The membrane technology appears as a convenient solution for millimeterwave appli cations. An alternative low-cost polymer membrane process is proposed in this paper, and extended to the design of high perfor mance antennas. Technological compatibility with standard process is also discused, and an original Flip-chip concept is proposed for membrane antennas.

INTRODUCTION

In the last few years, developments of millime terave applications have generated new requi rements for microwave integrated structures. Main efforts concerw the minimization of losses and parasitics effects at such frequencies, under low cost and high perforpances considerations.

In this way, some original configurations have been proposed, like the thin-film membrane technology suggested by Rebeiz et al [START_REF] Rebeiz | Monoli thic Millimeter-Wave Two-Dimensional Hom Imaging Arrays 11[END_REF] in 1990; the numerous investigations performed on such semi-transparent substrate have underlined its great advantages in terms of electrical per forqances (lossless microshielded structures), of modal dispersion (TEM mode up to 100GHz) and technological sensitivity ( oversized cir cuits). This is especifically attractive for planar anten na design because the radiation element can be located on membrane section, with high radia-tion efficiency and enlarged operating bandwith. Nevertheless, thin film membranes are usually processed using semi-conductor technologies, and are not really optimized in terms of cost and technological compatibility (interconnection). We have proposed in 1998 [START_REF] Ch | Ultra Low-Cost Membrane Technology For Millimeterave Applications[END_REF] an alternative low-cost membrane process based upon a po lymer thin-film dielectric sheet (BCB from Dow-Chemical). We demonsr¥ate at once in this paper that this solution can be efficiently used for r e design of optimum membrane antennas (in terms of bandwidth and interconnection pos sibilities). Furtherrore, it can be efficiently used for implementing antenna arrays, the upper mi croshielded cavity beeing considered as a de coupling str~cture between the radiating ele ments and the feeding network.

Lastly, we investigate on the possibility of de signing SMC (Surface Mounted Component) or Flip-Chip membrane antennas for mmwave applications. Previous verifications have been perforsed on usual Membrane transmission lines mounted on pre-formed alumina substrate. Experimental results are given and the technique is extended to the implementation of antennas in the fort of flip-chip devices.

-1-Technoloeical process

The generic sr«cr±re is depicted in figure 1 and is constituted of a spin-coated high performance dielectric film (tgo # le-4, Er=2. A good agreement is observed between theore tical and experimental results (figure 3); the ope rating bandwidth is of about 3 % for a return loss lower than -1 OdB. The upper copper cavity can be simultaneously used as a schielding structure, for minimizing parasitic interferences between the feeding line and the radiating fields, but also as a tuning element for controling the beamwidth radiation pattern.

• • • • • l• • • +• • • •l• • • • • l • \ • • • • -l --• • • l• • • • • l • • • • • • : •-• • • ., .. ,:••••••�---••:••••••••••• ....

-III -Leadin& to membrane antenna array

A) Basic Antenna arrays :

The BCB membrane technology offer quite a great flexibility for designing integrated mem brane antenna arrays, mainly because of the va rious possibilities concerning the feeding net work optimization. Indeed, different topologies are available (fig. 4), with specific advantages.

The first one (fig. 4a) makes for a minimization of dispersion and radiating element parasitic interferences, because of the membrane charac teristics (low peruittivity value). On the other hand, a combination of membrane and TFMS sections (fig. 4b) leads to a significant reduction of discontinuities effects (1 or 2 dB reduction of parasitic coupling and reflection ), thereby im proving cross-polarization levels. B ) Extension to Flip-chip membrane an tennas Some limitations can be encountered when designing planar membrane antennas; indeed, the bigger is the membrane section area (for implementing a complete array as mentionned on figure 4b for example), the bigger will be the problems concerning the membrane planarnty and homogeneity. In addition, the technological compatibility with exterzal functions (active MMIC functions, IF modules, biasing, .. ) is not really optimum, because bounding and intercon nections have to be considered. This contrmbutes to significant electrical distorsions, mainly at mmwave frequencies.

For these reasons, we propose the develope ment of flip-chip membrane antennas (or array). Indeed, flip-chip structures are drawing more an more attention for their several avantages ( compactness, small parasitic inductances and high yield rate) and will be probably rÁiversally used for active MMIC components in the future, as well as for sensitive passive functions like filters.

A

B IC ID JOO µm Height bumps

A-A' At first step, we have designed a back-to-back r §ansition from a coplanar waveguide to a mem brane microstrip line. The center membrane sec-

BCB TFMS Section

Upper shielding BCB

��J;{vm

Figure 1 :

 1 Figure 1 : Description of a polymer membrane structure ( Microshieled lower cavity only). We have made various investigations for defi ning different optimized membrane feeding structures. Thancks to the electrical characteris tics of the BCB dielectric film, quite good re sults have been obtained, and compared suc cessfully with 3D simulation results (HFSS). -II -Designing MMwave antenna on DCB membrane technolou Preliminary investigations have shown the va lidity of the BCB membrane technology for passive filters and couplers [3].

Figure 2 :

 2 Figure 2 : Membrane patch antenna on BCB substrate. Considering the different advantages offered by the BCB membrane structure ( i.e. low rela tive permittivity (<1.1), backside reflecting cavi-

Figure 3 :

 3 Figure 3 : Simulated and experimental results on return loss.( Antenna dimensions: W = 5mm, L = 3.7) Resonant frequency : Fr = 34.87 GHz Bandwith { TOS = 2 ) : Bp = 1.1 GHz

  Figure 4 :D!fferent topologies for antenna arays on BCB membrane substrate. ( □ Membrane sections)

Figure 5 :

 5 Figure 5 : "Flip-chip " Membrane : Description of the different transitions for a simple transmission line.

tion is directly report, as a membrane flip-chip component, on an alumina motherboard by means of a 50 µm height conductive glue (fig 5).

In practice, the brÀp heigth appears to be r•i ce the value we predicted. However, by inclu-. ding this technological dispersion, a good agreement between simulation and measrAEement•. (fig 6) has been achieved.

OdBr-:--e--:--.,.-r--,--":""""'. ::;;,,o_.,. .

---. r---r ----dB (Sil) -------�: -.• -: . :

oda, r-=:==�::::-:::.::-r:::,7

. . .

..

: : : : : :

( Effective bump heights taken into account)

We are presently developing flip-chip antennas to be morÂted on an alumina pre-forved subs trate, using similar transitions and feeding lines. Once again, several solutions exist for conr¨o ling the network as presented on figure 7.

CONCLUSION

This paper proposes an original technological solution in order to develop ver low-cost mem brane antennas for millimeterave applications.

In addition, r£e proposed antenna configurations are qr¿te ar°active regarding interconnexion possibilities, and design flexibilities.The flip chip membrane antenna concept is quite oroginal and could solve numerous problems commonly encorÃtered with antenna array designs. It can be extended to any passive membrane function and becomes a real oppor}unity for the standar dization of the membrane technology.


Bouzgou Kamel

Bestaoui Yasmina

Benchikh Laredj

Ibari Benaoumeur

Ahmed-Foitih Zoubir

Dynamic Modeling, Simulation and PID Controller of Unmanned Aerial Vehicle UAV

Keywords: Quadrotor, UAVs, Modeling, Control, PID Controller, Virtual Reality

This paper is focused on modeling and control of quadrotor; first modeling of moments and torque, second is of rotor,the result is the complete dinamincs of UAV.mathematical model is presented for a general study with disturbance and we take account all parameters of control. the PID controller is presented witout disturbance in dynamics equation for a linear model, where we can used in the control of a group of quadrotors for obstacle avoidance, and a group of quadrotors with a rigid and flexible links for transporting payloads in the free environment.finaly our results is simulated in the simulink and the virtual reality environment.

INTRODUCTION

The quadrotor,UAV (Unmanned aerial vehicles) is the autonomous unmanned flying machines offering us an additional degree of freedom and an unlimited workspace, many researchers are facing this new device, and several studies are led to discover the areas of application of quadrotors Modeling and control of quadrotor has been studied in several papers; mathematical and dynamic model in [START_REF] Bresciani | Modelling, identification and control of a quadrotor helicopter[END_REF][START_REF] Chovancová | Mathematical modelling and parameter identification of quadrotor (a survey)[END_REF] linear and nonlinear controller in [START_REF] Balas | Modelling and linear control of a quadrotor[END_REF]and for a quadrotor control with PID controller [START_REF] Atheer L Salih | Modelling and pid controller design for a quadrotor unmanned air vehicle[END_REF][START_REF] Lucas M Argentim | Pid, lqr and lqr-pid on a quadcopter platform[END_REF][START_REF] Bouabdallah | Pid vs lq control techniques applied to an indoor micro quadrotor[END_REF][START_REF] Szafranski | Different approaches of pid control uav type quadrotor[END_REF]but the most of them is not a complete modeling in all cases of quadrotor situation; the recent research is mainly based on the fact that the quadrotor is an element of agent set to schedule a mission that requires cooperation and the control of a group of quadrotor, several research are based on the complete modeling with more mathematical details [START_REF] Bouzgou | Geometric modeling and singularity of 6 dof fanuc 200ic robot[END_REF][START_REF] Kamel Bouzgou | Virtual reality simulation and singularity analysis of 3-rrr translational parallel robot[END_REF] This paper presents a complete modeling that is required for a projects that use the quadrotor as an elementary agent for cooperation. this paper is organized as follows:first,a mathematical model of quadrotor is presented .

. q = [X ω] T = [x y z φ θ ψ] T V = [V Ω] T = [v x v y v z p q r] T
where X = [x y z] T is the relative position of the center of mass of the UAV and ω = [φ θ ψ] T = [Roll P itch Y aw] T the orientaion of the quadrotor in E frame, with the constraint that: Assumption 1: it is assumed that roll, pitch and yaw angles are defined in bounded intervals as:

(-π 2 < φ < π 2 ), (-π 2 < θ < π 2 ) and (-π < ψ < π)
the Roll φ motion is given by the thrust imbalance between f 2 and f 4 forces around b 1 ,we can write the rotation matrix R φ as:

R φ =   1 0 0 0 cφ -sφ 0 sφ cφ  
the Pitch θ motion is given by the thrust imbalance between f 1 and f 3 around b 2 , the R θ is given by:

R θ =   cθ 0 sθ 0 1 0 -sθ 0 cθ  
and the Yaw ψ motion is given by the thrust imbalance of all moments around the z axis, the rotation matrix is:

R ψ =   cψ -sψ 0 sψ cψ 0 0 0 1  
we can now define the global rotaion matrix R of quadrotor in the inertial frame E with the product of three matrices as

R = R ψ R θ R φ : R =   cθcψ sφsθcψ -cφsψ cφsθcψ + sφsψ cθsψ sφsθsψ + cφsψ cφsθsψ -sφcψ -sθ sφcθ cφcθ   (1)
Where,R φ ,R θ ,R ψ and R matrices of rotation is defined in the special orthogonal group SO(3),which have the following properties:

SO(3) = R ∈ R 3×3 | R T R = I, det(R) = 1
Where: cx △ = cos x and sx △ = sin x The relation between ω and Ω is given by the following equation:

Ṙ = R Ω, (2)
where the hat map:

. :

R 3 → SO(3) (3)
xy = x × y, (4)
where x, y ∈ R 3 and:

x =   0 -x 3 x 2 x 3 0 -x 1 -x 2 x 1 0   (5)
with :x = [x 1 x 2 x 3 ] T to express the angular velocity of quadrotor in the fixed frame,we can use:

Ω = R T Ṙ ∨ (6)
where vee map is the inverse of hat map:

() ∨ : SO(3) -→ R 3 Ṙ = d dt (R ψ R θ R φ ). = Ṙψ R θ R φ + R ψ Ṙθ R φ + R ψ R θ Ṙφ . Ṙψ = R ψ 0 0 ψ T ∧ Ṙθ = R θ 0 θ 0 T ∧ Ṙφ = R φ φ 0 0 T ∧ R T = (R ψ R θ R φ ) T = R T φ R T θ R T ψ .
we replace in the equation ( 4) becomes:

Ω =   R T φ R T θ   0 0 ψ  ∧ R θ R φ + R T φ   0 θ 0   ∧ R φ +   φ 0 0   ∧   ∨
we can use this properties of the hat map:

RxR T = (Rx) ∧ the equation (4) becomes: Ω =     R T φ R T θ   0 0 ψ    ∧ +   R T φ   0 θ 0     ∧ +   φ 0 0   ∧   ∨ Ω =   p q r   =   1 0 -sθ 0 cφ sφcθ 0 -sφ cφcθ     φ θ ψ  (7) Ω = M ω =⇒ ω = M -1 Ω , M ∈ R 3×3 .
now we can rewrite the relation between q and V in the matrix form as: q = T.V

and

T = R 0 3×3 0 3×3 M -1
Assumption 2: we assume that the quadrotor as near of the nominal state, where we can rewrite the angular velocity in inertial frame as: M = I 3×3 therefore: ( φ = p , θ = q, ψ = r) the dynamics equation of the unmanned aerial vehicle UAV are as:

m v = f Re 3 -mge 3 (9)
M = J Ω + Ω × JΩ. ( 10
)
the translational and rotational kinetic energy of quadrotor is given by:

T trans = 1 2 m ẊT Ẋ (11)
T rot = 1 2 ΩT J Ω (12) and J = diag {J x J y J z } ∈ R 3×3
is the inertia matrix of the quadrotor in the body-fixed frame we assume symmetric. the potentiel energy of body is:

U = mgz (13)
the Lagrangian of the quadrotor is given:

L(q, q) = T trans + T rot -U L(q, q) = 1 2 m ẊT Ẋ + 1 2 ΩT J Ω -mgz d dt ( δL δ q ) -δL δq =
F τ F is the external force applied to the quadrotor . and τ is the generalized moments around the UAV.

given by F = F X + F t and τ = [τ φ τ θ τ ψ ] T where F X is the force of translational movement generated by propellers.and F t is the drag forces applied to the quadrotor

along b 1 b 2 b 3 vectors, f = 4 i=1 f i and F X = f Re 3 F X = f   cφsθcψ + sφsψ cφsθsψ -sφcψ cφcθ   ( 14
)
F t =   -K f x 0 0 0 -K f y 0 0 0 -K f z   Ẋ ( 15
)
where K f x , K f y , K f z are drag coefficients.

m v = f Re 3 + F t -mge 3 (16) m ẋ ẏ ż = f cψsθcφ + sψsφ sψsθcφ -sφcψ cθcφ + -K f x 0 0 0 -K f y 0 0 0 -K f z Ẋ -mg 0 0 1 (17) J Ω = Ω × JΩ + 4 i=1 J r (Ω × e 3 )Ω i + τ ( 18
)
Ix 0 0 0 Iy 0 0 0 Iz φ θ ψ = -ψ θIy + θ ψIz φ ψIx -φ ψIz -θ φIx + φ θIy + J r θ - φ 0 Ω r + τ φ τ θ τ ψ (19)   Ix φ Iy θ Iz ψ  =   θ ψ(Iz -Iy) φ ψ(Ix -Iz) θ φ(Iy -Ix)   +   Jr θΩr -Jr φΩr 0   + τ φ τ θ τ ψ (20) mẍ = (cφsθcψ + sφsψ)U 1 -K fx ẋ (21) mÿ = (cφsθsψ -sφcψ)U 1 -K fy ẏ (22) mz = (cφcθ)U 1 -mg -K fz ż (23)
I x φ = θ ψ(I y -I z ) -J r θΩ r + lU 2 (24)
I y θ = φ ψ(I z -I x ) -J r ψΩ r + lU 3 (25)
I z ψ = φ θ(I x -I y ) + U 4 (26)
where:

Ω r = ω 2 + ω 4 -ω 1 -ω 3 ( 27
)
the thrust generated by i-th propeller along b 3 is:

f i = b ω 2 i ( 28
)
b:is the thrust factor,and ω i is the speed of ith propeller. the rotor i also produce a moment given by: τ i = ±d ω 2 i and the total torque is:

τ Mi = 4 i=1 (-1) i d ω 2 i . (29)
d:is the drag factor. The quadrotor is defined as a six degrees of freedom, three for the position of center of mass, and three for rotation,which represents the output variables and also the final state of quadrotor in three dimensional space,but the system's input are just four, one thrust force and torque, we named by U i i = 1, ..., 4 variable inputs of UAV controller.

U 1 = 4 i=1 f 1 = f 1 + f 2 + f 3 + f 4 = b(ω 2 1 + ω 2 2 + ω 2 3 + ω 2 4 ) (30) U 2 = f 2 -f 4 = b(ω 2 2 -ω 2 4 ) (31)
U 3 = f 1 -f 3 = b(ω 2 1 -ω 2 3 ) (32) U 4 = 4 i=1 τ Mi = d(-ω 2 2 -ω 2 4 + ω 2 1 + ω 2 3 ) (33) τ =   τ φ τ θ τ ψ   =   lU 2 lU 3 U 4   =   lb(ω 2 4 -ω 2 2 ) lb(ω 2 3 -ω 2 1 ) d(ω 2 2 + ω 2 4 -ω 2 1 -ω 2 3 )   (34)     U 1 U 2 U 3 U 4     =     b b b b 0 -b 0 b -b 0 b 0 -d d -d d         ω 2 1 ω 2 2 ω 2 3 ω 2 4     ( 35
)
the determinant of the above 4 × 4 matrix is -8db 3 ,so it is invertible when d = 0 and b = 0.

    ω 2 1 ω 2 2 ω 2 3 ω 2 4     =     1 4b 0 -1 2b -1 4d 1 4b -1 2b 0 1 4d 1 4b 0 1 2b -1 4d 1 4b 1 2b 0 1 4d         U 1 U 2 U 3 U 4     (36)

Rotor dynamics

The quadrotor thrust and moments are produced by four DC-motors, the dynamics modeling consist the electrical and mechanical equations of motor and propeller. the motor is controlled by an electric voltage input;in order to have the induced voltage (BEMF) to generate the desired angular velocity,the equation of electrical model is:

v = Ri + L di dt + e ( 37
)
with e = K e .ω m , in the next, the differential term was neglected because the most motors used are a very small inductance, therefore:

v = Ri + K e ω m (38)
the mechanical model of motor is given by:

J m ωm = M m -M f ( 39
)
where M m = K m .i we used the propeller with a gearbox,then the friction torque of the motor is given by:

M f = dω 2 m ηr 3
and ω p = ωm r ,J pm = Jp ηr 2 therefore the equation (41) becomes:

( Jp ηr 2 + J m ) ωm = -KmKe R ω m -d ηr 3 ω 2 m + Km R v
we have the electric and mechanic power of motor are equals, therefore K m = K e , and we replace (

Jp ηr 2 + J m ) by J t we obtain: ωm = - K 2 m RJ t ω m - d ηr 3 J t ω 2 m + K m RJ t v (40)
this equation of second degree is too difficult for control, we used the taylor series in the first order at ω 0 value to linearize it :

f (ω m ) = f (ω 0 ) + f (ω 0 )(ω m -ω 0 )
where f (ω m ) = ωm , the linear equation of motor dynamic is written as:

ωm = -( 2dω 0 ηr 3 J t + K 2 m RJ t )ω m + K m RJ t v + dω 2 0 ηr 3 J t (41)
the equation has the linear form as: ωm = Aω m + Bv + C, the position of each term is shown with simulink model in fig. 3.

tan θ = ẍ + K f x m ẋ cψ + ÿ + K f y m ẏ sψ z + g + K f z m ż ( 42
) sin φ = ẍ + K f x m ẋ sψ + ÿ + K f y m ẏ cψ ẍ + K f x m ẋ 2 + ÿ + K f y m ẏ 2 + z + g + K f z m ż 2 (43)
in our case, we assumed that the disturbance is neglected,then

K f x = K f y = K f z =

Simulation results

we turne the PID parameters until we find values that converge to a stability situation, from a desired position and atitude of quadrotor, the values obtained are shown in the following table: it is necessary to define the four outputs (we have four con-trols) these outputs are chosen to be the translational position x,y and z and the yaw angle ψ. singularity in (42),can only appear when: K f z = 0 and z = -g,so is when the quadrotor is in free fall without wind disturbances. more simulation work VRML used for validation of results [START_REF] Bouzgou | Singularity analysis and illustration of inverse kinematic solutions of 6 dof fanuc 200ic robot in virtual environment[END_REF][START_REF] Kamel Bouzgou | Virtual reality simulation and singularity analysis of 3-rrr translational parallel robot[END_REF][START_REF] Bouzgou | Workspace analysis and geometric modeling of 6 dof fanuc 200ic robot[END_REF];our work is validated in a platform for virtual reality, why we designed an application that has the control interface and a simulation in VRML to show the appearance of the quadrotor in the 3D space.the IHM is shown in figure 7.

K pζ K iζ K dζ X 4

Conclusion

this paper includes the complete modeling of quadrotor, mathematical and dynamics model in the contraint envirenment, where is the strong point of that paper, Simulation results have been presented to show the powerful of the proposed controllers in an equilibrium point, these results will able to solve some problems of agents cooperative with a simple manipulation of position and atitude.

Figure 1 :

 1 Figure 1: Quadrotor Model in SO(3)

Figure 2 :

 2 Figure 2: Simulink model of DC-motor

  0 we study that in other papers,therefore: after we can inject the angle values founded to calculate the quadrotor position, the following figure shows the strategy of control structure.

Figure 3 :

 3 Figure 3: Block diagram of the used control structure

Figure 4 :

 4 Figure 4: Position regulation signals of Quadrotor.

Figure 5 :

 5 Figure 5: Atitude regulation signals of Quadrotor.

Figure 6 :

 6 Figure 6: Trajectory of Quadrotor in 3D.

Figure 7 :

 7 Figure 7: VR interface of Quadrotor in 3D.


In this paper, we present a new approach for maximizing the coverage path planning while minimizing the path length of an aerial robot in agriculture environment with concave obstacles. For resolving this problem, we propose a new cellular decomposition which is based on a generalization of the Boustrophedon variant, using Morse functions, with an extension of the representation of the critical points. This extension leads to a decrease of the number of cells after decomposition. The results show that this new cellular decomposition works well even with several concave obstacles inside the environment. Furthermore, for path planning, the cells are divided again into two classes, leading to have a cell set better suited for use of the traveling salesman problem (TSP) to get complete coverage. Genetic Algorithm (GA) and TSP algorithm are applied to obtain the shortest path. Then, an approach is also proposed to maximize the scanned area on the working area with obstacles. The proposed method can be applied in precision agriculture for monitoring insects and other crop pests. The effectiveness of the proposed method has been verified on Matlab/Simulink.

I. INTRODUCTION

Coverage path planning (CPP) is one of the most important tasks for robot motion and missions. CPP is the determination of a path that a robot must take in order to pass over each point in an environment while avoiding obstacles. Based on the path that has been generated, robots can accomplish their predefined tasks such as cleaning, painting, window cleaning, and inspection of complex structures. It will be easier if there is no obstacles on the working area, however obstacles are always present. In order to create way-points for the robot, first, the environment needs to be divided into smaller parts called cells. These cells are obstacles-free and the robot can freely move inside them. CPP has been extensively studied in recent years for applications such as vacuum cleaning robots [START_REF] Yuyi | Control system design for a surface cleaning robot[END_REF], painter robots [START_REF] Atkar | Uniform coverage of automotive surface patches[END_REF], path planning for autonomous underwater vehicles [START_REF] Paull | Path planning for autonomous underwater vehicles[END_REF], demining robots [START_REF] Acar | Path planning for robotic demining : Robust sensor-based coverage of unstructured environments and probabilistic methods[END_REF], lawn mowers [START_REF] Bosse | Coverage algorithms for an under-actuated car-like vehicle in an uncertain environment[END_REF].

Although the land resource on the Earth is limited, the human population is growing very fast on the globe. Thus, human being is now facing a very serious challenge of sustainable food and energy production. In that sense it is expected that field crop production is going to be more intensified, but with minimum unfavourable effect on environment. For improving the food production which can satisfy human needs, it is necessary to monitor the state of the crop such as the health and the pest of the plants. Several Informatique, Biologie Intégrative et Systèmes Complexes (IBISC), Universitéd ' Évry Val-d'Essonne, France. types of agricultural robots have been widely used to get the information of the crop as mobile robot, unmanned aerial vehicle (UAV), or the collaboration of both of them. Because of their high mobility and low cost, UAVs are now more and more expanding to commercial, scientific, recreational, and other applications, such as security and surveillance, product deliveries, aerial photography, especially in agriculture. Examples of applications of UAVs in agricultural studies are becoming more noticeable in the literature. In [START_REF] Hunt | Evaluation of Digital Photography from Model Aircraft for Remote Sensing of Crop Biomass and Nitrogen Status[END_REF], Hunt et al. evaluated an aerobatic model aircraft for acquiring highresolution digital photography to be used in estimating the nutrient status of corn and crop biomass of corn, alfalfa, and soybeans. In [START_REF] Ahmed | Multi-objective optimal path planning using elitist non-dominated sorting genetic algorithms[END_REF], Ahmed et al proposed a path planning which optimizes the turning curve of the UAV trajectory to reduce the distance as well as the cost but coverage was not considered. The contribution of this paper focuses on maximizing the coverage path planning and minimizing the path length of an aerial robot in agriculture environment with concave obstacles. The algorithm is depicted in Fig. 1. It is a combination of distance minimization and profit maximization. For accomplising this problem, one has to resolve three main tasks. The first task is dedicated for cellular decomposition(CD) (Fig. 1). A new cellular decomposition is based on a generalization of the Boustrophedon variant [START_REF] Choset | Coverage Path Planning : The Boustrophedon Cellular Decomposition[END_REF], using Morse functions [START_REF] Acar | Morse Decompositions for Coverage Tasks[END_REF], with an extension of the representation of the critical points. The feature of this extension is that the number of cells after decomposition is decreased compared to the decomposition methods in [START_REF] Choset | Coverage Path Planning : The Boustrophedon Cellular Decomposition[END_REF], [START_REF] Acar | Morse Decompositions for Coverage Tasks[END_REF]. With this new proposed approach, the environment can be divided into a number of separating obstacles and

Aerial Robot Coverage Path Planning approach with Concave

Obstacles in Precision Agriculture

The Hung Pham 1 and Yasmina Bestaoui 2 and Said Mammar 3 obstacle-free regions. In each obstacle-free region which can be called obstacle-free cell, the aerial robot can make back and forth motion to cover the whole working area. Based on the cells that have been decomposed by CD, the second task is aiming to generate way-points in each cell (WG in Fig. 1) and maximize the percentage of coverage (OSA in Fig. 1). The trade-off condition between percentage of coverage and number of way-points is the condition for ending the algorithm. Finally, in the third task, once the cells have been created by CD, the way-points have been created by WG and OSA, the cells are divided again into two classes, leading to have a cell set better suited for using the TSP algorithm to get complete coverage. GA and TSP algorithms with some modifications are applied to obtain the shortest path for the aerial robot to make complete coverage. The remainder of this paper is organized as follows. In section II, the system model used will be introduced. The approach and algorithm for the new cellular decomposition method are given in section III. Then, the use of genetic algorithm and travelling salesman problem with some additional constraints for minimizing the path length when moving between cells, optimization process of the percentage of coverage, and the trade-off between the percentage and the number of way-points will also be presented in this section. Section IV is dedicated for conclusion and future work.

II. SYSTEM MODEL

A. UAV platform

This paper considers a quad-rotor equipped with a video camera mounted on a gimbal stabilizer, which compensates for small rotation displacements that may be experienced by the UAV during navigation. The objective of the mission is to take images of the whole area of plants by a dedicated spectral camera. Based on spectrum study of images that have been taken, one can have the information about the health and pest of the plant. We assume that images are taken by special spectral camera which has the following parameters :

-Image frame is rectangle with the length and width are respectively L and H.

-Images have to be taken such that they have the same direction and the overlap regions on the horizontal and vertical sides as dL and dH (in Fig. 3) For the optimization problem, there is one constraint that all the center of the rectangles are in the obstacle-free region.

L H ( ) , ss XY 1 A 2 A 1 B 2 B 1 C 2 C 1 D 2 D dH dL ( ) , s s XH d H Y +- ( ) , ss XY L d L +-

Fig. 3: Picture frames

Without loss of generality, we consider that the working area is a rectangle. If the area is not a rectangle, it is possible to embed it into a rectangle area by adding more obstacles to the original area.

B. Add the geometry of the quad-rotor to each obstacle

In the process of performing its tasks, the UAV has to avoid the obstacles. Therefore, if we consider this UAV as a circle of radius r (in Fig. 4a), the distances from all the border points of obstacles to the center of UAV circle have to be greater or equal to r. For simplicity to consider the UAV as a point, it is necessary to augment the size of the obstacles by a radius dependent on the geometry of the drone and its footprint. At each border point of the obstacle, the operator draws a circle with radius r and considers the new obstacle is inside the red lines (in Fig. 4b). Several cellular decomposition methods have been studied as trapezoidal decomposition technique [START_REF] Latombe | Robotics : Vision, Manipulation and Sensors[END_REF], boustrophedon decomposition [START_REF] Choset | Coverage Path Planning : The Boustrophedon Cellular Decomposition[END_REF], and Morse-based cellular decomposition [START_REF] Acar | Morse Decompositions for Coverage Tasks[END_REF].

In the trapezoidal decomposition technique [START_REF] Latombe | Robotics : Vision, Manipulation and Sensors[END_REF], the free spaces are divided into trapezoidal cells. There are two parallel sides on each cell, and each cell can be covered by simple back and forth motion parallel to either side. Therefore, the whole working area can be totally scanned by visiting each cell in the adjacency graph. A vertical line is swept from left to right through the working area of the robot. Each time the line encounters a vertex of a polygon, one can say that an event occurs. Three types of events are defined : IN, OUT, and MIDDLE. In an IN event, the current cell is closed and two new cells are opened. When an OUT event occurs, the current two cells are closed and a new cell is created. Finally, MIDDLE event is the event that a current cell is closed and a new single cell is initiated. Consequently, by using these three events, the environment can be totally divided into separated trapezoidal cells.

A drawback of the trapezoidal decomposition is that it generates many cells that, intuitively, can be merged together to form bigger cells. This is clearly an inconvenient, as the more cells are present, the longer the final coverage path is. This happens because the trapezoidal decomposition creates only convex cells. However, non-convex cells can also be completely covered by simple motions. To overcome this limitation, the boustrophedon cellular decomposition was proposed. The boustrophedon decomposition is similar to the trapezoidal decomposition introduced above, but it only considers vertices where a vertical segment can be extended both above and below the vertex. The vertices where this occurs are called critical points.

The boustrophedon decomposition [START_REF] Choset | Coverage Path Planning : The Boustrophedon Cellular Decomposition[END_REF] addresses the issue of redundant movements by merging together cells between successive IN and OUT movements. The major difference is in the case of a MIDDLE event, where no new cell is formed, but the current cell is updated in width. So cells are only created on change of connectivity, as a result the number of cells are less than in the case of trapezoidal decomposition. The point which changes the connectivity of the cell is termed as a critical point. In the implementation of this algorithm, the MIDDLE event is replaced by two new events called -FLOOR and CEILING. The FLOOR event corresponds to vertices on top of obstacles, while CEILING event corresponds to vertices that are at the bottom of obstacles. Both an IN event as well as an OUT event are associated with FLOOR and CEILING pointers. So for a given cell, CEILING and FLOOR pointers points to the top and bottom of a cell.

In [START_REF] Acar | Morse Decompositions for Coverage Tasks[END_REF], Acar et al generalized the boustrophedon decomposition by proposing a cellular decomposition approach based on critical points of Morse functions. In this paper, the authors show that the boustrophedon decomposition is a particular case of Morse decomposition. With respect to the original boustrophedon decomposition, the Morsebased decomposition has the advantage of handling also nonpolygonal obstacles. By choosing different Morse functions, different cell shapes are obtained, e.g. circular or spiked cells.

A new approach for cellular decomposition will be proposed here. Based on the Morse method, critical points can be determined. However, Morse method does not categorize the type of critical points. In this approach critical points are categorized into four groups as follows (Fig. 6) :

- -At each critical point IN type 1, draw a horizontal line to the left till when that line touches the obstacle or the border of the area. -At each critical point OUT type 1, draw a horizontal line to the right till that line touches the obstacle or the border of the area. The results of the original boustrophedon method and the cellular decomposition of this approach are shown in Fig. 5a and5b.

Simulation results analysis

This approach has been run and verified in Matlab. From Fig. 5a and5b, one can see that with our approach, the area has been splitted into two cells compared with four cells of the Boustrophedon decomposition method. Area with complicated forms (convex, non-convex, polygon, and arbitrary shape) (Fig. 6) of obstacles has been splitted to only 9 cells by using the new proposed approach, compared with more than 20 cells if using Boustrophedon method.

B. Way-points generation and optimization of the percentage of coverage (WG-OSA in Fig. 1) 1) Way-points generation (WG):

As described in section II, picture frames which are taken by the camera mounted on Fig. 6: New approach for Boustrophedon decomposition the aerial robot are rectangular. Consequently, for gathering the whole area of the environment, frames need to be arranged as in Fig. 3. Centers of these rectangles are the positions of the UAV, in which the pictures will be taken. We can see that the set of all these points is way-points that the UAV has to visit.

Way-points are generated as follows :

-The first rectangle on the top left corner has the center with coordinate (X s ,Y s ). -Fill the whole working area with rectangles (Fig. 7) from the left to right and from the top to bottom such that two neighbor rectangles on horizontal direction have an overlapped rectangle area with dimension dH × L, and two neighbor rectangles on vertical direction have an overlapped rectangle area with dimension H × dL(Fig. 3). -Find coordinates of set X that consists of all the centers of rectangles which are located in the obstacle (Fig. 8).

2) Optimization of the percentage of coverage (OSA):

Optimization process is processed as follows : -Calculate the non-obstacle area of all rectangles that have been created with center in X. -Find the percentage of coverage and number of images needed for the coverage. -By varying the coordinate of the first image, means that by varying (X s ,Y s ) such that (X s ,Y s ) ∈ 1... H 2 ;1... L 2 , one can have the maximal coverage percent or minimal number or images needed to be taken.

Simulation results analysis

Simulations in this section have the following parameters : -Image dimension : 1200 × 700 -Picture frame dimension -H= 40; dH = 2; -L= 80; dL = 2; -trade-off function : coverage percentage × (1 -α)no way point × α where α = 0.2 Simulations in this section have the following parameters :

-Image dimension : 1200 × 700 -Picture frame dimension -H= 20; dH = 2; -L= 40; dL = 2; -trade-off function : coverage percentage × (1 -α)no way point × α where α = 0.2 The simulation results given in table I show that, in the case when the dimension of images is large (H = 40, dH = 2; L = 80, dL = 2) the number of way-points affects the percentage of coverage. However, with the same trade-off function, when the dimension of images is small (H = 20, dH = 2; L = 40, dL = 2), the percentage of coverage seems not to be affected by the number of way-points.

C. Optimized path planning generation (OPPG in Fig. 1) Theoretically, it is possible to find the shortest traveling distance for all the way-points generated in the previous section. To do that, firstly, the distance between each two points needs to be calculated and the path between this two points has to avoid the obstacles. Several methods can perform that such as PRM or Rapidly-exploring Random Tree(RRT) [START_REF] Adiyatov | Rapidly-exploring random tree based memory efficient motion planning[END_REF]. For instance, if there are n way-points, the number of paths to be found is n(n-1)

2 . After that we can use TSP for finding the shortest path between n way-points. However, the calculation time can be important.

For avoiding this disadvantage, this section provides another approach. Instead of applying TSP for all the way-points, first we will split the whole area to non-obstacle separated cell and each cell can be covered by simple back and forth motion (as has been proposed at the beginning of this section) by the UAV. Then, the TSP is applied only for finding the shortest path length for moving between cells.

In [START_REF] Choset | Coverage Path Planning : The Boustrophedon Cellular Decomposition[END_REF], the authors used the adjacency graph for finding the sequence of cells to be visited to cover the area. But this approach does not guarantee the order of visited cell gives the shortest path for changing between cells. In this section, we introduce a method for finding the cells' sequence with the shortest path length for changing cells. As seen in the previous section, the working area was splitted into obstacle-free cells, and the UAV can make the back and forth motion to cover cells. Consequently, for each cell, there will be two points : initial and final for the UAV coverage (Fig. 9). For example, for cell k , these two points are k a and k b . UAV can start covering the kth cell from point k a to k b or inverse from k b to k a .

Here each cell will form two nodes of the TSP such as two nodes cell 1 a and cell 1 b for cell 1 . Consequently, if the UAV is in node cell 1 a , it has to go through node cell 1 b and inverse. To do this, an additional constraint has to be added to the constraints of TSP for each cell :

x cellk a cellk b + x cellk b cellk a = 1 k = 1, 2,...,2n (1)
The UAV starts at initial position (START), scan the whole area and go back to the final position (STOP) for charging the battery as shown in Fig. 10. To ensure that there will always exist a solution for this optimization problem, the initial population for GA has the form as shown in Fig. 12. The flip, slide and swap operations are always operated on each pair of nodes x a and x b where x ∈ [1, n]. In each iteration, the random swap position of cell x a and x b is also processed.

The flip, slide and swap operations are described in Fig. 11b, 11c, 11a i, j such as i < j.

The advantages of using proposed flip, slide, swap operations as depicted above will make the search for optimal path faster because this search is performed only on the set of paths that satisfy the given request. Furthermore, the solution is always found even though with the number of cells is huge.

If proposed flip, slide, swap operations are not applied, the optimal path can only be found when the number of cell ≤ 16, and the simulation time is always greater than 2000 seconds.

Simulation results as showns in table III were performed on a Dell insprion 7559 with Core i7, 8Gb, Win-dows10, Matlab 2016b configurations specified in the table. The parameters for GA are : population size = 1000 ; number of iteration = 1e4; number of cell = 100 ;

Fig. 12: Genetic algorithm using in this approach

The algorithm is shown on Algorithm 1.

IV. CONCLUSION, AND FUTURE WORKS

In this paper, we have proposed a new approach for maximizing the coverage area and minimizing the path length of an aerial robot in agriculture environment with concave obstacles. Firstly, this paper developed a new approach for Based on the critical points as in [START_REF] Choset | Coverage Path Planning : The Boustrophedon Cellular Decomposition[END_REF] and [START_REF] Acar | Morse Decompositions for Coverage Tasks[END_REF], the extension of this paper is that the critical points have been categorized to several groups, and the cells have been created from these groups. This extension leads to a decrease of the number of cells after decomposition. The results show that this new cellular decomposition works well even with several concave obstacles inside the environment. Secondly, we have proposed a method for maximizing the percentage of coverage and we have given some comments for the trade-off between the percentage of coverage and the number of way-points for the UAV. The percentage of coverage is up to 97.9%.

Finally, by using GA with some modification on the swap, flip and slide operations and the TSP with some additional constraints, the shortest path for changing the cells has been developed. This algorithm guarantees that the solution always exist and the time for calculation is decreasing.

The results provided in this paper have been obtained under Matlab/Simulinks. The continuation of this work is to add the UAV equations of motion as constraints on the path. In the future work, firstly, the results will be verified in aerial platform. Then the efficiency of the algorithm will be improved to be real-time usabl.

Fig. 1 :

 1 Fig. 1: Proposed algorithm

Fig. 2 :

 2 Fig. 2: UAV with frame picture capture

Fig. 4 :

 4 Fig. 4: UAV footprint and augmentation of the size of obstacles

  Critical point IN type 1 (red circle) : the connectivity of a slice is increased by one and the obstacle is on the right. -Critical point IN type 2 (blue circle) : the connectivity of a slice is increased by one and the obstacle is on the left. -Critical point OUT type 1 (red square) : the connectivity of a slice is decreased by one and the obstacle is on the left. -Critical point OUT type 2 (blue square) : connectivity of a slice is decreased by one and the obstacle is on the right. The algorithm for decomposing the cluttered environment to cells is shown as follows : -Step 1 : Use a vertical line and sweep it from the left to the right of the environment, and find all the critical points. -Step 2 :

Fig. 5 :

 5 Fig. 5: Cellular decomposition methods

Fig. 7 :Fig. 8 :

 78 Fig. 7: Picture frames

Fig. 9 :

 9 Fig. 9: In/Out points in each cell

Fig. 10 :

 10 Fig. 10: Travelling Salesman problems with additional constraints for this problem

Fig. 11 :

 11 Fig. 11: Swap, flip, and slide operations for GA

TABLE I :

 I Some results for simulations in Matlab

	X s	Y s	Number of way-points	% coverage Trade-off function
	20	40 230	95.0196	30.0085
	20	38 229	94.6407	29.8860
	19	39 228	94.3653	29.8922
	1	1	275	96.3794	22.1035
	2	10 266	96.4520	23.9616
	3	13 266	96.3549	23.8839

TABLE II :

 II Some results for simulations in Matlab

	X s	Y s	Number of way-points	% coverage Trade-off function
	1	1	1044	97.5	-130.8
	1	7	1048	97.8	-131.4
	2	5	1030	97.9	-127.7
	10	20 1026	97.5	-127.2
	10	14 1031	97.9	-127.9
	9	14 1029	97.8	-127.6

TABLE III :

 III Time of simulation

	Simu.	With proposed operations	Without proposed operations
	1	732.8133 sec, solution found > 3000 sec, no solution
	2	743.4828 sec, solution found > 3000 sec, no solution
	3	742.2244 sec, solution found > 3000 sec, no solution
	4	742.6994 sec, solution found > 3000 sec, no solution
		START	1 a 1 b	a n	b n	STOP
			Cell 1	Cell n


Sara Ifqir
email: sara.ifqir@univ-evry.fr

Andreas Rauh
email: andreas.rauh@uni-rostock.de

Julia Kersten
email: julia.kersten@uni-rostock.de

Dalil Ichalal
email: dalil.ichalal@univ-evry.fr

Naima Ait-Oufroukh
email: naima.aitoufroukh@univ-evry.fr

Said Mammar
email: said.mammar@univ-evry.fr

Interval Observer-Based Controller Design for Systems with State Constraints: Application to Solid Oxide Fuel Cells Stacks

For many practical applications, it is necessary to design controllers that are, one the one hand, robust against bounded uncertainty in selected parameters and external disturbances. On the other hand, it is also quite common for such systems, that state constraints need to be accounted for which must not be exceeded for any possible parameter value. Although the task is quite challenging, several robust control design procedures were designed in recent years which allow for solving these problems, especially with respect to the latter aspect. In this paper, such robust control procedures are combined in a novel manner with an interval observer to additionally handle the influence of unknown but bounded measurement noise and to prevent the violation of state constraints in a provable fashion.

I. INTRODUCTION

Interval observers have been developed in recent years for several dynamic systems which are described by either finite-dimensional sets of state equations or (parabolic) partial differential equations [START_REF] Efimov | Interval State Observer for Nonlinear Time Varying Systems[END_REF], [START_REF] Ichalal | State Estimation of System with Bounded Uncertain Parameters: Interval Multimodel Approach[END_REF], [START_REF] Kharkovskaya | Design of Interval Observers and Controls for PDEs Using Finite-Element Approximations[END_REF], [START_REF] Raïssi | Some Recent Results on the Design and Implementation of Interval Observers for Uncertain Systems[END_REF]. In all cases, the design of these observers makes use of the derivation of suitable bounding systems which allow for the computation of guaranteed lower and upper state bounds despite set-valued uncertainty in system parameters, bounded external disturbances, and bounded (usually additive) measurement noise.

The computation of guaranteed lower and upper bounds for such estimates commonly relies on the system property of cooperativity which leads to a specific ordering relation in the following form: Lower and upper bounding systems can be described by independent sets of state equations [START_REF] Smith | Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems[END_REF], where each of them depends on selected extremal parameter combinations. Such independence can be achieved (for linear finite-dimensional dynamics), if the system matrix is Metzler, i.e., it consists only of non-negative off-diagonal elements [START_REF] Kaczorek | Positive 1D and 2D Systems[END_REF].

Such models are widely applicable in many disciplines of engineering, especially from the field of thermo-fluidic systems. To make a feedback controller for such cooperative systems robust against bounded uncertainty, it is desired that the error dynamics of the closed control loop remains cooperative and that the worst-case influence of uncertain parameters and bounded noise can be forecasted in a rigorous manner. Bounded noise usually influences the closed-loop dynamics implicitly via tolerance bounds for the state variables that are reconstructed by a model-based state observer.

This paper aims at designing robust interval observer-based state feedback controllers for cooperative systems given in the above-mentioned cooperative form so that besides insensitivity against uncertainty, the violation of user-defined upper state bounds is prevented with certainty.

Sec. II gives a brief overview of the task of temperature control for solid oxide fuel cells (SOFCs) for which the class of cooperativity-preserving robust controllers with state constraints is of huge practical relevance. The robust control design problem is solved in Sec. III by means of a novel linear matrix inequality (LMI) approach for the combined interval observer and cooperativity-preserving control synthesis. Sec. IV summarizes representative simulation results for a mathematical model of the thermal behavior of an SOFC stack module which is parameterized on the basis of measured data gathered on a test rig available at the Chair of Mechatronics at the University of Rostock. Finally, conclusions and an outlook on possible future research activities are given in Sec. V.

II. MODELING THE THERMAL BEHAVIOR OF SOFC STACK MODULES

SOFCs are typically operated at high temperature levels and provide the possibility to supply users simultaneously with thermal and electrical energy. Common applications of SOFCs can be found in the area of a decentralized power supply in which chemical energy contained in a fuel gas (such as hydrogen or hydro-carbonates) is directly converted into heat and electricity [START_REF]Modeling Solid Oxide Fuel Cells[END_REF], [START_REF] Pukrushpan | Control of Fuel Cell Power Systems: Principles, Modeling, Analysis and Feedback Design[END_REF].

In comparison with combustion engines, converting chemical energy through intermediate conversion stages into rotary mechanical energy, which is then transformed into electricity by electrical generator units, the type of energy conversion of SOFCs cells is not a-priori limited by the Carnot efficiency. Hence, significantly higher efficiency levels are, at least theoretically, possible. To exploit this advantage, it is, however, necessary to derive control strategies which prevent local overtemperatures in the interior of the SOFC stack with high reliability [START_REF] Rauh | Interval-Based Sliding Mode Control Design for Solid Oxide Fuel Cells with State and Actuator Constraints[END_REF], [START_REF] Rauh | Reliable Sliding Mode Approaches for the Temperature Control of Solid Oxide Fuel Cells with Input and Input Rate Constraints[END_REF]. Such overtemperatures are typically responsible for accelerated aging of the fuel cell stack and for degradation leading to the stack's destruction in the worst case.

To derive robust control strategies and associated reliable observers, which reconstruct the cell temperature distribution in an efficient model-based way by exploiting readily available measurements (i.e., the media temperatures at the gas inlet and outlet manifolds), it is inevitable to derive low-dimensional system models in terms of sets of ordinary differential equations (ODEs). Due to the complex structure of SOFC modules and due to limited possibilities for measurement of temperatures in its interior, suitable sets of state equations contain parameter values for which only worst-case interval bounds are known [START_REF] Rauh | An Interval Observer Approach for the Online Temperature Estimation in Solid Oxide Fuel Cell Stacks[END_REF].

Following the early lumping semi-discretization procedure summarized in [START_REF] Rauh | Reliable Control of High-Temperature Fuel Cell Systems Using Interval-Based Sliding Mode Techniques[END_REF], which exploits a spatial subdivision of the cuboid SOFC module into a finite number of n = L • M • N elements with piecewise homogeneous temperatures in each element (cf. Fig. 1), quasi-linear state equations

ẋ = A (x, p) • x + B (x, p) • ũ (1)
can be obtained, where the parameter vector p contained in both the system and input matrices allows for modeling of • heat transfer due to heat conduction and convection (comprising a linearized model for heat radiation), • enthalpy flows of the supplied gases, where χ in Fig. 1 denotes all relevant gas fractions at the anode and cathode side, • exothermic reaction enthalpies, and • Ohmic losses due to electric currents I I . One of the simplest, practically useful semi-discretizations is given by L = N = 1 with M = 3. This type of model approximates the SOFC stack temperature distribution in the direction of the parallel gas mass flows of both, the anode gas (i.e., the fuel gas mixture) and the cathode gas (typically preheated air). The resulting set of n ODEs is used in the following for the development of a novel interval observerbased reliable control approach.

For this case of n = 3 finite volume elements, the system and input matrices are given by

A (x, p) =   a11 a12 0 a21 a22 a23 0 a32 a33   and (2)
B (x, p) =   b11 b12 b13 b14 b21 0 0 b24 b31 0 0 b34   , (3)
where the state vector contains the element temperatures

x = ϑ (1,1,1) ϑ (1,2,1) ϑ (1,3,1) T . (4)
Moreover, the input vector

ũ = ϑA ϑAG,in ϑCG,in 1 3 I T (5)
consists of the ambient temperature ϑ A , the anode and cathode inlet temperatures ϑ AG,in and ϑ CG,in , as well as the stack current

I (1,1,1) = I (1,2,1) = I (1,3,1) = 1 3 I.
For a pure temperature control of the SOFC stack operated with a constant cathode gas mass flow, the only remaining control input is the cathode inlet temperature, while all other elements of ũ represent measurable disturbances with respect to the state vector x. Thus, B(x, p)ũ can be dissociated as follows

B(x, p)ũ =   b13 0 0   B u(t) +   b11 b12 b14 b21 0 b24 b31 0 b34   E   ϑA ϑAG,in 1 3 I   d(t)
. [START_REF] Kaczorek | Positive 1D and 2D Systems[END_REF] In this paper, we restrict our attention to the design of the controller with the input u = ϑ CG,in when d(t) = 0. This assumption is motivated by neglecting heat transfer through its encasing to the ambience with temperature ϑ A and bu accounting for the fact that the anode gas enthalpy flow is smaller than the one at the cathode by more than one order of magnitude. However, extensions of the proposed method to the case of both control and disturbance inputs are straightforward in the sense of input-to-state stability (ISS) during the design procedure. Enhancing the robustness of the proposed controller by accounting for variations of the disturbance inputs will be subject of future work.

In previous work, cf. [START_REF] Th | Thermal Behavior of High-Temperature Fuel Cells: Reliable Parameter Identification and Interval-Based Sliding Mode Control[END_REF], [START_REF] Rauh | Reliable Control of High-Temperature Fuel Cell Systems Using Interval-Based Sliding Mode Techniques[END_REF] and the references therein, an initial identification of the matrices (2) and (3) was performed so that the interval-valued enclosure for A (x, p) is strictly Metzler and the matrices B (x, p) and B(x, p) are elementwise non-negative. To obtain a reliable system model, accounting for the worst-case influence of bounded measurement tolerances on the output

y(t) = 0 0 1 • x(t) , (7)
representing the outlet temperature ϑ (1,3,1) , as well as for possibly neglected influence of heat convection and nonlinear radiation effects, the disturbance estimation scheme summarized in the following paragraphs has been developed. It estimates deviations between the assumed system dynamics (1) according to [START_REF] Th | Thermal Behavior of High-Temperature Fuel Cells: Reliable Parameter Identification and Interval-Based Sliding Mode Control[END_REF], [START_REF] Rauh | Reliable Control of High-Temperature Fuel Cell Systems Using Interval-Based Sliding Mode Techniques[END_REF] and the measured data in terms of an additive uncertainty [δ i ] for each element i ∈ {1, . . . , n = 3} of the derivative vector ẋ. This uncertainty is then mapped onto an additive interval-valued correction term for the system matrix according to

[A] = A ([x], p) +   [δ1] 0 0 0 [δ2] 0 0 0 [δ3]   , (8)
where A ([x] , p) represents an interval extension of the matrix A according to (2), which was evaluated in [START_REF] Rauh | An Interval Observer Approach for the Online Temperature Estimation in Solid Oxide Fuel Cell Stacks[END_REF] for the maximum admissible range of operating temperatures, however, with purely point-valued least-squares estimates for all system parameters p. Analogously, [ B] (and its submatrix [B] multiplied with the control signal u) is obtained by evaluating the respective symbolic expression for (3) over the complete range of possible temperatures. So that, we consider along the paper that the system as well as input matrices are bounded element-wise according to

A ≤ A(x(t), p) ≤ A 0 ≤ B ≤ B(x(t), p) ≤ B . (9)
Note that the state dependencies in A(x(t), p) and B(x(t), p) result from a polynomial approximation of the temperature dependencies of heat capacities and reaction enthalpies of the gas fractions contained in the anode and cathode gases.

Based on ( 9), an interval model for (1) can be expressed as

ẋ(t) = Ax(t) + Bu(t) ẋ(t) = A x(t) + Bu(t) (10)
where x(t), x(t) ∈ R n + are the upper and lower bounds of the state vector; u(t) ∈ R m + and y(t) ∈ [y(t), y(t)] are the element-wise non-negative inputs and the measurement vector with bounded uncertainty, respectively. The initial condition at the instant t 0 , x(t 0 ) = x 0 , is assumed to be bounded by two known upper and lower bounds according to

0 ≤ x 0 ≤ x0 ≤ x0 . (11)
In what follows, it is assumed that the available measurements are given by y(t) = Cx(t) with additive noise |ω(t)| ≤ Ω which is unknown but restricted to the a-priori known bound Ω ∈ R p + resulting in the linear output model

ym(t) = Cx(t) + ω(t) . (12)
In the sequel, we are interested in designing a control law ensuring stabilization of (1) based on the interval model [START_REF] Rami | Enlarging Ellipsoidal Invariant Sets for Constrained Linear Systems[END_REF]. As temperature measurements inside the stack are not available, an interval observer providing upper and lower estimates for the unknown state x(t) simultaneously is built and an alternative control law is proposed by using only these interval-valued estimates.

III. CONTROL DESIGN: AN INTERVAL OBSERVER-BASED

FRAMEWORK

To provide a feedback control law which can be implemented using only the interval-valued enclosures [A] and [B], we propose to combine the structure of a classical full state feedback controller with an interval observer in a novel fashion. The block diagram of the resulting control scheme is given in Fig. 2, where x(t) and x(t) are the upper and bounds for the estimated state trajectories. The performance requirements for the design of the controller include the following aspects:

1) The control input u(t) should be designed such that the positivity of the system (1) is not violated for all t ≥ 0, i.e., u(t) ≥ 0; 2) The interval inclusion

0 ≤ x(t) ≤ x(t) ≤ x(t) (13)
has to be satisfied; 3) The interval diameter x(t) -x(t) has to be minimized by tuning the controller parameters; 4) The steady-state value of x(t)+x(t) 2 is associated with a reference input vector r consisting of the same constant elements such that

x(t) + x(t) 2 → r as t → ∞ ; (14)
5) In order to avoid overheating phenomena of the SOFC stack, the controller must be able to prevent the stack from exceeding its maximum admissible upper temperature limit. Thus, this requirement can be described as

x(t) ≤ xmax, x(t) ≤ xmax , (15)
where x max is the maximum admissible temperature inside the SOFC stack. To achieve these control objectives, the following positive interval observer is proposed

ẋ(t) = (A -LC) x(t) + Bu(t) + Lym(t) + LΩ ẋ(t) = (A -LC)x(t) + Bu(t) + max{0, Lym(t) -LΩ} (16)
where x(t), x(t) ∈ R n . To preserve the order relationship [START_REF] Rauh | Reliable Sliding Mode Approaches for the Temperature Control of Solid Oxide Fuel Cells with Input and Input Rate Constraints[END_REF], let e(t) = x(t)x(t) and e(t) = x(t) -x(t) be the upper and lower estimation errors. With respect to these error vectors, we have the associated differential equations

ė(t) =(A -LC)e(t) + (B -B(x(t), p))u(t) + A -A(x(t), p) x(t) + Lω(t) + LΩ (17)
and

ė(t) =(A -LC)e(t) + (B(x(t), p) -B) u(t) + (A(x(t), p) -A)) x(t) + LCx(t) -max(0, LCx(t) -L(Ω -ω(t)) . (18)
Referring to the inequalities (9) and the fact that (1) is a positive dynamical system, i.e, x(t) ≥ 0, u(t) ≥ 0, then both inequalities

(B -B(x(t), p))u(t) + A -A(x(t), p) x(t) ≥ 0 (B(x(t), p) -B) u(t) + (A(x(t), p) -A) x(t) ≥ 0 (19)
hold. Furthermore, under the conditions L ≥ 0 and L ≥ 0, it follows that Lω(t) + LΩ ≥ 0 and LCx(t) -max(0, LCx(t) -L(Ω-ω(t)) ≥ 0. Moreover, if (A-LC) and (A-LC) are chosen to be Metzler matrices, then, the inclusion property ( 13) is satisfied provided that ( 11) is also satisfied. Now, consider the following interval observer-based controller

u(t) = K( x(t) -xss) + K(x(t) -x ss ) + uss , (20)
where u ss = N u r, x ss = N x r and x ss = N x r. N x , N x , N u , K, K ∈ R m×n are the observer-based controller gains to be designed. Define

ξ(t) = x T (t) x T (t) xT (t) xT (t) T .
Combining the open-loop model [START_REF] Rami | Enlarging Ellipsoidal Invariant Sets for Constrained Linear Systems[END_REF] as well as the positive interval observer (16) and substituting u(t) by ( 20) for r = 0 yield the 4n-dimensional closed-loop system

    ẋ(t) ẋ(t) ẋ(t) ẋ(t)     =     A 0 B K 0 A BK BK 0 0 A -LC + B K B K 0 0 BK A -LC + B K     •    x(t) x(t) x(t) x(t)    +    0 0 Lym(t) + LΩ max{0, Lym(t) -LΩ}    . (21)

With the help of the abbreviations

A = A 0 0 A , B = B B , K = K K C = C 0 0 C , L = L 0 0 L , D(t) =    0 0 Lym(t) + LΩ max{0, Lym(t) -LΩ}   , equation (21) is rewritten into ξ(t) = A BK 0 A -LC + BK ξ(t) + D(t) (22)
For stability analysis, we consider the block-diagonal quadratic ISS-Lyapunov function of the form

V (ξ(t)) = ξ T (t)P ξ(t) = ξ T (t) P1 0 0 P2 ξ(t) , (23)
where P 1 and P 2 are diagonal positive matrices.

Taking the time derivative of the Lyapunov function (23) along the trajectory of the closed-loop system (22), we get

V (ξ(t)) = ξ T (t) A BK 0 A -LC + BK T P + P A BK 0 A -LC + BK ξ(t) + D T P ξ(t) + ξ T (t)P D(t) . ( 24
)
Let γ and ε be two positive scalars. Then, by adding and subtracting the terms εξ T (t)P ξ(t) and -γ D T (t) D(t), (24) becomes

V (ξ(t)) = ξ T (t) D T (t) T Φ ξ(t) D(t) T -εV (ξ(t)) + γD T (t)D(t) (25)
with

Φ =      A BK 0 A -LC + BK T P + P A BK 0 A -LC + BK + εP P P -γI4n     
.

If the gain matrices K and L are found such that

Φ ≺ 0 (26)
is satisfied, the inequality

V (ξ(t)) < -εV (ξ(t)) + γD T (t)D(t) (27)
holds. Multiplying both sides of ( 27) by e εt and integrating from 0 to t, one obtains

V (ξ(t)) < -V (0)e -εt + γ t 0 e -ε(t-s) D(s) 2 2 ds . (28)
Recall that for any ξ(t) ∈ R 4n , the bounds

ε1 ξ(t) 2 2 ≤ V (ξ(t)) ≤ ε2 ξ(t) 2 2 , ∀ξ(t) ∈ R 4n (29)
hold, where

ε1 = λmin(P ), ε2 = λmax(P ) (30)
in which λ min (P ) and λ max (P ) denote the minimal and maximal eigenvalue of the matrix P , respectively. Then, the inequality (28) becomes

ξ(t) 2 2 ≤ ε2 ε1 ξ(0) 2 2 e -εt + γ εε1 D(t) 2 ∞ (31)
leading to

ξ(t) 2 ≤ ε2 ε1 ξ(0) 2e -ε 2 t + γ εε1 D(t) ∞ (32)
which implies that the system ( 22) is ISS if (26) is satisfied. The radius of the convergence region is upper bounded by γ εε1 D(t) 2 ∞ . Thus, the bilinear matrix inequality (BMI) (26) can be used to construct the optimal solution by minimization of γ ≥ 0. Although the proposed observer-based state feedback approach provides a controller that minimizes the norm of the disturbances, the state constraint [START_REF] Smith | Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems[END_REF] has not been taken into consideration yet. Let us consider the Lyapunov level set

Ω(P ) = {ξ(t) ∈ R 4n ξ T (t)P ξ(t) ≤ 1} , (33)
which is positively invariant with respect to the motion of the closed-loop system (22) (i.e., for every initial condition ξ(0) inside Ω(P ) the trajectory of ξ(t) remains in Ω(P ) for all t ≥ 0). Now, define the following set of state constraints

L(F, ϑ) = {ξ(t) ∈ R 4n F ξ(t) ≤ ϑ} , (34)
where F ∈ R q×4n and ϑ ∈ R q are given. According to the previous considerations, if Ω(P ) ⊂ L(F, ϑ) and P satisfies the condition (26), then for any initial condition inside the ellipsoid Ω(P ) we have ξ(t) ∈ L(F, ϑ) for all t ≥ 0.

From [START_REF] Rami | Enlarging Ellipsoidal Invariant Sets for Constrained Linear Systems[END_REF], we obtain the following useful Lemma. Lemma 1.

[10] The inclusion Ω(P ) ⊂ L(F, ϑ) is equivalent to the existence of positive scalars a 1 , . . . , a q > 0 such that

4aiϑi -FiPiF T i 2ai 2ai 1 ≥ 0 , ∀i ∈ {1, . . . , q} , (35)
where F i is the i th row of F . Remark. Regarding the result above, the state set (34) satisfying the constraints (15) can be chosen as

F = ✶n 0 0 0 0 0 ✶n 0 , ϑ = xmax xmax . (36)
The following theorem summarizes the discussion above. The operators ">", "≥" are applied element-wise. Theorem 1. The control law u(t) given in (20) with r = 0 is robustly stabilizing in an ISS sense, if there exist positive diagonal matrices P 1 , P 2 , and P 3 , full matrices W and Y, scalars γ > 0, a 1 , . . . , a q > 0 for given ε, η ∈ R + , F ∈ R q×4n , and ϑ ∈ R q such that the following LMIs hold true, min P1,P2,P3,W,Y,a1,...,aq

γ W > 0 , (37a)
Y > 0 , (37b)
P2A -YC + ηP2 ≥ 0 , (37c)
P2B = BP3 , (37d) 4aiϑi -FiPiF T i 2ai 2ai 1 ≥ 0, ∀i ∈ {1, . . . , 4n} , (37f)
where L = P -1 2 Y and K = P -1 3 W. Proof. The element-wise inequality (37a) guarantees the positivity of the controller gains, (i.e., K ≥ 0 and K ≥ 0), and therefore, the positivity of the control law u(t). Moreover, the element-wise inequality (37b) ensures the positivity of the interval observer gains. Furthermore, from [START_REF] Ifqir | Robust Interval Observer for Switched Systems with Unknown Inputs: Application to Dynamics Estimation[END_REF], the matrices (A -LC) and (A -LC) are Metzler if and only if there exists a positive scalar η such that

(A -LC) + ηI2n ≥ 0 . (38)
Multiplying the left-hand side of (38) by P 2 together with the change of coordinates Y = P 2 L, (37c) is directly obtained. The LMI (37f) guarantees the inclusion of the ellipsoid Ω(P ) in the set of trajectories L(F, ϑ). According to (26), the LMIs (37d) and (37e) guarantee ISS stability of the closed-loop system (22). After some calculations, the BMI (26) is rewritten into

     P 1 A + A T P 1 + εP 1 P 2 BK K T B T P 2 Ξ P 1 0 0 P 2 P 1 0 0 P 2 -γI 4n      ≺ 0,
where

Ξ = P 2 A + A T P 2 + εP 2 -P 2 LC -C T L T P 2 + P 2 BK + K T B T P 2 .
To deal with the bilinear term P 2 BK, the equality (37d) is introduced. Then, by letting W = P 3 K and Y = P 2 L, (37e) is obtained. This completes the proof. Finally, it is desired that the mean value x(t)+x(t) 2 tracks a reference input r such that x(t)+x(t) 2 → r as t → ∞. To find N x , N x , and N u , we consider the desired steady-state relationships:

ẋss = Axss + Buss = (A N x + BNu)r = 0 ẋss = A x ss + Buss = (A N x + BNu)r = 0 1 2 In 1 2 In xss x ss = 1 2 N x 1 2 N x ✶n ✶n r = r . (39)
Thus, the possible solutions can be found by solving the following set equality constraints

  A 0 B 0 A B 1 2 In 1 2 In 0     N x N x Nu   =   0 0 1   . (40)
Finally, the control law (20) is equivalent to

u(t) = K x(t) + K x(t) + (Nu -K N x -K N x )r . (41)
Thus, to ensure that u(t) remains positive, the matrix (N u -K N x -KN x ) should be positive element-wise according to

-K -K 1   N x N x Nu   ≥ 0 . (42)
Therefore, the steady-state controller design can be achieved by satisfying the linear constraints (40), (42).

IV. APPLICATION TO ROBUST TEMPERATURE CONTROL OF AN SOFC STACK The dynamic model according to Sec. II used for the interval observer-based control design has been validated using real measured data gathered during the heating phase of an SOFC stack module available at the Chair of Mechatronics at the University of Rostock with the parameters L = N = 1 and M = 3.

The observer-based controller gains K, K, L, and L are obtained by solving the convex optimization problem in Theorem 1 with C = 0 0 1 , x max = 1000 K, ε = 0.5 and η = 0.25. The measurement tolerance Ω is taken to be equal to 15 K. By applying Theorem 1, the following gains are obtained Corresponding simulation results are depicted in Figs. 3 and4. From these results, it can be concluded that the robust state feedback control design has been achieved successfully under consideration of the predefined state constraints. Furthermore, the estimates provided by the interval observer approach the desired steady-state operating point (r = 750 K). It should be noted that the reason why the average value of these estimates does not reach exactly the desired reference value r is because the dimension of the state space is greater than the number of control inputs (n = 3, m = 1) which reveals the underactuated character of the system.

V. CONCLUSIONS AND OUTLOOK ON FUTURE WORK

The novel interval observer-based state feedback controller presented in this paper allows for preventing the violation of state constraints for dynamic systems given in the form of linear, cooperative ODEs with interval-bounded parameter uncertainty. The applicability of the controller is demonstrated in simulations for the thermal behavior of a high-temperature SOFC stack, allowing for preventing overshooting safetycritical temperature bounds in a reliable manner. Future work will extend this research by accounting for the neglected disturbances in ( 6) and the dynamics of actuators (e.g., the lag behavior of the preheaters of the SOFC) in combination with corresponding range constraints for the admissible control signals and their admissible variation rates. Moreover, possibilities for the reduction of the control and observation effort by the development of partial state feedback controllers or pure output feedback strategies will be further investigated.

  Fig. Spatial semi-discretization of the fuel cell stack module.

Fig. 3 .

 3 Fig. 3. Time evolution of the upper and lower estimates of the states x 1 , x 2 and x 3 with their average values.

Fig. 4 .

 4 Fig. 4. Time evolution of the control action.


Zhang Jianhao

Roux Röhn John

Xavier Le

Alonso-Ramos Carlos

Berciano Mathias

Marris-Morini Delphine

He Sailing

Sun Zhipei

Vivien Laurent

Cassan Eric
email: eric.cassan@u-psud.fr

Jianhao Zhang

John Röhn

Xavier Le Roux

Carlos Alonso-Ramos

Mathias Berciano

Delphine Marris-Morini

Sailing He

Zhipei Sun

Laurent Vivien

Eric Cassan

Hybrid on silicon photonics for light amplification and Fanoenhanced electro-optical modulation at telecom wavelengths -Invited presentation

come   L'archive ouverte pluridisciplinaire

The challenges to be solved in silicon photonics at telecom wavelengths are still of several kinds. Among these, two important locks concern the realization of integrated optical amplifiers essential for re-amplifying attenuated signals, and the realization of fast electro-optical modulators (>>10GHz) with low power consumption (<1fJ/bit) [START_REF] Thomson | Roadmap on silicon photonics[END_REF]. The approaches used in the literature to address these two locks have led to a wealth of proposals and works, including the hybrid integration of III/V semiconductors on silicon by sticking vignettes or full plate, or more rarely by monolithic growth. This solution, based directly on pre-existing mature solutions, raises co-integration problems, both in terms of materials and photonic integration (coupling of Si guide light to III/V guides, etc.), but has already demonstrated several very solid proposals. However, it presents difficulties related to the complexity of processes and technologies, without integration into a complete CMOS process flow [START_REF] Thomson | Roadmap on silicon photonics[END_REF]. In this context, our presentation will summarize our recent contributions on the two key functions mentioned above. Our approach to light amplification is based on the integration of Erbium-doped materials and brings a strong innovation to the field through the integration of state-ofthe-art active materials combined with a proper choice of slot silicon nitride waveguides. We report ultra-high onchip optical gain in Erbium-based hybrid slot waveguides with a monolithic, CMOS-compatible and scalable atomic-layer deposition (ALD) process. The nature of ALD provides a versatile tool to atomic scale engineering of the gain layer properties and an efficient integration with SiN waveguides and other photonic structures. We demonstrate 20dB/cm of modal gain at =1.53µm, which opens up very interesting perspectives for the creation of light sources and their transcription into various integrated system-level functions [START_REF] Rönn | Ultra-high onchip optical gain in erbium-based hybrid waveguides[END_REF]. In another complementary direction, the signals thus produced or amplified must be encoded by electro-optical modulation for the realization of optical links. We started from the observation that beyond the physical mechanisms leading to index modulation by refraction, a significant part of the contribution to the energy dissipation of the modulators was due to the type of resonators or interferometers used. An electro-optical modulator exploiting the plasma effect (refraction on free carriers) and integrating a phase modulator into a Mach-Zehnder interferometer geometry, for example, consumes several ps/bit. This is mainly due to the size of the interferometer (several mm), which brings back high structural capacities. The use of nanoresonators, such as nanobeam cavities, provides a very significant downscaling factor but is still insufficient to achieve the targeted consumption levels (about 1 fs/bit). The reason lies in the classical form of the optical resonances used (Lorentzians), which leads to a necessary strong modulation to detune the resonance from its maximum. We have recently shown that a complete paradigm shift could be achieved in the consumption of plasma electro-optical modulators when an asymmetric Fano resonance is considered. We have specially proposed and designed an original Fano cavity structure based on the electromagnetic coupling of two pre-existing resonances in a single silicon guide geometry [START_REF] Zhang | Generating Fano Resonances in a Single-Waveguide Silicon Nanobeam Cavity for Efficient Electro-Optical Modulation[END_REF]. This single guide geometry offers a generic platform of Fano cavities directly applicable to the production of electro-optical modulators characterized by a set of merit factors superior to those obtained previously due to the very abrupt transmission of the Fano resonance. All these results will be detailed and supplemented at the conference by the most recent experimental results obtained.

Acknowledgements: ANR SITQOM, POPSTAR ERC


Nazim Agoulmine
email: nazim.agoulmine@ibisc.univ-evry.fr

Olfa Chabbouh
email: olfa.chabbouh@supcom.tn

Sonia Ben Rejeb
email: sonia.benrejeb@supcom.tn

Zied Choukair
email: z.choukair@supcom.tn

Offloading Decision Algorithm for 5G/HetNets Cloud RAN

Keywords: Cloud RAN, Cloud RRH, 5G/HetNet, mobile data offloading, energy-saving, QoS

The proliferation of mobile handsets has undoubtedly spurred on developers to build a large variety of applications that allow users to better exploit their powerful devices. However, running multiple sophisticated applications could result in poor performances and shortened battery lifetime. Mobile data offloading to edge cloud offers a promising solution to enhance both QoS and battery life of mobile terminals. In this paper, a novel multi parameters offloading decision algorithm based on cloud radio access network (C-RAN) is proposed. It takes decision about offloading mobile computation to Cloud-RRH in a 5G heterogeneous C-RAN. Unlike previous works, the proposed scheme incorporates a multitude of parameters in the offloading decision process while diminishing the mobile devise energy consumption and keeping a good user quality of experience by reducing application's response time. Simulation results show that the proposed algorithm is able to assure the computation of all the applications while respecting latency constraints and to extend the mobile battery lifetime.

I. INTRODUCTION

With the exponential growth of data traffic, it becomes more and more difficult for telecom operators to upgrade their Radio Access Networks (RAN) because the revenue is not growing at the same rate. To maintain profitability and growth, they must find solutions to reduce cost as well as to provide better services to the customers. In this context the Cloud RAN architecture was proposed. The concept was first introduced in [START_REF] Lin | Wireless network cloud: Architecture and system requirements[END_REF] and described in detail in [START_REF]C-RAN The Road Towards Green RAN[END_REF]. It consists of decoupling the Base Band Units (BBUs) from radio remote heads (RRHs) and move it to the cloud enabling a centralized processing and management. Traditional complicated base stations can be simplified to cost-effective and power-efficient radio units (RRHs). In addition, the centralized processing power enables more advanced and efficient network coordination and management.

A recent survey on computation offloading for mobile systems was described in [START_REF] Kumar | A Survey of Computation Offloading for Mobile Systems[END_REF]. Offloading may be performed at the levels of methods, tasks, applications, or virtual machines. Cloud computing is used to allow offloading at the virtual machine level [START_REF] Rim | Transparent method offloading for slim execution[END_REF] [START_REF] Yang | On effective offloading services for resource-constrained mobile devices running heavier mobile internet applications[END_REF][6] [START_REF] Chun | Augmented smartphone applications through clone cloud execution[END_REF]. It grants elastic resources and offloading to multiple servers. Several architectures and solutions have been proposed to improve offloading: they address different issues such as transparency to users, privacy, security, mobility, etc. For example, the femtocloud was proposed in the European TROPIC project [19] for LTE networks. This new paradigm consist of a distributed set of femtocell access points (FAPs) which combines the radio resource management of femtocell networks with cloud storage and computation capabilities in the same framework [START_REF] Cuervo | Maui: making smartphones last longer with code offload[END_REF]. Thus, the traffic offloading is reduced thanks to these additional cloud resources closer to mobile user. In order to optimize the Cloud RAN architecture and following the same intuition, we have proposed the Cloud-RRH. It consist of upgrading High-RRH capabilities by adding extra resources of computing and storage located in the cloud. Bringing resources closer to the user improves not only power consumption at the terminal side but also the other major issue, latency. Having a hierarchical cloud will allow offloading of applications, partially or totally, from user equipment to Cloud-RRH.

Traditional mobile computation offloading algorithms incorporates only energy consumption or latency in offloading decision. However, many other parameters should be considered in order to have the solution that best fit system conditions. In fact, several offloading decision schemes were proposed in the literature, but the originality of our algorithm is that it takes into consideration several parameters related to network state and mobile terminal mobility speed and capabilities while reducing the offloading decision process complexity.

The remainder of this paper is organized as follows. In the next section, previous works are discussed. Section III describes the system model and basic idea of our offloading decision algorithm for C-RANs. Then, simulation results are presented in section IV. Finally, Section V concludes this paper.

II. RELATED WORK

With the explosion of Internet data traffic, especially the growing portion of traffic going through mobile networks, mobile data offloading has become an important issue in cellular networks. A set of existing works concerning offloading are discussed in this section.

Various cloud offloading systems were proposed in the literature [START_REF] Cuervo | Maui: making smartphones last longer with code offload[END_REF] [START_REF] Chun | Clonecloud: elastic execution between mobile device and cloud[END_REF]. MAUI [START_REF] Cuervo | Maui: making smartphones last longer with code offload[END_REF] and ThinkAir [START_REF] Kosta | ThinkAir: Dynamic resource allocation and parallel execution in the cloud for mobile code offloading[END_REF] profile hardware components and make offloading to optimize energy usage of mobile devices. They only consider the energy consumption and ignore the other aspects of offloading. CloneCloud [START_REF] Chun | Clonecloud: elastic execution between mobile device and cloud[END_REF], was proposed in order to optimize applications partitioning between local execution and offloading with the purpose of minimizing execution time or energy consumption.

Besides mobile cloud frameworks, some other research works focus on offloading decision making issues. Authors in [START_REF] Yuh-Shyan | An Energy-Aware Data Offloading Scheme in Cloud Radio Access Networks[END_REF] have proposed an energy-aware data offloading scheme for C-RAN where the BBU make offloading decision considering the UEs' transmission rate and energy consumption of the cellular and WiFi networks. It uses the centralized characteristic of C-RAN to schedule Ues' offloading from RRH to Wi-Fi access point. The proposed scheme reduces the energy consumption and improves the throughput in the network. However, performance improvement are visible only if the data size is big enough.

Another offloading decision algorithm in the context of femtocloud paradigm is proposed in [START_REF] Oueis | Multi-parameter Decision Algorithm for Mobile Computation Offloading[END_REF]. It consists in performing a series of classifications that consider several parameters such as latency, battery level, UE memory, etc. The proposed algorithm divided the application tasks into "urgent tasks" and "not urgent task" and it sends offloadable tasks that are assigned as urgent to the femtocloud regardless the channel conditions. This affects the energy consumption of the mobile handset and violates latency constraints if total offloading is up 3%.

In [START_REF] Ting-Yi | Context-Aware Decision Engine for Mobile Cloud Offloading[END_REF], authors proposed a context-aware decision algorithm (CADA) in order to offload to the cloud servers. The decision engine is composed of four components: context-aware decision algorithm, context profiler, energy model, and context database. CADA uses the location and time-of-day to make the mobile offloading decisions of individual methods. That's require a profiler. However, it is a source of overhead and it consumes memory in order to store users' profiles.

Different factors have been involved in offloading decision for mobile cloud computing. In [START_REF] Huijun | Making Offloading Decisions Resistant to Network Unavailability for Mobile Cloud Collaboration[END_REF], an offloading decision model that takes network unavailability into consideration is proposed. Based on network connection states and durations that are recorded in a history buffer, the application partition that is aimed to give benefit in network with low availability is calculated. Then the decision about offloading is validated. The proposed algorithm enhances network performances in terms of execution time and energy consumption but the complexity of the system increase with the increase of the number of mobile users.

In summary, mobile cloud offloading issue was studied for different networks and several decision algorithm were proposed. They exploit multiple objective optimization techniques. However, regarding the network instability, decision needs to be refreshed whenever system conditions change.

In our work, we tried to avoid different limits that were highlighted in previous introduced works by considering a multi-parameter decision algorithm for offloading decision in a Cloud RAN in order to adapt the offloading decision to the current state of the system. Besides, we will consider application requirements and mobile available resources and mobility spped in offloading decision.

III. APPROACH : THE PROPOSED OFFLOADING DECISION ALGORITHM IN C-RAN

With the increasing of mobile data traffic, offloading to the cloud has become an important and popular issue in cellular network. In our approach, we aim at optimizing the QoS in terms of latency, throughput, application response time, etc. in Cloud Radio Access Networks. In this paper we will deal with energy efficiency and offloading.

In our architecture, we consider a Cloud RAN heterogeneous architecture composed of H-RRHs (High RRHs) which acts as macro cells and L-RRHs (Low RRHs) which acts as small cells. In our scenario, we introduced the Cloud-RRH which represents the edge cloud. While in a traditional C-RAN architecture all the RAN functionalities are centralized in BBU pools, we propose to flexibly split these functionalities between edge and central cloud. We also introduce additional computation and storage resources in the Cloud-RRH for computation offloading. These resources are represented by cloud containers. The infrastructure is represented in Figure 1.

Based on our proposed 5G/HetNets C-RAN architecture, we developed an offloading decision algorithm. The main objective of this algorithm is to ameliorate the user experience by optimizing the latency and reducing the energy consumption of mobile devices in order to extend battery lifetime and also to enhance the offloading data traffic for better wireless transmission quality.

A. Offloading interactions

The mechanism of offloading is depicted in Figure 2. When a mobile terminal receives a task, it starts by generating an Offloading Request packet and sends the request to RRH. The offloading request packet is shown as bellow:

-Offloading Req ( , CAPMT, , , ), where CAP MT is the Mobile Terminal (MT) capacity, CAP represents the capacity required by the received task, is the energy spent for local execution and B is the bandwidth between RRH and UE. If the application will be offloaded to the Cloud-RRH, the cloudlet manager sends a Resource Allocation Request packet to the serving virtual machine where it indicates the capacity required. After processing, the Offloading Response will be routed up to the mobile terminal.

B. System model and QoS parameters

The basic idea of Cloud RAN is to separate the BBUs from cell sites and group them into BBU pools in real-time cloud for a centralized processing and management. Thanks to network flexibility, C-RAN can support multi-standard operations and can be easily deployed with heterogeneous networks.

In this paper, as represented in Figure 3, we consider a 5G system in a C-RAN context with K users served by either a High RRH (H-RRH) or a Low RRH (L-RRH). We consider uplink connection between mobile terminal and the serving RRH with a bandwidth B. According to [START_REF] Muñoz | Joint Allocation of Radio and Computational Resources in Wireless Application Offloading[END_REF],

, which represents the time employed to send bits in the UL, only depends on the UP data rate and the number of bits to be transmitted, i.e., = / . Similarly, for the remote processing and the DL transmission from Cloud-RRH to the serving RRH, the time required is = / .

We adopted the following models for the power consumption at the mobile terminal in both UL and DL:

= ( , ) + ( , ) * (1)
= ( , ) + ( , ) * (2)
Where ( , ) , ( , ) , ( , ) and ( , ) are constants. These expressions are based on the measurements provided in [START_REF] Jensen | LTE UE Power Consumption Model: For System Level Energy and Performance Optimization[END_REF] which indicates that the UL power consumed by the user increases with the radiation power and a baseline power is consumed just for having the transmission chain switched on. While in the DL, the power consumed by the user terminal increases with the downlink data rate and a baseline power is consumed just for having the reception chain switched on.

The maximum rate supported by the channel with K users and that depends on the transmission power and the quality of the channel is given by Shannon's theorem:

( , ) = (1 + * ( , ) ) (3)
( , ) = (1 + * ( , ) ) (4)
Where and represent the channel gain normalized by the noise power in UP and DL and ( , ) and ( , ) are transmission power of the user and RRHce, respectively.

According to (1) and ( 2), the energy spent by MT (Mobile Terminal) in UP and DL is given by:

= ( , ) * + ( , ) * * (5) = ( , ) * + ( , ) * * ( 6
)
Using equation ( 3), = . So the energy consumed by the MT for offloading is given by:

= + = ( , ) * + ( , ) * * 2 -1 + ( , ) * + ( , ) * (7)
We considered that the energy spent by MT in local processing is proportional to the number of processed bits. The expression is given by the following equation: Where is a constant that accounts jointly for the Joules/cycle and cycles/bit for the processor at the MT and is the number of bits.

Concerning the latency, we considered and as the time needed to process one bit at the MT and at the RRHce respectively. The time required for offloading process to be completed is given by the sum of the time required to send bits from the mobile terminal to the serving RRHce through the UL, the time for the remote processor to execute the offloaded computation, and the time to send all the output bits through the DL. Latency expression for local processing and offloading are given by:

= * (9) = + * + (10)

C. Algorithm description

In this section, we will discuss the workflow of the proposed algorithm. The main objective of this algorithm is to decide if the application have to be processed locally or need to be offloaded to the Cloud-RRH in order enhance the QoE (Quality of Experience) while optimizing the network and MT resource utilization.

We propose to introduce a multitude of parameters in the offloading decision process without including them in a complex optimization problem. The algorithm of mobile application offloading decision is run, at each time slot, on the set of tasks generated by the launched applications. Figure 3 shows the general workflow of the proposed algorithm.

When a task is received and as a first step, we will consider the UE velocity and compare it to a velocity threshold: if the UE velocity is high the task will be executed locally in order to minimize network overhead which can lead to user experience deterioration. Otherwise, we will compare the latency using equations ( 9) and (10) because of the importance of latency regarding QoS. If the latency generated when the task is offloaded to Cloud-RRH is greater than the latency generated by local execution, the task should be computed locally at the mobile handset. However, mobile terminals resources are limited in terms of computational capacity. So, if the computational capacity required is greater than the predefined percentage of the total locally available capacity, the task have to be offloaded to the Cloud-RRH.

Then, if the latency generated by offloading is lower, we have to compare the consumed energy at the mobile handset in case of offloading, ,using equation ( 7) and , the consumed energy in case of local computation of this task using equation [START_REF] Cuervo | Maui: making smartphones last longer with code offload[END_REF]. If > , then it compares the latency generated by local computation and the maximum latency authorized by the application. If < the task is computed locally, otherwise, the task is offloaded.

Finally, if < it examines the channel conditions. The channel capacity can be calculated using Shannon theorem. Then, the channel coefficient is compared to the average channel coefficient calculated and updated over time. If the current channel realization is above this average, it is considered that the channel is in a relatively "good" state and the task is offloaded. Otherwise, it is calculated locally in order to prevent the system from applying costly offloading when channel conditions are not favorable.

IV. SIMULATION AND RESULTS

This section provides simulation results for the proposed method, when the goal is the minimization of the energy consumed by the MT and the application response time while considering a multitude of parameters concerning the network state and the handset characteristics.

In order to verify the performance of our approach, we consider an urban environment simulation scenario. We consider a heterogeneous C-RAN with 7 H-RRH and 4 L-RRH per cell. H-RRHs have a coverage of 500m and L-RRHs are 30m-radius [START_REF] Muñoz | Joint Allocation of Radio and Computational Resources in Wireless Application Offloading[END_REF]. System simulation parameters are listed in Table I. The values for and have been aligned with the measurements given in [START_REF]TROPIC: Distributed computing, storage and radio resource allocation over cooperative femtocells[END_REF] for energy and frequency characteristics of local computing in commercial handsets, as well as computation to data ratios in practical applications. The simulation aim to evaluate our proposed offloading decision algorithm compared to total offloading. Figure 4 illustrates the variation of the response time over the data size (ranging from 1 to 100 Kbits). We can observe that the proposed offloading scheme can ameliorate the user experience by reducing the response time. When the data size is small the results of the two schemes are close. However, when the data size becomes larger, the difference between the two schemes becomes lager. On average there is an improvement of 14% in response time. Thus, the proposed offloading decision algorithm can be useful for high resourcedemand applications. We can see that the proposed offloading decision algorithm can make the mobile handset consumes less energy. The difference is more important when the data size is big. Therefore, the proposed algorithm is able to augment the mobile handset battery lifetime while executing complex program applications compared to local execution and total offloading. CONCLUSION AND PERSPECTIVES Mobile data offloading has become an important issue for mobile cellular network in recent years. In this paper we propose an offloading decision algorithm which consider a multitude of parameters in order to make the offloading more efficient in heterogeneous C-RANs with Cloud-RRH. The proposed scheme helps to make the best usage of network and mobile terminal resources. We show the efficiency of our algorithm through numerical simulations. Results show that this algorithm can response time and the mobile terminal energy consumption and thus improve the user experience.

In future work, we will try to evaluate the cost depending upon energy consumption optimization. Furthermore, we will try to better investigate and evaluate the network performances by handling the interference and mobility management in C-RAN.

Fig. 1 .

 1 Fig. 1. Proposed C-RAN architecture

Fig. 2 .

 2 Fig. 2.Cloud RAN: Offloading mechanism

Fig. 3 .

 3 Fig. 3. Proposed offloading decision algorithm for C-RANs

Fig. 4 .

 4 Fig. 4. Application response timeFigure5shows the simulation results of the energy spent by the MT under the data size (ranging from 1 to 100 Kbits). We can see that the proposed offloading decision algorithm can make the mobile handset consumes less energy. The difference is more important when the data size is big. Therefore, the proposed algorithm is able to augment the mobile handset battery lifetime while executing complex program applications compared to local execution and total offloading.

Fig. 5 .

 5 Fig. 5. Total energy consumed by MT V.CONCLUSION AND PERSPECTIVES

TABLE I .

 I SIMULATION PARAMETERS

	Parameter		Value
	K(tx,1)		0.4W
	K(tx,2)		18
	K(rx,1)		0.4W
	K(rx,2)		2.86 10 -3 W/Mbps
			8.6 10 -8 J/bit
			10 -7 s/bit
			/2
	Bandwidth (B)		10 MHz
	Maximum latency authorized	4s
	by the application (	)
	Users mobility speed		3 Km/h < V ≤ 120 Km/h
	V th		5m/s