Donald L. Simon and Sanjay Garg

Glenn Research Center, Cleveland, Ohio

A Systematic Approach for Model-Based 

Aircraft Engine Performance Estimation

NASA/TM—2010-216077

January 2010

AIAA–2009–1872
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Donald L. Simon and Sanjay Garg

Glenn Research Center, Cleveland, Ohio

A Systematic Approach for Model-Based 

Aircraft Engine Performance Estimation

NASA/TM—2010-216077

January 2010

AIAA–2009–1872

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

Infotech@Aerospace (I@A) Conference

sponsored by the American Institute of Aeronautics and Astronautics

Seattle, Washington, April 6–9, 2009
Acknowledgments

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

The authors graciously acknowledge Jonathan Litt for his support and insightful discussions 

which led to the development of this work.

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National Technical Information Service

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Trade names and trademarks are used in this report for identifi cation 

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either expressed or implied, by the National Aeronautics and 

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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2010-216077 1 

A Systematic Approach for Model-Based 

Aircraft Engine Performance Estimation 

Donald L. Simon and Sanjay Garg 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract

A requirement for effective aircraft engine performance estimation is the ability to account for engine 

degradation, generally described in terms of unmeasurable health parameters such as efficiencies and flow 

capacities related to each major engine module. This paper presents a linear point design methodology for 

minimizing the degradation- induced error in model-based aircraft engine performance estimation 

applications. The technique specifically focuses on the underdetermined estimation problem, where there 

are more unknown health parameters than available sensor measurements. A condition for Kalman filterbased estimation is that the number of health parameters estimated cannot exceed the number of sensed 

measurements. In this paper, the estimated health parameter vector will be replaced by a reduced order 

tuner vector whose dimension is equivalent to the sensed measurement vector. The reduced order tuner 

vector is systematically selected to minimize the theoretical mean squared estimation error of a maximum 

a posteriori estimator formulation. This paper derives theoretical estimation errors at steady-state 

operating conditions, and presents the tuner selection routine applied to minimize these values. Results 

from the application of the technique to an aircraft engine simulation are presented and compared to the 

estimation accuracy achieved through conventional maximum a posteriori and Kalman filter estimation 

approaches. Maximum a posteriori estimation results demonstrate that reduced order tuning parameter 

vectors can be found that approximate the accuracy of estimating all health parameters directly. Kalman 

filter estimation results based on the same reduced order tuning parameter vectors demonstrate that 

significantly improved estimation accuracy can be achieved over the conventional approach of selecting a 

subset of health parameters to serve as the tuner vector. However, additional development is necessary to 

fully extend the methodology to Kalman filter-based estimation applications. 

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 

Fn net thrust 

Gh, Gz estimation bias matrices 

H matrix that relates health parameter effects to measured steady-state engine outputs 

Hq matrix that relates tuner vector effects to measured steady-state engine outputs 

Hz matrix that relates health parameter effects to unmeasured steady-state engine outputs 

HPC high pressure compressor 

HPT high pressure turbine 

I identity matrix 

K∞ Kalman filter gain 
NASA/TM—2010-216077 2 

LPC low pressure compressor 

LPT low pressure turbine 

MAP maximum a posteriori 

Ph, Pz health and auxiliary parameter covariance matrices 

ˆ ˆ, , , Ph k Pz k covariance matrices of estimated parameters 

P∞ Kalman filter state estimation covariance matrix 

Q, Qxh, Qxq process noise covariance matrices 

R measurement noise covariance matrix 

SmLPC LPC stall margin 

T40 combustor exit temperature 

T50 LPT exit temperature 

V* transformation matrix relating h to q

h health parameter vector 

q reduced order tuner vector 

uk actuator command vector 

vk measurement noise vector 

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

xk state vector 

xxh,k augmented state vector, T T T

k k ⎡x h ⎤ ⎣ ⎦

xxq,k reduced order augmented state vector, T T T

k k ⎡x q ⎤ ⎣ ⎦

yk vector of measured outputs 

zk vector of unmeasured (auxiliary) outputs 

εk residual vector (estimate minus its expected value) 

η efficiency health parameter 

γ flow capacity health parameter 

Subscripts 

k discrete time step index 

xh augmented state vector (x and h) 

xq reduced order state vector (x and q) 

ss steady-state value 

Superscripts 

† pseudoinverse 

^ estimated value 

~ error value 

– mean value 

T transpose 

Operators 

E[·] expected value of argument 

tr{·} trace of matrix 

SSEE(·) sum of squared estimation errors 
NASA/TM—2010-216077 3 

I. Introduction 

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. 1,2, 

and3). This technology, typically based on Kalman filter concepts, enables the estimation of unmeasured 

engine performance parameters which can be directly utilized by controls, prognostics and health 

management applications. A challenge that complicates this practice is the fact that an aircraft engine’s 

performance is affected by its level of degradation, generally described in terms of unmeasurable health 

parameters such as efficiencies and flow capacities related to each major engine module. Through 

Kalman filter-based estimation techniques, the level of engine performance degradation can be estimated, 

given that there are at least as many sensors as parameters to be estimated (Ref. 4). However, in an 

aircraft engine the number of sensors available is typically less than the number of health parameters, 

presenting an underdetermined estimation problem. A common approach to address this shortcoming is to 

estimate a subset of the health parameters, referred to as model tuning parameters. While this approach 

enables online Kalman filter-based estimation, it can result in “smearing” the effects of unestimated 

health parameters onto those that are estimated, and in turn reduces the accuracy of model-based 

estimation schemes. Recently, Litt (Ref. 5) presented an approach based on singular value decomposition 

that selects a model tuning parameter vector of low enough dimension to be estimated by a Kalman filter. 

The model tuning parameter vector, q, is constructed as a linear combination of all health parameters, h, 

given by 

 q = V*

h (1)

where the transformation matrix, V*

, is selected by applying singular value decomposition to capture the 

overall effect of the larger set of health parameters on the engine variables as closely as possible in the 

least squares sense. In this paper a new linear point design technique that applies a systematic approach to 

tuning parameter selection will be presented. This technique, like the one presented in Reference 5, 

defines a transformation matrix, V*

, used to construct a reduced order tuning parameter vector that is a 

linear combination of all health parameters, and of low enough dimension to enable Kalman filter 

estimation. The new approach selects a reduced order tuner vector, q, that minimizes the theoretical mean 

squared estimation error produced when applying maximum a posteriori (MAP) estimation. MAP 

estimation, sometimes referred to as regularization, is commonly applied for ground-based aircraft turbine 

engine gas path analysis (Refs. 6 and 7). It is applied to “snap shot” engine measurements collected at 

quasi-steady-state operating points during each flight, and takes advantage of prior knowledge of engine 

health parameter distributions. Unlike the Kalman filter, it is not a recursive estimator and it does not 

account for system dynamics, therefore the MAP estimator is not well suited for dynamic parameter 

estimation applications like in-flight aircraft engine performance estimation. However, the MAP 

estimator is capable of estimating more unknowns than available measurements due to its inclusion of a 

priori knowledge of the estimated parameters’ covariance. In this study, the MAP estimator formulation 

has been chosen to facilitate reduced order tuner parameter selection for two reasons. First, its theoretical 

estimation errors can be more readily derived than those of the Kalman filter. Second, the MAP 

estimation accuracy achieved through reduced order tuner estimation can be directly compared to the 

MAP estimation accuracy achieved by estimating the entire health parameter vector. The ability of the 

parameter estimation accuracy achieved through reduced order tuner estimation to approximate that of 

full order health parameter estimation will verify that a V*

 transformation matrix optimal for MAP 

estimation has indeed been produced. There is no known closed form solution for selecting V*

 to satisfy 

the objective of minimizing the theoretical MAP estimation error. Therefore, a multivariable iterative 

search routine is applied to perform this function. Once V*

 and the reduced order q vector are produced, 

they can be applied to model-based aircraft engine performance estimation applications based on either 

MAP estimation or Kalman filter estimation. 
NASA/TM—2010-216077 4 

The remaining sections of this paper are organized as follows. First, linear dynamic state space 

equations and steady-state measurement process equations for the engine are introduced, for both full 

order (health parameter vector) and reduced order (tuner parameter vector) formulations. The application 

of these equations to formulate the Kalman filter and the MAP estimator equations is then presented. This 

is followed by a derivation of the MAP estimator theoretical mean squared estimation error. The 

theoretical MAP estimation error information is directly used by the iterative search routine applied to 

optimally select the reduced order tuning parameter vector, which is described next. Example MAP and 

Kalman filter estimation results from the application of the new methodology to an aircraft turbofan 

engine simulation are then shown. After the example results, there is a discussion of findings and required 

future work. Finally, conclusions are presented. 

II. Problem Formulation 

A. Dynamic State Space Equations 

This subsection will first present the full order dynamic state space equations representative of engine 

dynamics around a linear design point, followed by a presentation of the reduced order dynamic state 

space equations. The latter can be applied to formulate the Kalman filter estimator. 

1. Full Order Dynamic State Space Equations 

The discrete linear time invariant engine state space equations about a linear design point are given as 

k k k kk 1

k k k kk

kkkk

x Ax Bu Lh w

y Cx Du Mh v

z Fx Gu Nh

+ = + ++

= +++

=++

(2)

where k is the time index, x is the vector of state variables, u is the vector of control inputs, y is the vector 

of measured outputs, and z is the vector of auxiliary (unmeasured) model outputs. The vector h 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, F, G, 

L, M, and N 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 (2) can be written as: 

{ {

[ ]{

[ ]{

, ,

,

,

1 , , 1 ,

,

,

0 0

xh xh k xh xh k

xh

xh k

xh

xh k

k k k

k xh xh k xh k xh k

k k h k

A x B w

k

k k k xh xh k k k

k

C x

k

k k xh xh k k

k

F x

x AL x B w

u Ax Bu w

h Ih w

x

y C M Du v C x Du v h

x

z F N Gu F x Gu

h

+

+

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤

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

⎡ ⎤

= + += + + ⎢ ⎥ ⎣ ⎦

⎡ ⎤

= += + ⎢ ⎥ ⎣ ⎦

123 123

123

14243

(3)

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. 
NASA/TM—2010-216077 5 

, ,

k

xh k

h k

w

w

w

⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ (4)

The eigenvalues of Axh consist of the original eigenvalues of A plus an additional dim(h) eigenvalues 

located at (1, 0) on the unit circle due to the augmentation. Thus, the new augmented system given in 

Equation (3) has at least as many eigenvalues located on the unit circle as there are elements of h. Once 

the h vector is appended to the state vector, it may be directly estimated by a Kalman filter, provided that 

the realization in Eq. (3) 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. 4). In an aircraft gas turbine 

engine there are usually fewer sensors than health parameters, presenting an underdetermined estimation 

problem. The conventional approach to address this problem is to estimate only a subset of h, referred to 

as tuning parameters. While this enables Kalman filter estimation, it does introduce error within the 

estimation process due to the effect of those health parameters excluded from estimation. In this paper a 

departure from the conventional approach of selecting a subset of h to serve as tuning parameters is taken. 

Instead of selecting the tuning parameter vector to simply consist of a subset of health parameters, this 

vector is chosen to consist of a linear combination of all health parameters. The tuning parameter will be 

of dimension equivalent to y in order to facilitate Kalman filter estimation, and will be optimally selected 

to reduce estimation error within the parameters of interest. The reduced order dynamic state space 

equations based on the reduced order tuner vector will be introduced in the next subsection. 

2. Reduced Order Dynamic State Space Equations 

Reduced order dynamic state space equations will be constructed based on the model tuning 

parameter vector, q, which is constructed as a linear combination of all health parameters, h, and given by 

* q Vh = (5)

where q ∈ Rm , h ∈ p R , m < p, and V* is an m × p transformation matrix of rank m, applied to construct 

the tuning parameter vector. An approximation of the health parameter vector, h

ˆ , can be obtained as 

ˆ *† hVq = (6)

where V*† is the pseudoinverse of V*

. Substituting Equation (6) into Equation (3) yields the following 

reduced order state space equations which can be used to formulate the Kalman filter 

{ {

{

, ,

,

*† 1 , , 1 ,

*† ,

*†

0 0

xq k xq xq xq k

xq xq k

xq xq

k k k

k xq xq k xq k xq k

k k q k

A x B w

k

k k k xq xq k k k

k

C x

k

k

k

F x

x x A LV B w

u Ax Bu w

q q I w

x

y C MV Du v C x Du v

q

x

z F NV

q

+

+

⎡ ⎤ ⎡ ⎤ ⎡⎤ ⎡ ⎤ ⎡ ⎤

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

⎡ ⎤

= + += + + ⎡ ⎤ ⎣ ⎦⎢ ⎥ ⎣ ⎦

⎡ ⎤ = ⎡ ⎤ ⎣ ⎦⎢ ⎥ ⎣ ⎦

14243 123

1442443

14243 {

,

,

k

+= + Gu F x Gu k xq xq k k

(7)

The state process noise, wxq, and its associated covariance, Qxq, for the reduced order system are 

calculated as 
NASA/TM—2010-216077 6 

, , * * ,

* *

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

⎡⎤ ⎡⎤⎡ ⎤ = = ⎢⎥ ⎢⎥⎢ ⎥ ⎣⎦ ⎣⎦⎣ ⎦

⎡ ⎤⎡ ⎤ = ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

(8)

B. Steady-State Measurement Process Equations 

The previous subsection presented state space equations for a dynamic system. Under steady-state 

operating conditions, these equations reduce to a steady-state measurement process. This subsection will 

present full order and reduced order steady-state measurement process equations that can be applied to 

formulate the MAP estimator. 

1. Full Order Steady-State Measurement Process Equations 

Under steady-state open-loop operating conditions, dynamic state space equations reduce to a 

measurement process of the following form where system outputs are a function of an arbitrary health 

parameter vector, h, as 

k k

z

y Hh v

z Hh

= +

= (9)

where H and Hz are appropriately dimensioned matrices relating the effects of the health parameter vector 

to the measured outputs, y, and the unmeasured auxiliary parameter outputs, z, respectively. The H and Hz

matrices can be derived from the dynamic state space equations given in Equation (2) by taking advantage 

of the following expected value properties at steady-state open-loop operating conditions 

[ ] [ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

1

0

0

0

k kss

k

k ss

k ss

k

k

k

E x Ex x

Eh h

Ey y

Ez z

E u

E w

E v

+ = =

=

=

=

=

=

=

(10)

where the subscript “ss” denotes steady-state operation. By taking expected values of Equation (2), xss, yss

and zss can be written as functions of the health parameter vector h 

[ ] [ ] [ ] [ ] [ ]

( )

1

1

k k k kk

ss ss

ss

E x AEx BEu LEh Ew

x Ax Lh

x I A Lh

+

−

=⋅ +⋅ +⋅ +

= +

= −

(11)
NASA/TM—2010-216077 7 

[ ] [ ] [ ] [ ] [ ]

( ( ) ) 1

k k k kk

ss ss

ss

H

ss

E y C Ex DEu M Eh Ev

y Cx Mh

y CI A L Mh

y Hh

−

=⋅ +⋅ + ⋅ +

= +

=− +

=

144424443

(12)

[ ] [ ] [ ] [ ]

( ( ) ) 1

z

k kkk

ss ss

ss

H

ss z

E z F Ex GEu N Eh

z Fx Nh

z FI A L Nh

z Hh

−

=⋅ +⋅ +⋅

= +

=− +

=

144424443

(13)

By including measurement noise effects, Equation (12) becomes y Hh v k k = + which, together with 

Equation (13), provide steady-state system equations of the form given in Equation (9). 

2. Reduced Order Steady-State Measurement Process Equations 

Substituting Equation (6) into Equation (9) yields the following reduced order steady-state 

measurement process equations 

{

*†

*†

q

k k

H

kqk

z

y HV q v

y Hq v

z HV q

= +

= +

=

(14)

C. Kalman Filter Formulation 

Next, the Kalman filter estimator formulation based on the reduced order dynamic state space 

equations will be presented. In this study, steady-state Kalman filtering is applied. In other words, the 

state estimation error covariance matrix and the Kalman gain matrix are converged and time invariant. 

Given the reduced order linear state space equations shown in Equation (7), the state estimation error 

covariance matrix, P∞, is calculated by solving the following Ricatti equation (Ref. 8) 

1 ( ) T TT T P A PA A PC C PC R C PA Q xq xq xq xq xq xq xq xq xq − ∞∞ ∞ ∞ ∞ =− + + (15)

The steady-state Kalman filter gain, K∞, can then be calculated as follows (Ref. 8) 

1 ( ) T T K PC C PC R xq xq xq − ∞∞ ∞ = + (16)

and, assuming steady-state, open-loop operation (u = 0), the Kalman filter estimator takes the following 

form 

x Ax K y CAx ˆ ˆ xq k xq xq k k xq xq xq k , ,1 = +− − ∞ − ( ˆ , 1 ) (17)
NASA/TM—2010-216077 8 

The reduced order state vector estimate, xxq ˆ , produced by Equation (17) can be used to produce an 

estimate of the state variables, health parameters, and the auxiliary parameter vector as follows 

, *†

*† ,

ˆ 0 ˆ ˆ 0

ˆ ˆ

k

xq k

k

k xq k

x I

x

h V

z F NV x

⎡ ⎤ ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦

= ⎡ ⎤ ⎣ ⎦

(18)

D. Maximum a Posteriori Estimator Formulation 

The maximum a posteriori (MAP) estimator can be formulated based on the steady-state 

measurement process equations given in Equations (12) and (13). The MAP estimator follows the closed 

form expression (Ref. 9) 

( ) 1 ˆ 1 T T 1 1 k k h h P HR H HR y − − − − = + (19)

where the matrix Ph, defined as [ ] T hhE , reflects a priori or historical knowledge of the covariance in the 

health parameters across all engines. The MAP estimate of the health parameter vector can be used to produce 

an estimate of the auxiliary parameter vector by premultiplying the right side of Equation (19) by Hz

( ) 1 1 1 1

ˆ ˆ

ˆ

k zk

T T k z h k

z Hh

z H P HR H HR y − − − −

=

= +

(20)

A MAP estimate of the reduced order tuner vector, q, can also be formulated based on the reduced 

order steady-state measurement process model given in Equation (14). This reduced order estimator is 

given as 

( ) 1 11 1 ˆ T T q P HR H HR kq q q q k y − −− − = + (21)

where Pq, defined as T E qq ⎡ ⎤ ⎣ ⎦ , is the covariance in the tuner vector. It is calculated as 

* *T P V PV q h = (22)

Once the estimate of ˆk q is obtained from Equation (21), estimates of the health parameter vector and the 

auxiliary parameter vector can be obtained as 

( )

( )

*†

1 *† 1 1 1

ˆ

*†

1 *† 1 1 1

ˆ

ˆ ˆ

ˆ

ˆ ˆ

k

k

k k

T T k q q qqk

q

kzk

T T z q q qqk

q

h Vq

h V P HR H HR y

z HV q

H V P HR H HR y

− −− −

− −− −

=

= +

=

= +

1444442444443

1444442444443

(23)
NASA/TM—2010-216077 9 

E. Analytical Derivation of MAP Estimation Error 

The estimation errors in ˆ

hk and ˆkz are defined as the difference between these estimates and their 

corresponding actual values 

ˆ

ˆ

k kk

k kk

hhh

zzz

= −

= −

%

% (24)

The estimation errors can be considered to consist of two components: an estimation error bias, and an 

estimation variance. Due to the underdetermined nature of the problem, the MAP estimator will be a 

biased estimator (i.e. the expected values of hk

% and kz% will be non-zero). The estimation error bias 

vector is equivalent to the mean estimation error vector defined as 

[] [ ]

ˆ

ˆ

k k kk

k k kk

h Eh Eh h

z Ez Ez z

= =− ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦

= =−

% %

% % (25)

The variance of the estimates can be found by constructing their respective estimation covariance 

matrices 

( )( )

( ) [ ] ( ) [ ]

ˆ,

ˆ,

ˆ ˆˆ ˆ

ˆ ˆˆ ˆ

T

k kk k h k

T

zk k k k k

P E h Eh h Eh

P E z Ez z Ez

⎡ ⎤ =− − ⎡⎤ ⎡⎤ ⎢ ⎥ ⎣⎦ ⎣⎦ ⎣ ⎦

⎡ ⎤ =− − ⎢ ⎥ ⎣ ⎦

(26)

Diagonal elements of the covariance matrices will reflect the variance in individual parameter estimates, 

while off diagonal elements reflect the covariance between parameter estimates. 

The overall sum of squared estimation errors (SSEE) can be obtained by combining the estimation 

error bias and estimation variance information as 

( ) { }

( ) { }

ˆ,

ˆ,

ˆ

ˆ

T k k k h k

T k k zk k

SSEE h h h tr P

SSEE z z z tr P

= +

= +

% %

% % (27)

where tr{·} represents the trace (sum of the diagonal elements) of the matrix. In this paper, minimization 

of the SSEE will be applied as the metric used to facilitate optimal tuner parameter selection. First, 

theoretical values for each component of the MAP SSEE, estimation error bias and estimation variance, 

will be derived assuming reduced order tuner estimation under steady-state, open-loop (u = 0) operating 

conditions. These derivations are provided in the following subsections. 

1. Analytical Derivation of MAP Estimation Error Bias 

The estimation error biases, hk

% and kz% , when applying reduced order tuner estimation, can be 

analytically derived for an arbitrary health parameter vector, h, at steady-state operating conditions. First, 

by taking expected values of both sides of Equation (23) the expected steady-state values of ˆ

hk and ˆkz

can be written as a function of h 
NASA/TM—2010-216077 10 

( ) [ ]

( )

( )

[ ] ( ) [ ]

( )

( )

1 *† 1 1 1

1 *† 1 1 1

1 *† 1 1 1

1 *† 1 1 1

1 *† 1 1 1

1 *† 1 1 1

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

T T k qq qq k

T T

ss q q q q ss

T T

ss q q q q

T T kz qq qq k

T T

ss z q q q q ss

T T

ss z q q q q

Eh V P H R H H R Ey

h V P HR H HR y

h V P H R H H R Hh

E z HV P H R H H R E y

z HV P H R H H R y

z HV P H R H H R

− −− −

− −− −

− −− −

− −− −

− −− −

− −− −

⎡ ⎤ =+ ⋅ ⎣ ⎦

= +

= +

=+ ⋅

= +

= + Hh

(28)

The steady-state estimation error biases, hss % and ss z% , can then be written as 

( )

( )

[ ]

( )

1 *† 1 1 1

ˆ

1 *† 1 1 1

1 *† 1 1 1

ˆ

ˆ

ˆ

ˆ

ˆ

ss

h

ss

ss k

ss

T T

qq qq

h

T T

qq qq

G

h

ss k

ss

T T

z qq qq z

z

h Eh h

h h

V P H R H H R Hh h

V P HR H HR H Ih

G h

z Ez z

z z

H V P H R H H R Hh H h

− −− −

− −− −

− −− −

= − ⎡ ⎤ ⎣ ⎦

= −

=+ −

⎛ ⎞ =+ − ⎜ ⎟ ⎝ ⎠

=

= −

= −

=+ −

%

144444424444443

14444444244444443

%

14444444244444443 {

( ) 1 *† 1 1 1

z

z

T T

z qq qq z

G

z

HV P H R H H R H H h

G h

− ⎛ ⎞ −− − =+ − ⎜ ⎟ ⎝ ⎠

=

144444444424444444443

(29)

The steady-state estimation error bias equations given in Equation (29) are functions of an arbitrary 

health parameter vector, h. As such they are 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 the health 

parameter vector h. The average sum of squared estimation error biases across a fleet of engines can be 

calculated as 
NASA/TM—2010-216077 11 

{ }

{ }

{ }

2

fleet

h

T T

ss ss ss ss

T T h h

T T h h

P

T h h h

h E h h E tr h h

E tr G hh G

tr G E hh G

tr G P G

⎡ ⎤ ⎡ ⎤ ≡ = ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦

= ⎡ ⎤ ⎣ ⎦

⎧ ⎫ ⎪ ⎪ = ⋅ ⎨ ⎡ ⎤ ⎣ ⎦ ⎬ ⎪ ⎪ ⎩ ⎭

=

% %% %%

14243

(30)

{ }

{ }

{ }

2

fleet

h

T T

ss ss ss ss

T T z z

T T z z

P

T zh z

z E z z E tr z z

E tr G hh G

tr G E hh G

tr G P G

≡ = ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦

= ⎡ ⎤ ⎣ ⎦

⎧ ⎫ ⎪ ⎪ = ⋅ ⎨ ⎡ ⎤ ⎣ ⎦ ⎬ ⎪ ⎪ ⎩ ⎭

=

% %% %%

14243

(31)

2. Analytical Derivation of MAP Estimation Variance 

Next, the variance of the health parameter and auxiliary parameter estimates is derived. First consider 

the health parameter estimate covariance matrices, ˆ

h k, P , defined as 

ˆ, ( )( ) ˆ ˆˆ ˆ

T k k

T

P E h Eh h Eh h k k kk k

ε ε

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

1442443 1442443 (32)

where the vector εk is defined as the residual between ˆ

hk at time k and its expected value. Since 

ˆ ˆ E h h k ss ⎡ ⎤ = ⎣ ⎦ , εk can be obtained by subtracting Equation (28) from Equation (23) 

() ()

( ) ( )

1 1 *† 1 1 1 *† 1 1 1

ˆ ˆ

1 *† 1 1 1

ˆ ˆ

ˆ ˆ

k ss

kk k

k ss

TT TT q q q q k q q q q ss

h h

T T q q q q k ss

h Eh

h h

V P HR H HR y V P HR H HR y

V P HR H HR y y

− − −− − −− −

− −− −

ε= − ⎡ ⎤ ⎣ ⎦

= −

=+ −+

=+ −

144444424444443 144444424444443 (33)

Making the substitution −= yyv sskk yields 
NASA/TM—2010-216077 12 

( ) 1 *† 1 1 1 T T k q q q qk V P HR H HRv − −− − ε= + (34)

The estimation covariance matrix ˆ

h k, P is then calculated as 

() ()

() ()

ˆ,

1 1 *† 1 1 1 *† 1 1 1

1 1 *† 1 1 1 *† 1 1 1

T

h k k k

T

TT TT T qq qq k qq qq k

R

T

TT TT

qq qq qq qq

P E

V P H R H H R E vv V P H R H H R

V P H R H H R RV P H R H H R

− − −− − −− −

− − −− − −− −

= εε ⎡ ⎤ ⎣ ⎦

⎛ ⎞ ⎛ ⎞ = + ⋅⋅ + ⎡ ⎤ ⎜ ⎟ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ = + ⎜ ⎟ ⎜ ⎟ + ⎝ ⎠ ⎝ ⎠

14243 (35)

Applying a similar derivation, the auxiliary parameter estimate covariance matrix, Pz k ˆ, , can be obtained as 

() () 1 1 *† 1 1 1 *† 1 1 1 ˆ,

T

T T T T P HV P H R H H R R HV P H R H H R zk z q q q q z q q q q

− − ⎛ ⎞ −− − −− − ⎛ ⎞ = + ⎜ ⎟ ⎜ ⎟ + ⎝ ⎠ ⎝ ⎠ (36)

The variance in the estimates ˆ

hk and ˆkz can be obtained from the diagonals of the covariance matrices 

produced by Equations (35) and (36) respectively. 

3. Analytical Derivation of MAP Sum of Squared Estimation Errors 

Once Equations (30), (31), (35), and (36) 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 information as previously shown in Equation (27). The health parameter sum of 

squared estimation errors, ( fleet) ˆ SSEE h , and the auxiliary parameter vector sum of squared estimation 

errors, SSEE z( ˆfleet ) , become 

( ) { }

{ }

( ) { }

{ }

2 fleet ˆ fleet ,

ˆ,

2 fleet ,ˆ fleet

ˆ,

ˆ

ˆ

h k

T h h h h k

z k

T z h z zk

SSEE h h tr P

tr G P G P

SSEE z z tr P

tr G P G P

= +

= +

= +

= +

%

%

(37)

From Equations (29), (35), and (36) it can be observed that both bias and variance are affected by the 

selection of the transformation matrix, V*

. The sum of squared estimation error terms derived in this 

section (Eq. (37)) give rise to an optimization problem: selecting V*

 to minimize the squared estimation 

error in the MAP-produced parameter estimates. These parameter estimates could include health 

parameter estimates, auxiliary parameter estimates, or a combination of parameters. Although there is no 

known closed form solution for optimally selecting the V*

 matrix to satisfy the objective of minimizing 

estimation errors, a multi-parameter iterative search method has been developed to perform this task, and 

will be described in the next section. 

F. Optimal Transformation Matrix Selection 

Prior to initiating the search for an optimal V*, specific system design information must be defined or 

obtained. This includes: 
NASA/TM—2010-216077 13 

• Specifying the auxiliary parameters, z, to be estimated 

• Generating system state space equations (Eq. (2)) at a fleet average (50 percent deteriorated) 

engine trim point. The result is expected values of zero for the health parameters and measured 

outputs, which is consistent with the theoretical derivations provided in this paper. 

• Defining measurement noise covariance matrix, R 

• Defining fleet average health parameter covariance, Ph

After the necessary system information has been obtained, the search for an optimal transformation 

matrix to minimize the MAP sum of squared estimation errors can commence. The process is performed 

using the lsqnonlin function of the Matlab (The MathWorks, Inc.) Optimization Toolbox. This function 

applies an iterative search to find the least squares solution of a user specified multivariable optimization 

problem. A further description of each step of this optimal iterative search is given below. 

1. Upon startup, an initial random guess of the transformation matrix, V*, is generated. 

2. The pseudoinverse of the transformation matrix, V*†, is calculated for implementation within the 

estimator, and for estimator error calculation purposes. V*† can be calculated by the Matlab pinv

function. 

3. Construct the reduced order steady-state measurement process model (Eq. (14)). 

4. Formulate the reduced order MAP estimator (Eq. (23)). 

5. Calculate sum of squared estimation errors, SSEE (Eq. (37)). 

6. 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 7 and proceed to step 8. 

b. If not converged, proceed to step 7 to update V*

. 

7. V*

 is updated via the Matlab lsqnonlin function, and the process returns to step 2. 

8. Upon convergence, the optimization routine returns the optimal value of V*

, and ends. 

Experience has shown that the transformation matrix returned by the optimization routine is not 

unique—different matrices can be found that provide equivalent estimation accuracy. Experience has also 

shown that the optimization routine will usually return a V*

 matrix that satisfies, or nearly satisfies (i.e. 

within 2 percent), the global minimum of the objective function. However, in order to guard against 

potential convergence to a local minimum, it is prudent to run the optimization routine multiple times, 

each time starting with a different initial guess for V*

. This is only to assure the designer that the global 

minimum is achieved, not to produce a consistent V*

. It should be emphasized that the optimal search for 

V* is only conducted offline during the estimator design process. This calculation is not conducted as part 

of the online estimator implementation, and thus places no additional computational burden upon it. 

III. Turbofan Engine Example 

A linearized cruise operating point extracted from the NASA Commercial Modular Aero-Propulsion 

System Simulation (C-MAPSS (Ref. 10)) high bypass turbofan engine model is used to evaluate the tuner 

selection methodology. The linear model has two state variables, ten health parameters, and three control 

inputs, all shown in Table 1. The model’s seven sensed outputs, and corresponding sensor noise standard 

deviation, are shown in Table 2. The auxiliary output parameters of interest to be estimated are shown in 

Table 3. 
NASA/TM—2010-216077 14 

TABLE 1.—STATE VARIABLES, HEALTH PARAMETERS, AND ACTUATORS 

State variables Health parameters Actuators 

Nf—fan speed ηFAN—Fan efficiency Wf—fuel flow 

Nc—core speed γFAN—Fan flow capacity VSV—variable stator vane 

ηLPC—LPC efficiency VBV—variable bleed valve 

γLPC—LPC flow capacity 

ηHPC—HPC efficiency 

γHPC—HPC flow capacity 

ηHPT—HPT efficiency 

γHPT—HPT flow capacity 

ηLPT—LPT efficiency 

γLPT—LPT flow capacity 

TABLE 2.—SENSED OUTPUTS AND STANDARD DEVIATION 

IN PERCENT OF OPERATING POINT TRIM VALUES 

Sensed output Standard deviation 

(%) 

Nf—fan speed ....................................................................... 0.25 

Nc—core speed ..................................................................... 0.25 

P24—HPC inlet total pressure .............................................. 0.50 

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

TABLE 3.—ESTIMATED AUXILIARY PARAMETERS 

Auxiliary parameter 

T40 ........................................................... Combustor exit temperature 

T50 ...................................................................... LPT exit temperature 

Fn .......................................................................................... Net thrust 

SmLPC ...................................................................... LPC stall margin 

Next, the estimation accuracy achievable by estimating the systematically selected reduced order 

tuner vectors will be assessed. This assessment will be conducted for both MAP estimation and Kalman 

filter estimation. 

A. MAP Estimation Accuracy Comparison 

A linear steady-state measurement process model-based on Equations (12) and (13) is used as the truth 

model for this application example. Deviations in all ten health parameter inputs are assumed to be 

uncorrelated and normally distributed about 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. The 

iterative search for a transformation matrix and associated reduced order tuner vector optimal for MAP 

estimation is run multiple times, each time starting with a different initial random guess of V*

. This is 

done so that the estimation consistency enabled by the q vectors returned by the iterative search routine 

can be assessed. First, four different q vectors are optimally selected to minimize the MAP estimated 

health parameter mean squared error. These four vectors are denoted as a, b, c and d. Then the process is 

repeated and four additional q vectors are selected, this time optimized for minimizing the MAP estimated 

auxiliary parameter mean squared error. These four vectors are denoted as e, f, g, and h. 

The health parameter estimation error results are summarized in Table 4 for MAP estimation of all 10 

health parameters directly through Equation (19), or indirectly through reduced order tuner vectors as 

given in Equation (23). This table shows the theoretically predicted and experimentally obtained percent 

squared estimation errors for each case. The experimental results are obtained through a Monte Carlo 

simulation analysis with random health parameter deviations normally distributed with covariance 
NASA/TM—2010-216077 15 

equivalent to Ph. A total of 100,000 test cases are produced and evaluated using each approach. It can be 

seen that the theoretically predicted and the experimentally obtained percent squared estimation errors 

exhibit good agreement. Based on Table 4 it can be seen that the four q vectors optimal for health 

parameter estimation, (vectors a, b, c and d), are fairly consistent in terms of the MAP estimation 

accuracy that they provide. Estimating all 10 health parameters directly results in a theoretical sum of 

squared estimation errors (SSEE) of 21.48, while estimating the reduced order (seven element) q vectors 

results in theoretical SSEE’s ranging from 21.73 to 21.95, approximately 1.1 to 2.2 percent more error 

than estimating all 10 health parameters. Also shown in Table 4 is the MAP health parameter estimation 

accuracy provided by the four q vectors optimally selected for auxiliary parameter estimation, vectors e, f, 

g and h. The estimation errors based on these tuners are larger and exhibit more variance. Their 

theoretical SSEE’s range from 23.90 to 26.13, or 11 to 22 percent larger than the error in estimating all 10 

health parameters directly. However, this result is not surprising considering that these tuners are 

optimized for reducing auxiliary parameter estimation error, not health parameter estimation error. 

TABLE 4.—MAP ESTIMATOR HEALTH PARAMETER PERCENT SQUARED ESTIMATION ERROR 

Estimated 

parameters 

Tuner 

vector 

Theory/ 

experiment 

ηFAN γFAN ηLPC γLPC ηHPC γHPC ηHPT γHPT ηLPT γLPT ( )ˆ SSEE h

10 health 

parameters N/A Theory 2.63 2.02 3.69 2.54 1.16 1.67 1.36 0.52 2.64 3.26 21.48 

Experiment 2.62 2.03 3.69 2.55 1.15 1.66 1.36 0.52 2.64 3.27 21.50 

7 element 

reduced 

order tuner 

vector 

selected to 

minimize h

estimation 

error 

a Theory 2.66 2.04 3.76 2.56 1.16 1.68 1.37 0.52 2.69 3.31 21.73 

Experiment 2.66 2.05 3.75 2.57 1.16 1.67 1.37 0.52 2.68 3.32 21.73 

b Theory 2.66 2.04 3.89 2.54 1.16 1.75 1.36 0.52 2.74 3.27 21.95 

Experiment 2.66 2.05 3.90 2.56 1.16 1.74 1.36 0.52 2.73 3.28 21.96 

c Theory 2.79 2.04 3.71 2.56 1.16 1.70 1.37 0.52 2.72 3.32 21.89 

Experiment 2.78 2.05 3.71 2.57 1.16 1.69 1.38 0.52 2.71 3.33 21.90 

d Theory 2.67 2.03 3.79 2.57 1.16 1.71 1.39 0.52 2.65 3.36 21.84 

Experiment 2.66 2.04 3.79 2.58 1.16 1.70 1.39 0.52 2.65 3.37 21.86 

7 element 

reduced 

order tuner 

vector 

selected to 

minimize z

estimation 

error 

e Theory 2.66 2.03 4.65 2.69 1.39 2.23 1.50 0.57 2.68 3.50 23.90 

Experiment 2.65 2.04 4.65 2.70 1.39 2.22 1.50 0.57 2.69 3.52 23.94 

f Theory 3.06 2.19 4.45 2.67 1.72 2.65 1.86 0.58 2.86 4.09 26.13 

Experiment 3.05 2.21 4.44 2.68 1.72 2.64 1.86 0.59 2.86 4.12 26.19 

g Theory 3.87 2.65 4.12 2.56 1.34 1.72 1.40 0.53 2.65 3.62 24.48 

Experiment 3.89 2.67 4.11 2.58 1.33 1.71 1.40 0.54 2.65 3.64 24.52 

h Theory 2.69 2.09 4.78 2.65 1.20 1.76 1.75 0.54 2.80 3.76 24.01 

Experiment 2.68 2.11 4.80 2.66 1.20 1.75 1.75 0.54 2.80 3.77 24.05 

Table 5 shows a comparison of theoretical and experimental auxiliary parameter estimation error 

results obtained via MAP estimation. The experimental results are based on the same 100,000 Monte 

Carlo simulation runs used to generate health parameter estimation results previously shown in Table 4. 

Those results based on the estimation of all 10 health parameters are obtained through Equation (20), 

while the results based on reduced order tuner estimation are obtained through Equation (23). Once again 

the theoretical and experimental estimation accuracy is found to exhibit good agreement. Auxiliary 

parameter estimates based on a MAP estimate of all 10 health parameters have a theoretical SSEE of 

363.86. Estimating the four q vectors optimized for health parameter estimation (vectors a, b, c, and d) 

have theoretical SSEE’s ranging from 366.13 to 371.16, a 0.6 to 2.0 percent increase in error compared to 

that obtained by estimating all 10 health parameters. This is comparable to the health parameter 

estimation accuracy found in Table 4. The four q vectors optimally selected for auxiliary parameter 

estimation (vectors e, f, g and h) result in theoretical SSEE’s ranging from 363.89 to 364.12, only 0.01 to 

0.07 percent more error than that found when estimating all 10 health parameters. These results reveal 

two significant findings. First, the q vectors optimally selected for auxiliary parameter estimation only 

come close to matching, not exceeding, the auxiliary parameter estimation accuracy obtained by 

estimating all 10 health parameters. Second, the q vectors optimally selected for health parameter 

estimation (vectors a, b, c, and d) appear to perform reasonably well in minimizing auxiliary parameter 

estimation errors. This last result is particularly significant because it implies that the application of a 
NASA/TM—2010-216077 16 

single estimator which satisfies the dual objective of estimating both health parameters and auxiliary 

parameters might be feasible. However, additional study beyond this single example is required to 

determine if this consistently holds true, particularly when considering estimation of different or 

expanded auxiliary parameter vectors. 

TABLE 5.—MAP ESTIMATOR AUXILIARY PARAMETER SQUARED ESTIMATION ERROR 

Estimated 

parameters 

Tuner 

vector 

Theory / 

experiment 

T40 

(°R) 

T50 

(°R) 

Fn 

(%) 

SmLPC 

(%) 

SSEE z( ˆ)

All 10 health 

parameters N/A Theory 193.49 167.15 0.96 2.26 363.86 

Experiment 195.61 167.63 0.96 2.27 366.47 

7 element 

reduced order 

tuner vector 

selected to 

minimize h error 

a Theory 193.55 170.10 0.97 2.27 366.90 

Experiment 195.63 170.18 0.97 2.28 369.06 

b Theory 194.73 173.19 0.97 2.27 371.16 

Experiment 196.56 173.09 0.97 2.27 372.89 

c Theory 193.52 172.54 0.97 2.27 369.30 

Experiment 195.59 172.40 0.97 2.28 371.24 

d Theory 194.04 168.85 0.96 2.27 366.13 

Experiment 195.94 168.97 0.97 2.28 368.16 

7 element 

reduced order 

tuner vector 

selected to 

minimize z error 

e Theory 193.49 167.16 0.96 2.28 363.89 

Experiment 195.61 167.66 0.97 2.29 366.52 

f Theory 193.49 167.15 0.97 2.52 364.12 

Experiment 195.61 167.63 0.97 2.52 366.74 

g Theory 193.49 167.15 1.10 2.27 364.01 

Experiment 195.61 167.66 1.10 2.28 366.65 

h Theory 193.49 167.15 0.97 2.31 363.93 

Experiment 195.60 167.65 0.98 2.32 366.55 

B. Kalman Filter Estimation Accuracy Comparison 

The results presented in the previous subsection demonstrate that an optimally selected reduced order 

tuner vector can be found that approaches the full order health parameter vector in terms of MAP 

estimator accuracy. However, readers are reminded that the actual objective for pursing this effort is to 

produce a reduced order vector that minimizes Kalman filter estimation errors when presented the 

underdetermined estimation problem of having fewer sensors than unknown health parameters. Towards 

that objective, the estimation accuracy of a Kalman filter using the previously selected reduced order 

tuner vectors is compared to the estimation accuracy obtained by applying the conventional approach of 

selecting a subset of health parameters to serve as Kalman filter tuners. These experimental estimation 

results are obtained through a Monte Carlo simulation analysis where the health parameter vectors are 

normally distributed in accordance with the covariance matrix, Ph. These health parameter test cases are 

provided as inputs to the C-MAPSS linear discrete state space model given in Equation (2), with an 

update rate of 15 ms. In order to provide the recursive Kalman filter estimator ample time to reach 

convergence each test case is 30 s in duration. The estimation errors are 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. A total of 375 30 s test 

cases are evaluated.

Table 6 shows a comparison of the experimental squared health parameter estimation errors obtained 

using a Kalman filter. This includes results obtained from applying the conventional approach of 

estimating a subset of health parameters (the seven health parameters denoted with an “*” in the table, 

which are identical to the subset of health parameters selected as tuners in Ref. 5), and from estimating 

the eight reduced order tuner vectors (vectors a through h) previously selected for MAP estimation. This 

table shows that the experimental health parameter SSEE obtained by estimating the subset of seven 

health parameters is 200.9. In this case, the estimated health parameters that comprise the subset are 

treated as tuning parameters that are adjusted by the Kalman filter to enable it to produce estimated 
NASA/TM—2010-216077 17 

outputs that match sensed engine outputs. As such the estimated values of these health parameters are no 

longer representative of the physical health parameters they are based on. In particular the tenth health 

parameter, LPT flow capacity, exhibits an extremely large difference between its actual value and its 

tuned value resulting in a percent squared error of 135.59 in this health parameter alone. The three health 

parameters that are not estimated, (i.e., ηFAN , γLPC, and ηLPT ), are treated as if their estimated values are 

zero. Thus they all exhibit squared estimation errors of approximately 4.0, which is the variance of their 

respective values based upon Ph. Based on Table 6 it can also be seen that the reduced order tuner vectors, 

q, selected to minimize MAP health parameter estimation errors (vectors a, b, c, and d) result in Kalman 

filter SSEE’s ranging from 17.55 to 38.28, a 91 to 81 percent improvement over Kalman estimation of a 

subset of health parameters. This improvement is significant and highly encouraging. The q vectors 

selected to minimize auxiliary parameter estimation errors (vectors e, f, g, and h) result in SSEE’s ranging 

from 39.68 to 152.21, an 80 to 24 percent improvement over Kalman estimation of a subset of health 

parameters. This too is a significant improvement, albeit not as large as that achieved by estimating 

vectors a through d. However, this is not particularly surprising as tuners e through h are optimized for 

auxiliary parameter estimation, not health parameter estimation. An observation that is noteworthy is the 

relatively large variation in estimation accuracy provided by the different reduced order vectors. While 

these vectors produce relatively consistent health parameter estimation accuracy when applied to MAP 

estimation, their individual Kalman estimation SSEE’s vary considerably. 

TABLE 6.—KALMAN FILTER HEALTH PARAMETER EXPERIMENTAL PERCENT SQUARED ESTIMATION ERRORS 

Estimated 

parameters 

Tuner 

vector 

ηFAN γFAN* ηLPC* γLPC ηHPC* γHPC* ηHPT* γHPT* ηLPT γLPT* ( )ˆ SSEE h

Subset of 7 health 

parameter tuners 

(denoted by *) 

N/A 3.44 7.07 12.23 4.22 0.40 0.63 33.12 0.20 4.00 135.5 200.90 

Tuner vector 

selected to 

minimize h error 

a 1.89 2.13 2.95 2.00 0.40 0.64 1.13 0.20 2.41 3.80 17.55 

b 5.54 6.20 4.76 6.28 0.39 0.44 2.01 0.19 5.00 7.47 38.28 

c 1.98 2.14 6.31 1.97 0.39 0.53 2.74 0.20 2.37 10.37 29.01 

d 1.82 1.86 2.49 1.74 0.39 0.53 2.48 0.20 4.21 9.41 25.11 

Tuner vector 

selected to 

minimize z error 

e 5.07 6.16 13.66 3.47 0.31 0.45 1.12 0.19 15.89 3.81 50.11 

f 20.16 42.28 18.00 37.45 0.29 0.32 2.47 0.19 21.41 9.65 152.21 

g 5.24 45.43 3.34 35.14 0.31 0.56 4.31 0.20 30.74 17.29 142.56 

h 2.88 5.91 6.49 3.74 0.37 0.53 1.56 0.18 11.47 6.55 39.68 

Table 7 shows the experimental auxiliary parameter estimation accuracy results obtained when 

implementing the Kalman filter using the q vectors. Applying the conventional Kalman filter approach of 

estimating a subset of health parameters results in an auxiliary parameter SSEE of 656.37. Applying 

Kalman filter estimation of the q vectors optimally selected for MAP health parameter estimation (vectors 

a, b, c, and d) results in auxiliary parameter SSEE’s ranging from 205.61 to 263.16, a 69 to 60 percent 

improvement over Kalman estimation of a subset of health parameter tuners. This is an encouraging 

finding, and demonstrates that a tuner vector can be applied which will significantly improve auxiliary 

parameter estimation accuracy. Applying Kalman filter estimation of the q vectors optimally selected for 

auxiliary parameter MAP estimation (vectors e, f, g, and h) results in auxiliary parameter SSEE’s ranging 

from 678.70 to 2114.59. This is actually 3 to 222 percent worse than simply choosing a subset of health 

parameters to serve as tuners. It is noted that these results are significantly worse than those obtained 

when applying vectors a, b, c, and d. These results reveal that estimating a reduced order q vector optimal 

for MAP estimation of auxiliary parameters does not guarantee good results when using a Kalman filter. 

In fact, this example shows that the Kalman filter auxiliary parameter estimation accuracy provided by 

these tuners is rather poor. 
NASA/TM—2010-216077 18 

TABLE 7.—KALMAN FILTER AUXILIARY PARAMETER 

EXPERIMENTAL SQUARED ESTIMATION ERRORS 

Estimated 

parameters 

tuner 

vector 

T40 

(°R) 

T50 

(°R) 

Fn 

(%) 

SmLPC 

(%) 

SSEE z( ˆ)

Subset of 7 health 

parameter tuners N/A 74.68 573.52 4.40 3.77 656.37 

Tuner vector selected 

to minimize h error 

a 77.12 125.81 1.00 1.68 205.61 

b 74.64 147.82 2.39 4.48 229.32 

c 77.64 182.70 1.10 1.72 263.16 

d 76.14 131.33 0.88 1.55 209.90 

Tuner vector selected 

to minimize z error 

e 75.61 800.83 2.94 3.47 882.86 

f 74.16 1221.61 17.79 27.05 1340.62 

g 74.48 1992.54 20.23 27.34 2114.59 

h 73.34 599.05 2.78 3.53 678.70 

IV. Discussion 

The process of selecting and evaluating the reduced order tuner vectors has revealed several 

significant findings. For MAP estimation applications, experimentally obtained mean squared estimation 

errors were found to be in agreement with theoretical predictions. This finding, combined with the fact 

that the MAP estimation accuracy of the reduced order tuner vectors approaches that obtained when 

estimating the entire health parameter vector verifies that the optimal iterative search routine is indeed 

performing as designed—it is returning a tuning vector optimal for MAP estimation. However, the 

ultimate intent is to apply these tuners to Kalman filter-based estimation. Implementation of the reduced 

order tuner vectors in a Kalman filter demonstrated that improved estimation accuracy, relative to the 

conventional approach of selecting a subset of health parameters to serve as tuners, could be obtained. 

However, this did not hold true in all cases. Specifically, the reduced order q vectors selected for MAP 

auxiliary parameter estimation resulted in degraded Kalman estimation performance. Furthermore, while 

there was not a significant difference in the estimation accuracy yielded by the individual tuners when 

used in MAP estimation, there was considerable difference between them when used in Kalman 

estimation. In summary, it has been shown that appropriately selected reduced order tuners can improve 

Kalman estimation accuracy over using a subset of health parameters, but tuners optimal for MAP 

estimation are not necessarily optimal for Kalman filter estimation applications. Based upon these 

findings, follow on work is being pursued to develop a methodology that will systematically select a 

reduced order tuner vector optimal for Kalman filter applications. This work uses an approach similar to 

the one presented in this paper for MAP estimator tuner selection. Namely, theoretical mean squared 

estimation error equations, consisting of squared bias and variance, are derived for the Kalman filter and 

applied within an iterative search routine designed to return an optimal reduced order tuner vector. Initial 

results have shown that reduced order tuner vectors can be found that consistently result in near optimal 

Kalman filter estimation accuracy. Furthermore, the Kalman filter estimation accuracy obtained by 

implementing these reduced order tuner vectors significantly exceeds that obtained by implementing the 

MAP-based reduced order tuner vectors as presented in this paper. The full results of this study will be 

presented in an upcoming conference paper to be published by the authors (Ref. 11). 

Conclusions 

A systematic approach to reduced order model tuning parameter selection for online engine 

performance estimation has been presented. This technique is specifically focused on the 

underdetermined aircraft engine parameter estimation problem where there are fewer sensor 
NASA/TM—2010-216077 19 

measurements than unknown health parameters that impact engine outputs. The technique creates and 

applies a linear transformation matrix, V*

, to select a reduced order vector of tuning parameters that are a 

linear combination of all health parameters, and of low enough dimension to be estimated by a Kalman 

filter. In order to facilitate reduced order tuner parameter selection, a multi-parameter iterative search was 

applied to optimally select V*

 to minimize the theoretical mean squared error of a maximum a posteriori

(MAP) estimator. Results have shown that while the transformation matrix returned by the optimization 

routine is not unique, the routine is effective in returning a transformation matrix that is near optimal for 

MAP estimation applications regardless of its initial starting guess of the matrix. This was theoretically 

predicted and experimentally validated through Monte Carlo simulation studies. It was also demonstrated 

that a reduced order tuner vector could be generated that performs reasonably well for the dual objective 

of minimizing both health and auxiliary parameter estimation errors. The estimation accuracy achieved 

when applying the reduced order tuner vectors for their intended application, Kalman filter-based 

estimation, was also evaluated experimentally. These studies revealed that the application of a reduced 

order tuner vector can yield significant improvements in Kalman filter estimation accuracy relative to the 

conventional approach of selecting a subset of health parameters to serve as tuners. However, there is no 

guarantee that a vector of reduced order tuners optimal for MAP estimation will be optimal when used for 

Kalman filter estimation. Based on these results it is concluded that the appropriate approach is to revise 

the methodology to perform the systematic selection of reduced order tuner parameters specifically for 

Kalman filter-based estimation applications. Follow on work towards that goal is currently being pursued. 

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, July 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. España, M.D., “Sensor Biases Effect on the Estimation Algorithm for Performance-Seeking 

Controllers,” J. of Propulsion and Power, Vol. 10, No. 4, 1994, pp. 527–532. 

5. Litt, J.S., “An Optimal Orthogonal Decomposition Method for Kalman Filter-Based Turbofan Engine 

Thrust Estimation,” J. of Engineering for Gas Turbines and Power, Vol. 130/011601–1, Jan. 2008. 

6. Doel, D.L., “An Assessment of Weighted-Least-Squares-Based Gas Path Analysis,” J. of Engineering 

for Gas Turbines and Power, Vol. 116, pp. 336–373, April 1994. 

7. Volponi, A.J., et al, “Gas Turbine Condition Monitoring and Fault Diagnostics”, Von Karman 

Institute for Fluid Dynamics, Lecture Series 2003–01, Jan. 2003. 

8. Simon, D., Optimal State Estimation, Kalman, H∞, and Nonlinear Approaches, John Wiley & Sons, 

Inc., Hoboken, NJ, 2006. 

9. Gelb, A., Applied Optimal Estimation, MIT Press, Cambridge, MA, 1974, pp. 104–105. 

10. Frederick, D.K., DeCastro, J.A., and Litt, J.S., “User’s Guide for the Commercial Modular AeroPropulsion System Simulation (C-MAPSS),” NASA/TM—2007-215026, Oct. 2007. 

11. Simon, D.L., and Garg, S., “Optimal Tuner Selection for Kalman Filter-Based Aircraft Engine 

Performance Estimation,” ASME Turbo Expo Conference, paper GT2009–59684, ASME, Orlando, 

FL, Jun. 8–12, 2009 (to be published). 
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14. ABSTRACT

A requirement for effective aircraft engine performance estimation is the ability to account for engine degradation, generally described in 

terms of unmeasurable health parameters such as efficiencies and flow capacities related to each major engine module. This paper presents a 

linear point design methodology for minimizing the degradation-induced error in model-based aircraft engine performance estimation 

applications. The technique specifically focuses on the underdetermined estimation problem, where there are more unknown health

parameters than available sensor measurements. A condition for Kalman filter-based estimation is that the number of health parameters 

estimated cannot exceed the number of sensed measurements. In this paper, the estimated health parameter vector will be replaced by a 

reduced order tuner vector whose dimension is equivalent to the sensed measurement vector. The reduced order tuner vector is 

systematically selected to minimize the theoretical mean squared estimation error of a maximum a posteriori estimator formulation. This 

paper derives theoretical estimation errors at steady-state operating conditions, and presents the tuner selection routine applied to minimize 

these values. Results from the application of the technique to an aircraft engine simulation are presented and compared to the estimation 

accuracy achieved through conventional maximum a posteriori and Kalman filter estimation approaches. Maximum a posteriori estimation 

results demonstrate that reduced order tuning parameter vectors can be found that approximate the accuracy of estimating all health 

parameters directly. Kalman filter estimation results based on the same reduced order tuning parameter vectors demonstrate that

significantly improved estimation accuracy can be achieved over the conventional approach of selecting a subset of health parameters to 

serve as the tuner vector. However, additional development is necessary to fully extend the methodology to Kalman filter-based estimation 

applications. 

15. SUBJECT TERMS

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

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