Fault Diagnosis and Fault-tolerant Control Strategy for the ...

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Fault Diagnosis and Fault-tolerant Control Strategy for the Aerosonde UAV François 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 unstable spiral 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 about 20% 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 is obtained 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-place actuators, are considered [4], [5], [6]. Besides, failures can be asymmetric, thus the equilibrium of moments are upset and couplings appear between the longitudinal and the lateral axis [7][8]. Sometimes the studied 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 proposed fault-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 faulty control 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 FTC strategy 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 , δf r , δf l , 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 gravity c.g. is fixed position. Let RE = (O, xE , yE , zE ) be a right-hand inertial frame such that zE 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, at t = 0 RE and Rb coincide. These frames are drawn in the Fig.16 in appendix A. The linear velocities µ = (u, v, w) and the angular velocities Ω = (p, q, r) are expressed in the body frame Rb where p, q, r are roll, pitch and yaw respectively. The orientation of the rigid body in RE is located with the bank angle ϕ, the pitch angle θ and the heading angle ψ. The transformation from Rb → RE is given by a transformation matrix

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TbE given in appendix A. According to Newton’s second law: FxRb − qw + rv m FyRb + pw − ru v˙ = m F Rb w˙ = z − pv + qu m u˙ =

(1)

Forces FxRb , FyRb , FzRb acting on the aircraft are expressed in Rb , they are due to gravity Fgrav , propulsion Fprop , and aerodynamic effects Faero . Let Rw = (c.g., xw , yw , zw ) be the wind reference frame where xw is aligned with the true airspeed V . The orientation of the body reference frame in the wind reference frame is located with the angle of attack α and the sideslipe β and it is drawn in Fig.16 in appendix A. The transformation from Rb → Rw is given by a transformation matrix Tbw given in appendix A. Furthermore, the aerodynamic state variables (V, α, β) and their time derivatives can be formulated using Tbw 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 the desired frame by means of the matrices TbE and Tbw or their inverse. Fgrav RE

=

Fprop Rb

=

Faero Rw

=

T 0 0 g  T kρ δx 0 0 V  T q¯S −CD Cy −CL



(2)

The model of the engine propeller is given in [17], ρ is the air density, k is a constant characteristic 1 of the propeller engine, q¯ = ρV 2 and S denote the aerodynamic pressure and a reference surface. The 2 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. S 2 C + CDδar δar + CDδal δal + CDδf r δf r + CDδf l δf l + CDδer δer + CDδel δel πb2 L

CD

=

CD0 +

Cy

=

Cyβ β + Cyδar δar + Cyδal δal + Cyδf r δf r + Cyδf l δf l + Cyδer δer + Cyδel δel

CL

=

CL0 + CLα α + CLδar δar + CLδal δal + CLδf r δf r + CLδf l δf l + CLδer δer + CLδel δel

(3)

Rb Rb b The relationships between the angular velocities, their derivatives and the moments (MR x , My , Mz )

applied to the aircraft originate from the general moment equation. J is the inertia matrix and × is the cross product.

        b p p MR p˙ x                −1  R b q˙  = J My  − q  × J q          Rb r r Mz r˙

(4)

The moments are expressed in Rb , they are due to aerodynamic effects and are modeled as follows:     b b b (5) = q¯S bCl c¯Cm bCn MR MR MR z y x

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c¯ and b denote respectively the mean aerodynamic chord and the wing span. The aerodynamic moment coefficients are expressed as a linear combination of state elements and control inputs as Cl

=

Cm

=

Cn

=

br bp + Clr + Clδar δar + Clδal δal + Clδer δer + Clδel δel + Clδf r δf r + Clδf l δf l (6) 2V 2V c¯q Cm0 + Cmα α + Cmq + Cmδar δar + Cmδal δal + Cmδer δer + Cmδel δel + Cmδf r δf r + Cmδf l δf l 2V bp br Cnβ β + Cnp + Cnr + Cnδar δar + Cnδal δal + Cnδer δer + Cnδel δel + Cnδf r δf r + Cnδf l δf l 2V 2V

Clβ β + Clp

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: ˙ bE T

=

TbE sk(Ω)

(7)

ξ˙

=

TbE υ

(8)

sk(Ω) is the skew-symmetric matrix such that sk(Ω)ǫ = Ω×ǫ for any ǫ ∈ R3 . However, the fault tolerant control problem is first an attitude control problem, thus the heading angle ψ 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 δf r δf l δer δel )

and Y = (ϕm θm Vm αm βm pm qm rm hm )

T

T

the control vector

the output measurement vector. Note that the height is

such as h = −z. From above, the model of the UAV which is detailed in appendix C can be written as ˙ X

=

f (X) + g(X)U

Y

=

CX

(9)

and the physical flight envelope of this UAV is defined as  XΦ = X ∈ R9 : Xmin ≤ X ≤ Xmax

(10)

For a given operating point {Xe , Ue }, where Ue denotes the trim positions of the controls, the linearized model of the aircraft can be written as x˙

=

Ax + Bu

y

=

Cx

(11)

For the equilibrium state Xe , we define the reduced flight envelope XR as the largest ellipsoid centered at Xe and contained in the physical flight envelope:

 with R = diag (min(Ximax

 XR = x ∈ R9 : xT Rx ≤ 1 (12)  − 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 onto R2 of this domain is illustrated in Fig.2.

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B. Control surface model On the other hand, the actuator travels are bounded Umin ≤ U ≤ Umax . For the ith 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 and ui+ ≥ 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 the ith 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 which the saturation levels are chosen between

x2 x2max XR Xe XΦ x2min x1min

Fig. 2.

x1max

x1

Physical and reduced flight envelopes

Uimin ui−

rotation axis

Uie ui+ Uimax Fig. 3.

The ith 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 saturation levels. 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 commonly 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 vector Uf (t) =

U f , where tf is the fault-time and U 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 to 600◦ /s which is the maximum speed of the actuators. Let Uh be the healthy controls, the state equation (9) in faulty mode becomes: ˙ = f (X) + g f (X)Uf + g h (X)Uh X

(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/input estimator 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

y

=

Cx

(17)

where x ∈ Rn is the state vector, u ∈ Rm is the known input vector, d ∈ Rℓ is the unknown input vector

and y ∈ Ro is the output vector. A, B, G and C are matrices with appropriate dimensions, C and G are assumed to be full rank. ˆ of the unknown input d and an estimation z of The UIDFO detailed in [23] provides an estimation d linear combination of state Tx. Theoretically, no boundedness conditions are required for the unknown inputs and their derivatives. z˙ =

ˆ Fz + Hy + TBu + TGd

ˆ d

γ(Wy − Ez) with γ ∈ R∗+

=

(18)

Matrices F, H, T, W and E are all design parameters that need to be set in order to satisfy the following conditions

   FT − TA + HC = 0       E = (TG)T P  

F is stable, with P solution of: PF + FT P = −Q and Q, a semi-positive definite matrix,

(19)

     ET = GT TT PT = WC      rank(TG) = rank(G) = ℓ

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 matrix system (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 suitable and the staircase form will be used to exhibit the Kroenecker form [24]. ˆ converges on d if the magnitude of the transfer function kH(s)k such that It is proved in [23] that d ˆ D(s) = H(s)D(s)

(20)

tends towards the ℓ × ℓ identity matrix in the bandwitdth of the UIDFO. According to [23], this can be reached if γ → +∞. However, the bandwidth increases as γ increases, which leads to an increase in the noise sensitivity. Considering that the maximum speed for the servocontrols is 600◦ .s−1 and that a rough estimation of the faulty actuator position is required by the FTC system, parameter γ 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 not measured, are used to design the diagnosis system. Since the actual control surface positions u they may be considered as unknown inputs. On the other hand, the controls u generated by the controller ¯ are equal to are known inputs. In the fault-free mode, it is assumed that the control surface positions u ¯ differs from u. their control values u. Otherwise, in the faulty mode, u 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. Let u ˆ be the input estimation vector. The diagnostic method consists of comparing the control vector u to the estimation vector u ˆ . A fault is detected if this difference, named residual, exceeds a given threshold. Unfortunately, the true airspeed measurement Vm is very noisy which deteriorates the unknown input estimation. To build the diagnosis system without using this measurement, a bank of 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. ¯1 = The first UIDFO computes an estimation u ˆ 1 of the right aileron and left ruddervator positions u (δ¯ar , δ¯el )T . It uses the reduced output measurement vector yr = y\{Vm } such as yr = Cr x and the

controls u2 = (δx , δal , δer )T generated by the controller. In the following, bi denotes the ith column of  T control matrix B in (11) matched with δx δar δal δf r δf l δer δel . The first UIDFO equations are: z˙ 1

=

(21)

γ1 (W1 yr − E1 z1 )     With B1 = b1 b3 b6 and G1 = b2 b7 . The second UIDFO estimates u ˆ 3 = (δˆal , δˆer )T . It uses the reduced output measurement vector yr and the u ˆ1

=

F1 z1 + H1 yr + T1 B1 u2 + T1 G1 u ˆ1

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controls u4 = (δx , δar , δel )T generated by the controller. This second UIDFO equations are: z˙ 2

With B2 = b1

b2

b7



F2 z2 + H2 yr + T2 B2 u4 + T2 G2 u ˆ3

(22)

γ2 (W2 yr − E2 z2 )   and G2 = b3 b6 .

u ˆ3 

= =

The state space and control matrices (9) are detailed in appendix C. For each UIDFO, condition (i) is achieved, indeed rank(Cr G1 ) = rank(G1 ) and rank(Cr G2 ) = rank(G2 ). Notice that the inputs in u ¯ 1 (resp u ¯ 3 ) 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 points {25m/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 occuring on the right aileron. A similar reasoning can be applied for the other control surfaces. When the control surface position δ¯ar ∈ u ¯ 1 is faulty, the

first UIDFO provides an accurate estimation δˆar of the actual faulty actuator position and δ¯ar = δˆar . But, as δ¯ar is faulty, it differs from its control value δar then δar 6= δ¯ar and δar 6= δˆar . For δ¯el ∈ u ¯ 1 , the

control surface position of a healthy actuator, it is equal to its control value, so δel = δ¯el . Moreover, the

first UIDFO provides a right estimation δˆel of the healthy actuator position then δˆel = δ¯el and δˆel = δel . As for the second UIDFO, it estimates u ˆ 3 by processing the reduced output measurements yr and the controlled input u4 . This latter contains δar which is a wrong information of position whereas δel is true. As a fault on the right aileron mainly affects the roll axis, UIDFO2 provides a false estimation δˆal of δ¯al and a right estimation δˆer of δ¯er . In order to avoid false alarms that may arise from the residuals transient behavior, these latter are integrated

yr

UIDFO1

u2

- selector - selector

u4

-

Fig. 4.

u ˆ1 + - R t+τ t u -6 1

|·|

6

reset

τ  -

-

reset u3 ? u ˆ 3 - ? R t+τ UIDFO2 t +

Bank of Unknown Input Decoupled Functional Observers

-+ --

-

µu1

Detection &

σu1 σu3

t

|·|

--+

Isolation Logic

-

µu3

-

uf

11

over a duration τ . Let σδar be a threshold and µδar be a logical state defined as Z t+τ ˆ δar (θ) − δar (θ)dθ > σδar ⇒ µδar = 1 otherwise µδar = 0

(23)

t

Then, to detect and to partially isolate the faulty control surface, an incidence matrix is defined as follows TABLE I T HE INCIDENCE MATRIX

Faulty control

µδar

µδel

µδal

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 frequency ω and amplitudes chosen in the kernel of the   reduced control matrix Bred = b2 b4 b5 b6 b7 . The kernel is obtained from the singular value decomposition of Bred . This null space exists if these actuators offer redundancies. Moreover, frequency

ω must be compatible with the servoactuator dynamics (about 600◦ /s). These excitation signals are given by:  uex = δarmax

δf rmax

δf lmax

δermax

δelmax

T

sin ωt

(24)

Note that to accelerate the isolation process, the throttle δx , which dynamic is slower than the control surfaces, is not used. When the right aileron is faulty, the controls cannot fulfill uex ∈ ker(Bred ) and this excitation disrupts the output measurements y which contain a signal with frequency ω. Detecting a faulty flap consists of detecting this signal component in the output measurements or in any signal processed with them e.g. the unknown input estimations. When the left aileron is faulty, since this control surface is not excited with uex , the excitation has no effect on the output measurements and these latter do not contain the frequency ω. Detecting a faulty aileron consists of showing the absence of this frequency in the output measurements or in any signal processed with them. Note that choosing the excitation signals in the kernel of control matrix B ensure the isolation of partial as well as of total loss of control surface efficiency. The frequency ω 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 position δˆar (t) contains a frequency ω which is detected with a coherent demodulation method. This frequency detection is achieved by multiplying δˆar (t) by a sinusoidal signal named carrier which frequency is ω. Next, the resulting signal is integrated over a duration 2kπ where k is a positive integer. Let s(∆) be the resulting signal and σ∆ a threshold. If s(∆) > σ∆ ∆= ω then s(∆) contains a frequency ω and the right aileron is faulty. Otherwise, the left aileron is 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 systems are 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 DOA with 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 be interesting to know the size of the DOA in faulty mode when using the nominal controller. 2) The synthesis problem, an FTC strategy should be designed to 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 is a problem of paramount importance. After a fault has occurred at time tF and before the nominal controller has been reconfigured to compensate for the effects of the fault at time tR , 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 denoted sat. For the ith control ui ∈ u: sat(ui ) = sign(ui )min(uisat , |ui |)

(25)

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For an operating point {Xe , Ue } and under a given saturated linear state feedback u = sat(Kx), e.g. K is the nominal controller, the closed loop system is given by: x˙ = Ax + Bsat(Kx)

(26)

Where x ∈ Rn , u ∈ Rm and K ∈ Rm×n the state feedback matrix.

Given two feedback matrices K, H ∈ Rm×n where hi denotes the ith row of H and assume that

|hi x| ≤ uisat , i = 1, . . . m. It is shown in [18] that the saturated state feedback sat(Kx) can be placed into the convex hull co of linear feedbacks  sat(Kx) ∈ co Di Kx + Di − Hx : i ∈ [1, 2m ]

(27)

where Di ∈ Rm×m are matrices whose diagonal elements are either 1 or 0 and D− i = I − Di with I the identity matrix. Further, L(K) is the region where the feedback control sat(Kx) is linear in x. L(K) := {x ∈ Rn : |ki x| ≤ uisat , i = 1, . . . , m}

(28)

For x0 = x(0) ∈ Rn and ψ(t, x0 ) the state trajectory of (26), the DOA of the origin is defined as n o S := x0 ∈ Rn : lim ψ(t, x0 ) = 0 t→∞

(29)

A measure of the size of this set is obtained with respect to a reference set XR e.g the reduced flight envelope (12). Assume that 0 ∈ XR and XR is a convex bounded set. For a positive real number α, ˜ denote α ˜ XR = {˜ α x : x ∈ XR }

(30)

For a set S ⊂ Rn , define the size of S with respect to XR as α ˜ R (S) := sup{˜ α>0:α ˜ XR ⊂ 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 : xT Px ≤ ρ˜

(32)

The optimization problem solved by Hu aims at finding the largest contractive invariant ellipsoid E(P, ρ˜)

such that the measure α ˜ R (E(P, ρ˜)) is maximized. Clearly, if E(P, ρ˜) is contractive and invariant, then it is inside the DOA. This optimization problem illustrated in Fig.5 is given by [18]: sup

α ˜

(33)

P>0,ρ,H ˜

s.t. :

a) αX ˜ R ⊂ E(P, ρ˜) T − b) (A + B(Di K + D− i H)) P + P(A + B(Di K + Di H)) < 0

c) E(P, ρ˜) ⊂ L(H)

and i ∈ [1, 2m ]

14

 −1 P 1 , Z = HQ, and zi the ith row of Z. To solve this optimization problem, it , Q = α ˜2 ρ˜ is transformed into a Linear Matrix Inequality (LMI) problem [18] Let γ˜ =

inf γ˜  γ˜ R a)  I

(34)

Q>0,Z

s.t. :

I Q



≥0

T T − b) QAT + AQ + (Di KQ + D− i Z) B + B(Di KQ + Di Z) < 0   u2i zi  ≥ 0 i = 1, . . . , m c)  sat T zi Q

and i ∈ [1, 2m ]

1 Let γ˜ ∗ the optimum of this problem with solutions Q∗ and Z∗ , then α ˜ ∗ = √ ∗ . Here, and without loss γ˜ of generality, ρ˜ was chosen to be equal to 1. 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 point Xe . Fault reconfiguration requires the following conditions: •

h a new operating point {Xfe , Uh e } must exist, Ue denotes the trim positions of the healthy controls,

•

at reconfiguration time tR , the state vector must belong to the physical flight envelope AND to the DOA of the UAV equipped with its fault tolerant controller.

The proposed FTC strategy consists of two stages. First, a new operating point {Xfe , Uh e } 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.

x2 α ˜ XR

XR

x2max Xe

XΦ x1min Fig. 5.

DOA, estimation of the DOA and measurement of its size

x2min

E(P, ρ˜) DoA

x1max x1

15

1) Operating point computation in faulty mode: An optimization method based on a sequential programing 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: h T h h J = (V − Ve )2 + (α − αe )2 + (β − βe )2 + (Uh j − Uje ) R(Uj − Uje )

(35)

where R 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 f f f (Xfe ) + g h (Xfe )Uh e + g (Xe )U = 0

•

•

solutions included in the physical flight envelope and in the control variation ranges,   X ≤X≤ Xmax min h  Uh ≤ U ≤ Uh max min

(36)

(37)

the flight level stage requires the following state variables set equal to zero, ϕfe = pfe = qef = ref = 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 at Xfe and contained in the physical flight envelope XΦ 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 the actuator 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.

x2 x2maxXR in fault-free mode

XΦ Xfe

x1min Fig. 6.

Physical and reduced flight envelopes

Xe

x2min

XR in faulty mode

x1max x1

16

2) Linear state feedback controller design: For this new operating point {Xfe , Uh e }, the faulty linearized model is written as: x˙ = Af x + Bf uh

(39)

The fault-tolerant state feedback controller Kf is subject to actuator saturations and the healthy control vector is: uh = sat(Kf x)

(40)

It is designed in order to maximize the DOA while the poles are placed 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 to cos ̺ and a time response approximately less than or equal to κ31 . First, it is necessary to compute the feedback matrix H such as the estimation of the DOA is maximized with respect to a reference set. This matrix is obtained by solving the optimization problem (41) [18]. This problem is similar to (33) with an extra optimization parameter M. It is solved using the LMI technique as defined by (34). sup

α ˜

(41)

P>0,ρ,H,M ˜

a) αX ˜ R ⊂ E(P, ρ˜)

s.t. :

T − m b) (Af + Bf (Di M + D− i H)) P + P(Af + Bf (Di M + Di H)) < 0 and i ∈ [1, 2 ]

c) E(P, ρ˜) ⊂ L(H)

The LMI formulation proposed hereafter is a continuation of Hu and Lin’s works. In the sequel, H is set equal to the optimal value obtained by solving (41). This guarantees a known DOA and allows to choose, from all the M 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: inf γ˜

Q>0,Z

s.t. 

a) 

γ˜ R

I

I

Q



≥0

T T m b) QAf T + Af Q + (D− + Bf (Di Z + D− i HQ + Di Z) Bf i HQ) < 0 and i ∈ [1, 2 ]   zi u2i  ≥ 0 i = 1, . . . , m c)  sat T zi Q

d) 2κ1 Q + QAf T + Af Q + Bf Z + ZT Bf T < 0 e) 2κ2 Q + QAf T + Af Q + Bf Z + ZT Bf T > 0   sin ̺(Af Q + QAf T + Bf Z + ZT Bf T ) cos ̺(Af Q − QAf T + Bf Z − ZT Bf T )  f)  cos ̺(QAf T − Af Q + ZT Bf T − Bf Z) sin ̺(Af Q + QAf T + Bf Z + ZT Bf T ) (42)

1 Let γ˜ ∗ the optimum of this problem with the solution Q∗ and Z∗ , then α ˜ ∗ = √ ∗ and Kf = Z∗ (Q∗ )−1 . γ˜

17

C. Fault tolerant control implementation A fault situation is defined by the faulty control surface and its 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 to all 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 DOA α ˜ R ≥ 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 by Ve = 25 m/s and he = 200 m. It is obtained with the trims δxe = 0.57, δare = δale = δf re = δf le = 0◦ , δere = δele = −3.9◦ . The physical flight envelope XΦ is such as −45◦ ≤ ϕ ≤ 45◦ , −15◦ ≤ θ ≤ 15◦ , 15 ms−1 ≤ V ≤ 50 ms−1 , −12◦ ≤ α ≤ 12◦ , −20◦ ≤ β ≤ 20◦ , −90◦ s−1 ≤ p, q, r ≤ 90◦ s−1 , 0 m ≤ h ≤ 3000 m. In these conditions, the reference set XR which is also the reduced flight envelope is given by (12). For this flight stage, with the nominal controller, the estimation of the size of the DOA with respect to XR is obtained by solving (34) and is equal to α ˜ ∗ = 1.1. Therefore the size of the estimation of the DOA is larger than

LMI region

the reduced flight envelope.

κ2

Fig. 7.

The LMI region

Im

̺ κ1

Re

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 becomes unstable. However, by relaxing the deflection constraints, an operating point always exists and the trims allowing to reach it are illustrated in Fig.8. Obviously, these new operating points impose new reference sets and new saturation levels. Regarding the

δal (°)

40

e

0.6

δ

X

e

(%)

0.8

0.4 −20

−10

0

10

δ stuck on [−20°,20°]

Trim in faulty mode in fault−free mode 20 * Trim Trim for δ stuck at 3° o

ar

0 −20 −20

20

ar

δfl (°) −10

0

10

−10

0

10

20

−10

0

10

20

0

δar stuck on [−20°,20°]

−2 −4 −20

20

δar stuck on [−20°,20°]

−3

δel (°)

(°)

20

e

(°) e

δ

fr

0

−4

e

e

−4

er

δ

10

2

−3

−5 −6 −20

−10

0

10

δ stuck on [−20°,20°] ar

Fig. 8.

0

ar

5

−5 −20

−10

δ stuck on [−20°,20°]

20

−5 −6 −20

δ stuck on [−20°,20°] ar

Trims of the healthy controls for right aileron fault positions ∈ [−20◦ , 20◦ ]

FTC controller, a damping ratio greater than 0.5 and a decay time less than 1 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 at 0◦ . 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 are plotted in Fig.9. As α ˜ ∗ > 1, the size of the estimation of the DOA is always larger than the reduced 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, at t = 41s, it initiates a turn and the right aileron locks at position 3◦ . 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, a 1s 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 the 1s 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 time tR = 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].

10

59.5

8

59

6

58.5

4

58

2

Imag axis

size of the estimation of the DOA

60

57.5

damping > 0.5

0.5

κ =−0.3 1

0

57

−2

56.5

−4

56

−6

55.5

−8 0.5

55 −20

−10

0

10

Right aileron fault position

20

−10 −10

−8

−6

−4

−2

0

Real axis

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

right aileron position (°) 5 0 real position control signal estimation 40.5

−5 −10 40

41

41.5

42

42.5

43

42.5

43

42.5

43

decision residual and fault detection signal decision residual

0.02

threshold σ

δ

0.01

ar

fault detection signal 0 40

40.5

41

41.5

42

residual for isolation and signal for isolation

−3

x 10

residual s(∆) 1 s time window threshold σ

10

∆

5 0 40

Fig. 10.

40.5

41

41.5 time (s)

42

Right aileron failure: the fault detection and isolation process

throttle with FTC throttle without FTC

throttle 1 0.5 0 40

41

42

43

44

45

46

47

48

49

50

48

49

50

right and left ailerons positions (°) 20 0 −20 40

41

42

43

44 45 46 47 right and left flap positions (°)

40 right control surface with FTC, left control surface with FTC,

20 0 40

without FTC without FTC

41

42

43 44 45 46 47 right and left ruddervator positions (°)

48

49

50

41

42

43

48

49

50

20 0 −20 40

Fig. 11.

44

45 time (s)

46

47

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 of the 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 for the right aileron failure and is illustrated in Fig.13. At t = 20s, the ruddervators turn down, next one of the two ruddervators locks at t = 22s

with FTC without FTC

true airspeed (m/s) 35 30 25 40

41

42

43

44

45

46

47

48

49

50

46

47

48

49

50

46

47

48

49

50

angle of attack (°) 10 5 0 40

41

42

43

44

45

sideslip (°) 40 20 0 −20 −40 40

41

42

43

44

45

time (s)

Bank angle (°) 100

0

−100 40

41

42

43

44

45

46

47

48

49

50

46

47

48

49

50

46

47

48

49

50

pitch angle (°) 50

0

−50 40

41

42

43

44

45

height (m) 220 200 180 160 40

41

42

43

44

45

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 to 0◦ . 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 time t = 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 a 1 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.

right ruddervator positions ° 5

0

−5 19.5

20

20.5

21

21.5

22

22.5

23

23.5

24

24.5

25

24

24.5

25

24

24.5

25

decision residual and fault detection signal 0.04

decision residual fault detection signal

0.02

threshold σδ

er

0 19.5

20

20.5

21

21.5

22

22.5

23

23.5

residual for isolation and signal for isolation

−3

x 10 10

residual S(∆) 1s time window threshold σ∆

5 0 19.5

Fig. 13.

20

20.5

21

21.5

22

22.5

23

23.5

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.

with FTC without FTC

pitch (°/s) 0.2

0

−0.2 15

20

25

30

35

40

30

35

40

30

35

40

true airspeed (m/s) 30 28 26 24 15

20

25

height (m) 220 200 180 160 15

20

25

time (s)

bank angle (°) 10 5 0 −5 15

20

25

30

35

40

30

35

40

30

35

40

sideslip (°) 2 1 0 −1 15

20

25

yaw (°/s) 10 5 0 −5 −10 15

20

25

time (s)

Fig. 14.

Right ruddervator failure: the UAV state vector in faulty mode with and without FTC strategy

24

A PPENDIX A R EFERENCE FRAMES AND TRANSFORMATION MATRICES The transformation matrix TbE : 



cos θ cos ψ

sin ϕ sin θ cos ψ − cos ϕ sin ψ

cos ϕ sin θ cos ψ + sin ϕ sin ψ

 TbE =   cos θ sin ψ − sin θ

sin ϕ sin θ sin ψ + cos ϕ cos ψ

 cos ϕ sin θ sin ψ − sinϕ cos ψ   cos ϕ cos θ

sin ϕ cos θ

(43)

Throttle 1

0.5 δer=−10°

0 20

25

30

Pitch angle (°)

δer=+5° 40 −9°≤δ ≤4°

35

45

er

data4 data5 data6 data7 data8

10 0 −10 20

25

30 35 true airspeed (m/s)

40

45

25

30

40

45

26 24 22 20 18 20

35 time (s)

Fig. 15.

Right ruddervator failure with FTC strategy

xb

x1

xb yE

θ

xE ψ

zb

ψ ϕ

x1 y12

yb

ϕ θ z2 z1

Fig. 16.

The reference frames Rb , RE and Rb , Rw .

yw

α

xw β

β

α zb z1

yb

25

The transformation matrix Tbw : 

cos α cos β

sin β

 Tbw =  − cos α sin β − sin α

cos β 0

sin α cos β



 − sin α sin β  

(44)

cos α

A PPENDIX B PARAMETERS OF THE AIRCRAFT MODEL The controls used in the Aerosonde toolbox match those of a classical aircraft: δa , δf , δe and δr 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

=

δf

=

δe

=

δr

=

δar − δal 2 δf r + δf l 2 δer + δel 2 δer − δel 2

(45)

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 with δe is equal to the pitch moment produced with δer and δel : cCmδel δel cCmδer δer + q¯S¯ q¯S¯ cCmδe δe = q¯S¯

(46)

Cmδe . 2 On the other hand, geometrical considerations resulting from the Data Compendium (DATCOM) method from (46) and due to aircraft’s symmetry: Cmδer = Cmδel =

[29] allow to draw up roughly most of the ungiven aerodynamic coefficients (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 effectiveness Clδf . Given, the dimensionless flap lift-force effectiveness CLδf , according to [29] and for two symmetric plain flaps: CLδf = 0.9Kf



∂CL ∂δf

′

2Sf lapped cos ΛHL S

(47)

Sw is the wettedsurface, ′ the other parameters are illustrated in Fig.17. Thanks to a nomogram, it is possible ∂CL . Next the dimensionless flap roll-moment effectiveness Clδf is calculated with to estimate Kf ∂δf  ′ Pn ∂CL Yi Si cos ΛHL i=1 Kf ∂δf Clδf = (48) Sw b

26

Similarly, knowing the dimensionless aileron roll-moment effectiveness Clδa , the dimensionless aileron-lift effectiveness CLδa can be calculated.

Yi Si

Sf lapped

ΛHL

c.g. δal

cf i

c¯f i

δar 0

Fig. 17.

Geometrical parameters used to calculate the aerodynamic coefficients

TABLE II C OEFFICIENTS IN THE AERODYNAMIC MODEL OF THE A EROSONDE , VALID WITHIN THE [15, 50ms−1 ] RANGE

Drag

value

Lateral force

value

Lift

value

C D0

0.0434

Cyβ

-0.83

C L0

0.23

0.0151

Cyp

0

CLα

5.616

0.0151

Cyr

0

CLq

CDδar = CDδal = CDδf r = CDδf l =

CDδa 2 CDδa 2 CDδ 2 CDδ

7.95

f

0.073

Cyδar = Cyδa

-0.075

CLδar

∗

f

0.073

Cyδal = −Cyδa

0.075

CLδal

∗

CDδer =

2 CDδe 2

CDδel =

CDδe 2

0.00675

Cyδf r

−

CLδf r =

0.00675

Cyδf l

−

CLδf l =

Cyδer = Cyδr

0.1914

Cyδel = −Cyδr

-0.1914

0.34 0.34 CLδf 2 CLδf

2 Cmδe CLδer = 2 Cmδe CLδel = 2

0.37 0.37 0.065 0.065

Roll

value

Pitch

value

Yaw

value

Clβ

-0.13

C m0

0.135

Cnβ

0.0726

Clp

-0.5

Cmα

-2.73

Cnp

-0.069

Clr

0.25

Cmq

-38.2

Cnr

-0.0946

0.021

Cnδar = Cnδa

0.0108

0.021

Cnδal = −Cnδa

-0.0108

0.023

Cnδf r

−

0.023

Cnδf l

−

-0.4995

Cnδer = Cnδr

-0.693

-0.4995

Cnδel = −Cnδr

0.693

Clδar = Clδa

-0.1695

Cmδar

∗

Clδal = −Clδa

0.1695

Cmδal

∗

Clδf r

∗

Clδf l

∗

-0.037

Cmδf r =

0.037

Cmδf l =

Clδer = Clδr

0.0024

Cmδer

Clδel = −Clδr

-0.0024

Cmδel

Cmδf 2 Cmδf

2 Cmδe = 2 Cmδf = 2

27

A PPENDIX C D ETAILED MODEL OF THE AIRCRAFT

ϕ˙

=

p + q sin ϕ tan θ + r cos ϕ tan θ

θ˙

=

q cos ϕ − r sin ϕ

V˙

=

−g(cos α cos β sin θ − sin β cos θ sin ϕ − sin α cos β cos θ cos ϕ) −

α˙

=

β˙

= −

q¯S kρ cos α cos β CD + δx m mV g(sin α sin θ + cos α cos θ cos ϕ) q¯S kρ sin α + q − (p cos α + r sin α) tan β − CL − δx V cos β mV cos β mV 2 cos β g(cos β cos θ sin ϕ + cos α sin β sin θ − sin α sin β cos θ cos ϕ) q¯S + p sin α − r cos α + Cy (49) V mV kρ cos α sin β δx mV 2 J˜11 [¯ q SbCl + (Jyy − Jzz )qr + Jzx pq] + J˜13 [¯ q SbCn + (Jxx − Jyy )pq − Izx qr]   2 2 cCm + (Jzz − Jxx )pr + Jxz (r − p ) J˜22 q¯S¯

p˙

=

q˙

=

r˙

=

J˜31 [¯ q SbCl + (Jyy − Jzz )qr + Izx pq] + J˜33 [¯ q SbCn + (Jxx − Jyy )pq − Izx qr]

Z˙E

=

−V cos α cos β sin θ + V sin ϕ cos θ sin β + V cos ϕ cos θ sin α cos β

Where Ix , Iy and Iz are the principal moments of inertia and Ixz = Izx are the products of inertia. J˜ij is the value in the ith row, j th column of the inverse of the inertia matrix. The state space and the control matrices for Ve = 25 m/s and he = 200 m. 

0   0    −0.0    −0.0   A =  0.39    0   0    0  −0.00 

0

0

0

0

1

0

0.0743

0

0

0

0

0

1

0

−9.81

−0.081

9.759

−0.00

0

0

0

−0.00

−0.031

−3.463

0

0

0.98

0

0

0

0

−0.535

0.0741

0

−0.99

0

0

0

−98.89

−21.23

0

10.96

0

0.00

−94.67

0

−0.00

−5.0143

0.00

0

0

0

31.452

0.094

0

−2.613

−25

0

25

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0   0    1.18   −0.003   B= 0    0   0    0  0

−0.23

−0.23

−1.12

−1.12

0.103

−0.208

−0.2081

−0.226

−0.226

−0.0398

−0.045

0.0459

0

0

0.117

−124.7

124.7

−27.1

27.1

5.2

0.725

0.725

0.8069

0.8069

−17.26

12.2

−12.2

1.85

−1.85

−23.9

0

0

0

0

0

0



 0    −0.0   −0.0   0    0   0.0    0   0

(50)



    0.103    −0.0398   −0.117    −5.2   −17.26    23.9   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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29

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François BATEMAN received his M.S. in electrical engineering in 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 Academy of 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 and conference 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.