Comparison of Adaptive Kalman Filters in Aircraft State Estimation
MERT SEVER1, TUNCAY YUNUS ERKEÇ1, CHINGIZ HAJIYEV2
1Atatürk Strategic Studies, and Graduate Institute,
Turkish National Defense University,
Levent, 34334, Istanbul,
TURKEY
2Faculty of Aeronautics and Astronautics,
Istanbul Technical University,
Ayazağa, 34469, Maslak, Istanbul,
TURKEY
Abstract: - Aircraft state estimation refers to the process of determining the current or future state of an aircraft,
such as its position, velocity, orientation, and other relevant parameters, based on available sensor data and
mathematical models. This information is crucial for safe and efficient flight operations, as well as for various
applications, including Guidance, Navigation, Control (GNC), and autonomous flight. Given the beginning
circumstances, the motion of the airplane was examined in this study by estimating the state vectors using the
Kalman Filter (KF) and the Adaptive Kalman Filters (AKF), as well as by comparing the various estimate
techniques.
Key-Words: - Aircraft, State Vectors, Kalman Filters, Flight Dynamics
Received: June 19, 2022. Revised: August 23, 2023. Accepted: September 25, 2023. Published: October 30, 2023.
1 Introduction
State estimation can be a challenging task due to the
complexity of the aircraft dynamics, sensor
limitations, and environmental factors. Advances in
sensor technology and estimation algorithms have
improved the accuracy and reliability of state
estimation in modern aircraft, contributing to safer
and more efficient flight operations, [1].
Key components of aircraft state estimation
include:
Sensor Data: Aircraft are equipped with various
sensors, such as GPS (Global Positioning System),
inertial measurement units (IMUs), altimeters,
airspeed indicators, and more. These sensors
provide data on the aircraft's physical state and
environment.
Mathematical Models: To estimate the aircraft's
state accurately, mathematical models are used.
These models incorporate the principles of physics
and aerodynamics to predict how an aircraft's state
will evolve. These models may include equations of
motion, atmospheric models, and sensor error
models.
Sensor Fusion: Since no single sensor is perfect
and sensor measurements can be noisy or subject to
errors, sensor fusion techniques are employed to
combine data from multiple sensors. Kalman filters
and extended Kalman filters are commonly used for
this purpose.
Filtering and Smoothing: Estimation methods
often involve filtering techniques, which provide
real-time estimates of the aircraft's state as new
sensor data becomes available. Smoothing
techniques, on the other hand, are used to refine the
state estimates using historical data.
Navigation: State estimation is fundamental to
aircraft navigation. It enables the aircraft to
determine its position and orientation accurately,
helping it follow a desired flight path, avoid
obstacles, and maintain proper altitude.
Control: Aircraft state estimation is critical for flight
control systems. By knowing the aircraft's state, the
control system can adjust, ensure stability, respond
to pilot inputs, and execute various flight
maneuvers.
Autonomous Flight: In the context of autonomous
flight, state estimation plays a crucial role in
enabling drones, UAVs (Unmanned Aerial
Vehicles), and autonomous aircraft to operate safely
and perform complex missions without direct
human intervention.
Safety and Redundancy: Aircraft state estimation
is an essential component of safety-critical systems.
Redundant sensors and estimation algorithms are
often employed to ensure that the aircraft can
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Mert Sever, Tuncay Yunus Erkeç, Chingiz Hajiyev
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continue to operate safely in the event of sensor
failures or other anomalies.
In addition, numerous approaches for estimating
aircraft state vectors are being worked on. For the
estimate of motion state vectors for various
platforms than airplanes, Kalman filters (KF) and
state vector estimations have demonstrated their
effectiveness and excel in terms of their high
accuracy, [2], [3], [4], [5], [6].
The Extended Kalman Filter (EKF) is a widely
used technique for state estimation in the presence
of noise and disturbances. However, as mentioned
before, it can face challenges when dealing with
practical usage where system and sensor noises
occur. In such cases, the Adaptive Kalman Filter
(AKF) has gained popularity for its ability to handle
varying noise and improve state estimates in the
presence of disturbances.
Here is an overview of the AKF and its
advantages:
Adaptability: The key advantage of the AKF over
the standard EKF is its adaptability. The AKF can
adjust its filter parameters (such as process and
measurement noise covariance matrices) based on
the evolving noise and uncertainties in the system.
This adaptability allows the filter to respond to
changing conditions and provide more accurate state
estimates.
Handling Changing Noise Levels: In practical
applications, the levels of system and sensor noise
can change over time due to various factors such as
environmental conditions, sensor degradation, or
system component wear and tear. The AKF is
designed to continuously update its noise models,
ensuring that it can effectively estimate the state
even when noise levels are not constant.
Robustness: By adapting to changing noise
characteristics, the AKF can provide more robust
and accurate state estimation in challenging
environments. It can effectively deal with
disturbances and sensor noise that may cause
problems for traditional fixed-parameter filters like
the EKF.
Improved Convergence: The adaptability of the
AKF can help improve convergence and reduce the
time it takes for the filter to provide accurate state
estimates, especially in situations where noise levels
change rapidly or unpredictably.
Reduced Tuning Requirements: Unlike traditional
Kalman filters, which often require manual tuning
of the noise covariance matrices, the AKF reduces
the need for extensive tuning. This can be
particularly advantageous in scenarios where
obtaining accurate noise models is challenging.
Real-Time Applications: The AKF is well-suited
for real-time applications, including autonomous
navigation, robotics, and aviation, where
adaptability and robustness are critical for safe and
accurate operation, [7].
It is important to note that the choice between
the EKF and the AKF depends on the specific
requirements and characteristics of the system, as
well as the nature of the noise and disturbances
encountered in practical applications. The AKF's
adaptability can be an asset in scenarios where noise
levels are dynamic and not easily predictable,
making it a suitable choice for applications where
maintaining accurate state estimates is essential.
A novel multiple fading factors Kalman filtering
technique is provided in the research "Multiple
Fading Factors Kalman Filter for SINS Static
Alignment Application" by, [8]. The fading factor
matrix is created by computing the unbiased
estimate of the innovation sequence covariance
using fenestration. The technique offers various
rates of fading for various filter channels by
adjusting the covariance matrix of prediction error
and fading factor matrix. The strap-down inertial
navigation devices are used using the suggested
approach. It is discovered that the suggested
technique has superior parameter estimation in real-
world settings and is more effective against noise.
A further work, [9], examined the same topic. A
sequential technique is presented to concurrently
calculate the orbit and attitude of a small spacecraft
based on magnetometer and gyro measurements,
[9]. A robust adaptive Kalman filter is developed to
reduce the effect of orbital errors on attitude
prediction.
For altering the measurement covariance matrix
®, Almagbile, Wang, and Al-Rawabdeh examined
the Sage Husa adaptive Kalman filter (SHAKF) and
innovation-based adaptive Kalman filter (IAKF)
techniques, [10].
A strong tracking variational Bayes adaptive
Kalman filter based on multiple damping factors
was proposed by Pan and colleagues in a study that
was published in 2020, [11]. This filter takes into
account the fact that if the system model or the
statistical properties of the noise are inaccurate, past
measurements will directly affect the accuracy of
the current state estimation and may even lead to
filtering bias.
The system errors have been increased beyond
the predetermined simulation duration, and the
filters' responses to the faults were observed. In the
event of a potential rise in inaccuracy in the
systems, it was attempted to show which filter
produced superior outcomes.
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In this study, the scaling adaptive Kalman filter,
residual adaptive Kalman filter, and conventional
Kalman filter are compared for the single and
double-sensor fault scenarios applied to aircraft
dynamics.
2 Problem Formulation
In this part, the mathematical model of the motion
of the aircraft is explained.
Longitudinal motion, [12].
Here, longitudinal velocity, w vertical velocity, q
pitch velocity, pitch angle and is the tilt angle
of the elevator.
Latitudinal motion, [12].
Here, is the yaw angle, p is the roll angular
velocity, r is the yaw angular velocity, is the roll
angle.
The longitudinal and latitudinal state vectors can be
obtained as follows
The longitudinal and latitudinal system matrices are
combined as following
The combined control vector is,
where and are the tilt angles of the ailerons
and rudder respectively
The combined control distribution matrix,
The equation of aircraft full motion can be written
in the form as follows.
After the discretization of Eq. (12) we
have
After mathematical transformations, the
mathematical model of the motion of the aircraft
was obtained as
Substituting
into equation 15, we can obtain the mathematical
model of the motion of aircraft as follows.
3 Kalman Filter for Aircraft State
Estimation
Kalman Filter is an estimation approach for linear
systems, [13]. As the model is linear, by processing
noisy measurements, KF estimates the state vectors
with high accuracy.
Estimation equation of the filter
The extrapolation equation is shown as
Innovation sequence
Z(k) is the measurement vector. Kalman gain matrix
can be expressed as
Predicted covariance matrix of estimation error,
Here Q is the system noise covariance matrix. The
covariance matrix of Estimation error is,
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Measurement matrix
Measurement error covariance matrix
Transition matrix of system noise
4 Adaptive Kalman Filter
4.1 Residual Adaptive Kalman Filter
The Residual Adaptive Kalman Filter is a filter that
adaptably calculates the R matrices during the
simulation. R matrix undergoes iterative alterations
in contrast to KF. This enhancement makes the filter
less susceptible to potential faults, [14].
Innovation
Estimation equation
Residual
Measurement error covariance matrix
Kalman gain matrix,
Here, according to previous studies, [15].
The rest of the KF equations are the same as in the
previous section.
4.2 Scaling Adaptive Kalman Filter
The multiple measurement noise scale factor
technique has been found to produce superior
outcomes in multivariate systems in earlier
investigations. The technique of choice fixes the
measurement noise-covariance matrix and Kalman
gain using a matrix known as the scale factor, [15].
Scale matrix S(k) is incorporated into the method in
contrast to the Kalman filter.
Here is the width of the moving window. The
diagonal elements of the scale matrix may not be
less than one, so the following rule is suggested to
avoid this situation.
and,
Here represents the i’th diagonal element of the
matrix S(k). By using scale factor Kalman gain
matrix can be expressed as,
As with section 3, the remaining Kalman filter
formulas are identical. Any system fault increases
the corresponding diagonal matrix element. Scale
factor increases lower the Kalman gain and the
impact of innovation on the state update process.
With this concept, estimates may be obtained with
more accuracy, [16].
5 Simulation Results and Discussion
In this study, the motion of the aircraft is simulated
by using MATLAB. The measurement vector is,
The measurements are simulated via the formulas
below,
Here is the standard deviation of the measurement
error.
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After a specified time, it's simulated that the pitch
angle sensor and roll rate sensor were broken.
5.1 Single Sensor Fault Results
Cases involving pitch angle fault and roll rate fault
are covered individually. State vector and scale
factor graphs for each normalized innovation under
each case are shown.
5.1.1 Noise Increment Fault Scenario
In this scenario, to simulate faulty measurements,
and are multiplied by 50 for each case.
i) Noise Increment Type Pitch Angle Gyro Fault
Fig. 1: Conventional KF normalized innovations in
the presence of pitch angle gyro noise increment
fault
As shown in Figure 1, because of the fault on
the pitch angle gyroscope, after 0.05 seconds, the
normalized innovation of the pitch angle (theta)
exceeds the threshold which is ± 3.
Pitch angle estimation results and pitch angle
scaling factor graph in the case of noise increment
fault are given in Figure 2 and Figure 3 respectively.
Fig. 2: Pitch angle estimation results in the presence
of pitch angle gyro noise increment fault
The root mean square errors (RMSE) for pitch
angle estimation in the presence of pitch angle gyro
noise increment fault are given in Table 1.
Table 1. RMSE for pitch angle in the case of noise
increment fault
Filter / State
KF
Residual AKF
Scaling AKF
Theta
0.711162
0.317198
0.040281
Fig. 3: Pitch angle scaling factor in the case of noise
increment fault
The pitch angle scaling factor rises as a result of
the pitch angle gyro fault, leading to more precise
pitch angle calculations.
ii) Noise Increment Type Rate Gyro Fault
The conventional KF normalized innovation results
for the case of roll rate gyro noise increment fault
are given in Figure 4.
Fig. 4: Conventional KF normalized innovations in
the presence of roll rate gyro noise increment fault
As can be seen in the figure, because of the fault
on the roll rate gyroscope, after 0.05 seconds of
simulation, the normalized innovation of the roll
angle (p) exceeds the threshold which is ± 3.
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Figure 5, Figure 6 show the roll rate estimation
results and roll rate scaling factor graph for the case
of noise increment fault respectively. The scaling
factor of the roll rate improves as a result of the roll
rate gyro's fault, allowing for more precise roll rate
calculations.
Fig. 5: Roll rate estimation results for noise
increment fault case
The roll rate root mean square errors in the case
of noise increment for KF, residual AKF, and
scaling AKF are given in Table 2.
Table 2. RMSE for roll rate in the case of noise
increment fault
Filter / State
KF
Residual AKF
Scaling AKF
p
0.364050
0.152013
0.074246
Fig. 6: Roll rate scaling factor graph for noise
increment fault case
Scaling AKF is the most effective method for
tolerating the system malfunction for noise
increment faulty systems, as demonstrated by
graphs and root mean square errors (Table 1, Table
2). Compared to KF, adaptive approaches produce
superior outcomes. Scaling AKF tolerates the
system error in the system better than the residual
AKF technique, according to comparisons between
adaptive approaches.
5.1.2 Bias Noise Fault Scenario
In this scenario, to simulate faulty measurements,
and are summed with 0.1 radians for each
case.
iii) Bias Noise Type Pitch Angle Gyro Fault
Pitch angle estimation results in case of bias noise
are given in Figure 7.
Fig. 7: Pitch angle estimation results in case of bias
noise
The pitch angle root mean square errors under
the condition of bias noise for KF, residual AKF,
and scaling AKF are given in Table 3.
Table 3. RMSE for pitch angle in the case of bias
noise fault
Filter / State
KF
Residual AKF
Scaling AKF
Theta
0.052242
0.050676
0.042016
.
iv) Bias Noise Type Roll Rate Gyro Fault
Roll rate estimation results in the case of bias noise
fault are given in Figure 8.
Fig. 8: Roll rate estimation results in the case of a
bias noise fault
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The roll rate root mean square errors under the
condition of bias noise for KF, residual AKF, and
scaling AKF are given in Table 4.
Table 4. RMSE for roll rate in the case of a bias
noise fault
Filter / State
KF
Residual AKF
Scaling AKF
p
0.587203
0.510237
0.423295
Scaling AKF is the most effective method for
tolerating system malfunction for bias noise faulty
systems, as demonstrated by graphs and root mean
square errors (Table 3, Table 4). Compared to KF,
adaptive approaches give better estimations. When
we rank the methods, scaling AKF achieved the best
result while conventional KF achieved the worst
result.
5.2 Double Sensor Fault Results
Pitch angle and roll rate sensors are faulty
simultaneously in this manner. At the end of the
specified time (0.05 seconds after the simulation
starts), distortion is introduced to the pitch angle and
rotation speed gyroscopes. Considering this
situation, the following scenarios are applied.
a. Noise increment fault for both sensors
b. Bias noise fault for both sensors
c. Noise increment fault for pitch angle gyro
and bias noise fault for roll rate gyro
5.2.1 Noise Increment Fault for Both Sensors
In this scenario, to simulate faulty measurements,
and are simultaneously multiplied by 50.
Figure 9, Figure 10, Figure 11, Figure 12 and
Figure 13 show the graphs for normalized
innovation, state estimation, and scale factor
respectively. Additionally, the RMSE results for the
noise increment type double sensor faults are
provided (Table 5).
Figure 9 presents the Conventional KF
normalized innovations for the case of double noise
increment sensor faults. Pitch angle gyro (theta) and
roll rate gyro (p) were found to be greater than the
threshold of 3 after the fault occurred at 0.05
seconds of the simulation.
Figure 10, Figure 12 and Figure 13 show the
results of the pitch angle and roll rate estimations as
well as scale factor graphs for the case of noise
increment type sensor faults. The results indicated
that the scaling factor grows as system malfunction
increases.
Fig. 9: Conventional KF normalized innovations for
the case of noise increment type double sensor faults
Fig. 10: Pitch angle estimation results for the case of
noise increment type double sensor faults
Fig. 11: Roll rate estimation results for the case of
noise increment type double sensor faults
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Fig. 12: Pitch angle scaling factor graph for the case
of noise increment type double sensor faults
Fig. 13: Roll rate scaling factor graph for the case of
noise increment type double sensor faults
Table 5 provides pitch angle and roll rate root
mean square errors for each filter in the case of
noise increment type double sensor faults scenario.
Table 5. RMSE of the pitch angle and roll rate in the
case of noise increment type double sensor faults.
Filter / State
KF
Residual AKF
Scaling AKF
Theta
0.716393
0.201855
0.066984
p
0.371447
0.222162
0.073049
As demonstrated by plots (Figure 9, Figure 10,
Figure 11, Figure 12, Figure 13) and root mean
square errors (Table 5 and Table 6), scaling AKF is
the best method for tolerating the failure in case of
noise increment type double sensor fault. Adaptive
approaches produce more precise estimates than
traditional KF. Scaling AKF is more tolerant of the
system flaw than other adaptive techniques.
5.2.2 Bias Noise Type Double Sensor Faults
In this scenario, to simulate faulty measurements,
and are simultaneously summed with 0.1
radians.
Figure. 14, Figure 15 and Figure 16 show the
graphs for state estimations. Additionally, the
RMSE results for the bias noise type double sensor
faults are provided (Table 6).
Figure 14 represents the pitch angle estimation
results for the case of bias noise type double sensor
faults.
Fig. 14: Pitch angle estimation results for the case of
bias noise type double sensor faults
Figure 15 represents the roll rate estimation
results for the case of bias noise type double sensor
faults.
Fig. 15: Roll rate estimation results for the case of
bias noise type double sensor faults
Table 6 provides pitch angle and roll rate root
mean square errors for each filter for the scenario of
bias noise type double sensor faults.
Table 6. RMSE of the pitch angle and roll rate for
the bias noise type double sensor faults.
Filter / State
KF
Residual AKF
Scaling AKF
theta
0.151124
0.123794
0.098763
p
0.173246
0.132962
0.088937
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5.2.3 Different Types of Double Sensor Faults
In this scenario, to simulate pitch angle gyro and roll
rate gyro measurements, multiplied by 50 and
is summed with 0.1 radians.
Figure 16 presents the Conventional KF
normalized innovations for the case of double
sensor faults, noise increment fault for pitch angle
gyro, and bias noise fault for roll rate gyro. The
pitch angle gyro (theta) and roll rate gyro (p) are
found to be greater than the threshold, which is 3.
As seen, the normalized innovations of the pitch
angle and roll angle channels exceed the threshold
which is ± 3 after the fault occurs at 0.05 seconds of
the simulation.
Fig. 16: Conventional KF normalized innovations
for the case of double sensor faults (noise increment
fault for pitch angle gyro and bias noise fault for roll
rate gyro)
Figure 17 and Figure 18 represent the pitch
angle and roll rate estimation results for the case of
double sensor faults (noise increment fault for pitch
angle gyro and bias noise fault for roll rate gyro).
Fig. 17: Pitch angle estimation results for the case of
double sensor faults (noise increment fault for pitch
angle gyro and bias noise fault for roll rate gyro).
Fig. 18: Roll rate estimation results for the case of
double sensor faults (noise increment fault for pitch
angle gyro and bias noise fault for roll rate gyro).
Figure 19 and Figure 20 show the results of the
pitch angle and roll rate estimations as well as scale
factor graphs. According to the obtained results, the
scaling factor increases as system malfunction
increases.
Fig. 19: Pitch angle scaling factor graph for the case
of double sensor faults (noise increment fault for
pitch angle gyro and bias noise fault for roll rate
gyro).
Fig. 20: Roll rate scaling factor graph for the case of
double sensor faults (noise increment fault for pitch
angle gyro and bias noise fault for roll rate gyro).
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Table 7 provides pitch angle and roll rate root
mean square errors for each filter in this scenario.
Table 7. RMSE of the pitch angle and roll rate for
the double sensor faults (noise increment fault for
pitch angle gyro and bias noise fault for roll rate
gyro)
Filter / State
KF
Residual AKF
Scaling AKF
Theta
0.0.644407
0.264420
0.216585
p
0.0.012244
0.015997
0.010125
When both noise approaches are compared, it
was found that noise increment is a more realistic
approach than bias noise for simulating the system
noise. The adaptive filters give better results for the
noise increment type scenario. Moreover, the results
revealed that scaling AKF is still the best filter for
not only both noise increment and bias noise
systems but also complex double-sensor fault
systems.
Increasing the vector sensitivity is crucial for
the direction control of fast-moving aircraft. An
increase in mistakes in aircraft status detection and
control is brought on by high-value error rates that
may arise in the estimate of aircraft orientation
states.
This study has established the significance of
adopting the scaling AKF estimate technique rather
than residual AKF and traditional KF, particularly in
aircraft orientation and control systems with high
noise ratios.
6 Conclusions
In this study, the motion of the airplane was
examined by estimating the state vector using the
Conventional Kalman Filter and Adaptive Kalman
Filters and comparing various estimation
techniques.
Utilizing the scaling adaptive Kalman filter,
residual adaptive Kalman filter, and conventional
Kalman filter, measurements were processed.
Investigations were conducted into the single-sensor
fault and double-sensor fault sensor failure
scenarios. For both single-sensor fault and double-
sensor fault scenarios, it was concluded that
predicted results by scaling AKF are more accurate
than those from the other two approaches.
Additionally, it was discovered that the KF and
residual AKF errors rise when a system fault occurs,
but the scaling AKF filter is adaptively self-
adjusting and is not as significantly impacted by the
increasing error as other systems. Scaling AKF
estimate remains more stable as a result. After
scaling the AKF technique, residual AKF provides
the second-best estimation.
This study may be used for UAV and aircraft
missions to improve system accuracy. Additionally, it
has been demonstrated via the use of this study that
scaling the AKF allows for the tolerability of large
system faults. The impact of KF and AKF approaches
on multi-satellite flight issues will be investigated in
the future.
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Space Technologies, June 7-9, 2023, Istanbul,
Turkey.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.14
Mert Sever, Tuncay Yunus Erkeç, Chingiz Hajiyev
E-ISSN: 2224-3488
138
Volume 19, 2023
