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 francois.bateman@lsis.org , 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

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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.