Donald L. Simon

Glenn Research Center, Cleveland, Ohio

Jeffrey B. Armstrong

ASRC Aerospace Corporation, Cleveland, Ohio

Sanjay Garg

Glenn Research Center, Cleveland, Ohio

Application of an Optimal Tuner Selection 

Approach for On-Board Self-Tuning Engine Models

NASA/TM—2012-217278

April 2012

GT2011–46408
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Donald L. Simon

Glenn Research Center, Cleveland, Ohio

Jeffrey B. Armstrong

ASRC Aerospace Corporation, Cleveland, Ohio

Sanjay Garg

Glenn Research Center, Cleveland, Ohio

Application of an Optimal Tuner Selection 

Approach for On-Board Self-Tuning Engine Models

NASA/TM—2012-217278

April 2012

GT2011–46408

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

Turbo Expo 2011

sponsored by the American Society of Mechanical Engineers (ASME)

Vancouver, British Columbia, Canada, June 6–10, 2011
Acknowledgments

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

gratefully acknowledge Jonathan Litt for his support and insightful discussions which led to the development of this work.

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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2012-217278 1

Application of an Optimal Tuner Selection Approach 

for On-Board Self-Tuning Engine Models

Donald L. Simon

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Jeffrey B. Armstrong

ASRC Aerospace Corporation

Cleveland, Ohio 44135

Sanjay Garg

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

An enhanced design methodology for minimizing the error 

in on-line Kalman filter-based aircraft engine performance 

estimation applications is presented in this paper. It specifically addresses the underdetermined estimation problem, in 

which there are more unknown parameters than available 

sensor measurements. This work builds upon an existing

technique for systematically selecting a model tuning parameter vector of appropriate dimension to enable estimation by 

a Kalman filter, while minimizing the estimation error in the 

parameters of interest. While the existing technique was 

optimized for open-loop engine operation at a fixed design 

point, in this paper an alternative formulation is presented that 

enables the technique to be optimized for an engine operating

under closed-loop control throughout the flight envelope. The 

theoretical Kalman filter mean squared estimation error at a 

steady-state closed-loop operating point is derived, and the 

tuner selection approach applied to minimize this error is 

discussed. A technique for constructing a globally optimal 

tuning parameter vector, which enables full-envelope application of the technology, is also presented, along with design 

steps for adjusting the dynamic response of the Kalman filter 

state estimates. Results from the application of the technique 

to linear and nonlinear aircraft engine simulations are 

presented and compared to the conventional approach of tuner 

selection. The new methodology is shown to yield a significant improvement in on-line Kalman filter estimation 

accuracy.

Introduction

An emerging approach in the field of aircraft engine controls and health management is the inclusion of real-time onboard adaptive models for the in-flight estimation of engine 

performance parameters (Refs. 1 to 3). These models, 

typically based on Kalman filters, enable the estimation of 

unmeasured engine performance parameters that can be used 

for diagnostics, prognostics, and controls 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. The level of engine 

performance degradation can be estimated using a Kalman 

filter, 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 under-determined 

estimation problem. The conventional approach to address this 

shortcoming is to estimate a subset of the health parameters, 

referred to as model tuning parameters. While this approach 

enables on-line Kalman filter-based estimation, it can 

introduce error in the accuracy of overall model-based 

performance estimation applications. In a departure from the 

conventional approach of selecting a subset of health parameters to serve as the tuner vector, Litt (Ref. 5) presented a novel

approach based on singular value decomposition that selects a 

model tuning parameter vector of low-enough dimension to be 

estimated by a Kalman filter. In this method, a 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. An enhancement 

to the work of Litt, presented by Simon and Garg (Ref. 6),

selects V* to minimize the theoretical mean squared estimation 

error in the parameters of interest at a steady-state open-loop

linear design point.

In this paper, several design enhancements to the optimal 

tuner selection methodology presented in Reference 6 are 
NASA/TM—2012-217278 2

discussed and presented. These include extending the optimal 

tuner selection methodology to encompass closed-loop control 

operating conditions, selection of a single tuner vector for 

application throughout the engine operating envelope as 

opposed to at a single design point, and design considerations 

to provide the desired Kalman filter dynamic response. The 

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

mathematical preliminaries are given regarding the problem 

formulation including the linear model representing system 

dynamics and the formulation of the Kalman filter. Next, 

several practical design considerations are discussed including 

an approach for selecting a tuner vector optimal for fullenvelope operation, referred to as a globally optimal tuner 

vector, and design steps for adjusting Kalman filter dynamic 

response. This is followed by an example application of the 

methodology to linear and nonlinear turbofan engine simulations. Finally, conclusions are presented.

Nomenclature

A, Ac, Axh, Axq,

B, Bce, Bco, Bxh, 

Bxq, C, Cc, Cxh, 

Cxq, D, F, Fxh, 

Fxq, G, L, M, N

system matrices

C-MAPSS Commercial Modular Aero-Propulsion 

System Simulation

Fn net thrust

HPC high pressure compressor

HPT high pressure turbine

I identity matrix

K∞ Kalman filter gain

LPC low pressure compressor

LPT low pressure turbine

Ph health parameter covariance matrix

PLA power lever angle

PWLKF piece-wise linear Kalman filter

P∞

Kalman filter state estimation covariance 

matrix

Q, Qxh, Qxq process noise covariance matrices

R measurement noise covariance matrix

SmLPC low pressure compressor stall margin

SSEE sum of squared estimation errors

V* transformation matrix relating hk to qk

WSSEE weighted sum of squared estimation errors

e error between control setpoint and feedback 

signal

h health parameter vector

m number of measured outputs

p number of health parameters

q Kalman filter tuning parameter vector

r control setpoint

u actuator command vector

v measurement noise vector

w, wh, wxh, wxq process noise vectors

x, xxh, xxq state vectors

y vector of measured outputs

z vector of unmeasured (auxiliary) outputs

Subscripts

c control parameter

k discrete time step index

o open-loop feedback signal

r feedback signal

xh augmented state vector (x and h)

xq reduced order state vector (x and q)

ss steady-state value

Superscripts

† pseudo-inverse

T transpose

- a priori estimate

+ a posteriori estimate

Diacritical marks

ˆ estimated value

 ̆ augmented

 ̃ error or residual

ˉ expected or mean value

Problem Formulation

The discrete linear time-invariant engine state space equations about a 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, z is the vector of auxiliary (unmeasured) model 

outputs, and h is the vector of engine health parameters. The 

vectors w and v are zero-mean white noise inputs, with 

covariance of Q and R, respectively. The matrices A, B, C, D, 
NASA/TM—2012-217278 3

F, G, L, M, and N are of appropriate dimension. From Eq. (2)

it can readily be observed that health parameter deviations

induce shifts in the engine state variables and outputs. As 

such, health parameter effects must be accounted for to 

achieve accurate engine performance estimation. Towards this 

requirement, Eq. (2) can be rewritten such that h is concatenated with x to form an augmented state vector, xxh, as shown 

in Eq. (3). Since engine performance deterioration is very 

slowly evolving relative to other engine dynamics, h is here 

modeled without dynamics.

 

[ ]



[ ]



, ,

,

,

1

1 ,

, ,

,

,

0 0

xh xh k xh xh k

xh

xh k

xh

xh k

k k k

k

k k h k

A x B w

xh xhk xh k xhk

k

k k k

k C

x

xh xhk k k

k

k k

k F

x

xh xhk k

x x AL B w

u

h h I w

Ax Bu w

x

y C M Du v h

C x Du v

x

z F N Gu h

F x Gu

+

+

       

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

= ++

 

= ++    

= ++

  =   +

 

= +

 





(3)

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

augmented state vector, [x

T hT

]

T

, with a covariance of Qxh. wxh

consists of the original state process noise, w, concatenated 

with the process noise associated with the health parameter 

vector, wh.

,

,

k

xh k

h k

w

w

w

  =    

(4)

Closed-Loop State-Space Model

This paper considers an engine operating under closed-loop 

control conditions. As a point of introduction, refer to the 

architecture shown in Figure 1 depicting the interaction 

between an aircraft engine, a controller, and a Kalman filter. 

Here, the controller accepts inputs consisting of an error 

signal, ek, (i.e., a residual between a commanded parameter, rk, 

and a sensed feedback parameter, yr,k), along with additional 

sensed measurements, yo,k. The controller processes these 

inputs to produce actuator commands, uk. 

As shown in Figure 1, engine operation is effected by both 

actuator commands and the engine’s level of deterioration 

denoted by the unknown set of health parameter inputs, hk. 

Controller Aircraft

Engine

rk ek

uk yk

yr,k

+

-

zk

hk

Kalman

Filter

ˆ ˆ ,,, ˆ

ˆ ˆ ,

k kk

k k

xyz

q h

yo,k

uk

Figure 1.—Aircraft engine, controller, and Kalman filter.

The Kalman filter processes the measurements, yk, 

and actuator commands, uk, to produce the estimates 

ˆ ˆˆ ˆ , , , , and ˆ k kkk k xyzq h .

The controller state space equations are given as

,1 , ,

,

,

c k c c k ce k co o k

k c ck

k k rk

x Ax B e B y

u Cx

e ry

+ = ++

=

= −

(5)

and the sensed feedback parameters, yr,k and yo,k, can be 

written as 

,

,

rk r k r k r k

ok o k o k o k

y Cx Du M h

y Cx Du M h

=++

=++ (6)

From Eqs. (5) and (6) the controller state variables become

( )

( )

( )

( ) ( )

,

,1 ,

,

...

...

...

k

o k

c k c c k ce k r k r k r k

e

co o k o k o k

y

c c k ce r co o k

ce r co o k ce r co o k

ce k

x Ax B r Cx Du M h

B Cx Du M h

Ax B C B C x

BD BD u BM BM h

B r

+ = + −−−

+ ++

= +− +

+− + +− +

+





(7)

Substituting uk = Ccxc,k (from Eq. (5)) into Eq. (7) yields

( )

( )

( )

1

2

1

, 1

,

...

...

c k ce r co o k

A

c ce r c co o c c k

A

ce r co o k ce k

L

x BC BC x

A B DC B DC x

BM BM h Br

+ =− +

+− +

+− + +







(8)

From Eqs. (2) and (5) the closed-loop engine state variables 

can be written as
NASA/TM—2012-217278 4

1 , ( )

k

k k c ck k

u

x Ax B C x Lh + =+ +  (9)

The quantities in Eqs. (8) and (9) can be augmented with the 

health parameters to obtain

1

,1 1 2 1 ,

1

0

0 0 0

k ck

c k c k ce k

k k

x A BC L x

x A A Lx Br

h h I

+

+

+

              = +               

(10)

For simplicity the following augmented matrices and vectors 

are defined

,

1 2

1

0

k

k

c k

c

ce

x

x

x

A BC

A

A A

B

B

L

L

L

  =    

  =    

  =    

  =    









(11)

allowing the closed-loop state space equations to be re-written 

in the following form using Eqs. (2), (5), (10), and (11)





1

1

,

,

0 0

k

k

k k

k

k

k k c ck k

u

k

c

C

k k c ck k

u

k

c

F

x A L x B r h I h

y Cx DC x Mh

x

C DC M

h

z Fx G C x Nh

x

F GC N

h

+

+

      

  =    +         

=+ +

   =        

=+ +

   =        





    









(12)

Reduced-Order State Space Model

To enable Kalman filter formulation when presented the 

underdetermined estimation problem a reduced-order state 

space model must be constructed. This is accomplished by 

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

combination of all health parameters, h, given by

* q Vh = (13)

where q ∈ m, h ∈ p, m < p, and V* is an m × p transformation matrix of rank m, applied to construct the tuning parameter vector. While q is constructed as a linear combination of 

health parameters, the elements of q do not have any physical 

meaning. Their purpose is to allow an accurate estimation of 

unmeasured engine parameters. However, given q, an 

approximation of the health parameter vector, h

ˆ , can be 

obtained as

ˆ *† h Vq = (14)

where V*† is the pseudo-inverse of V*. Substituting Eq. (14)

into Eq. (3) yields the following reduced order state space 

equations which will 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

k

k k q k

A x B w

xq xq k xq k xq k

k

k k k

k C x

xq xq k k k

k

k

k F x

x x A LV B w

u

q q I w

Ax Bu w

x

y C MV Du v

q

C x Du v

x

z F NV

q

+

+

        

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

= ++

  =    + +   

= ++

  =       

 





,

k

xq xq k k

Gu

F x Gu

+

= +

(15)

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

its associated covariance, Qxq, are calculated as

, , * *

,

* *

0 0

0 0

0 0

0 0

k

xq k xh k

h k

T

xq xh

I I w

w w

V V w

I I

Q Q V V

   =   =     

   =      

(16)

Kalman Filter Formulation 

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

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

state estimation error covariance matrix and the Kalman gain 

matrix are time invariant—instead of updating these matrices 

each time step they are pre-converged and held constant at 
NASA/TM—2012-217278 5

their final values. Given the reduced order linear state space 

equations shown in Eq. (15), the state estimation error 

covariance matrix, P∞, is calculated by solving the following 

Riccati equation (Ref. 7)

1 ( ) T TT

xq xq xq xq xq xq

T

xq xq xq

P A PA A PC C PC R

C PA Q

−

∞∞ ∞ ∞

∞

=− +

× +

 (17)

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

as follows (Ref. 7)

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

−

∞∞ ∞ = + (18)

The Kalman filter a priori and a posteriori estimates are given 

in Eq. (19) and Eq. (20) respectively (Ref. 7)

, ,1 1 ˆ ˆ xq k xq xq k xq k x Ax Bu − + = + − − (19)

, , ( , ) ˆˆ ˆ xq k xq k k xq xq k k x x K y C x Du + − − =+ − − ∞ (20)

The reduced order state vector a posteriori estimate, , ˆxq k x+ , 

produced by Eq. (20) can be used to produce an estimate of 

the augmented state vector and the auxiliary parameters as 

follows

, , *†

*†

,

0 ˆ ˆ 0

ˆ ˆ

xh k xq k

k xq k k

I x x V

z F NV x Gu

+

+

  =    

=   +  

(21)

Optimal Transformation Matrix Selection

As presented in Reference 6, the estimation accuracy of the 

Kalman filter is directly dependent on the selection of the

transformation matrix, V*

. This gives rise to the optimization 

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

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

minimizes the mean sum of squared estimation errors (SSEE) 

in the parameters of interest

( ) *

* arg min m p V

SSEE V × ∈ (22)

Alternatively, a weighted mean sum of squared estimation 

errors, WSSEE, can be applied to place more/less emphasis on 

certain parameters, or to account for variation in the engineering units of different parameters. Reference 6 presented 

derivations of health parameter and auxiliary parameter mean

SSEE as a function of V* at an open-loop steady-state operating point. While the functional design of a Kalman filter 

applied to an open-loop versus a closed-loop system is the 

same due to the separation principle (see Ref. 8), the transformation matrix, V*

, which is optimal under the two scenarios is 

different. Readers are referred to the Appendix for a complete 

derivation of the closed-loop SSEE as a function of V*

. As in 

the open-loop case presented in Reference 6, once the

theoretical derivation of the SSEE is obtained, an optimal 

iterative search can be applied to determine the V* matrix that 

produces the minimum estimation error. 

Practical Design Considerations 

The approach for selecting V* introduced in Reference 6 is 

designed to provide optimal estimation results at a single 

steady-state operating point. However, in real-world applications an aircraft engine will operate at and transition between a 

broad range of operating points. Some practical design 

considerations for selecting V* and designing a Kalman filter 

to provide full-envelope estimation accuracy and satisfactory 

transient estimation response are given in the following 

subsections.

Selecting V* for Full-Envelope Operation

A typical design approach for on-board adaptive aircraft 

engine models is to implement a piece-wise linear Kalman 

filter (Refs. 1, 2, and 4). This consists of designing individual 

Kalman filters at multiple operating points spanning the 

engine’s operating envelope, and then interpolating between 

points as the engine transitions between operating conditions. 

This is a suitable design approach if one follows the conventional technique of selecting a fixed subset of health parameters to serve as the Kalman filter tuner vector, q, throughout 

the entire engine operating envelope. However, if the method 

of Reference 6 is used to select a different V* (and thus a 

different q) for each design point comprising the piece-wise 

linear model, interpolation between the design points will 

yield meaningless results. Thus, an alternative strategy is 

necessary. This can be addressed by modifying the optimal 

iterative search routine to produce a single “globally optimal”

V* transformation matrix that minimizes the sum of theoretical 

SSEE’s (or WSSEE’s) calculated at a number of user-specified 

engine operating points. This procedure does add computational complexity to the design process, but, as the selection of 

V* is only performed off-line during the design phase, it does 

not add any computational burden to the Kalman filter 

implemented on-line. While the selection of a globally optimal 

V* matrix may result in sub-optimal estimation results at 

individual operating points, it will permit interpolation 

between operating points of a piece-wise linear Kalman filter

suitable for full-envelope operation. The next section will 

present estimation accuracy results from the application of this 

technique to an aircraft engine simulation.
NASA/TM—2012-217278 6

Adjusting Kalman Filter Process Noise to Provide 

Acceptable Dynamic Response

The optimal iterative search for V* introduced in Reference 6 is designed to minimize the Kalman filter mean 

squared estimation error at a steady-state operating point. The 

dynamic response of the Kalman filter state estimates is not 

considered in this process. This can lead to the selection of a

V* matrix that produces overly sensitive or overly sluggish

variations in the estimated state and tuning parameters when 

the engine experiences a transient. Typically, a Kalman filter 

designer directly specifies the state process noise covariance, 

Q, to provide the desired estimation response. However, the

optimal tuner selection approach performs a transformation 

that converts the designer-specified full-order process noise 

covariance matrix, Qxh, to a reduced-order process noise 

covariance matrix, Qxq (see Eq. (16)). The next section will 

discuss and present design steps that can be taken to improve 

the dynamic response of the Kalman filter.

Turbofan Engine Example

In this section the utility of the optimal tuner selection 

methodology is demonstrated by showing results from its 

application to the Commercial Modular Aero-Propulsion 

System Simulation (C-MAPSS), a NASA-developed highbypass turbofan engine simulation (Ref. 9). C-MAPSS is a 

transient nonlinear aerothermodynamic engine model 

developed for controls and diagnostics research and development purposes. It has two state variables (fan and core speed), 

and three actuators (fuel flow, variable stator vanes (VSV), 

and a variable bleed valve (VBV)). C-MAPSS also has ten 

adjustable efficiency and flow capacity health parameters that 

enable the simulation of engine performance deterioration and 

module performance faults. The state variables, actuators, and 

health parameters are listed in Table 1. In this study, a 

simplified version of the C-MAPSS engine controller 

comprised of a fuel flow controller and variable geometry 

open-loop schedules is considered. While C-MAPSS does

have control limit logic to prevent engine over speed and 

operating instabilities, that logic is not included here. The 

controller schedules fuel flow based on the error, ek, between 

commanded and sensed fan speed. VBV and VSV actuators 

are open-loop scheduled based on sensed fan speed and sensed 

core speed, respectively. Collectively, fuel flow, VBV, and 

VSV commands form the vector of control inputs, uk, 

provided as inputs to C-MAPSS. The C-MAPSS controller 

state variables and sensed feedback parameters are shown in

Table 2. 

For the purposes of this study, six sensed outputs and four 

unmeasured auxiliary outputs are defined. These parameters 

and their engineering units are listed in Table 3. 

TABLE 1.—STATE VARIABLES, ACTUATORS, 

AND HEALTH PARAMETERS

State 

variables

Actuators Health 

parameters

Nf – fan speed Wf – fuel flow Fan efficiency*

Nc – core speed VSV – variable stator vane Fan flow capacity*

VBV –variable bleed valve LPC efficiency

LPC flow capacity*

HPC efficiency*

HPC flow capacity*

HPT efficiency

HPT flow capacity*

LPT efficiency

LPT flow capacity

*Health parameters applied as tuners in conventional estimation approach

TABLE 2.—CONTROLLER STATE VARIABLES 

AND SENSED FEEDBACKS

Controller state variables

(xc)

Controller sensed feedbacks

xc(1) – fuel flow control state variable 1 yr – corrected fan speed

xc(2) – fuel flow control state variable 2 yo(1) – corrected fan speed

xc(3) – fuel flow control state variable 3 yo(2) – corrected core speed

xc(4) – VBV control state variable

xc(5) – VSV control state variable

TABLE 3.—SENSED OUTPUTS AND 

UNMEASURED AUXILIARY OUTPUTS

Sensed outputs 

(y)

Auxiliary parameters 

(z)

Nf – fan speed (rpm) T40 – Combustor exit temp. (○R)

Nc – core speed (rpm) T50 – LPT exit temperature (

○

R)

P24 – HPC inlet total pressure (psia) Fn – Net thrust (%)

T24 – HPC inlet total temp. (○R) SmLPC – LPC stall margin (%)

Ps30 – HPC exit static pressure (psia)

T48 – Exhaust gas temperature (

○

R)

Case 1: Kalman Filter Point Design

As an initial evaluation, the optimal tuner selection methodology was applied to design a Kalman filter for application at 

a single closed-loop operating point. Here, model tuning 

parameters were selected to minimize a weighted sum of 

squared estimation errors (WSSEE) in the four auxiliary 

parameters listed in Table 3. To serve as a comparison, two 

additional Kalman filters were designed. These included a 

Kalman filter designed applying the conventional approach of 

selecting a subset of health parameters to form the model 

tuning vector, and a Kalman filter designed applying the openloop optimal tuner selection approach presented in Reference 6. The health parameters selected to serve as the elements 

of the tuner vector in the conventional design approach are 

denoted with an “*” in Table 1. They were selected through an 

exhaustive search that considered all possible combinations 

and determined the subset of six health parameters that 
NASA/TM—2012-217278 7

to C-MAPSS (a linear state space point model and the full 

nonlinear version) at steady-state closed-loop conditions at a 

provided the minimum WSSEE. All three designs were applied 

cruise operating point of 35K ft, 0.65 Mach, and a power lever 

angle (PLA) setting of 60°. In each case the engine was 

subjected to sensor measurement noise with covariance R

(with elements of the y vector in corrected engineering units, 

ordered as shown in Table 3), and health parameter deterioration with covariance Ph (with elements of the h vector in 

percent, ordered as shown in Table 1), defined as follows

5.469 1.830 0.047 0.381 1.585 2.273

1.830 131.195 0.022 16.817 18.415 4.241

0.047 0.022 0.002 0.007 0.006 0.007

0.381 16.817 0.007 3.744 1.843 0.232

1.585 18.415 0.006 1.843 7.894 1.156

2.273 4.241 0.007 0.232 1.156 6.342

R

 −

 − 

− − = − −

 − −

2 10−





   ×



0.508 0.596 0.259 0.515 1.098 1.913 0.278 0.259 0.057 0.029

0.596 0.894 0.350 0.700 1.480 2.593 0.370 0.348 0.077 0.039

0.259 0.350 0.230 0.334 0.693 1.230 0.165 0.158 0.035 0.018

0.515 0.700 0.334 0.749 1.384 2.473 0.326 0.315

Ph

− −

− −

− −

−

=

0.070 0.035

1.098 1.480 0.693 1.384 3.213 5.670 0.691 0.669 0.147 0.075

1.913 2.593 1.230 2.473 5.670 10.468 1.182 1.165 0.258 0.132

0.278 0.370 0.165 0.326 0.691 1.182 0.206 0.179 0.037 0.019

0.259 0.348 0.158 0.315 0.669 1

−

− −

− −

− −

−−−−−− .165 0.179 0.183 0.035 0.018

0.057 0.077 0.035 0.070 0.147 0.258 0.037 0.035 0.018 0.004

0.029 0.039 0.018 0.035 0.075 0.132 0.019 0.018 0.004 0.012

 

 

− −

− −

  −−−−−−− −

Note that the R and Ph matrices shown above contain nonzero off diagonal elements. For R, this is due to the fact that 

the engine is operated closed-loop and parameter correction 

(Ref. 9) is applied. This results in non-zero covariance 

between each sensor measurement pair. To calculate R, a large 

simulated data set was generated by applying random noise to 

each sensor (i.e., the six sensors shown in Table 3 plus the 

inlet temperature and pressure sensors used for correction). 

The resulting covariance in the corrected sensor measurements 

yields R. The engine performance deterioration levels applied 

in this study are loosely based on historical aircraft engine 

data presented in Reference 10. This report shows that engine 

module deterioration occurs in a coupled fashion, resulting in 

some non-zero covariance between each health parameter pair. 

For this study, a routine was created to simulate random 

engine health parameter deterioration levels representative of 

the information shown in Reference 11. Based on a large 

simulated dataset generated by this routine, the health 

parameter covariance was calculated as Ph.

Table 4 shows a comparison of the theoretically predicted 

and experimentally obtained mean squared estimation errors 

for the three Kalman filter designs when applied to linear and 

nonlinear C-MAPSS operating in closed-loop. The experimental results were obtained through a Monte Carlo simulation 

analysis in which the health parameters varied over a random 

distribution in accordance with the covariance matrix, Ph

shown above. The test cases were concatenated to produce a 

single time history input that was provided to the C-MAPSS 

models. Each individual health parameter test case lasted 45 s.

The experimental errors shown in Table 4 are based on the last 

10.5 s of each 45 s test case. This allowed the engine model and 

the Kalman estimator to reach a quasi-steady-state operating 

condition prior to calculating the error. The corrected trim point 

values of T40, T50, Fn, and SmLPC are 2808 °R, 1222 °R, 

26.5 percent, and 11.3 percent respectively. For the construction of WSSEE, a weighting of 1.0 was applied to the T40 and 

T50 errors, and a weighting of 100.0 was applied to the Fn and 

SmLPC errors.

From Table 4 it can be seen that the mean squared estimation 

errors experimentally obtained from the linear model exhibit 

good agreement with the theoretically predicted errors. This 

result is encouraging as it validates the closed-loop optimal 

tuner selection methodology presented in this paper. It is also 

encouraging to find that the closed-loop optimal tuner selection 

approach provides superior estimation accuracy compared to the 

other two tuner selection approaches. A closer comparison of 

the closed-loop versus open-loop theoretical results reveals that 

the improvement offered by the closed-loop tuners is primarily 

due to a reduction in estimation variance. The mean squared 

estimation bias is nearly the same in both cases. This suggests 
NASA/TM—2012-217278 8

TABLE 4.—AUXILIARY PARAMETER 

MEAN SQUARED ESTIMATION ERRORS

Tuner

vector

Error T40 

(°R)

T50 

(°R)

Fn

(%)

Sm-LPC

(%)

WSSEE

Subset of 

health 

parameters

Theor. sqr. bias 20.48 0.69 0.003 0.031 24.51

Theor. variance 0.50 0.07 0.007 0.047 5.99

Theoretical 20.98 0.75 0.010 0.078 30.49

Exp. (linear) 20.16 0.78 0.010 0.075 29.43

Exp. (nonlinear) 24.52 1.39 0.012 0.197 46.73

Cruise 

optimal 

(open-loop)

Theor. sqr. bias 1.58 0.52 0.001 0.014 3.60

Theor. variance 0.75 0.11 0.005 0.029 4.28

Theoretical 2.32 0.62 0.007 0.043 7.89

Exp. (linear) 2.26 0.64 0.007 0.044 7.92

Exp. (nonlinear) 4.49 1.19 0.007 0.163 22.73

Cruise 

optimal 

(closed-loop)

Theor. sqr. bias 1.61 0.39 0.001 0.013 3.43

Theor. variance 0.36 0.07 0.000 0.002 0.64

Theoretical 1.97 0.45 0.002 0.015 4.06

Exp. (linear) 2.05 0.46 0.001 0.014 4.10

Exp. (nonlinear) 4.49 0.96 0.006 0.166 22.59

that the open-loop tuner selection approach, which is mathematically simpler and does not require access to detailed 

control design information, may be applied to provide 

comparable estimation accuracy, especially in applications 

that exhibit limited estimation variance. It is noted that the 

experimental estimation errors obtained based on nonlinear CMAPSS are larger than those based on linear C-MAPSS. This 

is attributed to differences between the nonlinear plant model 

and the linear model that the Kalman filter is based upon. In 

particular, the SmLPC mean squared estimation error is 

significantly larger in the nonlinear case. This error, which has 

a WSSEE weighting of 100, causes most of the increase 

observed in WSSEE over the linear and theoretical results. 

Case 2: Kalman filter Full-Envelope Design

The previous case presented estimation results at a single 

operating point. However, to be practical the methodology

must be applicable for constructing an estimator that can

provide accurate estimation performance as the engine 

transitions throughout the flight envelope. As previously 

discussed, this can be performed by selecting a single globally 

optimal V* matrix for application within a piece-wise linear 

Kalman filter (PWLKF) design, enabling estimation throughout the entire operating envelope. The optimal iterative search 

routine was modified to sum the theoretical WSSEE results

over multiple user-specified operating points. Through trial 

and error it was found that optimizing over a small number of 

operating points spanning the commonly encountered regions 

of the flight envelope provided reasonable estimation 

accuracy. To illustrate this technique, a single globally optimal 

V* matrix was produced based on the nine engine operating 

points shown in Figure 2. 

Table 5 shows the auxiliary parameter theoretical mean 

squared estimation errors obtained when applying the globally 

optimal V* matrix and tuner vector at the same closed-loop 

steady-state cruise operating condition used in Case 1. As a 

Figure 2.—Optimization points applied for globally optimal 

tuner selection

TABLE 5.—AUXILIARY PARAMETER THEORETICAL MEAN 

SQUARED ESTIMATION ERRORS AT STEADY-STATE 

CRUISE OPERATING POINT

Tuner vector T40 

(°R)

T50 

(°R)

Fn

(%)

SmLPC

(%)

WSSEE

Subset of health parameters 20.98 0.753 0.010 0.078 30.50

Cruise optimal (open-loop) 2.32 0.62 0.007 0.043 7.89

Cruise optimal (closed-loop) 1.97 0.454 0.002 0.015 4.06

Globally optimal (closed-loop) 1.97 0.466 0.002 0.015 4.11

comparison, the theoretical mean squared estimation errors 

based on the three model tuner vectors evaluated in Case 1 are 

also shown. The encouraging result is that application of the 

globally optimal tuner vector does not result in a significant 

loss in estimation accuracy. In fact, the WSSEE of the globally 

optimal tuner vector is only 1.2 percent larger than that of the 

tuner vector optimally selected for the given cruise operating 

point. Even more impressive is the fact that the evaluated 

cruise point is not one of the nine points used to determine the 

globally optimal V*

. 

As an additional comparison, the average theoretical estimation accuracy provided by the subset of health parameters, 

cruise optimal closed-loop, and globally optimal closed-loop 

tuner vectors was evaluated at 2000 operating points spanning 

the engine operating envelope. For this evaluation, separate 

Kalman filter point designs were constructed at each of the 

2000 operating points. These results are shown in Table 6, 

along with the standard deviation in the results (shown in 

parentheses). Here it can be observed that the Kalman filter 

design that applies the globally optimal tuner vector provides a

4.2 percent reduction in WSSEE compared to the cruise 

optimal tuner vector. Furthermore, the standard deviation in 

all squared estimation errors is relatively small, demonstrating 

little variance in the estimation error as the operating point is 

changed. These results are highly encouraging as they suggest 

that a fixed tuner vector, which is near optimal over a large 
NASA/TM—2012-217278 9

TABLE 6.—AUXILIARY PARAMETER THEORETICAL 

MEAN SQUARED ESTIMATION ERRORS (AND 

STANDARD DEVIATION) AVERAGED OVER 

2000 STEADY-STATE OPERATING POINTS

Tuner vector T40 

(°R)

T50 

(°R)

Fn

(%)

SmLPC

(%)

WSSEE

Subset of health 

parameters

20.03 0.762 0.012 0.054 27.36

(1.94) (0.155) (0.005) (0.016)

Cruise optimal 

(closed-loop)

1.90 0.711 0.003 0.014 4.27

(0.16) (0.531) (0.002) (0.003)

Globally optimal

(closed-loop)

1.90 0.634 0.003 0.013 4.09

(0.15) (0.298) (0.001) (0.002)

region of the flight envelope, can be found. However, 

additional evaluation is warranted to demonstrate that a 

globally optimal tuner vector can be generated and applied to 

other engine models in addition to C-MAPSS.

Case 3: Kalman Filter Design Considerations 

For Transient Operating Conditions

The optimal tuner selection strategy presented in this paper 

is designed to minimize the mean squared estimation error 

under steady-state operating conditions. However, it does not 

necessarily provide the desired accuracy when the engine is 

undergoing a transient. To illustrate this refer to Figure 3. The 

top half of this figure shows a comparison of nonlinear CMAPSS net thrust versus a PWLKF estimate of net thrust as 

the engine undergoes a power transient at a cruise condition 

(35K ft, 0.65 Mach). The lower half of the figure shows the 

corresponding variation in the estimates of the tuning 

parameters, q, produced by the Kalman filter. In this example, 

nonlinear C-MAPSS is a nominal (non-deteriorated) engine, 

whereas the PWLKF is based on a fleet average (50 percent

deteriorated) engine. Before and after the transient operating 

period, the actual and estimated net thrust exhibit good 

agreement. However, during the transient the estimate

produced by the Kalman filter responds more rapidly than the 

actual thrust; consequently the estimate is not as accurate. It

can also be observed that the tuner estimates undergo significant variations during the transient.

A design step that can be taken to adjust the dynamic response of the Kalman filter estimates is to modify the 

specified process noise covariance matrix, Qxh. Specifying 

smaller magnitude values will slow the dynamic response of 

the Kalman filter estimates. To illustrate this, the Qxh matrix 

was divided by 1106

, and the optimal tuner selection process 

was repeated. The identified vector of tuning parameters was 

then applied to design a new Kalman filter. The response of 

the new Kalman filter to the same power transient is shown in 

Figure 4. Here it can be observed that the estimation accuracy 

during the transient is improved, and the amount of variation 

in the tuning parameters during the transient is also reduced.

Figure 3.—Kalman filter response with original Qxh.

Figure 4.—Kalman filter response with adjusted Qxh.

Conclusions

A systematic approach to model tuning parameter selection 

for on-line Kalman filter based parameter estimation under 

closed-loop operating conditions has been presented, along 

with design considerations for applying the approach. This 

technique is specifically for the underdetermined parameter 

estimation problem where there are fewer sensor measurements than unknown health parameters that impact system

outputs. The approach creates and applies a linear transformation matrix, V*

, to select a vector of tuning parameters that are 

a linear combination of all health parameters. The tuning 

parameter vector is selected to be of low enough dimension to 

be estimated, while minimizing the mean-squared error of 

Kalman filter estimates. Evaluations based on an aircraft 
NASA/TM—2012-217278 10

engine linear point model demonstrate that the theoretically 

predicted and experimentally obtained estimation errors 

exhibit good agreement, thus confirming the theory that the 

tuner selection methodology is based upon. The technique was 

also found to provide acceptable estimation results when 

applied to a nonlinear aircraft turbofan engine simulation, 

although, as expected, some loss in estimation accuracy is 

incurred compared to the linear evaluation. Additionally, a 

technique for selecting a single globally optimal V* matrix 

applicable throughout an engine’s operating envelope has been 

demonstrated. This is necessary to enable interpolation 

between operating points within a piece-wise linear design

implementation. Theoretical evaluation results using an 

aircraft engine model showed that the application of a single

globally optimal V* matrix results in only a 1.2 percent

increase in mean squared estimation error compared to the 

implementation of an optimal V* matrix at a cruise operating 

point. While encouraging, additional evaluation is warranted 

to determine if this holds when the technique is applied to 

different engine operating points and different engine models.

Finally, it was also shown that, as in a conventional Kalman 

filter implementation, the system designer can adjust the

dynamic response of the Kalman filter estimates by adjusting 

the specified state vector process noise covariance matrix used 

in the filter design.
NASA/TM—2012-217278 11

Appendix: Derivation of Kalman Filter Mean Squared 

Estimation Error Under Closed-Loop Operating Conditions

This appendix provides a derivation of the Kalman filter 

mean squared estimation error at a closed-loop steady-state 

operating point. This information is used in the iterative search 

for an optimal V* matrix described in the paper. This appendix 

will first derive the control inputs, system outputs, and 

Kalman filter estimates under steady-state conditions. Next, 

the Kalman filter mean squared estimation error, which is 

comprised of a mean squared bias and a mean variance, will 

be derived. 

System Under Steady-State Conditions

For the linear system operating at the trim point under 

steady-state conditions rk = 0 and k k ss 1 x xx + = =   . We can 

leverage this information to write the x, y, and z equations as a 

function of h

( )

( )

( )

1

1

1

ss ss

ss

ss ss

ss

ss ss

ss

x Ax Lh

x I A Lh

y Cx Mh

y CI A L M h

z Fx Nh

z FI A L Nh

−

−

−

= +

= −

= +

  =− +    

= +

  =− +    

   

  



  



  

(A.1)

The steady-state actuator commands, uss, and the steady-state 

state vector, xss, can also be written as functions of h, so from 

Eqs. (5) and (A.1)

( )

[ ] [ ]( )

1

1

0 0

0 0

ss

ss

ss c ss c

x

ss ss

x

u C x C I A Lh

x I x I I A Lh

−

−

= = −      

= = −





  



  



(A.2)

Steady-state Kalman filter estimates can also be written as a 

function of h. To make this derivation we will begin by 

rewriting the a priori and a posteriori Kalman filter equations 

previously given in Eqs. (19) and (20) as follows

a priori equations

( ( ))

( )

1

, ,1 1

, , 1 1 ,1 1

ˆ

1

, ,1 1 ,1 1

1

, , 1

ˆ ˆ

ˆˆ ˆ

ˆ ˆ ( ) ˆ

ˆ ˆ

k

xq k xq xq k xq k

xq k xq xq k k xq xq k k

x

xq k

xq k xq xq k xq k xq xq k k

xq k

xq k xq xq xq k

xq

x Ax Bu

x A x K y C x Du

B u

x A x A K y C x Du

B u

x A I KC x

A K

+

−

− +

− −

− − −

− ∞− − −

−

− − −

− ∞− − −

−

− −

∞ −

∞

= +

= +− −

+

=+ −−

+

= −

+

 





1

1

k

xq xq

k

y B AKD

u

−

∞

−

      −    

(A.3)

a posteriori equations

( )

( ) ( )

( ) [ ]

, , ,

, ,

, ,

ˆˆ ˆ

ˆ ˆ

ˆ ˆ

xq k xq k k xq xq k k

xq k xq xq k k k

k

xq k xq xq k

k

x x K y C x Du

x I K C x K y Du

y x I KC x K KD u

+ − −

∞

+ −

∞ ∞

+ −

∞ ∞ ∞

=+ − −

=− + −

 

=− + −    

(A.4)

Next, the following expected value properties at steady-state 

operating conditions are defined 

1

1

, ,1 , ˆ ˆˆ

k k ss

k k ss

xq k xq k xq ss

Ey Ey y

Eu Eu u

Ex Ex x

−

−

− −−

−

   = =   

   = =   

  = =  

(A.5)

Substituting (A.5) into (A.3) allows the a priori Kalman filter 

equation under steady-state conditions to be written as

, , ( ) ˆ ˆ xq ss xq xq xq ss

ss

xq xq xq

ss

x A I KC x

y AK B AKD

u

− −

∞

∞ ∞

= −

 

+ −       



(A.6)

Then, substituting Eqs. (A.1) and (A.2) into (A.6), the steadystate Kalman filter a priori equation can be written as a 

function of h.
NASA/TM—2012-217278 12

( ) ( )

( )

( ( )) ( )

( )

1

, , 1

1

1

, 1

ˆ ˆ

0

ˆ

0

xq ss xq xq xq ss xq xq xq

c

xq ss xq xq xq xq xq

c

CI A L M

x A I KC x AK B AKD h

C IA L

CI A L M

x I A I KC AK B AKD h

C IA L

−

− −

∞ ∞∞ −

−

− −

∞ ∞∞ −

      − +     =− + −    

  −    

      − +     =− −   −  

  −    

  

 

  

 

(A.7)

Next, proceed to the a posteriori Kalman filter equations, which can also be written as a function of h

( ) [ ]

( )( ( )) ( )

( )

[ ] ( )

( )

, ,

1 1

1

, 1 1

,

ˆ ˆ

ˆ

0 0

ˆ

ss

xq ss xq xq ss

ss

xq ss xq xq xq xq xq xq

c c

xq ss

y x I KC x K KD

u

CI A L M CI A L M

x I KC I A I KC AK B AKD h K KD h

C IA L C IA L

x IK

+ −

∞ ∞ ∞

− −

− +

∞ ∞ ∞ ∞ ∞ ∞ − −

+

∞

 

=− + −    

       − + − +      =− − −   −    + −  

  − −         

= −

     

   

( )( ( )) [ ] ( )

( )

1

1

1

,

0

ˆ

xq

xq xq xq xq xq xq

c

G

xq ss xq

CI A L M

C I A I KC AK B AKD K KD h

C IA L

x Gh

−

−

∞ ∞ ∞ ∞∞ −

+

      − +       − −   − +−           −    

=

  

 



(A.8)

Steady-State Mean Squared Estimation Error 

Using Eqs. (21), (A.8), and (A.2), the steady-state mean 

estimation error bias is given as

[ ]( )

[ ]( )

, ,

, ,, , ,

1

*†

ˆ

1

*†

ˆ ˆ

0 0

0

0 0

0

xq ss xh ss

xh

ss

xh ss xh k xh k xh ss xh ss

ss

xq

x x

xq

G

ss

s

x

x Ex x x x

h

I I IA L G h h

V I

I I IA L G h

V I

x

h

+

+ +

−

−

 

= = − =−          

    − = −        

      − =   −          

=

  

 

 

 





 xh

s

G h

 

  =    

(A.9)

The steady-state auxiliary parameter estimation error bias can 

also be derived using Eqs. (15), (A.1), and (A.8)

( )

*†

,

1 *†

ˆ

ˆ

ˆ

z

ss k k

ss ss

xq ss ss

xq

G

z

z Ez z

z z

F NV x z

F NV G F I A L N h

G h

+

−

= −    

= −

=   −  

    = −− +            

=



  



(A.10)

Mean Squared Estimation Error Bias

The average sum of squared estimation error biases across a 

fleet of engines can be calculated from Eqs. (A.9) and (A.10)

as follows

{ }

{ }

{ }

2

, ,, , ,

h

T T

xh fleet xh ss xh ss xh ss xh ss

T T

xh xh

T T

xh xh

P

T

xh h xh

x E x x E tr x x

E tr G hh G

tr G E hh G

tr G P G

≡ =        

=    

    = ⋅          

=

  



(A.11)
NASA/TM—2012-217278 13

{ }

{ }

{ }

2

h

T T

fleet 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

≡ =        

=    

    = ⋅          

=

  



(A.12)

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

elements) of the matrix.

Estimation Error Variance

Next, derivations are presented for the covariance matrix of 

the augmented state estimate and of the auxiliary parameter

estimate, ˆ xh k ˆ , P and Pz k ˆ, respectively. These matrices will each 

be calculated as a function of the reduced-order state vector 

estimation covariance matrix, Pxq k ˆ ˆ, , which is defined as 

( )( ) , ,

ˆ ˆ, , ,, , ˆ ˆˆ ˆ

T xq k xq k

T

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

+ ++ +

ε ε

 

  =− −    

 

  (A.13)

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

at time k and its expected value. Since , , ˆ ˆ Ex x xq k xq ss

+ +   =   , εxq,k can 

be obtained by subtracting Eq. (A.8) from Eq. (20)

( ) [ ]

( ) [ ]

( )( )

,

,

,

,, ,

, ,

,

ˆ

,

ˆ

,1 1

ˆ

ˆ ˆ

ˆ ˆ

ˆ

ˆ

ˆ

xq k

xq ss

xq k

xq k xq k xq k

xq k xq ss

k

xq xq k

k

x

ss

xq xq ss

ss

x

xq xq xq k xq k

x

x Ex

x x

y I KC x K KD u

y I KC x K KD u

I KC A x Bu

+

+

−

+ +

+ +

−

∞ ∞ ∞

−

∞ ∞ ∞

+

∞ −−

ε= −    

= −

 

=− + −    

 

−− + −    

= − +







 [ ]

( )( ) [ ]

( ) ( )

( )

,

,

,

ˆ

,

ˆ

ˆ

,1 ,

ˆ

ˆ ˆ

xq k

xq ss

xq ss

k

k

x

ss

xq xq xq ss xq ss

ss

x

x

xq xq xq k xq ss

k

xq xq

y K KD u

y I KC A x Bu K KD u

I KC A x x

y y

K KD I KC B

+

−

+

∞ ∞

+

∞ ∞ ∞

+ +

∞ −

∞∞ ∞

 

+ −    

  − − + + −    

= − −

−

+− −    

 









1

ss

k ss

k ss

u u

u u −

 

  −  

  −  

(A.14)

While the actuators will exhibit some variance due to measurement noise in the control feedback sensors, their contribution to the overall estimation variance is not as large as that of 

the sensor measurements. Therefore, to help simplify this 

derivation we will assume that the actuator commands are held 

constant at uss (they have no variance). With this simplifying 

assumption uk = uk+1 = uss, allowing Eq. (A.14) to reduce to

xq k, (I KC A x x K y y xq xq xq k xq ss ) ( ˆ ˆ ,1 , ) ( k ss ) + +

∞ −∞ ε =− − + − (A.15)
NASA/TM—2012-217278 14

Next, we can substitute Eq. (A.15) into Eq. (A.13) and 

manipulate to produce the following Riccati equation:

ˆ ˆ, , ˆ ˆ

T

xq k xq xq xq xq k xq xq xq

T

P A KC A P A KC A

K RK

∞ ∞

∞ ∞

=    − −   

+

 (A.16)

The above Riccati equation can be solved to calculate the 

covariance in the state and health parameters estimates, or in 

the auxiliary parameter estimates:

ˆ ˆ ˆ, ˆ , *† *†

0 0

0 0

T

xq k xh k

I I

P P

V V

   =      

(A.17)

*† *†

ˆ, ˆ ˆ,

T

P F NV P F NV z k xq k =       (A.18)

Sum of Squared Estimation Errors

Once equations (A.11), (A.12), (A.17), and (A.18) are 

obtained, they may be used to analytically calculate the mean 

sum of squared estimation errors over all engines by combining the respective mean squared estimation error bias and

estimation variance information. The mean sum of squared

estimation errors of the augmented state vector, ˆ x h ˆ, SSEE , and 

the mean sum of squared estimation errors of the auxiliary 

parameter vector, SSEEzˆ , become

{ }

{ }

{ }

{ }

2

ˆ ˆ , ˆ ˆ , ,

ˆ ˆ ,

2

ˆ ˆ,

ˆ,

xh fleet x h xh k

T

xh h xh xh k

z fleet z k

T

z h z zk

SSEE x tr P

tr G P G P

SSEE z tr P

tr G P G P

= +

= +

= +

= +





(A.19)

A weighting, Wz, can be applied to calculate a “weighted” sum 

of auxiliary parameter squared estimation errors given as

ˆ ˆ { , } T WSSEE tr W G P G P z z z h z zk =   +   (A.20)

The SSEE or WSSEE equation, given in Eqs. (A.19) and 

(A.20) respectively, can be directly applied within the iterative 

search for an optimal V* matrix as denoted in Eq. (22).

References

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

(1989), “Estimating In-Flight Engine Performance Variations Using Kalman Filter Concepts,” AIAA-89-2584, 

AIAA 25th Joint Propulsion Conference.

2. Volponi, A., (2008), “Enhanced Self-Tuning On-Board 

Real-Time Model (eSTORM) for Aircraft Engine Performance Health Tracking,” NASA/CR—2008-215272.

3. Kumar, A., Viassolo, D., (2008), “Model-Based Fault 

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

4. España, M.D., (1994), “Sensor Biases Effect on the 

Estimation Algorithm for Performance-Seeking Controllers,” J. Propulsion and Power, 10, pp. 527-532.

5. Litt, J.S., (2008), “An Optimal Orthogonal Decomposition Method for Kalman Filter-Based Turbofan Engine 

Thrust Estimation,” Journal of Engineering for Gas Turbines and Power, Vol. 130 / 011601-1.

6. Simon, D.L., Garg, S., (2010), “Optimal Tuner Selection 

for Kalman Filter-Based Aircraft Engine Performance 

Estimation,” Journal of Engineering for Gas Turbines 

and Power, Vol. 132 / 0231601-1.

7. Simon, D., (2006), Optimal State Estimation, Kalman, 

H∞, and Nonlinear Approaches, John Wiley & Sons, 

Inc., Hoboken, NJ.

8. Kwakernaak, H., Sivan, R., (1972), Linear Optimal 

Control Systems, John Wiley & Sons, Inc., New York, 

NY.

9. DeCastro, J.A., Litt, J.S., Frederick, D.K., “A Modular 

Aero-Propulsion System Simulation of a Large Commercial Aircraft Engine,” AIAA-2008-4579, 44th

AIAA/ASME/SAE/ASEE Joint Propulsion Conference 

& Exhibit, Hartford, CT, July 21-23, 2008. 

10. Volponi, A.J., 1999, “Gas Turbine Parameter Corrections,” Journal of Engineering for Gas Turbines and 

Power, Vol. 121, pp. 613–621.

11. Sallee, G.P., (1978) “Performance Deterioration Based 

on Existing (Historical) Data – JT9D Jet Engine Diagnostics Program,” NASA Contractor Report CR-135448, 

United Technologies Corporation, Pratt & Whitney Aircraft Group Report PWA-5512-21.

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Application of an Optimal Tuner Selection Approach for On-Board Self-Tuning Engine 

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6. AUTHOR(S)

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

An enhanced design methodology for minimizing the error in on-line Kalman filter-based aircraft engine performance estimation 

applications is presented in this paper. It specific-ally addresses the underdetermined estimation problem, in which there are more unknown 

parameters than available sensor measurements. This work builds upon an existing technique for systematically selecting a model tuning 

parameter vector of appropriate dimension to enable estimation by a Kalman filter, while minimizing the estimation error in the parameters 

of interest. While the existing technique was optimized for open-loop engine operation at a fixed design point, in this paper an alternative 

formulation is presented that enables the technique to be optimized for an engine operating under closed-loop control throughout the flight 

envelope. The theoretical Kalman filter mean squared estimation error at a steady-state closed-loop operating point is derived, and the tuner 

selection approach applied to minimize this error is discussed. A technique for constructing a globally optimal tuning parameter vector, 

which enables full-envelope application of the technology, is also presented, along with design steps for adjusting the dynamic response of 

the Kalman filter state estimates. Results from the application of the technique to linear and nonlinear aircraft engine simulations are 

presented and compared to the conventional approach of tuner selection. The new methodology is shown to yield a significant improvement 

in on-line Kalman filter estimation accuracy. 

15. SUBJECT TERMS

Systems health monitoring; Gas turbine engines 

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