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Transactions on Fuzzy Systems
1
Abstract— This paper presents a fault-tolerant aircraft control
(FTAC) scheme against actuator faults. Firstly, the upper bounds
of the norms of the unknown functions are introduced, which
contain actuator faults and model uncertainties. Subsequently,
self-constructing fuzzy neural networks (SCFNNs) with adaptive
laws are capable of obtaining the bounds. The bound estimation
can reduce the computational burden with a lower amount of rules
and weights, rather than the dynamic matrix approximation.
Moreover, with the aid of SCFNNs, a multivariable sliding mode
control (SMC) is developed to guarantee the finite-time stability of
the handicapped aircraft. As compared to the existing intelligent
FTAC techniques, the proposed method has twofold merits: fault
accommodation can be promptly accomplished and decoupled
difficulties can be overcome. Finally, simulation results from the
nonlinear longitudinal Boeing 747 aircraft model illustrate the
capability of the presented FTAC scheme.
Index Terms—Fault-tolerant aircraft control; actuator faults;
self-constructing fuzzy neural network; finite-time stability;
multivariable sliding mode control.
I. INTRODUCTION
ITH a high degree of integrating automation
technologies, aerospace engineering systems have become
increasingly vulnerable to anomalies caused by structure
impairments, actuator/sensor faults, or other subsystem
malfunctions. Each of the in-flight failures can alter aircraft
characteristics, further undermining safety. Without any
appropriate reactions engaged in a timely fashion, even a
relatively minor error may develop into catastrophes.
Fault-tolerant aircraft control (FTAC) designs to maintain
flight safety can be essentially classified into passive and active
approaches [1-3]. Within a passive FTAC context, one flight
controller is usually developed with consideration for both
normal and faulty cases. The resulting control thereby makes
the closed-loop system invulnerable to the anticipated faults
without any control structure or parameter adjustment. This
type of FTAC provides accommodation for faults from a
“passive” viewpoint. On the other hand, the principle of active
FTAC is to reconfigure the flight controller in response to the
knowledge of the current state of the aircraft. Thus, the term
“active” implies that corrective actions are triggered to handle
the identified system/component malfunctions.
The past decades have witnessed the development of various
FTAC technologies. 1) With respect to passive FTAC, the
eigenvalue assignment technique [4] and multi-objective
optimization approach [5, 6] are exploited for preserving the
asymptotic stability of the handicapped aircraft and an
acceptable level of performance. However, feasible solutions
may not be found if excessive quantities of fault scenarios are
prescribed in the design phase of passive FTAC. 2) Active
FTAC systems are developed based on a variety of control
technologies. To mention a few, model predictive control (MPC)
[7], backstepping control [8, 9], adaptive control [10-14],
sliding mode control (SMC) [15-17], and linear parameter
varying (LPV) control [18] techniques are exploited to
reconfigure the control corresponding to in-flight faults. Within
an active FTAC scheme, the accuracy of fault detection and
diagnosis (FDD) and switching time of reconfigured control
have a predominant impact on fault tolerant performance [2, 3].
Additionally, by resorting to fuzzy techniques, several results
in the literature are available not only to improve the safety of
other engineering systems [19, 20], but also to advance the state
of the art of FTAC designs. In [21], a fuzzy model reference
learning control technique is deployed to counteract the effects
of aileron stuck failures. Moreover, an expert supervisory
mechanism enables the flight safety without explicit FDD
results. A sequential adaptive fuzzy inference system (SAFIS),
which can update the rules, is adopted to approximate the
aircraft dynamics [22]. As a consequence, the SAFIS-aided
FTAC allows the aircraft to successfully land in spite of
actuator failures. As reported in [23], fuzzy logic systems (FLSs)
are employed to estimate amplitudes of actuator gain and bias
faults. Then, the resulting adaptive controller attempts to
guarantee the asymptotic stability of the near-space vehicle
(NSV) subject to actuator malfunctions. The use of a Takagi-
Sugeno (T-S) fuzzy model is established to describe the NSV
dynamics [24-26]. An adaptive control approach is applied to
alleviate adverse impacts of actuator faults [24, 26], while an
adaptive observer is developed to identify sensor faults [25]. In
[27], type-2 fuzzy logic control and SMC methods are
combined to cope with aircraft actuator faults. The basic idea
in [27] is to separate the FTAC into the pitch, roll, yaw, and
altitude channels. More recently, the findings in [28, 29] show
that the unknown nonlinear functions can be estimated by FLSs,
while the asymptotic stability of the faulty aircraft can be
maintained using SMC techniques.
Fault-Tolerant Aircraft Control Based on Self-
Constructing Fuzzy Neural Networks and Multivariable
SMC under Actuator Faults
Xiang Yu, Senior Member, IEEE, Yu Fu, Peng Li, and Youmin Zhang, Senior Member, IEEE
W
This work was supported in part by Natural Sciences and Engineering
Research Council of Canada, in part by the National Natural Science
Foundation of China under Grant 61403407, Grant 61573282, and Grant
61603130, and in part by SPNSF under Grant 2015JZ020.
Xiang Yu is with the Department of Mechanical, Industrial and Aerospace
Engineering, Concordia University, Montreal, Quebec, H3G 1M8, Canada (E-
mail: xiangyu1110@gmail.com).
Yu Fu is with the State Grid Zhangzhou Electric Power Supply Company.
She was with the Department of Mechanical, Industrial and Aerospace
Engineering, Concordia University, Montreal, Quebec, H3G 1M8, Canada (E-
mail: yufu1013@gmail.com).
Peng Li is with the College of Mechatronics Engineering and Automation,
National University of Defense Technology, Changsha, 410073, China (E-
mail: lipeng_2010@163.com).
Youmin Zhang (Corresponding Author) is with the Department of
Mechanical, Industrial and Aerospace Engineering, Concordia University,
Montreal, Quebec, H3G 1M8, Canada (E-mail: ymzhang@encs.concordia.ca).
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Transactions on Fuzzy Systems
2
Although extensive design activities are conducted for
aircraft safety, there are some difficulties that need to be
addressed. 1) Model variations and FDD accuracy are well
recognized as two important factors affecting FTAC
performance [2, 3]. Any real continuous functions on a compact
set can be approximated to an arbitrary accuracy using a FLS.
Hence, a FLS has potential for estimating aircraft dynamics and
aircraft faults. The adaptation capability of FLS with only
output weights being updated is limited due to the fact that the
regressors are fixed [23, 28-31]. By taking advantage of neural
networks (NNs) [32], self-constructing fuzzy systems are
capable of updating fuzzy rules under system operating
conditions [22, 33, 34]. Nevertheless, the computational cost is
substantially increased as the quantity of rules and weights
increases. As aforementioned, determining how to exploit a
fuzzy system with a high level of adaptation capability and a
low level of computation burden is very challenging for FTAC
design. 2) Time available for fault recovery depends solely on
fault nature and flight conditions [2, 3]. In flight, the time frame
of faulty aircraft developing into an irreversible state is
typically a few seconds. More specifically, actuator
malfunctions can quickly drive the faulty aircraft out of control
without prompt reactions exposed. Thus, for preventing aircraft
breakup, fault accommodation must be accomplished in a
timely manner. However, the existing FTAC based on fuzzy
strategies can only guarantee the asymptotic stability of the
handicapped aircraft [24, 26-29]. 3) In most of FTAC systems
based on both fuzzy and SMC techniques, the design problem
is often formulated as the decoupled problem with m single-
input structures. By contrast, aircraft aerodynamics exhibits
strong couplings. For instance, in addition to contributing to
rolling maneuvers, ailerons can affect pitching and yawing
motions. Thus, accounting for multivariable situations may be
more appropriate for FTAC design rather than the decoupled
treatment.
In an attempt to overcome the discussed difficulties, this
paper presents new developments in the integration of self-
constructing fuzzy neural network (SCFNN) and multivariable
SMC methods into a FTAC system against actuator faults.
Since the proposed FTAC can actively counteract actuator
faults, it can be seen as an active FTAC scheme. The major
contributions are briefly outlined by three aspects.
1) SCFNNs, which can be continuously running to update both
the structures and parameters, are incorporated into adaptive
techniques. Consequently, the upper bounds of the norms of
unknown functions including actuator fault amplitudes and
model uncertainties can be captured. When comparing to
the previous studies [23, 27-29], the proposed algorithm
with the learning property in response to actuator faults can
achieve superior approximation performance and facilitate
fault accommodation. Furthermore, estimating the bounds
helps in reducing the computational burden with a lower
amount of rules and weights, as opposed to approximating
the overall dynamics [22, 33, 34].
2) A SMC approach is deployed in the proposed FTAC
scheme. The trajectory of the faulty system can be steered
to the equilibrium within finite time as long as the sliding
surface is reached. Thus, the resulting FTAC can ensure the
finite-time stability of the aircraft, even under conditions
involving actuator faults and model uncertainties. This
feature sets this study apart from the similar works [24, 26-
29], based on which the stability of post-fault aircraft is
asymptotically guaranteed. Hence, the integration of finite-
time SMC allows the developed scheme to improve flight
safety.
3) The FTAC based on SCFNNs and SMC is designed for
multivariable situations. The so-called multivariable SMC
is formed by vector expression, which is successfully
incorporated in the FTAC design. In contrast to [27], the
decoupled issue can be avoided in the proposed algorithm
by incorporating the multivariable SMC approach.
Therefore, the proposed design becomes more proper for the
cases where strong couplings are inherent to aircraft
aerodynamics.
The remainder of this paper is arranged as follows. Aircraft
longitudinal model and actuator fault model are presented in
Section II. The principle of SCFNN is described in Section III.
A FTAC scheme is developed to counteract actuator failures
within finite time in Section IV, where the SCFNNs and the
multivariable SMC are integrated. In Section V, the
performance of the proposed FTAC scheme is evaluated
through simulation studies based on a longitudinal model of
Boeing 747 aircraft. Section VI includes a discussion of the
conclusions.
TABLE I
NOMENCLATURE
Symbols
Interpretations
,
, and
Mean chord length, reference surface area, and
dynamic pressure
Inertial coefficient
-axis engine position
and
Center of gravity positions
, , and
True airspeed, angle of attack (AOA), and pitch
angular rate
and
Inner elevator deflection and outer elevator deflection
and
Stabilizer deflection and thrust
and
Total mass and gravity acceleration
Total lift coefficient
Lift coefficient for the rigid aircraft at zero stabilizer
angle
Effective factor of the elevator
Pitch moment coefficient
Pitch moment coefficient for the rigid aircraft at zero
stabilizer angle
Drag coefficient at a fixed Mach number
II. AIRCRAFT LONGITUDINAL MODEL
A. Aircraft Dynamics
Even though the analysis and the design approaches are not
limited to a specific type of aircraft in this work, it is
advantageous to work with a specific aircraft system to explain
the concepts and to validate the design procedure. The Boeing
747 series 100/200, as one of the most popular and widely used
wide-body commercial jet airliners, is used as an example in
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Transactions on Fuzzy Systems
3
this research to illustrate the FTAC design procedure.
According to [35], the body-axes longitudinal motion of the
Boeing 747 without considering flexible effects can be
represented as:
, (1)
. (2)
The body-axis aerodynamic forces and moments are described
as:
, (3)
, (4)
. (5)
The aerodynamic coefficients for the longitudinal motion can
be expressed as:
, (6)
, (7)
.
(8)
Furthermore, the aerodynamic coefficients can be
approximated as polynomial functions of AOA and velocity
over the flight regime [36]:
, (9)
, (10)
, (11)
, (12)
, (13)
where
(14)
Remark 1: From Eq. (6), it is known that the lift coefficient
is based on the effects of the pitch angular rate, the elevator
deflections, and the basic component
, respectively. Eq. (7)
indicates that the drag coefficient
greatly relies on the effect
of the Mach number. As can be observed from Eq. (8), the
essential factors affecting the pitch moment coefficient
contain the pitch angular rate, the inner elevator deflection, the
outer elevator deflection, the stabilizer deflection, and the basic
component
, respectively.
Substituting Eqs. (3)-(8) into Eqs. (1)-(2) gives:
, (15)
. (16)
The aircraft parameters cannot be obtained precisely, leading
to the challenges for flight control design. In common practice,
there exist additive parameter perturbations
to the nominal
values:
. (17)
By defining
and
, the
longitudinal motion equations can be simplified as:
, (18)
where
and
are smooth nonlinear
functions of .
and
stand for the nominal terms of
and
, while
and
denote the uncertain terms
(modeling errors/uncertainties) of
and
, respectively.
Remark 2: Linearized models based on small perturbation
theory are often used at the flight control design stage. Even
though the control design is relatively simple using the
linearized model, the performance may be greatly degraded
when the resulting control is engaged in a realistic environment.
Furthermore, LPV [18] and T-S fuzzy [22-26] modelling
techniques are recently applied to approximate aircraft
nonlinear dynamics. The basic idea is to linearize the aircraft
model at specific operating points and establish the relationship
between these points. Nonetheless, model approximation
accuracy and computational burden are recognized as major
challenges. In this study, the nonlinear model as Eq. (18) is
established to describe the nonlinear aircraft characteristics.
Due to the lack of modeling technologies and experimental data,
aerodynamic coefficients and relevant parameters cannot be
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Transactions on Fuzzy Systems
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obtained precisely. By considering this fact, model uncertainty
is included as well in Eq. (18).
B. Actuator Fault Model
Actuators that can generate appropriate forces and moments
are key components in any aircraft. Desired maneuvers can be
completed if actuation systems work under a normal condition.
On the contrary, poor performance and even instability are
induced by actuator malfunctions. Gain fault appearing on an
actuator is thought as a multiplicative-type fault, which
deteriorates actuator effectiveness. Actuator bias fault as an
additive-type fault creates a specific drift from the true
amplitude. Since both gain fault and bias fault are concerned in
this study, the model of actuator faults is represented as:
, (19)
where
is used to describe the gain fault
and
denotes the bias fault, respectively. Note
that
for .
Remark 3: It is reported that the leakage of hydraulic fluid
can be the root cause of degrading the actuator effectiveness
[37]. Therefore,
in Eq. (19) can be seen as
the indicators of actuators effectiveness, where
. In addition, a flight actuation system consists of an actuator
controller, an actuator, and embedded sensors. The sensor fault
in an actuator system can attribute to actuator bias faults. If the
amplitude sensor encounters a bias fault, the measured
amplitude is the actual amplitude plus the bias value. The
sensed amplitude is mandated to follow the referenced signal.
However, the actual value of the actuator amplitude is deviated
from the one as required by the flight control. Hence,
in Eq. (19) can represent bias faults of the inner
elevator, outer elevator, and stabilizer, respectively.
Therefore, the corresponding expression for the aircraft
longitudinal motion can be further represented upon Eq. (18) by
taking into consideration of the actuator faults as follows:
. (20)
Assumption 1: It is assumed that the following inequalities
hold:
, (21)
, (22)
where
is the pseudo inverse of
,
and
are unknown
positive parameters, respectively.
Remark 4: The term
is pertinent to the
model uncertainties and the bias faults of actuators. A close
look at
reveals that this term is
associated with the uncertainty of control input matrix and the
gain faults of actuators. The condition,
, implies that
dominates the function
. This condition, in turn, ensures that the
configured actuation systems possess adequate authority to
counteract the considered faults.
C. Problem Statement
Even though the aircraft encounters actuator faults and model
uncertainties, a FTAC system based on SCFNNs and
multivariable SMC with adaptation techniques is proposed in
this paper such that: 1) aircraft states can track the reference
signals; and 2) the stability of the closed-loop system can be
guaranteed within finite time.
III. SELF-CONSTRUCTING FUZZY NEURAL NETWORKS
SCFNN possesses the learning ability of NNs to tune the
shape of the fuzzy membership functions and the output
weights. In this study, the purpose of SCFNNs is to capture
online the upper bounds (
and
) of the norms of unknown
terms. It should be emphasized that the estimation process of
and
is time varying.
A. SCFNN Architecture
The SCFNN, sketched in Fig. 1, is comprised of four layers.
Layer 1 receives the input variables. The membership values
are calculated in Layer 2 such that the degree to which an input
value associates with a fuzzy set can be determined.
Precondition matching is carried out in Layer 3. The
preconditions of the fuzzy rules are specified by the links before
Layer 3, while the consequences are described by the
succeeding links. Layer 4 is regarded as the output layer.
Fig. 1. Illustration of SCFNNs.
The rule base is:
IF
is
, THEN
, (23)
where , ,
,
denotes the membership value of the ith input variable in the
rule , and
is the output action strength related to the rule .
The fuzzy basis function (FBF),
, is represented as:
, (24)
where
,
,
,
,
, and ,
respectively. and denote the FBF center vector and the
width vector, respectively.
For ease of notation, the output weight matrix and the
regressor vector are specified as:
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Transactions on Fuzzy Systems
5
, (25)
. (26)
The SCFNN output with N fuzzy rules are thereby described
in a vector form:
, (27)
where
and
are the estimates of and , and
stands
for the approximation error.
By adopting the SCFNN,
can be approximated in a
manner nearly identical to that described for
:
, (28)
where
and
denote the output weighting matrix
and regressor vector,
specifies the approximation error,
and represent the FBF center vector and the width vector,
respectively.
Assumption 2: With respect to the SCFNN of
, all
parameters are bounded on
, and
. (29)
Focusing on the SCFNN of
, all the parameters are
bounded on
, and
. (30)
B. Self-Constructing Mechanism
The approximation error in general depends on the number
of fuzzy rules (N in this paper). A small number N usually
results in low accuracy. In contrast, the reduction of the
approximation error becomes negligible if the number N is
adequately large. The role of self-constructing mechanism is to
generate or delete rules in terms of the novelty of correction
observation
to the existing FBFs. With considerations
analogous to [34, 38], a new rule is created when the distance
between a new input signal and the current clusters is too far,
while a redundant rule is removed when the fuzzy rule is
insignificant. Note that the SCFNN starts with no fuzzy rule.
Hence,
,
,
, and
,
respectively. The system model presented in this study is
continuous. However, the SCFNN needs sampled data to
accomplish self-constructing. Thus, “ ” represents the
previous sampling interval in the sequel [30, 31]. Without loss
of generality, suppose that there exist
FBFs to be
adjusted before the current input
arrives, i.e.,
,
,
, and
.
Define a measure between
and the existing FBFs as:
, (31)
where
,
, and
,
respectively. The following is to present the criteria of rule
generation and removal.
1) Rule Generation
Find the nearest fuzzy rule as:
. (32)
A new FBF is required:
, (33)
from the condition:
, (34)
where
is a predefined threshold to be chosen as
,
represents the initial width of the generated FBF,
and
, respectively.
2) Rule Removal
Find the redundant FBFs as:
,
. (35)
If the following condition satisfies:
, (36)
the redundant fuzzy rule is eliminated:
, (37)
where
is a pre-specified threshold under which
the fuzzy rule is determined inappropriate and
,
respectively.
For the sake of brevity, the self-constructing mechanism to
produce or delete rules in terms of the novelty of correction
observation
to the existing FBFs is omitted herein.
Remark 5: The past few years have witnessed the
development of learning approaches. A self-learning fuzzy
logic system with reinforcement learning techniques can
capture the desirability of states and adjust the fuzzy rules
accordingly. One of the main hurdles is that the determination
of fuzzy rules greatly relies on pure experiments [39, 40]. A
