Two-Stage Kalman Filter Based Estimation of Boeing 747
Actuator/Control Surface Stuck Faults
AKAN GUVEN, CHINGIZ HAJIYEV
Faculty of Aeronautics and Astronautics
Istanbul Technical University
Maslak, 34469, Istanbul
TURKEY
Abstract: - This research aims to construct a two-stage Kalman filter (TSKF) that is available to estimate the
control effectiveness of the actuator on behalf of an actuator stuck fault incident occurring on Boeing-747
commercial airplane. The actuator faults can be diagnosed via TSKF that maintains the states and stuck
positions or control loss by two section encapsulated estimation algorithms. The performance of the TSKF
algorithm is tested. The source of accidents can be as a result from a control surface stuck such an aileron,
rudder, elevator; also, it can be present and appear as bird strike that could tear some part of the control surfaces
located on the wings or tail of the airplane. In this study, there is a stuck fault on the rudder control surface and
the proposed algorithm introduces the value of the stuck of the broken control surface and it is achieved that
utilizing TSKF performs satisfying estimation values which are verified as well on lateral dynamics of the
airplane.
Key-Words: - Actuator, Control Surface, Fault Tolerant Control, Estimator, Two-Stage Kalman Filter
Received: May 11, 2022. Revised: February 15, 2023. Accepted: March 14, 2023. Published: April 25, 2023.
1 Introduction
During the design phase of any mechanism, the
most critical point is to assemble a system that
guarantees safety for its customers. Control systems
of aircrafts shall be based on solid foundations
providing the ability of handling possible risky
situations. The primary purpose here is to minimize
the effects arising from those faults, and if possible,
completely avoiding those effects would be the best
case. Fault Tolerant Control (FTC) is an essential
method for aircrafts to maintain a safe flight in the
events of unknown disturbances, uncertainties or
unplanned system compound, and also actuator or
sensor failures.
The Boeing 747 is an ideal test bed and an
excellent example for any of the commercial
aircrafts currently flying, by having a wide lineup of
characteristics (spoilers, control surface variations,
four turbofan engines...) [1].
The fault-tolerant control plays an essential
role in aerospace applications and is studied by
several researchers as to deal with difficulties. A
fault tolerant control is described as having the
ability to keep utilizing its purpose even afterwards
the fault. The solutions to fault can be categorized
into two sections which are passive and active
controls. Former solution, passive control consists
of managing the system process with the wrecked
controller; the capability of the remnant system
effectualness is linked to the main default control
law. The passive control laws are commonly based
on robust control laws that are proper for apparent
structural faults. Those types of failures are
remarked as uncertainties on peripheral radius of the
original model. Yet, a lot of faults cannot be
overcome by defining them as uncertainties. For this
reason, solid definitions must be made first hand by
constructing control structure.
On the active control side, the control
infrastructure is reestablished instantly post fault
accrue or switch to a predefined control law [2].
An active fault-tolerant control form that is
tolerant to various levels of actuator faults is applied
in this study.
The active fault-tolerant systems include two
essential nodes:
1. Fault detection and isolation (FDI) or
system identification and
2. Control restructure
FDI notation is followed by detection and
identification of faults just as time goes on instantly.
The measurements taken by the FDI action has to
meet the requirements of qualified active fault-
tolerant control stem. Several model-based FDI
approaches are present to detect and locate the
faulty sensor or actuator by analytical redundancy
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algorithms, state-estimation and system
identification methods [2].
The actuators are healthy in proceeding the control
commands if they perform the track as required
from input signals they are %100 running by the
controller structure within normal activity. As
treating a faulty case as such a small fraction loss of
control surface, wrecked hydraulics, stuck at valve,
shortage at electrical servos, seen on actuators, they
are not eligible to accomplish the tasks from inputs
entirely. The problems pointed out would mean
actuator control effectiveness loss. The proceedings
in fault-tolerant control accepted the actuator fault
parameter as the control effectiveness factor which
is calculated by the Kalman filter [2]. It is also
remarked that the faulty actuator’ s control
effectiveness factor is suggested to be the same in
the entities of the related control distribution vector.
Another notion is problems seen on control
surfaces by having partial loss or harmed/broken
parts even if the actuator is in a healthy condition.
When encountered with such an incident, the
corresponding loss of effectiveness of the
actuator/control surface, formerly mentioned
solution, is not capable of detection. Technically it
can be accepted as a measurement for determined
surface faults like a control surface part loss, icing
of control surface in winter conditions, those
accounted as altering control effectiveness factors
on the actuator [3].
Research on the FTC system builds a struct that
is able to cope with faults on sensors and actuators
that are observed at the same time. The proposed
FTC system includes a sensor FDD system and a
controller that can be reconfigured. The detection
and estimation of the sensor faults are gained by an
adaptive three-step unscented Kalman filter that is
capable of estimating the values of state and fault
with unbiasedness. The unbiased state estimation
data is sent to a reconfigurable controller when there
is a sensor fault. In presence of an actuator fault, an
incremental backstepping approach is added to
adjust the controller, by help of this process sensor
and actuator fault detection and control is available
at the same time [4].
The two-stage Kalman filter (TSKF) produces
the estimation values of the system state values and
faulty control surface or actuator’s related control
distribution matrix terms [2-4]. This method is
applicable for actuator/surface faults that lead to
altered control effectiveness factors.
TSKFs are used in aerospace widely for
estimation of system states and model parameters
[2-10, 13]. In [5], an estimation algorithm based on
TSKF was developed for wind speed and
Unmanned Aerial Vehicle (UAV) motion
parameters. In [6-8] the vector measurement-based
algorithms are used with the Kalman filter to form
complete two-stage attitude filters.
The connection between the control commands
created by the controller and the physical actions in
the systems are expressed by actuators [9]. Ailerons
and rudders will be used as the actuators in this
study and Kalman filter technique will be applied to
detect and minimize the effects of those actuator
faults observed in Boeing 747. The fault / faults are
going to be estimated with the help of the residuals,
and they will be analyzed with respect to the
selected confident level thresholds by making the
algorithm available to detect errors simultaneously
with a fast-working observer. Two stage Kalman
filter will be used for estimating the faulty states
even if there are malfunctions on the system and this
method sustains a value (residual) to take remedial
action which is going to be configured by the
controller in order to cover flight safety. Some
cumulative studies led the filter to become an
adaptive two-stage Kalman filter [10]. The TSKF
introduced in [11] is implemented for a linear
aircraft model and also covariance-based forgetting
factors [12] are presented in order to estimate the
state and control influence simultaneously.
In this study, the non-linear Simulink model of
Boeing 747 is linearized as a first step, then, the
lateral state-space model is obtained and transferred
into a code in the MATLAB environment. The
actuator fault detection and isolation processes are
again implemented in MATLAB and this procedure
is applied step by step, focusing on an actuator fault.
It is shown that TSKF algorithm has detected the
fault and made fault isolation possible that brought
out the faulty actuator and estimated both faulty
parameters and states firmly.
2 Mathematical Model of Boeing 747
A proper model of the system dynamics is necessary
during the implementation of Kalman filtering
technique [6]. For this reason, model information
for Boeing 747 is also included. Since the studies
will be carried out on the linearized model in order
not to be affected by the nonlinear dynamics, some
conditions are defined for the linearization process.
Being in a trim point, the values km/h as true
air speed and ft as altitude have been chosen
and linearization is done within those conditions.
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Further information can be seen at Table 1 in
Appendix 1.
2.1 Non-Linear Mathematical Model of
Boeing 747 in Simulink
Equations of motion were obtained by making the
assumptions that the earth is standing still and flat,
the airplane is rigid and not damaged, the mass of
the aircraft does not change within time, and the
moments of inertia , are equal to zero. The
equations obtained based on [8] are as follows:
External forces:
Outer moments:
The forces and moments calculated as in above are
required for mathematical modeling.
The Simulink model for Boeing 747 is
obtained with the help of the open-source library
Airlib, created by MATLAB developers. By using
the models in this library, possible behavior of a
system in case of any fault can be observed, thus,
various improvements can be made to eliminate the
effects of those faults. Details of the model of
Boeing 747 can be examined from Figure 1.
The main purpose of using the Simulink model
of Boeing 747 provided by Airlib is to apply a
linearization process based on previously defined
Fig. 1 Non-Linear Mathematical Model of Boeing 747 in Simulink [18]
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trim and steady-state flight conditions by obtaining
nonlinear system dynamics. In addition, the model
is also used to examine how the system behaves in
case of any fault. The faulty conditions here are
analyzed by creating residuals while considering the
difference between faulty and fault-free conditions,
and a threshold value is defined for the faulty states.
2.2 Linearized State-Space Model of Boeing
747 in MATLAB
In this work, obtaining a proper lateral orientation in
case of a fault will be the main field of study. The
linearization process is done in MATLAB, with the
help of a linear analysis tool. The model is
linearized through the non-linear Boeing 747 model
in flight conditions at Mach km/h and
ft 0.5 km altitude and steady-state flight.
The lateral dynamics of the aircraft as follows:
The system transition matrix:
Control distribution matrix:
Measurement matrix:
Feedforward matrix:
The latera states and control inputs are:
2.3 Controllability, Observability and
Stability Properties of Boeing 747
For controllability check from MATLAB, rank
(ctrb(A, B)) = 4. Therefore, Boeing-747 commercial
airplane is controllable which is equal to the number
of states which is 4 lateral states, sideslip angle, yaw
rate, roll rate and roll angle.
For observability check from MATLAB,
rank(obsv(A,B)) = 4 Therefore, Boeing-747 is
observable which is equal to the number of states
which is 4 lateral states, sideslip angle, yaw rate, roll
rate and roll angle.
For stability check The eigenvalues are
gathered from MATLAB eig(A):
Dutch roll mode
Spiral mode
The eigenvalues are on the left-hand plane so our
aircraft is laterally stable.
3 Estimation of Actuator/Control
Surface Stuck Faults
Detection of the grade at which the control surface
is stuck and the size at which control effectiveness
vanishes is the main objective of fault
parameterization in a linear estimator design [13].
Rudders and ailerons are the control surfaces at the
trailing edges of the tail and wings respectively.
Rudder, which will be one of the main objects of
our study is the component controlling the rotation
of an aircraft, referred to as yaw angle. In aviation
history, serious financial loss and personal
casualties resulted from a degeneration of the
control system performance which may be caused
by the faults of an aileron actuator, including motor
/ sensor coil break, cylinder leakage, and amplifier
gain reduction [14]. Thus, in order to avoid such
risks, a rudder stuck fault will be implemented on
the aircraft and the roll and yaw movement will be
affected by the faulty actuator, then, isolation and
reconfiguration process will be applied. The other
actuator faults to be investigated are the faults
observed in the ailerons that are used to turn the
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aircraft right or left by running asymmetrically by
the pilot input from pedals.
The Kalman-filter is established in case of
achieving a residual between the actual value and
estimated values of states to accomplish fault
detection via exceeding the threshold value. A two-
stage Kalman filter algorithm is performed in case
of control surface/actuator faults.
If there is a process noise the Kalman filter
relies on measurements for future state estimates.
However, if there is measurement noise in the
system, this time Kalman filter relies more on state
estimations to produce state predictions accurately.
The Q and R matrices are process and noise
covariance matrices respectively and they are
selected as random Gaussian noise with zero mean
and they have no coupling between to not have an
effect on state estimations.
The Two stage Kalman filter has two stages
considered as from the name, first it estimates the
actuator control loss and stuck degree and later on it
estimates the states [15].
3.1 Problem Formulation
Considering the open-loop aircraft model that has
been linearized around a trim operating point and a
parametrization of two different category of actuator
faults, the following model is achieved as in discrete
time as the execution of adaptive TSKF:
Here , and = state,
control input and output variables, respectively; and
and = bias vectors of dimension ,
representing the faults are entered actuators.
Suggested noise and are white Gaussian
noise sequences, zero-mean uncorrelated.
Here , and , and
Kronecker delta. The initial states and
are suggested as uncorrelated with the white
noise processes and and have covariances
and respectively.
The components
of describe the percentage reduction in the
control effectiveness when the terms
are considered together, where
Estimator design model Eq. (11-13) is inherited
from Eq. (15), with a set of new bias components
[10]
added to note the degrees at which control surfaces
are stuck. A combination consists of three terms
forms stuck fault model from Eq. (11-13):
We point out the conditions formed by the value of
fault parameters. The nominal case is represented by
Loss of control effectiveness as percentage
in the actuator is represented by
and a control surface stuck fault of magnitude
degrees is shown by
The linear adaptive TSKF [10] can be used for
estimation both for and by a general form of
to
Where which already models stuck
fault at the same time. Since
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By this method, it is crystal clear that the inputs of
must alter and develop in
time independently which is named as “persistently
excited” to make possible the estimator to figure out
among and also and their components [8].
3.2 Two-stage Kalman Filter Algorithm
The TSKF algorithm in [10] can be found as below:
The algorithm estimates both actuator fault
parameters and system states as well, is as follows:
Bias free state estimator:
where and are calculated is in
equations (36), (37), (39), and
as in (35):
The covariance of the filter and its residual
vector of the filter are obtained as below:
Bias estimator:
where is calculated as in Eq. (38).
Coupling equations:
Error and state covariance estimates,
compensated:
3.2 The Forgetting Factor
The Kalman filter is a recursive technique and
measures the value of the current variables by
analyzing previous steps, so the forgetting factor
technique can be used in this algorithm. The
forgetting factor reduces the weight of the previous
data when a new data is obtained, thus, it accelerates
the convergence to the actual value. [16]
4 Simulation Results
Rudder control loss of %100 and stuck fault at 2
degrees are introduced. Our first case consists of a
stuck fault at a time interval between 200 – 600
seconds at rudder. The fault detection stuck fault
case relates to the control surface which cannot
move and respond to pilot or flight control inputs.
From Figure 2, the control surface rudder is
stuck at 2 degrees so the control loss at the actuator
is 100% as can be shown between the implemented
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time interval. Stuck actuator fault with magnitude 2
(deg) has occurred in the rudder because
and . When we can obtain
the estimated stuck magnitude in the corresponding
control channel. The estimation error seems
very oscillating, but yet it is relatively small and
close to zero approximately.
The fault detection and isolation are done by
TSKF and the estimated surface stuck faults are
converged to actual surface stuck faults, which
confirms that the presented algorithm works reliably
and well.
Actual and estimated state variables in case of
rudder stuck fault are given in Figures 3 - 6.
Fig. 2 Actual inputs and estimations when a stuck
actuator fault in rudder occurred at 200 seconds till
600seconds
Fig. 3 Actual and estimated sideslip angle
Fig. 4 Actual and estimated yaw rate
Fig. 5 Actual and estimated roll rate
Fig. 6 Actual and estimated roll(bank) angle
We can imply from Figures 3 – 6 that all the
states are broken due to the fault in the rudder
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actuator and all are affected because of rudder stuck
fault, thus, estimations are also broken between the
time period from 200 to 600 seconds. Since there is
no fault in sensors, measurements are satisfactory,
but Kalman filter estimations are affected by rudder
stuck fault. It can be clearly seen from Figure 2 that
the actuator faults are detected and isolated well
using TSKF.
4 Conclusion
The two-stage Kalman filter is used to estimate the
control effectiveness of the actuator on behalf of an
actuator stuck fault incident occurring on Boeing-
747 commercial airplane. The actuator faults can be
diagnosed via TSKF that maintains the states and
stuck positions or control loss by two sections that
include encapsulated estimation algorithm
The simulation results show that the TSKF
algorithm performed well and estimated both the
faulty parameters and states as desired.
For the following study, a remedial control
action which is going to be taken by flight computer
for reconfiguration purposes of the flight control is
planned to be established. By estimating the grade
of the control loss of effectiveness and stuck degree
of faulty actuator, the required control action can be
taken place by remaining control accommodation.
This control enhancement is going to be ready for
maintaining flight safety whether there is a fault
occurring during flight envelope and reconfigurable
control actions can be structed by the information of
the Kalman filter sustains.
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Appendix
Table 1: Aerodynamic coefficients and trim
conditions for Boeing-747 [17]
Initial Flight
Condition of Boeing
747
Cruise(low)
State and Control
Value
ft/s
rad
rad
rad
rad
rad/s
rad/s
rad/s
rad
ft
ft
ft
rad
rad
rad
rad
Geometry and
Inertias
Value
Lateral Directional
aerodynamic stability
coefficients
Value
: roll moment
caused from sideslip
angle derivative
: roll moment
caused from roll rate
derivative
: roll moment
caused from yaw rate
derivative
: roll moment
caused from aileron
deflection
: rol moment
caused from rudder
deflection
: yaw moment
caused from sideslip
angle derivative
: yaw moment
caused from roll rate
derivative
: yaw moment
caused from yaw rate
derivative
: yaw moment
caused from aileron
deflection
: yaw moment
caused from rudder
deflection
: side force caused
from the side slip
derivative
: side force caused
from roll rate
derivative
: side force caused
from yaw rate derivate
: side force caused
from aileron deflection
: side force caused
from rudder deflection
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.
Conflicts of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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
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