1 Introduction

Time to failure (TTF) estimation plays an important role in performing Predictive Maintenance [1,2,3,4, 45] or prognostics. The precise TTF estimation ensures predictive maintenance or prognostic performed “just-in-time” [27]. Generally speaking, TTF estimation can be achieved using data-driven methods or physics-based approaches. Physics-based approaches consist of reliability analysis [6] and knowledge-based methods from physical sciences [7,8,9,10]. These approaches assist exploring the underlying physical mechanisms but they demand a great amount of domain knowledge. Moreover, it is difficult to deploy these methods as they tend to rely on difficult to obtain data on component damage or material properties.

With the development and integration of data acquisition devices into complex systems, data-driven methods [13, 14] are now starting to complement the traditional methods for building prognostic models [5]. Data-driven methods use pattern recognition and machine learning algorithms [15,16,17,18,19,20,21,22] to detect changes of the states [28, 29, 44]. Recent research has demonstrated that classification systems could be a feasible solution in identifying the likelihood of component failures in a timely manner. However, none of the existing techniques is able to provide sufficiently precise TTF estimates to optimize predictive maintenance for identifying the optimal time for performing the maintenance action [23, 24].

For instance, the methodology proposed in [11] helps build classifiers (models) for prognostics. These models continuously evaluate the probabilities of a component failure within a pre-specified alert target window (e.g., between 1 and 20 hours in advance of a functional failure), but are unable to estimate a relatively accurate TTF. When a classifier detects patterns in the data that are characteristic of an incipient failure, it generates an alert indicating that the suspected component is likely to fail within the alert target window and without being able to specify the exact number of hours of operation left. With this approach, the larger the alert target window, the larger the imprecision on the TTF estimates. In some specific applications, it is reasonable to try to increase precision by reducing the width of the target window. However, this is generally not suitable as it could prevent the end users from getting alerts as early as possible. In turn, this would reduce the opportunity for optimization. A too narrow target window may also have detrimental effects on the performance of the predictive models. For instance, when a component has various failure modes, each following the own time frame, there is a risk that a model specific to a narrow target window would only be able to detect a fraction of these failure modes.

Estimating TTF technically can be seen as a regression problem and as such, regression analysis and time-series forecasting methods could be used to develop models that are able to directly estimate TTF. To be successful, such models need to accurately map all the subtle changes in the data to specific life reduction estimates. These models also have to account for the fact that with complex components, we often observe significant variations in actual time to failure. Obviously, building such models is a challenging task. On the top of this, data from real world problems is typically characterized by issues such as irregular sampling intervals, sensor measurement errors, and small signal/noise ratio. It is generally hopeless to develop a global regression model for TTF from sensor data. To integrate classification and regression [25, 26] we developed an on-demand regression [12, 44] to improve the precision of the TTF estimation. On-demand regression method relies on a comprehensive data mining methodology to build a classification system capable of identifying incipient component failures and providing rough TTF estimates.

Even though on-demand regression helps improve the precision of TTF estimation largely, its accuracy relies on the quantity but also the quality of system history data. Another principal drawback of data-driven methods is that the model inference process is usually opaque to users [30]; as a result, they may not suitable for some domain problems in which reasoning transparency is required. Physics-based approaches typically involve building models (or mathematical functions) to describe the physics of the system states and failure modes; they incorporate physical understanding of the system into the estimation of system state and/or TTF [31,32,33]. They may not be suitable for some applications where the physical parameters and fault modes may vary under different operation environments [34]. Firstly it is usually difficult to tune the derived models in situ to accommodate time-varying system dynamics. Secondly, physics-based approaches cannot be used for complex systems in which internal state variables are not available to directly measure using general sensors. In this case, inference is requested based on indirect measurements using techniques such as extended Kalman filtering (EKF) and particle filtering (PF).

PF-based approaches have been used for many applications [35,36,37,38,39] such as predictive maintenance, in which the PF is employed to update the nonlinear prediction model and the identified model is applied for forecasting system states. From the published results it is obvious that FP-based approach, a Sequential Monte Carlo (SMC) statistic method [40, 41] is a good solution for addressing the issues in data-driven and physics-based approach. In this work, we apply the PF method to Auxiliary Power Unit (APU) prognostics by estimating remaining start cycles of a Starter. This paper presents the developed PF-based methods for estimating TTF for prognostics along with the experimental results obtained from APU Starter prognostic. This paper extends a preliminary work [42, 43] by introducing the new evaluation results from experiments and providing more technical details. The main contributions are as follows: (1) Building on previous work, we present a detailed technique for implementing PF-based prognostic technologies for real- world applications such as APU prognostics; (2) To our best knowledge, this is the first work to apply FP-based method to APU prognostics to estimate remaining useful starting cycles; (3) we conduct a comprehensive performance comparison to other machine learning-based TTF methods such as Random Forest, SVC (SMOReg), on-demand regression, and so on.

The rest of this paper is organized as follows. Section 2 briefly describes the principle of PF technique; Section 3 presents the algorithm and its implementation; Section 4 provides some experimental results from a case study. Section 5 discusses the results and future work. The final Section concludes the paper.

2 Principle of particle filtering

In predicting system states, if internal state variables are inaccessible to direct measurement using general sensors, it is necessary to infer those states from indirect measurements. Bayesian learning provides a framework for resolving this issue. Given a general discrete-time state estimation problem, the unobservable state vectorX k ∈ R n evolves according to the system model expressed in (1)

$$ X_{k} =f\left( {X_{k-1} } \right)+w_{k} $$
(1)

where f : R n → R n is the system state transition function and w k ∈ R n is a noise whose known distribution is independent of time. At each discrete time instant, an observation Y k ∈ R p becomes available. It is related to the unobservable state vector via (2).

$$ Y_{k} =h\left( {X_{k} } \right)+v_{k} $$
(2)

where, h : R n → R p is the measurement function and v k ∈ R p is another noise whose known distribution is independent of the system noise and time. The Bayesian learning approach for system state estimation is to recursively estimate the probability density function (pdf) of the unobservable state X k based on a sequence of noisy measurements Y 1:k , k= 1, …, K. If X k has an initial density p(X 0), the probability transition density can be presented by p(X k | X k− 1). The inference of the probability of the states X k depends on the marginal filtering density p(X k | Y 1:k ). Assuming that the density p(X k− 1|Y k− 1) is available at step k-1, the prior density of the state at step k can be estimated via the transition density p(X k | X k− 1),

$$ p\left( {X_{k} \vert Y_{1:k-1} } \right)=\int {p\left( {X_{k} \vert X_{k-1} } \right)} p\left( {X_{k-1} \vert Y_{1:k-1} } \right)\,\,dX_{k-1} $$
(3)

Correspondingly, the marginal filtering density is computed via the Bayes’ theorem,

$$ p\left( {X_{k} \vert Y_{1:k} } \right)=\frac{p\left( {Y_{k} \vert X_{k} } \right)p\left( {X_{k} \vert Y_{1:k-1} } \right)}{p\left( {Y_{k} \vert Y_{1:k-1} } \right)} $$
(4)

where, the normalizing constant is determined by

$$ p\left( {Y_{k} \vert Y_{1:k-1} } \right)=\int {p\left( {Y_{k} \vert X_{k} } \right)p\left( {X_{k} \vert Y_{1:k-1} } \right)\,\,d} X_{k} $$
(5)

Equations (3)–(5) form the solution to the Bayesian recursive state estimation problems. In the case of the linear system with Gaussian noise, the above method equals to the Kalman filter. For nonlinear/non-Gaussian systems, there may be no closed-form solutions and thus numerical approximations are usually used.

The PF, sequential important sampling (SIS), is used as an algorithm to solve the recursive Bayesian filtering problem via Monte Carlo simulations. Therefore the posterior density functionp(X k | Y 1:k ) can be represented by a set of random samples (particles) \(\mathrm {x}_{\mathrm {k}}^{\mathrm {i}}\left (\mathrm {i = 1,2,\mathellipsis ,N} \right )\) and their associated weightswki(i = 1,2,…,N).

$$ p\left( x_{k}\thinspace\vert\thinspace Y_{1:k}\right) \approx {\sum}_{i = 1}^{N} {{w_{k}^{i}}\delta \left( x_{k}-{x_{k}^{i}} \right)} ,\thinspace \thinspace {\sum}_{i = 1}^{N} {w_{k}^{i}} = 1 $$
(6)

The \({\mathrm {w}}_{\mathrm {k}}^{\mathrm {i}}\), known as importance weight, is the approximation of the probability density of the corresponding particle. In a nonlinear/non-Gaussian system, the \(\mathrm {w}_{\mathrm {k}}^{\mathrm {i}}\) of a dynamic set of particles can be recursively updated through (7).

$$ \mathrm{w}_{\mathrm{k}}^{\mathrm{i}}\mathrm{\propto }\mathrm{w}_{\mathrm{k-1}}^{\mathrm{i}}\frac{\mathrm{p}\left( \mathrm{y}_{\mathrm{k}}\thinspace\vert\thinspace \mathrm{x}_{\mathrm{k}}^{\mathrm{i}}\right) \mathrm{p}\left( \mathrm{x}_{\mathrm{k}}^{\mathrm{i}}\thinspace\vert\thinspace \mathrm{x}_{\mathrm{k-1}}^{\mathrm{i}}\right) }{\mathrm{q}\left( \mathrm{x}_{\mathrm{k}}^{\mathrm{i}}\thinspace\vert\thinspace {\mathrm{x}_{\mathrm{k-1}}^{\mathrm{i}}\mathrm{,}\mathrm{y}_{\mathrm{k}}}\right) } $$
(7)

where, \(q({x_{k}^{i}}\vert x_{k-1}^{i}y_{k})\) is a proposal function. There are various ways of estimating the importance density function. One common way is to select \(q\left ({x_{k}^{i}}\thinspace \vert \thinspace {x_{k-1}^{i},y_{k}}\right ) =p({x_{k}^{i}}\vert x_{k-1}^{i})\) so that

$$ {w_{k}^{i}}\propto w_{k-1}^{i}p\left( y_{k}\thinspace\vert\thinspace {x_{k}^{i}}\right) $$
(8)

3 Implementation: a case study

This section demonstrates the implementation of PF-based method for estimating TTF through a case study, APU starter prognostics. First some background of APU and operational data are briefed, and then an algorithm of PF implementation is introduced. Over past decade, we have used APU as a case study to demonstrate the progressive development of the predictive modeling technology, from binary classification, to two stage classification, to on-demand regression and to PF-based prognostics. This is an effective way to evaluate the progress of the developed technologies. The following is an overview of the data collected over 10 years.

3.1 APU data

The data have been collected for a fleet of 35 commercial aircraft over a period of 10 years. Only ACARS (Aircraft Communications Addressing and Reporting System) APU starting reports have been made available. The APU diagram is shown in Fig. 1. APU is a small gas turbine engine that delivers mechanical shaft power for the aircraft electrical network and produces bleed air for the main engines starting and air conditioning systems. The APU contains the six main components: power section, load compressor, accessory gearbox, starter, electronic control box (ECB), and Air intake flap.

Fig. 1
figure 1

The schema of the APU

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We generated a dataset based on the APU schema from the data collected in an airline operator. The dataset consists of 18 attributes (5 symbolic, 11 numeric, and 2 for timestamps of the event). More than 161000 observations are available. Only a subset of these observations is relevant to learning predictive models. In this work, we use engine operating hours as the time unit. Our analysis has been based on data generated between 250 operating hours prior to a failure. A comprehensive search in the maintenance database revealed information on 83 occurrences of APU starter motor replacements. One replacement is assumed as a failure event. When an engine suffered consecutive failures in a short period of time, the aforementioned above interval is constrained to ensure that each observation is included only once. The data from 68 failures are used for training and data from the remaining 15 failures are reserved for testing.

In the collected data for each APU operation cycle, there are existing six variables related to APU performance:

  • T 1:ambient air temperature,

  • P 1 : ambient air pressure

  • E G T p e a k : peak value of exhaust gas temperature in starting process,

  • N p e a k : rotational speed at the moment of E G T p e a k

  • t s t a r t : time duration of starting process,

  • E G T s t a b l e : exhaust gas temperature when air conditioning is enable after starting with 100%.

There are 3 parameters related to starting cycles:

  • S n : APU serial number,

  • h o p : cumulative count of APU operating hours,

  • cyc : cumulative count of starting cycles.

In this work, remaining useful cycle (RUC) relevant to TTF is defined as the difference between cyc 0 and cyc. cyc 0 is the cycle count when a failure happened and a repair was undertaken. When RUC is equal to zero (0), it means that APU failed.

3.2 APU data correction

The APU data was collected at a wide range of ambient temperatures from − 20° to 40° and pressures relevant to the airport elevations from sea level to 3557 ft. Since the ambient conditions greatly impact on gas turbine engine performance, it requires a correction from the actual ambient conditions to the sea level condition of international standard atmosphere (ISA) to make the engine parameters comparable. To improve the data quality, the data correction is proposed based on the March number similarity from gas turbine engine theory. Two main parameters (E G T p e a k (noted as EP) and N p e a k (noted as NP)) are corrected using (9) and (10).

$$ \mathrm{NP=}\frac{N_{peak}}{\Theta}^{a_{N}} $$
(9)
$$ EP\mathrm{=}\frac{EGT_{peak}}{\Theta}^{a_{EGT}} $$
(10)

Where, NP stands for the APU rotational speed corresponding to EGT peak values after correction. N p e a k is the data collected from aircraft under various environmental temperatures; Θ is the ratio of actual ambient temperature to ISA temperature, T 1/T I S A . Here, the empirical exponents a E G T and a N are normally determined by running a calibrated thermodynamic computing model provided by engine manufacturers.

3.3 PF algorithm implementation

This section presents an implementation of PF-based algorithm for TTF estimation. Two key parameters related to degradation of APU system states are NP and EP. In terms of statistical analysis, these two parameters are identical in experiencing two phases: normal and degraded operation. Figures 2 and 3 are the examples of moving average of EP and NP during evolution of APU degradation respectively. The moving average \(\mu _{X_{RUC}}\) and the standard deviation \(\sigma _{X_{RUC}}\) are relatively stable in the normal phase, but increasing or decreasing dramatically in the degraded phase. In the normal operation phase, EP and NP satisfy a stationary Gaussian \(N(\mu _{nor}\sigma _{nor}^{2})\). In this phase, the health of a starter is indicated by a signal, a relative constant value equivalent to μ n o r . On the other hand, the noise signal is a stationary white noise with variance of \({\sigma }_{nor}^{2}\). In the degraded phase, EP and NP are a non-stationary distribution. In this phase, the degradation level is indicated by the estimate of the measurements; and the noise signal is a non-stationary white noise with a variance. Therefore, we can apply PF algorithm to filter out the white noise and identify the degradation trend. To this end, we developed APU states estimation models for EP and NP as shown as follows:

$$\begin{array}{@{}rcl@{}} &&\overline{EP}_{k}, \overline{NP}_{k} x_{1_{k}} \,=\, x_{1_{k-1}}\left( \!\frac{x_{3_{k}}}{x_{3_{k-1}}}\!\right)\exp \left[x_{2_{k}}\left( \mathit{RUC}_{k}\right.\right. \end{array} $$
(11)
$$\begin{array}{@{}rcl@{}} &&\left.\left.-\mathit{RUC}_{k-1}\right)\right],\\ &&\lambda_{k}: x_{2_{k}} = x_{2_{k-1}} + \omega_{2_{k^{\prime}}} \end{array} $$
(12)
$$\begin{array}{@{}rcl@{}} &&C_{k}: x_{3_{k}} = x_{3_{k-1}} + \omega_{3_{k^{\prime}}} \end{array} $$
(13)
$$\begin{array}{@{}rcl@{}} &&EP_{k}, EP_{k} Y_{k} = x_{1_{k}} + \nu_{k}. \end{array} $$
(14)

where, the subscript k represents the k th time step and R U C k represents the starting cycle in this k th time step. There are four states, \(\bar {NP}\), λ, C, \(\bar {EP}\) and two measurements, N P and E P,in this system state model. These states and measurements are also denoted as x 1, x 2, x 3 and y respectively. ω 2 and ω 3 are independent Gaussian white noise processes, and v is approximate by the standard deviation of R U C in the collected dataset.

Fig. 2
figure 2

An example of moving average for EP statistical analysis

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Fig. 3
figure 3

An example of moving average for NP statistical analysis

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The system states, \(\bar {NP}\), \(\bar {EP}\) represent the starter signal. As described in (11), its value at time step k is determined from the system states at the previous time step. The second system state λ represents the starter degradation rate. It is located in the exponential part of (10). Therefore, the starter degradation rate between two adjacent starting cycles is indicated by e λ. The higher λ is, the faster a starter degrades along with an exponential growth. When λ = 0, no degradation develops between two starting cycles. The third system state C represents a discrete change of the starter degradation between two adjacent starting cycles. During the PF iterations, the system states are estimated in the framework of recursive Bayesian by constructing the conditional pdf based on the measurements. Consequently, APU starter prognostic is implemented by λ estimation. Once the measurement stops, both λ and C are stabilized with their most recent values. Thus the future degradation trend is expressed as an exponential growth of e λ. The PF-based algorithm is implemented as shown in Table 1.

Table 1 The implementation of PF algorithm
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4 Experiments and results

4.1 TTF modeling experiments

By implementing PF algorithms for EP and NP, we can use λ to perform prognostics for APU starter. The idea is that λ is fixed at its most recent values updated by the available measurements. Then the future degradation trend is expressed as an exponential changes, e λ started from the most recent \(\bar {EP} \bar {NP}\) estimations. The experiments were mainly conducted to learn the weight parameters for PF algorithm and to predict or estimate the E P and NP using learned parameters. The triggering point for prediction is determined based on the statistical analysis given a failure mode. In Figs. 4 to 7, “negative” numbers express the remaining cycles to failure; the “star dots” represents the measurements; the red points are estimation from PF model during learning phase; and black line is prediction of EP and NP from the learnt PF models.

Fig. 4
figure 4

PF TTF estimation result for EP (Triggered RUC= -100) (Unit for EP is Celsius Degree, for RUC is cycles)

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Fig. 7
figure 5

PF TTF estimation result for NP (Triggered RUC = − 75)

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Figures 4 and 5 are experimental results for EP estimation. They show the PF results when the prognostics is triggered at 650C and 750C for EP predictions corresponding to RUC at − 100 and − 50 starting cycles prior to the failure respectively.

Fig. 5
figure 6

PF TTF estimation result for EP (Triggered RUC= − 50) (Unit for EP is Celsius Degree, for RUC is cycles)

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From the results, the APU starter prognostic can be easily performed by setting up a threshold for\(\thinspace \bar {EP}. \)From the Fig. 4, \(\bar {EP}\) threshold is set at 650C. In other words, when EP estimation from the learnt PF model reached 650C, it starts to use EP prediction to estimate the RUC and the triggered EP will be used as onset point of RUC estimation. If the EP prediction is reaching 650C, it means APU starter should be changed or replacement within 100 RUCs. Similarly, Fig. 5 shows the result of prediction triggered at EP= 750C correspond RUC = − 50. From that point, the trained PF model starts to predict the EP for APU Starter prognostics.

We also implement PF algorithm for system state NP. The λ is a consistent in normal operation phase. Then the degradation trend is expressed as an exponential changes, e λ, started from the most recent \(\bar {NP}\) estimations. The experiments were mainly conducted to learn the weight parameters for PF algorithm and to estimate the N P using learned parameters. The triggering point for prediction is determined based on the statistical analysis given a failure mode.

Figures 6 and 7 show the results when NP is triggered at 38% and 33% of NP p e a k for NP prediction corresponding to RUC at − 125 and − 75 starting cycles prior to the failure or replacement, respectively. As we mentioned, \(\bar {NP}\) is the rotational speed of APU when EGT reach its peak value during start. The unit is %. From the results, the TTF estimation of APU starter can be easily performed by setting up a threshold for\(\bar {NP}\). From Fig. 6, \(\bar {NP}\) threshold is set at 38%. In other words, when NP estimation from the learnt PF model reached 38% of NP p e a k , it starts to use NP prediction to perform prognostics and the RUC corresponding to triggered NP will be used as onset point of RUC estimation. If the NP prediction is reducing to 38% of NP p e a k , it means APU starter should be changed or replacement within 125 RUCs. Figure 7 shows the result of prediction triggered at NP= 33% of NP p e a k correspond RUC = − 75. From that point, the trained PF model starts to predict the NP.

Fig. 6
figure 7

PF TTF estimation result for NP (Triggered RUC= − 125)

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4.2 Experimental results from PF-based methods

As we described above, we could built two types of PF models for TTF estimation in this case study: NP-based and EP-based RUC estimation. To evaluate the generic estimation precision, we use 68 time series (failure data) from the collected APU data (See Section 3.1) as training dataset and 15 time series (failures) as the test dataset. We compute mean error (Err), standard deviation (std), and mean square error (MES) for each failure time series based on estimates from NP and EP model by (15), (16) and (17). These statistical evaluation metrics are computed based on actual RUC value and estimated RUC value. Figure 8 shows an example of the relationship between actual RUC and estimated RUC from testset #7.

$$ Err= \frac{1}{N}\sum\limits_{i = 1}^{N} \vert e_{i} \vert $$
(15)
$$ std=\sigma =\sqrt{\sum\limits_{i-1}^{N} \frac{(x_{i}-\mu)^{2}}{N}} $$
(16)
$$ MSE=\sum\limits_{i = 1}^{N}\sqrt{e_{i}} $$
(17)

where, e i is the individual prediction error; x i is the estimated value; and N is the number of examples in the test data.

Fig. 8
figure 8

An example of the estimated RUC vs. actual RUC (NP Model)

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These analysis results are summarized in Table 2.

Table 2 Error analysis for RUC estimation on test dataset running on NP and EP models
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5 Evaluation Results from PF-based methods

To comprehensively evaluate the performance of PF-based methods, we conducted extensive experiments for comparing the results with that from recent regression-based algorithms available in Weka machine-learning platform, including SVM-based regression (SMOReg), Neural Networks such as Multilayer Perception and other regression methods (Random Forest, Simple Liner Regression, and Isotonic Regression). We deployed the developed on-demand [44] regression method in experiment as well. Within the on-demand regression method, first, the classification-based prognostic model analyzes the sensor data to determine if there is a risk for component failure. When such a risk has been confirmed, regression is launched for TTF estimation. To help mitigate the potential negative effects of the issues mentioned in the previous section, several regression models are built, but only the most relevant one is selected based on evaluation performance of each individual model based on MSE metrics, and is applied for TTF estimation at any given time step. In such a way, the performance of TTF estimation is greatly improved. Therefore, if our PF-based methods can outperform the on-demand regression method, it should be identified as a robust and high-performance method for TTF estimation. We run these algorithms on the same testing dataset that was used to evaluate the PF-based algorithms (NP PF model and EP FP model). We computed the same performance metrics. In comprehensive experiments, the on-demand regression was our previous work [44]. Other algorithms are performed individually. In other words, we merely deployed a single model for each algorithm for running testing dataset. Therefore, an individual algorithm did not perform well comparing to on-demand regression method. In this work, the intent is to compare the developed FP-based method to the recent algorithms available from the community. The results in Table 3 demonstrated that the FP-based method for TTF estimation is much more efficient than other individual regression-based algorithms. As we stated in [44], TTF estimation is a huge challenge for development of prognostic technologies. Individual machine learning algorithm sometimes is not suitable for TTF estimation directly.

Table 3 The comprehensive comparison to other algorithm or methods
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6 Discussion

The experimental results above demonstrated that the PF-based method is useful and effective for estimating TTF for prognostics. In the case study, the RUC can be estimated by monitoring engine exhaust gas temperature (EP) or engine speed (NP).

Since there is existing a large variance in various failure models, the precise RUC prediction for a particular APU Starter is challenged. However, our methods suggested that once the estimated \(\bar {EP}\) is 50 °C higher than μ n o r , the APU starts the degradation phase. From the results it is clearly that once the estimated \(\bar {NP}\) is 38% higher than μ n o r , the APU starts the degradation phase. Such information is useful for making decision on predictive maintenance.

From Table 2, the results of NP and EP for RUC estimation are close to each other. This assumes that the APU starter degradation follows a certain exponential growth pattern when we implemented PF-based prognostic for APU starter. This may not be effective for repetitive fluctuations of the starter degradation.

From Table 3, it is obvious that NP FP-based TTF estimation model outperformed all other algorithms even on-demand regression method; and it achieved better result than EP PF-based model. This indicates that PF-based method is robust and can achieve a relative high performance in estimating TTF comprising to other recent algorithms.

As the future work we should integrate data-driven TTF estimation techniques with PF-based TTF estimation algorithms to develop a PF-based model fusion method for improving the precision of TTF estimation. In this work, we implemented the PF-based algorithm for NP and EP respectively. We may develop multiple dimensional PF algorithms to integrate two systems states into a global PF model for TTF estimation.

7 Conclusion

In this paper we presented the developed PF-based method for estimating TTF for prognostics, and applied it to APU prognostics to estimate RUC for APU starter. The developed PF algorithm was implemented using sequential importance sampling to meet the needs of applications. To demonstrate the effectiveness, we conducted the experiments with 10 years historic operational data provided by an airline operator. The comprehensive experiments were conducted to compare the PF method results to other recent approaches such as on-demand regression. From the experimental results, it is obvious that the developed PF-based technique for TTF estimation is useful for performing predictive maintenance. It is promising that the developed methods can provide a relatively precise TTF estimation for the monitored components or machinery systems and it outperformed other machine learning algorithms even on-demand regression. As the future work, we will focus on multiple dimensional PF algorithms and PF-based model fusion methods to continue improving the precision of TTF estimation.