Jonathan S. Litt

U.S. Army Research Laboratory, Glenn Research Center, Cleveland, Ohio

An Optimal Orthogonal Decomposition

Method for Kalman Filter-Based Turbofan

Engine Thrust Estimation

NASA/TM—2005-213864

October 2005

ARL–TR–3487

GT2005–68808 U.S. ARMY

RESEARCH LABORATORY
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Jonathan S. Litt

U.S. Army Research Laboratory, Glenn Research Center, Cleveland, Ohio

An Optimal Orthogonal Decomposition

Method for Kalman Filter-Based Turbofan

Engine Thrust Estimation

NASA/TM—2005-213864

October 2005

National Aeronautics and

Space Administration

Glenn Research Center

Prepared for the

Turbo Expo 2005

sponsored by the American Society of Mechanical Engineers

Reno, Nevada, June 6–9, 2005

ARL–TR–3487

GT2005–68808 U.S. ARMY

RESEARCH LABORATORY
Acknowledgments

The author wishes to thank Tak Kobayashi for his inspirational discussions and insights that led to the

development of this work.

Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at http://gltrs.grc.nasa.gov
NASA/TM—2005-213864 1

An Optimal Orthogonal Decomposition Method for Kalman Filter-Based 

Turbofan Engine Thrust Estimation 

Jonathan S. Litt 

U.S. Army Research Laboratory 

Glenn Research Center 

Cleveland, Ohio 44135 

 

Abstract 

A new linear point design technique is presented for the 

determination of tuning parameters that enable the optimal 

estimation of unmeasured engine outputs such as thrust. The 

engine’s performance is affected by its level of degradation, 

generally described in terms of unmeasurable health parameters 

related to each major engine component. Accurate thrust 

reconstruction depends upon knowledge of these health 

parameters, but there are usually too few sensors to be able to 

estimate their values. In this new technique, a set of tuning 

parameters is determined which accounts for degradation by 

representing the overall effect of the larger set of health 

parameters as closely as possible in a least squares sense. The 

technique takes advantage of the properties of the singular 

value decomposition of a matrix to generate a tuning parameter 

vector of low enough dimension that it can be estimated by a 

Kalman filter. A concise design procedure to generate a tuning 

vector that specifically takes into account the variables of 

interest is presented. An example demonstrates the tuning 

parameters’ ability to facilitate matching of both measured and 

unmeasured engine outputs, as well as state variables. 

Additional properties of the formulation are shown to lend 

themselves well to diagnostics. 

Introduction 

In-flight estimation of unmeasurable turbofan engine 

outputs such as thrust is difficult because the values depend on 

the degradation level of the engine, which is often not known 

accurately. Degradation is generally defined in terms of offnominal values of health parameters such as efficiency and 

flow capacity, related to each major engine component. It is 

possible to estimate these health parameter deviations, given 

that there are at least as many sensors as parameters to be 

estimated. In standard engine installations, however, there are 

typically fewer sensors than health parameters, making accurate 

estimation impossible. An approach used in this situation is to 

select a subset of health parameters to estimate, assuming the 

others remain unchanged. If any of the unaccounted-for health 

parameters deviate from nominal, their effect will be captured 

to some extent in the estimated subset. As a result, the 

estimated values will no longer represent the true health 

parameter deviations. There are examples in the literature of a 

subset of health parameter “tuners” being used to reconstruct 

performance variables such as thrust (refs. 1 to 3), but this 

approach of health parameter subset selection is much better 

established as a diagnostic tool for gas path analysis (refs. 4 to 

6), where studies have determined which health parameters 

give good indications of certain faults for particular types of 

turbine engines (refs. 7 to 9). 

When a Kalman filter is used to estimate the subset of 

health parameters, the estimates of measured outputs will 

usually be good, i.e., the sensed outputs and the recreated 

values obtained using the health parameter estimates will 

match, even if the health parameter estimates themselves are 

inaccurate. However, good estimation of sensed outputs does 

not guarantee that the estimation of unmeasured outputs will be 

accurate. Since thrust is affected by the level of degradation, 

poor health parameter estimation can result in poor thrust 

reconstruction. It might be possible to determine a subset of 

health parameters that produces good thrust reconstruction even 

when all health parameters deviate, but this is a timeconsuming, empirical, trial-and-error process that gives no 

guarantee about the optimality of the result given the potential 

range of health parameter deviations and operating conditions. 

The main issue that affects the estimation accuracy is that 

the total influence of the health parameters needs to be 

approximated using fewer variables. The selection of a subset 

of health parameters is not a general approach to solving this 

problem as long as all health parameters may deviate. We 

would like to derive a set of tuning parameters (not necessarily 

a subset of health parameters) that is smaller in dimension than 

the set of health parameters, but retains as much information as 

possible from that original set. 

The following sections of this paper describe and formulate 

the thrust estimation problem mathematically, and then outline 

a new approach to the solution using singular value 

decomposition to obtain an optimal set of tuners. Following 

that, a concise design process is described that generates a set 

of optimal tuners at each operating (linearization) point, 

explicitly including all variables of interest; these tuners are 

incorporated into the Kalman filter to provide the optimal 

estimates. The technique is then demonstrated through an 

example. After the example, there is general discussion about 

the technique that covers other applications, such as its use as a 

diagnostic tool. Finally conclusions are presented. 

Nomenclature 

A, AAug, B, system matrices 

BAug, C, CAug, 

D, E, EAug, F, 

G, L, M, N 

e, w noise vectors 

FAN Fan 

FG Gross Thrust 

FN Net Thrust 
NASA/TM—2005-213864 2

HPC High Pressure Compressor 

HPT High Pressure Turbine 

LPC Low Pressure Compressor 

LPT Low Pressure Turbine 

LS Least Squares (subscript) 

p vector of health parameters 

P17 Bypass discharge pressure 

P25 HPC inlet pressure 

PS3 Combustor inlet static pressure 

Q, R noise covariance matrices 

q vector of optimal tuners 

SMW12 Fan stall margin 

SMW2 LPC stall margin 

SMW25 HPC stall margin 

T25 HPC inlet temperature 

T3 Combustor inlet temperature 

T49 LPT inlet temperature 

TMHS23 LPC metal temperature 

TMHS3 HPC metal temperature 

TMHS41 HPT nozzle metal temperature 

TMHS42 HPT metal temperature 

TMHS5 LPT metal temperature 

TMBRNC Combustor case metal temperature 

TMBRNL Combustor liner metal temperature 

U orthogonal matrix obtained using SVD 

U*

 optimal transformation matrix 

u vector of control inputs

ui ith column of U

V orthogonal matrix obtained using SVD 

V*

 optimal transformation matrix 

vi ith column of V

VBV Variable bleed valve 

VSV Variable stator vane 

WF36 Fuel flow 

WR2A Total FAN corrected flow 

x state vector 

xAug augmented state vector 

XN12 Low-pressure spool speed 

XN25 High-pressure spool speed 

z vector of auxiliary outputs 

δ vector of health parameter-induced shifts 

Σ singular value matrix obtained using SVD 

σi ith singular value 

Problem Development 

For a linear point design or a piece-wise linear state 

variable model, the equations of interest are 

x Ax Bu Lp e

y Cx Du Mp w

z Ex Fu Np

=+++

=+ + +

=++



(1) 

where 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 (unmeasurable or at least unmeasured) modelbased outputs. The vector p represents the engine health 

parameters, which induce shifts in other variables as the health 

parameters move away from their nominal values. The vector e

represents white process noise with covariance Q, and w

represents white measurement noise with covariance R. The 

matrices A, B, C, D, E, F, L, M, and N are of appropriate 

dimension. 

As the health parameter vector p is an unknown input to 

the system, we would like to be able to estimate it since it 

affects unmeasurable parameters such as thrust. The health 

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

modeled without dynamics. With this interpretation, we can 

represent equation (1) as 

[ ]

[ ]

00 0 Aug Aug Aug

Aug Aug

Aug Aug

x AL x B

ue A x B ue

p p

x

y C M Du w C x Du w

p

x

z E N Fu E x Fu

p

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ = + += + +

⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

⎡ ⎤

= + += + + ⎢ ⎥ ⎣ ⎦

⎡ ⎤

= += + ⎢ ⎥ ⎣ ⎦





. (2) 

This new system has at least as many eigenvalues at the origin 

as there are elements of p (the eigenvalues of AAug are the 

eigenvalues of A plus an additional dim(p) zeros due to the 

augmentation). Once the p vector is appended to the state 

vector, it may be directly estimated, provided that the 

realization in equation (2) is observable. 

Using this formulation, the number of health parameters 

that can be estimated is limited to the number of sensors, the 

dimension of y (ref. 10). This is easily seen by examining the 

observability criterion (ref. 11), which states that for 

observability, the matrix [(λI-AAug)

T CAugT

]

T

 must be full rank 

for each eigenvalue, λ, of AAug, i.e., 

{eig( )}, rank dim( ) Aug

Aug Aug

Aug

I A

A x

C

⎛ ⎞ ⎡ ⎤ λ −

∀λ ∈ = ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦

. 

Since the matrix AAug in equation (2) clearly has at least as 

many zero eigenvalues as there are health parameters, the 

observability criterion matrix reduces to T T T A C Aug Aug ⎡− ⎤ ⎣ ⎦ for 

each zero eigenvalue, implying that for observability, CAug must 

have at least as many rows (i.e., there must be at least as many 

sensors) as AAug has rows of zeros. 

Since there are usually fewer sensors than health 

parameters, the problem becomes one of choosing the best set 

of tuners for the application. This is addressed next. 

Problem Formulation 

In this work, the objective is to determine a tuning vector 

of low enough dimension to be estimated, that represents as 

much of the health parameter information as possible in a 

known way. This tuning vector should permit shifts in the 

variables of interest, caused by health parameter deviations, to 

be represented as closely as possible in a least squares sense. 
NASA/TM—2005-213864 3

We may define δ as the vector of shifts due to the health 

parameters, 

L

M p Gp

N

⎡ ⎤

⎢ ⎥ δ= = ⎢ ⎥

⎢ ⎥ ⎣ ⎦

. (3) 

Given that the number of elements in the tuning vector 

(tuners) is less than the number of health parameters, we know 

that, except for specific cases, the estimated tuners will not 

represent true health parameter values. Therefore, there is no 

reason to model the tuning vector as a subset of existing health 

parameters and, in fact, δ can be matched more closely without 

this constraint. The key is to find a matrix U*

 and tuning vector 

q that correspond to G and p in equation (3), of small enough 

dimension that the tuning vector can be estimated, and the 

product approximates δ in a least squares sense. That is, 

ˆ * ,,, n k δ= ≈δ= ∈ ∈ < Gp U q p q k n \ \ and 

ˆ ˆ ( )( ) T T J = δ−δ δ−δ =δ δ   (4) 

is minimized. 

For the tuning vector q to contain as much of the 

information contained in p as possible, p is mapped into q

through a transformation V*

 such that q=V*p where V* is full 

rank. Thus equation (4) becomes 

* *

** **

ˆ ˆ ( )( ) ( )( )

( )( )

T T

T T

J Gp U q Gp U q

p G UV G UV p

= δ−δ δ−δ = − −

=− − . (5) 

Since G may have rank as large as n (full rank) and the inner 

dimension of U*

V* is only k, U*V* will not be full rank. Thus 

equation (5) may be rewritten in vector 2-norm notation, 

2 * *

2 * * rank( )

min

UV k

J UV p

=

= δ− . (6) 

Another interpretation of equation (5) is that we want to 

approximate G by a lower rank matrix through the 

minimization of the Frobenius norm (the square root of the sum 

of the squares of each matrix element), i.e., 

* *

* * rank( )

min F UV kJ G UV == − . 

The solution to both of these minimization problems is obtained 

using Singular Value Decomposition (SVD) (ref. 12), and we 

shall show that SVD leads to an optimal solution of the form 

U*

V*.

Singular Value Decomposition 

The Singular Value Decomposition (SVD) of any m×n

matrix G, with m≥n, may be defined as 

T G UV = Σ (7) 

where U and VT

 are orthonormal square matrices, m m U × ∈ \ , 

n n V × ∈ \ , and Σ is a matrix of the same dimensions as G, with 

the upper portion a diagonal matrix of the singular values of G

and the lower portion all zeroes. That is 

1

2

0 0

0

0

0 0

0 00

n

⎡σ ⎤ ⎢ ⎥ σ ⎢ ⎥

Σ = ⎢ ⎥ ⎢ ⎥

σ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

"

#

# %

"

"

where σ1≥ σ2≥ ...≥ σn≥ 0, UUT

=Im, and VVT=In. 

Some algebraic manipulation shows that equation (7) is 

equivalent to 

1

n T

iii i

G uv == σ ∑ (8) 

where ui is the ith column of U and vi is the ith column of V

( T

i v the ith row of VT

). Equation (8) is called the rank one 

decomposition of G because G is represented by the sum of 

rank one matrices. 

If the rank of an m×n matrix H is k<n, its SVD is 

11 11

1 1

0

0 0

0

0 0

T k

kk m kk n

k k

T k

kk m k k n

T

k kk

H u uu u v vv v

U V

Uu u Vv v

U V

+ +

+ +

⎡ ⎤ Σ = ⎡ ⎤ ⎡ ⎤ ⎣ ⎦⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤ Σ = ⎡ ⎤ ⎡ ⎤ ⎣ ⎦⎢ ⎥⎣ ⎦ ⎣ ⎦

= Σ

"" ..."

	
 	


" " (9) 

where Uk consists of the first k columns of U, Vk consists of the 

first k columns of V, and Σk is the upper left diagonal k×k block 

of Σ containing the non-zero singular values. 

The linear least squares problem 

2

2 rank( )

min

H k

Hp

=

δ −

can be solved using equation (9) as 

†

1

1 1

1

0

0 0

LS

T k

kk m k k n

T

kk k

p H

Vv v U u u

V U

−

+ +

−

= δ

⎡ ⎤ Σ = ⎡ ⎤⎢ ⎥⎡ ⎤ δ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

=Σ δ

" " (10) 

where the pseudo-inverse of the non-square, non-full rank, 

block diagonal singular value matrix Σ in equation (9) is that 

matrix’s transpose with the non-zero block inverted. 
NASA/TM—2005-213864 4

Additionally, any full rank m×n matrix G, with m≥n, is 

most closely approximated in the Frobenius norm sense by a 

matrix H of rank k<n, i.e., 

rank( )

min F F H kGH GH

=

−= − , 

when H is given by 

 

1

k T T

iii k k k i

H uv U V ==σ = Σ ∑ , (11) 

the first k terms of equation (8). 

The Frobenius norm is equal to the square root of the sum 

of the squares of the singular values of the matrix, (which is 

equal to the square root of the sum of the squares of each 

matrix element) so 2 2 G H− = σ + +σ F k n +1 " is minimal 

since the singular values are ordered from largest to smallest 

and the difference consists of only the smallest ones. 

Thus, equations (9) and (11) are the same, and both imply 

equation (10). Moreover, the solution has the structure of the 

original U*

V* formulation proposed in equation (5), even 

though this structure was not assumed in the derivation of these 

results, it simply falls out as a consequence of the SVD 

approach (see reference 12 or a similar text for derivations, 

proofs, definitions of norms, etc.). 

This demonstrates that using the SVD to obtain the U*

V*

formulation will indeed give the optimal approximation to δ in 

a least squares sense. The procedure for coming up with the 

optimal matrices and their use will be described next. 

Design Procedure 

The design procedure follows directly from the above 

derivation. 

1. For each linearization point, create G by stacking L, M, and 

N as in equation (3). The L and M matrices must be 

included for the accuracy of the Kalman filter estimates, 

the variables used to create N are left to the user’s 

discretion. 

2. Compute the magnitude of each row gi of G as gigi

T

 for 

scaling. Premultiply G by 

1

1 1 0 0

0 0

0 0

T

T

m m

g g

W

g g

−

⎡ ⎤

⎢ ⎥

=

⎣ ⎦

% . 

This is important because there may be several orders of 

magnitude difference between the values in different rows 

of G, and, since the SVD minimizes the difference between 

the elements of G and U*

V* in a least squares sense 

(Frobenius norm), all rows should be about the same 

magnitude to prevent the large matrix elements from 

dominating. Recognizing, however, that some variables 

need to be estimated more accurately than others, 

additional weights can be incorporated into the diagonal 

elements of W to account for this relative importance. This 

is all done after any other scaling or balancing used to 

develop the initial realization (eq. 1). 

3. Compute the SVD as in equation (7) from the scaled G, 

WG = (WU)ΣVT

. 

4. Unscale WU by premultiplying by W -1. This brings the 

magnitude of each row back to its appropriate level and 

makes it compatible with the rest of the realization. 

5. Generate U*

 and V* by selecting only the k most significant 

terms of equation (8). At most, k will equal the number of 

sensors less the number of zero eigenvalues in the original 

A matrix. Incorporate the k×k singular value matrix into U*

such that * U =Uk k Σ and * T V V= k . Thus 

( )

*

** * *

*

ˆ

ˆ

ˆ

ˆ

T

kk k

L

M

N

L

G M GU V

N

U L

UV U V M

U N

⎡ ⎤

⎢ ⎥ = ≈= Σ ⎢ ⎥

⎢ ⎥ ⎣ ⎦

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

where * * ˆ UV L L L = ≈ , * * ˆ UV M M M = ≈ , and 

* * ˆ UV N N N = ≈ . Thus we have generated a common right 

factor for the approximation of all three matrices L, M, and 

N. 

6. Using the rows of U*

 that correspond to the original L, M, 

and N, set up the Kalman filter using the new augmented 

system equations 

*

*

*

0 0 0

L

M

N

x xB A U

u e

q q

x

y C U Du w

q

x

z E U Fu

q

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

⎡ ⎤

= ++ ⎡ ⎤ ⎣ ⎦⎢ ⎥ ⎣ ⎦

⎡ ⎤

= + ⎡ ⎤ ⎣ ⎦⎢ ⎥ ⎣ ⎦





. (12) 

This system will generally be observable if q is of small 

enough dimension. 

Turbofan Engine Example 

A large commercial turbofan engine model linearized at a 

cruise operating point is used to evaluate the concept. The 

model has nine state variables, 10 health parameters, three 

control inputs, all shown in table 1, and seven sensors, shown 

in table 2. The auxiliary outputs of interest are shown in table 3. 

The linear model is used as the “truth” model for this example. 

It must be noted that for this work, the P25 measurement in the 

conventional sensor suite is replaced by P17. Without this 

substitution, thrust estimation is poor. This indicates that the 
NASA/TM—2005-213864 5

ability to estimate unmeasured outputs is strongly influenced by 

sensor location. The weights on net and gross thrust (step 2 in 

the design procedure) are doubled because of the relative 

importance of these auxiliary variables. The model is run openloop, so all control inputs remain at 0, i.e., they do not deviate 

from the trim value for the linear model and no actuator bias is 

present. All 10 health parameters are shifted randomly up to 

3 percent in the direction of degradation, and these shifts result 

in shifts in the measured variables. The linear truth model and 

Kalman filter estimator use the same A, B, C, and D matrices, 

but the truth model has all 10 health parameters entering 

through the L, M, and N matrices. The Kalman filter is set up 

using the system in equation (12) with a seven-element q

vector. 

To demonstrate the optimality of the SVD-based tuner 

approach, it was compared to the Kalman filter approach that 

uses health parameter-based tuners. To make the comparison as 

meaningful as possible, the health parameter tuners were 

chosen specifically to be compatible with the metric used by 

the SVD approach (eq. 4). Structurally the Kalman filter is 

designed to estimate seven selected health parameter tuners 

(while assuming the others remain constant) and the 

corresponding columns of L, M, and N are retained. An optimal 

selection process based on orthogonal least squares (OLS) 

(refs. 13 and 14) is used to determine which health parameters 

to estimate. Briefly, the scaled G matrix (from step 2 in the 

design procedure, the same one as used for the SVD-based 

tuners) is multiplied by a random health parameter vector p

whose 10 elements take on values from 0 to 3 percent in the 

appropriate direction (positive for the HPT and LPT flow 

capacities, negative for all others). OLS is used to choose the 

seven columns of G that best align with the shift vector δ = Gp. 

This is repeated thousands of times for different random p

vectors and the results are tallied. The individual columns that 

are selected most often determine the health parameters that on 

average best capture the degradation-induced shifts. The most 

often selected columns are indicated by asterisks (*) in table 1.

Of the 120 (10 choose 7) possible sets of columns, this exact set 

was selected about 9 percent of the time. 

TABLE 1.—STATE VARIABLES, HEALTH PARAMETERS, 

AND ACTUATORS 

State Variables Health Parameters Actuators 

XN12 FAN efficiency‡ WF36 

XN25 FAN flow capacity*‡§ VBV 

TMHS23 LPC efficiency*‡ VSV 

TMHS3 LPC flow capacity‡§ 

TMBRNL HPC efficiency*‡ 

TMBRNC HPC flow capacity*‡§ 

TMHS41 HPT efficiency*‡§ 

TMHS42 HPT flow capacity*‡§ 

TMHS5 LPT efficiency 

 LPT flow capacity*§ 

Results

To demonstrate the ability of the optimal tuners to 

approximate a variety of health parameter-induced shifts well, 

all 10 health parameters are shifted randomly. The results 

TABLE 2.—SENSOR SETS AND SENSOR STANDARD 

DEVIATION (STD. DEV. AS % OF STEADY-STATE 

VALUES AT FULL POWER) 

Conventional Sensor 

Set 

New Sensor Set Standard 

Deviation (%) 

XN12 XN12 0.25 

XN25 XN25 0.25 

-------- P17 0.50 

P25 ------- 0.50 

T25 T25 0.75 

PS3 PS3 0.50 

T3 T3 0.75 

T49 T49 0.75 

TABLE 3.—AUXILIARY OUTPUTS 

TO BE ESTIMATED IN FLIGHT 

Auxiliary Outputs 

WR2A 

FN 

FG 

SMW12 

SMW2 

SMW25 

shown are for a representative random sequence, many 

simultaneous random shifts were performed at this operating 

point with similar results. Figure 1 shows the state variable 

estimation over a time sequence containing 10 such sets of 

random shifts. Generally the variables are reconstructed 

accurately with the exception of TMBRNC, combustor case 

metal temperature, which does not seem to have much of an 

effect on any other variables.

Figure 2 shows the output variables for the same sequence 

of health parameter shifts. These also demonstrate good 

matching, which is to be expected when using a Kalman filter. 

Figure 3 shows the auxiliary parameters. For all cases of 

health parameter shifts both estimators performed well. 

Because of the large gains on some of the auxiliary variables, 

the estimates needed to be low-pass filtered to attenuate the 

magnified noise level. The estimation performance on net and 

gross thrust and two of the three stall margins, as well as total 

fan corrected flow was consistently accurate. The only auxiliary 

parameter whose estimate was consistently poor is LPC stall 

margin, SMW2. Adjustment of the weighting matrix W in 

step 2 of the design procedure made little difference in the 

results. 

In an effort to obtain a better LPC stall margin estimate, 

new U*

 and V* were determined, this time using SMW2 as the 

only auxiliary parameter, i.e., N in equation (3) has only one 

row. If SMW2 requires different information than the other 

variables for accurate estimation, generating a new G matrix 

would result in a different SVD (the SVD of a matrix is unique) 

possibly prioritizing the tuners such that SMW2 could be 

reconstructed. Even with the new G matrix though, the 

estimation of SMW2 is poor. This may be due to the L and M

matrices dominating the generation of the U*

 and V* matrices, 

but it is more likely a sensor placement issue (in fact, replacing 

P17 with P25 greatly improves the estimation of SMW2, even 

without additional weighting). 
NASA/TM—2005-213864 6

0 100 200 −60

−50

−40

−30

−20

−10

0

10

XN12

0 100 200 −150

−100

−50

0

50

100

XN25

0 100 200 −2

−1.5

−1

−0.5

0

0.5

1

TMHS23

0 100 200 −8

−6

−4

−2

0

2

TMHS3

0 100 200 −20

0

20

40

60

80

TMBRNL

0 100 200 −120

−100

−80

−60

−40

−20

0

20

TMBRNC

0 100 200 −10

0

10

20

30

40

50

60

Time (sec)

TMHS41

0 100 200 0

20

40

60

80

100

Time (sec)

TMHS42

0 100 200 −20

0

20

40

60

80

100

Time (sec)

TMHS5

Health Parameter−based Estimate

SVD−based Estimate

Actual

Figure 1.—State variables: scaled actual and estimates. 

0 20 40 60 80 100 120 140 160 180 200 −100

−50

0

50

XN12

0 20 40 60 80 100 120 140 160 180 200 −200

0

200

XN25

0 20 40 60 80 100 120 140 160 180 200 −0.5

0

0.5

P17

0 20 40 60 80 100 120 140 160 180 200 −20

0

20

T25

0 20 40 60 80 100 120 140 160 180 200 −50

0

50

100

PS3

0 20 40 60 80 100 120 140 160 180 200 −100

0

100

200

T3

0 20 40 60 80 100 120 140 160 180 200 −20

−10

0

10

Time (sec)

T49

SVD−based Estimate

Health Parameter−based Estimate

Actual

Figure 2.—Output parameters: scaled actual and estimates. 

0 20 40 60 80 100 120 140 160 180 200 −100

−50

0

WR2A

0 20 40 60 80 100 120 140 160 180 200 −1000

−500

0

500

FN

0 20 40 60 80 100 120 140 160 180 200 −2000

−1000

0

FG

0 20 40 60 80 100 120 140 160 180 200 −1

0

1

2

SMW12

0 20 40 60 80 100 120 140 160 180 200 −5

0

5

SMW2

0 20 40 60 80 100 120 140 160 180 200 −4

−2

0

2

Time (sec)

SMW25

Health Parameter−based Estimate

SVD−based Estimate

Actual

Figure 3.—Auxiliary parameters: scaled actual and filtered 

estimates. 

Likewise, a slight improvement in net and gross thrust 

estimation was achieved by the SVD-based tuners when the 

auxiliary parameters consisted of only those two variables. 

In order to compare the performance of the tuners 

statistically, runs containing a sequence of 400 simultaneous 

random shifts of all 10 health parameters were performed. After 

each set of shifts the engine simulation was allowed to reach 

steady state and data from each variable were gathered and 

averaged. This information was used to compute an average 

percent error for each variable in steady state over the entire 

sequence. The quantity was computed as 

400

1

400

1

100 %

400

100

400

i

i

i i

i

i i

i

=

=

= ∑

≈ ∑

actual deviation -estimated deviation

error

trim value

actual -estimate

actual

. 

The comparison of the percent estimation error of the state 

variables, outputs, and auxiliary outputs based on this run for 

the two sets of tuners are presented in table 4, table 5, and 

table 6, respectively. It can be seen that both sets of tuners 

produce similar results in most cases, but where the error is 

relatively large (TMBNC in table 4 and SMW2 and SMW25 in 

table 6), the SVD-based tuners are significantly better. Results 

for the estimated degradation-induced shifts in each variable, 

δ = Gp (eq. 3), were similar. The value of the metric (eq. 4) for 

the scaled shifts in the two cases was 2.187 for the SVD-based 
NASA/TM—2005-213864 7

tuners and 6.016 for the health parameter-based tuners. It must 

be noted that minimization of the metric does not in itself 

guarantee minimization of the total least squares error without 

appropriate weighting in step 2 of the design procedure. In this 

example, the SVD-based tuners give a value of 42.6865 for the 

sum of the squared percent estimation error of the variables in 

table 4, table 5, and table 6, compared to 95.4661 for the health 

parameter-based tuners. 

TABLE 4.—PERCENT ERROR OF STATE 

VARIABLE ESTIMATES 

State Variables SVD-based Tuners Health Parameterbased Tuners 

XN12 0.0318 0.0317 

XN25 0.0269 0.0267 

TMHS23 0.0129 0.0130 

TMHS3 0.0363 0.0363 

TMBRNL 0.0278 0.0306 

TMBRNC 0.4083 1.1256 

TMHS41 0.0300 0.0313 

TMHS42 0.0267 0.0274 

TMHS5 0.3208 0.1942 

TABLE 5.—PERCENT ERROR OF OUTPUT ESTIMATES 

Outputs SVD-based Tuners Health Parameterbased Tuners

XN12 0.0317 0.0318 

XN25 0.0286 0.0285 

P17 0.0634 0.0633 

T25 0.0859 0.0857 

PS3 0.1027 0.1027 

T3 0.0990 0.0990 

T49 0.0748 0.0748 

TABLE 6.—PERCENT ERROR OF AUXILIARY 

OUTPUT ESTIMATES 

Auxiliary 

Outputs SVD-based Tuners Health Parameterbased Tuners

WR2A 0.1081 0.0787 

FN 0.2704 0.2745 

FG 0.1225 0.1169 

SMW12 0.4348 0.3469 

SMW2 6.1605 9.3142 

SMW25 2.0326 2.6733 

Additionally, estimation performance was evaluated for 

comparable sequences of 400 random simultaneous shifts in all 

10 health parameters with six sensors and with eight sensors. 

For the eight-sensor case, P25 was added to the “new sensor 

set” of seven shown in table 2. The tuners used were the first 

eight health parameters, indicated by a (‡) in table 1, (LPT 

efficiency and flow capacity were not used), selected using 

OLS. As expected, the estimate of SMW2, which was poor 

with only seven sensors, was much more accurate, as was the 

estimate of TMBRNC. For eight sensors, the value of the 

metric for the SVD-based tuners was 2.0883 and 6.5243 for the 

health parameter-based tuners; this is essentially the same as for 

the seven sensor case, with the difference being attributed to the 

random factors of sensor noise and the health parameter shifts. 

The sum of the squared percent estimation error for the SVDbased tuners was 6.0438 compared to 7.1943 for the health 

parameter-based tuners. For the six sensor case, the T3 sensor 

was removed from the “new sensor set” shown in table 2. The 

tuners used were the flow capacities of each component plus 

HPT efficiency, indicated by (§) in table 1, which were selected 

using engineering judgment. It was possible to get reasonably 

good matching of the auxiliary outputs with this set of tuners, 

but no set investigated produced accurate reconstruction of the 

state variables. The value of the metric for the SVD-based 

tuners in the six-sensor case was 9.4068 and 12.4667 for the 

health parameter-based tuners. The big increase in the metric 

values over the cases with more sensors indicates that 

significant information has been lost by approximating the G

matrix by a rank six matrix. The relative size of the dropped 

singular values gives an indication of this. In this case, the sum 

of the squared percent estimation error for the SVD-based 

tuners was 53.8702 compared to 78.1314 for the health 

parameter-based tuners.

Further Application of Tuners 

The example shows that the optimal tuner selection method 

obtained using SVD provides accurate estimates of most 

variables of interest, including auxiliary parameters such as 

thrust. The SVD formulation has several attractive properties 

that make its use in this application highly desirable beyond 

what has already been discussed. In addition to providing the 

least squares solution to the approximation of the shift vector δ, 

other features include 1) the approximation error to the matrix 

G is known; 2) fault detection is possible through the tuners’ 

ability to capture sudden shifts, even if the tuners do not 

correspond to individual health parameters; and 3) the 

orthogonality of the V matrix suggests a way to isolate a 

component fault, while at the same time some of the true health 

parameter shifts may be able to be reconstructed accurately, in 

some cases. The following sections provide more detail on each 

of these topics. 

Approximation Error of G

Since 

11 1 1

ˆ nk n n TT T T

iii iii iii iii i i ik ik

G uv uv uv G uv = = =+ =+=σ =σ + σ =+ σ ∑∑ ∑ ∑ (13)

the approximation error of G is the last term of equation (13), 

namely 

1

ˆ n T

iii i k

G G G uv = +=−= σ ∑  . 

The error in the approximation of G depends on both the 

number of terms dropped in the approximation and on the size 

of the singular value in each dropped term. (Singular values 

give an indication of the linear independence of the columns of 

G, so if the singular values in the dropped terms are very small, 

not much information is lost, but on the other hand, if the 

columns of G are not linearly independent, the accurate 

estimation of the health parameters is impossible, regardless of 

the number of sensors.) The fact that the error is known allows, 
NASA/TM—2005-213864 8

for instance, the calculation of accuracy bounds on the health 

parameter-induced shifts, δ, and on the estimated variables 

themselves. 

The error introduced into the system equations (eq. 12) 

through the approximation of G represents modeling error. The 

process noise term, e, in the state equation, is used at least 

partially to represent model uncertainty, so a level can be 

established by multiplying the known approximation error G

by a range of degradation values p. Since the level of 

degradation is unknown a priori, it can be considered a random 

variable and thus so can Gp . The covariance of this product 

could be used to help determine the process noise covariance

matrix, Q, for the Kalman filter, which is often considered a 

design parameter. 

Fault Detection 

Even though the tuners do not represent individual health 

parameters, component faults can be detected by sudden shifts. 

The objective of fault detection in this context is to be able to 

distinguish an abrupt shift representing a fault, from the general 

“background level” of degradation caused by use and wear, 

represented by slowly varying q during normal operation. Since 

q = V*

p, the shifts are mapped into q as long as they are not 

limited to the null space of V*

, that is, as long as they do not lie 

completely within the span of 1

T

k n v v + ⎡ ⎤ ⎣ ⎦ " . However, since 

this space contains the least significant directions of possible 

shift (since only the smallest singular values are dropped) and 

the tuners will, in general, cut across rather than align with the 

influence of each health parameter, additional shifts due to 

faults ∆q=V*

∆p will very likely be non-zero for any significant 

shift in an element of p. 

TABLE 7.—HEALTH PARAMETER SHIFTS 

REPRESENTING COMPONENT FAULTS 

Time (sec) Health Parameters Shift 

0-20 FAN efficiency -3% 

20-40 FAN flow capacity -3% 

40-60 LPC efficiency -3% 

60-80 LPC flow capacity -3% 

80-100 HPC efficiency -3% 

100-120 HPC flow capacity -3% 

120-140 HPT efficiency -3% 

140-160 HPT flow capacity 3% 

160-180 LPT efficiency -3% 

180-200 LPT flow capacity 3% 

To demonstrate this, component faults of 3 percent 

magnitude were injected, sustained for 20 seconds, and 

removed, in succession through each of the 10 health 

parameters (efficiency then flow capacity for each component 

in the order shown in table 1), as shown in table 7. Figure 4 

shows the estimated tuners during this sequence of shifts. As 

the health parameters shift, the tuners generally shift as well, so 

changes can be determined and thus faults can be detected. The 

fan efficiency fault is hard to determine as the tuner shifts that 

correspond to it are small, but several tuners clearly shift in 

Figure 4.—Estimated q signals and V*p, demonstrating fault 

detection. 

response to a fault in fan flow capacity. When looking at all 

tuners together, it is reasonably clear in most cases when a fault 

has occurred. Different sets of tuners result from different 

weightings of the G matrix (step 2 of the design procedure), 

and it is possible to have at least one of the SVD-based tuners 

align well with an actual health parameter. 

One thing that is clear from figure 4 is that in this example,

the elements of q are tracked accurately, and this is a 

consequence of the procedure used to select them. This is not 

the case in general when the tuners are modeled as actual health 

parameters. Figure 5 shows the Kalman filter estimates of the 

health parameter-based tuners with the same input sequence as 

in figure 4. In figure 5, the tuners clearly indicate abrupt shifts 

due to faults, but often do not match the health parameters they 

represent because of the influence of unmodeled health 

parameters. Techniques exist to select the health parameters for 

estimation that are least affected by the umodeled health 

parameters and sensor uncertainty (refs. 15 to 18). From a 

diagnostics point of view this is fine, but there is no guarantee 

that those health parameters will facilitate good reconstruction 

of auxiliary parameters such as thrust. 

The ability to estimate the state and auxiliary variables in 

the presence of actuator bias is not addressed here. From 

equation (1) it is clear that the effect of actuator bias is no 

different than that of health parameter shifts, so as long as the 

tuners can capture this effect, the results should not deteriorate 

significantly. As with component degradation and faults, 

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

q1

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

q2

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

q3

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

q4

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

q5

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

q6

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

Time (sec)

q7

Estimate

Actual
NASA/TM—2005-213864 9

Figure 5.—Estimated health parameter-based tuners. 

constant actuators biases should be generally represented by the 

elements of q, and any sudden actuator shifts due to bias should 

be captured as shifts in the elements of q. 

Fault Isolation 

By construction, the vector of estimated tuners * q Vp ˆ ≈ , 

(since G is not perfectly recreated by U*

V*, the estimated q will 

not exactly equal V*

p, even in a noise-free environment) and 

the columns of V are orthonormal. Of course V is a square n×n

matrix and V*

 is k×n, but if k is close to n, in some cases V*TV*

≈In. The implication of this is that * ˆ T Vq p ≈ , so even if the 

magnitude of the shift is not determined accurately, the shift 

might be able to be isolated to the affected health parameters 

(or at least the affected component) by identifying those 

elements of the * ˆ T V q vector with the largest shifts. Analysis of 

the V*

 matrix will determine the feasibility of the approach on a 

case by case basis. 

It is interesting to note that the * ˆ T Vq p ≈ approximation is 

actually the least squares estimate of p from equation (3). This 

is clear from the fact that 

*

†1 *

ˆ ˆ ˆ

ˆˆ ˆ

T

k k

T

LS k k k

U q U q Gp U V p

p V U U q Vq V q −

δ= = Σ ≈ = Σ

⇒ =Σ Σ = =

(14) 

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

FAN Eff

Fault Signal

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

FAN Flow

Fault Signal

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

LPC Eff

Fault Signal

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

LPC Flow

Fault Signal

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

HPC Eff

Fault Signal

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

HPC Flow

Fault Signal

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

HPT Eff

Fault Signal

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

HPT Flow

Fault Signal

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

Time (sec)

LPT Eff

Fault Signal

0 20 40 60 80 100 120 140 160 180 200 −0.05

0

0.05

Time (sec)

LPT Flow

Fault Signal

Estimate

Actual

Figure 6.—Fault signal * ˆ ˆ T pVq ≈ and actual health parameter 

shift, demonstrating fault isolation. 

where † Σ is the transpose of Σ with the square singular value 

block inverted. Note that once U is unscaled in step 4 of the 

design procedure it is no longer orthonormal, so U -1 rather than 

UT

 is used in equation (14). 

Figure 6 shows the recreated set of health parameters using 

* ˆ ˆ T pVq ≈ . It is clear that shifts in certain variables are easily 

isolated while others are mapped back as shifts smeared across 

more than one health parameter. For this example, it seems that 

health parameter shifts in the high-pressure shaft components 

(HPC and HPT) are easier to isolate than those on the lowpressure shaft. For instance, the plots for FAN efficiency and 

LPT efficiency are very similar, meaning that it would be hard 

for a diagnostic system to distinguish between them. Efficiency 

and flow capacity of the LPC are clearly interrelated, with one 

being nearly the opposite of the other; for cases such as this, 

one might constrain the problem by assuming that efficiency 

and flow capacity corresponding to a single component shift 

together, somewhat simplifying the isolation logic. The issue 

always exists of simultaneous shifts in the health parameters of 

multiple components confounding the problem. 

It is interesting to compare the results in figure 6 with 

those in figure 5, which shows the Kalman filter estimates of 

seven health parameter-based tuners. The quality of the 

reconstruction is similar in that the same three parameters 

(FAN flow capacity, HPC efficiency, and HPT flow capacity) 

0 20 40 60 80 100 120 140 160 180 200 −0.1

0

0.1

FAN flow

0 20 40 60 80 100 120 140 160 180 200 −0.1

0

0.1

LPC eff

0 20 40 60 80 100 120 140 160 180 200 −0.1

0

0.1

HPC Eff

0 20 40 60 80 100 120 140 160 180 200 −0.1

0

0.1

HPC Flow

0 20 40 60 80 100 120 140 160 180 200 −0.1

0

0.1

HPT eff

0 20 40 60 80 100 120 140 160 180 200 −0.1

0

0.1

HPT Flow

0 20 40 60 80 100 120 140 160 180 200 −0.1

0

0.1

Time (sec)

LPT Flow

Estimate

Actual
NASA/TM—2005-213864 10

were estimated accurately in both cases and those estimates 

corrupted by unmodeled shifts in health parameters were the 

same in both cases, but the approach demonstrated in figure 6 

has the benefit that the remaining three health parameters are 

reconstructed as well, albeit corrupted by shifts in other 

variables. One observation is that the reconstruction of each of 

these remaining health parameters (FAN efficiency, LPC flow 

capacity, and LPT efficiency) is strongly related to the 

reconstruction of some other health parameter, underscoring the 

lack of observability. Still, a diagnostic system might be able to 

use this information to isolate faults. 

If actuator bias, ubias, were to be added on top of the health 

parameter deviations, it would certainly make the fault isolation 

task more challenging. An approach to address this is to define 

G such that 

bias bias

L B

p p MD G

u u

N F

⎡ ⎤⎡⎤ ⎡⎤ ⎢ ⎥ δ= = ⎢⎥ ⎢⎥ ⎢ ⎥⎣⎦ ⎣⎦ ⎢ ⎥ ⎣ ⎦

. 

This explicitly accounts for actuator bias, but the increase in the 

number of columns of G means that the U*

V* approximation 

will be less accurate, since the size of q is fixed. A consequence 

of this is that V*TV*

 is further from In (in the sense of the 

Frobenius norm) and thus the [pT

 ubiasT]

T

≈V*Tq approximation 

is not as good as when fewer faults are accounted for. 

Conclusions 

A new optimal linear point design approach to determine 

engine deterioration tuning parameters that enables the accurate 

reconstruction of unmeasured engine outputs was presented. It 

was designed specifically for the case where there are too few 

sensors to estimate the true engine health parameters. The 

tuning parameters are determined using singular value 

decomposition, which was shown to generate the best 

approximation to the influence of the full set of health 

parameters in a least squares sense, using a set small enough to 

be estimated. A concise design procedure to generate the tuners 

was described. An example demonstrated that the method 

worked well in reconstructing the unmeasured parameters of 

interest. It was shown to perform better against its metric than a 

health parameter-based Kalman filter specifically designed for 

the metric used, and it will generally result in a smaller total 

squared estimation error with appropriate weighting 

incorporated into the metric. This highlights the freedom 

inherent in the SVD-based approach as a result of the 

weighting, while the selection of possible subsets of health 

parameters for tuners is limited. Still, the choice of weights in 

the SVD approach is not arbitrary since the variables interact, 

and allowing poor state estimation, for instance, will ultimately 

hurt the accuracy of the auxiliary variables. It was also shown 

that the ability to estimate well is sensitive to sensor placement. 

As a side benefit of the estimator formulation process, the 

tuners have desirable properties for diagnostics because of the 

orthogonal decomposition. In particular, the reconstruction of 

the complete set of health parameters compares favorably to the 

estimation of only a subset by a health parameter-based Kalman 

filter. The effect of actuator bias was not investigated with 

respect to the optimal tuners determined for health parameter 

degradation, however a consistent approach to account for it 

explicitly was proposed. 

Areas of future research involve the generation of bounds 

on the accuracy of the low rank matrix approximation and their 

effect on estimation, the continuity of the SVD for piece-wise 

linear implementations, and further investigation into the fault 

detection and isolation properties of the SVD formulation. 

References 

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

1989, “Estimating In-Flight Engine Performance 

Variations Using Kalman Filter Concepts,” AIAA–89–

2584. 

2. Turevskiy, A., Meisner, R., Luppold, R.H., Kern, R.A., and 

Fuller, J.W., 2002, “A Model-Based Controller for 

Commercial Aero Gas Turbines,” ASME Paper GT2002–

30041. 

3. Kobayashi, T., Simon, D.L., Litt, J.S., 2005, “Application 

of a Constant Gain Extended Kalman Filter for In-Flight 

Estimation of Aircraft Engine Performance Parameters,” 

ASME Paper GT2005–68494. 

4. Brotherton, T., Volponi A., Luppold, R., Simon, D.L., 

2003, “eSTORM: Enhanced Self Tuning On-board Realtime Engine Model,” Proceedings of the 2003 IEEE 

Aerospace Conference. 

5. Kobayashi, T., Simon, D.L., 2003, “Application of a Bank 

of Kalman Filters for Aircraft Engine Fault Diagnostics,” 

ASME Paper GT2003–38550. 

6. Kobayashi, T., Simon, D.L., 2004, “Evaluation of an 

Enhanced Bank of Kalman Filters for In-Flight Aircraft 

Engine Sensor Fault Diagnostics,” ASME Paper GT2004–

53640. 

7. Stamatis, A., Mathioudakis, K., Papailiou, K.D., 1990, 

“Adaptive Simulation of Gas Turbine Performance,” 

Journal of Engineering for Gas Turbines and Power, 112, 

pp. 168–175. 

8. Tsalavoutas, A., Mathioudakis, K., Stamatis, A., Smith, 

M., 2001, “Identifying Faults in the Variable Geometry 

System of a Gas Turbine Compressor,” Journal of 

Turbomachinery, 123, pp. 33–39. 

9. Ogaji, S.O.T., Sampath, S., Singh, R., Probert, S.D., 2002, 

“Parameter Selection for Diagnosing a Gas-Turbine’s 

Performance-Deterioration,” Applied Energy, 73, 

pp. 25–46. 

10. España, M.D., 1994, “Sensor Biases Effect on the 

Estimation Algorithm for Performance-Seeking 

Controllers,” Journal of Propulsion and Power, 10, 

pp. 527–532. 

11. Callier, F.M., and Desoer, C.A., 1991, Linear Systems 

Theory, Springer-Verlag, New York, p. 240. 

12. Stewart, G.W., 1973, Introduction to Matrix Computations, 

Academic Press, New York, pp. 322–324. 

13. Chen, S., Billings, S.A., Luo, W., “Orthogonal Least 

Squares Methods and Their Application to Non-Linear 
NASA/TM—2005-213864 11

System Identification,” 1989, International Journal of 

Control, 50, pp. 1873–1896. 

14. Chen, S., Cowan, C.F.N., Grant, P.M., 1991, “Orthogonal 

Least Squares Learning Algorithm for Radial Basis 

Function Networks,” IEEE Transactions on Neural 

Networks, 2, pp. 302–309. 

15. Stamatis, A., Mathioudakis, K., Papailiou, K.D., 1992, 

“Optimal Measurement and Health Index Selection for Gas 

Turbine Performance Status and Fault Diagnosis,” Journal 

of Engineering for Gas Turbines and Power, 114, pp. 209–

216. 

16. Grönstedt, T.U.J., 2002, “Identifiability in Multi-Point Gas 

Turbine Parameter Estimation Problems,” ASME Paper 

GT2002–30020. 

17. Pinelli, M., Spina, P.R., 2002, “Gas Turbine Field 

Performance Determination: Sources of Uncertainties,” 

Journal of Engineering for Gas Turbines and Power, 124, 

pp. 155–160. 

18. Mathioudakis, K., Kamboukos, Ph., 2004, “Assessment of 

the Effectiveness of Gas Path Diagnostic Schemes,” 

ASME Paper GT2004–53862. 
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October 2005

NASA TM—2005–213864

ARL–TR–3487

GT2005–68808

E–15234

17

An Optimal Orthogonal Decomposition Method for Kalman Filter-Based

Turbofan Engine Thrust Estimation

Jonathan S. Litt

Engine thrust estimation; Aircraft gas turbine engine; Kalman filter; Gas path analysis;

Diagnostics; Engine deterioration

Unclassified -Unlimited

Subject Category: 07

WBS–22–303–30–72

1L161102AF20

Prepared for the Turbo Expo 2005 sponsored by the American Society of Mechanical Engineers, Reno, Nevada, June 6–9,

2005. Responsible person, Jonathan S. Litt, organization code RIC, 216–433–3748.

A new linear point design technique is presented for the determination of tuning parameters that enable the optimal

estimation of unmeasured engine outputs such as thrust. The engine’s performance is affected by its level of degradation,

generally described in terms of unmeasurable health parameters related to each major engine component. Accurate thrust

reconstruction depends upon knowledge of these health parameters, but there are usually too few sensors to be able to

estimate their values. In this new technique, a set of tuning parameters is determined which accounts for degradation by

representing the overall effect of the larger set of health parameters as closely as possible in a least squares sense. The

technique takes advantage of the properties of the singular value decomposition of a matrix to generate a tuning parameter

vector of low enough dimension that it can be estimated by a Kalman filter. A concise design procedure to generate a

tuning vector that specifically takes into account the variables of interest is presented. An example demonstrates the tuning

parameters’ ability to facilitate matching of both measured and unmeasured engine outputs, as well as state variables.

Additional properties of the formulation are shown to lend themselves well to diagnostics.