Takahisa Kobayashi

QSS Group, Inc., Cleveland, Ohio

Donald L. Simon and Jonathan S. Litt

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

Application of a Constant Gain Extended

Kalman Filter for In-Flight Estimation of

Aircraft Engine Performance Parameters

NASA/TM—2005-213865

September 2005

ARL–TR–3489

GT2005–68494 U.S. ARMY

RESEARCH LABORATORY
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Takahisa Kobayashi

QSS Group, Inc., Cleveland, Ohio

Donald L. Simon and Jonathan S. Litt

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

Application of a Constant Gain Extended

Kalman Filter for In-Flight Estimation of

Aircraft Engine Performance Parameters

NASA/TM—2005-213865

September 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–3489

GT2005–68494 U.S. ARMY

RESEARCH LABORATORY
Acknowledgments

This research was funded by the NASA Aviation Safety and Security Program as a task under the Propulsion

System Safety Technologies Element.

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Available electronically at http://gltrs.grc.nasa.gov
NASA/TM—2005-213865 1

Application of a Constant Gain Extended Kalman Filter for In-Flight 

Estimation of Aircraft Engine Performance Parameters 

Takahisa Kobayashi 

QSS Group Inc. 

Cleveland, Ohio 44135 

Donald L. Simon and Jonathan S. Litt 

U.S. Army Research Laboratory 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

An approach based on the Constant Gain Extended Kalman 

Filter (CGEKF) technique is investigated for the in-flight 

estimation of non-measurable performance parameters of 

aircraft engines. Performance parameters, such as thrust and 

stall margins, provide crucial information for operating an 

aircraft engine in a safe and efficient manner, but they cannot 

be directly measured during flight. A technique to accurately 

estimate these parameters is, therefore, essential for further 

enhancement of engine operation. In this paper, a CGEKF is 

developed by combining an on-board engine model and a single 

Kalman gain matrix. In order to make the on-board engine 

model adaptive to the real engine’s performance variations due 

to degradation or anomalies, the CGEKF is designed with the 

ability to adjust its performance through the adjustment of 

artificial parameters called “tuning parameters.” With this 

design approach, the CGEKF can maintain accurate estimation 

performance when it is applied to aircraft engines at offnominal conditions. The performance of the CGEKF is 

evaluated in a simulation environment using numerous 

component degradation and fault scenarios at multiple 

operating conditions. 

Introduction 

Aircraft gas turbine engines can be operated with enhanced 

efficiency while ensuring safe operation if performance 

parameters, such as thrust and compressor stall margin, are 

known. Since these performance parameters are not directly 

measurable during flight, they can only be controlled indirectly 

through the utilization of measurable variables in the feedback 

control architecture. This aspect of indirectly controlling 

performance parameters introduces uncertainty in the engine 

operation and thus necessitates that additional safety margins 

be preserved. As a result, aircraft engines are typically operated 

at conditions where more than sufficient safety margins are 

available, while efficiency is compromised. However, under 

extreme conditions and/or events, engine safety margins may 

also be compromised. 

Accurate in-flight estimation of aircraft engine 

performance parameters is, therefore, desired to advance the 

feedback control strategy and consequently enhance aircraft 

engine safety and efficiency. Some researchers have 

investigated the application of on-board engine models for the 

estimation of performance parameters (refs. 1 to 3). An onboard model can be a linear or nonlinear representation of the 

physical aircraft engine, and it can compute non-measurable 

parameters. Since an on-board model represents the “nominal” 

engine, it must be adapted to the performance of the real engine 

as it deviates from the nominal baseline with time. 

A well-known approach for adapting an on-board model to 

an off-nominal engine is to estimate health parameters using a 

Kalman filter (refs. 4 to 6). Health parameters, such as 

efficiency and flow capacity, represent performance 

deteriorations of engine components. They deviate from the 

nominal baseline gradually with time due to normal usage and 

also abruptly due to component fault events. A challenging 

aspect of this estimation approach is that the number of health 

parameters is, in general, greater than the number of available 

sensors. Therefore, only a subset of health parameters can be 

estimated assuming the others remain unchanged. In reality, all 

health parameters will change, and this calls into question what 

will happen to the adapted on-board model if those health 

parameters not being estimated deviate from the nominal. 

In previous research (ref. 7), it was shown that a Kalman 

filter, designed to estimate a subset of health parameters, could 

be used to accurately estimate engine sensor outputs even if all 

health parameters were subject to change. The Kalman filter 

maintained its accurate sensor matching performance by 

attributing the shifts in any health parameters to a subset of 

health parameters being estimated. Consequently, those 

estimated health parameters no longer represented the actual 

health parameters; they became tuning parameters to facilitate 

the output matching performance of the Kalman filter. 

In this paper, the approach of tuning is extended to the 

estimation of engine performance parameters. To accurately 

estimate engine performance parameters, a constant gain 

extended Kalman filter (CGEKF) is used. The CGEKF 

approach (refs. 8 and 9) is similar to the general extended 

Kalman filter approach (ref. 10) in which a nonlinear model of 

the plant is used as a part of the Kalman filter. However, unlike 

the general approach, the linearization of the plant model and 

the computation of Kalman gains are done off-line. A CGEKF 

is formed by combining pre-computed single or multiple 
NASA/TM—2005-213865 2

Kalman gain matrices and the nonlinear model. A significant 

advantage of the CGEKF approach is that a large nonlinear 

operational range can be covered while the computational 

burden is much less than that of the general extended Kalman 

filter approach. Its stability, however, must be carefully 

evaluated since the problem is inherently nonlinear. 

In the following sections of this paper, the CGEKF design 

approach is described, followed by its application to a nonlinear 

simulation of a large commercial aircraft engine model. Then, 

the stability and estimation performance of the CGEKF are 

evaluated in a simulation environment using numerous 

component degradation and fault scenarios at multiple 

operating conditions. Finally, a discussion regarding further 

design enhancement is given, followed by conclusions. 

Nomenclature 

BST Booster 

FAN Fan 

HPC High Pressure Compressor 

HPT High Pressure Turbine 

LPT Low Pressure Turbine 

P17 Bypass discharge pressure 

P25 HPC inlet pressure 

PLA Power Lever Angle 

PS3 Combustor inlet static pressure 

T25 HPC inlet temperature 

T3 Combustor inlet temperature 

T4 Combustor temperature 

T49 LPT inlet temperature 

TMHS23 BST metal temperature 

TMHS3 HPC metal temperature 

TMHS41 HPT nozzle metal temperature 

TMHS42 HPT metal temperature 

TMHS5 LPT metal temperature 

TMHSBC Combustor case metal temperature 

TMHSBL Combustor liner metal temperature 

VBV Variable bleed valve 

VSV Variable stator vane 

WF36 Fuel flow 

XN12 Low-pressure spool speed, measured 

XN25 High-pressure spool speed, measured 

XNH High-pressure spool speed, state variable 

XNL High-pressure spool speed, state variable 

d Tuning parameter vector 

dref Subset of href

e Environmental parameter vector 

h Health parameter vector 

href Reference health condition vector 

ucmd Control command vector 

v Sensor noise vector 

x State variable vector 

y Sensor output vector 

z Engine performance parameter vector 

Development of Constant Gain Extended 

Kalman Filter 

The overall architecture of the propulsion system 

integrated with a Constant Gain Extended Kalman Filter 

(CGEKF) is shown in figure 1. In the general control approach, 

the Digital Engine Control Unit (DECU) adjusts its actuator 

commands based on the sensor feedback measurements. Engine 

outputs such as spool speeds, pressures, and temperatures are 

measured by the sensors. Based on the information provided by 

the available sensors, the DECU operates the engine to meet the 

power demand (thrust) from the aircraft while maintaining the 

engine operation within the mechanical (e.g., spool speeds), 

thermal (e.g., combustor temperature), and aerodynamic (e.g., 

stall margins) limits. Since performance parameters, such as 

thrust, stall margins, and combustor temperature, are not 

directly measured during flight, they can only be inferred from 

the available sensor output values. Because of this aspect of 

indirectly controlling the performance parameters, it is difficult 

to operate the engine in an optimal fashion in terms of safety 

and efficiency. 

As shown in the figure, the CGEKF uses two sets of input 

signals: sensor measurements and control commands. Given 

these inputs, the CGEKF estimates the state variables, sensor 

outputs, and non-measurable performance parameters. If the 

CGEKF is able to accurately estimate performance parameters, 

those estimated values can be fed into the DECU, as indicated 

by the dashed line, for the enhancement of engine operation. 

A critical aspect in designing a CGEKF is to make it robust 

to off-nominal engine behavior caused by both degradation and 

anomalies. Component degradation is a natural phenomenon 

which occurs to all aircraft engines as a result of normal usage. 

The level of component degradation worsens gradually with 

time, and consequently the engine performance deviates 

gradually from its nominal level. Component faults similarly 

result in off-nominal engine performance, but they occur 

abruptly due to anomalous events such as foreign or domestic 

object damage. Sensors and actuators may also exhibit 

anomalous behavior due to the presence of bias and drift. 

Without robustness to component degradation and anomalies, it 

is possible to encounter a scenario where incorrect estimation 

by the CGEKF leads mission-critical elements (e.g., control 

system, flight crew) to take incorrect action. 

In this paper, a CGEKF is designed so that it is robust to 

component degradation and component faults. Its robustness to 

actuator biases is also investigated. In the presence of a sensor 

fault, the CGEKF tends to accurately estimate all sensor 

outputs, including the faulty measurement, without recognizing 

that a fault exists. In the meantime, as a result of a sensor fault, 

the estimation accuracy of the performance parameters 

becomes poor. As such, the CGEKF is not able to handle sensor 

faults by itself, and therefore CGEKF performance against 

sensor faults is not investigated in this paper, assuming that 

sensor faults are handled by an in-flight sensor fault detection 

and isolation system. In the following section, the design 

process of the CGEKF is discussed in detail. 
NASA/TM—2005-213865 3

CGEKF Design Approach 

The design approach of a CGEKF is exactly the same as 

that of general linear Kalman filter. First, a nonlinear plant 

model is linearized at operating points, and then Kalman gains 

are computed based on the linear representations of the plant 

model. When implemented, however, Kalman gains are 

integrated with the nonlinear model instead of linear models. 

The nonlinear model of an aircraft engine is represented by the 

following equations: 

( )

( , , , )

( , , , )

, , ,

z g x h u e

y g x h u e v

x f x h u e

z cmd

y cmd

cmd

=

= +

 =

 (1) 

where x, h, u, and e represent the vectors of state variables, 

health parameters, control command inputs, and environmental 

parameters, respectively. A health parameter, such as 

compressor efficiency or flow capacity, indicates the “health” 

of an engine component. For given input values, the nonlinear 

functions f, gy, and gz generate the vectors of state derivatives 

x , sensor outputs y, and non-measurable performance 

parameters z, respectively. The sensor outputs are corrupted by 

a white noise vector v. By linearizing the engine model, the 

following state-space equations are obtained: 

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ss z ss ref z cmd ss

ss y ss ref y cmd ss

ss ref cmd ss

z z C x x N h h D u u

y y C x x M h h D u u v

x A x x L h h B u u

− = − + − + −

− = − + − + − +

 = − + − + −

 (2) 

where A, B, Cy, Cz, Dy, Dz, L, M, and N are the state-space 

matrices with appropriate dimensions. The subscript “ss” 

indicates the steady-state point at which the engine model is 

linearized. The vector href represents a reference health 

condition. The health parameters deviate gradually or abruptly 

from this reference over the lifetime of engine operation. As 

can be seen in the above equations, health parameter deviations 

cause the state derivatives, sensor outputs, and performance 

parameters to deviate from their nominal condition values. One 

approach to make the estimation capability of a Kalman filter 

adaptive to health parameter deviations is to estimate the health 

parameter vector in addition to the state variable vector. If all 

health parameters can be accurately estimated, the state 

variables, sensor outputs, and performance parameters are also 

accurately estimated regardless of the health parameter 

deviations. However, to estimate all health parameters, the 

number of available sensors must be at least equal to the 

number of health parameters (ref. 11). In general, this condition 

is not met, especially for commercial aircraft engines. 

If the objective of utilizing a Kalman filter is to accurately 

estimate sensor outputs regardless of health parameter 

deviations, this objective can be met by estimating a subset of 

health parameters. In reference 7, it was shown that a Kalman 

filter designed to estimate a subset of health parameters actually 

“tuned” these selected health parameters instead of estimating 

their actual values in the presence of deviations in all health 

parameters. Through this tuning, the Kalman filter maintained 

its sensor output estimation accuracy. The estimated health 

parameters, however, no longer represented the actual health 

condition; they simply became tuning parameters to facilitate 

the output matching performance of the Kalman filter. In the 

present research, this tuning approach is investigated further to 

determine whether it will enable the Kalman filter to accurately 

estimate performance parameters in addition to sensor outputs. 

The linear Kalman filter equation that incorporates the 

tuning parameters (a subset of health parameters) is given as 

follows: 

( ) ( )

( )

[ ] ( ) z cmd ss ref

ss

ss z d

y cmd ss ref

ss

ss aug

cmd ss ref

ss

aug

D u u d d

x x z z C N

D u u d d

x x

y y C

u u K y y

B

d d

x x

A

d

x

⎥ + −

⎦

⎤ ⎢

⎣

⎡

−

− − =

⎥ + −

⎦

⎤ ⎢

⎣

⎡

−

− − =

⎥ − + −

⎦

⎤ ⎢

⎣

⎡ ⎥ +

⎦

⎤ ⎢

⎣

⎡

−

− = ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

ˆ ˆ

ˆ ˆ

ˆ 0

ˆ ˆ





 (3) 

where 

[ ] aug y d

d

aug C C M

A L

A ⎥ =

⎦

⎤ ⎢

⎣

⎡ = , 0 0

The vectors xˆ and yˆ represent the estimates of the state 

variables and sensor outputs, respectively. The vector d

contains the tuning parameters, and dref is a subset of href

y

Digital

Engine

Control

Unit

CGEKF

Actuators Sensors

ucmd utrue

v

Engine

Power Demand

from Aircraft

yˆ, Thrust, Stall Margin y

Digital

Engine

Control

Unit

CGEKF

Actuators Sensors

ucmd utrue

v

Engine

Power Demand

from Aircraft

yˆ, Thrust, Stall Margin

Figure 1.—Architecture of Propulsion System Integrated with CGEKF. 
NASA/TM—2005-213865 4

corresponding to those tuning parameters. The matrices Ld, Md, 

and Nd are selected columns of L, M, and N respectively, and 

their columns correspond to the tuning parameters. The matrix 

K represents the Kalman gain. In order for the Kalman gain to 

converge, the matrix pair (Aaug, Caug) must be observable. 

The linear Kalman filter is composed of a linear plant 

model, represented by state-space matrices and steady-state 

value vectors, and the corresponding Kalman gain. To construct 

a CGEKF, the linear plant model is simply replaced by the 

nonlinear model of equation (1), as shown in the following 

CGEKF equation: 

( ) ( )

ˆ (ˆ, , , )

ˆ (ˆ, , , )

ˆ 0

ˆ ˆ, , ,

z g x d u e

y g x d u e

K y y

f x d u e

d

x

z cmd

y cmd

cmd

=

=

⎥ + −

⎦

⎤ ⎢

⎣

⎡ =

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡





 (4) 

As can be seen in the above equation, the tuning parameters are 

embedded in the Kalman filter design. If sensor outputs deviate 

from nominal condition values due to component degradation 

and/or faults, the Kalman filter will attribute the cause of sensor 

output deviations (i.e., deviations in health parameters) to the 

tuning parameters, so that the residuals ( ) y − yˆ will remain 

small. 

An advantage of the CGEKF approach (eq. 4) over the 

linear Kalman filter approach (eq. 3) is its capability to cover a 

wide operating range. For the linear Kalman filter approach, a 

single Kalman filter cannot cover the typical range of operating 

conditions that aircraft engines will experience. Consequently, 

piece-wise Kalman filters must be designed at different 

operating conditions, and then they must be interpolated 

through scheduling parameters which indicate the specific 

condition at which an engine is operating at a given time. For 

the CGEKF approach, a single or a few Kalman gain matrices 

are sufficient to cover the entire engine operating envelope as 

indicated in reference 9. This is a significant advantage in terms 

of design complexity. 

Moreover, the CGEKF approach can capture the 

nonlinearity of aircraft engine operation under the influence of 

anomalies. For instance, when an engine experiences 

component faults, the control system adjusts the actuator 

positions based on the sensor feedback measurements in order 

to meet its objective (e.g., to maintain fan speed at the 

commanded value). Because of this control adjustment, in 

addition to component faults, the engine moves to a new 

operating condition which may be a significant deviation from 

the nominal condition. Piece-wise linear Kalman filters, 

designed along the operating line of the nominal condition, can 

not capture such off-nominal behavior of an aircraft engine due 

to closed-loop control effects in the presence of anomalies 

(ref. 7). 

The CGEKF approach, however, also has disadvantages. 

The CGEKF is much more computationally intensive than the 

linear approach since the nonlinear plant model must be 

executed in real-time. Moreover, the nonlinear approach may 

not be as numerically robust as the linear approach as 

mentioned in reference 9. Therefore, the stability of the 

CGEKF must be carefully evaluated. In the following sections, 

the CGEKF approach is applied to the nonlinear simulation of a 

large commercial aircraft engine, and its performance and 

stability are evaluated. 

Application of CGEKF Technique to an 

Aircraft Engine Simulation 

In this section, the CGEKF technique is applied to an 

aircraft engine simulation. A description of the engine 

simulation is given first. Then the CGEKF design-specific 

issues are discussed. 

Engine Model 

The engine model used in this paper is a nonlinear 

simulation of an advanced high-bypass turbofan engine, a 

typical power plant for large commercial aircraft. This engine 

model has been constructed as a Component Level Model 

(CLM), which consists of the major components of an aircraft 

engine. The CLM represents highly complex engine physics 

while having the capability to run in real-time. Engine 

performance deviations from the nominal baseline are modeled 

by adjustments to efficiency and flow capacity scalars of the 

following five components: Fan (FAN), Booster (BST), HighPressure Compressor (HPC), High-Pressure Turbine (HPT), 

and Low-Pressure Turbine (LPT). A total of 10 adjustments are 

called health parameters. These health parameters are used to 

represent component degradation and faults. The engine state 

variables, health parameters, actuator variables, environmental 

parameters, and performance parameters are shown in table 1. 

There are two sets of sensor suites to be used in this study. 

These two sets are shown in table 2 along with the standard 

deviation of the sensor noise given in percent of steady-state 

values at the ground maximum power condition. 

TABLE 1.—ENGINE MODEL VARIABLES 

State Variables XNL, XNH, TMHS23, TMHS3 

TMHSBL, TMHSBC, TMHS41 

TMHS42, TMHS5 

Health 

Parameters 

FAN efficiency, FAN flow capacity 

BST efficiency, BST flow capacity 

HPC efficiency, HPC flow capacity 

HPT efficiency, HPT flow capacity 

LPT efficiency, LPT flow capacity 

Actuators WF36, VBV, VSV 

Environmental 

Parameters 

Altitude, Mach Number 

Ambient Temperature 

Performance 

Parameters 

Thrust, FAN stall margin, BST stall 

margin, HPC stall margin, T4 

Sensor set no. 1 is the standard sensor suite available to the 

digital engine control unit of this engine. Sensor set no. 2 is 

composed of sensor set no. 1 and one additional sensor: P17. 
NASA/TM—2005-213865 5

TABLE 2.—SENSOR SETS AND SENSOR STANDARD 

DEVIATION (σ IN PERCENT OF STEADY-STATE 

VALUES AT GROUND MAXIMUM 

POWER CONDITION) 

Sensor Set no. 1 Sensor Set no. 2 σ (%) 

XN12 XN12 0.25 

XN25 XN25 0.25 

------- P17 0.50 

P25 P25 0.50 

T25 T25 0.75 

PS3 PS3 0.50 

T3 T3 0.75 

T49 T49 0.75 

During a preliminary study of a linear Kalman filter design, it 

was found that the P17 sensor is crucial to relate FAN 

component performance to thrust. Without this sensor, a linear 

Kalman filter was not able to generate an accurate thrust 

estimate when the health parameters for the FAN component 

were shifted. Therefore, there is a limit on the achievable 

accuracy for thrust estimation if sensor set no. 1 is used. Sensor 

set no. 2 is used to investigate the benefits of having the P17 

sensor. 

Engine control is performed based on sensor set no. 1 (i.e., 

the P17 sensor has no influence on control action). For the 

major portion of a typical flight profile, the corrected lowpressure spool speed is regulated by the control system. In the 

current control architecture, the power level angle (PLA) is 

converted to desired corrected low-pressure spool speed (an 

indicator of thrust). The control system adjusts three actuation 

values to cause the corrected measured low-pressure spool 

speed to match the desired value. 

Kalman Gain Design Condition 

A steady-state Kalman gain is computed based on statespace matrices derived at a specific operating point. Single or 

multiple Kalman gain matrices generated off-line are then used 

to construct a CGEKF. In this paper, a single Kalman gain 

matrix is used. The major significance of the CGEKF approach 

is that, as shown by Safonov et al. (ref. 8), a non-divergent 

nonlinear estimator can be designed for systems with 

substantial nonlinearity even when a relatively crude linear 

representation is used for computing a Kalman gain. The nondivergence proof in reference 8 is derived based on the 

assumption that the plant model varies as a function of known 

or observed variables. In the present paper, the true plant varies 

as a function of health parameters, but the plant model uses a 

subset of health parameters which are “tuned” rather than 

“estimated.” Therefore, the operating condition of the plant 

model will differ from that of the true plant, thus violating the 

previously stated assumption. However, the powerful nondivergence aspect of the CGEKF approach is investigated 

empirically by covering a wide operating range just using a 

single Kalman gain matrix. The design operating point for 

computing the Kalman gain is set to the ground maximum 

power condition as was done by Sugiyama (ref. 9), while the 

reference health condition (href in eq. 2) is set to the nominal or 

“healthy” baseline. It should be noted that the selection of 

design condition is left to engineering judgment, and the 

performance of the CGEKF will vary with selected design 

conditions. 

Selection of Tuning Parameters 

The selection of tuning parameters is a critical part in the 

CGEKF design process. It is desirable to find a subset of health 

parameters which can capture the major influence of all 10 

health parameters. The selection of tuning parameters for the 

case of engine-to-engine variation (i.e., relatively small 

deviations in all health parameters) was investigated by Brunell 

et al. (ref. 12). In this reference, an index, which indicates the 

level of observability and sensitivity, was determined for each 

health parameter through singular value decomposition of statespace matrices (eq. 2). Based on the index values, a subset of 

health parameters to be estimated was selected. The same 

technique was applied in the present paper; however, some 

issues were encountered. First, a subset of health parameters 

selected from this approach varied significantly with operating 

point. Since a wide operating range is covered by a CGEKF, it 

is difficult to determine a “global” subset of health parameters 

from the current approach. Another issue that needs to be 

considered is the highly nonlinear aspect of the present 

problem. In the present problem where component degradation 

and faults are considered, all health parameters are shifted 

beyond the level of engine-to-engine variation. When a subset 

of health parameters is estimated in the presence of such 

relatively large shifts, the estimated values often become large 

and exceed the linear range. The approach based on singular 

value decomposition of state-space matrices is not able to 

capture this nonlinear aspect of the problem. Therefore, some 

engineering judgment is still needed to select tuning 

parameters. 

Table 3 shows the two sets of tuning parameters for the 

CGEKF designed with sensor sets no. 1 and 2. It can be seen 

that the CGEKF design with sensor set no. 1 does not use the 

health parameters of the FAN component. During a preliminary 

evaluation of the CGEKF performance, it was noted that the 

use of either FAN health parameter as a tuning parameter 

causes instability when component degradation is present. 

Without the P17 sensor, it appears that the CGEKF is not able 

to converge either FAN health parameter to a steady-state 

condition, and consequently the system becomes unstable. 

TABLE 3.—TUNING PARAMETERS FOR 

TWO CGEKF DESIGNS 

CGEKF Design with 

Sensor Set no. 1 

CGEKF Design with 

Sensor Set no. 2 

BST flow FAN flow 

HPC efficiency BST flow 

HPC flow HPC efficiency 

HPT efficiency HPC flow 

HPT flow HPT efficiency 

LPT flow HPT flow 

 LPT flow 
NASA/TM—2005-213865 6

ucmd

Engine

(CLM)

y

h

On-Board Model

(CLM)

K

yˆ

zˆ

v

e

CGEKF

bias

-

ucmd

Engine

(CLM)

y

h

On-Board Model

(CLM)

K

yˆ

zˆ

v

e

CGEKF

bias

-

Figure 2.—Simulation Setup for CGEKF Evaluation. 

Implementation of CGEKF in the Simulation Environment 

The simulation setup to run the CGEKF is shown in 

figure 2. The nonlinear engine model (CLM) is used to 

represent both actual engine and on-board model. Off-nominal 

health conditions of an actual engine are modeled by shifting 

all 10 health parameters of the nonlinear engine model from the 

healthy baseline. It is assumed that the health condition of the 

actual engine is unknown. The reference health condition of the 

on-board model is set to the healthy baseline at which a Kalman 

gain was computed. (Again, the selection of the reference 

health condition is up to the designer.) The CGEKF is 

implemented by integrating the “healthy” on-board model and 

the single Kalman gain matrix. When the CGEKF is applied to 

the nonlinear engine model at off-nominal health conditions, 

the on-board model is adapted to the performance of the 

“degraded” engine through the adjustment of tuning 

parameters. In addition to component degradation, actuator 

biases can also be injected to simulate off-nominal operation. 

The nonlinear engine model and the CGEKF are set to run in a 

discrete fashion with a 0.02-second time step. 

Performance Evaluation of CGEKF 

In this section, the stability and the estimation accuracy of 

the CGEKF are evaluated. The evaluation is conducted first at 

the nominal condition and then at various off-nominal 

conditions. As shown in figure 2, off-nominal conditions are 

modeled by the following two types of engine variation 

sources: 1) component degradation, and 2) actuator biases. For 

each engine variation source type, 100 cases are created by 

randomly shifting the vector elements of health parameters and 

actuator biases. For each case, the nonlinear engine model and 

the CGEKF are run through ground-idle, takeoff, climb and 

then cruise phases of a typical flight (Altitude: 0.0 ~ 35,000 ft, 

Mach: 0.0 ~ 0.84). The flight duration is 2500 seconds. 

Stability of the CGEKF is evaluated through a total of 

200 simulation runs (2 engine variation source types 

× 100 cases) of the typical flight profile. 

Estimation accuracy of the CGEKF at off-nominal 

conditions is also evaluated using the same 100 cases for each 

engine variation source type. For each run, a 100-second time 

history of the estimation error is taken from each of the groundidle, climb, and cruise phases. Using 100-second time histories 

from 100 cases, the mean and the maximum estimation errors 

are computed. 

At the end of this section, the responses of the CGEKF to 

abrupt shifts in health parameters (component faults) are 

investigated. 

Evaluation 1: Engine at the Healthy Condition 

The CGEKF is first applied to the nonlinear engine model 

at the healthy condition (no component degradation). The 

estimation performance of the CGEKF of this nominal case 

establishes the baseline with which the performance at offnominal cases can be compared. For the evaluation of the 

estimation accuracy, the following absolute percent estimation 

errors are computed for the sensor outputs and the engine 

performance parameters: 

true

true error

true

true error

z

z z z

y

y y

y

− = ⋅

− = ⋅

ˆ 100

ˆ 100

 (5) 

where the vectors ytrue and ztrue are true engine outputs (no 

noise) and performance parameters, respectively, generated by 

the engine model. A 100-second time history of the absolute 

estimation error is taken from each of the ground-idle, climb, 

and cruise phases of the typical flight profile. Then, the mean 

and the maximum of absolute percent estimation errors are 

computed for each 100-second segment. The result of the 

estimation performance is shown in table 4 for the two CGEKF 

designs using sensor sets no. 1 and 2. 

It can be seen that both CGEKF designs accurately 

estimate the sensor outputs. Since both the nonlinear engine 

model and CGEKF are set to the nominal health condition, the 

estimation error is due to the sensor noise. The largest 

maximum sensor output estimation errors for sensor set no. 1 

and 2 are 0.44 percent (T3) and 0.52 percent (T49), 

respectively. For the performance parameters, both CGEKF 

designs accurately estimate thrust, FAN stall margin, and T4. It 

can be seen that the estimates of BST and HPC stall margins 

are not as accurate as those of the other performance 

parameters. 

The estimation accuracy varies with the operating 

condition, and this variation is due to two factors. First of all, 

the true values used in equation (5) for normalization vary with 

operating condition. For instance, XN12 is higher at cruise than 

at ground-idle. Thus, a smaller percent estimation error in the 

cruise phase versus the ground-idle phase does not mean that 

the actual error is smaller. The second factor is that the Kalman 

gain was computed at a specific operating point: ground 

maximum power condition. None of the three operating 

conditions used in the evaluation exactly match the design 

point. The ambient condition of the ground-idle is identical to 
NASA/TM—2005-213865 7

TABLE 4.—MEAN AND MAXIMUM ABSOLUTE PERCENT ESTIMATION ERROR 

FOR HEALTHY ENGINE (MAXIMUM ERROR IN PARENTHESES) 

 CGEKF Design with 

Sensor Set no. 1 

 CGEKF Design with 

Sensor Set no. 2 

 ground climb cruise ground climb cruise 

XN12 0.05 

(0.25) 

0.02 

(0.08) 

0.02 

(0.09) 

 0.06 

(0.28) 

0.02 

(0.07) 

0.02 

(0.08) 

XN2 0.02 

(0.09) 

0.03 

(0.09) 

0.02 

(0.07) 

 0.02 

(0.08) 

0.03 

(0.10) 

0.02 

(0.08) 

P17 ------- ------- ------- 0.01 

(0.02) 

0.04 

(0.13) 

0.04 

(0.15) 

P25 0.02 

(0.04) 

0.06 

(0.22) 

0.06 

(0.27) 

 0.03 

(0.08) 

0.07 

(0.25) 

0.08 

(0.31) 

T25 0.02 

(0.07) 

0.02 

(0.07) 

0.02 

(0.10) 

 0.02 

(0.08) 

0.02 

(0.08) 

0.03 

(0.11) 

PS3 0.09 

(0.37) 

0.08 

(0.29) 

0.08 

(0.31) 

 0.07 

(0.19) 

0.06 

(0.22) 

0.05 

(0.19) 

T3 0.08 

(0.44) 

0.08 

(0.32) 

0.08 

(0.34) 

 0.09 

(0.49) 

0.09 

(0.34) 

0.09 

(0.36) 

T49 0.08 

(0.41) 

0.06 

(0.24) 

0.07 

(0.27) 

 0.11 

(0.52) 

0.08 

(0.29) 

0.08 

(0.35) 

 

Thrust 0.08 

(0.43) 

0.03 

(0.14) 

0.03 

(0.12) 

 0.19 

(0.64) 

0.05 

(0.22) 

0.06 

(0.21) 

FAN Stall 

Margin 

 0.03 

(0.13) 

0.16 

(0.57) 

0.15 

(0.63) 

 0.10 

(0.40) 

0.21 

(0.71) 

0.21 

(0.89) 

BST Stall 

Margin 

 0.30 

(1.09) 

1.19 

(4.74) 

1.21 

(5.17) 

 0.47 

(1.69) 

1.38 

(5.36) 

1.46 

(6.04) 

HPC Stall 

Margin 

 1.04 

(4.97) 

1.17 

(4.77) 

1.24 

(5.19) 

 0.93 

(4.62) 

1.20 

(4.80) 

1.24 

(5.06) 

T4 0.11 

(0.55) 

0.08 

(0.34) 

0.09 

(0.36) 

 0.13 

(0.65) 

0.10 

(0.39) 

0.10 

(0.41) 

the design point, but the power setting of the design point is 

closest to that of the climb phase. 

Evaluation 2: Engine at Degraded Conditions 

All health parameters of an aircraft engine deviate from 

the healthy baseline with time due to wear and tear on engine 

components. In the real environment, the level of such 

deviations will differ from engine to engine depending on how 

and where an engine was operated. In this section, the CGEKF 

is applied to the nonlinear engine model at degraded 

conditions. One hundred degradation cases are created by 

randomly shifting all 10 health parameters. The deviation 

values are uniformly distributed over the range from 1 to 

5 percent, and this level of deviation is beyond the typical 

level of engine-to-engine variation due to manufacturing 

tolerance. With a given degradation, the nonlinear engine 

model is run through the typical flight profile described 

earlier. The given degradation is fixed throughout the flight. 

Since degradation causes shifts in the sensor outputs and the 

control system uses the sensor outputs to adjust its commands, 

the closed-loop system response is unique for each 

degradation case. 

When applied to a degraded engine, the estimates of the 

CGEKF upon initialization do not match the actual values 

because the CGEKF has been designed for an engine at the 

healthy condition. This mismatch results in large residuals 

( y − yˆ ), and the CGEKF tries to reduce the residuals by 

adjusting tuning parameters. If the CGEKF is able to maintain 

small residuals throughout the flight profile, it means that the 

influence of component degradation can be captured by tuning 

parameter adjustments at any point along the operating path. 

For 100 degradation cases, the CGEKF was stable throughout 

the flight profile. Thus, the influence of component 

degradation was successfully captured to the level where 

stability is maintained. 

The CGEKF maintained its stability under the influence 

of component degradation. However, it is reasonable to expect 

that the level of estimation accuracy will become worse, 

compared to the estimation accuracy of the nominal condition 

case, unless the influence of component degradation is 

completely accounted for. To investigate estimation 

performance, the estimation accuracy of the CGEKF is 

evaluated through the following process. From each 

degradation case, a 100-second time history of the absolute 

estimation error in equation (5) is taken from each of the 

ground-idle, climb, and cruise phases. Then, the mean and the 

maximum of absolute percent estimation errors of 

100 degradation cases, each of which consists of a 100-second 

time history, are computed for each flight segment. The result 

of the estimation performance is shown in table 5 for the two 

CGEKF designs using sensor sets no. 1 and 2. 

Again, both CGEKF designs are successful in accurately 

estimating sensor outputs. The largest maximum sensor output 
NASA/TM—2005-213865 8

estimation errors for sensor set no. 1 and 2 are 0.80 percent 

(T25) and 0.79 percent (T25), respectively. The benefit of 

having the P17 sensor can be seen in the estimation accuracy 

of thrust and FAN stall margin. Especially for thrust 

estimation, the level of estimation accuracy under degraded 

conditions is comparable to that of the healthy engine case. As 

mentioned earlier, the P17 sensor is crucial for relating the 

FAN performance to thrust. Without this sensor, the 

estimation accuracy that the CGEKF can achieve is limited. 

The estimation accuracy for BST and HPC stall margins 

gets worse compared to the healthy engine case. The difficulty 

of estimating these stall margins may be due to the fact that 

the sensors and the tuning parameters being used are not 

appropriate. In the current CGEKF approach, 10 health 

parameters are mapped into a subset of health parameters 

(tuning parameters) through the matching of sensor output 

estimates to the measurements. Since the influence of 

10 health parameters can not be fully captured by the tuning 

parameters, some information is lost in the reconstruction of 

the performance parameter estimates. It is considered that the 

lost information is critical for the estimation of BST and HPC 

stall margins. 

Although some of the performance parameters are not 

accurately estimated, those estimates with high accuracy have 

some potential to improve the engine performance through 

their utilization in the control algorithm. For instance, with the 

spool-speed control used in this study, degradation causes 

deviation in thrust from the nominal target value. This thrust 

deviation is about 2 percent on average at steady-state cruise 

conditions. Since the average thrust estimation error for the 

CGEKF design with sensor set no. 2 (0.07% in table 5) is 

much smaller than the average thrust deviation, direct control 

of estimated thrust can result in better thrust tracking 

performance than the spool-speed control. Through direct 

thrust control, benefits such as avoiding asymmetric thrust can 

be achieved. 

Evaluation 3: Healthy Engine with Biased Actuators 

In the real environment, there are many elements, other 

than those health parameters considered in this paper, which 

deviate from their nominal baseline conditions over the 

lifetime of engine operation. Some examples of such elements 

are seal leakage, combustor efficiency, and actuator biases. 

(Sensor biases are not considered here, as discussed earlier.) 

These elements can cause the engine to operate at off-nominal 

conditions, but they are not directly accounted for in the 

current CGEKF design process. In this section, actuator biases 

are used to evaluate the CGEKF performance at off-nominal 

conditions other than those “anticipated” off-nominal 

conditions due to component degradation and/or faults. 

One hundred cases of actuator anomalies are created by 

randomly selecting the bias values for three actuators. The 

bias values are uniformly distributed over the range of 

±10 percent of the steady-state actuator command values at the 

ground-idle condition. With a given set of actuator biases, the 

TABLE 5.—MEAN AND MAXIMUM ABSOLUTE PERCENT ESTIMATION ERROR FOR 

100 CASES OF DEGRADED ENGINES (MAXIMUM ERROR IN PARENTHESES) 

 CGEKF Design with 

Sensor Set no. 1 

 CGEKF Design with 

Sensor Set no. 2 

 ground climb cruise ground climb cruise 

XN12 0.05 

(0.24) 

0.02 

(0.09) 

0.02 

(0.10) 

 0.05 

(0.27) 

0.02 

(0.07) 

0.01 

(0.07) 

XN2 0.02 

(0.10) 

0.03 

(0.15) 

0.02 

(0.10) 

 0.02 

(0.10) 

0.02 

(0.16) 

0.02 

(0.09) 

P17 ---- ---- ---- 0.01 

(0.03) 

0.04 

(0.17) 

0.05 

(0.21) 

P25 0.02 

(0.07) 

0.07 

(0.30) 

0.06 

(0.28) 

 0.03 

(0.09) 

0.06 

(0.33) 

0.07 

(0.31) 

T25 0.04 

(0.11) 

0.42 

(0.80) 

0.34 

(0.67) 

 0.04 

(0.12) 

0.42 

(0.79) 

0.34 

(0.66) 

PS3 0.08 

(0.36) 

0.07 

(0.34) 

0.07 

(0.29) 

 0.08 

(0.32) 

0.04 

(0.31) 

0.04 

(0.18) 

T3 0.08 

(0.45) 

0.08 

(0.37) 

0.08 

(0.37) 

 0.09 

(0.50) 

0.08 

(0.39) 

0.09 

(0.39) 

T49 0.08 

(0.45) 

0.06 

(0.27) 

0.06 

(0.30) 

 0.10 

(0.56) 

0.07 

(0.30) 

0.07 

(0.35) 

 

Thrust 1.47 

(2.96) 

3.44 

(6.88) 

2.91 

(5.80) 

 0.20 

(0.66) 

0.08 

(0.43) 

0.07 

(0.31) 

FAN Stall 

Margin 

 0.77 

(1.47) 

5.92 

(12.29) 

4.88 

(10.15) 

 0.11 

(0.47) 

0.74 

(2.12) 

0.77 

(2.13) 

BST Stall 

Margin 

 0.32 

(1.58) 

2.08 

(11.61) 

1.72 

(10.62) 

 0.47 

(1.69) 

2.13 

(12.47) 

1.83 

(11.58) 

HPC Stall 

Margin 

 1.07 

(5.98) 

3.11 

(10.11) 

2.49 

(8.57) 

 0.94 

(5.45) 

3.10 

(10.22) 

2.48 

(8.58) 

T4 0.10 

(0.57) 

0.10 

(0.40) 

0.09 

(0.39) 

 0.12 

(0.68) 

0.10 

(0.44) 

0.10 

(0.43) 
NASA/TM—2005-213865 9

nonlinear engine model is run through the typical flight profile 

described earlier. The biases are kept constant throughout the 

flight. 

When applied to the healthy engine with biased actuators, 

the estimates of the CGEKF upon initialization do not match 

the actual values because the CGEKF has been designed for 

an engine at the nominal condition where no actuator bias is 

present. Again, this mismatch results in large residuals 

( y − yˆ ), and the CGEKF tries to reduce the residuals by 

adjusting tuning parameters. If the CGEKF is able to maintain 

small residuals throughout the flight profile, it means that the 

influence of actuator biases can be captured by tuning 

parameter adjustments at any point along the operating path. 

For 100 cases of actuator anomalies, the CGEKF was stable 

throughout the flight profile. 

The estimation accuracy of the CGEKF is evaluated using 

the same process described in the previous section for the 

degraded engine case. From each actuator anomaly case, a 

100-second time history of the estimation errors is taken from 

each of the ground-idle, climb, and cruise phases. Then, the 

mean and the maximum of absolute percent estimation errors 

of 100 anomaly cases, each of which consists of a 100-second 

time history, are computed for each flight segment. The result 

of the estimation performance is shown in table 6 for the two 

CGEKF designs using sensor sets no. 1 and 2. 

Both CGEKF designs are successful in accurately 

estimating sensor outputs. This ability to accurately estimate 

sensor outputs in the presence of actuator biases was also 

observed in reference 7 where the piece-wise linear Kalman 

filters with tuning parameters were designed at a cruise 

condition. The largest maximum estimation errors for sensor 

set no. 1 and 2 are 0.5 percent (T49) and 0.63 percent (T49), 

respectively. 

The level of estimation accuracy for thrust, FAN stall 

margin, and T4 is still fairly good. For actuator anomaly cases, 

the benefit of having the additional P17 sensor can not be 

observed. This is understandable since the P17 sensor 

measures bypass pressure while the actuators mainly influence 

core operation. The estimation accuracy for BST and HPC 

stall margins are much worse than in the degraded engine 

case. As discussed in the previous section, the combination of 

sensors and tuning parameters being used may not be 

appropriate for the estimation of BST and HPC stall margins. 

Moreover, these stall margins are very sensitive to the VBV 

and VSV actuator positions, and the influence of the actuator 

biases can not fully be captured by the tuning parameters. 

It should be noted that the actuator bias terms could have 

been treated as tuning parameters, as long as the number of 

tuning parameters does not exceed the number of sensors. 

Through such an implementation, the estimation accuracy of 

TABLE 6.—MEAN AND MAXIMUM ABSOLUTE PERCENT ESTIMATION ERROR FOR 

100 CASES OF HEALTHY ENGINE WITH BIASED ACTUATORS 

(MAXIMUM ERROR IN PARENTHESES) 

 CGEKF Design with 

Sensor Set no.1 

 CGEKF Design with 

Sensor Set no.2 

 ground climb cruise ground climb cruise 

XN12 0.05 

(0.26) 

0.02 

(0.09) 

0.02 

(0.11) 

 0.06 

(0.30) 

0.02 

(0.09) 

0.02 

(0.10) 

XN2 0.02 

(0.09) 

0.03 

(0.17) 

0.02 

(0.11) 

 0.02 

(0.09) 

0.03 

(0.18) 

0.03 

(0.14) 

P17 ---- ---- ---- 0.01 

(0.02) 

0.04 

(0.14) 

0.04 

(0.18) 

P25 0.02 

(0.07) 

0.06 

(0.30) 

0.06 

(0.31) 

 0.03 

(0.11) 

0.07 

(0.35) 

0.08 

(0.37) 

T25 0.02 

(0.12) 

0.02 

(0.12) 

0.03 

(0.14) 

 0.02 

(0.13) 

0.03 

(0.12) 

0.03 

(0.15) 

PS3 0.08 

(0.40) 

0.08 

(0.34) 

0.09 

(0.36) 

 0.07 

(0.32) 

0.06 

(0.33) 

0.06 

(0.26) 

T3 0.08 

(0.45) 

0.08 

(0.35) 

0.08 

(0.34) 

 0.09 

(0.49) 

0.09 

(0.36) 

0.09 

(0.36) 

T49 0.08 

(0.50) 

0.06 

(0.25) 

0.07 

(0.29) 

 0.11 

(0.63) 

0.08 

(0.30) 

0.08 

(0.37) 

 

Thrust 0.59 

(1.64) 

0.40 

(0.96) 

0.89 

(1.89) 

 0.76 

(2.06) 

0.47 

(1.15) 

1.01 

(2.22) 

FAN Stall 

Margin 

 1.00 

(2.11) 

0.48 

(1.55) 

0.89 

(2.43) 

 1.08 

(2.49) 

0.37 

(1.37) 

0.65 

(2.12) 

BST Stall 

Margin 

 6.54 

(17.05) 

7.49 

(23.36) 

12.16 

(42.54) 

 6.53 

(17.58) 

7.52 

(23.83) 

12.18 

(43.34) 

HPC Stall 

Margin 

 25.02 

(54.43) 

4.99 

(15.33) 

10.64 

(26.15) 

 25.02 

(54.24) 

5.00 

(15.39) 

10.64 

(25.97) 

T4 0.11 

(0.68) 

0.10 

(0.49) 

0.10 

(0.45) 

 0.14 

(0.80) 

0.11 

(0.53) 

0.11 

(0.49) 
NASA/TM—2005-213865 10

the CGEKF may be improved. However, it is impossible to 

treat all elements that can deviate from the nominal as tuning 

parameters. The result in this section reveals the performance 

of the CGEKF when it encounters unanticipated off-nominal 

conditions. 

Evaluation 4: CGEKF Response to Component Faults 

The CGEKF was evaluated in the previous sections using 

constant off-nominal engine variation sources; component 

degradation and actuator biases were constant throughout the 

flight profile. In this section, the CGEKF response to 

component faults is investigated. A component fault is 

modeled by abruptly shifting the health parameters of an 

engine component. Similar to the case of component 

degradation, the sensor outputs will shift from their nominal 

condition values when an engine experiences component 

faults. The shifts in sensor outputs cause the control system to 

adjust its commands, so the closed-loop system will move to a 

new operating condition through a transient period. During the 

transient operation, the CGEKF must remain stable. 

This test is done at ground-idle, climb, and cruise 

operating conditions. Each of the 10 health parameters is 

abruptly shifted by 5 percent from the healthy baseline 

sequentially over a time interval of 5 seconds. A 5 percent 

abrupt shift in all health parameters represents quite severe 

damage to an engine, and such a scenario may never happen in 

the real environment. However, this scenario of severe 

damage is used to check the stability of the CGEKF. 

At three operating conditions, both CGEKF designs with 

sensor sets no. 1 and 2 were stable. Figure 3 shows the time 

history of the responses generated by the nonlinear engine 

model and two CGEKF designs with sensor sets no. 1 and 2 at 

the cruise condition. In this particular case, the health 

parameters of the FAN, BST, and HPC components were 

abruptly shifted sequentially in the time interval between 10 

and 12 seconds, and the health parameters of the HPT and 

LPT components were abruptly shifted sequentially in the 

time interval between 60 and 61 seconds. At the cruise 

condition, the control system tries to maintain the corrected 

low-spool speed at the demanded condition. As discussed 

before, the CGEKF designed with sensor set no. 1 is not able 

to accurately estimate thrust and FAN stall margin when the 

health parameters for the FAN component are shifted. The 

CGEKF design with sensor set no. 2 exhibits accurate 

estimation performance. This difference in the estimation 

accuracy indicates the significance of the P17 sensor for 

estimating thrust. 

Discussion 

The evaluation of the CGEKF performance in the 

previous section indicated the effectiveness of using tuning 

parameters for the adaptation of the on-board model to aircraft 

engines at off-nominal conditions. Through the adjustment of 

tuning parameters, the CGEKF accurately estimated sensor 

outputs in the presence of component degradation or actuator 

anomalies. This ability to maintain accurate estimation 

performance at off-nominal conditions was also extended to 

the estimation of the performance parameters. Without using 

tuning parameters, any mismatch between the actual engine 

and the on-board model will corrupt the estimation 

performance. To validate this point, the evaluation in the 

previous section was repeated for the “general” CGEKF which 

was designed with no tuning parameters. The general CGEKF 

performed better than the “adaptive” CGEKF when it was 

applied to the nonlinear engine model at the healthy condition. 

However, when the general CGEKF was applied to the 

nonlinear engine model at off-nominal conditions, its 

estimation performance became significantly worse: both 

sensor outputs and performance parameters were poorly 

estimated. Unlike the adaptive CGEKF, the general design 

does not have any freedom to adjust itself in the presence of 

component degradation or anomalies, and thus the influence of 

such off-nominal engine-variation sources directly effect 

estimation performance. 

Although the CGEKF demonstrated its capability to 

maintain stability and estimate both measurable and nonmeasurable engine parameters, further investigation is needed 

0 20 40 60 80 100 120

Thrust

0 20 40 60 80 100 120

FAN Stall Margin

0 20 40 60 80 100 120

BST Stall Margin

0 20 40 60 80 100 120

HPC Stall Margin

0 20 40 60 80 100 120

T4

Time (sec)

Nonlinear Engine Model

CGEKF with Sensor Set #1

CGEKF with Sensor Set #2

Figure 3.—Responses of Nonlinear Engine Model and 

CGEKF Designs to Component Faults. 
NASA/TM—2005-213865 11

to improve the CGEKF performance. During the evaluation of 

the CGEKF, it was noted that the accuracy of performance 

parameter estimation is influenced by the sensors and tuning 

parameters used. In the current problem where in-flight 

application is pursued, the sensors to be used may be limited 

to those sensors currently implemented. However, the designer 

does have freedom in the selection of tuning parameters. The 

tuning parameters used in this paper were selected based on 

engineering judgment, and the non-divergence property was 

shown empirically to be maintained with appropriate tuning 

parameters. It is desirable to develop an analytical approach to 

find an optimal set of tuning parameters that will maintain the 

non-divergence property and minimize the estimation errors 

for a given set of sensors. 

Other areas for investigation that may help to improve the 

estimation accuracy include the optimal selection of the 

Kalman gain design point, or the utilization of multiple 

Kalman gain matrices. It was observed in the evaluation 

results that the estimation accuracy of the CGEKF varies with 

operating condition. The variation in the estimation accuracy 

is partially due to the fact that a single Kalman gain matrix 

designed at a specific operating condition was used. In this 

paper, the ground maximum power condition was used as the 

design point. It is reasonable to consider that this selected 

design point may not be the optimal point at which estimation 

errors over the flight envelope are minimized. Thus, it is 

desirable to repeat the evaluation process using Kalman gains 

computed at different design points and then to assess the 

influence of the design points on the CGEKF performance. 

Through such an investigation, the optimal design point may 

be determined. An alternative approach which may also 

improve the CGEKF performance is the utilization of multiple 

Kalman gain matrices. By using multiple Kalman gain 

matrices, each of which provides best estimation performance 

at its design point, the CGEKF performance over the entire 

flight envelope may be improved. Since this approach 

increases the design complexity, such as selection of 

appropriate design points and interpolation of multiple 

Kalman gain matrices, the trade-offs between the increased 

complexity and the performance improvement must be 

studied. 

Finally, it should be noted that the performance level that 

the CGEKF can achieve is limited by the available sensors. 

For instance, it was shown that the CGEKF was not able to 

accurately estimate thrust without the P17 sensor. If the ability 

to accurately estimate thrust outweighs the cost of sensor 

installation, the P17 sensor may be added or may replace an 

existing engine sensor. 

Conclusion 

An approach based on the Constant Gain Extended 

Kalman Filter (CGEKF) technique was investigated for the inflight estimation of non-measurable performance parameters 

of aircraft engines. A CGEKF was developed by combining an 

on-board engine model and a single Kalman gain matrix. In 

order to make the on-board engine model adaptive to a real 

engine whose performance will deviate from the nominal 

baseline over its lifetime of operation, the CGEKF was 

designed with the ability to adjust its performance through the 

adjustment of artificial parameters called “tuning parameters.” 

With this design approach, the CGEKF attributes any 

component performance deviations due to degradation and/or 

faults to tuning parameters so that it can maintain accurate 

estimation performance at off-nominal health conditions. 

The evaluation of the CGEKF performance was 

conducted in a simulation environment using a nonlinear 

simulation of a large commercial aircraft engine model. The 

stability and estimation accuracy of the CGEKF were 

evaluated at multiple operating points using component 

degradation, actuator biases, and component faults. The 

CGEKF exhibited excellent stability. Despite the fact that the 

Kalman gain was designed at a specific operating point, the 

CGEKF was stable throughout a typical flight profile. The 

CGEKF was also able to estimate the performance parameters 

with fairly good accuracy at off-nominal health conditions. 

Some performance parameters such as thrust and combustor 

temperature were estimated with higher accuracy than the 

others. It is considered that the difference in the estimation 

accuracy is strongly affected by the sensors and tuning 

parameters being used. To further improve the estimation 

performance of the CGEKF, a technique to select an optimal 

set of tuning parameters is desired. 

References 

1. Adibhatla, S., and Lewis, T.J., 1997, “Model-Based 

Intelligent Digital Engine Control (MoBIDEC),” AIAA 

Paper AIAA–97–3192. 

2. Lietzau, K., and Kreiner, A., 2001, “Model Based Control 

Concepts for Jet Engines,” ASME Paper 2001–GT–0016. 

3. 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 

GT–2002–30041. 

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

L.J., 1989, “Estimating In-Flight Engine Performance 

Variations Using Kalman Filter Concepts,” AIAA Paper 

AIAA–89–2584. 

5. Kerr, L.J., Nemec, T.S., and Gallops, G.W., 1991, “RealTime Estimation of Gas Turbine Engine Damage Using a 

Control Based Kalman Filter Algorithm,” ASME Paper 

91–GT–216. 

6. Brotherton, T., Volponi, A. Luppold, R., and Simon, D.L., 

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

Aerospace Conference. 

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

Enhanced Bank of Kalman Filters for In-Flight Aircraft 

Engine Sensor Fault Diagnostics,” ASME Paper 

GT2004–53640. 

8. Safonov, M.G., and Athans, M., 1978, “Robustness and 

Computational Aspects of Nonlinear Stochastic 
NASA/TM—2005-213865 12

Estimators and Regulators,” IEEE Transactions on 

Automatic Control, 23, pp. 717–725. 

9. Sugiyama, N., 2000, “System Identification of Jet 

Engines,” Journal of Engineering for Gas Turbines and 

Power, 122, pp. 19–26. 

10. Gelb, A., Kasper, J.F. Jr., Nash, R.A. Jr., Price, C.F., and 

Sutherland, A.A. Jr., 1974, “Applied Optimal 

Estimation,” The MIT Press, Cambridge, Massachusetts. 

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

Estimation Algorithm for Performance-Seeking 

Controllers,” Journal of Propulsion and Power, 10, 

pp. 527–532. 

12. Brunell, B.J., Viassolo, D.E., and Prasanth, R., 2004, 

“Model Adaptation and Nonlinear Model Predictive 

Control of an Aircraft Engine,” ASME Paper GT2004–

53780. 
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

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E–15235

18

Application of a Constant Gain Extended Kalman Filter for In-Flight Estimation

of Aircraft Engine Performance Parameters

Takahisa Kobayashi, Donald L. Simon, and Jonathan S. Litt

Aircraft engines; Constant gain extended Kalman filter; Engine thrust estimation; Adaptive

on-board engine model; Flight safety

Unclassified -Unlimited

Subject Category: 07

WBS–22–079–30–05

1L161102AF20

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

2005. Takahisa Kobayashi, QSS Group, Inc., 21000 Brookpark Road, Cleveland, Ohio 44135; and Donald L. Simon and

Jonathan S. Litt, U.S. Army Research Laboratory, NASA Glenn Research Center. Responsible person, Donald L. Simon,

organization code RIC, 216–433–3740.

An approach based on the Constant Gain Extended Kalman Filter (CGEKF) technique is investigated for the in-flight

estimation of non-measurable performance parameters of aircraft engines. Performance parameters, such as thrust and

stall margins, provide crucial information for operating an aircraft engine in a safe and efficient manner, but they cannot

be directly measured during flight. A technique to accurately estimate these parameters is, therefore, essential for further

enhancement of engine operation. In this paper, a CGEKF is developed by combining an on-board engine model and a

single Kalman gain matrix. In order to make the on-board engine model adaptive to the real engine’s performance

variations due to degradation or anomalies, the CGEKF is designed with the ability to adjust its performance through the

adjustment of artificial parameters called “tuning parameters.” With this design approach, the CGEKF can maintain

accurate estimation performance when it is applied to aircraft engines at off-nominal conditions. The performance of the

CGEKF is evaluated in a simulation environment using numerous component degradation and fault scenarios at multiple

operating conditions.