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Proceedings of ASME Turbo Expo 2015: Turbine Technical Conference and Exposition GT2015 June 15 – 19, 2015, Montréal, Canada
GT201 5 - 43744
SENSOR SELECTION FOR AIRCRAFT ENGINE P ERFORMANCE ESTIMATIO N AND GAS PATH FAULT DIAGNOSTICS Donald L. Simon NASA Glenn Research Center 21000 Brookpark Road Cleveland, OH , 44135 ABSTRACT and isolation of gas path system faults affecting engine This paper presents analytical techniques for aiding system perfor mance , which are typically relatively rapid or abrupt in designers in making aircraft engine health management sensor nature [ 1 , 2 ] . A notional illustration of the observed selection decisions . The presented techniques , which are based measurement shifts caused by gradual deterioration compared on linear estimation and probability theory, are tailored for gas to an abrupt fault is shown in Figure 1 .
turbine engine performance estimation and gas path fault diagnostics applications. They enable quantification of the Rapid shift performance estimation and diagnostic accuracy offered by (potentially due to different candidate sensor suites. For performance estimation, a fault event) sensor selection metric s are presented for two types of estimators including a Kalman filter and a ma ximum a posteriori estimator. For each type of performance estimator, Gradual deterioration sensor selection is based on minimizing the theoretical sum of squared estimation errors in health parameters representing performance deterioration in the major rotating modules of the engine . For gas path fault diagnostic s, the sensor selection metric is set up to maximize correct classification rate for a Measurement shift Gradual deterioration diagnostic strategy that performs fault classification by Measurement shift identifying the fault type that most closely matches the Measurement shift moving average observed measurem ent signature in a weighted least squares sense. Results from the application of the sensor selection Time (flights) metrics to a linear engine model are presented and discussed.
Figure 1 . Gradual versus rapid performance shifts.
Given a baseline sensor suite and a candidate list of optional sensors, an exhaustive search is performed to determine the optimal sensor suites for performance estimation and fault Although performance estimation and gas path fault diagnostics. For any given sensor suite, Monte Carlo sim ulation diagnostics typically apply different algorithmic approaches, results are found to exhibit good agreement with theoretical both are conducted using the same engine sensor measurement predictions of estimation and diagnostic accuracies. data — primarily data acquired from the available engine control sensor suite. In gen eral, adding additional engine sensors will INTRODUCTION improve performance estimation and diagnostic accuracy, but Aircraft operators rely on engine performance estimation this does add to the overall engine life cycle cost . Therefore, the and gas pat h fault diagnostics to ensure the safe and efficient decision to add sensors should be made judiciously.
operation of their gas turbine engine assets. Performance Several researchers have presen ted sensor selection estimation enables the estimation and trending of g radual approaches for engine health management applications.
Mushini and Simon (no relation to the author) proposed a performance deterioration that the engine will experience over time due to fouling, corrosion , and erosion of turbomachinery sensor selection approach for Kalman filter - based performance components . G as path fault diagnostics enables the detection estimation applications [ 3 ]. In this work, a performance metri c This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited. 1 C - MAPSS 40k Commercial Modular Aero - Propulsion System wa s defined as a function of the steady state error covariance Simulation 40k and the cost of the selected sensors . T hree separate metric s D M Mahalanobis distance were considered for searching for the optimal sensor suite , FPR false positive rate including a random search, a genetic algorithm search , and an H influence coefficient matrix relating changes in exhaustive search. The study by Mushini and Simon assumed health parameter s to changes in sensed that the estimation problem was over - determined (i.e., there are measurement s more sensors than unknown parameters to be estimated), which H f fault influence coefficient matrix relating faults to is usually not the case for engine performance estimation changes in sensed measurements applicatio ns. Borguet and L é onard approached the problem of I identity matrix sensor selection for engine performance estimation within the MAP maximum a posteriori N number of fault types scope of linear information theory [ 4 ]. They defined PMC probability of misclassification performance metrics based on the Fisher information matrix , P h health parameter covariance matrix and an exhaustive search was conducted to identify the best R measurement noise covariance matrix sensor suite. Sowers et al . introduced a systematic framework SSEE sum of squared estimation errors for automating sensor selection decisions for diagnostic T fault detection thre s hold applications. This framework enables incorporation of factors TPR true positive rate * of merit commonly considered in the sensor selection process V transformation matrix relating h to q including diagnostic accuracy, diagnostic criticality, and cost WSSE weighted sum of squared errors WSSM weighted sum of squared measurements [ 5 ]. The framework relies on the end user to specify the merit h health parameter vector function used by the optimal search algorithm . Kamboukos et f fault vector al . proposed sensor selection for performance estim ation k number of additional sensors to add applications based on the condition number of the influence m number of tuning parameters matrix that relates changes in health parameters to changes in n Number of additional sensors to choose from sensed measurements [ 6 ]. Here, a determined health parameter p number of health parameters estimation problem was considered where there are as many q reduced order tuning parameter vector sensors as parameters to be estimated.
u actuator command vector The contribution of this paper will be to introduce separate v measurement noise vector sensor selection metrics for performance estimation and fault w k , w h,k , w xh,k process noise vectors x state vector diagnostic applications. In terms of performance estimation , the y measurement vector problem is assumed to be underdetermined (i.e., fewer sensors gamma function than unknown health parameters to be estimated) , and two lower incomplete gamma function separate estimators will be considered — one applying a Kalman ε residual vector (estimate minus its expected value) filter designed for processing dynamic sensed measurement standard normal distribution function information , and a second applying a maximu m a posteriori λ mean value of the WSS M signal estimator for processing quasi - steady - state measurement data.
th μ i mean value of i sensed measurement In terms of fault diagnostics, a single fault diagnostic strategy applying a weighted least squares hypothesis test will be Subscripts considered. a fault type index b misclassified fault type index The remainder of this paper is organized as follows. First, k sample index metrics are defined through analytical derivations of the xh augmented state vector ( x and h ) performance estimation accuracy and gas path fault diagnostic xq reduced order state vector ( x and q ) accuracy based on linear system theory . These analytical functions can be di rectly used to theoretically predict the Su per scripts estimation or diagnostic accuracy offered by a given sensor † pseudo - inverse suite . Next, example application of the sensor selection ^ estimated value technique s is presented by applying the approach es to a linear ~ error value engine model . Theoretically predicted results are calculated and – mean value compared against empirical results obtained through Monte Operators Carlo simulation analysis. This is followed by discussions and E [•] expected value of argument conclusions.
tr {•} trace of a matrix NOMENCLATURE SENSOR SELECTION METRICS A , A xh , A xq , B , system m atrices As previously mentioned, aircraft engine performance B xh , B xq , C , C xh , estimation and gas path fault diagnostics pose different problem C , D , L , M xq formulations. Analytical formulations of each are introduced CCR correct classification rate This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited. 2 below along with derivations of performance estimation and w B x L A x k k k 1 u diagnostic accurac y for a given sensor suite . The performance k w h I h 0 0 k h k k , 1 estimation problem assumes the application of two separate B x w A xh k xh , xh estimators — a linear Kalman filter and a maximum a posteriori k xh , estimator, while the gas path fault diagnostic problem assumes w u B x A k xh k xh k xh xh , , the application of a single f ault isolator applying a weighted ( 2 ) least squares hypothesis test .
x k
v Du M C y
k k k Kalman Filter - Based Health Parameter Estimation h k C xh In the aircraft engine community, Kalman filters are x k xh , commonly applied for on-board performance estimation or v Du x C k k k xh xh , post-flight analysis of full-flight streaming measurement data.
In this subsection, Kalman filter health parameter estimation The vector w is zero - mean white noise associated with the accuracy is discussed following a derivation previously xh T T T augmented state vector, [ x h ] . w consists of the original introduced by Simon and Garg as part of an optimal tuner xh selection methodology for Kalman filter-based performance state process noise, w , concatenated with the process noise associated with the health parameter vector, w .
estimation applications [7]. This optimal tuner selection h O nce the h vector is appended to the state vector as shown methodology is designed to minimize the Kalman mean in Eq. ( 2 ) , it may be directly estimated by applying a Kalman squared estimation error in the parameters of interest when filter as long as the system is observable. However, the number facing underdetermined estimation problems, but can readily be of health parameters that can be estimated is limited to the extended to also calculate the mean squared estimation error offered by different sensor suites, as was shown in Ref. [8]. number of sensors, the dimension of y [ 9 ] , and typically an aircraft gas turbine engine has fewer sensors than health The formulation begins by considering the following parameters. To enable Kalman filter formulation for an discrete linear time-invariant state space equations representing underdetermined estimation problem, a reduced - order state engine dynamics about an operating point space model is constructed. This is accomplished by defining a mod el tuning parameter vector, q , which is a linear combination w h L u B x A x k k k k k 1 ( 1 ) of all health parameters, h , given by v h M u D x C y k k k k k * h V q ( 3 ) where k is the sample index, x is the vector of state variables, u is the vector of control inputs, and y is the vector of measured m p * where q , h , m < p , and V is an m p p outputs. The vector h , where h , represents the engine transformation matrix of rank m , which relates h to q . G iven an health parameters, which induce shifts in other variables as the estimate of q (i.e., ˆ ), an approximation of the health q health parameters deviate from their nominal values. The Δ ˆ symbols denote parameter deviations relative to the linear parameter vector, , can be obtained as h operating point trim condition. The vectors w and v are *† uncorrelated zero - mean white noise input sequences. The ˆ ( 4 ) q V h ˆ matrices A , B , C , D , L , and M are of appropriate dimensions.
Through algebraic manipulation , Eq. ( 1 ) can be re - written such * † * where V is the pseudo - inverse of V . Substituting Eq. ( 4 ) into that h is concatenated with x to form an augmented state vector, Eq. ( 2 ) yields the following reduced - order state space x , as shown in Eq. ( 2 ) . Since engine performance deterioration xh equations , which may be used to formulate a Kalman filter is very slowly evolving relative to other engine dynamics, h is here modeled without dynamics. Here, and throughout the remainder of this section , the symbols are omitted for simplicity.
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited. 3 *† accuracy o f any unmeasured performance parameter s such as w B x LV A x k k k 1 u thrust , airflows, or metal temperatures.
k w q I q 0 0 k q k k , 1 B x w A x xq k xq , k xq , 1 k xq xq , Maximum A P osteriori Health Parameter Estimation Maximum a posteriori (MAP) estimation is commonly w u B x A k xq k xq k xq xq , , applied for ground - based aircraft engine gas path analysis . It is ( 5 ) based on quasi - steady - state engine snapshot measurements x acquired in flight [ 2 , 10 ]. Unlike the Kalman filter, which is a k *†
v Du MV C y
k k k recursive estimator designed to process dy namic measurement q k C xq data, the MAP estimator provides a point estimate based on an x k xq , assumed quasi - steady - state measurement process. The MAP v Du x C k k k xq xq , estimator incorporates a priori knowledge regarding the distribution of the parameters to be estimated, which enables it to provide an estimate when facing underdetermined estimation The reduced - order equations introduced in Eq. ( 5 ) will problems. To introduce the MAP estimator , consider the enable a Kalman filter to be formulated that can estimate the T T T following linear steady - state measurement process augmented state vector , [ x q ] . The resulting Kalman filter - produced tuner parameter vector estimate, q ˆ , can be inserted v h H y ( 7 ) ˆ into Eq. ( 4 ) to produce an estimated health parameter vector, h .
However, this does not circumvent the underdetermined nature where H is an influence coefficient matrix that relates the ˆ ˆ of the h estimation problem, and the fact that the produced h effects of the health parameter vector changes , Δ h , to changes estimates will contain errors is unavoidable . However, (i.e., residuals) in the sensed measurement vector , Δ y . Here, v , estimation accuracy is directly dependent on the available is zero - mean white noise with covariance R . As with the * sensor suite and the selection of the transformation matrix, V .
previously introduced Kalman filter equations , t he Δ symbols This gives rise to an optimization problem of selecting the best denote parameter deviations relative to the operating point trim * sensor suite and the corresponding V that minimize s the condition at which Eq. ( 7 ) was generated . For simplicity, the Δ estimation error in the parameters of interest. For a g iven sensor symbols are omitted throughout the remainder of this section on * suite , a n optimal iterative search can be conducted to select a V the MAP estimator and the terms y and h are used to indicate matrix that minimizes the theoretical mean sum of squared measurement and health parameter changes, respect ively . The estimation errors ( SSEE ) in the parameters of interest maximum a posteriori (MAP) estimator follows the closed form expression *
V SSEE min arg
( 6 ) p m * V T T 1 1 1 1 ˆ
y R H H R H P h
h * where the above statement indicates the V matrix that G h ( 8 ) * minimizes the SSEE function. Once V is obtained, it can be ˆ y G h h inserted into Eq. ( 5 ) to construct the reduced - order state space * equations . Here, it is important to emphasize that the V matrix w here P is a matrix containing a priori knowledge of the h and q vector are unique to each sensor suite considered.
expected health parameter covariance. As with the Kalman Therefore, Eq. ( 6 ) is individually applied to each sensor suite, filter introduced above, the MAP estimator produce s a biased and the suite that provides the lowest SSEE is identified as estimate due to the underdetermined nature of the estimation optimal.
problem. However, its accuracy depend s on the available sensor Due to page limitations, a complete derivation of the suite, thus giving rise to a sensor selection problem. As with the Kalman filter SSEE metric is not provided in this document.
Kalm an filter, the MAP health parameter estimation error will However , readers are referred to Ref. [ 7 ] for this derivation.
be defined in terms of the sum of squared estimation errors Some notable aspects regarding the derivation are that it ( SSEE ), which consists of the sum of two components: mean focuses on linear Kalman filter estimation accura cy under squared bias and variance , as defined below.
steady - state operating conditions , and that the error of each estimated parameter comprises mean squared bias and variance MAP Estimation Mean Squared Bias. The bias of an terms. Additionally, the derivation incorporates user - specified a estimat or is the expected difference between the estimator’s priori knowledge regarding the health parameter cov a riance estimated value and the true value of the parameter being matrix reflecting the expected distribution in the health estimated. For the MAP estimator, the estimated health parameters to be estimated. While this paper will only consider ~ Kalman filter health parameter estimation accuracy, the parameter bias vector , h , is defined as technique can be readily extended to optimize the estimation This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited. 4 ~ Then, by substituting Eq. ( 12 ) into Eq. ( 11 ) the co variance ˆ
h h E h
matrix of the MAP estimate becomes T h y G E h
E P
ˆ h ˆ h T T
G vv G E h h T T ( 13 )
G vv E G
( 9 )
h v Hh G E
h h h R y T RG G h h
v G h I H G E
h h
v E G h E I H G
h h Diagonal elements of will reflect the va riance of individual P ˆ h h 0 health parameter estimates, while off diagonal elements reflect
h I H G
h the covariance between estimates.
The overall sum of squared estimation errors ( SSEE ) can where the operator E [ ● ] represents the expected value of the be obtained by combining the estimation mean squared bias and argument , and the expected value properties E [ h ]= h and E [ v ]=0 variance informati on as are leveraged in Eq. ( 9 ) . The estimation error bias equation given in Eq. ( 9 ) is a function of an arbitrary health parameter T T ˆ
( 14 )
RG G tr I H G P I H G tr h SSEE
ve ctor h . The mean sum of squared bias es across a fleet of h h h h h engines is given as Mean squared bias and variance are equally weighted in the above equation. However, end user s may weight them ~ ~ ~ ~ ~ T T 2 h h tr E h h E h differently if they so choose.
T T
I H G hh I H G tr E
Weighted Least Squares Single Fault Diagnostic h h Approach ( 10 ) T T Gas path fault diagnostics poses a different problem than
I H G hh E I H G tr
h h that of performance estimation. Unlike performance P h deterioration, which is assumed to occur gradually and affect all T
I H G P I H G tr
health parameters simultaneously and somewhat independently , h h h gas path f aults ar e assumed to primarily occur abruptly and in isolation. In other words, it is rare to have multiple unrelated where tr { ● } represents the trace (sum of the diagonal elements) T gas path system faults occurring simultaneously. Applying th e of the matrix. Here, the E [ hh ] reduces to the health parameter single fault assumption transforms gas path fault diagnostics covariance matrix, P , which is leveraged in Eq. ( 10 ) .
h from an underd etermined to an overdetermined estimation problem . This subsection will present a single fault isolator that MAP Estimation Variance. The variance of the MAP applies a weighted least squares hypothesis test to diagnose estimate is found by construct ing the estimation covariance faults. Additionally, the accuracy offered by this diagnostic matrix , P , which is defined as ˆ h approach is analytically derived.
The fault diagnostic approach considered in th is study, like T the previously described MAP estimation approach, is ground - ˆ ˆ ˆ ˆ
h E h h E h E P
( 11 ) ˆ h based , and designed to process snapshot engine measurements acquired in flight. To introduce the diagnostic approach , first ˆ where the vector ε is the residual between h and its expected a ssume the following linear steady - state sensor measurement process value. By combining Eq. ( 7 ) and Eq. ( 8 ) , ε can be written as v f H y ˆ ˆ ( 15 )
h E h f
y G E y G
h h w here ΔΔ y is a vector of residuals reflecting recent shifts in
v Hh G E v Hh G
h h ( 12 ) engine sensor measurements , for example, the change v E G h E H G v G Hh G measurements have undergone within the past one or two h h h h h 0 flights. Also shown in Eq. ( 15 ) is f , a vector of gas path fault magnitudes, and H f , a fault influence coeffi cient matrix relating v G h fault magnitudes to sensor measurement residuals.
Furthermore, v denotes zero - mean normally distributed sensor This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited. 5 measurement noise of covariance R . The measurement After WSS E ’s are calculated for each potential fault type residuals , ΔΔ y , are regularly updated as new snapshot data they are compared, and the hypothesized fault type that become available . They are referred to as “delta - delta” produces the minimum WSS E is classified as the fault cause.
measurement shift s , as they will reflect fault induced shift s Theoretical predictions of fault detection and fault classification relative to the gradual deterioration induced shift s the engine performance fo r the single fault isolator are given below.
has experienced up until the time of fault initiation [ 2 ] . Since faults are assumed to occur abruptly and cause relatively large Fault Detection Performance . F or any diagnostic measurement shifts , the ΔΔ y residuals will be small in the case system, f ault detection performance is directly related to the of no fault, and larger once a fault has occurred (See Fig. 1). applied fault detection threshold. Larger thresholds will result For simplicity, the ΔΔ symbols are omitted throughout the in fewer false alarms in the absence of a fault (false positives), remainder of this section and the term y is used to indicate but will also result in fewer true detections when a fault is recent observed shifts in the sensor measurements . Given Eq. actually present (true positives) , while the opposite is true for ( 15 ) , a fault detection and classification ( isolation ) approach smaller thresholds . In order to facilitate a common basis of can be formulated. Here, it is assumed that f ault detection is comparison, each sensor suite considered in this study applies a performed by calculating and monitoring a weighted sum of WSS M fault detection threshold, T , necessary to achieve a user - squared measurement ( WSS M ) signal: specified target false positive rate ( FPR ). The FPR of a system monitoring a WSS M signal for fault detection purposes can be T 1 approximated if it is assumed th at all sensed measurements are y R y WSSM ( 16 ) independent in addition to being zero mean and normally distributed . With this simplification, the distribution of the If the WSS M signal exceeds an est ablished detection threshold WSS M signal under the no - fault case will be the sum of the ( T ), a fault is assumed to be present and the diagnostic logic squares of k independent s tandard normal random variables , proceeds in attempting to isolate the most plausible single fault which is a chi square distribution with k degrees of freedom.
root cause for the fault . Here, fault classification is performed The cumulative distribution function of a chi square by applying a weighted least squares approach. Each possible distribution is given as [ 11 ] ga s path fault type is evaluated individually, and the hypothesized fault whose signature best matches the observed T k measurement residuals in a weighted least squares sense is , th classified as the fault. For the l fault type , the estimated fault 2 2
, k T CDF
( 21 ) magnitude is c alculated as k 1 T T 1 1 ˆ
y R H H R H f ( 17 ) l f l f l f l , , , where (·) is the gamma function and (·) is the lower th where H is the column of the H matrix corresponding to the l f, l f incomplete gamma function. The above equation reflects the ˆ th fault type, and the scalar f is the estimated magnitude of the l probability that a random sample of the WSS M signal is less l than the threshold, T , when no fault is present (i.e., the true fault type that produces the best match of the observed vector negative rate). Therefore, the false positive rate is given as of sensor measurement residuals, y , in a weighted least ˆ squares sense. The resulting f estimate is then combined with
, 1 , k T CDF k T FPR
l ˆ H to produce an estimated measurement residual vector, y , f,l l T k , th for the l fault type : ( 22 ) 2 2 1 k ˆ ˆ f H y ( 18 ) l f l , ˆ When a fault occurs , the WSS M signal will be distributed The difference between y and y defines the estimation error l ~ as a non - central chi - squared distribution. This distribution will th y vector for the l fault type , , defined as l be a function of: 1) the detection threshold, T ; 2) the number of sensors, k ; and 3) the mean value of the WSS M signal . The ~ y y y ˆ ( 19 ) l l mean value of the WSSM signal for a fault of given type and k magnitude is defined as λ , where , wher e μ is the th i i The weighted sum of squared errors for the l hypothesized i 1 fault type is calculated as th mean value of the i sensor in the presence of the fault . Given this information, t he true positive rate ( TPR ) can be calculated T ~ ~ 1 y R y WSSE ( 20 ) l l l This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited. 6 from the cumula tive distribution function of the non - central expect ed sensor measurement vector for fault type b , scaled to chi - square distribution as [ 11 ] : be the same weighted length as y b as shown in Eq. ( 26 ) j T 1 T k y R y
j , a a
2 2 2 H y ( 26 ) b b
1 e k T TPR , ,
( 23 ) T 1 H R H k 0 j j ! b b
j
The above equations allow the probability of misclassification The above equation reflects the probability that a random PMC for each fault pair to be calculated . The overall b|a sample of the WSS M signal is greater than the threshold, T , probability of misclassification for fault type a can be when a fault of magnitude λ is present. Given Eqs. ( 22 ) and approximated by summing all fault pair combina tions: ( 23 ) , overall FPR and TPR for individual fault types can be approximated for a ny given sensor suite. N PMC PMC a b a | ( 27 ) b 1 a b Fault Classification Performance . In this study an approximation of the theoretical misclassification rate is where N is the number of different fault types. Once PMC is produced by considering the probability of misclassification a obtained, an approximation of the correct classification rate between fault pair combinations (i.e., making the assumption (CCR) for fault type a of the considered fault magnitude can be that only two fault classes exist) given that a fault has been found by combining the fault’s TPR (given by Eq. ( 23 ) ) and its correctly detected . The two - class misclassification rate results PMC (given by Eq. ( 27 ) ) as across all fault pairs are then summed to estimate an overall misclassification rate. Calculating the two - class
misclassification rate is readily tractable compared to multi - PMC TPR CCR 1
( 28 ) a a a class misclassification rate given three or more faults. While this simplification does not enable an exact calculation of the The average CCR for the diagnostic system considering all fault overall misclassification rate for a given fault type, it is types thus becomes effective for identifying fault pairs at high risk of misclassifica tion. Let us consider a fault of a given type, a , and N CCR a ( 29 ) CCR magnitude, f . From Eq. ( 15 ) , the expected sensed measurement a a N 1 vector under this condition become s f H y . The a a f a , th probability that a sensor measurement vector observation, y , where CCR is the correct classification rate for the a fault a collected when fault f is present is misclassified as fault type b type, and N is the total number of fault types . The CCR shown a (assumed to be of equivalent probability and resulting in in Eq. ( 29 ) serves as a m etric that can be used to estimate and equivalent sensor measurement covariance as fault type a ) is compare the diagnostic performance offered by different given as [ 12 ] candidate sensor suites.
LINEAR TURBOFAN ENGINE MODEL EXAMPLE 1 D PMC 1 ( 24 ) M a b | In this section, an example application of the previously introduced metrics is given. This is done by applying the metrics to a linear point model and linear influence coefficient where PMC is the probability of misclassifying fault type a b|a matrices extracted from the NASA Commercial Modular Aero - as b , is the standard normal distribution function, and D is M Propulsion System Simulation 40k (C - MAPSS 40k ) turbofan the Mahalanobis distance defined as engine simulation [ 13 ] at standard day sea level static conditions (i.e., air temperature = 59F, altitude = 0, and Mach = T 1 0) and an intermediate power setting . The linear model , which
y y R y y
b a b a is used for Kalman filter estimation , and is of the format shown or n mi D ( 25 ) M in Eq. ( 1 ) , has seven state variables and three control inputs T 1
y y R y y
(actuator commands ) , as shown in Table 1 , and ten health b a b a parameters , as shown in Table 2 . The linear model has six baseline sensors, and four additional (optional) sensors , which The above expression accounts for the fact that the least are shown in Table 3 along with their corresponding standard squares estimation approach is able to produce bi - directional deviations . Here, the sensor noise is assumed to be fault estimates of either a positive or negative magnitude. The uncorrelated, zero - mean and normally distributed. The linear sign that produces the minimum distance will have the largest influence coefficient matrix to be use d in MAP estimation (i.e. , contribution to the misclassification rate. In Eq . ( 25 ) , y is the b the H matrix given in Eq. ( 7 ) ), and the linear fault influence This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited. 7 coefficient matrix to be used in gas path fault diagnostics (i.e. , Table 1 . State variables and control inputs the H f ma trix given in Eq. ( 15 ) ), are generated from State Control i nputs ( u ) C - MAPSS40k at t he same o perating point as the linear model .
variables ( x ) However, these matrices are generated assuming that fan speed Nf – fan speed Wf – fuel flow is held constant. As such, fan speed (Nf) is replaced by fuel Nc – core speed VSV – variable stator vane flow (Wf) as one of the six baseline sensors when performing Hs_LPC – LPC metal temp VBV – variable bleed valve MAP estimation or gas path fault diagnostics.
The optional sensors shown in Table 3 are evaluated for the Hs_HPC – HPC metal temp estimation accuracy or diagnostic improvement they provide if Hs_burner – burner metal temp added individually or in combination to the bas eline sensor Hs_HPT – HPT metal temp suite. Given a set of n additional se nsors to choose from, and a Hs_LPT – LPT metal temp target number, k , of additional sensors, the total number of sensor suite combinations will be: Table 2 . Health parameters ( h ) n ! n Health p arameters ( 30 )
! ! k n k k
1 η Fan efficiency FAN 2 γ Fan flow capacity FAN Therefore, the number of sensor combinations when adding 1, 2, 3, or 4 sensors to the baseline 6 sensors are: 3 η Low pressure compressor (LPC) efficiency LPC 4 γ Low pressure compressor (LPC) flow capacity LPC Baseline sensors 1 combination 5 η High pressure compressor (HPC) efficiency HPC Baseline + 1 sensor (n = 4, k = 1) 4 combinations 6 γ High pressure compressor (HPC) flow capacity HPC Baseline + 2 sensors (n = 4, k = 2) 6 combinations 7 η High pressure turbine (HPT) efficiency HPT Baseline + 3 sensors (n = 4, k = 3) 4 combinations 8 γ High pressure turbine (HPT) flow capacity HPT Baseline + 4 sensors (n = 4, k = 4) 1 combinations 9 η Low pressure turbine (LPT) efficiency LPT Total sensor combinations 16 combinations 10 γ Low pressure turbine (LPT) flow capacity LPT The subsections below will present and discuss results from the application of the sensor selection metrics for performance Table 3 . Sensed outputs and standard deviation as percent of estimation and gas path fault d iagnostic s .
operating point trim values Standard Sensed output Sensor Selection for Performance Estimation d eviation Performance estimation accuracy is assessed based on the * Nf – fan speed (rpm) 0.360 rpm health parameter mean squared estimation error offered by Nc – core speed (rpm) 1.23 rpm different sensor suites. The linear engine model contains 10 health parameters as shown in Table 2 , which represent Ps30 – HPC exit static pressure 0.333 psia Baseline efficiency and flow capacity scalars associated with each major Sensors T30 – HPC exit total tem p 0.273 ºR rotating module of the engine. In this study, d eviations in all P50 – LPT exit total pressure 0.021 psia health parameters are assumed to be uncorrelated , and randomly shifted from their trim conditions with a standard T50 – LPT exit total temp 0.259 ºR deviation of 2% . Since a parameter’s variance is equal to its P14 – Bypass duct total pressure 0.022 psia standard deviation squared, the health parameter covariance Additional T14 – Bypass duct total temp 0.117 ºR matrix, P , is defined as a diagonal matrix with all diagonal h (Optional) eleme nts equal to 4.0 . The subsections below will present P25 – HPC inlet total pressure 0.031 psia Sensors health parameter estimation results first assuming application of T25 – HPC inlet total temp 0.132 ºR a Kalman filter estimator and then the MAP estimator.
* Note: For the MAP estimator and gas path fault diagnostics, fan speed (Nf) serves as the engine power reference parameter and is Kalman Filter Sensor Selection Results . For each of replace d in the list of six baseline sensors by fuel flow (Wf), which has the 16 candidate sensor suites , the Kalman filter SSEE metric a standard deviation of 9.03 pounds per hour (pph) shown in Eq. ( 6 ) is applied to calculate the theoretical health parameter SSEE offered by each of the 16 candidate sensor suites. Additionally, a Monte Carlo simulation analysis is conducted to verify the theoretically predicted results. This is based on 200 health parameter vector combinations randomly This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited. 8 selected in accordance with the defined hea lth parameter Table 4 . Kalman filter performance estimation accuracy sensors added Theoretical Health Parameter Mean Squared Estimation Errors (% squared) covar iance matrix, P h . These random health parameter vectors to baseline and random measurement noise, v , are substituted into Eq. ( 1 ) η γ η γ η γ η γ η γ SSEE FAN FAN LP C LP C HP C HP C HP T HP T LP T LP T T14 T25 P14 P25 # Sensors 6 2.53 1.66 3.34 3.07 0.26 1.51 0.95 0.04 1.00 2.85 17.21 to produce sensed measurement tes t cases used for Monte Carlo 7 x 0.23 0.15 3.43 3.29 0.25 1.47 0.96 0.04 1.02 2.83 13.66 evaluation. The resulting mean squared estimation error results 7 x 0.17 0.20 3.47 3.22 0.26 1.48 0.95 0.04 1.01 3.01 13.81 7 x 2.54 1.67 3.23 0.05 0.26 0.70 0.83 0.04 0.86 2.65 12.83 are shown in Table 4 . The top half of the table shows 7 x 2.50 1.65 1.94 0.69 0.22 1.29 0.80 0.04 0.83 2.62 12.58 theoretically predicted results while the bottom half shows 8 x x 0.15 0.15 9.09 0.34 0.74 2.25 1.52 0.04 1.62 6.54 22.45 8 x x 0.23 0.15 3.33 0.05 0.27 0.73 0.82 0.04 0.86 2.67 9.14 results obtained via the Monte Carlo simulation analysis. Each 8 x x 0.24 0.15 1.94 0.69 0.22 1.29 0.79 0.04 0.82 2.60 8.78 row corresponds to one of the 16 candidate sensor suites. In the 8 x x 0.13 0.16 3.83 0.05 0.31 0.82 0.94 0.04 0.99 3.18 10.44 cases of Baseline + 1, +2, or +3 o ptional sensors, the sensor 8 x x 0.14 0.17 2.07 0.74 0.23 1.37 0.82 0.04 0.87 2.82 9.27 8 x x 2.51 1.66 0.03 0.05 0.01 0.06 0.80 0.04 0.83 2.64 8.60 suite that provides the minimum SSEE is highlighted in red 9 x x x 0.16 0.15 3.43 0.05 0.27 0.74 0.99 0.04 1.05 3.19 10.07 font. In general, adding sensors reduces the SSEE . The results 9 x x x 0.16 0.16 0.64 0.27 0.08 0.44 0.80 0.04 0.84 2.72 6.13 9 x x x 0.24 0.15 0.03 0.05 0.01 0.06 0.79 0.04 0.83 2.61 4.79 also show that specific additional sensors are highly beneficial 9 x x x 0.14 0.17 0.03 0.05 0.01 0.06 0.81 0.04 0.85 2.80 4.95 10 x x x x 0.16 0.16 0.02 0.05 0.01 0.06 0.70 0.04 0.76 2.51 4.47 in improving the estimation accuracy of i ndividual health parameters. For example, adding sensors such as P 14 or T 14 sensors added Monte Carlo Health Parameter Mean Squared Estimation Errors (% squared) to baseline improves the estimation accuracy of fan efficiency (η ) and FAN η FAN γ FAN η LP C γ LP C η HP C γ HP C η HP T γ HP T η LP T γ LP T SSEE T14 T25 P14 P25 # Sensors fan flow capacity (γ ), while adding P25 improves estimation FAN 6 2.26 1.48 3.48 2.90 0.28 1.55 1.06 0.04 1.14 3.18 17.35 7 x 0.23 0.15 3.52 2.95 0.27 1.46 1.04 0.04 1.11 3.17 13.94 accuracy of LPC flow capacity (γ LPC ) . It is also encour aging to 7 x 0.17 0.20 3.48 3.18 0.27 1.52 1.08 0.04 1.14 3.37 14.44 note that the theoretical and simulation results exhibit good 7 x 2.29 1.51 3.34 0.05 0.27 0.73 0.99 0.04 1.05 2.92 13.19 7 x 2.26 1.49 1.99 0.70 0.23 1.33 0.91 0.04 0.96 2.98 12.90 agreement . The overall estimation accuracy is very similar and 8 x x 0.15 0.15 8.16 0.87 0.74 3.04 1.77 0.04 1.89 6.92 23.74 the combination of sensors identified as optimal is identical 8 x x 0.23 0.15 3.46 0.05 0.28 0.75 0.98 0.04 1.03 2.97 9.94 (theoretical vs. Monte Carlo simulation ) for each candidate 8 x x 0.24 0.15 2.03 0.72 0.23 1.35 0.90 0.04 0.94 3.00 9.60 8 x x 0.13 0.16 3.91 0.05 0.31 0.84 1.14 0.04 1.21 3.67 11.47 number of sensors . M inor differences between theoretical and 8 x x 0.15 0.17 2.05 0.73 0.23 1.36 0.93 0.04 0.98 3.20 9.84 Monte Carlo results are likely due to the number of Monte 8 x x 2.30 1.52 0.03 0.05 0.01 0.06 0.91 0.04 0.96 3.04 8.90 9 x x x 0.16 0.15 3.30 0.05 0.26 0.71 1.20 0.04 1.28 4.52 11.66 Carlo trials conducted . If the number of trials were increased , 9 x x x 0.16 0.15 0.84 0.34 0.10 0.57 0.93 0.04 0.98 3.18 7.29 the differences between analytical and simulation results should 9 x x x 0.24 0.15 0.03 0.05 0.01 0.06 0.91 0.04 0.95 3.03 5.45 9 x x x 0.14 0.17 0.03 0.05 0.01 0.06 0.95 0.04 1.00 3.30 5.75 diminish . Based on this analysis, the sensor selection decisions 10 x x x x 0.16 0.15 0.02 0.05 0.01 0.06 0.80 0.04 0.86 2.84 4.98 for Kalman filter estimation accuracy would be: Table 5 . MAP estimator performance estimation accuracy Baseline + 1 sensor, choose: T25 sensors added Theoretical Health Parameter Mean Squared Estimation Errors (% squared) to baseline Baseline + 2 sensors, choose: T25 and P25 η γ η γ η γ η γ η γ SSEE FAN FAN LP C LP C HP C HP C HP T HP T LP T LP T T14 T25 P14 P25 # Sensors 6 2.45 1.64 2.81 3.09 0.27 1.35 0.88 0.04 1.06 2.78 16.35 Baseline + 3 sensors, choose: T25, P25, and P14 7 x 0.48 0.15 2.81 3.08 0.27 1.34 0.88 0.04 1.04 2.78 12.86 7 x 1.12 1.23 2.81 3.07 0.27 1.34 0.88 0.03 1.02 2.78 14.54 7 x 2.45 1.64 2.75 0.04 0.27 0.91 0.75 0.04 0.90 2.62 12.36 MAP Estimator Sensor Selection Results . Next, sensor 7 x 2.45 1.64 1.92 0.64 0.22 1.31 0.71 0.04 0.86 2.57 12.36 selection is conducted assuming that a MAP estimator is 8 x x 0.15 0.14 2.81 3.04 0.27 1.33 0.87 0.02 0.97 2.77 12.38 8 x x 0.48 0.15 2.75 0.04 0.27 0.90 0.74 0.03 0.89 2.62 8.87 applied for health parameter estimation. Here, the metric 8 x x 0.48 0.15 1.92 0.63 0.22 1.31 0.71 0.03 0.85 2.57 8.87 previously introduced in Eq. ( 14 ) is used to theoretically predict 8 x x 1.12 1.23 2.75 0.04 0.27 0.90 0.74 0.03 0.86 2.61 10.55 8 x x 1.12 1.23 1.92 0.63 0.22 1.31 0.71 0.03 0.82 2.57 10.55 the health parameter SSEE accuracy offered by each of the 8 x x 2.45 1.64 0.02 0.04 0.01 0.05 0.71 0.04 0.86 2.57 8.40 candidate sensor suites. Additionally, a Monte Carlo simulation 9 x x x 0.15 0.14 2.75 0.02 0.27 0.89 0.74 0.02 0.80 2.61 8.39 9 x x x 0.15 0.14 1.92 0.61 0.22 1.29 0.70 0.02 0.76 2.56 8.39 study is performed to verify the theoretical results. Here, 9 x x x 0.48 0.15 0.02 0.04 0.01 0.05 0.71 0.03 0.85 2.57 4.91 * 400, 000 health parameter vectors are randomly generated in 9 x x x 1.12 1.23 0.02 0.04 0.01 0.05 0.71 0.03 0.82 2.57 6.59 10 x x x x 0.15 0.14 0.02 0.02 0.01 0.03 0.70 0.02 0.76 2.56 4.43 accordance with P . These health parameters along with h sensors added random sensor measurement noise are substituted into Eq. ( 7 ) Monte Carlo Health Parameter Mean Squared Estimation Errors (% squared) to baseline to produce sensed measurement test cases, which are then η γ η γ η γ η γ η γ SSEE FAN FAN LP C LP C HP C HP C HP T HP T LP T LP T T14 T25 P14 P25 # Sensors processed to produce health parameter estimates using Eq. ( 8 ) .
6 2.46 1.64 2.81 3.10 0.27 1.34 0.88 0.04 1.06 2.78 16.36 7 x 0.48 0.15 2.81 3.09 0.27 1.33 0.88 0.04 1.04 2.78 12.86 The resulting theoretical and Monte Carlo simulation health 7 x 1.13 1.23 2.81 3.08 0.27 1.34 0.88 0.03 1.02 2.78 14.55 parameter mean squared estimation errors are shown in Table 5 .
7 x 2.46 1.64 2.75 0.04 0.27 0.91 0.75 0.04 0.90 2.62 12.37 7 x 2.46 1.64 1.91 0.64 0.22 1.31 0.71 0.04 0.86 2.57 12.36 Here, the theoretical and Monte Carlo simulation results exhibit 8 x x 0.15 0.14 2.81 3.05 0.27 1.32 0.87 0.02 0.97 2.77 12.38 very good agreement, which is expected given the large number 8 x x 0.48 0.15 2.75 0.04 0.27 0.90 0.74 0.03 0.89 2.62 8.86 8 x x 0.48 0.15 1.91 0.63 0.22 1.30 0.71 0.03 0.85 2.57 8.86 8 x x 1.12 1.23 2.75 0.04 0.27 0.90 0.74 0.03 0.86 2.62 10.55 * The disparity in the number of Monte Carlo trials conducted for the MAP 8 x x 1.12 1.23 1.91 0.63 0.22 1.30 0.71 0.03 0.82 2.57 10.54 estimator versus the Kalman filter is due to the nature of the two estimators. 8 x x 2.46 1.64 0.02 0.04 0.01 0.05 0.71 0.04 0.86 2.57 8.41 9 x x x 0.15 0.14 2.75 0.02 0.27 0.88 0.74 0.02 0.80 2.61 8.38 The MAP estimator only requires a single steady - state sample for each random 9 x x x 0.15 0.14 1.91 0.61 0.22 1.29 0.70 0.02 0.76 2.56 8.38 health parameter vector con sidered. Conversely, the Kalman filter, which is a 9 x x x 0.48 0.15 0.02 0.04 0.01 0.05 0.71 0.03 0.85 2.57 4.91 dynamic recursive estimator, requires a sufficient quantity of measurement data 9 x x x 1.12 1.23 0.02 0.04 0.01 0.05 0.71 0.03 0.82 2.57 6.60 at each health condition to ensure convergence to a steady - state solution. This 10 x x x x 0.15 0.14 0.02 0.02 0.01 0.03 0.70 0.02 0.76 2.56 4.43 limited the practical number of Monte Carol t rials for the Kalman filter.
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited. 9 of Monte Carlo trials runs. The sensor suites identified as gas path faults considered. Therefore, in order to present a optimal for the MAP estimator agree with those previously more interesting sensor selection problem, the sensor identified in Table 4 for the Kalman filter. Furthermore, the measurement noise was increased by a factor of four and the mean squared estimation errors of individual health parameters diagnostic assessment was repeated. The ensuing correct and the overall health parameter SSEE for most sensor suites classification rate results are shown in Table 7 . The top half of exhibit fairly good agreeme nt between the MAP estimator and the table shows theoretically predicted results while the bottom the Kalman filter. This is not unexpected since both estimators half shows results obtained via the Monte Carlo simulation are designed to minimize the mean sum of squared estimation analysis. Here, the theoretical results slightly under - predict the errors, and in this study both incorporate the same a priori correct classification rates found via Monte Carlo analysis. This knowledge regarding health parameter c ovariance, P , and is due to the simplification made in deriving the theoretical h make the same assumptions regarding sensor measurement c orrect classification rate , which essentially establishes a covariance, R . theoretical lower bound on th is rate. Based on the theoretical analysis, the sensor selection decisions for gas path fault Gas Path Fault Diagnostics Sensor Selection Results diagnostics would be: For the gas path fault diagnostics sensor selection problem setup, it is assumed that the engine may encounter eig ht Baseline + 1 sensor, choose: T25 different gas path fault types consisting of turbomachinery Baseline + 2 sensors, choose: T25 and T14 faults (implemented via health parameter perturbations) and Baseline + 3 sensors, choose: T25, T14, and P25 actuator biases. The eight faults along with the parameter perturbations applied within C - MAPSS40k to generate the fault The Monte Carlo simulation analysis shows the same optimal influence coefficient matrix are shown in Table 6 . For this sensor suites except for the Baseline + 3 sensor case, where P14 study, all faults are assumed to occur in isolation and to be of would be substituted in place of P25.
equivalent probability of occurrence.
Table 7 . Gas path fault diagnostic accuracy Table 6 . Gas Path Faults sensors added Theoretical Correct Classification Rate (CCR) % to baseline Fault Fault H ealth parameters and No Fault Fan LPC HPC HPT LPT Wf VSV VBV T14 T25 P14 P25 # Sensors ID t ype actuator biases Fault CCR 6 73.3 74.4 99.2 100.0 98.8 92.1 60.6 78.4 99.0 84.6 1 Fan fault η FAN = - 1%, γ FAN = - 2% 7 x 80.6 74.6 99.2 100.0 99.8 90.6 60.7 78.6 99.0 85.5 7 x 81.6 75.2 99.2 100.0 99.9 90.6 60.8 79.5 99.0 85.8 2 LPC fault η = - 1%, γ = - 2% LPC LPC 7 x 76.3 87.1 99.9 100.0 98.8 90.6 70.4 88.8 99.0 89.0 3 HPC fault η = - 1%, γ = - 2% 7 x 77.0 94.2 99.9 100.0 98.8 90.6 73.3 98.8 99.0 91.6 HPC HPC 8 x x 86.3 75.4 99.2 100.0 100.0 89.0 60.7 79.7 99.0 86.3 4 HPT fault η = - 2%, γ = +1% HPT HPT 8 x x 82.9 87.2 99.9 100.0 99.8 89.1 70.1 88.9 99.0 89.7 8 x x 83.5 94.3 99.9 100.0 99.8 89.1 72.9 98.8 99.0 92.3 5 LPT fault η = - 2%, γ = +1% LPT LPT 8 x x 83.8 87.6 99.9 100.0 99.9 89.1 70.1 89.3 99.0 90.0 8 x x 84.4 94.3 99.9 100.0 99.9 89.1 72.9 98.8 99.0 92.4 6 Wf bias Wf bias = - 2% 8 x x 77.5 95.9 99.9 100.0 98.8 89.1 76.3 98.8 99.0 92.0 7 VSV bias VSV bias = - 1 degree stroke 9 x x x 88.0 87.7 99.9 100.0 100.0 87.5 69.7 89.4 99.0 90.3 9 x x x 88.4 94.4 99.9 100.0 100.0 87.5 72.4 98.8 99.0 92.7 8 VBV bias VBV bias = +20% 9 x x x 83.9 95.9 99.9 100.0 99.8 87.5 75.7 98.8 99.0 92.7 9 x x x 84.7 96.0 99.9 100.0 99.9 87.5 75.8 98.8 99.0 92.8 10 x x x x 88.7 96.0 99.9 100.0 100.0 86.0 75.2 98.8 99.0 93.1 For each of the 16 candidate sensor suites, the correct sensors added Monte Carlo Correct Classification Rate (CCR) % classification rate metric given in Eq. ( 29 ) is applied to to baseline No Fault calculate the theoretical correct classification rate offered by Fan LPC HPC HPT LPT Wf VSV VBV P14 T14 P25 T25 Fault CCR # Sensors each candidate sensor suite. In making this assessment, the 6 77.3 79.2 99.3 100.0 98.8 92.1 78.1 82.5 99.0 88.4 7 x 83.7 79.5 99.3 100.0 99.8 90.5 77.2 82.5 99.0 89.1 applied WSS M signal fault detection threshold is s et to give a 7 x 83.9 80.0 99.3 100.0 99.8 90.4 77.1 82.9 99.0 89.2 theoretical false positive rate of 0.01 (1%) as defined via Eq.
7 x 77.3 87.8 99.9 100.0 98.8 90.4 79.6 89.3 99.0 90.4 7 x 77.7 94.7 100.0 100.0 98.8 90.3 80.4 98.9 98.9 92.6 ( 22 ) . This threshold will change based on the number of sensors 8 x x 87.8 80.2 99.3 100.0 100.0 89.1 76.3 83.1 99.0 89.5 included in each candidate sensor suite. Additionally, a Monte 8 x x 84.0 88.1 99.9 100.0 99.8 89.1 78.7 89.3 99.0 91.1 Carlo simulation analysis was conducted to verify the 8 x x 84.1 94.7 100.0 100.0 99.8 89.0 79.4 98.9 98.9 93.2 8 x x 84.0 88.6 99.9 100.0 99.8 89.0 78.5 89.6 99.0 91.2 theoretically p redicted CCR results. This is done using Eq. ( 15 ) 8 x x 84.3 94.8 100.0 100.0 99.8 89.0 79.6 99.0 99.0 93.3 to generate 8 0,000 no fault cases and 10,000 fault cases for 8 x x 77.5 96.0 100.0 100.0 98.8 88.9 80.6 98.9 99.0 92.6 each individual fault type, all corrupte d by random 9 x x x 88.1 88.7 99.9 100.0 100.0 87.5 77.9 89.7 99.0 91.5 9 x x x 88.3 94.8 100.0 100.0 100.0 87.7 78.8 99.0 99.0 93.5 measurement noise, v . This data set is then analyzed by 9 x x x 84.2 96.0 100.0 100.0 99.8 87.5 80.0 98.9 99.0 93.3 applying the single fault diagnostic logic shown in Eqs. ( 16 ) - 9 x x x 84.3 96.1 100.0 100.0 99.8 87.4 80.0 99.0 99.0 93.3 ( 20 ) to detect and classify the occurrence of any faults. Initial 10 x x x x 88.3 96.1 100.0 100.0 100.0 86.0 79.3 99.0 99.0 93.6 diagnostic analysis revealed that even the baseline 6 sensor measurement suite performed extremely well in diagnosing the This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited. 10 DISCUSSION different engine operating points to assess how this affects The sensor selection metrics introduced in this paper were sensor selection results. Furthermore, equal importance is shown to perform well in identifying optimal sensor suites from placed on each parameter to be estimated and each fault type to a performance estimation and diagnostic accuracy perspect ive . be diagnosed. A natural extension to the m etrics is to place a Although not specifically shown in this paper, the resulting user - specified weighting on the different parameters or faults sensor suites identified to be optimal are expected to change if based on their criticality or frequency of occurrence. Finally, different assumptions are made regarding the design inputs the estimation and diagnostic accuracy is only one piece of the such as sensor measurement noise, health parameter overall sensor selection decision process. O ther factors of merit covariance, fau lt types and magnitudes, and the engine model include criteria such as sensor weight, reliability, and overall that the metrics are applied to. life cycle cost. Those factors should also be considered as part A notable finding in this work was the relative agreement of the sensor selection process.
between the Kalman filter and MAP estimator in terms of the predicted SSEE results and the sensor suites identified to be CONCLUSIONS optimal. A s previously noted, this is not unexpected given that The sensor selection metrics introduced in this paper both estimators are set up to minimize mean squared estimation prov ide analytical tools to assist engine health management error and , in the given example application , both incorporate system designers in making sensor selection decisions. The the same a priori knowledge regarding health parameter metrics are easy to use , and are specifically tailored towards covariance and sensor measurement noise covariance. estimation and diagnostic approaches commonly applied to However, an advantage of the MAP estimator metric is that it aircraft engines . The y can be readily applied for assess ing the offers a closed - form solution while the Kalman filter SSEE benefits of adding or removing currently available engine * metric requires solution of the V transformat ion matrix via an sensors, or assessing the benefits of newly developed sensors as optimal iterative search . As such, the Kalman filter metric can they become available. Through Monte Carlo simulation be prone to convergence to local minima. To guard against such analysis, the metrics were verified to perform well in occurrences, a recommended approach is to cross - check identifying optimal sensor suites when evaluated using linear Kalman filter results using the MA P metric to ensure that system information. For both Kalman filter and maximum a similar sensor suites and SSEE values are predicted . posteriori health parameter estimation, the corresponding T he performance estimation metrics exhibited very good sensor selection metrics were found to perform very well in agreement between their theoretically predicted estimation satisfying their intended objective — identifying the sensor suite accuracy and that obtained via Monte Carlo simulation that minimizes the mean sum of squared estimation errors . The analysis. However, the theoretica l gas path fault diagn ostic gas path fault diagnostic sensor selection metric based on metric was found to under - predict the CCR found via Monte theoretical correct classification rate also performed well in its Carlo analysis. This is due to the two - fault class obj ective of identifying sensor suites that provide the best misclassification assumption made in deriving the metric.
diagnostic performance . Due to a simplification made in the While this simplification does make the derivation tractable, it theoretical derivation , the metric was found to under - predict the can lead to inaccurate results, especially when faults are prone true correct classification rate . However, it does provide a to misclassification as more than one fault type. Another theoretical lower bound on correct classification performance simplification made in this derivation is to assume that all offered by a given sensor suite . Additionally, it is effective for sensor residual measurement s are independent. This assumption identifying fault pairs at risk for misclassification and making does n ot usually hold for gas turbine engine applications , as sensor selection decisions to address such risks . Recommend ed some amount of covariance usually exists between sensor follow on work is to couple these accuracy metrics with residual measurements . For example, they are corrected using additional figures of merit pertinent for sensor selection the same parameters and generate d using the same reference decision. This includes considering the individual criticality of model . An approach to ad dress this is to define sensor the performance parameters to be estimated or the fault types to measurement probability density functions in multi - parameter be diag nosed, and to also couple these metrics with additional space and then perform multidimensional integration to assess metrics reflecting the life cycle cost of adding specific sensors.
detection and classification performance. However, this would add much more complexity. The given metr ic based on the ACKNOWLEDGMENTS properties of the chi square distribution and the non - central chi This work was conducted under the NASA Aviation Safety square distribution is more simplistic, but should be verified by Program, Vehicle Syst ems Safety Technologies Project .
additional analysis such as the Monte Carlo simulation analysis conducted in this paper .
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This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited. 12