Xin ZHOU,Feng LU,Jinquan HUANG

College of Energy and Power Engineering,Jiangsu Province Key Laboratory of Aerospace Power System,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

KEYWORDS Component-level model;Condition monitoring;Fault diagnosis;Measurement reconstruction;Turbofan engines;Unscented Kalman filter

Abstract Aero-engine gas path health monitoring plays a critical role in Engine Health Management(EHM).To achieve unbiased estimation,traditional filtering methods have strict requirements on measurement parameters which sometimes cannot be measured in engineering.The most typical one is the High-Pressure Turbine(HPT)exit pressure,which is vital to distinguishing failure modes between different turbines.For the case of an abrupt failure occurring in a single turbine component,a model-based sensor measurement reconstruction method is proposed in this paper.First,to estimate the missing measurements,the forward algorithm and the backward algorithm are developed based on corresponding component models according to the failure hypotheses.Then,a new fault diagnosis logic is designed and the traditional nonlinear filter is improved by adding the measurement estimation module and the health parameter correction module,which uses the reconstructed measurement to complete the health parameters estimation.Simulation results show that the proposed method can well restore the desired measurement and the estimated measurement can be used in the turbofan engine gas path diagnosis.Compared with the diagnosis under the condition of missing sensors,this method can distinguish between different failure modes,quantify the variations of health parameters,and achieve good performance at multiple operating points in the flight envelope.

1.Introduction

Modern aircraft gas turbine engines are highly complex systems which are required to provide reliable power generation over thousands of flight cycles while being subjected to a broad range of operating loads and conditions including extreme temperature and strong vibration environments.Over repeated flight cycles,the functions of the engine components will degenerate,and even malfunction.These problems are usually caused by the compressor fouling,Foreign Object Damage(FOD),blade erosion and corrosion,increase of the bladetip clearance in the turbine,labyrinth seal leakage,wear and erosion,and corrosion in the hot sections due to long-term use.1,2Generally these deteriorations are reflected by variations of efficiencies and flow capacities of the components,which are the so-called‘health parameters'.Though health parameters cannot be measured directly,their degradations will cause changes in some observable parameters,such as temperature,pressure and rotational speed.The health degradation estimation and condition monitoring play an important role in the engine gas path health management system in order to enhance the reliability,availability and safety of the gas turbine engine.3,4In addition,the fault detection mechanism also has great significance for the on-condition maintenance and fault tolerant control.5

To achieve a real-time gas path fault diagnosis,a large number of studies have been performed,including modelbased approaches(such as filters and observers)and datadriven approaches(such as neural networks and machine learning).Some efforts have been made in the data-driven approaches.Xu and Shi proposed a new multi-class fault diagnosis algorithm based on H-SVM and applied it to aeroengines to fast diagnose the multi-class single faults and combination faults for the gas path components.6A data-driven fault detection and degradation estimation scheme is developed for GTE diagnostics based on an Adaptive Neuro-Fuzzy Inference System(ANFIS)by Hanachi et al.7Compared to data-driven approaches,model-based approaches utilize all model information available, and offer better estimation accuracy. The model-based method has been employed for engine gas-path fault diagnosis since 1970s,and the unmeasured component performance shifts are estimated from the residuals between the engine model outputs and sensed signals.A well-developed model-based strategy for health monitoring systems is the Kalman Filter(KF)strategy.In recent decades,variants of Kalman filter have been widely used in gas turbine engine analyses.Simon et al applied constrained Kalman filtering,along with the constraint tuning on the basis of measurement residuals,to estimating engine health parameters.8,9Kobayashi and Simon proposed a constant gain extended Kalman filter for in-flight estimation of non-measurable performance parameters of aircraft engines and a baseline system for online diagnosis.10,11Naderi et al developed a nonlinear multiple model fault detection and isolation scheme for health monitoring of jet engines.12An improved hybrid architecture composed of a nonlinear onboard engine model and piecewise linear Kalman filter for intelligent engine control is designed by Guo,the outputs of the nonlinear on-board engine model being used for the baseline of the piecewise linear Kalman filter,while engine performance deterioration factors are estimated on-line by the measured engine output deviations.13Yang et al proposed a kind of robust observer called UIO(Unknown Input Observer)for model-based FDI systems to locate or isolate the gas path fault.14Furthermore,several other methods are also applied to the gas path diagnosis and health parameters estimation.Yang et al applied a nonlinear,non-Gaussian Particle Filter(PF)for engine health parameter estimation.15Chang et al developed a new gas-path health monitoring architecture using a Sliding Mode Observer(SMO)to enhance robustness against uncertainties.16

With the increasing requirements of modern aero-engine control systems,the number of control variables and sensors needed is also increasing.17The fault-tolerant control and health management of aero-engines both depend on accurate and reliable sensor readings,while the number of sensors available for installation is limited because most of the sensors work in severe high temperature and strong vibration environments.4,18Many researchers have worked out the solutions to the problem of limited sensors.Simon and Garg proposed a systematic approach to select an optimal suite of sensors for on-board aircraft gas turbine engine health estimation and evaluated two different model tuning parameter vector selection approaches:the conventional approach of selecting a subset of health parameters to serve as the tuning parameters,and an alternative approach that selects tuning parameters as a linear combination of all health parameters.19They also presented a linear point design methodology to solve the underdetermined estimation problem,which uses a multivariable iterative search routine that seeks to minimize the theoretical mean-squared estimation error.20,21For performance estimation,sensor selection metrics are presented for two types of estimators including a Kalman filter and a Maximum A Posteriori(MAP)estimator based on linear estimation and the probability theory by Simon and Aidan,which can minimize the theoretical Sum of Squared Estimation Errors(SSEE)of health parameter estimation.22

However,these methods decease the sensor numbers by means of reducing the dimension of the state parameters to be estimated,and many of them are based on the linearized model. Therefore, higher precision for linear models is required and the scope of application is limited.In most situations,unbiased estimates of all health parameters cannot be achieved due to the dimensionality reduction model.For the reasons mentioned above,the main contribution of this paper is to propose a new model-based sensor measurement reconstruction method to achieve turbofan engine fault diagnoses with a sensor missing in the case that an abrupt failure occurs in a single turbine component.This article mainly focuses on measurement reconstruction and diagnosis in the absence of high pressure turbine exit pressure sensor(P43).Assumptions are made according to different failure hypotheses,and the thermodynamic parameters of the desired cross section that lacks of a sensor are calculated by the mathematical model of engine components using the available information,and the estimated parameters will be used as the measurements for the filter instead of the real sensor signals to complete the diagnosis.The advantage of this method is that the state variable is not required to be dimensionally reduced,and it can also be applied to the nonlinear model.The simulation results show that this method can effectively solve the problem of the health parameter coupling under the condition of sensor loss,and achieve good diagnostic performance in the whole flight envelop.

This paper is organized as follows:Section 2 formulates the problem and analyzes why the HPT exit pressure is important to the estimation;the measurement reconstruction method of the HPT exit pressure is described in Section 3;Section 4 introduces the structure of the diagnosis system and explains how the reconstructed measurement is used; some simulation results and analyses are given in Section 5,while Section 6 concludes the paper.

2.Problem formulation

The aero-engine state estimation is usually achieved through the residuals between the engine measurements and the model outputs.According to Kalman filter theory,unbiased estimation can be obtained if and only if the dimension of the measured parameters is more than or equal to that of the estimated parameters.However,in practical applications,the type and number of sensors are limited by various factors,and their selection is crucial to the design of the filter.Usually,optional measurement parameters include the following parameters or part of them:high-pressure shaft speed,lowpressure shaft speed,fan exit temperature and pressure,compressor exit temperature and pressure,HPT exit temperature and pressure,Low-Pressure Turbine(LPT)exit temperature and pressure.Except the two necessary speed sensors,the remaining sensors can be selected according to the demand.

Due to installation restrictions of sensors,some of the above parameters are often unmeasurable,such as the HPT exit parameters;however,they determine the accuracy of the estimation.After repeated simulation experiments,an interesting fact is found;that is,in the absence of the HPT exit pressure sensor,the estimation of certain health parameters will deviate,sometimes diverge.Even after some sensors are added in other cross sections so that the total sensor number can meet the unbiased estimate request, the situation cannot be alleviated.

This phenomenon can be illustrated by the following example in the circumstance that an abrupt failure occurs in the HPT.Assuming the nonlinear aero-thermodynamic model of a turbofan engine is given by Eq.(1):

where k is the time index,y the measured output,x∈R8×1the state vector representing the engine health condition,and u∈R2×1is the control input including the fuel flow Wfand the nozzle throttle area A8.The noise terms w and v represent the process inaccuracies and measurement inaccuracies in the model.The sampling frequency is 50 Hz.During the lifespan of the engine,the components may inevitably experience faults and degeneration,leading to changes in the thermodynamic performance of the turbofan engine.The typical feature of these degradations is that the flow capacity or efficiency of the four rotating components may change.To quantitatively describe the degree of engine failure or degeneration,the following health parameters are defined.

where subscript i indicates different components,ηiand Wiare the actual values of efficiency and flow capacity of the corresponding components under the current working condition,while ηi*and Wi*are the ideal values of the efficiency and flow capacity of the corresponding components under the fault-free condition.The eight health parameters and their acronyms used in this paper are listed in Table 1.

Table 1 Turbofan engine model health parameters.

Take a dual-spool turbofan engine as an example.The locations and the functions of the sensors used in this paper,as well as the health parameters of each rotating component,are briefly described.Fig.1 shows the schematic diagram of the turbofan engine.As can be seen,the main components of the engine are:an inlet,a fan,a compressor,a combustor,an HPT,an LPT,a mixing chamber,a bypass,and a nozzle.The eight health parameters which need to be estimated for the engine fault diagnosis are marked in the figure,where subscript i=1,2,3,4 denotes the fan,the compressor,the HPT and the LPT,respectively.Table 1 provides their meanings and abbreviations.These unmeasurable health parameters are thermodynamically related with the measurable parameters.Sensors are installed on different cross sections of the engine to measure these thermodynamic parameters.The numbers at the bottom of Fig.1 indicate the serial numbers of the cross sections. The acronyms and the cross section subscripts together describe the positions and types of the sensors,where P represents the pressure sensor and T the temperature sensor.The available sensors and their standard deviations are listed in Table 2.

Table 2 lists ten measurement parameters commonly used in experiments or simulations.Researchers usually choose eight or more of them to achieve accurate estimation of the health parameters.Among them,rotational speed sensors are necessary.In addition,one or two temperature and/or pressure sensors should be installed on each cross section.Considering the locations and the working environment of the sensors,the exit parameters of the fan and the compressor are much easier to measure than the exit parameters of turbines.Different combinations of measurement parameters have different effects on the estimation effect.Some parameters are usually not measurable in practical applications;however,they are of great importance in distinguishing different failure modes,the most typical of which is P43.To verify this phenomenon,some comparative experiments were carried out.For different failure modes,the typical nonlinear filtering algorithm-UKF was used to estimate the health state of the turbofan engine at several working conditions such as on the ground or in the high altitude,through two kinds of combination of measurement parameters:I.NL,NH,T22,P22,T3,P3,P43,T43,T5,P5and II.NL,NH,T22,P22,T3,P3,T43,T5,P5,respectively.23The component-level-model used for the simulation is written in C++language and packaged with Dynamic Link Library(DLL)for use in the Matlab environment.Other models mentioned in the following part of this paper are also used in this way.

Fig.1 Schematic representation of a gas turbine engine.

Table 2 Available measurements of turbofan engine.

5%abrupt failure is injected into SE3,SW3,SE4,SW4,respectively,and the ground(design point)and high-altitude(H=4000 m,Ma=1.7)states are selected for the health parameter estimation.At the design point,the sum of the relative errors of the measurement parameter combination I for the eight health parameter estimation results are:0.3503%,0.4113%,0.3347%,0.2420%,respectively;while the sum of the relative errors of the measurement parameter combination II for the estimation are:4.3355%,4.1456%,4.8420%,and 4.8018%,respectively.In the high-altitude environment,two sets of relative errors are:I.0.2387%,0.3636%,0.3937%,0.3211% and II. 7.8927%, 1.4597%, 8.1649%, 1.6854%.Among them,SE3,SE4and SW4have the largest deviations in the estimation.With measurement combination I(including P43),the estimation errors of SE3,SE4and SW4do not exceed 0.18%,0.13%and 0.11%,while with measurement combination II(without P43),the maximum estimation errors of SE3,SE4and SW4can reach 1.5%,3.83%and 2.79%,which are almost dozens of times as much as the former.

As the simulation results indicate,if P43is included in the measurement parameters,the filter can estimate the variations of the health parameters well when the failure occurs.Nevertheless,when P43is not included in the measurement parameters,the estimation will be biased or even diverge,which is mainly manifested as the coupling of SE3,SE4and SW4.A simple explanation can be given by the condition number of the matrix.The condition number characterizes the sensitivity of the matrix calculation to errors.Generally,a matrix with a large condition number has several vectors with strong relativity.For a given matrix A ∈Rm×n,the conditional number is defined as:

where σmaxand σminare the maximum and minimum singular values of A,respectively.

A sensor analysis can be carried out through the condition number of the deviation matrix.Firstly,the deviation matrix of the sensor measurement parameters under the given health parameter mutation(-1%)is established.Since eight health parameters are to be estimated,eight measurement parameters are needed.Each group of sensors corresponds to an eight dimensional deviation matrix.The row elements are the variations of all the measurement parameters with respect to the nominal state when any single health parameter changes abruptly;and the column elements are the variations of a measurement parameter with respect to the nominal state when different health parameters have faults.Table 3 lists the deviation matrix.

According to the deviation matrix,the condition numbers of each sensor matrix are obtained for a special combination of sensors respectively,as shown in Table 4.

As shown in Table 4,all the first few groups of sensors with a small condition number have a P43sensor,while the condition numbers corresponding to those groups of sensors without P43are large, which illustrates its significance to parameter estimation.The P43sensor is located between the high and low pressure turbines,and its measurement carries some important information that differentiates the faults between the HPT and the LPT.Therefore the absence of it will lead to parameter coupling.After several simulations,it can be found that coupling always occurs in these three parameters:HPT efficiency,LPT efficiency,and LPT flow capacity.This means when one of the SE3,SE4or SW4fails,the diagnostic system is unable to determine the fault component or quantify the changes of the health parameters without P43.The estimated values of these three parameters are often divergent;consequently,the model cannot run properly.For the otherfive parameters SE1,SW1,SE2,SW2,SW3,the absence of P43does not affect their estimation results.However,the system may still give the wrong or divergent results of SE3,SE4or SW4,even when it has already acquired accurate estimation of the other five parameters.According to the Kalman filtering theory,the filter adjusts the estimation based on the error between the measured parameters.Although the deviation of the measurements between the model and the real engine is small,this situation cannot be alleviated.To solve the problem of inaccurate estimates in the absence of P43,a model-based HPT exit pressure measurement reconstruction method is proposed in this paper to realize accurate estimation.

Table 3 Sensor measurement parameters deviation matrix under given health parameter change(10-4).

Table 4 Sensor combinations and their condition numbers.

3.P43 reconstruction method based on component model

3.1.P43 signal reconstruction

The gas turbine engine studied in this paper is an axial,dualspool,low-bypass turbofan engine.The main components and the schematic diagram have been shown in Fig.1.Each component has its corresponding input parameters and output parameters.24In Fig.1,the air flow passing through the fan is divided into two streams:one flows into the core engine path,and the other into the bypass duct.The air flowing into the core machine mixes with the fuel in the combustor to produce high temperature gas to drive the turbine.The gas flowing out of the LPT is mixed with the air from the bypass in the mixing chamber.The mixed gas is guided into the Laval nozzle which has a variable throttle area.The dot line in the figure represents the flow path of the cooling air and the bleeding air.The notations and subscripts used in the modeling are listed in Table 5.

3.2.Gas turbine engine component level model(CLM)description

The basic idea of calculating P43measurement is to design signal reconstruction schemes based on the assumptions of different component failure cases by using other effective sensor measurements.The premise of the assumptions is that only one turbine fails,either the HPT or the LPT,and T43is measurable.When the LPT fails,a forward algorithm is designed to estimate P43,which mainly uses the compressor exit measurements(P3,T3)and the HPT component model to calculate the HPT exit parameters.When the HPT fails,a backward algorithm is designed using the LPT exit measurements(P5,T5)and the LPT component model to calculate its inlet parameters reversely.In addition,since the presence or absence of P43does not affect the estimation effect of SW3,the HPT failure is mentioned here only for efficiency.In this way,a suitable algorithm can be selected to solve the missing P43signal according to different fault conditions.

3.2.1.Turbine component calculation

The forward and backward algorithms mentioned below are both developed based on the turbine component model.A brief introduction of the aerodynamics calculation process of the turbine is provided here.Parameters describing the characteristics of the turbine can be obtained by interpolating the performance map,where the pressure ratio is one of the guess variables of the common operation equations.Air flow and efficiency are calculated by the corrected rotating speed and pressure ratio through the performance maps,as shown in Eq.(4).

Table 5 Subscripts used in modeling.

where n is the corrected rotating speed,W the mass flow,π the pressure ratio,fmap1the characteristic of n versus π,fmap2the characteristic of n versus W and C the coefficient of the mass flow or the efficiency.

The output parameters are calculated by the input parameters and the pressure ratio:

where S is the entropy of the gas,H the enthalpy of the gas,fS2Hthe function calculating the gas enthalpy from its entropy,fH2Tthe function calculating gas temperature from its enthalpy,and fris the fuel-air ratio.

3.2.2.Forward calculation

The forward calculation of the P43sensor signal is based on the HPT component model.As is mentioned above,the precondition of using the forward algorithm to estimate P43is that there is no failure or degradation in the HPT efficiency,which means SE3=1.Since it has been analyzed that the absence of P43measurement will not affect the estimation of SW3,the estimated SW3and the undegraded SE3are regarded as the input parameters of the HPT component.The other measurements needed are P3,T3,and T43.The calculation process is shown in Fig.2.

First,the input parameters of the HPT,P4and T4,are calculated by the model of the combustor and the compressor exit temperature and pressure which are measured by real sensors.In the calculation of the combustor,it is not necessary to use the initial guess parameters(rotational speed and pressure ratio)of the engine.Instead,the exit parameters of the combustor can be directly calculated by Eq.(6)without any iteration.

where σBis the total pressure recovery coefficient of the combustor,Hμthe heat value of the fuel,and ηcombis the combustion efficiency.

At this point,the HPT inlet pressure and temperature are obtained,and the problem of solving P43turns into solving πHPT.The component model of the HPT is considered as a nonlinear function as shown in Eqs.(4)and(5).The pressure ratio can be solved with Newton's iterative method,with the HPT inlet temperature and pressure,exit temperature,high pressure shaft rotor speed and component characteristic known.Given the initial πHPTas the guess value,which is usually selected as the pressure ratio calculated by the complete engine model in the last sampling step,the HPT component model runs once to obtain the corresponding T43,which will be compared with the actual sensor signal.Then the initial guess is modified according to the error:

Fig.2 Structure of forward calculation algorithm.

where k means the number of iterations,T43,HPTis the HPT exit temperature calculated by the component model,T43,sensoris the HPT exit temperature measured by the sensor,and h equaling 0.0001 is the disturbance step.The iteration process will continue till the error is smaller than the given precision,which equals 0.0015 in this paper,or the number of iterations reaches the maximum.Then P43can be calculated by P4and πHPTaccording to the first formula in Eq.(5).

3.2.3.Backward calculation

The backward calculation of P43is based on the LPT component model.The premise of using the backward algorithm to estimate P43is that the LPT has no fault;that is,when calculating P43backward,these conditions are satisfied:SE4=1 and SW4=1,as the input of the LPT component.The other measurements needed are T43,P5and T5.The calculation process is shown in Fig.3.

Similar to the last section,the problem of solving P43is transformed into solving the LPT pressure ratio.With the LPT inlet temperature,exit temperature,pressure,low pressure shaft rational speed and component characteristics given,the Newton iterative method is also used to solve πLPT.Different from the forward algorithm,it uses the output parameters to calculate the input parameters reversely.Given the initial πLPTas the guess value,which is usually selected as the pressure ratio calculated by the complete engine model in the last sampling step,a corresponding P43can be calculated by the first formula in Eq.(5)through measured P5.This P43is used to complete the component calculation so that a corresponding T5is obtained based on the guess πLPT,the measured T43and the undegraded health parameters according to Eqs.(4)and(5).This estimated T5will be compared with the actual measurement so that the initial guess is modified according to the error:

where T5,LPTis the LPT exit temperature calculated by the component model,T5,sensorthe LPT exit temperature measured by the sensor,and h equaling 0.0001 is the disturbance step.The iteration process will continue till the error is smaller than the given precision or the number of iterations reaches the maximum.Then P43can be calculated by the real P5and the estimated πLPT.

Fig.3 Procedure of backward calculation algorithm.

In addition,it should be noted that the available sensor measurement parameters used to solve P43are with noise which might be amplified after iterative calculations so that the noise level of the estimated P43may be out of acceptable range.Further filtering might be affected if this signal is used directly;therefore,a low pass filter can be utilized for simple processing to reduce the noise level.

4.Fault diagnosis using reconstructed P43 measurement

4.1.Health parameters estimation based on reconstructed measurement

As already mentioned in Section 2,in the absence of the P43signal,all parameters can be estimated well except the three health parameters of the HPT efficiency,the LPT flow capacity and efficiency.In addition,the reconstruction of the P43measurement based on the turbine component model according to different fault modes is introduced in detail in Section 3.To solve the problem of fault diagnosis with the P43sensor missing,a novel method of health parameter estimation using the reconstructed P43measurement is proposed;that is,apart from the traditional filtering process,a correction process is added.The correction module is essentially another nonlinear filter,which uses the reconstructed P43measurement to rectify the inaccurate health parameters.The structure of the diagnosis system is shown in Fig.4.

As shown in Fig.4,the same input acts on the turbofan engine and the engine models to produce the corresponding outputs respectively,and the residuals of these two outputs are sent into nonlinear filter I to estimate health parameters.Due to the lack of the real P43sensor,the available sensor measurements P3,T3,T43,P5,T5are needed for P43reconstruction using the HPT or the LPT component model.The main logic is:considering the order of the gas flowing pass the components,it can be supposed first that the high pressure turbine efficiency is trouble-free,and P43is solved through the forward algorithm and regarded as a real sensor signal to complete the estimation by nonlinear filer II in the correction module. If the result is consistent with the hypothesis, the calculated P43and the health parameters are reliable;otherwise,the HPT fails and P43is solved through the backward algorithm and used to complete the estimation by nonlinear filter II.If the result is consistent with the assumption,P43and the health parameters are credible;otherwise,two turbine components fail simultaneously.

The process of measurement estimation is performed independently without affecting the operation of the complete engine model.Finally,the corrected health parameters are used as inputs of the engine model to make sure that it can reflect the degradation of the real engine.In Fig.4,the solid lines represent the estimation process while the dash lines represent the correction process.The procedure of the correction module is shown in Fig.5 and described in detail in the following part.

As shown in Fig.5,when the sampling data of time k is coming,a one-step UKF filtering estimation is performed through the existing sensor information to obtain the estimation of the other five health parameters except the three coupling parameters (SE3, SE4, and SW4). According to the description in Section 2,these five parameters are considered to be well-estimated and can be used for further calculations of measurement reconstruction or parameter correction.First,assumptions need to be made according to different failure modes when reconstructing the missing P43.We first suppose the high pressure turbine efficiency is trouble-free;then T3,P3,T43,the model input,and SW3estimated by the first filter,are used as the inlet parameters of the HPT component model(including the combustor),as shown in Fig.3,to reconstruct P43through the forward calculation algorithm.Nonlinear filter II in the correction module uses the residual between the reconstructed P43and the output of the real-time engine model to estimate SE3,SE4,and SW4.Then,we need to determine whether the hypothesis is consistent with the estimation results:if the difference between the SE3estimated by the correction filter and the value in the normal condition is less than the given threshold,the HPT efficiency is considered to be faultless;that is,the hypothesis is reasonable.Deservedly,SE3,SE4,and SW4estimated by the correction module are taken as the final results.

Fig.4 Turbofan engine diagnosis system based on the nonlinear filter using reconstructed measurement.

Fig.5 Procedure of health parameter correction module of the filter.

Otherwise,the original assumption is incorrect.In other words,the high pressure turbine efficiency has degradation,thus the forward algorithm is no longer applicable, and instead,the backward algorithm is used to calculate P43.Suppose the LPT is fault-free.Then πLPTis solved iteratively by the LPT component model using T43,T5,P5to further calculate P43.Likewise,the estimated P43is considered as the real sensor signal for nonlinear filter II to perform filtering to correct SE3,SE4and SW4,and determine whether the hypothesis is consistent with the reality.If the differences between SE4and SW4estimated by the correction filter and the value in the normal condition are smaller than the given threshold, the hypothesis is reasonable.If not,the assumption is incorrect;that is,the HPT and the LPT have faults simultaneously,implying a serious malfunction of the engine.This estimation method and diagnosis logic are designed for single turbine abrupt failures.If the HPT and the LPT both have faults,further inspections and maintenance are required.For the diagnosable failure cases,the conjunct estimation of nonlinear filters I and II is used as the final health parameter estimation at the moment.

4.2.Unscented Kalman filter

The fault diagnosis method proposed in this paper is based on the Unscented Kalman Filter(UKF).Julier et al.proposed the UKF based on the Unscented Transform(UT)according to the basic idea of deterministic sampling.25,26UT transformation is a method to calculate the statistical properties of random variables after nonlinear transformation through a set of weighted Sigma points and the UKF is a non-linear Kalman filter that uses UT to approximate the characteristics of the nonlinear variables.A brief description of the UKF filtering process is provided here.Suppose the nonlinear equation of the turbofan engine is given by Eq.(1),in which wkand vkare system noise and measurement noise,respectively,which are unrelated to each other and satisfy:E(wkwkT)=Qk,E(vkvkT)=Rk.The UKF filter recursion formula can be expressed as:

Step 1:Filter initialization

where^x0and Px0are the mean and the variance of the initial state,Wmand Wcare the weight factors of first and the second orders,respectively,λ=α2(n+κ)-n is the scaling factor,κ is another scaling factor which is usually zero,and α is the zoom factor whose value ranges between 0 to 1.Parameter β depicts the information of the state distribution,typically equals 2 for the Gaussian system.

Step 2:Sigma point calculation

where subscript k-1 indicates the last sampling step.

Step 3:Time update

where χi,k|k-1and γi,k|k-1are the values of the sigma point propagated from the state function and the measurement function,^xk|k-1indicates an estimation of xkat time k based on the information available up to and including time k-1,and Pk|k-1is the forecast error covariance.

Step 4:Measurement update

where Pxkykis the cross covariance,Pykthe innovation covariance,Kkthe Kalman gain matrix,^xkthe state estimation,and Pxkthe data-assimilation error-covariance.

5.Simulations and analyses

To verify the validity of the proposed measurement reconstruction and fault diagnosis method,simulation experiments are carried out.The simulations are based on the statistical data of changes in the gas path health parameters of the turbofan engine which has completed a certain number of operating cycles in the MAPSS simulation platform.27According to thereference,28the failure levels can be classified as shown in Table 6.

Table 6 Gas path components failure classification.

5.1.P43 signal reconstruction

The effectiveness of the proposed method of calculating measurement P43described in Sections 3.2.2 and 3.2.3 is firstly verified.The simulation of both algorithms for the fan and the compressor at 4500 operating cycles and for the HPT and the LPT at 6000 operating cycles is carried out at the design point of the engine,namely,under the ground condition,and the rotor speed equals 100%of the design speed.The available sensors are NL,NH,T22,P3,T43,T5,and P5.Fig.6 displays the results:(a)shows the P43calculation result when the fan and the compressor have malfunctions;(b)and(c)show the P43calculation result when the HPT and the LPT have malfunctions respectively.The black bars represent the real P43measurement,the blue bars the P43obtained through the forward algorithm,and the red bars the P43obtained through the backward algorithm.Fig.7 presents the simulation results of P43estimation in the dynamic process of severe failures occurring to different health parameters.To make the results more intuitive,the sensor measurements used to estimate P43do not contain noise.

Fig.6 Comparison of real and estimated P43 signals under different failure modes.

Fig.7 P43 dynamic estimation without considering noise under different failure modes.

In Fig.6,the error between the real P43measurement and that calculated by the backward algorithm is no larger than 0.2%.As for the forward algorithm,this error is smaller than 0.3%except in the LPT efficiency fault mode,which is 0.83%.These errors are caused by factors such as the convergence accuracy of the iterative algorithm or the measurement noise.For filtering,the error of this magnitude is acceptable,which will be illustrated in the following simulations.In Fig.7,both the forward and the backward algorithms can achieve satisfactory estimation of P43measurement.When a fault occurs,it can be regarded as a small step acting on the system.Though the estimated P43may have deviations,it will soon converge to the truth value.P43is well tracked throughout the whole dynamic process using the backward algorithm.The relative error does not exceed 0.21%,and the maximum error is the estimated error at the moment when the fault occurs.The forward estimation is slightly delayed in some cases,as shown in Fig.7(b)and(f);nevertheless,the maximum error is no larger than 0.48%.The steady-state errors of both algorithms are zero;namely,the real P43could be completely restored if the sensor noise was not taken into account.In the actual situation,the influence of the P43estimation noise on the subsequent health parameter estimation can be eliminated by adjusting the noise covariance matrix of nonlinear filter II.

To further verify the validity of the measurement reconstruction method in a wider range of working conditions,P43estimation during acceleration and deceleration processes are performed at the ground operation point and the highaltitude operation point respectively without considering noise.The corrected rotational speed increases from 70%of the design speed to 100%during the acceleration and reduces from 100%to 80%during the deceleration.At the highaltitude point,the flight height is 10 km and the Mach number is 1.5.The simulation results are shown in Fig.8,where(a)shows the estimations of P43by the forward and the backward algorithms in the healthy state,and(b)to(i)are the estimations of P43when the eight health parameters suffer severe failures in turn.

Fig.8 P43 estimations during acceleration and deceleration processes under different working conditions.

As can be seen from Fig.8,both the forward and the backward algorithms can reliably track the missing P43signals in a broad range of the operating conditions of the engine.The same as in previous experiments,the steady-state error is zero.In Fig.8(b)and(d),fluctuations may appear when the system is switching between the transient and dynamic states at the high-altitude point.This is probably because the number of iterations has reached the maximum but not converged when the status of the system is changing.Except for this,the tracking accuracy of P43is high enough for filtering.The relative errors of the two algorithms in all states are calculated:during the dynamic process,the maximum relative error of the forward algorithm does not exceed 1.352%at the ground point,and 1.553%at the high-altitude point;while that of the backward algorithm does not exceed 0.047%at the ground point,and 0.568%at the high-altitude point.

5.2.Fault diagnosis based on the reconstructed signal

To test the validity of the reconstructed P43measurement for the UKF-based health parameter estimation logic,each rotating component of the turbofan engine is injected with a severe failure as classified in Table 6;namely,a 5%abrupt failure.The turbofan engine works at the design point.Fig.9 shows the comparison results of the fault diagnosis before and after adding the correction module.Among them,(a),(c),(e),and(g)are the filtering results of the abrupt failure of SE3,SW3,SE4,and SW4in the absence of the P43sensor respectively;and(b),(d),(f),and(h)are the filtering results of the proposed algorithm in the corresponding fault mode.In addition,the diagnosis results with multi-component failures are presented in Fig.10:(a)medium fault on SE2and severe fault on SE3;(b)medium fault on SW3and severe fault on SE1;(c)medium fault on SW1and severe fault on SE4;(d)medium fault on SW4and severe fault on SE2.

Before adding the correction module,when any one of the turbine component health parameters fails,SE3,SE4and SW4will be coupled during the estimation,although acceptable estimation results of other parameters can be obtained.In Fig.9(a),(e),and(g),the actual situation is that,though only one health parameter fault occurs,the diagnosis is that SE3,SE4,and SW4all deviate.In Fig.9(c),although the change of SW3is accurately estimated,as the simulation time passes,all SE3,SE4,and SW4diverge.The most obvious improvement of the proposed correction process is to eliminate the coupling between the parameters.In addition,it has a good performance in quantifying the shifts of health parameters in the case when an abrupt failure occurs in the turbine components.Particularly when the HPT fails,the estimation accuracy of the health parameters and the speed of fault diagnosis have both been improved,and fluctuations in the estimated parameters are not that significant,as shown in Fig.9(a)-(d).In the case of the LPT fault,although the fluctuations of the health parameters are larger,the mean value of the estimation results is distributed near the true value anyway;namely,the localization and quantification of the fault are realized.Furthermore,the algorithm for the estimation of P43makes assumptions only based on whether the turbine component is faulty or not,hence the influence of the health status of the remaining rotating components still needs to be verified.Fig.10 illustrates the consequence of multi-component faults diagnoses.It can be seen that,as long as the two turbines do not fail simultaneously,nonlinear filter II can achieve nice correction without affecting the work of nonlinear filter I.It is indicated that the proposed diagnosis strategy can deal with the diagnosis when a single turbine component and other compression components fail at the same time,which further extends the application of the method.

In addition,to verify the effectiveness of the algorithm in the whole envelope,a 10-second(501 sampling steps)abrupt fault diagnosis experiment under different operation conditions has been conducted.The simulation is performed with measurement combination containing real P43signals,reconstructed P43signals,and no P43signals respectively.The comparison is presented in Table 7.The operating points are illustrated by flight height,Mach number,and high-pressure shaft corrected rotational speed.Three typical operating points were selected,which were ground 1(H=0 m,Ma=0,NH,cor=100%),ground 2(H=0 m,Ma=1,NH,cor=99%),and a high-altitude point (H=4000 m, Ma=1.7, NH,cor=97%), respectively. The Root Mean Square Error(RMSE)is used to evaluate the estimation effect:

where M is the sample point number,xithe actual value of the state variables,^xithe estimated value of the state variables,and subscript i represents the i-th sample point.To eliminate the influence of other measurement parameters on the estimation and meet the requirement that the number of measurement parameters is larger than or equal to the number of estimation parameters, as many measurements as possible are used,including NL,NH,T22,P22,T3,P3,T43,T5,and P5.In view of the randomness of noise,the simulation is performed 10 times in each condition,and the average result is given.

In Table 7,the best diagnosis results are always obtained with the real P43signal,because the filter takes advantage of the effective information carried by P43to distinguish different fault modes.It is worth noting that as the working point deviates from the design point, the estimation accuracy will decrease due to the decrease of model accuracy,which is true for all three sensor combinations.In the absence of the P43sensor,the estimation will deviate when any of the turbine fails.Although at some operating points,the measurement combination without P43can also calculate the health parameters well,for example,when SE4is slightly malfunctioned at the high-altitude operating point.While in most cases,the diagnosis results are not satisfactory,and with the increase in the level of failures,the estimation accuracy will deteriorate.According to simulation experiences,RMSE exceeding 0.05 often means that the estimation is divergent,thus this kind of measurement combination is not reliable.When using the signal reconstruction method proposed in this paper for diagnosis,the diagnostic accuracy is lower than that obtained by real signals since there is still an error between the reconstructed measurement and real measurement.However,the RMSEs are not larger than 0.04 at different operating points,under different failure modes and fault levels within the flight envelope,which illustrates the stability and universality of the algorithm.

Fig.9 Comparison of proposed algorithm with reconstructed P43 and original algorithm without P43 in abrupt failure cases.

Fig.10 Turbofan engine diagnoses with multi-component failures.

Table 7 The diagnosis RMSEs comparison using different P43 signals in abrupt failure cases at different operating points.

6.Conclusions

(1)The importance of the HPT exit pressure sensor signal in the fault diagnosis system is first analyzed,and a new parameter estimation strategy is designed.For different failure modes,the forward algorithm and the backward algorithm are developed based on corresponding component models and other effective sensor measurements to estimate the missing measurements alternatively.The calculation is performed through the Newton iterative method.

(2)The traditional nonlinear filter is improved by adding the signal estimation module and the health parameter correction module.In the correction module,the reconstructed P43is used to complete the estimation and determine whether it is consistent with the hypothesis to correct the coupled health parameters so as to realize the fault location and diagnosis.

(3)The diagnosis structure is applied to the UKF-based filter.The accuracy of the measurement reconstruction is tested,and then the diagnostic method based on the reconstructed P43is verified in single component and multi-component abrupt fault cases,and applied to multiple operating points in the flight envelope.The simulation results show that P43can be calculated well under the condition that one of the turbine has malfunction in both steady and dynamic processes and the reconstruction measurement can be used for parameter estimation.The diagnosis system realizes the distinction between different failure modes,the elimination of coupling parameters and the estimation of the change of health parameters and has good performance under various working conditions.

(4)The fault diagnosis method based on measurement reconstruction proposed in this paper avoids dimension reduction and model linearization and provides a new direction for similar research.

Acknowledgement

The authors are grateful to the anonymous reviewers for their critical and constructive review of the manuscript.This work is supported by the Fundamental Research Funds for the Central Universities(NO.NS2018018).