Donald L. Simon

Glenn Research Center, Cleveland, Ohio

Aidan W. Rinehart

Vantage Partners, LLC, Brook Park, Ohio

Sensor Selection for Aircraft Engine Performance 

Estimation and Gas Path Fault Diagnostics

NASA/TM—2016-218926

January 2016

GT2015–43744
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Donald L. Simon

Glenn Research Center, Cleveland, Ohio

Aidan W. Rinehart

Vantage Partners, LLC, Brook Park, Ohio

Sensor Selection for Aircraft Engine Performance 

Estimation and Gas Path Fault Diagnostics

NASA/TM—2016-218926

January 2016

GT2015–43744

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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NASA/TM—2016-218926 1

Sensor Selection for Aircraft Engine Performance

Estimation and Gas Path Fault Diagnostics

Donald L. Simon

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Aidan W. Rinehart

Vantage Partners, LLC

Brook Park, Ohio 44142

Abstract

This paper presents analytical techniques for aiding system 

designers in making aircraft engine health management sensor 

selection decisions. The presented techniques, which are based 

on linear estimation and probability theory, are tailored for gas 

turbine engine performance estimation and gas path fault 

diagnostics applications. They enable quantification of the 

performance estimation and diagnostic accuracy offered by 

different candidate sensor suites. For performance estimation, 

sensor selection metrics are presented for two types of 

estimators including a Kalman filter and a maximum a 

posteriori estimator. For each type of performance estimator, 

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 diagnostics, the sensor selection 

metric is set up to maximize correct classification rate for a

diagnostic strategy that performs fault classification by 

identifying the fault type that most closely matches the 

observed measurement signature in a weighted least squares 

sense. Results from the application of the sensor selection 

metrics to a linear engine model are presented and discussed. 

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 

diagnostics. For any given sensor suite, Monte Carlo 

simulation results are found to exhibit good agreement with

theoretical predictions of estimation and diagnostic accuracies.

Introduction

Aircraft operators rely on engine performance estimation 

and gas path fault diagnostics to ensure the safe and efficient 

operation of their gas turbine engine assets. Performance 

estimation enables the estimation and trending of gradual 

performance deterioration that the engine will experience over 

time due to fouling, corrosion, and erosion of turbomachinery 

components. Gas path fault diagnostics enables the detection 

and isolation of gas path system faults affecting engine 

performance, which are typically relatively rapid or abrupt in 

nature (Refs. 1 and 2). A notional illustration of the observed 

measurement shifts caused by gradual deterioration compared 

to an abrupt fault is shown in Figure 1. 

Although performance estimation and gas path fault 

diagnostics typically apply different algorithmic approaches, 

both are conducted using the same engine sensor measurement 

data—primarily data acquired from the available engine 

control sensor suite. In general, adding additional engine 

sensors will improve performance estimation and diagnostic 

accuracy, but this does add to the overall engine life cycle 

cost. Therefore, the decision to add sensors should be made 

judiciously. 

Several researchers have presented sensor selection 

approaches for engine health management applications. 

Mushini and Simon (no relation to the author) proposed a 

sensor selection approach for Kalman filter-based 

performance estimation applications (Ref. 3). In this work, a 

performance metric was defined as a function of the steady 

state error covariance and the cost of the selected sensors. 

Three separate metrics were considered for searching for the 

optimal sensor suite, including a random search, a genetic 

algorithm search, and an exhaustive search. The study by 

Mushini and Simon assumed that the estimation problem was

over-determined (i.e., there are more sensors than unknown 

parameters to be estimated), which is usually not the case for 

engine performance estimation applications. Borguet and 

Léonard approached the problem of sensor selection for

engine performance estimation within the scope of linear 

information theory (Ref. 4). They defined performance 

metrics based on the Fisher information matrix, and an 

exhaustive search was conducted to identify the best sensor 

suite. Sowers et al. introduced a systematic framework for 

automating sensor selection decisions for diagnostic 

applications. This framework enables incorporation of factors 

of merit commonly considered in the sensor selection process

including diagnostic accuracy, diagnostic criticality, and cost

(Ref. 5). The framework relies on the end user to specify the 

merit function used by the optimal search algorithm. 

Kamboukos et al. proposed sensor selection for performance
NASA/TM—2016-218926 2

Time (flights)

Measurement shift

Measurement shift 

Measurement shift moving average

Gradual deterioration

Gradual deterioration

Rapid shift 

(potentially due to 

a fault event)

Figure 1.—Gradual versus rapid performance shifts.

estimation applications based on the condition number of the 

influence matrix that relates changes in health parameters to 

changes in sensed measurements (Ref. 6). Here, a determined 

health parameter estimation problem was considered where 

there are as many sensors as parameters to be estimated. 

The contribution of this paper will be to introduce separate 

sensor selection metrics for performance estimation and fault 

diagnostic applications. In terms of performance estimation, 

the problem is assumed to be underdetermined (i.e., fewer 

sensors than unknown health parameters to be estimated), and 

two separate estimators will be considered—one applying a 

Kalman filter designed for processing dynamic sensed 

measurement information, and a second applying a maximum 

a posteriori estimator for processing quasi-steady-state 

measurement data. In terms of fault diagnostics, a single fault 

diagnostic strategy applying a weighted least squares 

hypothesis test will be considered. 

The remainder of this paper is organized as follows. First, 

metrics are defined through analytical derivations of the 

performance estimation accuracy and gas path fault diagnostic 

accuracy based on linear system theory. These analytical 

functions can be directly used to theoretically predict the 

estimation or diagnostic accuracy offered by a given sensor 

suite. Next, example application of the sensor selection 

techniques is presented by applying the approaches to a linear

engine model. Theoretically predicted results are calculated 

and compared against empirical results obtained through 

Monte Carlo simulation analysis. This is followed by 

discussions and conclusions. 

Nomenclature

A, Axh, Axq, B, Bxh, Bxq, 

C, Cxh, Cxq, D, L, M

system matrices

CCR correct classification rate

C-MAPSS40k Commercial Modular Aero-Propulsion 

System Simulation 40k

DM Mahalanobis distance

FPR false positive rate

H influence coefficient matrix relating

changes in health parameters to changes 

in sensed measurements

Hf fault influence coefficient matrix relating 

faults to changes in sensed measurements

I identity matrix

MAP maximum a posteriori

N number of fault types

PMC probability of misclassification

Ph health parameter covariance matrix

R measurement noise covariance matrix

SSEE sum of squared estimation errors

T fault detection threshold

TPR true positive rate

V* transformation matrix relating h to q

WSSE weighted sum of squared errors

WSSM weighted sum of squared measurements

h health parameter vector

f fault vector

k number of additional sensors to add

m number of tuning parameters

n Number of additional sensors to choose 

from

p number of health parameters

q reduced order tuning parameter vector

u actuator command vector

v measurement noise vector

wk, wh,k, wxh,k process noise vectors

x state vector

y measurement vector

Γ gamma function

γ lower incomplete gamma function

ε residual vector (estimate minus its 

expected value) 

Φ standard normal distribution function

λ mean value of the WSSM signal

μi mean value of i

th sensed measurement

Subscripts

a fault type index

b misclassified fault type index

k sample index

xh augmented state vector (x and h)

xq reduced order state vector (x and q)

Superscripts

† pseudo-inverse

^ estimated value

~ error value

– mean value
NASA/TM—2016-218926 3

Operators

E[•] expected value of argument

tr{•} trace of a matrix

Sensor Selection Metrics

As previously mentioned, aircraft engine performance 

estimation and gas path fault diagnostics pose different 

problem formulations. Analytical formulations of each are 

introduced below along with derivations of performance 

estimation and diagnostic accuracy for a given sensor suite. 

The performance estimation problem assumes the application 

of two separate estimators—a linear Kalman filter and a 

maximum a posteriori estimator, while the gas path fault 

diagnostic problem assumes the application of a single fault 

isolator applying a weighted least squares hypothesis test. 

Kalman Filter-Based Health Parameter Estimation

In the aircraft engine community, Kalman filters are 

commonly applied for on-board performance estimation or 

post-flight analysis of full-flight streaming measurement data. 

In this subsection, Kalman filter health parameter estimation 

accuracy is discussed following a derivation previously 

introduced by Simon and Garg as part of an optimal tuner 

selection methodology for Kalman filter-based performance 

estimation applications (Ref. 7). This optimal tuner selection 

methodology is designed to minimize the Kalman mean 

squared estimation error in the parameters of interest when 

facing underdetermined estimation problems, but can readily 

be extended to also calculate the mean squared estimation 

error offered by different sensor suites, as was shown in 

Reference 8.

The formulation begins by considering the following 

discrete linear time-invariant state space equations

representing engine dynamics about an operating point

k k k k k

k k k k k

y C x D u M h v

x A x B u L h w

∆ = ∆ + ∆ + ∆ +

∆ +1 = ∆ + ∆ + ∆ +

(1)

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 

outputs. The vector h, where h ∈ p

, represents the engine 

health parameters, which induce shifts in other variables as the 

health parameters deviate from their nominal values. The Δ 

symbols denote parameter deviations relative to the linear 

operating point trim condition. The vectors w and v are 

uncorrelated zero-mean white noise input sequences. The 

matrices A, B, C, D, L, and M are of appropriate dimensions. 

Through algebraic manipulation, Equation (1) can be rewritten 

such that h is concatenated with x to form an augmented state 

vector, xxh, as shown in Equation (2). Since engine 

performance deterioration 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. 

 

[ ]



xh xh k k k

k k

x

k

k

C

k

xh xh k xh k xh k

w

h k

k

k

x B

k

k

A

k

k

C x Du v

Du v

h

x

y C M

A x B u w

w

w

u

B

h

x

I

A L

h

x

xh k

xh

xh k xh xh k xh

= + +

 + +



 



 =

= + +





 



  +



 



  +



 









 



  =



 





+

+

,

, ,

1 ,

1

,

, ,

0 0



 

(2)

The vector wxh is zero-mean white noise associated with the 

augmented state vector, [xT hT]

T. wxh consists of the original 

state process noise, w, concatenated with the process noise 

associated with the health parameter vector, wh.

Once the h vector is appended to the state vector as shown 

in Equation (2), it may be directly estimated by applying a 

Kalman filter as long as the system is observable. However, 

the number of health parameters that can be estimated is 

limited to the number of sensors, the dimension of y (Ref. 9), 

and typically an aircraft gas turbine engine has fewer sensors 

than health parameters. To enable Kalman filter formulation 

for an underdetermined estimation problem, a reduced-order 

state space model is constructed. This is accomplished by 

defining a model tuning parameter vector, q, which is a linear 

combination of all health parameters, h, given by

q V h* = (3)

where q ∈ m, h ∈ p, m < p, and V* is an m × p

transformation matrix of rank m, which relates h to q. Given 

an estimate of q (i.e., qˆ ), an approximation of the health 

parameter vector, h

ˆ , can be obtained as

h V qˆ ˆ *† = (4)

where V*† is the pseudo-inverse of V*. Substituting Equation (4) 

into Equation (2) yields the following reduced-order state space 

equations, which may be used to formulate a Kalman filter
NASA/TM—2016-218926 4

 

[ ]



xq xq k k k

k k

x

k

k

C

k

xq xq k xq k xq k

w

q k

k

k

x B

k

k

x A

k

k

C x Du v

Du v

q

x

y C MV

A x B u w

w

w

u

B

q

x

I

A LV

q

x

xq k

xq

xq k xq k xq xq k xq

= + +

 + +



 



 =

= + +





 





 +



 



  +



 









 





 =



 





+

+

+

,

*†

, ,

,

*†

1

1

,

, 1 , ,

0 0



  

(5)

The reduced-order equations introduced in Equation (5) will 

enable a Kalman filter to be formulated that can estimate the 

augmented state vector, [xT qT]

T. The resulting Kalman filterproduced tuner parameter vector estimate, qˆ , can be inserted 

into Equation (4) to produce an estimated health parameter 

vector, h

ˆ However, this does not circumvent the 

underdetermined nature of the h

ˆ estimation problem, and the 

fact that the produced h

ˆ estimates will contain errors is 

unavoidable. However, estimation accuracy is directly 

dependent on the available sensor suite and the selection of the 

transformation matrix, V*

. This gives rise to an optimization 

problem of selecting the best sensor suite and the 

corresponding V* that minimizes the estimation error in the 

parameters of interest. For a given sensor suite, an optimal 

iterative search can be conducted to select a V* matrix that 

minimizes the theoretical mean sum of squared estimation 

errors (SSEE) in the parameters of interest

( )

V m p

SSEE V ∈R ×

*

*

arg min (6)

where the above statement indicates the V* matrix that 

minimizes the SSEE function. Once V* is obtained, it can be 

inserted into Equation (5) to construct the reduced-order state 

space equations. Here, it is important to emphasize that the V*

matrix and q vector are unique to each sensor suite considered.

Therefore, Equation (6) is individually applied to each sensor 

suite, and the suite that provides the lowest SSEE is identified 

as optimal.

Due to page limitations, a complete derivation of the 

Kalman filter SSEE metric is not provided in this document. 

However, readers are referred to Reference 7 for this 

derivation. Some notable aspects regarding the derivation are

that it focuses on linear Kalman filter estimation accuracy 

under steady-state operating conditions, and that the error of 

each estimated parameter comprises mean squared bias and 

variance terms. Additionally, the derivation incorporates userspecified a priori knowledge regarding the health parameter 

covariance matrix reflecting the expected distribution in the 

health parameters to be estimated. While this paper will only 

consider Kalman filter health parameter estimation accuracy, 

the technique can be readily extended to optimize the 

estimation accuracy of any unmeasured performance 

parameters such as thrust, airflows, or metal temperatures. 

Maximum A Posteriori Health Parameter Estimation

Maximum a posteriori (MAP) estimation is commonly 

applied for ground-based aircraft engine gas path analysis. It is 

based on quasi-steady-state engine snapshot measurements 

acquired in flight (Refs. 2 and 10). Unlike the Kalman filter, 

which is a recursive estimator designed to process dynamic 

measurement data, the MAP estimator provides a point 

estimate based on an assumed quasi-steady-state measurement 

process. The MAP 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 problems. To introduce the MAP 

estimator, consider the following linear steady-state 

measurement process

∆y = H∆h + v (7)

where H is an influence coefficient matrix that relates the 

effects of the health parameter vector changes, Δh, to changes

(i.e., residuals) in the sensed measurement vector, Δy. Here, v, 

is zero-mean white noise with covariance R. As with the 

previously introduced Kalman filter equations, the Δ symbols 

denote parameter deviations relative to the operating point

trim condition at which Equation (7) was generated. For 

simplicity, the Δ symbols are omitted throughout the 

remainder of this section on the MAP estimator and the terms 

y and h are used to indicate measurement and health parameter 

changes, respectively. The maximum a posteriori (MAP) 

estimator follows the closed form expression

( )

h G y

h P H R H H R y

h

G

T T

h

h

=

= + − − − −

ˆ

ˆ 1 1 1 1

 (8)

where Ph is a matrix containing a priori knowledge of the 

expected health parameter covariance. As with the Kalman 

filter introduced above, the MAP estimator produces a biased 

estimate due to the underdetermined nature of the estimation 

problem. However, its accuracy depends on the available 

sensor suite, thus giving rise to a sensor selection problem. As 

with the Kalman filter, the MAP health parameter estimation 

error will be defined in terms of the sum of squared estimation 

errors (SSEE), which consists of the sum of two components: 

mean squared bias and variance, as defined below.
NASA/TM—2016-218926 5

MAP Estimation Mean Squared Bias

The bias of an estimator is the expected difference between 

the estimator’s estimated value and the true value of the 

parameter being estimated. For the MAP estimator, the 

estimated health parameter bias vector, h

~ , is defined as

[ ]



( )

[( ) ]

( )



[ ] [ ]

(G H I)h

G H I E h G E v

E G H I h G v

E G Hh v h

E G y h

h E h h

h

h

h

h

h h

y

h

h

h

= −

= − +

= − +

















= + −

















= −

= −

0

ˆ

ˆ ~

 (9)

where the operator E[●] represents the expected value of the 

argument, and the expected value properties E[h]=h and 

E[v]=0 are leveraged in Equation (9). The estimation error 

bias equation given in Equation (9) is a function of an 

arbitrary health parameter vector h. The mean sum of squared 

biases across a fleet of engines is given as 

{ }

[ {( ) ( ) }]

( ) [ ]( )

{( ) ( ) } T

h h h

T

h

P

T h

T

h T h

T T

tr G H I P G H I

tr G H I E hh G H I

E tr G H I hh G H I

h E h h E tr h h

h

= − −





















= − −

= − −



  =

  =



~ ~ ~ ~~ 2

(10)

where tr{●} represents the trace (sum of the diagonal 

elements) of the matrix. Here, the E[hhT] reduces to the health 

parameter covariance matrix, Ph, which is leveraged in 

Equation (10).

MAP Estimation Variance

The variance of the MAP estimate is found by constructing 

the estimation covariance matrix, hPˆ , which is defined as

( [ ])( [ ]) 















= − −

ε ε

T

hP E h E h h E h  ˆ ˆ ˆ ˆ ˆ (11)

where the vector ε is the residual between h

ˆ and its expected 

value. By combining Equation (7) and Equation (8), ε can be 

written as

( [ ])

[ ]

( ) [ ( )]



[ ] [ ]

G v

G Hh G v G H E h G E v

G Hh v E G Hh v

G y E G y

h E h

h

h

h

h h h

h h

h h

=

= + − −

= + − +

= −

ε = −

0

ˆ ˆ

(12)

Then, by substituting Equation (12) into Equation (11) the 

covariance matrix of the MAP estimate becomes

[ ]

[ ]

[ ]

T h h

T

h

R

T h

T

h

T h

T

h

G RG

G E vv G

E G vv G

P E

=

=

=

= εε



ˆ

(13)

Diagonal elements of hPˆ will reflect the variance of 

individual health parameter estimates, while off diagonal 

elements reflect the covariance between estimates. 

The overall sum of squared estimation errors (SSEE) can be 

obtained by combining the estimation mean squared bias and 

variance information as 

( ) {( ) ( ) } { }T

h h

T SSEE h

ˆ = tr GhH − I Ph GhH − I + tr G RG (14)

Mean squared bias and variance are equally weighted in the 

above equation. However, end users may weight them 

differently if they so choose.

Weighted Least Squares Single Fault Diagnostic 

Approach

Gas path fault diagnostics poses a different problem than 

that of performance estimation. Unlike performance 

deterioration, which is assumed to occur gradually and affect

all health parameters simultaneously and somewhat 

independently, gas path faults are assumed to primarily occur

abruptly and in isolation. In other words, it is rare to have 

multiple unrelated gas path system faults occurring 

simultaneously. Applying the single fault assumption 

transforms gas path fault diagnostics from an underdetermined 

to an over-determined estimation problem. This subsection 

will present a single fault isolator that applies a weighted least 

squares hypothesis test to diagnose faults. Additionally, the 
NASA/TM—2016-218926 6

accuracy offered by this diagnostic approach is analytically 

derived.

The fault diagnostic approach considered in this study, like 

the previously described MAP estimation approach, is groundbased, and designed to process snapshot engine measurements 

acquired in flight. To introduce the diagnostic approach, first 

assume the following linear steady-state sensor measurement 

process 

∆∆y = H f f + v (15)

where ΔΔy is a vector of residuals reflecting recent shifts in

engine sensor measurements, for example, the change 

measurements have undergone within the past one or two 

flights. Also shown in Equation (15) is f, a vector of gas path 

fault magnitudes, and Hf, a fault influence coefficient matrix 

relating fault magnitudes to sensor measurement residuals. 

Furthermore, v denotes zero-mean normally distributed sensor 

measurement noise of covariance R. The measurement 

residuals, ΔΔy, are regularly updated as new snapshot data 

become available. They are referred to as “delta-delta” 

measurement shifts, as they will reflect fault induced shifts

relative to the gradual deterioration induced shifts the engine 

has experienced up until the time of fault initiation (Ref. 2). 

Since faults are assumed to occur abruptly and cause relatively 

large measurement shifts, the ΔΔy residuals will be small in 

the case of no fault, and larger once a fault has occurred (see 

Fig. 1). For simplicity, the ΔΔ symbols are omitted throughout 

the remainder of this section and the term y is used to indicate

recent observed shifts in the sensor measurements. Given 

Equation (15), a fault detection and classification (isolation)

approach can be formulated. Here, it is assumed that fault 

detection is performed by calculating and monitoring a 

weighted sum of squared measurement (WSSM) signal:

WSSM y R y T −1 = (16)

If the WSSM signal exceeds an established detection threshold 

(T), a fault is assumed to be present and the diagnostic logic 

proceeds in attempting to isolate the most plausible single 

fault root cause for the fault. Here, fault classification is 

performed by applying a weighted least squares approach. 

Each possible gas path fault type is evaluated individually, and 

the hypothesized fault whose signature best matches the 

observed measurement residuals in a weighted least squares 

sense is classified as the fault. For the l

th fault type, the 

estimated fault magnitude is calculated as

f (H R H ) H R y T

f l f l Tf l l 1 ,

1

, 1 , ˆ − − − = (17)

where Hf,l is the column of the Hf matrix corresponding to the 

l

th fault type, and the scalar l f

ˆ is the estimated magnitude of 

the l

th fault type that produces the best match of the observed 

vector of sensor measurement residuals, y, in a weighted least 

squares sense. The resulting l f

ˆ estimate is then combined 

with Hf,l to produce an estimated measurement residual vector, 

l yˆ , for the l

th fault type:

y H f l f l ˆ ˆ = , (18)

The difference between ylˆ and y defines the estimation error 

vector for the l

th fault type, yl

~ , defined as

yl = yˆl − y ~ (19)

The weighted sum of squared errors for the l

th hypothesized 

fault type is calculated as

l T

l WSSEl y R y ~ −1~ = (20)

After WSSE’s are calculated for each potential fault type 

they are compared, and the hypothesized fault type that 

produces the minimum WSSE is classified as the fault cause.

Theoretical predictions of fault detection and fault 

classification performance for the single fault isolator are 

given below.

Fault Detection Performance

For any diagnostic system, fault detection performance is 

directly related to the applied fault detection threshold. Larger 

thresholds will result in fewer false alarms in the absence of a 

fault (false positives), but will also result in fewer true 

detections when a fault is actually present (true positives), 

while the opposite is true for smaller thresholds. In order to 

facilitate a common basis of comparison, each sensor suite 

considered in this study applies a WSSM fault detection 

threshold, T, necessary to achieve a user-specified target false 

positive rate (FPR). The FPR of a system monitoring a WSSM

signal for fault detection purposes can be approximated if it is 

assumed that all sensed measurements are independent in 

addition to being zero mean and normally distributed. With 

this simplification, the distribution of the WSSM signal under 

the no-fault case will be the sum of the squares of k

independent standard normal random variables, which is a chi 

square distribution with k degrees of freedom. The cumulative 

distribution function of a chi square distribution is given as

(Ref. 11)
NASA/TM—2016-218926 7

( )





 



 Γ





 





γ

=

2

2

,

2 , k

k T

CDF T k (21)

where Γ(·) is the gamma function and γ(·) is the lower 

incomplete gamma function. The above equation reflects the 

probability that a random sample of the WSSM signal is less 

than the threshold, T, when no fault is present (i.e., the true 

negative rate). Therefore, the false positive rate is given as 

( ) ( )





 



 Γ





 





γ

= −

= −

2

2

,

2 1

, 1 ,

k

k T

FPR T k CDF T k

(22)

When a fault occurs, the WSSM signal will be distributed as 

a non-central chi-squared distribution. This distribution will be 

a function of: 1) the detection threshold, T; 2) the number of 

sensors, k; and 3) the mean value of the WSSM signal. The 

mean value of the WSSM signal for a fault of given type and 

magnitude is defined as λ, where ∑

=

λ = μ

k

i

i

1

2 , where μi is the 

mean value of the i

th sensor in the presence of the fault. Given 

this information, the true positive rate (TPR) can be calculated 

from the cumulative distribution function of the non-central 

chi-square distribution as (Ref. 11):

( ) ( ) ( ) ( ) 



















Γ +

λ γ +

λ = − ∑

∞

=

−λ

0

2

2

2 , 2

!

2 , , 1

j

j

k j

j k T

j

TPR T k e (23)

The above equation reflects the probability that a random 

sample of the WSSM signal is greater than the threshold, T, 

when a fault of magnitude λ is present. Given Equations (22)

and (23), overall FPR and TPR for individual fault types can 

be approximated for any given sensor suite. 

Fault Classification Performance

In this study an approximation of the theoretical 

misclassification rate is produced by considering the 

probability of misclassification between fault pair 

combinations (i.e., making the assumption that only two fault 

classes exist) given that a fault has been correctly detected. 

The two-class misclassification rate results 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-class misclassification rate given 

three or more faults. While this simplification does not enable 

an exact calculation of the overall misclassification rate for a 

given fault type, it is effective for identifying fault pairs at 

high risk of misclassification. Let us consider a fault of a 

given type, a, and magnitude, fa. From Equation (15), the 

expected sensed measurement vector under this condition 

becomes ya = Hf,a fa. The probability that a sensor 

measurement vector observation, y, collected when fault fa is 

present is misclassified as fault type b (assumed to be of 

equivalent probability and resulting in equivalent sensor 

measurement covariance as fault type a) is given as (Ref. 12)





 



 PMCb a = − Φ ⋅ DM 2

1

| 1 (24)

where PMCb|a is the probability of misclassifying fault type a

as b, Φ is the standard normal distribution function, and DM is 

the Mahalanobis distance defined as

( ) ( )

( ) ( )





























+ +

− −

− −

=

−

−

a b

T a b

a b

T a b

M

y y R y y

or

y y R y y

D min

1

1

(25)

The above expression accounts for the fact that the least 

squares estimation approach is able to produce bi-directional 

fault estimates of either a positive or negative magnitude. The 

sign that produces the minimum distance will have the largest 

contribution to the misclassification rate. In Equation (25), yb

is the expected sensor measurement vector for fault type b, 

scaled to be the same weighted length as yb as shown in 

Equation (26)

b T

b

a T

a b b

H R H

y R y

y H

1

1

−

−

= (26)

The above equations allow the probability of misclassification 

PMCb|a for each fault pair to be calculated. The overall 

probability of misclassification for fault type a can be 

approximated by summing all fault pair combinations:

∑

≠

=

=

N

b a

b

PMCa PMCb a

1

| (27)

where N is the number of different fault types. Once PMCa is 

obtained, an approximation of the correct classification rate 

(CCR) for fault type a of the considered fault magnitude can 

be found by combining the fault’s TPR (given by Eq. (23)) 

and its PMC (given by Eq. (27)) as
NASA/TM—2016-218926 8

( ) CCRa = TPRa × 1− PMCa (28)

The average CCR for the diagnostic system considering all 

fault types thus becomes

∑

=

=

N

a

a

N

CCR CCR

1

(29)

where CCRa is the correct classification rate for the ath fault 

type, and N is the total number of fault types. The CCR shown 

in Equation (29) serves as a metric that can be used to estimate 

and compare the diagnostic performance offered by different 

candidate sensor suites.

Linear Turbofan Engine Model Example

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 

matrices extracted from the NASA Commercial Modular 

Aero-Propulsion System Simulation 40k (C-MAPSS40k) 

turbofan engine simulation (Ref. 13) at standard day sea level 

static conditions (i.e., air temperature = 59 °F, altitude = 0, and 

Mach = 0) and an intermediate power setting. The linear 

model, which is used for Kalman filter estimation, and is of 

the format shown in Equation (1), has seven state variables 

and three control inputs (actuator commands), as shown in 

Table 1, and ten health parameters, as shown in Table 2. The 

linear model has six baseline sensors, and four additional 

(optional) sensors, which are shown in Table 3 along with 

their corresponding standard deviations. Here, the sensor noise 

is assumed to be uncorrelated, zero-mean and normally 

distributed. The linear influence coefficient matrix to be used 

in MAP estimation (i.e., the H matrix given in Equation (7)), 

and the linear fault influence coefficient matrix to be used in 

gas path fault diagnostics (i.e., the Hf matrix given in Equation

(15)), are generated from C-MAPSS40k at the same operating 

point as the linear model. However, these matrices are 

generated assuming that fan speed is held constant. As such, 

fan speed (Nf) is replaced by fuel flow (Wf) as one of the six 

baseline sensors when performing MAP estimation or gas path 

fault diagnostics. 

The optional sensors shown in Table 3 are evaluated for the 

estimation accuracy or diagnostic improvement they provide if 

added individually or in combination to the baseline sensor 

suite. Given a set of n additional sensors to choose from, and a 

target number, k, of additional sensors, the total number of 

sensor suite combinations will be:

!( )!

!

k n k

n

k

n

− =













 (30)

Therefore, the number of sensor combinations when adding 1, 

2, 3, or 4 sensors to the baseline 6 sensors are:

• Baseline sensors 1 combination

• Baseline + 1 sensor (n = 4, k = 1) 4 combinations

• Baseline + 2 sensors (n = 4, k = 2) 6 combinations

• Baseline + 3 sensors (n = 4, k = 3) 4 combinations

• Baseline + 4 sensors (n = 4, k = 4) 1 combinations

• Total sensor combinations 16 combinations

TABLE 1.—STATE VARIABLES AND CONTROL INPUTS

State variables

(x)

Control inputs

(u)

Nf – fan speed Wf – fuel flow

Nc – core speed VSV – variable stator vane

Hs_LPC – LPC metal temperature VBV –variable bleed valve

Hs_HPC – HPC metal temperature

Hs_burner – burner metal temperature

Hs_HPT – HPT metal temperature

Hs_LPT – LPT metal temperature

TABLE 2.—HEALTH PARAMETERS (h)

1 ηFAN Fan efficiency

2 γFAN Fan flow capacity

3 ηLPC Low pressure compressor (LPC) efficiency

4 γLPC Low pressure compressor (LPC) flow capacity

5 ηHPC High pressure compressor (HPC) efficiency

6 γHPC High pressure compressor (HPC) flow capacity

7 ηHPT High pressure turbine (HPT) efficiency

8 γHPT High pressure turbine (HPT) flow capacity

9 ηLPT Low pressure turbine (LPT) efficiency

10 γLPT Low pressure turbine (LPT) flow capacity

TABLE 3.—SENSED OUTPUTS AND STANDARD DEVIATION 

AS PERCENT OF OPERATING POINT TRIM VALUES

Sensed output Standard deviation

Baseline

sensors

Nfa – fan speed (rpm) 0.360 rpm

Nc – core speed (rpm) 1.23 rpm

Ps30 – HPC exit static pressure 0.333 psia

T30 – HPC exit total temperature 0.273 °R

P50 – LPT exit total pressure 0.021 psia

T50 – LPT exit total temperature 0.259 °R

Additional

(optional)

sensors

P14 – Bypass duct total pressure 0.022 psia

T14 – Bypass duct total temperature 0.117 °R

P25 – HPC inlet total pressure 0.031 psia

T25 – HPC inlet total temperature 0.132 °R

a

Note: For the MAP estimator and gas path fault diagnostics, fan 

speed (Nf) serves as the engine power reference parameter and is 

replaced in the list of six baseline sensors by fuel flow (Wf), which 

has a standard deviation of 9.03 pounds per hour (pph)

The subsections below will present and discuss results from 

the application of the sensor selection metrics for performance 

estimation and gas path fault diagnostics.
NASA/TM—2016-218926 9

Sensor Selection for Performance Estimation

Performance estimation accuracy is assessed based on the 

health parameter mean squared estimation error offered by 

different sensor suites. The linear engine model contains 10 

health parameters as shown in Table 2, which represent 

efficiency and flow capacity scalars associated with each 

major rotating module of the engine. In this study, deviations 

in all health parameters are assumed to be uncorrelated, and 

randomly shifted from their trim conditions with a standard 

deviation of ±2 percent. Since a parameter’s variance is equal 

to its standard deviation squared, the health parameter 

covariance matrix, Ph, is defined as a diagonal matrix with all 

diagonal elements equal to 4.0. The subsections below will 

present health parameter estimation results first assuming 

application of a Kalman filter estimator and then the MAP 

estimator.

Kalman Filter Sensor Selection Results

For each of the 16 candidate sensor suites, the Kalman filter 

SSEE metric shown in Equation (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 selected in accordance with the 

defined health parameter covariance matrix, Ph. These random 

health parameter vectors and random measurement noise, v, 

are substituted into Equation (1) to produce sensed 

measurement test cases used for Monte Carlo evaluation. The 

resulting mean squared estimation error results are shown in

Table 4. The top half of the table shows theoretically predicted 

results while the bottom half shows results obtained via the 

Monte Carlo simulation analysis. Each row corresponds to one 

of the 16 candidate sensor suites. In the cases of Baseline + 1, 

+2, or +3 optional sensors, the sensor suite that provides the 

minimum SSEE is highlighted in bold font. In general, adding 

sensors reduces the SSEE. The results also show that specific 

additional sensors are highly beneficial in improving the 

estimation accuracy of individual health parameters. For 

example, adding sensors such as P14 or T14 improves the 

estimation accuracy of fan efficiency (ηFAN) and fan flow 

capacity (γFAN), while adding P25 improves estimation 

accuracy of LPC flow capacity (γLPC). It is also encouraging to 

note that the theoretical and simulation results exhibit good 

agreement. The overall estimation accuracy is very similar and 

the combination of sensors identified as optimal is identical 

(theoretical vs. Monte Carlo simulation) for each candidate 

number of sensors. Minor differences between theoretical and 

Monte Carlo results are likely due to the number of Monte 

Carlo trials conducted. If the number of trials were increased, 

the differences between analytical and simulation results 

should diminish. Based on this analysis, the sensor selection 

decisions for Kalman filter estimation accuracy would be:

• Baseline + 1 sensor, choose: T25

• Baseline + 2 sensors, choose: T25 and P25

• Baseline + 3 sensors, choose: T25, P25, and P14

MAP Estimator Sensor Selection Results

Next, sensor selection is conducted assuming that a MAP 

estimator is applied for health parameter estimation. Here, the 

metric previously introduced in Equation (14) is used to 

theoretically predict the health parameter SSEE accuracy 

offered by each of the candidate sensor suites. Additionally, a 

Monte Carlo simulation study is performed to verify the 

theoretical results. Here, 400,0001 health parameter vectors 

are randomly generated in accordance with Ph. These health 

parameters along with random sensor measurement noise are 

substituted into Equation (7) to produce sensed measurement 

test cases, which are then processed to produce health 

parameter estimates using Equation (8). The resulting 

theoretical and Monte Carlo simulation health parameter mean 

squared estimation errors are shown in Table 5. Here, the 

theoretical and Monte Carlo simulation results exhibit very 

good agreement, which is expected given the large number of 

Monte Carlo trials runs. The sensor suites identified as optimal 

for the MAP estimator agree with those previously identified 

in Table 4 for the Kalman filter. Furthermore, the mean 

squared estimation errors of individual health parameters and 

the overall health parameter SSEE for most sensor suites 

exhibit fairly good agreement between the MAP estimator and 

the Kalman filter. This is not unexpected since both estimators 

are designed to minimize the mean sum of squared estimation 

errors, and in this study both incorporate the same a priori

knowledge regarding health parameter covariance, Ph, and 

make the same assumptions regarding sensor measurement 

covariance, R.

 1 The disparity in the number of Monte Carlo trials conducted for the 

MAP estimator versus the Kalman filter is due to the nature of the 

two estimators. The MAP estimator only requires a single steadystate sample for each random health parameter vector considered. 

Conversely, the Kalman filter, which is a dynamic recursive 

estimator, requires a sufficient quantity of measurement data at each 

health condition to ensure convergence to a steady-state solution. 

This limited the practical number of Monte Carol trials for the 

Kalman filter.
NASA/TM—2016-218926 10

TABLE 4.—KALMAN FILTER PERFORMANCE ESTIMATION ACCURACY

No. 

sensors

Sensors added to 

baseline

Theoretical Health Parameter Mean Squared Estimation Errors 

(% squared)

P14 T14 P25 T25 η

FAN γFAN ηLPC γLPC ηHPC γHPC ηHPT γHPT ηLPT γLPT SSEE

6 2.53 1.66 3.34 3.07 0.26 1.51 0.95 0.04 1.00 2.85 17.21

7 x 0.23 0.15 3.43 3.29 0.25 1.47 0.96 0.04 1.02 2.83 13.66

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

7 x 2.50 1.65 1.94 0.69 0.22 1.29 0.80 0.04 0.83 2.62 12.58

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

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

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

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

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

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

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

No. 

sensors

Sensors added to 

baseline

Monte Carlo Health Parameter Mean Squared Estimation Errors 

(% squared)

P14 T14 P25 T25 η

FAN γFAN ηLPC γLPC ηHPC γHPC ηHPT γHPT ηLPT γLPT SSEE

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

7 x 0.17 0.20 3.48 3.18 0.27 1.52 1.08 0.04 1.14 3.37 14.44

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

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

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

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

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

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

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

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

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
NASA/TM—2016-218926 11

TABLE 5.—MAP ESTIMATOR PERFORMANCE ESTIMATION ACCURACY

No.

sensors

Sensors added to

baseline

Theoretical Health Parameter Mean Squared Estimation Errors

(%squared)

P14 T14 P25 T25 η

FAN γFAN ηLPC γLPC ηHPC γHPC ηHPT γHPT ηLPT γLPT SSEE

6 2.45 1.64 2.81 3.09 0.27 1.35 0.88 0.04 1.06 2.78 16.35

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

7 x 2.45 1.64 1.92 0.64 0.22 1.31 0.71 0.04 0.86 2.57 12.36

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

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

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

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

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

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

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

No.

sensors

Sensors added to

baseline

Monte Carlo Health Parameter Mean Squared Estimation Errors

(%squared)

P14 T14 P25 T25 η

FAN γFAN ηLPC γLPC ηHPC γHPC ηHPT γHPT ηLPT γLPT SSEE

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

7 x 1.13 1.23 2.81 3.08 0.27 1.34 0.88 0.03 1.02 2.78 14.55

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

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

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

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

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

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

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

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

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

Gas Path Fault Diagnostics Sensor Selection Results

For the gas path fault diagnostics sensor selection problem 

setup, it is assumed that the engine may encounter eight 

different gas path fault types consisting of turbomachinery 

faults (implemented via health parameter perturbations) and 

actuator biases. The eight faults along with the parameter 

perturbations applied within C-MAPSS40k to generate the 

fault influence coefficient matrix are shown in Table 6. For 

this study, all faults are assumed to occur in isolation and to be 

of equivalent probability of occurrence. 

TABLE 6.—GAS PATH FAULTS

Fault ID Fault type Health parameters and actuator biases

1 Fan fault ηFAN = –1%, γFAN = –2%

2 LPC fault ηLPC = –1%, γLPC = –2%

3 HPC fault ηHPC = –1%, γHPC = –2%

4 HPT fault ηHPT = –2%, γHPT = +1%

5 LPT fault ηLPT = –2%, γLPT = +1%

6 Wf bias Wf bias = –2%

7 VSV bias VSV bias = –1° stroke

8 VBV bias VBV bias = +20%
NASA/TM—2016-218926 12

For each of the 16 candidate sensor suites, the correct 

classification rate metric given in Equation (29) is applied to 

calculate the theoretical correct classification rate offered by 

each candidate sensor suite. In making this assessment, the 

applied WSSM signal fault detection threshold is set to give a 

theoretical false positive rate of 0.01 (1 percent) as defined via 

Equation (22). This threshold will change based on the 

number of sensors included in each candidate sensor suite.

Additionally, a Monte Carlo simulation analysis was 

conducted to verify the theoretically predicted CCR results. 

This is done using Equation (15) to generate 80,000 no fault 

cases and 10,000 fault cases for each individual fault type, all 

corrupted by random measurement noise, v. This data set is 

then analyzed by applying the single fault diagnostic logic 

shown in Equations (16)-(20) to detect and classify the 

occurrence of any faults. Initial diagnostic analysis revealed 

that even the baseline 6 sensor measurement suite performed 

extremely well in diagnosing the gas path faults considered.

Therefore, in order to present a more interesting sensor 

selection problem, the sensor measurement noise was 

increased by a factor of four and the diagnostic assessment 

was repeated. The ensuing correct classification rate results 

are shown in Table 7. The top half of the table shows 

theoretically predicted results while the bottom half shows 

TABLE 7.—GAS PATH FAULT DIAGNOSTIC ACCURACY

No.

sensors

Sensors added to

baseline

Theoretical Correct Classification Rate (CCR) %

P14 T14 P25 T25 Fan LPC HPC HPT LPT Wf VSV VBV No Fault Fault CCR

6 73.3 74.4 99.2 100.0 98.8 92.1 60.6 78.4 99.0 84.6

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

7 x 76.3 87.1 99.9 100.0 98.8 90.6 70.4 88.8 99.0 89.0

7 x 77.0 94.2 99.9 100.0 98.8 90.6 73.3 98.8 99.0 91.6

8 x x 86.3 75.4 99.2 100.0 100.0 89.0 60.7 79.7 99.0 86.3

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

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

8 x x 77.5 95.9 99.9 100.0 98.8 89.1 76.3 98.8 99.0 92.0

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

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

No.

sensors

Sensors added to

baseline

Monte Carlo Correct Classification Rate (CCR)%

P14 T14 P25 T25 Fan LPC HPC HPT LPT Wf VSV VBV No Fault Fault CCR

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

7 x 83.9 80.0 99.3 100.0 99.8 90.4 77.1 82.9 99.0 89.2

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

8 x x 87.8 80.2 99.3 100.0 100.0 89.1 76.3 83.1 99.0 89.5

8 x x 84.0 88.1 99.9 100.0 99.8 89.1 78.7 89.3 99.0 91.1

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

8 x x 84.3 94.8 100.0 100.0 99.8 89.0 79.6 99.0 99.0 93.3

8 x x 77.5 96.0 100.0 100.0 98.8 88.9 80.6 98.9 99.0 92.6

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

9 x x x 84.2 96.0 100.0 100.0 99.8 87.5 80.0 98.9 99.0 93.3

9 x x x 84.3 96.1 100.0 100.0 99.8 87.4 80.0 99.0 99.0 93.3

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
NASA/TM—2016-218926 13

results obtained via the Monte Carlo simulation analysis. 

Here, the theoretical results slightly underpredict the correct 

classification rates found via Monte Carlo analysis. This is due 

to the simplification made in deriving the theoretical correct 

classification rate, which essentially establishes a theoretical 

lower bound on this rate. Based on the theoretical analysis, the 

sensor selection decisions for gas path fault diagnostics would 

be:

• Baseline + 1 sensor, choose: T25

• Baseline + 2 sensors, choose: T25 and T14

• Baseline + 3 sensors, choose: T25, T14, and P25

The Monte Carlo simulation analysis shows the same 

optimal sensor suites except for the Baseline + 3 sensor case, 

where P14 would be substituted in place of P25. 

Discussion

The sensor selection metrics introduced in this paper were 

shown to perform well in identifying optimal sensor suites 

from a performance estimation and diagnostic accuracy 

perspective. Although not specifically shown in this paper, the 

resulting sensor suites identified to be optimal are expected to

change if different assumptions are made regarding the design 

inputs such as sensor measurement noise, health parameter 

covariance, fault types and magnitudes, and the engine model 

that the metrics are applied to.

A notable finding in this work was the relative agreement 

between the Kalman filter and MAP estimator in terms of the 

predicted SSEE results and the sensor suites identified to be 

optimal. As previously noted, this is not unexpected given that

both estimators are set up to minimize mean squared 

estimation error and, in the given example application, both 

incorporate the same a priori knowledge regarding health 

parameter covariance and sensor measurement noise 

covariance. However, an advantage of the MAP estimator 

metric is that it offers a closed-form solution while the 

Kalman filter SSEE metric requires solution of the V*

transformation matrix via an optimal iterative search. As such, 

the Kalman filter metric can be prone to convergence to local 

minima. To guard against such occurrences, a recommended 

approach is to cross-check Kalman filter results using the 

MAP metric to ensure that similar sensor suites and SSEE

values are predicted. 

The performance estimation metrics exhibited very good 

agreement between their theoretically predicted estimation 

accuracy and that obtained via Monte Carlo simulation 

analysis. However, the theoretical gas path fault diagnostic 

metric was found to underpredict the CCR found via Monte 

Carlo analysis. This is due to the two-fault class 

misclassification assumption made in deriving the metric. 

While this simplification does make the derivation tractable, it 

can lead to inaccurate results, especially when faults are prone 

to misclassification as more than one fault type. Another 

simplification made in this derivation is to assume that all 

sensor residual measurements are independent. This 

assumption does not usually hold for gas turbine engine 

applications, as some amount of covariance usually exists

between sensor residual measurements. For example, they are 

corrected using the same parameters and generated using the 

same reference model. An approach to address this is to define 

sensor measurement probability density functions in multiparameter space and then perform multidimensional 

integration to assess detection and classification performance. 

However, this would add much more complexity. The given 

metric based on the properties of the chi square distribution 

and the non-central chi square distribution is more simplistic, 

but should be verified by additional analysis such as the 

Monte Carlo simulation analysis conducted in this paper. 

A couple of recommendations for follow-on work are 

suggested. First, the presented analytical metrics are based on 

linear theory while aircraft engines exhibit nonlinear behavior. 

As such, a recommendation to extend the analysis to fullenvelope engine operation would be to repeat the analysis at 

different engine operating points to assess how this affects 

sensor selection results. Furthermore, equal importance is 

placed on each parameter to be estimated and each fault type 

to be diagnosed. A natural extension to the metrics is to place 

a user-specified weighting on the different parameters or faults 

based on their criticality or frequency of occurrence. Finally, 

the estimation and diagnostic accuracy is only one piece of the 

overall sensor selection decision process. Other factors of 

merit include criteria such as sensor weight, reliability, and 

overall life cycle cost. Those factors should also be considered 

as part of the sensor selection process. 

Conclusions

The sensor selection metrics introduced in this paper 

provide analytical tools to assist engine health management 

system designers in making sensor selection decisions. The 

metrics are easy to use, and are specifically tailored towards 

estimation and diagnostic approaches commonly applied to 

aircraft engines. They can be readily applied for assessing the 

benefits of adding or removing currently available engine 

sensors, or assessing the benefits of newly developed sensors 

as they become available. Through Monte Carlo simulation 

analysis, the metrics were verified to perform well in 

identifying optimal sensor suites when evaluated using linear 

system information. For both Kalman filter and maximum a 

posteriori health parameter estimation, the corresponding 

sensor selection metrics were found to perform very well in 

satisfying their intended objective—identifying the sensor 
NASA/TM—2016-218926 14

suite that minimizes the mean sum of squared estimation 

errors. The gas path fault diagnostic sensor selection metric 

based on theoretical correct classification rate also performed 

well in its objective of identifying sensor suites that provide 

the best diagnostic performance. Due to a simplification made 

in the theoretical derivation, the metric was found to 

underpredict the true correct classification rate. However, it 

does provide a theoretical lower bound on correct 

classification performance offered by a given sensor suite. 

Additionally, it is effective for identifying fault pairs at risk 

for misclassification and making sensor selection decisions to 

address such risks. Recommended follow on work is to couple 

these accuracy metrics with additional figures of merit 

pertinent for sensor selection decision. This includes 

considering the individual criticality of the performance 

parameters to be estimated or the fault types to be diagnosed, 

and to also couple these metrics with additional metrics 

reflecting the life cycle cost of adding specific sensors. 

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