An adaptive-order particle filter for remaining useful life prediction of aviation piston pumps

Tongyang LI,Shaoping WANG,Jian SHI,Zhonghai MA

School of Automation Science and Electrical Engineering,Beihang University,Beijing 100083,China

1.Introduction

An aircraft hydraulic power supply system provides high-pressure fluid for the actuation system,braking system,landing gear system,and other sub-function systems.As the power source of an aircraft hydraulic system,an aviation piston pump’s performance in fluences flight safety directly.Therefore,an aircraft prognostics and health management(PHM)system appears to keep high reliability and long life of an aerial piston pump.In PHM technologies,an accurate estimation of the remaining useful life(RUL)is the most difficult issue because it is related to the failure physics and stress spectrum imposed on a hydraulic pump.Since the structure of an aviation piston pump is very complicated,its failure generation and development are comprehensively affected by inner frictional pairs with uncertain characteristics.Statistically,an aviation piston pump shows variant degradation paths under divers iform operating conditions,which unavoidably brings about a great deal of uncertainties and difficulties in the analytic solution of the RUL.Although the life of a certain type of aviation piston pump can be obtained through tens of thousands of hours of experiments under a fixed spectrum,it is difficult to give the exact RUL under an arbitrary condition.A prognostic estimation method of RUL is imminently needed,which will highly benefit the reduction of costs by providing the possibility to de fine predictive maintenance strategies and prolonging useful life.

Over past decades,a lot of research has been conducted in estimating the RUL of machinery.The methods can be generally divided into two categories1:data-driven methods and model-based methods.Typical data-driven methods based on machine learning are artificial neural networks(ANNs)and the hidden semi-Markov model(HSMM).Zangenehmadar and Moselhi2used an ANN to assess the RUL of pipelines successfully in which more than 80000 groups of data were used for training.Dong et al.3–5applied the HSMM function for machine health prognosis and veri fied the method by using data from a real hydraulic pump health monitoring application case study.In fact,the data used for training were far more than the sample size of a certain type of aviation piston pump.Several thousand of hours were taken to obtain only one set of lift-cycle data of an aviation piston pump.6He et al.7presented a health monitoring and prognostic method using the PSOSVM to predict the RUL for an axial piston pump.The small sample problem is what makes an aviation piston pump distinguish from traditional machinery.Model-based methods like physics-of-failure(POF)and filter-based methods are also widely utilized for prediction of component life.Liu et al.8developed a failure physics model for the creep fatigue of a piston,and the degradation mechanism was analyzed for prognosis.Lamoureux et al.9defined a health indicator to describe the degradation of an aircraft engine fuel pumping unit by using a linear regression method.To build a model by POF,the mechanism needs to be known firstly.Thereby,this kind of method can hardly be used in a complicated system like an aviation piston pump whose failure mechanism is still under research.Filter-based functions take advantages in combining system models with experimental data.For linear systems with Gaussian noise,Kalman filter(KF)is a commonly used prognostic technique,and its effectiveness has been proven in many works.10–13Extended KF(EKF)and unscented KF(UKF)methods are modified KFs to cope with non-linear systems while limitation isshown in somesy stems with high nonlinearity.

According to the fact that the sample size of an aviation piston pump is very small,a data-driven method would not be a good choice for life prediction.Among model-based methods,particle filter(PF)has shown great advantages as an efficient prognostics tool in handling the uncertainty and noise affecting measurements.14A dual-particle- filter method was used to estimate the state of charge for power Li-ion batteries.15To address the particle impoverishment problem,a modified particle filter,named intelligent particle filter(IPF),was proposed by Yin and Zhu.16Miao et al.17introduced an improved PF algorithm–unscented particle filter(UPF)into battery RUL prediction,and the analytical results showed that UPF could predict the actual RUL with an error less than 5%.Zio et al.18–21improved the method a lot by applying PF functions in different degrading systems,and the framework to estimate the RUL of nonlinear components provides ideas for the prognosis of pump systems.However,to adopt the method,a degrading model should be built.Some parts of a piston pump has been modeled like a friction mechanism model of oil between the valve plate and the cylinder block in axial piston pumps22and a wear mechanism model of friction pairs23while a physical model that can be used for prediction has not been proposed.24,25For a system that is difficult to be modeled,an empirical model or a model built according to historical data is used to describe the degrading process.Fagogenis et al.26proposed an auto-regressive(AR)model with an RUSBoost classifier,and a CMAPSS dataset provided by the NASA AMES research center was used to verify the performance of computing the RUL of turbofan engines.A gray prognostic model based on the Markov process was used for a gas turbine compressor’s state estimation.27These kinds of models compromise the merits of data-driven models and have less demand in data quantity.

To address the problem mentioned above,a novel particl efilter based prognostic method for an aviation piston pump’s RUL prediction,named adaptive-order particle filter(AOPF),is proposed in this work.Though the wear mechanism of the pump is complex,the return oil flow has been proven to be a logical characteristic of the pump internal wear status.23The degradation of the pump is re flected by the increase of the flow while the flow shows a non-smooth characteristic that violent fluctuation occurs randomly.The main task is to handle the uncertainties which are classi fied in three categories.28The first one is the uncertainty of future degradation progress of an aviation piston pump which may be caused by unknown load spectrum and random environment factors and will result in different degrading paths,because of which model should be modi fied timely in order to maintain accuracy.Secondly,in modeling,an incomplete data set and some sufficient but not necessary assumptions and simpli fications may prevent a prognostic model from precision.A model with biased parameters will in fluence the performance in prediction.Thirdly,data collected by sensors and acquisition systems are often accompanied by measurement noise.29–31

The rest of this paper is organized as follows:Section 2 describes the particle filtering framework for estimating the RUL and the proposed AOPF technique is described in detail;in Section 3,the application of the method is verified by an experimental test,and a comparison with traditional methods is discussed;in Section 4,some conclusions and remarks are drawn.

2.Adaptive-order particle filter based prognostics

2.1.Degrading characteristics of aviation piston pumps

There are four main friction pairs in a typical aviation piston pump:pair of the cylinder block and the valve plate,pair of the swash plate and the slipper,pair of the slipper and the piston,and pair of the piston and the cylinder block.Under normal circumstances,oil film exists between the friction pairs.It is shown that wear occurs and becomes serious with an increase of the serving time.22The wear process is accompanied by emerging abrasives.The accumulative amount of wear particles can indicate the degree of wear to some extent.However,the sizes of the abrasives are usually too small to be detected.There is no sensor installed to detect wear particles as well.Consequently,abrasives’increase cannot be utilized as a degrading index of the pump.The leakage caused by wear can also lead to a decrease of the outlet flow rate.In an actual hydraulic system of an aircraft,the outlet flow is controlled to be at a stable value,which means that the decline of the outlet flow cannot be monitored.In fact,the leakage oil is usually discharged out of the pump through the return oil outlet.By collecting the data of the return oil flow,the leakage of the pump can therefore be reflected.

The mechanism of wear is very complicated that it cannot be described only by establishing the model of each friction pair.As is mentioned above,abrasives occur randomly during the wear.The abrasives produced by one friction pair will transmit to another friction pair.A mutual reinforcement of wear then happens which leads to the uncertainties of the degrading path.It is obvious that the trend of the return oil flow is increased for wear can only become more serious.What should be taken care of is that short-term declines often appear in the increasing trend,which cannot be explained by measurement noises.The phenomenon has been described as a process with independent incensement by Wang et al.23On this basis,a pump can be modeled with a system noise whose distribution obeys the process with independent in censement.

2.2.Basic algorithm of particle filter based prognostics

Particle filter has become an attractive state estimation method for its capability of accounting for the randomness of the process and the noise affecting measurements.The main idea of PF is that particles are used to describe a probability distribution instead of an analytical expression.To de fine a system,consider a state sequence x0:k={x0，x1，...，xk}and its corresponding measurement sequence y0:k={y0，y1，...，yk},where k denotes a time metric.The state space model is commonly defined as

where f and h represent the state transition function and measurement function,respectively.xk∈Rnxis the state to be estimated,and yk∈Rnyis the observation,where nxand nyare the dimensions of the states and the observations.uk∈Rnxand vk∈Rnyare independent,identically distributed noises with known probability densities.

According to the Bayes’rule,the posterior probability density can be described as

where the requirement is that p(y)＞0,and p(x)is the probability of x.The state estimation problem is to solve the probability density function(PDF)contains all the information about the state xk,which is inferred from the observationsis the initial distribution of the states and is known previously.The PDFis obtained recursively from the prediction step and the update step.

In the prediction step,the system model is used to obtain the prior PDF of the state at time k by the Chapman–Kolmogorov equation as follows:

In the update step,a new observation ykis available and can be used to update the prior PDF of the state via Bayes’rule as follows:

and p(yk|xk)can be obtained by Eq.(2).

For the system defined by Eqs.(1)and(2),f and h are usually nonlinear,which leads to a difficulty in obtaining an analytic solution for the posterior distributionA PF approximates it with a set of samples (particles)where N is the particle number.The initial particles are drawn from p(x0).To address the problem of sampling from the posterior distribution,importance sampling is employed.Samples can be easily drawn from an importance distribution q(x0:k|y0:k),and the importance weight for each particle can be calculated by

where δ is the Dirac function.

2.3.Adaptive-order particle filter

2.3.1.Particle filter with a high-order model

Note that the state space model given by Eqs.(1)and(2)is described by a Markov model of first order.Actually,the system model may be given by

in which the system is described as a p order model.The PF framework still takes hold and is proven as follows.

In the prediction step,the prior PDF of the state at time k is

In the update step,with a new observation yk,the posterior PDF of the state can be calculated by Eqs.(5)and(6).Thus,the Bayesian solution is capable to be used in a high-order model.The proof of the applicability of importance sampling is as follows.

The original definition of the importance weight of a particle is

where σ(x0:t)is the joint probability distribution as

which means that the importance weight is the ratio between two distributions.It can be deduced as

Substitute Eq.(12)into Eq.(13),and then

For each particle,the importance can be obtained by

Then the importance sampling process is proven to be effective in a high-order model.The res ampling step changes the importance sampling weight to be equal by adjusting the number of the particles like

where j is the new particle index after resampling,and xij∈ {xi，i=1，2，...，N}.Substituting Eq.(17)into Eq.(16)yields

By a condensation filter,32take a sub-optimal of q asThe weighting function would be like

The posterior distribution can be approximated by Eq.(8).

2.3.2.Adaptive-order particle filter prediction framework

A general PF uses a fixed dynamic model to estimate system states.In fact,it is difficult to build a dynamic model for a piston pump analytically.The complicated system’s degrading mechanism is still unknown whose model is usually built empirically or built by data-driven models.Models built like these might not be guaranteed to accord with the actual degrading process,which means that updating the model is necessary under such a circumstance.

According to the return oil flow data shown in Fig.1,the curves of return oil flow are not monotonously increasing so that sometimes a decline appears and lasts for a few hours.The most importance in a PF is to ensure the accuracy of the prior distribution.However,a first-order data-driven model could not deal with such un smooth data.Take the hypothetical sequence shown in Fig.2 for an example.If the dynamic model is first-order,the estimated value is more likely to show at xk+1.If a second-order model is taken into consideration,seems to be a more reasonable value.A highorder model could contain more recent information.A shortterm trend by the latest states is more likely to reflect the possible states of next time steps.Note that when the parameters of all the high-order variables are equal to zero,the model is a first-order model.

Based on the theory mentioned above,we propose an adaptive-order particle filter method,which can be applied to a dynamic model with a current time state related to the states at not only last time step but also several time steps previously which can be described by Eqs.(9)and(2).The adaptive order means that the order number of the dynamic model is updated at each time step adaptively.The main difference between a general PF and a high-order particle filter is the prediction step during that to predict the state at time k,if the order of the dynamic model is Ok=p,then the states of the particlesshould be stored without aresampling step.{O0:k}is the order sequence of the updated dynamic model.Thus a recursive function instead of a onestep function is applied to sampling particles from the suboptimal of q,which is

An empirical or data-driven model is built according to the historical data firstly.To fit a new observation sequence,parameters of the model are usually modi fied accordingly.The model may already be accurate enough to describe the trend until time k,when a new data comes.Without changing the form of the original model,the adaptive-order framework re-determines the order number of the model by calculating the accumulated errors.If a higher-order model is calculated to be more fitted,the order number is then updated.By rebuilding the model in a higher-order form,the short-term trend is likely to be more accurate.The adaptive-order framework is shown in Fig.3.

Fig.1 Return oil flow.

Fig.2 Hypothetical sequence.

In the proposed model,the adaptive-order model is embedded in the PF measurement equation.The flow chart of the proposed updated-order particle filter is depicted in Fig.4,and its main steps are as follows:

(1)Initialization step. The initial modelis built by the historical data,in which p0is the order number of the model.The number of particles N is set.The particles and their weights are initialized as

where u0is the obeyed uniform distribution.(2)For k=p，p+1，···

(a) Calculate the prior estimation by the current modelwhich is described by the particles:is the updated state model.

(b) When a new observation ykarrives,update the weight of each particle:

Fig.3 Adaptive-order framework.

Fig.4 Flow chart of the AOPF.

(d) Resampling the particles by the weight.

(e) Update the order number of the model and get a new current model

(3)Output the predicted states sequence:whereis the estimated value of

The parameters of the initial model will not be as important as the form of the model for that the parameters from the historical data will be replaced by the parameters for the new data sequence by the model updating procedure.The model form f0:kis not modified and will be used in the prediction step.When there is no observation arriving at time k+1,the model is fixed,and the weights of the particles will not change.To get the RUL,the current dynamic model is the only dependence,which means that the recursive state in the future xk+tis obtained by

where t is the step of time that xk+t≥Threshold(return oil flow).The distribution of a piston pump’s RUL TRULis defined as

3.Example verification

In this section,data collected by a full-life experiment for a certain type of piston pump are used to demonstrate the forecasting performance of the proposed AOPF algorithm.The return flow data were measured by a turbine flow sensor installed on the return oil pipe of a pump.The work operation of the pump was full output flow by which an accelerated wear process was adopted so that the cost could be reduced.The flow data were recorded every hour by the same measurement system.

where

and

The data shown in Fig.5 are the experimental results of the return oil flow of a certain pump.When the return oil flow reaches 2.8 L/min,the wear condition of the pump is considered to be the threshold of the failure of wear according to the design parameters.The trend of the return oil flow is increasing except some parts of reduction,which means that the leakage of the pump is increasing and gradually leads to an incapability for the pump to provide high-pressure hydraulic power for the actuation system.

Fig.5 Experimental results of the return oil flow.

Fig.6 shows the short-term prediction results by the GM(1,1)and the proposed AOPF method.Both of the algorithms show good performance in the one-step prediction.Each point of the curves stands for a prediction obtained by the prior data.The gray method fits well while the AOPF shows a better result which is demonstrated in Fig.7.The red curve and the blue curve indicate the mean square errors(MSEs)of gray prediction and AOPF prediction,respectively.To get Fig.7,each point is calculated by accumulating all the MSE of the previous points.Obviously,the MSE of the AOPF prediction is lower than that of the gray prediction after 47 time steps.Raw data of the return oil flow illustrate a large fluctuation at the beginning during that from 20 hours the return oil flow continues to fail which impacts the performance of the method.However,after a few steps,the AOPF performs a good anti-interference characteristic.The blue curve keeps lower than the red curve,which means that the error of the AOPF prediction is smaller than that of the gray prediction and the noise is depressed after enough prior knowledge accumulated.

The adaptive-order number of each step is displayed in Fig.8.The order numbers show a polarization that most of the order numbers are very small and large order numbers usually appear and last for some time.Comparing Figs.6–8,a strong connection appears between the large order number and the decline of the raw data.When the trend is rising,the adaptive order number is stabilized at a low level.To adapt the unusual trend,the AOPF matches with a higher-order model adaptively.

The characteristics of the adaptive-order numbers shown in Fig.8 provide an optimization strategy.In a rising stage,the best order number occurs between 0 and 40.Then if the trend of the sequence is determined,the computation burden can be largely declined.The strategy shows a greater significance with an increase of the time.In a later period,the resolving time is impacted mainly by the time on MSE calculation.The strategy could keep the computing time rise steadily.

However,many functions can provide a high precision in the short-term prediction.What challenges is the long-term prediction.In order to verify the performance of a long-term prediction,the raw data are divided into two sets.One is used for model adaption and the other for proving.Taking the first 600 points as the training data,a gray model sequence is generated.Meanwhile,a gray model initialized by the first 4 points are updated adaptively by the AOPF.Long-term prediction results by GM(1,1)and the proposed AOPF method are shown in Fig.9.The actual failure time of the pump is 1186 hours.By gray forecasting with GM(1,1),the time that reaches the threshold is 1103 hours,which means that the error is 73 hours.Whereas the life predicted by the AOPF is 1144 hours and the error is 42 hours,which means that the error is reduced by 42.5%.Note that the precision of the AOPF is affected by the number of particles.More particles lead to a heavier computation burden.In the proposed stage,the number of the particles is 300.In the long-term prediction,the measurement error and the system error are accumulated in the datadriven model.The AOPF can promote the long-term prediction accuracy through filtering the noise of each measurement point.

Fig.6 Short-term prediction comparison.

Fig.7 MSEs of return oil flow comparison between grey prediction and AOPF prediction.

Fig.8 Adaptive-order number of the AOPF.

Fig.9 Long-term prediction comparison.

The order number influences the accuracy of the prediction.Although the AOPF only takes the model which fits best,the relationship between the order number and the performance of that is expected to be known so that an optimization could be applied for a reduction of time cost.By fixing the order number of the AOPF,the results of a 7-order model and a 20-order model are shown in Fig.10,and the performance is listed in Table 1.The 20-order model shows a better performance than that of the 7-order model.The errors of the 7-order model and the 20-order model are 68 hours and 55 hours,respectively.As it is analyzed above,a higher-order model may fit the actual data better.

Fig.10 Different order models’prediction results comparison.

Table 1 Performances of 7-order and 20-order models.

Fig.11 Different order number models’MSEs of the return oil flow.

For an intuitive understanding of the relationship between the order number and the performance,Fig.11 shows the MSEs of different order number models at time 600 h.The order number is from 7 to 60.With the order number increasing,the trend of the MSE is decreasing.The lowest point of the curve occurs near 30 after which the curve shows a fluctuation around 5.5×10-3L/min.It is not that the higher the order number is,the lower the error is.By taking an appropriate order,the error can be reduced by more than 50%which will increase the precision of the RUL prediction accordingly.

4.Conclusions

An AOPF method is proposed in this paper to improve the prediction accuracy of an aviation piston pump’s RUL.In the AOPF,the accuracy of long-term prediction is promoted via changing the order number of a model adaptively.With this strategy,the model is modified with a new observation arriving at every time step,so that the information of the new point and the empirical knowledge can be well fused.Compared to a general PF,the state prior distribution in the AOPF is calculated recursively.In the AOPF,the order number is determined by the minimum mean square error between estimated states and raw data.It shows that the MSE between future data and predicted states is reduced by the proposed method.The experimental results indicate that the AOPF reduces the error by 42.5%in contrast with the traditional grey forecasting method.

However,there are still some improvements that could be made on the AOPF.Firstly,the experimental results show that the overall trend of the MSE is decreasing.When the order number is near 30,the MSE reaches a minimum value and after that,the MSE begins to stabilize.To get a minimum MSE,the MSEs of all order number models should be computed,which leads to a heavy computation burden.A method of optimization may be applied to improve the adaptive strategy.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China(Nos.51620105010,51575019),National Basic Research Program of China(No.2014CB046400),and Program 111 of China.

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