 4.

Estimation errors of four algorithms.

In Table 1, the effective sample size N_eff, the mean and variance of RMSE, and the computation time T are given for 100 and 500 particles in the different levels of noise. Even for a small number of particles $N = 100$ , the EMPF performs significantly better than the PF with 500 particles. Compared to the PF, the effective sample size of the EMPF is much larger, which means the particle degeneracy is effectively relieved. Comparison of mean and variance of RMSE, the EMPF has obvious advantages in terms of accuracy and stability. The EMPF is less sensitive with respect to the level of noise by comparing the RMSE of filters with different process noise $σ_{ω} = 10$ and $σ_{ω} = 20$ .

Table 1.

Performance comparison of four algorithms for UNGM with different process noises.

Algorithm	−8% on W₂			Multiple-component fault
Algorithm	RMSE	T (s)	t (s)	RMSE	T (s)	t (s)
EKF	0.0442	1.32	0.032	0.0452	1.31	0.034
PF	0.0458	63.76	7.268	0.0476	63.48	6.358
PSOPF	0.0361	91.64	1.650	0.0388	93.53	1.496
EMPF	0.0338	94.91	0.949	0.0337	94.63	0.763

EKF: extended Kalman filter; PF: particle filter; PSOPF: particle swarm optimization algorithm based particle filter; EMPF: EM-based particle filtering; RMSE: root mean square error.

Notice that the performance of the EMPF is not much different from PSOPF because the two filters both introduce intelligent optimization algorithms into resampling procedure. The difference is that the PSOPF moves the particles according to optimal solutions of individual and population, whereas the EMPF imitates the interaction force between charged particles to move the particles. Due to considering the effect of attraction and the repulsion which not only inhibits the particle degradation but also ensures the diversity, the EMPF performs a little better than PSOPF which does not consider the particle diversity. The most important thing is that the inappropriate selection of inertia coefficient w and learning factors $c_{1}, c_{2}$ in the PSOPF will decrease the filtering accuracy and convergence rate, while the EMPF does not need any empirical parameters.

Application of EMPF for aircraft engine gas-path abrupt fault diagnosis

In this section, the performance of EMPF algorithm is applied for a turbofan engine by selecting eight performance parameters which are the efficiency variation coefficients and the flow capacity variation coefficients of the four rotating components including fan, compressor, high-pressure turbine, and low-pressure turbine. Figure 5 shows the principle of gas-path component fault diagnosis based on EMPF algorithm. The diagnosis principle is to estimate the degradations of the components’ performance parameters through comparing the error between real engine outputs and model prediction. Therefore, the engine gas-path component fault diagnosis can be transformed into the state estimation problem of the components’ performance parameters.

Figure 5.

The principle of engine gas-path fault diagnosis based on EMPF algorithm.

Aircraft engine gas-path abrupt fault model

The nonlinear equations modeling the turbofan engine can be described as follows

\begin{matrix} x_{k} = f (x_{k - 1}, u_{k - 1}) + ω_{k - 1} \\ z_{k} = h (x_{k}, u_{k}) + υ_{k} \end{matrix}

(16)

where control vector $u = [\begin{matrix} W f_{b} & A_{8} \end{matrix}]^{T}$ includes fuel flow and nozzle throat area. State vector is $x = [N_{L} N_{H} Δ E_{1} Δ W_{1} Δ E_{2} Δ W_{2} Δ E_{3} Δ W_{3} Δ E_{4} Δ W_{4}]^{T}$ including low-pressure spool speed $N_{L}$ , high-pressure spool speed $N_{H}$ , and the degradation of fan, compressor, and high- and low-pressure turbine which are denoted as performance parameters as follows

\begin{matrix} Δ E_{i} = \frac{E_{i}}{E_{i, d}} - 1 \\ Δ W_{i} = \frac{W_{i}}{W_{i, d}} - 1 \end{matrix}, i = 1, 2, 3, 4

(17)

where $E_{i}$ and $W_{i}$ represent actual characteristics of every rotating component efficiency and flow capacity, respectively, while $E_{i, d}$ and $W_{i, d}$ represent the design characteristics. It indicates that the corresponding components malfunction when the actual characteristics deviate from the design characteristics in a certain degree. Measurement vector is $z = [N_{L} N_{H} T_{22} P_{22} T_{3} P_{3} T_{43} P_{43} T_{5} P_{5}]^{T}$ , where the signs denote low- and high-pressure spool speed, the exit temperature and pressure of fan, compressor, and high- and low-pressure turbine. $ω_{k}$ and $υ_{k}$ represent process noise and measurement noise, respectively.

The vector RNG in equation (11) is related to the upper or lower bound of state variables. Refer to the statistical data of the performance degeneration of gas-path components after cycling work of a turbofan engine on the NASA MAPSS simulation platform,²⁰ and the bounds are given as U_k = [1.1, 1.1, 0.005, 0.005, 0.005, 0.005, 0.005, 0.03, 0.005, 0.01] and L_k = [0.9, 0.9, −0.04, −0.03, −0.06, −0.09, −0.03, −0.005, −0.01, −0.005], where the first two data are the range of normalized low-pressure and high-pressure rotor speed, and other data are the range of performance variation coefficient. In order to be comparable, in the simulation, the bound conditions are considered in state variable estimations of all algorithms.

Simulations including single-component fault and multiple-component fault are conducted at ground steady operating point $H = 0 km$ , $Ma = 0$ , $W f_{b} = 100 %$ , and $A_{8} = 0.2597 m^{2}$ of the turbofan engine.

Performance parameter estimation with Gaussian noise

Fault modes are given according to Borguet and Leonard.²¹ $W_{2}$ fault occurs with $- 8 %$ step change at the 2nd second as a single-component fault mode, and $E_{1}$ , $W_{2}$ , and $W_{3}$ faults occur with $- 3 %$ , $- 8 %$ , and $2 %$ step changes at the 2nd second as multiple-component fault. $ω$ and $υ$ are white Gaussian independent noise, $ω ~ N (0, Q)$ and $υ ~ N (0, R)$ , where $Q = 0 . 004^{2} I_{10 \times 10}$ and $R = 0 . 003^{2} I_{10 \times 10}$ . In the simulation of state estimation for engine gas-path components using the EKF, PF, PSOPF, and EMPF algorithm, the number of simulation steps is $M = 500$ , the sampling time period is 0.02 s, and the number of particles is $N = 50$ . The results are shown in Figures 6 and 7. Figure 8 shows the sum of errors of eight parameters at every moment. And Table 2 shows the comparison of estimation accuracy, computation time, and abrupt fault diagnosis time. The simulation time of 500 steps is defined as computation time T. The time from fault exited to performance parameters estimated is denoted as abrupt fault diagnosis time $t = m (T / M)$ , where m is the simulation step number required to diagnose the fault. Here, when the estimated parameters remain in its $\pm 0.005$ range of the degradation, the abrupt fault is considered to be diagnosed.

Figure 6.

Estimation results of four algorithms with Gaussian noise under the single-component fault: (a) EKF, (b) PF, (c) PSOPF, and (d) EMPF.

Figure 7.

Estimation results of four algorithms with Gaussian noise under the multiple-component fault: (a) EKF, (b) PF, (c) PSOPF, and (d) EMPF.

Figure 8.

Estimation errors of four algorithms with Gaussian noise: (a) single-component fault and (b) multiple-component fault.

Table 2.

Performance comparison of four algorithms with Gaussian noise.

Algorithm	−8% on W₂			Multiple-component fault
Algorithm	RMSE	T (s)	t (s)	RMSE	T (s)	t (s)
EKF	0.0551	1.33	0.045	0.546	1.33	0.051
PF	0.0469	63.41	9.512	0.0462	63.26	5.314
PSOPF	0.0369	91.45	1.634	0.0395	90.56	1.721
EMPF	0.0347	94.18	1.036	0.0341	93.64	0.936

EKF: extended Kalman filter; PF: particle filter; PSOPF: particle swarm optimization algorithm based particle filter; EMPF: EM-based particle filtering; RMSE: root mean square error.

As it can be seen from Figures 6 and 8(a), in the Gaussian system, due to the existence of linear error, the parameters estimated with the EKF fluctuate greatly, and its steady-state accuracy is lower than the PF. However, the PFs track the abrupt process slowly due to complex computation and the ignorance of the latest measurements. So, as shown in Table 2, the RMSE of the two algorithms is close. The intelligent optimization algorithm is introduced into the PSOPF and the EMPF, which alleviates the particle degeneracy phenomenon and improves the steady-state accuracy of estimation. Meanwhile, with the consideration of the latest measurements in the optimization process, both of the improved filters can track the abrupt fault very well, which improves the dynamic estimation accuracy. As a result, the RMSE reduces greatly. However, the estimation error of the high-pressure turbine efficiency $Δ E_{3}$ with the PSOPF is quite large, which may lead to misdiagnosis of faults, while the EMPF can identify correlation among different parameters well. As shown in Figures 7 and 8(b), under the multiple-component fault diagnosis, the EMPF algorithm maintains the estimation accuracy at quite a high level and has good stability. However, the process of particle optimization based on the EM requires iteratively calling the nonlinear engine model, which results in the increase in the computation.

Performance parameter estimation with non-Gaussian noise

In general, the real engine works in a complex and changeable environment, the noise of which contains not only internal white noise but also nonstationary colored noise; therefore, the process noise and the measurement noise are not necessarily white Gaussian noise.²² In order to simulate the real engine, this article selects Gamma distribution function as the non-Gaussian noise for simulation evaluation: $ω ~ Q \times Γ (r, λ), υ ~ R \times Γ (r, λ)$ . In this simulation, $r = 0.25, λ = 0.5$ , which means the mean and variance are 0.5 and 1, respectively. The experiments take gas-path component fault same as the one with Gaussian noise for simulation. The results are shown in Figures 9 and 10. Figure 11 shows the sum of errors of eight parameters at every moment. And Table 3 shows the comparison of estimation accuracy, computation time, and abrupt fault diagnosis time.

Figure 9.

Estimation results of four algorithms with non-Gaussian noise under the single-component fault: (a) EKF, (b) PF, (c) PSOPF, and (d) EMPF.

Figure 10.

Estimation results of four algorithms with non-Gaussian noise under the multiple-component fault: (a) EKF, (b) PF, (c) PSOPF, and (d) EMPF.

Figure 11.

Estimation errors of four algorithms with non-Gaussian noise: (a) single-component fault and (b) multiple-component fault.

Table 3.

Performance comparison of four algorithms with non-Gaussian noise.

Noise characteristics	Methods	N	N_eff	RMSE		T (s)
Noise characteristics	Methods	N	N_eff	Mean	Variance	T (s)
$\begin{matrix} σ_{ω} = 10 \\ σ_{υ} = 1 \end{matrix}$	EKF	–	–	18.2915	44.2548	0.0021
	PF	100	11.975	4.7835	2.6071	0.1543
	PF	500	18.442	4.2551	2.0014	0.5772
	PSOPF	100	19.215	2.8363	1.2865	0.4823
	EMPF	100	21.143	2.1715	1.0807	0.5135
$\begin{matrix} σ_{ω} = 20 \\ σ_{υ} = 1 \end{matrix}$	EKF	–	–	23.8362	90.4595	0.0019
	PF	100	12.344	6.1254	3.0201	0.1236
	PSOPF	100	19.049	3.4511	0.9245	0.4801
	EMPF	100	20.471	2.8152	0.9107	0.5230

EKF: extended Kalman filter; PF: particle filter; PSOPF: particle swarm optimization algorithm based particle filter; EMPF: EM-based particle filtering; RMSE: root mean square error.

Figures 9 –11 and Table 3 show that for non-Gaussian system, the estimation accuracy of the EKF decreases greatly and thus the misdiagnosis happens easily. Although the RMSEs of the PF, the PSOPF, and the EMPF increase slightly, the performance parameters of the engine are still tracked well, which means that the three methods have a better applicability within non-Gaussian noise. Among them, due to the remission of the particle degeneracy and sample impoverishment phenomenon and the consideration of the latest measurements, the EMPF algorithm maintains a higher filtering accuracy and tracks the abrupt fault in a very short time.

Conclusion

An improved particle filtering algorithm is proposed for the gas-path fault diagnosis of aircraft engine. Considering the particle degeneracy and sample impoverishment phenomenon of the normal particle filtering and the slow response of estimation for abrupt fault, the intelligent optimization algorithm has been introduced and the EM-based particle filtering is proposed.

The estimation effectiveness of the four nonlinear filters, the EKF, the PF, the PSOPF, and the EMPF, are compared for UNGM and engine gas-path fault model. Compared to the EKF, the PF, the PSOPF, and the EMPF are not only applicable for the Gaussian system but also applicable for non-Gaussian system. Because of the introduction of the intelligent optimization algorithms and the consideration of the latest measurements, the estimation effect of the performance parameters with the PSOPF and the EMPF is better than the PF. In addition, while the EM drives the particles to move toward the high-likelihood region, it also keeps them in a certain distance, which ensures the particles’ diversity. The simulation results show that the estimation effect with the EMPF is a little better than the one with the PSOPF. The EMPF does not need any empirical parameters which cause the decrease in PSOPF accuracy and slow convergence. In summary, the EMPF is an effective filtering method suitable for gas-path abrupt fault diagnosis of nonlinear engine system.

Footnotes

Academic Editor: Nam-Ho Kim

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (51276087).

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An improved particle filtering algorithm for aircraft engine gas-path fault diagnosis

Abstract

Keywords

Introduction

Particle filtering

Improved particle filtering algorithm

EM algorithm

Improved EMPF algorithm

Comparison of filters for univariate nonstationary growth model

Application of EMPF for aircraft engine gas-path abrupt fault diagnosis

Aircraft engine gas-path abrupt fault model

Performance parameter estimation with Gaussian noise

Performance parameter estimation with non-Gaussian noise

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References