Donald L. Simon and Sanjay Garg

Glenn Research Center, Cleveland, Ohio

A Systematic Approach to Sensor Selection for 

Aircraft Engine Health Estimation

NASA/TM—2009-215839

December 2009

ISABE–2009–1125
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Donald L. Simon and Sanjay Garg

Glenn Research Center, Cleveland, Ohio

A Systematic Approach to Sensor Selection for 

Aircraft Engine Health Estimation

NASA/TM—2009-215839

December 2009

ISABE–2009–1125

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

19th ISABE Conference (ISABE 2009)

sponsored by the International Society for Air Breathing Engines

Montreal, Canada, September 7–11, 2009
Acknowledgments

This research was conducted under the NASA Aviation Safety Program, Integrated Vehicle Health Management Project.

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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2009-215839 1

A Systematic Approach to Sensor Selection for 

Aircraft Engine Health Estimation

Donald L. Simon and Sanjay Garg 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

A systematic approach for selecting an optimal suite of 

sensors for on-board aircraft gas turbine engine health 

estimation is presented. The methodology optimally chooses 

the engine sensor suite and the model tuning parameter vector 

to minimize the Kalman filter mean squared estimation error 

in the engine’s health parameters or other unmeasured engine 

outputs. This technique specifically addresses the 

underdetermined estimation problem where there are more 

unknown system health parameters representing degradation 

than available sensor measurements. This paper presents the 

theoretical estimation error equations, and describes the 

optimization approach that is applied to select the sensors and 

model tuning parameters to minimize these errors. Two 

different model tuning parameter vector selection approaches 

are evaluated: the conventional approach of selecting a subset 

of health parameters to serve as the tuning parameters, and an 

alternative approach that selects tuning parameters as a linear 

combination of all health parameters. Results from the 

application of the technique to an aircraft engine simulation 

are presented, and compared to those from an alternative 

sensor selection strategy. 

Nomenclature 

A, Axh, Axq

B, Bxh, Bxq, 

C, Cxh, Cxq, 

D, F, Fxh, Fxq, 

G, L, M, N

system matrices 

C-MAPSS Commercial Modular Aero-Propulsion 

System Simulation 

Gxh, Gx, Gh, Gz estimation bias matrices 

HPC high pressure compressor 

HPT high pressure turbine 

I identity matrix 

K∞ Kalman filter gain 

LPC low pressure compressor 

LPT low pressure turbine 

Ph, Pz health and auxiliary parameter 

covariance matrices 

kqx kz khx ˆˆ , PPP ˆ, ,ˆ ˆ ,, covariance matrices of estimated 

parameters 

Q, Qxh, Qxq process noise covariance matrices 

R measurement noise covariance matrix 

V*

 transformation matrix relating hk to qk

Wz auxiliary parameter weighting matrix 

hk health parameter vector 

m dimension of tuning parameter vector 

p dimension of health parameter vector 

qk Kalman filter tuning parameter vector 

r number of sensors to choose from 

s number of additional sensors to add 

uk actuator command vector 

vk measurement noise vector 

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

xk state vector 

xxh,k augmented state vector (xk and hk) 

xxq,k reduced order augmented state vector 

(xk and qk) 

yk vector of measured outputs 

zk vector of unmeasured (auxiliary) 

outputs 

εxq,k residual vector (estimate minus its 

expected value) 

Subscripts 

k discrete time step index 

ss steady-state 

xh augmented state vector (x and h) 

xq reduced order augmented state vector (x

and q) 

Superscripts 

† pseudo-inverse 

^ estimated value 

~ error value 

– mean value 

T transpose 

Operators 

E[·] expected value of argument 

tr{·} trace of matrix 

SSEE(·) sum of squared estimation errors 

Introduction 

An emerging approach in the field of aircraft engine 

controls and health management is the inclusion of real-time 
NASA/TM—2009-215839 2

on-board models for the in-flight estimation of engine 

performance variables (Refs. 1, 2, and 3) that can be directly 

utilized by controls, prognostics and health management 

applications. This includes the estimation of engine health 

parameters, such as efficiencies and flow capacities, that 

reflect the level of degradation in each major engine module, 

and might indicate the existence of faults. 

Aircraft operators conventionally apply a ground-based 

engine condition monitoring process known as gas path 

analysis to estimate and trend the health parameters of 

individual engines (Refs. 4 and 5). This analysis is typically 

conducted based on steady-state snapshot measurement data 

collected each flight, and uses linear estimation techniques 

such as weighted least squares (Ref. 4) or maximum a 

posteriori estimation (Ref. 5). With the emergence of onboard model-based technology, the real-time continuous 

estimation of engine health parameters is becoming possible. 

The conventional approach used for real-time estimation is 

based on Kalman filter concepts. As will be explained, a 

requirement for Kalman filter estimation is that there be at 

least as many sensors as health parameters. However, in an 

aircraft engine this is typically not the case, presenting an 

underdetermined estimation problem. The steady-state gas 

path analysis approaches discussed in References 4 and 5 are 

capable of estimating more health parameters than sensed 

measurements due to the inclusion of a priori knowledge 

regarding health parameter variations. This enables estimation, 

albeit biased estimation, when faced with an underdetermined 

estimation problem. For Kalman filter estimation applications, 

a common approach to address the underdetermined 

estimation shortcoming is to only estimate a subset of the 

health parameters, referred to as model tuning parameters, and 

to assume that other health parameters remain constant. While 

this approach enables on-line Kalman filter-based estimation, 

it also results in biased estimation, as the effects of the 

unestimated health parameters will be reflected in those that 

are estimated. 

Two steps that can be taken to improve the accuracy of onboard models are to optimize the selection of the model tuning 

parameter vector, and to add additional gas path measurement 

sensors. In order to optimize overall performance estimation 

accuracy, it is desirable to develop and apply a systematic 

approach for combined model tuning parameter and sensor 

selection. 

In a departure from the conventional technique of selecting 

a subset of health parameters to serve as the model tuning 

parameter vector, Litt (Ref. 6) and the current authors (Refs. 7 

and 8) have presented methodologies that create a tuning 

parameter vector as a linear combination of all health 

parameters that is of appropriate dimension to enable Kalman 

filter estimation. These studies have shown that an optimally 

selected model tuning parameter vector can significantly 

reduce the estimation error in the engine performance 

parameters of interest (Ref. 8). 

Several sensor selection approaches have been investigated 

within the engine health monitoring community. In Reference 9, 

an information theoretic approach to aircraft engine sensor 

selection is performed. This method selects sensor suites to 

optimize a metric defined from the Fisher information matrix. 

Reference 10 presents a sensor selection approach that seeks to 

optimize the diagnostic information contained in the Jacobian 

matrix. The Jacobian matrix, sometimes referred to as the 

influence coefficient matrix, relates the effects of health 

parameter deviations to corresponding deviations in sensed 

engine outputs. In Reference 11 a sensor selection performance 

metric is defined as a function of the steady-state covariance of 

Kalman filter estimates, which also includes sensor cost. In 

References 12 and 13 a performance metric is defined to 

maximize the detectability and discriminability of user-specified 

fault types. Sensor cost and fault criticality are also considered 

in this approach. 

This work extends previous research by applying the dual 

approach of tuning parameter and sensor selection for 

minimizing on-board Kalman filter estimation errors. The 

remaining sections of this paper are organized as follows. 

First, the mathematical formulation of the linear Kalman 

estimator is presented, and theoretical estimation error 

equations are introduced assuming open-loop, steady-state 

operating conditions. Next, the estimation error information is 

utilized within the methodology for tuner parameter and 

sensor selection. The methodology is then applied to an 

aircraft turbofan engine simulation. Theoretical and 

experimental results, including comparisons with another 

sensor selection method, are presented to illustrate the 

advantages of utilizing this methodology for underdetermined 

health parameter estimation problems. Finally, conclusions are 

presented.

Problem Formulation 

The discrete linear time-invariant engine state space 

equations about an operating point are given as 

k k k kk 1

k k k kk

x Ax Bu Lh w

y Cx Du Mh v

+ = +++

=+ + + (1) 

where k is the time 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 represents the engine health parameters. 

As the health parameters deviate from their nominal values, 

they induce shifts in other variables. The vectors w and v are 

uncorrelated zero-mean white noise input sequences. The 

matrix Q will be used to denote the covariance of w, and R to 

denote the covariance of v. The matrices A, B, C, D, L, and M

are of appropriate dimension. The health parameters, 

represented by the vector h, are unknown inputs to the system. 

They may be treated as a set of biases, and are thus modeled 

without dynamics. With this interpretation Equation (1) can be 

written as: 
NASA/TM—2009-215839 3

{ {

[ ]{

, ,

,

1

1 ,

, ,

,

0 0

xh xh k xh xh k

xh

xh k

k k k

k

k k h k

A x B w

xh xh k xh k xh k

k

k kk

k C

x

xh xh k k k

x x AL B w

u

h h I w

Ax Bu w

x

y C M Du v h

C x Du v

+

+

⎡ ⎤ ⎡⎤ ⎡ ⎤ ⎡⎤ ⎡ ⎤ ⎢ ⎥ ⎢⎥ = ++ ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎣ ⎦ ⎣⎦ ⎣ ⎦ ⎣⎦ ⎣ ⎦

= ++

⎡ ⎤

= ++ ⎢ ⎥ ⎣ ⎦

= ++

123 123

123

(2) 

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

augmented state vector, [x

T hT

]

T

, with a covariance of Qxh. The 

vector 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, it may be 

directly estimated, provided that the realization in Equation (2) 

is observable. Using this formulation, the number of health 

parameters that can be estimated is limited to the number of 

sensors, the dimension of y (Ref. 14). An aircraft gas turbine 

engine typically has fewer sensors than health parameters, thus 

presenting an underdetermined estimation problem. 

This paper presents a systematic methodology for the 

selection of additional engine sensors and model tuning 

parameters. These selections are performed to minimize the 

estimation error in engine health parameters. The following 

subsections will cover the steps in the problem setup. This 

includes construction of the reduced-order state space model 

applied for underdetermined estimation, formulation of the 

Kalman filter estimator, derivation of the mean sum of 

squared estimation errors, and optimal selection of the reduced 

order tuner vector to minimize the estimation error. 

Reduced-Order State Space Model 

To enable Kalman filter formulation when faced with an 

underdetermined estimation problem, a reduced-order system 

model is applied. First a reduced model tuning parameter 

vector, q, is constructed from the health parameter vector, h, 

as 

* q Vh = (3) 

where q ∈ Rm, h ∈ Rp

, m < p, and V* is an m × p

transformation matrix of rank m, relating h to q. In this 

formulation q and the sensor measurement vector, y, are 

defined to be of equivalent dimension (i.e., y ∈ Rm). The 

transformation matrix V*

 can be defined to construct q as a 

subset of h (the conventional approach to tuner selection), or 

V* can be defined to construct q as a linear combination of all 

health parameters. An approximation of the health parameter 

vector, h

ˆ , can be obtained as 

ˆ *† h Vq = (4) 

where V*† is the pseudo-inverse of V*

. Substituting Equation 

(4) into Equation (2) yields the following reduced order state 

space equations that will be used to formulate the Kalman 

filter 

{ {

{

, ,

,

*†

1

1 ,

, ,

*†

,

0 0

xq k xq xq xq k

xq

xq k

k k k

k

k k q k

A x B w

xq xq k xq k xq k

k

k kk

k C x

xq xq k k k

x x A LV B w

u

q q I w

Ax Bu w

x

y C MV Du v

q

C x Du v

+

+

⎡ ⎤ ⎡⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢⎥ = ++ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

= ++

⎡ ⎤

= ++ ⎡ ⎤ ⎣ ⎦⎢ ⎥ ⎣ ⎦

= ++

14243 123

14243

(5) 

The state process noise, wxq, and its associated covariance, Qxq, 

for the reduced order system are calculated as 

, , * *

,

* *

0 0

0 0

0 0

0 0

k

xq k xh k

h k

T

xq xh

I I w

w w

V V w

I I

Q Q V V

⎡⎤ ⎡⎤⎡ ⎤ = = ⎢⎥ ⎢⎥⎢ ⎥ ⎣⎦ ⎣⎦⎣ ⎦

⎡ ⎤⎡ ⎤ = ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

(6) 

Kalman Filter Formulation 

In this study, steady-state Kalman filtering is applied. 

This means that while the Kalman filter is a dynamic system, 

the state estimation error covariance matrix and the Kalman 

gain matrix are invariant—instead of these matrices being 

updated each time step, they are pre-converged and held 

constant. Given the reduced order linear state space 

equations shown in Equation (5), and assuming steady-state, 

open-loop operation (u = 0), the Kalman filter estimator 

becomes (Ref. 15) 

, , 1 ( ) , 1 ˆ ˆ ˆ xq k xq xq k k xq xq xq k x Ax K y CAx = +− − ∞ − (7)

where K∞, is the steady-state Kalman filter gain. The reduced 

order augmented state vector estimate, xxq ˆ , produced by 

Equation (7) can be used to obtain an estimate of the 

augmented state vector as 

, , *†

ˆ 0 ˆ ˆ ˆ 0

k

xh k xq k

k

x I

x x

h V

⎡ ⎤ ⎡ ⎤ = = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ (8)
NASA/TM—2009-215839 4

Using Equation (8), an estimate of the entire health parameter 

vector can be obtained when faced with the underdetermined 

estimation problem. 

Kalman Filter Estimation Error 

The estimation errors in x ,kxh ˆ are defined as the difference 

between estimated and actual values 

, ,, ˆ xh k xh k xh k x% = − x x (9)

When facing an underdetermined estimation problem, it will 

be impossible for the Kalman filter estimator to completely 

restore all information when transforming qˆ into h

ˆ . 

Therefore, the Kalman filter will be a biased estimator (i.e., 

the expected values of xh k, x% will be non-zero). The estimation 

errors can be considered to consist of two components: an 

estimation error bias, and an estimation variance. Complete 

derivations of the Kalman filter estimation error bias and 

variance under steady-state, open-loop operating conditions 

are provided in Reference 8. Abbreviated derivations are 

provided in the following subsections. 

Estimation Error Bias 

As shown in Reference 8 the steady-state augmented state 

estimation error bias for an arbitrary health parameter vector, 

h, is given as 

( )

( )

( )

, , ,

1

*†

1

1

ˆ

0

0

xh

ss

xh ss xh k xh k

ss

xq xq xq

G

ss x

xh

h ss

x

x Ex x

h

I

I A KC A

V

K CI A L M h

IA L

I

x G

h Gh

h G

−

∞

−

∞

−

⎡ ⎤

== − ⎢ ⎥ ⎡ ⎤ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ − +

⎣ ⎦

=× − + ⎡ ⎤ ⎣ ⎦

⎡ ⎤ − − ⎢ ⎥

⎣ ⎦ ⎢ ⎥ ⎣ ⎦

⎡ ⎤ ⎡ ⎤ == = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦

% % %

K

K

14444444244444443

%

%

(10)

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

argument. The estimation error bias equation, Equation (10), is 

a function of an arbitrary h. As such it is representative of the 

parameter estimation error biases in a single engine, at a given 

point in its lifetime of use where its deterioration is 

represented by h. The average sum of squared estimation error 

biases across a fleet of engines can be calculated as 

{ }

2

,fleet , ,

T

xh xh ss xh ss

T

xh h xh

x Ex x

tr G P G

≡ ⎡ ⎤ ⎣ ⎦

=

% %%

(11)

where the matrix Ph, defined as T E hh ⎡ ⎤ ⎣ ⎦ , reflects a priori or 

historical knowledge of the covariance in the health 

parameters across all engines. If available, it can be used to 

predict the sum of squared estimation errors biases as shown 

in (11).

Estimation Variance 

Reference 8 also presents the derivation of the augmented 

state estimate covariance matrix, ˆ xˆh k, P . This matrix can be 

calculated as a function of the reduced-order state vector 

estimation covariance matrix, Pxˆ ˆq k, , defined as 

( )( ) , ,

ˆ ˆ, , ,, , ˆ ˆˆ ˆ

T xq k xq k

T

P E x Ex x Ex xq k xq k xq k xq k xq k

⎡ ⎤ ⎢ ⎥ =− − ⎡ ⎤ ⎡⎤ ⎢ ⎥ ⎣ ⎦ ⎣⎦ ⎢ ⎥ ⎢⎣ ⎥⎦

14424431442443 ε ε

(12)

where the vector εxq,k is defined as the residual between , ˆxq k x

at time k and its expected value. Under steady-state operating 

conditions the following Ricatti equation can be solved for 

Pxˆ ˆq k, : 

ˆ ˆ,

ˆ ˆ,

xq k xq xq xq

T

xq k xq xq xq

T

P A KC A

P A KC A

K RK

∞

∞

∞ ∞

= − ⎡ ⎤ ⎣ ⎦

× − ⎡ ⎤ ⎣ ⎦

+

K

(13)

Once Pxˆ ˆq k, is obtained, it can be used to calculate ˆ xˆh k, P , the 

covariance of , ˆxh k x , defined as 

ˆ ˆ, ( , ,, , ) ( )

ˆ ˆ ˆ , *† ˆ, *†

ˆ ˆˆ ˆ

0 0

0 0

T

xh k xh k xh k xh k xh k

T

xq k xh k

P E x Ex x Ex

I I

P P

V V

⎡ ⎤ =− − ⎡ ⎤ ⎡⎤ ⎢ ⎣ ⎦ ⎣⎦ ⎥ ⎣ ⎦

⎡ ⎤⎡ ⎤ = ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

(14)

The augmented state vector estimation covariance given in 

Equation (14) can be partitioned into covariance information 

for the original state vector, Pxˆ,k (upper left corner of ˆ xˆh k, P ), 

and the health parameter vector, ˆ

h k, P (lower right corner of 

ˆ xˆh k, P ) 

ˆ,

ˆ ˆ , ˆ,

x k

xh k

h k

P

P P

⎡ ⎤

= ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

L

L (15)

The variance in , ˆxh k x can be obtained from the diagonal of the 

covariance matrix produced by (14). 
NASA/TM—2009-215839 5

Sum of Squared Estimation Errors 

Once Eqs. (11) and (14) are obtained, they may be used to 

analytically calculate the mean sum of squared estimation 

errors over all engines by combining the respective estimation 

error bias and estimation variance. The mean augmented state 

vector sum of squared estimation errors, SSEE x( ˆxh,fleet) , 

becomes 

( ) { }

{ }

2

,fleet ,fleet ˆ ˆ ,

ˆ ˆ ,

ˆxh xh xh k

T

xh h xh xh k

SSEE x x tr P

tr G P G P

= +

= +

%

(16) 

Similarly, the mean health parameter sum of squared 

estimation errors is 

( ) fleet { } ˆ,

ˆ T

hh h h k SSEE h tr G P G P = + (17)

From the previous equations it can be observed that both 

estimation bias and variance are affected by the selection of 

the sensor suite and the model tuning parameter vector, q, 

since both contain C and V*

. The sum of squared estimation 

errors presented in this section gives rise to an optimization 

problem: selecting the sensor suite and the tuner vector to 

minimize the SSEE in the Kalman filter estimates. The 

approach applied for selecting these parameters will be 

described in the next section. 

Optimal Sensor and Tuner Selection 

Prior to initiating the search for an optimal sensor suite, 

specific system design information must be defined or 

obtained. This includes: 

• Specifying the number of sensors to be added to the 

baseline sensor suite, and the candidate list of sensors to 

choose from. 

• Generating system state space equations at a fleet average 

(50 percent deteriorated) engine trim point. Thus at the 

trim point, the mean value of h is zero, and the variations 

in h for an individual engine are equally likely to be 

positive or negative. 

• Defining measurement noise covariance matrix, R. 

• Defining augmented state process noise covariance 

matrix, Qxh. 

• Defining average health parameter covariance, Ph, for a 

fleet of engines. 

After the necessary system information has been obtained, 

the search for the sensor suite and/or tuner vector to minimize 

the Kalman filter sum of squared estimation errors can 

commence. This study assumes that a baseline sensor suite is 

given, for example a suite of engine control sensors, and it is 

desired to assess the estimation accuracy improvement that 

can be gained by adding additional sensors. Other factors such 

as sensor cost, weight, or reliability are not considered, but 

could be included by applying a similar strategy to that 

described in References 12 and 13, which combines the effects 

of sensor cost and diagnostic effectiveness into a single merit 

value. 

If the target number of sensors (baseline + additional) is 

greater than or equal to the number of health parameters, and 

the system is fully observable, the tuner vector will consist of 

all health parameters, and V*

 simply becomes the identity 

matrix. Conversely, in the case where there are fewer sensors 

than health parameters a reduced order tuner vector will be 

necessary, and consequently V*

 will not equal the identity 

matrix. The following subsections will first describe sensor 

selection for a fully observable system, followed by a 

description of combined sensor and tuner selection for rank 

deficient cases. In each case the selections will be performed 

to minimize the Kalman filter SSEE. 

Optimal Sensor Selection 

(Fully Observable System With V*

 = I) 

For the fully observable estimation case, the relative merits 

of different candidate sensors suites are assessed by applying 

an exhaustive brute force search. Given a set of r additional 

sensors to choose from, and a target number, s, of additional 

sensors, the total number of sensor suite combinations will be: 

( )

!

! !

r r

s s r s

⎛ ⎞ ⎜ ⎟ =

⎝ ⎠ − (18)

One requirement that must be satisfied in this search is that the 

dimension of the sensor suite (baseline + additional sensors) 

cannot be less than the dimension of h. The matrix V*

 is set 

equal to the identity matrix, and the exhaustive search 

evaluates the SSEE (Equation (17)) for each candidate sensor 

suite. The sensor suite that produces the minimum SSEE is 

identified and returned as the optimal choice. 

Combined Sensor and Tuner Selection 

(q Defined as a Subset of h) 

For rank deficient systems, combined sensor and tuner 

selection will be required. In this study two different tuner 

selection approaches are considered. These are the 

conventional approach of selecting q vectors that are subsets 

of h, and the new approach described in Reference 8 that 

selects q vectors that are combinations of all elements of h. 

For the case where q is defined as a subset of h, an exhaustive 

search of all sensor and tuner vector combinations is applied. 

Given the following information: 

• a set of r additional sensors to choose from, and a target 

number, s, of additional sensors, and 

• a p × 1 health parameter vector, and a sensor suite and 

consequent q vector of size m × 1, 

the total number of sensor and tuner combinations is 
NASA/TM—2009-215839 6

( ) ( ) 14243 14243 sensor combinations tuner combinations

!!

!

!!

! Combinations

mpm

p

srs

r

− × − = (19)

For each sensor suite and tuner vector combination, the SSEE 

(Eq. (17)) is calculated. The sensor suite and tuner vector 

combination found to produce the lowest SSEE is selected as 

the optimal design. 

Combined Sensor and Tuner Selection 

(q Defined as a Combination of All Elements of h) 

The second tuner selection approach considered defines q

as a combination of all elements of h. This approach also 

applies an exhaustive search of all possible sensor 

combinations. However, the corresponding optimal search for 

the V*

, that minimizes the SSEE for a given sensor suite is 

performed by applying a multiparameter optimal iterative 

search as described in Reference 8. This is done using the 

lsqnonlin function of the Matlab (The MathWorks, Inc.) 

Optimization Toolbox. The steps in the combined optimal 

sensor and tuner selection search are: 

A. Given a set of r additional sensors to choose from, and a 

target number, s, of additional sensors, the number of 

sensor suite combinations considered will be as shown in 

Equation (18). 

B. For each sensor suite, the optimal iterative search is 

applied to find the corresponding tuning parameter vector 

that minimizes the SSEE. The steps in this process are: 

1. Generate an initial random guess of V*

. 

2. Construct a reduced order state-space model based on 

V*

, (Eq. (5)). 

3. Formulate Kalman filter (Eq. (7)). 

4. Calculate the SSEE (Eq. (17)). 

5. On each iteration, the change in SSEE relative to the 

previous iteration is assessed to determine if 

convergence within a user specified tolerance has been 

achieved. 

a. If converged, skip step 6 and proceed to step 7. 

b. If not converged, proceed to step 6 to update V*

. 

6. V* is updated via the Matlab lsqnonlin function, and the 

process returns to step 2. 

7. Upon convergence, the routine returns the optimal 

value of V*, and ends. 

Upon completion of the search, the sensor suite and tuner 

vector combination found to produce the lowest SSEE is 

selected as the optimal combination. 

Optimal Sensor and Tuner Selection for Auxiliary 

(Unmeasured) Output Estimation 

The previous section of this paper presented an approach 

for the selection of sensors and model tuning parameters for 

the minimization of health parameter estimation errors. If 

desired, the process can be reformulated for minimizing the 

estimation error of unmeasured, or auxiliary, engine outputs 

such as thrust or stall margin. Linear equations representing 

auxiliary outputs as functions of the state variables, actuator 

commands, and health parameters can be written as 

kkkk z = Fx Gu Nh + + (20)

where z denotes the auxiliary output vector, and F, G, and N

are appropriately sized matrices. Following a theoretical 

derivation similar to the one presented in this paper for h

estimation errors, the Kalman filter auxiliary output sum of 

squared estimation errors (SSEE) can be obtained as 

( ) { }

{ }

2

fleet fleet ,ˆ

ˆ,

ˆ z k

T

z h z zk

SSEE z z tr P

tr G P G P

= +

= +

%

(21)

The above equation can be used in place of Equation (17) 

within the methodology to find a design that is optimal for 

auxiliary parameter estimation. A complete derivation of 

Equation (21) is given in Reference 8. 

Turbofan Engine Example 

A linearized cruise operating point extracted from the 

NASA Commercial Modular Aero-Propulsion System 

Simulation (C-MAPSS) (Ref. 16) high-bypass turbofan engine 

model is used to evaluate the systematic sensor and tuner 

selection methodology. The linear model has two state 

variables and three control inputs (shown in Table 1), and 10 

health parameters (shown in Table 2). The engine is assumed 

to have five baseline sensors, and six additional (optional) 

sensors are to be assessed for the estimation accuracy 

improvement they will provide if added individually, or in 

combination, to the baseline sensor suite. All sensors, along 

with their sensor noise standard deviations, are shown in 

Table 3. In this study the model is assumed to run open-loop, 

so all control inputs remain at 0, i.e., they do not deviate from 

their trim values for the linear model, and no actuator bias is 

present. Deviations in all ten health parameters are assumed to 

be uncorrelated, and randomly shifted from their trim 

conditions with a standard deviation of ±0.02 (±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 

0.0004. 

TABLE 1.—STATE VARIABLES AND ACTUATORS

State variables Actuators 

Nf—fan speed Wf—fuel flow 

Nc—core speed VSV—variable stator vane 

VBV—variable bleed valve 
NASA/TM—2009-215839 7

TABLE 2.—HEALTH PARAMETERS

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 

(percent) 

Baseline 

Sensors 

Nf—fan speed 0.25 

Nc—core speed 0.25 

T24—HPC inlet total temperature 0.75 

Ps30—HPC exit static pressure 0.50 

T48—Exhaust gas temperature 0.75 

Additional 

(Optional) 

Sensors 

P24—HPC inlet total pressure 0.50 

T30—HPC exit total temperature 0.75 

P45—LPT inlet total pressure 0.50 

P50—LPT exit total pressure 0.50 

T50—LPT exit total temperature 0.75 

P15—Bypass duct total pressure 0.50 

Theoretical Kalman Filter Estimation Errors 

The methodology described in the previous section was 

applied to assess the theoretical Kalman filter estimation 

errors that can be obtained through optimal sensor and tuner 

selection for the given test case. This assessment was 

performed for the baseline sensor suite containing five 

sensors, and sensor suites of successively increasing size, 

ranging from 6 to 11 sensors. For sensor suites consisting of 

10 or 11 sensors, the tuner vector was simply set to the entire 

10 element health parameter vector (i.e., q = h and V*

 = I). For 

rank deficient cases (i.e., fewer sensors than health 

parameters), combined sensor and tuner selection was 

performed. For these cases, theoretical mean SSEE’s were 

formulated by applying the following two tuner selection 

approaches: 

• Selecting q as a subset of h to minimize the health 

parameter (h) SSEE. 

– Selecting q as a combination of all elements of h to 

minimize the health parameter (h) SSEE. 

TABLE 4.—HEALTH PARAMETER THEORETICAL MEAN SQUARED ESTIMATION ERRORS 

(q SELECTED AS A SUBSET OF h)

a

No. 

sensors 

Sensors added 

to baseline 

Health parameter theoretical mean squared estimation errors 

(percent squared) 

P24 

T30 

P45 P50 

T50 

P15 

ηFAN γFAN ηLPC γLPC ηHPC γHPC ηHPT γHPT ηLPT γLPT SSEE 

5 4.00 5.01 4.00 6.09 4.00 0.79 3.31 0.20 4.00 4.00 35.40 

6 x 4.00 2.01 4.00 2.80 4.00 0.74 3.26 0.18 1.02 4.00 26.01 

7 x x 4.00 2.02 4.00 2.82 0.41 0.78 1.28 0.17 1.02 4.00 20.51 

8 x x x 4.00 2.02 4.00 2.90 0.41 0.80 0.26 0.18 0.48 0.23 15.28 

9 x x x x 1.06 0.50 4.00 3.20 0.38 0.64 0.25 0.19 0.45 0.24 10.90 

10 x x x x x 0.98 0.48 3.38 0.83 0.38 0.62 0.24 0.19 0.43 0.24 7.77 

11 x x x x x x 0.85 0.48 3.16 0.79 0.37 0.57 0.23 0.15 0.31 0.19 7.11 a

Bold italic font denotes health parameter used as a tuning parameter.

TABLE 5.—HEALTH PARAMETER THEORETICAL MEAN SQUARED ESTIMATION ERRORS 

(q IS A COMBINATION OF ALL ELEMENTS OF h) 

No. 

sensors 

Sensors added 

to baseline 

Health parameter theoretical mean squared estimation errors 

(percent squared) 

P24 

T30 

P45 P50 

T50 

P15 

ηFAN γFAN ηLPC γLPC ηHPC γHPC ηHPT γHPT ηLPT γLPT SSEE 

5 3.03 2.00 3.43 2.55 2.87 0.76 1.79 0.22 2.34 3.31 22.30 

6 x 3.02 1.95 3.42 2.54 2.65 0.75 1.45 0.21 2.27 0.26 18.53 

7 x x 3.01 1.95 3.41 2.54 0.40 0.75 0.26 0.20 2.27 0.26 15.06 

8 x x x 2.77 1.37 3.29 2.26 0.39 0.70 0.26 0.19 0.47 0.25 11.95 

9 x x x x 0.99 0.50 3.37 2.23 0.38 0.62 0.26 0.19 0.45 0.24 9.22 

10 x x x x x 0.98 0.48 3.38 0.83 0.38 0.62 0.24 0.19 0.43 0.24 7.77 

11 x x x x x x 0.85 0.48 3.16 0.79 0.37 0.57 0.23 0.15 0.31 0.19 7.11 

Theoretical estimation errors for the two cases are 

summarized in Table 4 and Table 5. The SSEE values were 

obtained by taking the trace of the health parameter estimation 

error covariance matrix as shown in Equation (17). Theoretical 

mean squared estimation errors for individual health 

parameters were obtained from the diagonal of that same 

matrix. Readers will notice that the bottom two rows of Table 

4 and Table 5 are identical. These rows correspond to the 10 

and 11 element sensor suite cases. In these cases the tuner 

vector is simply set equal to the health parameter vector 
NASA/TM—2009-215839 8

(q = h), providing equivalent tuners and estimation accuracy 

for both tuner selection approaches. 

Several observations can be made based on the data 

contained in these tables. First, as expected, the estimation 

accuracy improves as additional sensors are added. Once the 

tenth sensor is added and the system is no longer rank 

deficient, the estimates are unbiased and the estimation errors 

are entirely due to the variance in the Kalman filter estimate. It 

can also be observed that the sensor suites chosen in Table 4 

and Table 5 are not consistent—different sensors are selected 

depending upon the tuner selection approach applied. 

Another observation drawn from these data is that for 

underdetermined estimation scenarios the approach that 

selects tuner parameters as a linear combination of all health 

parameters (Table 5) provides improved SSEE results 

compared to the conventional approach of selecting tuners that 

are a subset of the health parameter vector (Table 4). The 

improvement is more pronounced when fewer sensors are 

available. As more sensors are added, the benefit of the new 

tuner selection approach becomes progressively less. 

Readers are reminded that the optimization strategy seeks to 

determine the sensor suite and tuner vector combination that 

minimizes the mean sum of squared estimation errors in the 

entire health parameter vector. The estimation accuracy of 

individual health parameters can vary widely. This is 

illustrated in Figure 1 for three of the cases shown in the Table 

4 and Table 5 including: 

a) Five sensors with q defined as a subset of h; first row 

of Table 4 (black bars) 

b) Seven sensors with q defined as a subset of h; third 

row of Table 4 (grey bars) 

c) Seven sensors with q defined as a combination of all 

elements of h; third row of Table 5 (white bars) 

This figure shows the mean squared estimation error for each 

of the 10 health parameters as opposed to the SSEE for the 

entire health parameter vector. 

For the two cases in which q is selected as a subset of h

(cases a and b), any instance where the mean squared 

estimation error is exactly equal to 4.0 (shown in squared 

percentage units) is indicative of a health parameter excluded 

from the tuner vector. This is expected given the defined Ph

for this example. In general, case c has the best estimation 

accuracy, but this is not true in all cases. For example, case b

shows improved estimation accuracy for the γHPT and ηLPT

health parameters. However, the overall SSEE for case b

(20.51) is much worse than that for case c (15.06). 

Experimental Kalman Filter Estimation Errors 

Simulation studies were conducted to experimentally 

validate the theoretically predicted Kalman filter health 

parameter estimation errors. This experimental validation was 

conducted for the same three cases described above (cases 

Figure 1.—Individual health parameter theoretical mean 

squared estimation error 

a, b, and c). A separate Kalman filter was designed for each 

case. The experimental results were obtained through a Monte 

Carlo simulation analysis where the health parameters were 

randomly distributed in accordance with the defined 

covariance matrix, Ph. The test cases were concatenated to 

produce a single time history input that was provided to the CMAPSS linear discrete state space model given in Equation 

(1), with an update rate of 15 ms. Each individual health 

parameter test case lasted 30 sec. A total of 375 30 sec test 

cases were evaluated, resulting in an 11,250 s input time 

history. 

The theoretical and experimental estimation error results, 

shown in Table 6, exhibit good agreement. A visual 

illustration of the effect that sensor and tuner selection has on 

Kalman filter health parameter estimation accuracy can be 

seen in Figures 2 to 4, which show actual and estimated results 

for three of the 10 health parameters (ηHPT, γHPT and ηLPT). The 

three subplots shown in each figure correspond to the three 

different Kalman filter designs (cases a, b, and c). Each plot 

shows a 300 sec segment of the evaluated test cases. The step 

changes that can be observed in each plot every 30 sec 

correspond to a transition to a different health parameter 

vector. True model health parameter inputs are shown in 

black, and Kalman filter estimates are shown in red. These 

plots corroborate the information shown in Table 6. Figure 2 

shows that for ηHPT (HPT efficiency) estimation, case c has the 

best estimation accuracy, followed by case b, and finally case 

a. Figure 3 shows that for γHPT (HPT flow capacity) all three 

Kalman filters are able to provide good estimates. Figure 4 

shows that for ηLPT (LPT efficiency), the estimates in case b

are superior to those in case c. Case a excludes ηLPT from q, 

and thus its corresponding ηLPT estimate in the top subplot of 

Figure 4 is invariant. 
NASA/TM—2009-215839 9

TABLE 6.—COMPARISON OF THEORETICAL AND EXPERIMENTAL MEAN SQUARED ESTIMATION ERRORS

Case 

Sensors added 

to baseline Mean 

squared 

error 

Health parameter mean squared estimation errors 

(percent squared) 

P24 

T30 

P45 P50 

T50 

P15 

ηFAN γFAN ηLPC γLPC ηHPC γHPC ηHPT γHPT ηLPT γLPT SSEE 

a Theoretical 4.00 5.01 4.00 6.09 4.00 0.79 3.31 0.20 4.00 4.00 35.40 

Experimental 3.56 5.38 3.80 6.40 4.32 0.75 3.42 0.20 4.13 3.88 35.84 

b x x Theoretical 4.00 2.02 4.00 2.82 0.41 0.78 1.28 0.17 1.02 4.00 20.51 

Experimental 3.56 1.83 3.80 2.77 0.41 0.76 1.25 0.18 1.00 3.88 19.43 

c x x Theoretical 3.01 1.95 3.41 2.54 0.40 0.75 0.26 0.20 2.27 0.26 15.06 

Experimental 2.57 1.99 3.28 2.44 0.40 0.72 0.26 0.20 2.19 0.26 14.33 

Figure 2.—ηHPT estimation (sensor and tuner selection 

comparison). 

Figure 3.—γHPT estimation (sensor and tuner selection 

comparison). 

Figure 4.—ηLPT estimation (sensor and tuner selection 

comparison). 
NASA/TM—2009-215839 10

These experimental results validate the theoretical 

developments in this paper for the sensor and tuner selection 

methodology. Furthermore the results show that although 

overall (global) health parameter vector estimation accuracy is 

optimized, the relative estimation accuracy of individual 

elements of the health parameter vector may vary. 

Discussion 

The results presented in this paper suggest that the sensor 

and tuner selection process cannot be performed devoid of 

corresponding knowledge regarding the intended end use or 

application. This was illustrated by the different sensor suites 

selected for health parameter estimation (see Tables 4 and 5) 

when different tuner vector formulations were considered. A 

further illustration of this fact can be shown by optimizing 

sensor suite and tuner vector selection for auxiliary 

(unmeasured) engine output estimation (Eq. (21)), and 

comparing the results to those obtained when optimizing for 

health parameter estimation (Eq. (17)). To conduct this 

comparison the following two sensor and tuner selection 

problems were established (similar to the one previously 

described for case c health parameter estimation) for auxiliary 

output estimation: 

Case d: 

• Assume that the five baseline sensors previously shown in 

Table 3 are available. 

• Select the same two additional sensors optimal for h

estimation in case c (T30 and P45). 

• Optimally select the tuner vector (where q is defined as a 

combination of all elements of h) that minimizes the 

auxiliary parameter SSEE (Eq. (21)) in the following four 

outputs: 

– T40: combustor temperature 

– T50: LPT exit temperature 

– Fn: net thrust 

– SmLPC: LPC stall margin 

Case e: 

• Assume that the five baseline sensors previously shown in 

Table 3 are available. 

• Consider all possible combinations of two additional 

sensors from the six optional sensors shown in Table 3. 

• For each sensor combination, optimally select the tuner 

vector (where q is defined as a combination of all h) that 

minimizes the SSEE in the auxiliary parameters of 

interest (same auxiliary parameters shown for case d). 

• Choose the sensor suite and tuner vector combination that 

minimizes the SSEE as the optimal design. 

The sensor selection and auxiliary output estimation results 

for case c, d, and e are shown in Table 7. In comparing case c

and d, it can be seen that given the same sensor suite (baseline 

+ T30 and P45), choosing a tuner vector optimal for z

estimation instead of h estimation reduces the z SSEE by 

approximately 30 percent. If combined sensor and tuner 

selection is applied (case e) an 87 percent reduction can be 

gained compared to case c. 

TABLE 7.—COMPARISON OF z ESTIMATION RESULTS 

WHEN OPTIMIZING FOR h VERSUS z ESTIMATION

Case and 

optimization 

objective 

Sensors added 

to baseline SSEE (z)

results 

P24 T30 P45 P50 T50 P15 

case c: h estimation x x 188.1 

case d: z estimation x x 132.8 

case e: z estimation x x 24.9 

The sensors selected through the approach presented in this 

paper can also be compared to those selected through the 

Systematic Sensor Selection Strategy (S4) presented in 

Reference 13. Reference 13 also used C-MAPSS to perform 

sensor selection. One example shown in Reference 13 

assumed a baseline suite of commanded fuel flow and six 

sensed measurements, and the merits of adding additional 

sensors were considered. The six baseline and five optional 

sensors used in that study are shown in Table 8. In order to 

compare results, the approach presented in this paper (where q

is a combination of all elements of h for rank deficient 

problems) was applied to this same test case. 

TABLE 8.—SENSORS USED IN S4 COMPARISON TEST CASE 

Sensed output Standard deviation 

(percent) 

Baseline 

Sensors 

Nf—fan speed 0.25 

Nc—core speed 0.25 

T24—HPC inlet total temperature 0.75 

Ps30—HPC exit static pressure 0.50 

T30—HPC exit total temperature 0.75 

T48—Exhaust gas temperature 0.75 

Additional 

(Optional) 

Sensors 

P15—Bypass duct total pressure 0.50 

T21—LPC inlet total temperature 0.75 

P24—HPC inlet total pressure 0.50 

T30—HPC exit total temperature 0.75 

P50—LPT exit total pressure 0.50 

T50—LPT exit total temperature 0.75 

The results obtained by applying the two strategies are 

shown in Table 9. Both approaches produced identical 

selections for the 8 and 10 element sensor suite cases, and 

both select T21 as the last sensor added (11 element sensor 

suite case). However, they produce different selections for the 

seven and nine element sensor suite cases. The fact that the 

two approaches do not produce identical results in all cases is 

not surprising as there are fundamental differences in the 

objectives of the two approaches. The work in Reference 13 

focused on the diagnosis of five fault types assumed to occur 

in isolation, whereas this paper focuses on the estimation 
NASA/TM—2009-215839 11

accuracy of 10 health parameters that are uncorrelated and can 

vary simultaneously. Also, the focus of Reference 13 was on 

the fault detection and isolation capabilities offered by the 

selected sensor suite. Estimating or quantifying the fault 

magnitude was not considered. Furthermore, in Reference 13 

sensor selection was based on two operating points—takeoff 

and cruise. The different sensor selection results, while 

perhaps not unexpected, do illustrate the importance of 

performing sensor selection with the intended application in 

mind. 

TABLE 9.—COMPARISON OF SENSOR SUITES SELECTED 

APPLYING S4 VERSUS THIS PAPER’S APPROACH 

No. 

sensors 

Sensors added to baseline 

(S4) 

Sensors added to baseline 

(this paper's approach) 

P15 

T21 

P24 P50 

T50 

P15 

T21 

P24 P50 

T50 

6 

7 x x 

8 x x x x 

9 x x x x x x 

10 x x x x x x x x 

11 x x x x x x x x x x 

Conclusions 

A systematic approach to combined sensor suite and model 

tuning parameter selection for on-line Kalman filter-based 

health parameter estimation has been presented. The Kalman 

filter estimation accuracy improvement that could be gained 

by adding additional sensors was theoretically predicted and 

experimentally validated through simulation studies. Of the 

two model-tuning parameter selection approaches considered, 

the new approach of selecting the model tuner vector as a 

linear combination of all health parameters was found to 

provide improved estimation accuracy over the conventional 

approach of selecting a subset of health parameters to serve as 

model tuner vector. The new technique was demonstrated to 

consistently produce smaller sum of squared health parameter 

estimation errors, and in most cases produce smaller errors in 

individual health parameters estimates as well. The 

methodology presented in this paper is general, and can 

facilitate combined sensor and tuner parameter selection for 

different objective functions such as health parameter 

estimation or auxiliary output estimation. The systematic 

sensor and tuning parameter selection methodology is 

envisioned to be a valuable tool for system designers to assess 

the estimation accuracy enabled by different design choices. 

Areas for future work include extending the technique to 

select sensors and tuning parameters optimal over a range of 

operating conditions, and evaluating the technique on a nonlinear engine model, under both steady-state and transient 

operating conditions. 

References 

1. Luppold, R.H., Roman, J.R., Gallops, G.W., and Kerr, L.J., 

“Estimating In-Flight Engine Performance Variations Using 

Kalman Filter Concepts,” AIAA–89–2584, AIAA 25th Joint 

Propulsion Conference, Monterey, CA, Jul. 10–13, 1989. 

2. Volponi, A., “Enhanced Self-Tuning On-Board Real-Time Model 

(eSTORM) for Aircraft Engine Performance Health Tracking,” 

NASA/CR—2008-215272, Jul. 2008. 

3. Kumar, A., and Viassolo, D., “Model-Based Fault Tolerant 

Control,” NASA/CR—2008-215273, Sep. 2008. 

4. Doel, D.L., “An Assessment of Weighted-Least-Squares-Based 

Gas Path Analysis,” J. of Engineering for Gas Turbines and 

Power, Vol. 116, pp. 366–373, Apr. 1994. 

5. Volponi, A.J., “Foundations of Gas Path Analysis (part i and ii),” 

in Gas Turbine Condition Monitoring and Fault Diagnosis, Von 

Karman Institute Lecture Series no. 1, Von Karman Institute for 

Fluid Dynamics, Brussels, Belgium, Jan. 2003. 

6. Litt, J.S., “An Optimal Orthogonal Decomposition Method for 

Kalman Filter-Based Turbofan Engine Thrust Estimation,” 

Journal of Engineering for Gas Turbines and Power, Vol. 130 

/011601–1, Jan. 2008. 

7. Simon, D.L., and Garg, S., “A Systematic Approach for ModelBased Aircraft Engine Performance Estimation,” AIAA 

Infotech@Aerospace Conference, Seattle, WA, Apr. 6–9, 2009. 

8. Simon, D.L., and Garg, S., “Optimal Tuner Selection for Kalman 

Filter-Based Aircraft Engine Performance Estimation,” ASME 

Turbo Expo Conference, paper GT2009–59684, Orlando, FL, 

Jun. 8–12, 2009. 

9. Borquet, S., and Léonard, O., “The Fisher Information Matrix as a 

Relevant Tool for Sensor Selection in Engine Health Monitoring,” 

International Journal of Rotating Machinery, Vol. 2008, article 

ID 784749, 2008. 

10. Kamboukos, Ph., Oikonomou, P., Stamatis, A., and Mathioudakis, 

K., “Optimizing Diagnostic Effectiveness of Mixed Turbofans by 

Means of Adaptive Modelling and Choice of Appropriate 

Monitoring Parameters,” Ageing Mechanisms and Control 

Specialists’ Meeting on Life Management Techniques for Ageing 

Air Vehicles, Manchester, UK, Oct. 2001. 

11. Mushini, R., and Simon, D., “On Optimization of Sensor 

Selection for Aircraft Gas Turbine Engines,” Proceedings of the 

18th International Conference on Systems Engineering (ICSEng 

’05), pp. 9–14, Las Vegas, NV, Aug. 2005. 

12. Sowers, T.S., Kopasakis, G., and Simon, D.L., “Application of the 

Systematic Sensor Selection Strategy for Turbofan Engine 

Diagnostics,” ASME Turbo Expo Conference, paper GT2008–

50525, Berlin, Germany, Jun. 9–13, 2008. 

13. Sowers, T.S., Fittje, J.E., Kopasakis, G., and Simon, D.L., 

“Expanded Application of the Systematic Sensor Selection Strategy 

for Turbofan Engine Diagnostics,” ASME Turbo Expo Conference, 

paper GT2009–59251, Orlando, FL, June 8–12, 2009. 

14. España, M.D., “Sensor Biases Effect on the Estimation Algorithm 

for Performance-Seeking Controllers,” Journal of Propulsion and 

Power, Vol. 10, No. 4, 1994, pp. 527–532. 

15. Simon, D., Optimal State Estimation, Kalman, H∞, and Nonlinear 

Approaches, John Wiley & Sons, Inc., Hoboken, NJ, 2006. 

16. Frederick, D.K., DeCastro, J.A., and Litt, J.S., “User’s Guide for 

the Commercial Modular Aero-Propulsion System Simulation (CMAPSS),” NASA/TM—2007-215026, Oct. 2007.
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A Systematic Approach to Sensor Selection for Aircraft Engine Health Estimation 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

A systematic approach for selecting an optimal suite of sensors for on-board aircraft gas turbine engine health estimation is presented. The 

methodology optimally chooses the engine sensor suite and the model tuning parameter vector to minimize the Kalman filter mean squared 

estimation error in the engine’s health parameters or other unmeasured engine outputs. This technique specifically addresses the 

underdetermined estimation problem where there are more unknown system health parameters representing degradation than available 

sensor measurements. This paper presents the theoretical estimation error equations, and describes the optimization approach that is applied 

to select the sensors and model tuning parameters to minimize these errors. Two different model tuning parameter vector selection 

approaches are evaluated: the conventional approach of selecting a subset of health parameters to serve as the tuning parameters, and an 

alternative approach that selects tuning parameters as a linear combination of all health parameters. Results from the application of the 

technique to an aircraft engine simulation are presented, and compared to those from an alternative sensor selection strategy. 

15. SUBJECT TERMS

Aircraft engines; Systems health monitoring; Gas turbine engines; Kalman filtering; State estimation 

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17 

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email:help@sti.nasa.gov) 

a. REPORT

U 

b. ABSTRACT

U 

c. THIS 

PAGE

U 

19b. TELEPHONE NUMBER (include area code)

443-757-5802 

Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. Z39-18