1
Fault Diagnosis and Fault-tolerant Control
Strategy for the Aerosonde UAVFrançois Bateman†, Hassan Noura, Mustapha Ouladsine
Abstract
In this paper a Fault Detection and Diagnosis (FDD) and a Fault-tolerant Control (FTC) system for
an Unmanned Aerial Vehicle subject to control surface failures are presented. This FDD/FTC technique is
designed considering the following constraints: the control surface positions are not measured and some
actuator faults are not isolable. Moreover, the aircraft has an unstablespiral mode and offers few actuator
redundancies. Thus, to compensate for actuator faults, the healthy controls may move close to their saturation
values and the aircraft may become uncontrollable, this is critical due to its open-loop unstability. A nonlinear
aircraft model designed for FTC researches has been proposed,it describes the aerodynamic effects produced
by each control surface. The diagnosis system is designed with a bank of Unknown Input Decoupled
Functional Observers (UIDFO) which is able to estimate unknown inputs. It is coupled with an active
diagnosis method in order to isolate the faulty control. Once the fault diagnosed, an FTC based on state
feedback controllers aims at sizing the stability domain with respect to the flight envelope and actuator
saturations while setting the dynamics of the closed-loop system. The complete system was demonstrated
in simulation with a nonlinear model of the aircraft.
Index Terms
Aircraft dynamics and control; fault detection and diagnosis; actuator saturation; domain of attraction;
fault-tolerant flight control.
I. INTRODUCTION
In spite of a wide range of applications and good market prospects, the integration of Unmanned Aerial
Vehicles (UAVs) in the civilian airspaces depends on both the progress made with regards to their reliability
and a greater social acceptance. Reliability studies [1] have shown that Flight Control Systems (FCS) are
involved in about20% of the mishaps. Such failures are critical [2], also to enhance UAV reliability, it is
recommended in [1] to implement Self Repairing "Smart" FCS.In this paper, FDD and FTC are understood
in this sense. FTC are control systems that possess the ability to accommodate failures automatically in
† Corresponding author [email protected] , F. Bateman is with the LSIS, UMR CNRS 6168, Paul Cézanne University,
Marseille, France.
H. Noura is with the United Arab Emirates University.
M. Ouladsine is with the LSIS, UMR CNRS 6168, Paul Cézanne University, Marseille, France.
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order to maintain system stability and a sufficient level of performance. FTC are classified into passive and
active methods. The first one are equivalent to robust control methods and does not require FDD. They are
designed in order to guarantee an acceptable degree of performance in fault-free case and to accommodate
a priori known faults. The drawback is that fault tolerant isobtained by reducing the performances in
fault-free mode. By contrast, active methods react to the occurrence of system faults on-line in real-time
in an attempt to maintain the overall system stability and performance. To do that, an FDD module which
provides information about the fault is required. When a fault occurs, this latter is detected, diagnosed
and a new controller is implemented. This controller can be designed on-line or precomputed. This latter
technique is the one adopted in this paper. An exhaustive bibliographical review for FDD and FTC is
presented in [3].
FDD and FTC applied to aircraft flight control surface failures have received considerable attention in the
literature. As far as these topics are concerned, recent works have dealt with realistic scenarios: nonlinear
aircraft models and severe faults, such as locked-in-placeactuators, are considered [4], [5], [6]. Besides,
failures can be asymmetric, thus the equilibrium of momentsare upset and couplings appear between
the longitudinal and the lateral axis [7][8]. Sometimes thestudied aircraft are open-loop unstable [9].
Regarding the FDD problem, the actuator positions are not always measured (e.g. for small UAVs) and
must be estimated. Moreover aircrafts are often over-actuated and faults on their control surfaces cannot
be isolated [7], [10]. However, the FTC and FDD problems are rarely considered together [5][6].
Actuator saturations are also of paramount importance, especially in faulty mode, where to compensate
for the fault, the healthy actuator strokes may be significantly reduced and may affect the control stability. In
this respect, some FTC are designed in order to avoid actuator saturations [11]. Even though many papers
consider the actuator saturations to redistribute the remaining controls to achieve the desired moments
[4][8][12], to our knowledge, no work has dealt with the FTC design by considering the effects of actuator
saturations on the stability domain defined with respect to the flight envelope [13]. This latter being
defined as the state space region in which the aircraft was designed to fly. The proposed approach has
to be considered in the following perspective: when a control surface locks at a non-neutral position, the
state vector moves away from its equilibrium value with the risk to leave the flight envelope OR the region
of stability which depends on the actuator saturations. In either cases, the aircraft is theoretically lost. As
a rule, the state space region for which the aircraft can be safeguarded is defined as the intersection of
the flight envelope AND the region of stability. The proposedfault-tolerant controller aims at sizing the
stability domain to be at least larger than a given flight envelope. Thus, this latter defines the state space
region for which the aircraft can be safeguarded.
This paper tackles a joint FDD-FTC strategy for an unstable open-loop aircraft with redundant actuators
subject to asymmetric failures, actuator saturations are tacken into account. Practically, a bank of Unkown
Input Decoupled Functional Observers identifies the faultycontrol surface and estimates its lock position.
These data are used to define a new operating point and to select a precomputed controller which maximizes
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the DOA and guarantees a fast and soft transient. The remainder of this paper is organized as follows:
Section II describes the nonlinear model of the aircraft. Section III presents the FDD system where an
active diagnosis method is discussed. In Section IV, the FTCstrategy is designed and results are presented
in Section V.
II. AIRCRAFT MODEL
A. Aircraft dynamic model
Fig. 1. The UAV Aerosonde
The aircraft studied in this paper and shown in Fig.1 is the UAV Aerosonde for which the Aerosim
MATLAB toolbox [14] was developed. Note that its inverted V-tail is common to many UAVs like the AAI
RQ7A Shadow 200 or the BAI Scimitar [15]. In this paper, each control surface may lock at an arbitrary
position. In this perspective, the UAV model has been adapted to consider the aerodynamic effects produced
by each control surface. The controls are shown in Fig.1, they are fully indenpendent:δx is the throttle,
δar, δal, δfr, δfl, δer, δel control the right and left ailerons, the right and left flaps and the right and left
inverted V tail control surfaces respectively. In the sequel, these latter controls are named ruddervators
because they combine the tasks of the elevators and rudder. So, as the aircraft is open-loop unstable and
because the other controls offer few redundancies, faults on these controls are particularly critical.
The following dynamic model of the aircraft is presented in the case of a rigid-body aircraft, the weight
m is constant and the centre of gravityc.g. is fixed position. LetRE = (O,xE,yE, zE) be a right-hand
inertial frame such thatzE is the vertical direction downwards the earth,ξ = (x, y, z) denotes the position
of c.g. in RE . Let Rb = (c.g.,xb,yb, zb) be a right-hand body fixed frame for the UAV, att = 0 RE
andRb coincide. These frames are drawn in the Fig.16 in appendix A.The linear velocitiesµ = (u, v, w)
and the angular velocitiesΩ = (p, q, r) are expressed in the body frameRb wherep, q, r are roll, pitch
and yaw respectively. The orientation of the rigid body inRE is located with the bank angleϕ, the pitch
angleθ and the heading angleψ. The transformation fromRb → RE is given by a transformation matrix
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TbE given in appendix A. According to Newton’s second law:
u =FRbx
m− qw + rv
v =FRby
m+ pw − ru (1)
w =FRbz
m− pv + qu
ForcesFRbx , FRb
y , FRbz acting on the aircraft are expressed inRb, they are due to gravityFgrav, propulsion
Fprop, and aerodynamic effectsFaero. Let Rw = (c.g.,xw,yw, zw) be the wind reference frame where
xw is aligned with the true airspeedV . The orientation of the body reference frame in the wind reference
frame is located with the angle of attackα and the sideslipeβ and it is drawn in Fig.16 in appendix A.
The transformation fromRb → Rw is given by a transformation matrixTbw given in appendix A.
Furthermore, the aerodynamic state variables(V, α, β) and their time derivatives can be formulated
usingTbw from µ [16]. For the sake of clarity, the forces are written in the reference frame where their
expressions are the simplest. They are transformed into thedesired frame by means of the matricesTbE
andTbw or their inverse.
FgravRE =
(
0 0 g)T
FpropRb =
(
kρ
Vδx 0 0
)T
(2)
FaeroRw = qS
(
−CD Cy −CL
)T
The model of the engine propeller is given in [17],ρ is the air density,k is a constant characteristic
of the propeller engine,q =1
2ρV 2 andS denote the aerodynamic pressure and a reference surface. The
aerodynamic force coefficients are expressed as linear combination of the state elements and control inputs.
The calculation of these aerodynamic coefficients is detailed in appendix B.
CD = CD0 +S
πb2C2
L + CDδarδar + CDδal
δal + CDδfrδfr + CDδfl
δfl + CDδerδer + CDδelδel
Cy = Cyββ + Cyδarδar + Cyδal
δal + Cyδfrδfr + Cyδfl
δfl + Cyδerδer + Cyδelδel (3)
CL = CL0 + CLαα+ CLδarδar + CLδal
δal + CLδfrδfr + CLδfl
δfl + CLδerδer + CLδelδel
The relationships between the angular velocities, their derivatives and the moments(MRbx ,MRb
y ,MRbz )
applied to the aircraft originate from the general moment equation.J is the inertia matrix and× is the
cross product.
p
q
r
= J−1
MRbx
MRby
MRbz
−
p
q
r
× J
p
q
r
(4)
The moments are expressed inRb, they are due to aerodynamic effects and are modeled as follows:(
MRbx MRb
y MRbz
)
= qS(
bCl cCm bCn
)
(5)
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c and b denote respectively the mean aerodynamic chord and the wingspan. The aerodynamic moment
coefficients are expressed as a linear combination of state elements and control inputs as
Cl = Clββ + Clp
bp
2V+ Clr
br
2V+ Clδar
δar + Clδalδal + Clδerδer + Clδelδel + Clδfr
δfr + Clδflδfl (6)
Cm = Cm0 + Cmαα+ Cmq
cq
2V+ Cmδar
δar + Cmδalδal + Cmδerδer + Cmδelδel + Cmδfr
δfr + Cmδflδfl
Cn = Cnββ + Cnp
bp
2V+ Cnr
br
2V+ Cnδar
δar + Cnδalδal + Cnδerδer + Cnδelδel + Cnδfr
δfr + Cnδflδfl
Equations (3) and (6) make obvious the aerodynamic forces and moments produced by each control surface.
This is useful to model the fault effects and the redundancies provided by the healthy control surfaces.
It is also necessary to be able to track the flight path relative to earth. The kinematic relations are given
by:
TbE = TbEsk(Ω) (7)
ξ = TbEυ (8)
sk(Ω) is the skew-symmetric matrix such thatsk(Ω)ǫ = Ω×ǫ for anyǫ ∈ R3. However, the fault tolerant
control problem is first an attitude control problem, thus the heading angleψ and the cartesian coordinates
x, y variations are not studied in the sequel.
Let X = (ϕ θ V α β p q r z)T the state vector,U = (δx δar δal δfr δfl δer δel)
T the control vector
andY = (ϕm θm Vm αm βm pm qm rm hm)T the output measurement vector. Note that the height is
such ash = −z. From above, the model of the UAV which is detailed in appendix C can be written as
X = f(X) + g(X)U (9)
Y = CX
and the physical flight envelope of this UAV is defined as
XΦ =
X ∈ R9 : Xmin ≤ X ≤ Xmax
(10)
For a given operating pointXe,Ue, whereUe denotes the trim positions of the controls, the linearized
model of the aircraft can be written as
x = Ax+Bu (11)
y = Cx
For the equilibrium stateXe, we define the reduced flight envelopeXR as the largest ellipsoid centered
at Xe and contained in the physical flight envelope:
XR =
x ∈ R9 : xTRx ≤ 1
(12)
with R = diag(
(min(Ximax−Xie , |Ximin
−Xie |))−2
)
and i = 1 . . . 9. Later, this set will be used
as a reference set to estimate the Domain of Attraction (DOA)of the UAV in closed-loop. A projection
ontoR2 of this domain is illustrated in Fig.2.
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B. Control surface model
On the other hand, the actuator travels are boundedUmin ≤ U ≤ Umax. For theith control Ui, the
saturation levels are shown in Fig.3, and are defined as
ui+ = Uimax− Uie
ui− = Uimin− Uie (13)
With these notations,ui− ≤ 0 andui+ ≥ 0. These saturation levels are asymmetric. The method developed
by Hu and Lin in [18] to estimate the DOA presented further in the paper, allows only to deal with symmetric
saturation levels. Thus, for theith control, the saturation level is defined as
uisat= min(ui+, |ui−|) (14)
Consequently, the considered saturation range[−uisat,+uisat
] is reduced, therefore the estimation of the
DOA will be conservative. Unlike Hu and Lin’s theory in whichthe saturation levels are chosen between
x1
x2
x1maxx1min
XR
x2max
XΦ
x2min
Xe
Fig. 2. Physical and reduced flight envelopes
rotation axisui−
ui+
Uie
Uimin
Uimax
Fig. 3. Theith control surface in its trim position, the minimum and maximum deflections and the saturation levels
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±1, we have matched their results to consider other saturationlevels.
In the fault-free mode, the control surface deflections are constrained: asymmetrical aileron deflections
produce the roll control, pitch is achieved through deflecting both ruddervators in the same direction
and yaw is achieved through deflecting both ruddervators in opposite direction. Notice that flaps have
symmetrical deflection, they are only used to produce a lift increment during takeoff and a drag increment
during landing.
To design the autopilot, state feedback controllers based on eigenstructure assignment methods are com-
monly used to set aircraft handling qualities [19]. These methods allow to set the modes of the closed-loop
aircraft with respect to the standards [20] and to decouple some state elements from some modes. It is
based on the fault-free linearized model (11). The nominal control law is given by:
u = Kx (15)
C. Fault model
Faults considered in this work are stuck control surfaces. For t ≥ tf , the faulty control vectorUf (t) =
U f , wheretf is the fault-time andU f are the stuck control surface positions. For the simulations, a fault is
modeled as rate limiter response to a step. The slew rate is chosen equal to600/s which is the maximum
speed of the actuators. LetUh be the healthy controls, the state equation (9) in faulty mode becomes:
X = f(X) + gf (X)Uf + gh(X)Uh (16)
III. FAULT DIAGNOSIS
In this part, a fault actuator diagnosis system is designed.Failures may affect the ailerons and the
ruddervators. This diagnosis system could be achieved by measuring the actuator positions which requires
potentiometers, wiring and data acquisition boards. However, for a mini UAV, due to the lack of space and
to avoid increasing the weight, this solution is not realistic.
Without these measurements, control surface positions appear as unknown inputs which have to be
estimated using the measured outputs. The use of observers designed to estimate the unknown inputs
offers an interesting alternative solution. The Interacting Multiple Model approach consists in banks of
model-based state observer. Each observer is designed to a particular fault status of the system and the
overall state estimation is a combination of these models [21]. To deal with stuck control surfaces, various
works have used banks of augmented state vector Kalman filters to estimate unknown inputs considered
as constant state variables [7][22]. Even though this approach is convenient to estimate stuck actuator
positions, it does not allow to capture the fault transient.However, due to the open-loop unstability,
control surface failures have to be detected, isolated and estimated quickly. In this connection, Xiong [23]
proposed a combined reduced-order function of state/inputestimator called Unknown Input Decoupled
Functional Observer (UIDFO). This observer is able to estimate time varying inputs without differentiating
the measured outputs which are corrupted by noise.
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In this paper, the fault actuator diagnosis system is designed with a bank of UIDFOs and is implemented on
the nonlinear model of the aircraft. However control surfaces, such as ailerons, provide redundant effects.
Therefore, faults on these controls can not be isolated. Thus, an active diagnosis strategy is studied. It
consists of exciting the control surfaces with external additive signals in order to form the fault signatures
[7].
A. The Unknown Input Decoupled Functional Observer
In this part, results established in [23] are recalled. The following dynamic system driven by both known
and unknown inputs is considered
x = Ax+Bu+Gd (17)
y = Cx
wherex ∈ Rn is the state vector,u ∈ R
m is the known input vector,d ∈ Rℓ is the unknown input vector
andy ∈ Ro is the output vector.A, B, G andC are matrices with appropriate dimensions,C andG are
assumed to be full rank.
The UIDFO detailed in [23] provides an estimationd of the unknown inputd and an estimationz of
linear combination of stateTx. Theoretically, no boundedness conditions are required for the unknown
inputs and their derivatives.
z = Fz+Hy +TBu+TGd (18)
d = γ(Wy −Ez) with γ ∈ R∗+
MatricesF,H,T,W andE are all design parameters that need to be set in order to satisfy the following
conditions
FT−TA+HC = 0 F is stable,
E = (TG)TP with P solution of:PF+ FTP = −Q
andQ, a semi-positive definite matrix,
ET = GTTTPT = WC
rank(TG) = rank(G) = ℓ
(19)
These matrices exist if and only if
(i) rank(CG) = rank(G),
(ii) all unstable transmission zeros of system(A,G,C) are unobservable modes of(A,C).
With respect to condition (i), the ranks of matrices are obtained by computing their singular values.
The higher those singular values are, the less the controls are redundant, the better are the estimations.
Regarding condition (ii), the transmission zeros of matrixsystem(A,G,C) and the unobservable modes
of observability pencil(A,C) are the finite eigenvalues of these matrix pencils respectively. Theoretically,
they are obtained by computing the Kronecker’s forms of these pencils. However, due to the numerical
9
unreliability of the computation, this form is not suitableand the staircase form will be used to exhibit the
Kroenecker form [24].
It is proved in [23] thatd converges ond if the magnitude of the transfer function‖H(s)‖ such that
D(s) = H(s)D(s) (20)
tends towards theℓ × ℓ identity matrix in the bandwitdth of the UIDFO. According to[23], this can be
reached ifγ → +∞. However, the bandwidth increases asγ increases, which leads to an increase in the
noise sensitivity. Considering that the maximum speed for the servocontrols is600.s−1 and that a rough
estimation of the faulty actuator position is required by the FTC system, parameterγ is chosen in order
to find a compromise between a gain in the bandwidth close to one and an appropriate bandwidth.
B. FAULT DIAGNOSIS SYSTEM
1) Bank of UIDFOs for detection and partial isolation of actuator failures: The UIDFOs described above
are used to design the diagnosis system. Since the actual control surface positionsu are not measured,
they may be considered as unknown inputs. On the other hand, the controlsu generated by the controller
are known inputs. In the fault-free mode, it is assumed that the control surface positionsu are equal to
their control valuesu. Otherwise, in the faulty mode,u differs fromu.
Assuming that all the output measurement vector is used, conditions (i) and (ii) are satisfied, and a unique
UIDFO can estimate the unknown right and left aileron and ruddervator positions. Letu be the input
estimation vector. The diagnostic method consists of comparing the control vectoru to the estimation
vector u. A fault is detected if this difference, named residual, exceeds a given threshold. Unfortunately,
the true airspeed measurementVm is very noisy which deteriorates the unknown input estimation. To build
the diagnosis system without using this measurement, a bankof UIDFOs shown in Fig.4 is designed, it
requires a reduced number of UIDFOs. Each UIDFO is designed using the fault-free model in order to
estimate a subset of unknown inputs with respect to (i) and (ii). That means that the bank provides an
accurate estimation of all the unknown inputs only in the fault-free mode.
The first UIDFO computes an estimationu1 of the right aileron and left ruddervator positionsu1 =
(δar, δel)T . It uses the reduced output measurement vectoryr = y\Vm such asyr = Crx and the
controlsu2 = (δx, δal, δer)T generated by the controller. In the following,bi denotes theith column of
control matrixB in (11) matched with(
δx δar δal δfr δfl δer δel
)T
.
The first UIDFO equations are:
z1 = F1z1 +H1yr +T1B1u2 +T1G1u1 (21)
u1 = γ1(W1yr −E1z1)
With B1 =(
b1 b3 b6
)
andG1 =(
b2 b7
)
.
The second UIDFO estimatesu3 = (δal, δer)T . It uses the reduced output measurement vectoryr and the
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controlsu4 = (δx, δar, δel)T generated by the controller. This second UIDFO equations are:
z2 = F2z2 +H2yr +T2B2u4 +T2G2u3 (22)
u3 = γ2(W2yr −E2z2)
With B2 =(
b1 b2 b7
)
andG2 =(
b3 b6
)
.
The state space and control matrices (9) are detailed in appendix C. For each UIDFO, condition (i) is
achieved, indeedrank(CrG1) = rank(G1) and rank(CrG2) = rank(G2). Notice that the inputs in
u1 (resp u3) are those that present the fewest redundancies. On the other hand, The staircase forms of
pencils(
A G1 Cr
)
,(
A G2 Cr
)
and(
A Cr
)
have been computed using the GUPTRI algo-
rithm (Generalized Upper Triangular) [24]. For various operating points25m/s, 200m,40m/s, 200m,
25m/s, 1000m, 40m/s, 1000m and for various flight stages (flight level, climb, descent, turn), the
matrix pencil structures are invariant. Moreover the system matrices and the observability pencil have no
unstable transmission zero and no unobservable mode respectively. Therefore (ii) in §III-A is also fulfilled.
The functioning of the bank is described for a fault occuringon the right aileron. A similar reasoning
can be applied for the other control surfaces. When the control surface positionδar ∈ u1 is faulty, the
first UIDFO provides an accurate estimationδar of the actual faulty actuator position andδar = δar. But,
as δar is faulty, it differs from its control valueδar then δar 6= δar and δar 6= δar. For δel ∈ u1, the
control surface position of a healthy actuator, it is equal to its control value, soδel = δel. Moreover, the
first UIDFO provides a right estimationδel of the healthy actuator position thenδel = δel and δel = δel.
As for the second UIDFO, it estimatesu3 by processing the reduced output measurementsyr and the
controlled inputu4. This latter containsδar which is a wrong information of position whereasδel is true.
As a fault on the right aileron mainly affects the roll axis, UIDFO2 provides a false estimationδal of δal
and a right estimationδer of δer.
In order to avoid false alarms that may arise from the residuals transient behavior, these latter are integrated
+
-
-+
-?
-
6
-
-
yr∫ t+τ
t
∫ t+τ
t
σu1
-
-
-
-
+
-
+
--
u2
u4
u1
u3
selector-- selector
u1
u3
| · |
| · |UIDFO1
UIDFO2
-
-
-
6
?
reset
reset
- t
-τ
µu1-
-
-
Detection
Isolation
Logic
&
uf
µu3
σu3
Fig. 4. Bank of Unknown Input Decoupled Functional Observers
11
over a durationτ . Let σδarbe a threshold andµδar
be a logical state defined as∣
∣
∣
∣
∫ t+τ
t
δar(θ)− δar(θ)dθ
∣
∣
∣
∣
> σδar⇒ µδar
= 1 otherwiseµδar= 0 (23)
Then, to detect and to partially isolate the faulty control surface, an incidence matrix is defined as follows
TABLE I
THE INCIDENCE MATRIX
Faulty control µδar µδelµδal
µδer
right aileron 1 0 1 0
left aileron 1 0 1 0
right ruddervator 0 1 0 1
left ruddervator 0 1 0 1
The incidence matrix shows that faults on right and left ailerons (resp. right and left ruddervators) cannot
be isolated. This is due to the redundancies which make that the UIDFOs cannot estimate all the unknown
inputs using the reduced measurement vector. To overcome this problem an active fault diagnosis strategy
aiming at discriminating the faulty control is proposed in the sequel.
2) Active diagnosis: faulty aileron:At time tD, a fault is detected on the ailerons, next an active
diagnosis procedure is triggered. For a faulty aileron, this consists of exciting the right aileron, the flaps
and the ruddervators with sinusoidal signals at a frequencyω and amplitudes chosen in the kernel of the
reduced control matrixBred =(
b2 b4 b5 b6 b7
)
. The kernel is obtained from the singular value
decomposition ofBred. This null space exists if these actuators offer redundancies. Moreover, frequency
ω must be compatible with the servoactuator dynamics (about600/s). These excitation signals are given
by:
uex =(
δarmaxδfrmax
δflmaxδermax
δelmax
)T
sinωt (24)
Note that to accelerate the isolation process, the throttleδx, which dynamic is slower than the control
surfaces, is not used.
When the right aileron is faulty, the controls cannot fulfilluex ∈ ker(Bred) and this excitation disrupts
the output measurementsy which contain a signal with frequencyω. Detecting a faulty flap consists of
detecting this signal component in the output measurementsor in any signal processed with theme.g. the
unknown input estimations.
When the left aileron is faulty, since this control surface isnot excited withuex, the excitation has no
effect on the output measurements and these latter do not contain the frequencyω. Detecting a faulty
aileron consists of showing the absence of this frequency inthe output measurements or in any signal
processed with them.
Note that choosing the excitation signals in the kernel of control matrixB ensure the isolation of partial
as well as of total loss of control surface efficiency. The frequencyω does not appear in the output
measurements if and only if the controls are operating safely.
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When the right aileron is faulty, its estimated positionδar(t) contains a frequencyω which is detected
with a coherent demodulation method. This frequency detection is achieved by multiplyingδar(t) by a
sinusoidal signal named carrier which frequency isω. Next, the resulting signal is integrated over a duration
∆ =2kπ
ωwherek is a positive integer. Lets(∆) be the resulting signal andσ∆ a threshold. Ifs(∆) > σ∆
thens(∆) contains a frequencyω and the right aileron is faulty. Otherwise, the left aileronis declared to
be faulty. A similar approach is implemented to isolate the faulty ruddervator.
IV. FAULT TOLERANT CONTROL STRATEGY
The studies dealing with faulty systems should consider their actuator deflection ranges and their physical
limits. In the presence of an actuator fault, the healthy actuator saturation levels influence the efficiency
of the control laws. It is all the more true that these systemsare open-loop unstable since the actuator
saturation levels determine partially the stable state space region henceforth referred as the domain of
attraction (DOA). Yet, when an actuator fault occurs, the state vector moves away from its operating point
with the risk to leave the DOA OR the physical domain (the flight envelope for an aircraft). Therefore,
two problems have to be considered:
1) The analysis problem which consists in estimating the DOAwith respect to the a priori known
reduced flight envelope (10). In the fault-free case and in the faulty case, this problem allows to
compare the sizes of these two sets. For example, it should beinteresting to know the size of the
DOA in faulty mode when using the nominal controller.
2) The synthesis problem, an FTC strategy should be designedto increase the DOA by considering the
healthy actuator saturation levels.
A. Estimation of the domain of attraction
Considering the actuator saturations to estimate the DOA isa problem of paramount importance. After
a fault has occurred at timetF and before the nominal controller has been reconfigured to compensate
for the effects of the fault at timetR, the system operates under the feedback control designed for normal
conditions which provides inappropriate closed-loop control signals. To compensate for severe faults, the
healthy actuator control signals may saturate and the system may become uncontrollable [25]. Because of
the unstable mode, the state vector moves away from its operating point and may leave the DOA, leading
to the loss of the system.
The problem of estimating the DOA for general linear system under saturated linear feedback was
presented by Hu and Lin [18]. The definitions and the main results are recalled below, they are required
to understand the design of the FTC presented in Section IV-B. The saturation function is denotedsat.
For theith control ui ∈ u:
sat(ui) = sign(ui)min(uisat, |ui|) (25)
13
For an operating pointXe,Ue and under a given saturated linear state feedbacku = sat(Kx), e.g.K
is the nominal controller, the closed loop system is given by:
x = Ax+Bsat(Kx) (26)
Wherex ∈ Rn, u ∈ R
m andK ∈ Rm×n the state feedback matrix.
Given two feedback matricesK,H ∈ Rm×n wherehi denotes theith row of H and assume that
|hix| ≤ uisat, i = 1, . . .m. It is shown in [18] that the saturated state feedbacksat(Kx) can be placed
into the convex hullco of linear feedbacks
sat(Kx) ∈ co
DiKx+Di−Hx : i ∈ [1, 2m]
(27)
whereDi ∈ Rm×m are matrices whose diagonal elements are either1 or 0 andD−
i = I−Di with I the
identity matrix.
Further,L(K) is the region where the feedback controlsat(Kx) is linear inx.
L(K) := x ∈ Rn : |kix| ≤ uisat
, i = 1, . . . ,m (28)
For x0 = x(0) ∈ Rn andψ(t,x0) the state trajectory of (26), the DOA of the origin is defined as
S :=
x0 ∈ Rn : lim
t→∞
ψ(t,x0) = 0
(29)
A measure of the size of this set is obtained with respect to a reference setXR e.g the reduced flight
envelope (12). Assume that0 ∈ XR andXR is a convex bounded set. For a positive real numberα, denote
αXR = αx : x ∈ XR (30)
For a setS ⊂ Rn, define the size ofS with respect toXR as
αR(S) := supα > 0 : αXR ⊂ S (31)
For this problem, the reduced flight envelope presented above plays the role of the reference set. Let
P ∈ Rn×n be a positive-definite matrix and the ellipsoid
E(P, ρ) =
x ∈ Rn : xTPx ≤ ρ
(32)
The optimization problem solved by Hu aims at finding the largest contractive invariant ellipsoidE(P, ρ)such that the measureαR(E(P, ρ)) is maximized. Clearly, ifE(P, ρ) is contractive and invariant, then it
is inside the DOA. This optimization problem illustrated inFig.5 is given by [18]:
supP>0,ρ,H
α (33)
s.t. : a)αXR ⊂ E(P, ρ)
b) (A+B(DiK+D−i H))TP+P(A+B(DiK+D
−i H)) < 0 and i ∈ [1, 2m]
c) E(P, ρ) ⊂ L(H)
14
Let γ =1
α2, Q =
(
P
ρ
)
−1
, Z = HQ, andzi the ith row of Z. To solve this optimization problem, it
is transformed into a Linear Matrix Inequality (LMI) problem [18]
infQ>0,Z
γ (34)
s.t. : a)
γR I
I Q
≥ 0
b) QAT +AQ+ (DiKQ+D
−i Z)
TB
T +B(DiKQ+D−i Z) < 0 and i ∈ [1, 2m]
c)
u2
isatzi
ziT Q
≥ 0 i = 1, . . . ,m
Let γ∗ the optimum of this problem with solutionsQ∗ andZ∗, thenα∗ =1√γ∗
. Here, and without loss
of generality,ρ was chosen to be equal to1.
B. Fault tolerant control strategy
At time tF , a control surface locks, the equilibrium of forces and moments is broken and the state vector
moves away from the nominal operating pointXe. Fault reconfiguration requires the following conditions:
• a new operating pointXfe,U
he must exist,Uh
e denotes the trim positions of the healthy controls,
• at reconfiguration timetR, the state vector must belong to the physical flight envelopeAND to the
DOA of the UAV equipped with its fault tolerant controller.
The proposed FTC strategy consists of two stages. First, a new operating pointXfe,U
he is calculated,
taking into account the nonlinear characteristics of the UAV, the state variable limitations and the control
saturations. To compute this new operating point, the faulty control surface and its position must be known.
This information is provided by the diagnosis system studied in §III-B. Furthermore, the fault-free deflection
constraints of the healthy control surfaces are released and each one of the healthy actuators is trimmed
separately. Then, for this new operating point, a pre-computed state feedback controller is implemented.
This controller is designed by taking into account the actuator saturations, it aims at maximizing the DOA
while ensuring a fast and soft transient toward the new operating point.
x1
x2
x1maxx1min
XR x2max
XΦ x2min
E(P, ρ)αXR
Xe
DoA
Fig. 5. DOA, estimation of the DOA and measurement of its size
15
1) Operating point computation in faulty mode:An optimization method based on a sequential program-
ing quadratic algorithm was presented in [26] to compute a new operating point. It consists of re-allocating
the healthy controls in order to
• keep the new operating point close to the fault-free operating point. This is done by minimizing the
cost function:
J = (V − Ve)2 + (α− αe)
2 + (β − βe)2 + (Uh
j −Uhje)
TR(Uhj −Uh
je) (35)
whereR is a weighting matrix used to balance the demands on the healthy actuators. This cost
function has to satisfy the following constraints:
• the equilibrium equation,
f(Xfe) + gh(Xf
e)Uhe + gf (Xf
e)Uf = 0 (36)
• solutions included in the physical flight envelope and in thecontrol variation ranges,
Xmin ≤ X ≤ Xmax
Uhmin ≤ Uh ≤ Uh
max
(37)
• the flight level stage requires the following state variables set equal to zero,
ϕfe = pfe = qfe = rfe = 0 (38)
It is worth noticing that the new trims determine new saturation levels as it is illustrated in Fig.3 while
the new operating point determines the reduced flight envelope in faulty mode which is the largest ellipsoid
centered atXfe and contained in the physical flight envelopeXΦ as it is shown in Fig.6. From now on, the
reduced flight envelope in faulty mode is the reference set used to design an FTC which aims at maximizing
the DOA, the size of this latter being partially defined by theactuator saturations as it is shown in Hu and
Lin’s works.
Notice that, if no operating point satisfying an equilibrium exists, the aircraft will be lost. In this case,
degraded modes of operation such as a descent at minimum kinetic energy may be considered.
x1
x2
x1maxx1min
x2maxXΦ
x2min
XeXf
e
XR in fault-free mode
XR in faulty mode
Fig. 6. Physical and reduced flight envelopes
16
2) Linear state feedback controller design:For this new operating pointXfe,U
he, the faulty linearized
model is written as:
x = Afx+Bfuh (39)
The fault-tolerant state feedback controllerKf is subject to actuator saturations and the healthy control
vector is:
uh = sat(Kfx) (40)
It is designed in order to maximize the DOA while the poles areplaced in an LMI region as illustrated
in Fig.7. In faulty mode, this strategy aims at increasing the chances of saving the UAV while the current
state vector is steered toward the new equilibrium with a damping factor greater than or equal tocos
and a time response approximately less than or equal to3
κ1
. First, it is necessary to compute the feedback
matrix H such as the estimation of the DOA is maximized with respect toa reference set. This matrix
is obtained by solving the optimization problem (41) [18]. This problem is similar to (33) with an extra
optimization parameterM. It is solved using the LMI technique as defined by (34).
supP>0,ρ,H,M
α (41)
s.t. : a)αXR ⊂ E(P, ρ)
b) (Af +Bf (DiM+D−i H))TP+P(Af +Bf (DiM+D
−i H)) < 0 and i ∈ [1, 2m]
c) E(P, ρ) ⊂ L(H)
The LMI formulation proposed hereafter is a continuation ofHu and Lin’s works. In the sequel,H is
set equal to the optimal value obtained by solving (41). Thisguarantees a known DOA and allows to
choose, from all theM satisfying (41.b), the one which satisfies dynamic performance: here, controlling
the damping and the decay rate. To do so, LMI region constraints (42.d), (42.e), (42.f) are added to (41)
[27] which is transformed into LMIs and becomes:
infQ>0,Z
γ
s.t.
a)
γR I
I Q
≥ 0
b) QAfT +AfQ+ (D−
i HQ+DiZ)TBf
T +Bf (DiZ+D−i HQ) < 0 and i ∈ [1, 2m]
c)
u2
isatzi
ziT Q
≥ 0 i = 1, . . . ,m
d) 2κ1Q+QAfT +AfQ+BfZ+ Z
TBf
T< 0
e) 2κ2Q+QAfT +AfQ+BfZ+ Z
TBf
T> 0
f)
sin (AfQ+QAfT +BfZ+ ZTBf
T ) cos (AfQ−QAfT +BfZ− ZTBf
T )
cos (QAfT −AfQ+ ZTBf
T −BfZ) sin (AfQ+QAfT +BfZ+ ZTBf
T )
(42)
Let γ∗ the optimum of this problem with the solutionQ∗ andZ∗, thenα∗ =1√γ∗
andKf = Z∗(Q∗)−1.
17
C. Fault tolerant control implementation
A fault situation is defined by the faulty control surface andits lock position. This information is provided
by the fault diagnostic system. When a fault occurs, a new operating point and a feedback matrix must
be computed for each fault situation. The operating points in faulty mode are pre-computed and tabulated.
It is also possible to compute them online as it is proposed in[26]. The time required to compute the
feedback matrices (42) is incompatible with the short time available to make a decision. Thus, to design
the FTC controller, all feedback matrices corresponding toall fault situations are pre-computed. Next, for
each fault situation, a matched fault tolerant controller is selected in a bank.
However, and in order to reduce the number of controllers, for each control surface, the range of fault
positions is split into sectors. In each sector a unique controller must satisfy the LMI region constraints
and must guarantee a measure of the DOAαR ≥ 1. This last condition imposes the DOA to be greater
than or equal to the reduced flight envelope.
V. SIMULATION RESULTS
In this study, the UAV is supposed to fly level. Its operating point is defined byVe = 25 m/s andhe =
200 m. It is obtained with the trimsδxe= 0.57, δare = δale = δfre = δfle = 0, δere = δele = −3.9. The
physical flight envelopeXΦ is such as−45 ≤ ϕ ≤ 45, −15 ≤ θ ≤ 15, 15 ms−1 ≤ V ≤ 50 ms−1,
−12 ≤ α ≤ 12, −20 ≤ β ≤ 20, −90s−1 ≤ p, q, r ≤ 90s−1, 0 m ≤ h ≤ 3000 m. In these
conditions, the reference setXR which is also the reduced flight envelope is given by (12). Forthis flight
stage, with the nominal controller, the estimation of the size of the DOA with respect toXR is obtained
by solving (34) and is equal toα∗ = 1.1. Therefore the size of the estimation of the DOA is larger than
the reduced flight envelope.
Re
Im
LM
Ire
gio
n
κ2 κ1
Fig. 7. The LMI region
18
A. FTC design for the right aileron
The right aileron may lock at any position in[−20, 20]. Whatever the fault amplitude, with the
nominal controller, (34) has no solution and the UAV becomesunstable. However, by relaxing the deflection
constraints, an operating point always exists and the trimsallowing to reach it are illustrated in Fig.8.
Obviously, these new operating points impose new referencesets and new saturation levels. Regarding the
−20 −10 0 10 200.4
0.6
0.8
δar
stuck on [−20°,20°]
δ Xe (
%)
−20 −10 0 10 20−20
0
20
40
δ ale (
°)
δar
stuck on [−20°,20°]
−20 −10 0 10 20−5
0
5
δ fre (
°)
δar
stuck on [−20°,20°]−20 −10 0 10 20−4
−2
0
2δ fl e (
°)
δar
stuck on [−20°,20°]
−20 −10 0 10 20−6
−5
−4
−3
δ ere (
°)
δar
stuck on [−20°,20°]−20 −10 0 10 20−6
−5
−4
−3
δ ele (
°)
δar
stuck on [−20°,20°]
Trim in faulty modeTrim in fault−free modeTrim for δ
ar stuck at 3°
o
*
Fig. 8. Trims of the healthy controls for right aileron fault positions∈ [−20, 20]
FTC controller, a damping ratio greater than0.5 and a decay time less than1 s are desired so = 60 and
κ1 = −0.3 while κ2 is chosen to be equal−30. For a small aircraft and for a non terminal flight phase,
these values are compatible with the standards [28]. The feedback matrix which maximizes the DOA of
the saturated linear state feedback faulty system while it guarantees the poles in the desired LMI region is
given by (42). A unique controller is designed for the case where the right aileron is stuck at0. It aims
at accommodating all fault positions included in[−20, 20]. To prove its efficiency, faults are simulated
on the whole interval[−20, 20]. For the various linearized faulty model, the size of the estimation of
the DOA α∗ and the map of the poles of the faulty closed-loop systems areplotted in Fig.9. Asα∗ > 1,
the size of the estimation of the DOA is always larger than thereduced flight envelope. This means that
the physical domain determines the critical limit of use of the aircraft. Weird as it may look, the size
of the DOA in faulty mode is greater than those in fault-free mode. In fact, the deflection constraints
are relaxed and each healthy control surface produces roll,pitch and yaw moments (6) which extend the
aircraft manoeuvrability.
19
B. Right aileron failure
The UAV flies level, att = 41s, it initiates a turn and the right aileron locks at position3. The top plot
on Fig.8 shows the aileron position, the aileron control signal and the aileron position estimation. On the
middle plot, the error between these last two signals is continuously computed and integrated to produce
a decision residual, next this latter is compared to a threshold in order to generate an alarm. When the
fault is detected, a1s time window is set and sinuoidal signals are added to the control signals. The top
plot shows that the aileron position estimation contains a sinusoidal component which is detected with the
coherent demodulation process illustrated on the bottom plot. All the estimation positions provided by the
UIDFOs are sampled and averaged for the1s time window duration. Once the faulty control has been
isolated, the FTC uses the faulty control position estimation.
Without the FTC strategy, Fig.11 and Fig.12 show that the aircraft is lost. To compensate for the
fault, the throttle turns off, the angle and angular velocities oscillate, the height decreases and the true
airspeed increases. On the contrary, the proposed FTC strategy maintains the aircraft close to the fault-free
operating point. The FTC controller is triggered at timetR = 42.1s, the state variables are steered toward
their equilibrium in about one second with a good damping ratio. The reader can see the presence of the
sinusoidal excitation used for the isolation in the roll in the time interval[41.1s, 42.1s].
−20 −10 0 10 2055
55.5
56
56.5
57
57.5
58
58.5
59
59.5
60
Right aileron fault position
size
of t
he e
stim
atio
n of
the
DO
A
−10 −8 −6 −4 −2 0−10
−8
−6
−4
−2
0
2
4
6
8
10
0.5
0.5
Real axis
Imag
axi
s
damping> 0.5
κ1=−0.3
Fig. 9. Size of the DOA and map of the poles for right aileron fault positions∈ [−20, 20] with a unique fault-tolerant-controller
20
40 40.5 41 41.5 42 42.5 43
−10
−5
0
5
right aileron position (°)
40 40.5 41 41.5 42 42.5 430
0.01
0.02
decision residual and fault detection signal
40 40.5 41 41.5 42 42.5 43
0
5
10
x 10−3 residual for isolation and signal for isolation
time (s)
decision residual
threshold σδ
ar
fault detection signal
residual s(∆)1 s time windowthreshold σ
∆
real positioncontrol signalestimation
Fig. 10. Right aileron failure: the fault detection and isolation process
40 41 42 43 44 45 46 47 48 49 500
0.5
1
throttle
40 41 42 43 44 45 46 47 48 49 50−20
0
20right and left ailerons positions (°)
40 41 42 43 44 45 46 47 48 49 500
20
40right and left flap positions (°)
40 41 42 43 44 45 46 47 48 49 50−20
0
20right and left ruddervator positions (°)
time (s)
throttle with FTCthrottle without FTC
right control surface with FTC,left control surface with FTC,
without FTCwithout FTC
Fig. 11. Right aileron failure: the controls in faulty mode with and without FTC strategy
21
C. Failures on right ruddervator
The ruddervators are of great importance to the stability ofthe plane. If either one of these two control
surfaces locks at any position in the interval∈ [−20, 20], no operating point exist for this aircraft which
control surface deflections are constrainted. By relaxing them, an operating point exists for fault positions
in the [−9, 3] interval. Compared to the ailerons case, this interval is significantly reduced. This is due
to the fact that the ruddervators have to control both the pitch and the yaw axis. Thus, when one of these
two control surfaces is stuck, the remaining healthy controls provide few redundancies to accommodate
for the fault. As a consequence, the FTC can only operate for fault positions contained in this reduced
interval. The FDD process is similar to the one presented forthe right aileron failure and is illustrated
in Fig.13. At t = 20s, the ruddervators turn down, next one of the two ruddervators locks att = 22s
40 41 42 43 44 45 46 47 48 49 50
25
30
35true airspeed (m/s)
40 41 42 43 44 45 46 47 48 49 500
5
10
angle of attack (°)
40 41 42 43 44 45 46 47 48 49 50−40
−20
0
20
40sideslip (°)
time (s)
with FTCwithout FTC
40 41 42 43 44 45 46 47 48 49 50−100
0
100Bank angle (°)
40 41 42 43 44 45 46 47 48 49 50−50
0
50pitch angle (°)
40 41 42 43 44 45 46 47 48 49 50160
180
200
220height (m)
time (s)
Fig. 12. Right aileron failure: the UAV state vector in faulty mode with and without FTC strategy
22
and the fault position is equal to0. The top plot on this figure shows the actual ruddervator position, its
estimation and the ruddervator control signal. The middle plot shows the residual which is compared with a
threshold in order to produce a fault detection signal (23).At detection timet = 22.2s, a sinusoidal signal
is added to the left ruddervator (to all the controls surfaces except the right ruddervator) and a coherent
demodulation is started for a1 s time duration. As it is illustrated on Fig. 13 and Fig.14, the ruddervator
position estimation, the pitch and the yaw contain a sinusoidal component, this latter is detected and the
right ruddervator is declared to be faulty. The FTC dedicated to accommodate for this fault is selected.
Its design is similar to those of the ailerons. It aims at enlarging the DOA with respect to a reference set
while placing the poles in the aforementioned LMI region. Without the FTC strategy, Fig. 14 shows that
the bank angle differs from zero, the height decreases and the trajectory is uncontrolled. In opposite, the
proposed FTC strategy allows to keep the state variables close to their nominal values. Fig.15 shows that
right ruddervator stuck positions included in the[−9, 3] interval can be compensated for. However, for
faults outside this interval (dashed and dotted lines) the FTC strategy is ineffective and the aircraft is lost.
19.5 20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25
−5
0
5right ruddervator positions °
19.5 20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25
0
0.02
0.04
decision residual and fault detection signal
decision residual
fault detection signal
threshold σδer
19.5 20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25
0
5
10
x 10−3 residual for isolation and signal for isolation
residual S(∆)1s time windowthreshold σ∆
Fig. 13. Right ruddervator failure: the FDD process
VI. CONCLUSION
This paper illustrates a Fault-Diagnosis and Fault-Tolerant Control strategy applied to the Aerosonde
UAV. A nonlinear model of the aircraft which describes the aerodynamic effects produced by each control
surface has been proposed. The FDD system is designed with a model based approach in order to
diagnose control surface failures while control surface positions are not measured. Moreover, due to the
redundancies, faults on the ailerons (rudderevators) are not isolable. Thus an active diagnosis strategy has
been implemented in order to identify the faulty control surface. The information provided by the FDD
23
system is used to calculate a new operating point and to select a pre-computed fault-tolerant controller
in a bank of controllers. The design of these fault tolerant controllers takes explicitly into account the
physical limits of the system, particularly the control surface deflections. This design aims at maximizing
the domain of attraction while it guarantees the dynamic performances. The efficiency of the method has
been illustrated through simulations for faults on an aileron and ruddervator. The limits of the method have
also been highlighted.
15 20 25 30 35 40−0.2
0
0.2pitch (°/s)
15 20 25 30 35 4024
26
28
30true airspeed (m/s)
15 20 25 30 35 40
160
180
200
220height (m)
time (s)
with FTCwithout FTC
15 20 25 30 35 40−5
0
5
10bank angle (°)
15 20 25 30 35 40−1
0
1
2sideslip (°)
15 20 25 30 35 40−10
−5
0
5
10yaw (°/s)
time (s)
Fig. 14. Right ruddervator failure: the UAV state vector in faulty mode with and without FTC strategy
24
APPENDIX A
REFERENCE FRAMES AND TRANSFORMATION MATRICES
The transformation matrixTbE :
TbE =
cos θ cosψ sinϕ sin θ cosψ − cosϕ sinψ cosϕ sin θ cosψ + sinϕ sinψ
cos θ sinψ sinϕ sin θ sinψ + cosϕ cosψ cosϕ sin θ sinψ − sinϕ cosψ
− sin θ sinϕ cos θ cosϕ cos θ
(43)
20 25 30 35 40 45
0
0.5
1
Throttle
20 25 30 35 40 4518
20
22
24
26
true airspeed (m/s)
time (s)
δer
=−10°
δer
=+5°
−9°≤δer
≤4°
data4data5data6data7data8
20 25 30 35 40 45
−10
0
10
Pitch angle (°)
Fig. 15. Right ruddervator failure with FTC strategy
xE
yE
z1
x1
y12ψ ψ
xb
z2
θ
θ
yb
zb
ϕ
ϕ
xw
yw
z1
x1
ybβ β
xb
zb
α
α
Fig. 16. The reference framesRb, RE andRb, Rw .
25
The transformation matrixTbw:
Tbw =
cosα cosβ sinβ sinα cosβ
− cosα sinβ cosβ − sinα sinβ
− sinα 0 cosα
(44)
APPENDIX B
PARAMETERS OF THE AIRCRAFT MODEL
The controls used in the Aerosonde toolbox match those of a classical aircraft:δa, δf , δe andδr control
the ailerons, the flaps, the elevators and the rudder respectively. These controls are obtained by mixing the
control surfaces and are not suitable for an FTC problem. Indeed, the faults considered are asymmetric
control surface failures and from this point of view, the dimensionless aerodynamic coefficients of each
control surface must be taken into account.
δa =δar − δal
2
δf =δfr + δfl
2(45)
δe =δer + δel
2
δr =δer − δel
2
On the one hand, some aerodynamic coefficients are obtained from those provided with the Aerosonde
model by equalizing the forces and the moments produced by the actual and the virtual controls with
respect to the deflection constraints (45). These coefficients are reported in table II. For example, the pitch
moment produced withδe is equal to the pitch moment produced withδer andδel:
qScCmδeδe = qScCmδerδer + qScCmδelδel (46)
from (46) and due to aircraft’s symmetry:Cmδer = Cmδel =Cmδe
2.
On the other hand, geometrical considerations resulting from the Data Compendium (DATCOM) method
[29] allow to draw up roughly most of the ungiven aerodynamiccoefficients (fields marked with an asterisk
in table II). It is assumed that all the control surfaces are plain flaps.
The following example shows how to calculate the dimensionless flap roll-moment effectivenessClδf .
Given, the dimensionless flap lift-force effectivenessCLδf , according to [29] and for two symmetric plain
flaps:
CLδf = 0.9Kf
(
∂CL
∂δf
)
′
2Sflapped
ScosΛHL (47)
Sw is the wetted surface, the other parameters are illustratedin Fig.17. Thanks to a nomogram, it is possible
to estimateKf
(
∂CL
∂δf
)
′
. Next the dimensionless flap roll-moment effectivenessClδf is calculated with
Clδf =
∑ni=1
Kf
(
∂CL
∂δf
)
′
YiSi cosΛHL
Swb(48)
26
Similarly, knowing the dimensionless aileron roll-momenteffectivenessClδa , the dimensionless aileron-lift
effectivenessCLδa can be calculated.
ΛHL
δal δar
c.g.
Yi
Sflapped
cficfi
Si
0
Fig. 17. Geometrical parameters used to calculate the aerodynamic coefficients
TABLE II
COEFFICIENTS IN THE AERODYNAMIC MODEL OF THEAEROSONDE, VALID WITHIN THE [15, 50ms−1] RANGE
Drag value Lateral force value Lift value
CD00.0434 Cyβ -0.83 CL0
0.23
CDδar=
CDδa2
0.0151 Cyp 0 CLα 5.616
CDδal=
CDδa2
0.0151 Cyr 0 CLq 7.95
CDδfr=
CDδf
20.073 Cyδar
= Cyδa -0.075 CLδar∗ 0.34
CDδfl=
CDδf
20.073 Cyδal
= −Cyδa 0.075 CLδal∗ 0.34
CDδer =CDδe
20.00675 Cyδfr
− CLδfr=
CLδf
20.37
CDδel=
CDδe2
0.00675 Cyδfl− CLδfl
=CLδf
20.37
Cyδer = Cyδr 0.1914 CLδer =Cmδe
20.065
Cyδel= −Cyδr -0.1914 CLδel
=Cmδe
20.065
Roll value Pitch value Yaw value
Clβ -0.13 Cm00.135 Cnβ 0.0726
Clp -0.5 Cmα -2.73 Cnp -0.069
Clr 0.25 Cmq -38.2 Cnr -0.0946
Clδar= Clδa -0.1695 Cmδar
∗ 0.021 Cnδar= Cnδa 0.0108
Clδal= −Clδa 0.1695 Cmδal
∗ 0.021 Cnδal= −Cnδa -0.0108
Clδfr∗ -0.037 Cmδfr
=Cmδf
20.023 Cnδfr
−
Clδfl∗ 0.037 Cmδfl
=Cmδf
20.023 Cnδfl
−
Clδer = Clδr 0.0024 Cmδer =Cmδe
2-0.4995 Cnδer = Cnδr -0.693
Clδel= −Clδr -0.0024 Cmδel
=Cmδf
2-0.4995 Cnδel
= −Cnδr 0.693
27
APPENDIX C
DETAILED MODEL OF THE AIRCRAFT
ϕ = p+ q sinϕ tan θ + r cosϕ tan θ
θ = q cosϕ− r sinϕ
V = −g(cosα cosβ sin θ − sinβ cos θ sinϕ− sinα cosβ cos θ cosϕ)−qS
mCD +
kρ cosα cosβ
mVδx
α =g(sinα sin θ + cosα cos θ cosϕ)
V cosβ+ q − (p cosα+ r sinα) tanβ −
qS
mV cosβCL −
kρ sinα
mV 2 cosβδx
β =g(cosβ cos θ sinϕ+ cosα sinβ sin θ − sinα sinβ cos θ cosϕ)
V+ p sinα− r cosα+
qS
mVCy (49)
−kρ cosα sinβ
mV 2δx
p = J11 [qSbCl + (Jyy − Jzz)qr + Jzxpq] + J13 [qSbCn + (Jxx − Jyy)pq − Izxqr]
q = J22[
qScCm + (Jzz − Jxx)pr + Jxz(r2− p
2)]
r = J31 [qSbCl + (Jyy − Jzz)qr + Izxpq] + J33 [qSbCn + (Jxx − Jyy)pq − Izxqr]
ZE = −V cosα cosβ sin θ + V sinϕ cos θ sinβ + V cosϕ cos θ sinα cosβ
WhereIx, Iy and Iz are the principal moments of inertia andIxz = Izx are the products of inertia.Jij
is the value in theith row, jth column of the inverse of the inertia matrix.The state space and the control matrices forVe = 25 m/s andhe = 200 m.
A =
0 0 0 0 0 1 0 0.0743 0
0 0 0 0 0 0 1 0 0
−0.0 −9.81 −0.081 9.759 −0.00 0 0 0 −0.0
−0.0 −0.00 −0.031 −3.463 0 0 0.98 0 −0.0
0.39 0 0 0 −0.535 0.0741 0 −0.99 0
0 0 0 0 −98.89 −21.23 0 10.96 0
0 0 0.00 −94.67 0 −0.00 −5.0143 0.00 0.0
0 0 0 0 31.452 0.094 0 −2.613 0
−0.00 −25 0 25 0 0 0 0 0
(50)
B =
0 0 0 0 0 0 0
0 0 0 0 0 0 0
1.18 −0.23 −0.23 −1.12 −1.12 0.103 0.103
−0.003 −0.208 −0.2081 −0.226 −0.226 −0.0398 −0.0398
0 −0.045 0.0459 0 0 0.117 −0.117
0 −124.7 124.7 −27.1 27.1 5.2 −5.2
0 0.725 0.725 0.8069 0.8069 −17.26 −17.26
0 12.2 −12.2 1.85 −1.85 −23.9 23.9
0 0 0 0 0 0 0
(51)
ACKNOWLEDGMENT
The first author would like to acknowledge the support provided by the French Air Force Academy and
Pr. T. Hermas for proofreading the initial manuscript.
28
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François BATEMAN received his M.S. in electrical engineeringin 1992 from the Ecole Normale
Supérieure de Cachan, France and the Ph.D from Paul Cezanne University, Marseille, France, in
2008. He is currently teaching in the French Air Force Academyof Salon de Provence and leads his
research activities in the Paul Cezanne University. Bateman’s research interests are in fault diagnosis
and fault accommodation and application of these to aircraft and helicopters.
Hassan Noura received his Master and the PhD Degrees in Automatic Control from the University
Henri Poincaré, Nancy 1, France in 1990 and 1993. He obtained his Habilitation to supervise
Researches in Automatic Control at the University Henri Poincaré, Nancy1 in March 2002. He
was Associate Professor in this university from 1994 to 2003. In September 2003, he got a Professor
position at the University Paul Cézanne, Aix-Marseille III, France. He participated and led research
projects in collaboration with industries in the fields of fault diagnosis and fault tolerant control. He
has authored and co-authored one book and over 90 journal andconference papers.
Mustapha OULADSINE received his Ph.D. in 1993 in the estimation and identification of nonlinear
systems from the Nancy University (France). In 2001, he joined the LSIS in Marseille (France).
His research interests include estimation, identification,neural networks, control, diagnostics, and
prognostics; and their applications in the vehicle, aeronautic, and naval domains. He has published
more than 80 technical papers.
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