Donald L. Simon

Glenn Research Center, Cleveland, Ohio

Jeffrey B. Armstrong

ASRC Aerospace Corporation, Cleveland, Ohio

An Integrated Approach for Aircraft Engine 

Performance Estimation and Fault Diagnostics

NASA/TM—2012-217725

October 2012

GT2012–69905
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Donald L. Simon

Glenn Research Center, Cleveland, Ohio

Jeffrey B. Armstrong

ASRC Aerospace Corporation, Cleveland, Ohio

An Integrated Approach for Aircraft Engine 

Performance Estimation and Fault Diagnostics

NASA/TM—2012-217725

October 2012

GT2012–69905

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

Turbo Expo 2012

sponsored by the American Society of Mechanical Engineers (ASME)

Copenhagen, Denmark, June 11–15, 2012
Acknowledgments

This research was conducted under the NASA Aviation Safety Program, Vehicle Systems Safety Technologies Project.

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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2012-217725 1

An Integrated Approach for Aircraft Engine 

Performance Estimation and Fault Diagnostics

Donald L. Simon

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Jeffrey B. Armstrong

ASRC Aerospace Corporation

Cleveland, Ohio 44135

Abstract

A Kalman filter-based approach for integrated on-line

aircraft engine performance estimation and gas path fault 

diagnostics is presented. This technique is specifically 

designed for underdetermined estimation problems where 

there are more unknown system parameters representing 

deterioration and faults than available sensor measurements. A 

previously developed methodology is applied to optimally 

design a Kalman filter to estimate a vector of tuning parameters, appropriately sized to enable estimation. The estimated

tuning parameters can then be transformed into a larger vector 

of health parameters representing system performance 

deterioration and fault effects. The results of this study show 

that basing fault isolation decisions solely on the estimated 

health parameter vector does not provide ideal results. 

Furthermore, expanding the number of the health parameters 

to address additional gas path faults causes a decrease in the 

estimation accuracy of those health parameters representative

of turbomachinery performance deterioration. However, 

improved fault isolation performance is demonstrated through 

direct analysis of the estimated tuning parameters produced by 

the Kalman filter. This was found to provide equivalent or 

superior accuracy compared to the conventional fault isolation

approach based on the analysis of sensed engine outputs, 

while simplifying online implementation requirements. 

Results from the application of these techniques to an aircraft 

engine simulation are presented and discussed.

Introduction

Aircraft engine performance trend monitoring and gas path 

fault diagnostics are closely related technologies that assist 

operators in managing the health of their gas turbine engine 

assets. Today, these functions are primarily performed offboard the aircraft within engine health management ground 

stations, which provide fleet-wide engine health management 

functionality (Ref. 1). This is accomplished by processing 

snapshot engine measurement data acquired in-flight. A toplevel description of the aircraft engine performance trend 

monitoring and gas path fault diagnostics process is given in 

Reference 2, and a representation of this process is shown in 

Figure 1. The process begins by calculating residuals between 

measured engine outputs and a reference model. Performance 

trend monitoring is conducted by analyzing the sensor 

residuals to assess performance changes that a gas turbine 

engine will naturally incur over time due to turbomachinery 

deterioration. Typically, this is performed by estimating and 

trending health parameters reflective of gradual performance 

deterioration within each major engine module. Since the 

unknown health parameters typically outnumber the available 

sensor measurements, this poses an underdetermined estimation problem. Gas path fault diagnostics is the process of 

detecting and isolating the occurrence of faults impacting 

engine flow-path performance. It is performed by detecting 

any rapid or abrupt engine performance changes, and then 

isolating the most likely cause. Often a single fault isolation 

approach is applied based on the assumption that engine gas 

path faults usually occur in an isolated fashion. The observed 

shift in sensed engine outputs is compared against a collection 

of known fault type signatures, and the best match is then 

isolated as the root cause for the event. An illustration of 

gradual versus rapid performance shifts is shown in Figure 2. 

Additional details on conventional engine performance trend 

monitoring and fault diagnostics can be found in References 3

to 6.

An emerging approach in the field of aircraft engine controls 

and health management is the inclusion of real-time on-board 

models for the in-flight estimation of engine performance 

variations (Refs. 7 to 9). This technology, typically based on 

Kalman filter concepts, enables the estimation of unmeasured 

engine performance parameters that can be directly utilized for 

controls and health management applications. Recently, NASA 

proposed a model-based gas path health management architecture for on-board performance trend monitoring and gas path 

fault diagnostics (Refs. 10 and 11). This architecture, shown in

Figure 3, is similar in design to the ground-based architecture 

shown in Figure 1, and provides the same performance estimation and fault diagnostic functionality. The 

benefits of the on-board architecture include the real-time 

continuous monitoring of engine performance and the 

early diagnosis of fault conditions. This architecture 
NASA/TM—2012-217725 2

Engine

measurement

data

Compare to 

reference 

model

Reference 

(nominal) 

model

Fault 

isolation

Performance trend monitoring

Reconcile 

& report 

results

y +

-

Δy

No

Engine

operating 

conditions

Fault

detection

Record 

no fault

Fault diagnostics

Yes

Figure 1.—Ground station performance trend monitoring and gas path diagnostic process.

Gradual 

deterioration

Gradual 

deterioration

Rapid shift 

(potentially due to 

a fault event)

Figure 2.—Gradual versus rapid performance shifts

Control

logic

Sensor

Measurements

Actuator

Commands

Fault diagnostics

Transmission to

ground station

Engine performance and diagnostic information

Real-time adaptive

performance model 

(Performance trend monitoring)

Performance 

baseline model

Model tuning

parameters +

-

Compare to 

baseline model

∆y

Figure 3.—On-board performance trend monitoring and gas path diagnostic process (original proposed design).
NASA/TM—2012-217725 3

Control

logic

Sensor

Measurements

Actuator

Commands

Fault diagnostics

Transmission to

ground station

Engine performance and diagnostic information

Real-time adaptive

performance model 

(Performance trend monitoring)

Tuning parameter

archive

Model tuning

parameters +

-

Compare to 

baseline model

∆q

Figure 4.—On-board performance trend monitoring and gas path diagnostic process (simplified proposed design).

contains two engine models designed to operate in parallel. 

The first model, referred to as the real-time adaptive performance model, applies a Kalman filter to estimate a vector of 

adjustable model tuning parameters that enable the model to 

track engine performance over the engine’s lifetime of use. To 

address the underdetermined estimation problem, a previously 

developed optimal tuner selection strategy is applied to design 

the Kalman filter (Refs. 12 and 13). This enables the Kalman 

filter to estimate a reduced-order tuner vector that can be 

directly transformed into a vector of estimated health parameters reflecting gas path deterioration and/or individual fault 

magnitudes. The second model, referred to as the performance 

baseline model, serves as a baseline of recent engine performance. Residuals between the engine and performance 

baseline model outputs are monitored for fault detection and 

isolation purposes.

A criticism of the on-board architecture shown in Figure 4

is its complexity as it requires the implementation of two 

separate on-board engine models. This may not be feasible

given on-board processor limitations. In an attempt to design a 

simpler architecture, this paper will evaluate fault diagnostics 

based on the model tuning parameters produced by the 

Kalman filter-based real-time adaptive performance model as 

shown in Figure 4. Such an approach, if successful, bypasses

the need for the on-board performance baseline model. 

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

some mathematical preliminaries are given regarding the 

problem formulation including the linear model representing 

system dynamics and the formulation of the Kalman filter. 

The optimal tuner selection methodology that enables the 

estimation of health parameters when facing underdetermined 

estimation problems is also introduced. Next, two fault 

isolation approaches are presented. The first approach bases

fault isolation decisions directly on the health parameter 

estimates. The second approach applies a weighted least 

squares single fault isolation strategy that seeks to determine 

the fault type that best matches the observed signature in the 

Kalman filter estimated model tuning parameters. This is 

followed by an example application of the techniques to a 

linear turbofan engine model. Finally, conclusions are

presented.

Nomenclature

A, Axh, Axq

B, Bxh, Bxq,

C, Cxh, Cxq,

D, L, M

system matrices

C-MAPSS Commercial Modular Aero-Propulsion 

System Simulation

DM Mahalanobis distance

H fault influence matrix 

HPC high pressure compressor

HPT high pressure turbine

I identity matrix

K∞ Kalman filter gain

LPC low pressure compressor

LPT low pressure turbine

P covariance matrix

PMC probability of misclassification

P∞ Kalman filter state estimation covariance 

matrix

Q, Qxh, Qxq process noise covariance matrices

R measurement noise covariance matrix

SSEE sum of squared estimation errors

V* transformation matrix relating h to q

VSV variable stator vane
NASA/TM—2012-217725 4

VBV variable bleed valve

Wf fuel flow

WSSR weighted sum of squared residuals

h health parameter vector

q Kalman filter tuning parameter vector

u actuator command vector

v measurement noise vector

w process noise vector

x state vector

xxh augmented state vector (x and h)

xxq reduced-order state vector (x and q)

y vector of measured outputs

Subscripts

f fault

h health parameter vector

k discrete time step index

q tuner vector

xh augmented state vector (x and h)

xq reduced-order state vector (x and q)

ss steady-state value

Superscripts

† pseudo-inverse

^ estimated value

~ error value

T transpose

Problem Formulation

Problem formulation will begin by reviewing the steps 

necessary to design a Kalman filter applying the optimal tuner 

selection methodology described in Reference 12. First, 

consider the discrete linear time-invariant engine state space 

equations about a linear design point 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, where h ∈ p

, represents the engine 

health parameters, which induce shifts in other variables as the 

health parameters deviate from their nominal values. The 

vectors w and v are uncorrelated zero-mean white noise input

sequences. 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 dimensions. Through algebraic manipulation Equation (1) can be re-written 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.

 

[ ]



, ,

,

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 k k

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

+

+

       

   = ++             

= ++

 

= ++    

= ++

 



(2)

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

augmented state vector, [x

T hT

]

T

. wxh consists of the original 

state process noise, w, concatenated with the process noise 

associated with the health parameter vector, wh.

,

,

k

xh k

h k

w

w

w

  =    

(3)

The covariance of wxh, denoted as Qxh, is necessary for 

Kalman filter formulation as will be discussed later in the 

paper.

Reduced-Order State Space Model

To enable Kalman filter formulation for an underdetermined 

estimation problem, a reduced-order state space model must 

be 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 Vh = (4)

where q ∈ m, h ∈ p

, m < p, and V* is an m × p transformation matrix of rank m, which relates h to q. Although q is a 

linear combination of health parameters, the elements of q do 

not have any physical meaning. Their purpose is to allow an 

accurate estimation of unmeasured engine parameters. 

However, given an estimate of q (i.e., qˆ ), an approximation of 

the health parameter vector, h

ˆ , can be obtained as
NASA/TM—2012-217725 5

ˆ *† h Vq = ˆ (5)

where V*† is the pseudo-inverse of V*

. Substituting Equation

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

space equations that will be used to formulate the Kalman 

filter

 



, , 1 ,

,

*† 1

1 ,

, ,

*†

,

0 0

xq k xq xq k xq xq k

xq

xq k

k k k

k

k k q k

x A x B w

xq xq k xq k xq k

k

k k k

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

+

+

+

        

   = ++               

= ++

  =    + +   

= ++

  



(6)

For the reduced-order system, the state process noise, wxq, and 

its associated covariance, Qxq, are a function of the full-order 

state process noise and covariance (i.e., wxh and Qxh) and the 

transformation matrix, V*

, , * * ,

* *

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

   =   =     

   =      

(7)

Here, as in many Kalman filter designs, the specification of

system process noise covariance, Qxh, can be treated as a 

design parameter. Increasing the process noise will cause the 

ensuing Kalman filter to place more emphasis on the measurements, and less on the process model. An evaluation of the 

effect of Qxh specification on the estimation accuracy and 

responsiveness of a Kalman filter applying the optimal tuner 

selection methodology can be found in Reference 13. 

Kalman Filter Formulation 

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

means that although the Kalman filter is a dynamic system, the 

state estimation error covariance matrix and the Kalman gain 

matrix are invariant; instead of updating these matrices each 

time step, they are held constant. Given the reduced-order 

linear state space equations shown in Equation (6), the state 

estimation error covariance matrix, P∞, is calculated by 

solving the following Riccati equation (Ref. 14)

1 ( )

T

xq xq

TT T

xq xq xq xq xq xq xq

P A PA

A PC C PC R C PA Q

∞ ∞

−

∞∞ ∞

=

− ++



(8)

The steady-state Kalman filter gain, K∞, can then be calculated 

as follows (Ref. 14)

( ) 1 T T K PC C PC R xq xq xq

−

∞∞ ∞ = + (9)

and, assuming steady-state, open-loop operation (u = 0), the 

Kalman filter estimator takes the following form

, , 1 ( , 1 ) ˆ ˆˆ ˆ ˆ

k

xq k xq xq k k xq xq xq k

k

x

x Ax K y CAx

q − ∞ −

 

== + −     (10)

Using Equation (5), the reduced-order state vector estimate,

xq xˆ , produced by Equation (10) can be transformed into an 

estimate of the augmented state vector as follows



,

, , *† *†

ˆ

ˆ 0 0 ˆ ˆ ˆ ˆ 0 0 ˆ

xq k

k k

xh k xq k

k k

x

x I I x

x x

h V V q

     = =     =          (11)

Optimal Transformation Matrix Selection

As presented in Reference 12, the estimation accuracy of 

the Kalman filter is directly dependent on the selection of the 

transformation matrix, V*

. This gives rise to the optimization 

problem of selecting V* to minimize the estimation error in the 

parameters of interest. This can be accomplished by conducting an optimal iterative search to select a V* matrix that 

minimizes the mean sum of squared estimation errors (SSEE) 

in the parameters of interest

( ) *

* arg min m p V

SSEE V × ∈ (12)

Readers are referred to Reference 12 for the derivation of 

health parameter mean SSEE as a function of V* under openloop steady-state operating conditions. 

Analysis of Kalman Filter Parameter Estimates 

for Gas Path Fault Isolation

As opposed to performing fault isolation by analyzing 

sensor measurements directly, this study proposes the 

monitoring of the estimated parameters produced by the 

Kalman filter. Like the sensed measurement illustration shown 

in Figure 2, Kalman filter-produced estimated parameters are 

expected to undergo a discernable change when the engine 

experiences a gas path fault. However, monitoring the Kalman 

filter outputs directly helps simplify the diagnostic system 

design. While it will require the archival of past tuning 

parameter values in order to calculate the magnitude of any 
NASA/TM—2012-217725 6

recent abrupt/rapid change, it eliminates the need to implement and maintain a complete reference model for diagnostic 

purposes (e.g., the Performance Baseline Model previously 

shown in Figure 3). Although diagnostics typically consist of 

the two-step process of fault detection and fault isolation (see 

Figure 1), this study will exclusively focus on the fault 

isolation aspect of the problem while assuming that fault 

detection has already successfully occurred. While not 

explicitly analyzed or demonstrated, it is anticipated that event 

detection techniques similar to those commonly applied to 

monitor for changes in sensed engine outputs (e.g., see 

Refs. 15 to 17) can be applied to detect event changes in 

estimated Kalman filter outputs as well. The fault isolation 

techniques considered in this study consist of two separate 

approaches designed to analyze Kalman filter-produced 

parameter estimates. The first approach is designed to analyze 

the health parameter estimates, ˆ

h (given by Eq. (11)), while 

the second approach is designed to analyze the tuning 

parameter estimates, qˆ (given by Eq. (10)). These two 

approaches are further discussed in the sections below.

Fault Isolation Through ĥ Analysis

The first fault isolation approach performs fault isolation

based on the estimated health parameters that undergo the 

largest magnitude change (relative to recently archived 

values). For example in a turbofan engine, if the largest 

estimated health parameter changes are observed in fan 

efficiency and fan flow capacity a fan fault is declared. In 

addition to turbomachinery module faults, the capability to 

isolate other gas path system fault types may be addressed by 

incorporating additional health parameters (e.g., health 

parameters representing actuator and sensor biases). Doing so 

increases the number of health parameters, resulting in the 

following updated state-space equations

1

,

,

k

k kk f k

f k

k

kkk f k

f k

h

x Ax Bu L L w h

h

y Cx Du M M v h

+

 

=++ +      

 

=+ +    +   

(13)

where hf represents non-turbomachinery module faults, and 

the matrices Lf and Mf reflect the effects of those additional 

faults in the system states and sensed outputs, respectively. As 

the optimal tuner selection methodology enables the estimation of any number of health parameters, estimation of the 

expanded health parameter vector is feasible. However, this 

will require selection of a V* matrix optimized, and appropriately sized, for the expanded health parameter configuration.

Fault Isolation Through qˆ Analysis

While the previously described fault isolation approach 

based on ˆ

h analysis is simple and intuitive as it provides a 

direct estimate of fault magnitude, it does have limitations. It 

is predicated on the assumption that when a fault occurs, the 

resulting change in health parameter estimates will be sparsely 

concentrated in those health parameters associated with the 

fault. However, due to the underdetermined nature of the 

estimation problem the occurrence of a fault will often result 

in a “smearing” of estimates across the entire vector of 

estimated health parameters (Ref. 18). Furthermore, the 

underdetermined nature of the Kalman estimation problem 

worsens as the dimension of hf expands to incorporate more 

fault types. Borguet and Léonard presented an encouraging 

sparse isolation approach for underdetermined least squares 

estimation problems (Ref. 19). However, it is unclear if this 

approach can be extended to online Kalman filter estimation 

applications. Instead of basing fault isolation decisions on

those health parameters that undergo the largest change, an 

alternative fault isolation strategy is to monitor for specific 

fault signatures contained within the estimated reduced-order 

tuner vector, qˆ , produced by the Kalman filter. This technique is analogous to the single-fault isolation strategy that is 

commonly applied to sensed engine outputs. To apply this

technique it is first necessary to determine the qˆ signature 

that corresponds to each fault type. This derivation will be 

made assuming quasi-steady-state open-loop operating 

conditions and assuming that the Kalman filter has been 

designed based upon the reduced-order state-space model 

equations shown in (6). These derivations will take advantage 

of the following expected value properties at steady-state 

open-loop operating conditions

1

,

, ,1 ,

0

0

0

ˆ ˆˆ

k k ss

k

fk f

k ss

k

k

k

xq k xq k xq ss

Ex Ex x

h h

E h h

Ey y

E u

E w

E v

Ex Ex x

+

−

   = =   

 

  =

 

  =  

  =  

  =  

  =  

  = =  

(14)

By taking expected values of both sides of the ensuing 

Kalman filter shown in Equation (10), the expected value of 

xq k x , ˆ can be obtained as a function of yss
NASA/TM—2012-217725 7

( )

( )

( )

, , 1

, 1

, , ,

1

,

ˆ ˆ

ˆ

ˆ ˆ ˆ

ˆ

xq k xq xq k

k xq xq xq k

xq ss xq xq ss ss xq xq xq ss

xq ss xq xq xq ss

Ex AEx

K Ey C AEx

x Ax K y CAx

x I A KC A K y

−

∞ −

∞

−

∞ ∞

   =   

+ −        

= +−

=− +



(15)

Next, the steady-state engine sensed outputs as a function of 

the fault condition will be derived. The derivation will be 

based upon the system equations shown in Equation (13) with 

the expanded health parameter vector to encompass additional 

fault types. Taking expected values produces

( )

1

,

1

k kk

k

f k

f k

ss ss f

f

ss f

f

E x AE x BE u

h

L L E Ew h

h

x Ax L L h

h

x IA LL h

+

−

    = +    

 

+ +            

 

= +      

  = −      



(16)

( ( ) )

,

1

y

kkk

k

f k

f k

ss ss f

f

ss f f

f

G

ss y

f

E y CE x DE u

h

M M E Ev h

h

y Cx M M h

h

y CI A L L M M h

h

y G h

−

   = +   

 

+ +            

 

= +      

 

=− +          

  =    





(17)

Substituting Equation (17) into Equation (15) enables the 

calculation of the reduced-order state vector as a function of 

the fault condition

( ) 1

, ˆxq ss xq xq xq y

f

h

x I A KC A KG h

−

∞ ∞

 

=− +    

(18)

Recall that xq x is the concatenation of x and q (see Eqs. (6)

and (10)). Therefore, based on Equation (18), an expression 

for the steady-state reduced-order tuner vector as a function of 

the fault condition can be obtained

[ ]( ) 1

ˆ 0

ˆ

q

ss xq xq xq y

f

G

ss q

f

h

q I I A KC A KG h

h

q G h

−

∞ ∞

 

= −+    

  =    



(19)

The above equation can be used to construct a fault influence 

coefficient matrix, denoted as H, that relates fault effects to 

Kalman filter tuning parameter estimates under steady-state 

conditions. This is performed off-line (i.e., during the system 

design phase) by individually perturbing health parameters, or 

combinations of health parameters, representing each system 

fault type to determine the corresponding change induced in 

the elements of the vector ˆss q . Dividing the change in ˆss q by 

the inserted fault magnitude produces a column of the fault 

influence coefficient matrix. Once a fault influence matrix 

column has been calculated for each fault type they can be 

concatenated to form the matrix. This gives rise to the 

following quasi-steady-state equation representing qˆ as a 

function of the fault condition

ˆ q q Hf v = + (20)

where the vector f represents system faults and vq represents 

random uncertainty in the qˆ estimates with covariance Pq. Pq

can be analytically derived based upon the system matrices, 

measurement noise covariance, and the Kalman gain as shown 

in Reference 12. It is important to emphasize that the dimension and elements of the fault vector f do not have to be 

equivalent to those of [h hf]

T

. For example, it may be desirable 

to model a turbomachinery module fault as a combined shift in

efficiency and flow capacity health parameters as opposed to 

separating them into two separate faults. Given Equation (20), 

a least squares fault isolation approach can be formulated.

Here, each fault type is evaluated individually, and the

hypothesized fault type that best matches the observed qˆ

signature in a weighted least squares sense is isolated as the 

fault type. For example, for the i

th fault type the estimated fault 

magnitude is calculated as

( ) 1 ˆ 1 1 ˆ T T

i iq i iq f HPH HPq − − − = (21)

where Hi is the column of the H matrix corresponding to the i

th

fault type, and the scalar ˆ

i f is the estimated magnitude of the 

i

th fault type that produces the best match of the observed qˆ

signature in a weighted least squares sense. The resulting ˆ

i f

estimate is then used to calculate the estimation error residual 

vector for the i

th fault type as
NASA/TM—2012-217725 8

( )

( )

1 1 1

1 1 1

ˆ ˆ ˆˆ

ˆ ˆ

ˆ ˆ

ˆ

i i

i i

T T

i iq i iq

T T

i iq i iq

q qq

qHf

q H HPH HPq

I H HPH HP q

− − −

− − −

= −

=− ⋅

= −

  = −    



(22)

The weighted sum of squared residuals for the i

th hypothesized 

fault type is calculated as

1 ˆ ˆT WSSR q P q i iq i− =   (23)

After WSSR’s are calculated for each potential fault type they 

are compared, and the hypothesized fault type that produces 

the minimum WSSR is isolated as the fault cause. 

Probability of Fault Misclassification

This subsection will introduce a stochastic approach for 

calculating the probability of fault misclassification for the 

single fault weighted least squares isolation approach previously presented. This information will be incorporated into the 

optimal iterative search routine designed to select the transformation matrix V*

. Doing so will make tuning parameter 

selection a dual-objective function of minimizing estimation 

error within the parameters of interest while also minimizing 

the probability of misclassification when performing fault 

isolation. Several simplifying assumptions are made in 

producing this calculation. First, the calculation applied to 

approximate the misclassification rate only considers the

probability of misclassification between fault pair combinations (i.e., makes the assumption that only two fault classes 

exist). 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. Second, misclassification 

rates are based on an assumed mean fault magnitude for each 

fault type. While this assumption does not fully account for 

the fact that faults can be of varying severity or magnitude it is 

effective for identifying the misclassification probability for 

faults of average severity. Let us consider a fault of a given 

type, a, and magnitude, fa. From Equation (20) the expected 

estimated tuner vector under this condition becomes

a aa q Hf = . The probability that a qˆ observation collected 

when fault fa is present is misclassified as fault type b

(assumed to be of equivalent probability and equivalent qˆ

covariance as fault type a) is given as (Ref. 20)

( ) 1

| 2 1 PMCb a = −Φ ⋅ DM (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

( ) ( )

( ) ( )

1

1

min or

T

ab q ab

M

T

ab q ab

qq Pqq

D

qq Pqq

−

−

  − −  

= −−  

 

+ +  

(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), qb

is the estimated tuner vector for fault type b, scaled to be the 

same weighted length as qa as shown in the equation below.

1

1

T

aq a

b b T

bq b

qP q

q H

HPH

−

− = (26)

The above equations allow approximate measures for the 

probability of misclassification PMCb|a for each fault pair to 

be included within the optimization routine designed to select 

the V* matrix. 

Turbofan Engine Example

A linear model extracted from the NASA Commercial 

Modular Aero-Propulsion System Simulation (C-MAPSS) 

high-bypass turbofan engine model (Ref. 21) is used to 

evaluate the methodology. The linear model has two state 

variables and three control actuators as shown in Table 1. Also 

shown in Table 1 are 13 health parameters used for the 

purposes of this study. This includes 10 efficiency and flow 

capacity health parameters associated with the major rotating 

modules of the engine, plus 3 actuator bias health parameters. 

The model’s six sensed outputs, and corresponding sensor 

noise standard deviation, are shown in Table 2. The gas path 

fault types and corresponding health parameter magnitude 

shifts applied to simulate these fault types are shown in

Table 3. The values shown in Table 3 reflect rapid/abrupt 

deltas, or shifts, in these parameters relative to a recently 

archived baseline. These values were manually chosen 

through trial and error to present challenging, yet representative, fault isolation scenarios. 
NASA/TM—2012-217725 9

TABLE 1.—STATE VARIABLES, HEALTH 

PARAMETERS, AND ACTUATORS

State 

variables

Actuators Health 

parameters

Nf – fan speed Wf – fuel flow Fan efficiency

Nc – core speed VSV – variable stator 

vane

Fan flow capacity

VBV –variable bleed 

valve

LPC efficiency

LPC flow capacity

HPC efficiency

HPC flow capacity

HPT efficiency

HPT flow capacity

LPT efficiency

LPT flow capacity

Wf bias

VSV bias

VBV bias

TABLE 2.—SENSED OUTPUTS AND STANDARD DEVIATION 

AS PERCENT OF OPERATING POINT TRIM VALUES

Sensed 

output

Standard deviation,

%

Nf – fan speed 0.005

Nc – core speed 0.005

T24 – HPC inlet total temperature 0.015

Ps30 – HPC exit static pressure 0.010

T30 – HPC exit total temperature 0.015

T48 – Exhaust gas temperature 0.015

TABLE 3.—GAS PATH FAULT 

TYPES AND MAGNITUDES

Fault type Health parameter 

shifts

Fan –2% ηfan and –2% γfan

LPC –4% ηLPC and –4% γLPC

HPC –2% ηHPC and –2% γHPC

HPT –2% ηHPT and +2% γHPT

LPT –2% ηLPT and +2% γLPT

Wf bias –2% Wf bias

VSV bias –2% VSV bias

VBV bias –10% VBV bias

The linear model is used as the truth model for this application example. The model is run open-loop, so all control 

inputs remain at 0, unless an actuator bias fault is present. In 

emulating turbomachinery performance deterioration effects, 

deviations in the first ten health parameters are considered. 

Variations in these 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. 

For evaluation purposes, the optimal tuner selection methodology was applied to produce three separate Kalman filter 

designs for application at a single open-loop operating point. 

The first Kalman filter was designed to minimize the sum of 

squared estimation errors (SSEE) in the first 10 health parameters listed in Table 1. In constructing this Kalman filter, the

system state space equations were assumed to be of the form 

shown in Equation (1), and the three health parameters 

associated with actuator bias were not included in the design. 

Furthermore, fault probability of misclassification was not 

considered in the selection of the V* matrix. The second Kalman 

filter was designed in accordance with Equation (13) assuming 

all 13 health parameters (the 10 representing turbomachinery 

deterioration plus the three actuator bias health parameters). The 

second Kalman filter required selection of a V* matrix with 

three additional columns to accommodate the additional health 

parameters. Here, V* matrix selection was performed to 

minimize the sum of squared estimation errors (SSEE) in the 

first 10 health parameters listed in Table 1. While the second 

Kalman filter design does provide an estimate of all 13 health 

parameters, the accuracy of the 3 health parameters reflecting 

actuator bias was not considered during the V* matrix selection 

process.1 The associated fault isolation strategy was to identify

the health parameter(s) that exhibited the largest change in 

accordance with the eight fault types listed in Table 3. The third 

Kalman filter was designed in accordance with Equation (1) to 

enable an estimate of the first 10 health parameters shown in 

Table 1 to be produced. Optimal tuner selection was performed 

with the two-fold objective of minimizing the SSEE in the first 

10 health parameters and minimizing the probability of fault 

misclassification between all fault pairs. A summary of the three 

Kalman filter designs is shown in Table 4.

TABLE 4.—SUMMARY OF KALMAN FILTER DESIGNS

Kalman 

filter

design

Size of h used 

in design of 

Kalman filter

V*

optimization

objective

Fault isolation 

strategy

1 10 health 

parameters

Minimize ˆ

h SSEE qˆ least squares

2 13 health 

parameters

minimize ˆ

h SSEE

(in elements 1 to 10 

only)

ˆ

h maximum 

element(s)

3 10 health 

parameters

Minimize ˆ

h SSEE

and minimize 

PMCb|a

qˆ least squares

 1

This decision was made in an attempt to strike a balance between the 

dual objectives of performance estimation and fault diagnostics. 

Accurate turbomachinery deterioration estimation is necessary for 

performance trend monitoring purposes. Conversely, from a fault 

isolation standpoint accurate actuator bias estimation is secondary to 

the ability to discriminate between different fault types. 
NASA/TM—2012-217725 10

First, an evaluation of the ˆ

h estimation accuracy provided by

the three Kalman filter designs was conducted. Theoretically 

predicted results are based on the analytical derivations found in 

Reference 12. Experimental results were obtained through a 

Monte Carlo simulation analysis where the first 10 health 

parameters varied over a random distribution in accordance with 

the covariance matrix, Ph. The test cases were provided to the 

C-MAPSS linear discrete state space model given in Equation

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

parameter test case lasted 30 s, and a total of 200 test cases were 

evaluated. The experimental estimation errors were determined 

by calculating the mean squared error between estimated and 

actual values during the last 10 s of each 30 s test case. The 

error calculation is based on only the last 10 s so that engine 

model outputs and Kalman estimator outputs have reached a 

quasi-steady-state operating condition prior to calculating the 

error. This ensures that the experimental results are consistent 

with the theoretically predicted estimation errors that are

derived assuming steady-state operation.

The theoretically predicted and the experimentally obtained

ˆ

h estimation error results are shown in Table 5. The best 

estimation accuracy is provided by Kalman filter design 1. 

This is expected as this design has been optimized for 

estimating the first 10 health parameters. This produces 

a theoretical SSEE of 17.111. For the second and third Kalman 

filter designs, the theoretical ˆ

h SSEE increases to 20.648 and 

19.848, respectively. For the second design the increase is due 

to adding three additional parameters to the health parameter 

vector. This increases the underdetermined nature of the 

estimation problem, causing a decrease in estimation accuracy. 

In this case, the design choice to perform V* selection to

minimize the estimation error in the first 10 health parameters 

was found to be helpful, but this design could not replicate the 

accuracy of the first Kalman filter. The third Kalman filter, 

like the first Kalman filter, is designed assuming 10 system 

health parameters. However, the dual optimization objectives 

of minimizing the ˆ

h SSEE and the probability of misclassification applied in selecting V* causes an increase in ˆ

h SSEE.

Next, an evaluation of the fault isolation accuracy provided 

by the three Kalman filter designs was assessed. Experimental 

results were obtained through simulation analysis where eight 

gas path fault cases of the type and magnitude shown in 

Table 3 were individually analyzed. These eight test cases 

were provided to the C-MAPSS linear discrete state space 

model shown in Equation (13), with 13 health parameters and 

an update rate of 15 ms. Each test case was 300 s in duration, 

and a fault isolation assessment is conducted at each 15 ms 

time step (20,000 per test case). Only the health parameters 

associated with the fault under consideration were varied 

during each case. All remaining health parameters were left 

unchanged at their default values of zero. 

The ensuing fault isolation results are shown in Table 6

through Table 8. Table 6 shows the resulting fault isolation 

confusion matrix when using Kalman filter design 1 and the 

qˆ analysis fault isolation approach. The elements along the 

diagonal of the confusion matrix represent correct isolation 

percentages. Perfect fault isolation performance is achieved 

for HPC, HPT, LPT, and Wf faults. Of the remaining faults, 

LPC, VSV and VBV are the most challenging to isolate. The 

overall correct isolation rate for this design is 90.5 percent.

TABLE 5.—HEALTH PARAMETER PERCENT SQUARED ESTIMATION ERRORS

Estimator Error type h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 SSEE

Kalman 

filter

design 1

Theor. squared bias 0.007 0.065 0.112 0.474 0.073 0.124 0.044 0.033 0.144 0.035 1.111

Theor. variance 3.131 1.523 3.304 1.725 0.034 0.232 0.701 0.000 2.355 2.995 16.000

Theor. squared error 3.138 1.588 3.416 2.200 0.107 0.355 0.745 0.034 2.499 3.029 17.111

Exper. squared error 3.477 1.495 3.847 1.940 0.107 0.351 0.689 0.033 2.457 2.796 17.192

Kalman 

filter

design 2

Theor. squared bias 0.006 0.023 0.113 0.406 0.073 0.019 0.029 0.021 0.168 0.043 0.901

Theor. variance 3.146 2.084 3.306 1.834 0.039 1.780 0.843 1.219 2.411 3.084 19.747

Theor. squared error 3.152 2.107 3.419 2.240 0.112 1.799 0.871 1.240 2.579 3.128 20.648

Exper. squared error 3.388 2.196 3.836 2.039 0.110 1.833 0.776 1.402 2.619 3.001 21.200

Kalman 

filter 

design 3

Theor. squared bias 0.051 0.101 0.117 0.269 0.074 0.129 0.039 0.032 0.170 0.014 0.996

Theor. variance 3.683 1.812 3.974 1.834 0.048 0.324 0.885 0.000 2.514 3.779 18.852

Theor. squared error 3.733 1.913 4.091 2.103 0.122 0.453 0.924 0.032 2.684 3.793 19.848

Exper. squared error 4.181 1.747 4.886 1.861 0.125 0.472 0.838 0.032 2.594 3.426 20.163
NASA/TM—2012-217725 11

TABLE 6.—CONFUSION MATRIX 

(KF DESIGN 1, qˆ LEAST SQUARES)

Isolated fault state 

(%)

Fan LPC HPC HPT LPT Wf VSV VBV

True condition

Fan 99.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0

LPC 0.1 77.8 12.0 6.4 0.0 1.3 0.3 2.2

HPC 0.0 0.0 100.0 0.0 0.0 0.0 0.0 0.0

HPT 0.0 0.0 0.0 100.0 0.0 0.0 0.0 0.0

LPT 0.0 0.0 0.0 0.0 100.0 0.0 0.0 0.0

Wf 0.0 0.0 0.0 0.0 0.0 100.0 0.0 0.0

VSV 2.1 2.8 1.6 1.5 3.3 2.3 82.5 3.9

VBV 3.2 8.2 2.2 5.8 5.3 4.1 6.1 65.1

Overall correct isolation rate (%) = 90.5%

TABLE 7.—CONFUSION MATRIX 

(KF DESIGN 2, MAXIMUM ˆ

h )

Isolated fault state 

(%)

Fan LPC HPC HPT LPT Wf VSV VBV

True condition

Fan 14.8 45.2 8.5 21.2 10.4 0.0 0.0 0.0

LPC 0.0 82.3 14.9 2.7 0.0 0.0 0.0 0.0

HPC 0.0 0.8 99.2 0.0 0.0 0.0 0.0 0.0

HPT 0.0 0.1 0.0 99.9 0.0 0.0 0.0 0.0

LPT 11.6 17.0 0.0 2.6 68.8 0.0 0.0 0.0

Wf 0.0 0.9 0.0 0.0 0.0 99.1 0.0 0.0

VSV 0.1 54.0 42.4 3.1 0.5 0.0 0.0 0.0

VBV 0.2 92.5 5.9 1.2 0.2 0.0 0.0 0.0

Overall correct isolation rate (%) = 58.0%

TABLE 8.—CONFUSION MATRIX 

(KF DESIGN 3, qˆ LEAST SQUARES)

Isolated fault state 

(%)

Fan LPC HPC HPT LPT Wf VSV VBV

True condition

Fan 97.6 0.0 0.0 0.0 2.2 0.0 0.0 0.2

LPC 0.1 90.7 3.6 0.1 0.0 0.9 0.0 4.6

HPC 0.0 0.0 100.0 0.0 0.0 0.0 0.0 0.0

HPT 0.0 0.0 0.0 100.0 0.0 0.0 0.0 0.0

LPT 0.0 0.0 0.0 0.0 100.0 0.0 0.0 0.0

Wf 0.0 0.0 0.0 0.0 0.0 100.0 0.0 0.0

VSV 1.3 2.8 1.9 2.1 3.5 1.6 85.6 1.3

VBV 6.8 11.2 0.3 0.2 2.1 3.9 0.1 75.6

Overall correct isolation rate (%) = 93.7%

Table 7 shows fault isolation results obtained applying 

Kalman filter design 2 and making fault isolation decisions 

based on observed health parameter shifts. Here, an estimated 

magnitude for each of the five turbomachinery module faults 

was calculated as the root-sum-squared value of the module’s 

efficiency and flow capacity ˆ

h values. Actuator bias fault 

magnitudes were calculated by taking the absolute value of the 

corresponding ˆ

h values. Fault isolation was then made by 

selecting the fault type with the largest estimated magnitude.

Figure 5.—Example of Kalman filter design 2 health 

parameter estimation during fan fault (top) and HPT 

fault (bottom).

The results in Table 7 show that while some faults such as 

HPC, HPT, and Wf are isolated with a high degree of 

accuracy, the overall fault isolation accuracy of 58 percent is 

relatively poor. In fact, faults such as Fan, VSV, and VBV are 

usually misclassified. This suggests that simply adding 

additional health parameters as part of the optimal tuner 

design process does not guarantee that the estimation accuracy 

of those health parameters will be adequate. In fact, in this 

case, which is attempting to simultaneously estimate 13 health 

parameters based on only 6 sensor measurements, there is a 

high degree of “smearing” of the estimates. This in turn results 

in unsatisfactory fault isolation performance. An illustration of 

the health parameter estimation results is shown in Figure 5

for a fan fault and an HPT fault. In the fan fault case (top 

plot), the Kalman filter is unable to provide good health 

parameter estimation, and consequently a fault misclassification will occur. In the HPT fault case (bottom plot), some 

smearing of the estimates across multiple health parameters is 

observed, but the largest estimated deviations are in health 

parameters 7 and 8 (HPT efficiency and flow capacity). Here, 

correct classification as an HPT fault will occur.

Table 8 shows confusion matrix results obtained using 

Kalman filter design 3 and the qˆ analysis single fault 

isolation approach. Recall that this Kalman filter was designed 

with the objective of minimizing ˆ

h SSEE and the probability 

of fault misclassification. The overall correct isolation rate is 

93.7 percent, which is an improvement over the results shown 

in Table 6 and Table 7. This is encouraging as it demonstrates 

that selection of the V* matrix can have a positive impact on 

diagnostic performance. However, it is also acknowledged that 

this improvement in diagnostic performance comes at the 
NASA/TM—2012-217725 12

expense of a slight degradation in performance estimation 

accuracy as previously shown in Table 5.

As a final comparison, an assessment of the fault isolation 

performance achievable through the analysis of sensor 

residuals is assessed. This serves as a baseline, and is representative of the fault diagnostic approaches shown in Figure 1

and Figure 3 designed to process ∆y information. Here, the 

fault isolation approach is designed to analyze the estimated 

ˆ

y values produced by Kalman filter design 1.2 This approach 

is the same as that described for processing the qˆ estimates, 

but the confusion matrix and the measurement covariance 

matrix are updated to reflect sensed engine outputs. The 

results from this assessment, shown in Table 9, reveal that this 

approach provides fault isolation performance very similar to 

that shown in Table 6. This is very encouraging as it suggests 

that analyzing the estimated tuning parameters produced by 

the Kalman filter can yield isolation results of equivalent or 

better performance than the conventional approach of 

analyzing sensor residuals. Furthermore, the qˆ analysis 

approach does not require a separate reference baseline model 

for residual calculation purposes. Conversely, ∆y analysis 

would require a reference or baseline model against which 

measurement residuals could be calculated. 

TABLE 9.—CONFUSION MATRIX 

(KF DESIGN 1, yˆ LEAST SQUARES)

Isolated Fault State 

(%)

Fan LPC HPC HPT LPT Wf VSV VBV

True Condition

Fan 99.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0

LPC 0.1 76.1 13.1 7.2 0.1 1.7 0.3 1.6

HPC 0.0 0.0 100.0 0.0 0.0 0.0 0.0 0.0

HPT 0.0 0.0 0.0 100.0 0.0 0.0 0.0 0.0

LPT 0.0 0.0 0.0 0.0 100.0 0.0 0.0 0.0

Wf 0.0 0.0 0.0 0.0 0.0 100.0 0.0 0.0

VSV 2.3 2.9 1.6 1.6 3.4 2.3 83.1 2.8

VBV 4.0 8.2 2.8 6.4 6.5 5.2 7.4 59.5

Overall Correct Isolation Rate (%) =89.7%

Conclusions

The Kalman filter-based approach for combined aircraft 

engine performance estimation and gas path fault diagnostics

has been shown to simplify implementation requirements 

without sacrificing performance. This simplification comes 

about by eliminating the need to maintain a second reference 

model for diagnostic purposes. Instead of analyzing sensor 

residuals to diagnose faults, it has been demonstrated that 

 2 Estimated sensor measurements, ˆ

y , produced by the Kalman filter 

are used instead of actual sensor measurements, y, to conduct this 

comparison. The estimated values track measured values well and are 

less noisy, thus providing superior isolation performance.

direct analysis of the deviations in Kalman filter estimates can 

be performed for fault diagnostic purposes. In the context of 

this effort, two variations in the fault isolation approach were 

evaluated. Both approaches are based upon a previously 

developed optimal tuner selection methodology for underdetermined Kalman filter estimation applications. This methodology enables estimation of a set of model tuning parameters 

that can be transformed into a vector of health parameters 

reflecting engine performance deterioration. The first fault 

isolation approach expands the number of health parameters to 

encompass additional (actuator) fault types, and then conducts 

fault isolation based upon the estimated health parameter(s) 

that exhibit the largest change. The second fault isolation 

approach relies on the analytically derived open-loop quasisteady-state signature that a fault will produce in Kalman filter 

estimated tuning parameters. Here, fault isolation is performed 

by identifying the fault type signature that best matches the 

observed tuning parameters in a least squares sense. As a 

comparison, these two fault isolation techniques were 

compared against the conventional technique of performing 

fault isolation based upon sensor measurements. Simulation 

studies demonstrated that the first design did not provide ideal 

results. Due to the underdetermined nature of the estimation 

problem, individual faults were found to produce a “smearing” 

of estimates across multiple health parameters, including 

parameters not associated with the fault, which led to fault 

misclassification. Furthermore, adding additional health 

parameters to the design to account for actuator faults was 

found to compound the underdetermined nature of the 

problem. The fault isolation results were not ideal, and 

furthermore the estimation accuracy in the subset of health 

parameters reflective of performance deterioration worsened 

when the additional actuator bias health parameters were 

added to the design. Conversely, the second design was found 

to provide fault isolation accuracy comparable to that of the 

conventional approach of analyzing changes in engine sensed 

outputs. Furthermore, it was shown that an approximate 

measure for the probability of fault misclassification can be 

analytically formulated and incorporated into the optimal 

iterative search applied to select the Kalman filter tuning 

parameters, which was found to provide improved fault 

isolation performance. This modification did result in a 

decrease in performance deterioration estimation accuracy. 

The merits of its application are thus dependent upon end-user 

objectives.

The initial findings of this study have been encouraging. 

Follow-on evaluations are recommended to incorporate 

additional fault types and to consider larger variations in fault 

magnitudes. Additionally, evaluations are needed to assess the 

technique over a broader range of operating conditions. Due to 

system nonlinearities, it is reasonable to assume that the fault 

signatures contained within the estimated tuning parameters 

will vary as a function of engine operating point. If so, the 
NASA/TM—2012-217725 13

proposed fault isolation approach will either require a means 

to archive/retrieve fault influence coefficient matrix information as a function of operating point, or it will be limited to 

performing fault isolation analysis at fixed operating conditions. Additional studies, such as closed-loop nonlinear 

simulation studies and actual aircraft engine data studies, are 

warranted to fully assess these design issues.

References

1. Society of Automotive Engineers E-32, (2005), “A Guide 

to the Development of a Ground Station for Engine Condition Monitoring,” SAE Aerospace Information Report 

4175A.

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An Integrated Approach for Aircraft Engine Performance Estimation and Fault Diagnostics 

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Simon, Donald, L.; Armstrong, Jeffrey, B. 

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14. ABSTRACT

A Kalman filter-based approach for integrated on-line aircraft engine performance estimation and gas path fault diagnostics is presented. 

This technique is specifically designed for underdetermined estimation problems where there are more unknown system parameters 

representing deterioration and faults than available sensor measurements. A previously developed methodology is applied to optimally 

design a Kalman filter to estimate a vector of tuning parameters, appropriately sized to enable estimation. The estimated tuning parameters 

can then be transformed into a larger vector of health parameters representing system performance deterioration and fault effects. The results 

of this study show that basing fault isolation decisions solely on the estimated health parameter vector does not provide ideal results. 

Furthermore, expanding the number of the health parameters to address additional gas path faults causes a decrease in the estimation 

accuracy of those health parameters representative of turbomachinery performance deterioration. However, improved fault isolation 

performance is demonstrated through direct analysis of the estimated tuning parameters produced by the Kalman filter. This was found to 

provide equivalent or superior accuracy compared to the conventional fault isolation approach based on the analysis of sensed engine 

outputs, while simplifying online implementation requirements. Results from the application of these techniques to an aircraft engine 

simulation are presented and discussed. 

15. SUBJECT TERMS

Systems health monitoring; Gas turbine engines 

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