Alex Hamdan
email: alex.hamdan@hds.utc.fr

Reine Talj

Véronique Cherfaoui

A Centralized Multilayer LPV /H ∞ vs Decentralized Sliding Mode Control Architectures, for the Lane Keeping and Stability of an Intelligent Vehicle

Keywords: Semi-autonomous vehicle, ADAS systems, Centralized LPV /H ∞ control technique, Decentralized Sliding Mode control technique, Decision-making, Human-machine cooperative control

This paper deals with the development of two Advanced Driving Assistance systems (ADASs) architectures involving the Active Front Steering (AFS) and the Direct Yaw Control (DYC) for the Lane Keeping and the lateral stability enhancement in Semi-Autonomous Vehicles. The objective of these systems is to assist and help the driver to keep the lane and to ensure lateral stability. To do that, a new Centralized and Decentralized architectures of ADAS system are developed. The different layers of each architecture are detailed including the control layer where the LPV /H ∞ and the Super-Twisting Sliding Mode (ST SM) control techniques are used for the development of ADAS controllers respectively. Then, a decision making layer monitors the driver's behavior and the lateral stability to adjust the different controllers. Both ADAS system architectures are validated on Matlab/Simulink for a defined scenario with a complete nonlinear model of the vehicle validated on "SCANeR Studio" (OKtal) professional simulator. Finally, a comparison is done between them to show the difference in performance and the effectiveness of both strategies of control on the assistance objective and guaranteeing vehicle's stability.

INTRODUCTION

Active safety is an important issue that should be considered while driving on the road. In 2013, the US Department of Transportation "National Highway Traffic Safety Administration NHTSA" has classified the automation of vehicles into 5 levels to characterize their capabilities [START_REF] Favarò | Examining accident reports involving autonomous vehicles in california[END_REF]]. These levels start at having some assistance features and end at the full-autonomous vehicle passing by the semi-autonomous ones. Level 2 of automation consists of assisting the driver by several automated functionalities especially for active safety purpose. These systems are called Advanced Driving Assistance Systems (ADASs). They are integrated in the vehicle to protect the passengers in case of accident. There are the Passive systems (air bags, seat belts, etc.), and the Active systems that help and assist the driver in some driving tasks especially when he is tired or distracted (Lane keeping system, Emergency braking, Lane departure avoidance, etc.). In addition, many Advanced Driving Assistance System (ADAS) have been proposed and marketed, such as: Active Front Steering (AFS) basically to enhance vehicle's maneuverability; Direct Yaw Control (DYC) or Electronic Stability Program (ESP) to improve vehicle's lateral stability; Active Suspensions (AS) to improve comfort and rollover avoidance. These systems influence the vehicle's behavior and enhance safety on the road. Moreover, the development of electronics and sensors devices (LIDAR, GPS, etc.) supports the integration of these systems into the vehicle. The authors in [START_REF] Kukkala | Advanced driver-assistance systems: A path toward autonomous vehicles[END_REF] classified the ADAS systems to many categories based on their types, utilities, limitations, etc. In addition, many advanced studies are presented in the literature to deal with the development of these systems especially the Lane Keeping system. A Fuzzy Takagi-Sugeno control method is used in [START_REF] Soualmi | Fuzzy takagi-sugeno lq controller for a shared control of vehicle[END_REF] to develop a lane keeping assistance system where a decision algorithm is integrated to manage the authority between the driver and the controller depending on the lateral deviation error. The authors in [START_REF] Nguyen | Online adaptation of the authority level for shared lateral control of driver steering assist system using dynamic output feedback controller[END_REF] have developed a shared steering controller for lane keeping maneuver where the driver's activity and behavior are considered in the computation of the control input of assistance system, in order to manage conflict between both agents. An assistance steer-by-wire system is presented in [START_REF] Perozzi | On nonlinear control for lane keeping assist system in steer-bywire road wheeled vehicles[END_REF], where the fusion of two steering inputs is done considering the driver's availability via monitoring system. The authors in [START_REF] Sentouh | Cooperative steering assist control system[END_REF] applied the H 2 approach to calculate the assistance torque through a first order filter coordination variable. A weighting approach presented in [START_REF] Borroni | A weighting approach to the shared-control of lateral vehicle dynamics[END_REF] is used to blend the two control inputs by using a fusion parameter α adjusted manually or automatically depending on the driving situations. A similar approach is developed in [START_REF] Li | Shared steering control of tire blowout for ground vehicles[END_REF] where the computation of α takes into consideration many factors: avoidance of lane departure, excessive steering, etc, to deal with lane following in case of tire blowout. Note that the main difference between the presented works cited above is how to determine the fusion parameter α in order to realize the assistance objective. All these interesting studies have motivated us to design new centralized and decentralized ADAS System architectures including the Active Front Steering (AFS) and the Direct Yaw Control (DYC) for the lane keeping purpose using steer-by-wire system and for the vehicle lateral stability improvement. However, this method can be adapted to overcome more complex maneuvers. Thus, in the present work, a new centralized LPV /H ∞ architecture of the ADAS System given in the Fig. 1, is developed to assist the driver in an intuitive way during a lane keeping maneuver while guaranteeing vehicle's stability. The paper's contributions are as follows:

• A new centralized and decentralized ADAS System architectures by using: the LPV /H ∞ and the Super-Twisting Sliding Mode (ST SM) control techniques respectively to stay within the lane and maintaining the lateral stability of the vehicle. • A decision layer based on the driver's behavior and the lateral stability index for the adjustment of different controllers.

DYC M 𝑧 Tb 𝑟𝑟 Tb 𝑟𝑙 + - β 𝑟𝑒𝑓 =0 β Fig. 1. Centralized LPV /H ∞ ADAS system architecture.
The paper structure is as follow: in Section 2 a vehicle bicycle model is given for the control synthesis. The centralized and decentralized ADAS system architectures including the different layers are detailed in the Sections 3 and 4 respectively. Then, in Section 5, the simulation results and validation for a given scenario are described to show the performance of both proposed ADAS systems. Finally, the conclusions and the perspectives for future work are given in Section 6.

CONTROL SYNTHESIS VEHICLE MODEL

The vehicle bicycle model used for control synthesis is a linear simplified LT I vehicle model. It is an LPV model with two variables being the vehicle side-slip angle β and the yaw rate ψ. This model is used to develop the different controllers of this work and it is given by the following system:

Plant P :              β = - C f +C r mV x β -(1 + l f C f -l r C r mV 2 x ) ψ + C f mV x δ sw-c , ψ = - l f C f -l r C r I z β - l 2 f C f +l 2 r C r I z V x ψ + l f C f I z δ sw-c + 1 I z M z , (1)
where β and ψ are respectively the vehicle side-slip angle and the vehicle yaw rate. m is the vehicle mass, l f ,r are the distances from the vehicle center of gravity to the front and rear axles respectively, C f ,r are the front and rear tire cornering stiffness, I z is the vehicle yaw moment of inertia, V x is the vehicle longitudinal speed and finally δ sw-c and M z are the steering angle and the yaw moment representing the AFS and the DYC inputs respectively. The state space representation of the Plant P can be formalized as in (2), where X = [β , ψ] T is the state vector, U = [δ sw-c , M z ] T is the control input.

CENTRALIZED ARCHITECTURE

This section provides a full description of the centralized multilayer ADAS architecture given in the Fig. 1 The state representation of the Plant P of the generalized plant ∑ g is formalized in (2), where X = [β , ψ] T is the state vector, the actual yaw rate ψ and the side-slip angle β are also the outputs to be controlled, U = [δ sw-c , M z ] T is the vector of control inputs.

The remaining subsystems of ∑ g i.e. the weighting functions W ψ (ρ 1 ), W β (ρ 2 ), W δ (ρ 1 ), and W M z (ρ 2 ) of Fig. 2 are defined to characterize the performance objectives Z 1 , Z 2 , and the actuators' constraints Z 3 and Z 4 . The general form of these weights [START_REF] Doumiati | Robust lpv control for vehicle steerability and lateral stability[END_REF]] is given as: -W ψ (ρ 1 ) weights the yaw rate control objective:

W ψ (ρ 1 ) = ρ 1 s/M 1 + 2π f 1 s + 2π f 1 A 1 , (3)
where M 1 is sufficiently high for a large robustness margin, and A 1 is the tolerated tracking error on e ψ . W ψ (ρ 1 ) is used to reduce the yaw rate error. W ψ (ρ 1 ) is linearly parametrized by the varying parameter ρ 1 , where ρ 1 ∈ ρ 1 ≤ ρ 1 ≤ ρ 1 (ρ 1 and ρ 1 are constants representing the lower and higher values of ρ 1 ).

When ρ 1 = ρ 1 , the performance objective e ψ is prioritized and the ADAS system (AFS) is switched-on; on the contrary, when ρ 1 = ρ 1 , e ψ is relaxed, so the driver navigates correctly and the

Ẋ = β ψ =   - C f +C r mV x -(1 + l f C f -l r C r mV 2 x ) - l f C f -l r C r I z - l 2 f C f +l 2 r C r I z V x   A β ψ + C f mV x 0 l f C f I z 1 I z B δ sw-c M z (2) Yaw rate generator 𝐾 𝐿𝑃𝑉/𝐻 ∞ (ρ 1 , ρ 2 ) Vehicle LTI Model (Plant P) ሶ Ψ 𝑟𝑒𝑓 + - 𝛅 𝑠𝑤-𝑐 ሶ Ψ 𝑍 1 𝑍 2 𝑒 ሶ Ψ 𝙒 𝛅(ρ 𝟏 ) 𝙒 ሶ Ψ(ρ 𝟏 ) Σ 𝑔 𝙒 β (ρ 𝟐 ) 𝙒 𝑀 𝑧 (ρ 𝟐 ) ሶ Ψ + - β 𝑟𝑒𝑓 =0 β 𝑍 3 𝑍 4 M 𝑧 β 𝑒 β
Fig. 2. Control layer architecture.

ADAS system (AFS) for trajectory following is switched-off.

-W β (ρ 2 ) weights the side-slip angle control objective:

W β (ρ 2 ) = 1 ρ 2 s/M 2 + 2π f 2 s + 2π f 2 A 2 . ( 4
)
where M 2 , A 2 and f 2 have similar meanings as M 1 , A 1 and

f 1 . W β (ρ 2 ) is similar to W ψ (ρ 1 ). The main difference is that W β (ρ 2
) is inversely dependent on the varying parameter ρ 2 . This is because ρ 2 is inversely related to the lateral stability index SI, that means the lateral stability is prioritized when ρ 2 = ρ 2 and vice-versa. This issue is explained later in the decision layer.

-W δ (ρ 1 ) weights the steering control input, δ sw-c :

W δ (ρ 1 ) = 1 ρ 1 s + 2π f 3 /M 3 ε u s + 2π f 3 , (5)
where M 3 is sufficiently high for a large robustness margin, ε u is concerned with the noise rejection at high frequencies and f 3 is the filter's frequency. This filter forces the steering system to act at this frequency in order to avoid driver annoyance. This filter design is inspired from Atoui et al. (2021). W δ (ρ 1 ) depends on ρ 1 , which allows to promote or penalize the steering depending on all possible situations. For instance, when driver's behavior is wrong, AFS is promoted to realize the lane keeping (when ρ 1 = ρ 1 ) and help the driver. However, if the driver acts appropriately, AFS is penalized (relaxed) (when ρ 1 = ρ 1 ) and in this case, the driver is the only responsible of driving action.

-W M z (ρ 2 ) weights the braking control input, M z :

W M z (ρ 2 ) = ρ 2 10 -3 s/(2π f 4 ) + 1 s/(κ2π f 4 ) + 1 , (6)
where f 4 is the braking actuator cut-off frequency and κ to handle the braking actuator limitations (see [START_REF] Doumiati | Integrated vehicle dynamics control via coordination of active front steering and rear braking[END_REF]).

When ρ 2 = ρ 2 , the braking control signal is promoted, on the contrary, when ρ 2 = ρ 2 , the braking input is penalized. This filter is designed depending on the vehicle's lateral stability.

After determining the subsystems of Fig. 2, the LPV /H ∞ control technique is applied to minimize the controlled outputs Z 1 , Z 2 , Z 3 and Z 4 of the generalized plant ∑ g for any exogenous input. More information about the optimal LPV /H ∞ theory is given in [START_REF] Sename | Robust control and linear parameter varying approaches: application to vehicle dynamics[END_REF]. The Matlab function "sysic" (Robust Control Toolbox) interconnects the subsystems of ∑ g . Remember that ∑ g is a LPV [Apkarian et al. (1995)] formulated as:

Σ g (ρ) : ẋ z y = A(ρ) B 1 (ρ) B 2 (ρ) C 1 (ρ) D 11 (ρ) D 12 (ρ) C 2 D 21 0 x w U , (7)
where ρ = {ρ 1 , ρ 2 }, x includes the state variables of Plant P and the weighting functions, w = [ ψre f , β re f ] T is the exogenous input vector representing the reference of the outputs, U = [δ sw-c , M z ] T represents the control inputs, y = [ ψ, β ] T is the measurement vector fed-back to the controller, and

z = [Z 1 , Z 2 , Z 3 , Z 4 ]
T is the weighted controlled output vector. Note that the matrices B 2 , and D 12 depend on ρ, which is not compatible with H ∞ requirements for polytopic systems. However, this problem is solved using some filter on the control input [Apkarian and Gahinet (1995)].

Problem resolution: LMI based LPV /H ∞ :

The LMI based LPV /H ∞ problem consists in finding the controller K LPV /H ∞ (ρ 1 , ρ 2 ), in order to minimize the H ∞ norm of the closed-loop LPV system formed by the equations ( 7) and ( 8). Note that the controller K LPV /H ∞ (ρ 1 , ρ 2 ) is given as:

K LPV /H ∞ (ρ 1 , ρ 2 ) : ẋc u = A c (ρ) B c (ρ) C c (ρ) 0 x c y , (8)
Many approaches exist in the literature to solve this problem such as: polytopic, gridding and Linear Fractional Transformation LFT [START_REF] Zin | Sur la commande robuste de suspensions automobiles en vue du contrôle global de châssis[END_REF]]. In the present work, a polytopic approach (see [START_REF] Scherer | Multiobjective output-feedback control via lmi optimization[END_REF]) has been used for controller synthesis. Using the Bounded Real Lemma (BRL) extended to LPV systems and after a change of basis presented in [START_REF] Scherer | Multiobjective output-feedback control via lmi optimization[END_REF], a non conservative LMI is formulated in (9) and a Semi-Definite Program (SDP) has been applied to solve these inequalities equations (see [START_REF] Doumiati | Integrated vehicle dynamics control via coordination of active front steering and rear braking[END_REF]]), while minimizing

γ for ρ ∈ Ω = [ρ 1 , ρ 1 ]x[ρ 2 , ρ 2 ].
The polytopic approach aims to find the A(ρ), B(ρ) and C(ρ) by using a common Lyapunov function i.e X(ρ) > 0 and Y (ρ) > 0 at each vertex of the polytope function of ρ ∈ Ω.

Noting that the number of vertex is 4 (2 n ) where n is the number of parameters ρ i . Thus, the solution is given by the resolution of system (10) at each vertex of the convex hull Ω

ω 1 = (ρ 1 , ρ 2 ), ω 2 = (ρ 1 , ρ 2 ), ω 3 = (ρ 1 , ρ 2 ), ω 4 = (ρ 1 , ρ 2 ) :        C c (ρ) = C(ρ)M(ρ) -T B c (ρ) = N(ρ) -1 B(ρ) A c (ρ) = N(ρ) -1 ( A(ρ) -Y (ρ)A(ρ)X(ρ) -N(ρ)B c (ρ). -C 2 X(ρ) -Y (ρ)B 2 (ρ)C c (ρ)M(ρ) -T )M(ρ) -T , ( 10
) where M(ρ)N(ρ) T = I -X(ρ)Y (ρ) with M(ρ) and N(ρ) are given by the user. More details about the computation solution have been presented in [START_REF] Scherer | Multiobjective output-feedback control via lmi optimization[END_REF]. Referring to the polytopic approach, the final controller K LPV /H ∞ (ρ 1 , ρ 2 ) is the weighted summation of each convex controller calculated on each vertex of the polytope [Apkarian et al. (1995)] such as:

K LPV /H ∞ (ρ 1 , ρ 2 ) = α 1 K H ∞ (ω 1 ) + α 2 K H ∞ (ω 2 ) +α 3 K H ∞ (ω 3 ) + α 4 K H ∞ (ω 4 ), (11)
    A(ρ)X + XA(ρ) T + B 2 C(ρ) + C(ρ) T B T 2 ( * ) T ( * ) T ( * ) T A(ρ) + A(ρ) T YA(ρ) + A(ρ) T Y + B(ρ)C 2 +C T 2 B(ρ) T ( * ) T ( * ) T B 1 (ρ) T B 1 (ρ) T Y + D T 21 B(ρ) T -γI ( * ) T C 1 (ρ)X + D 12 C(ρ) C 1 (ρ) D 11 (ρ) -γI     < 0 and X(ρ) I I Y (ρ) > 0. ( 9
)
where

∑ i=4 i=1 α i (ρ 1 , ρ 2 ) = 1; α i (ρ 1 , ρ 2 ) > 0.
Depending on the driving situation given in Fig. 3, the different polytopic coordinates α i (ρ 1 , ρ 2 ) weight the controller on each vertex where each vertex represents an objective, in order to build the final controller of our system. They are calculated by using the Matlab function"polydec" (Robust Control Toolbox):

α 1 = ρ 1 -ρ 1 ρ 1 -ρ 1 . ρ 2 -ρ 2 ρ 2 -ρ 2 ; α 3 = ρ 1 -ρ 1 ρ 1 -ρ 1 . ρ 2 -ρ 2 ρ 2 -ρ 2 ; α 2 = ρ 1 -ρ 1 ρ 1 -ρ 1 . ρ 2 -ρ 2 ρ 2 -ρ 2 ; α 4 = ρ 1 -ρ 1 ρ 1 -ρ 1 . ρ 2 -ρ 2 ρ 2 -ρ 2 . ( 12
)
𝐾 𝐻∞ (ω 1 )

𝐾 𝐿𝑃𝑉/𝐻∞ ρ 1 ρ 2 𝐾 𝐻∞ (ω 3 ) 𝐾 𝐻∞ (ω 2 ) 𝐾 𝐻∞ (ω 4 ) Normal driving situation Critical lateral Stability (Via DYC) ADAS is On (Via AFS) Critical lateral Stability + ADAS is On (Via AFS+ DYC) ρ 1 ρ 1 ρ 2 ρ 2 Fig. 3. Polytopic H ∞ controller
Yaw rate generator at a look-ahead distance:

The reference yaw rate generator is developed at a look-ahead distance l s in front of the vehicle, in order to generate a coherent reference yaw rate ψre f to the MIMO LPV /H ∞ controller for the trajectory following purpose. This controller aims that ψ follows ψre f in order to keep the lane with a high accuracy. For that, the reference yaw rate generator uses the current vehicle's speed V x and the information from the map matching e y-l s to calculate ψre f . Refer to [START_REF] Tan | Design of a high-performance automatic steering controller for bus revenue service based on how drivers steer[END_REF], ψre f can be approximated as:

ψre f = -2V x e y-l s l 2 s ( 13
)
where e y-l s is the vehicle lateral error at a look-ahead distance l s .

Decision Layer: ρ 1 and ρ 2 calculations

The decision layer is developed to monitor the controller according to the driving situations. This layer delivers the scheduling parameters ρ 1 and ρ 2 depending on the driver's behavior (λ ) and the lateral stability index (SI) respectively. Before the calculation of ρ 1 and ρ 2 , let us introduce the definition of both criteria λ and SI of this layer. Driver's behavior λ : λ is function of the lateral error e y of the vehicle w.r.t the trajectory at the center of gravity of the vehicle (CG) and the driver's availability (DA). DA ∈ [0, 1] is a dynamic variable related to the driver. DA=1 corresponds to a full driver's confidence.

It can be calculated based on different factors: driver's eyes analysis, driver's head position, level of driver's sleepiness, etc. Therefore, the calculation of diver's availability is not in the scope of this work and it is considered as an input given by a diagnosis module to this layer. λ is expressed as:

λ = |e y | + (1 -DA) (14)
λ has two rules depending on e y and DA, given as follows: * When λ ≤ λ , that means |e y | ≤ e y and DA=1, the ADAS system AFS is switched-off (no need to assistance). * When λ ≥ λ , that means |e y | ≥ e y and/or DA is simply low (DA=0), so the ADAS system AFS should be switched-on, in order to compensate the driver's error and unavailability.

According to this analysis, the scheduled gain ρ 1 feeds the LPV /H ∞ controller the sufficient information about the weights to be pushed or attenuated. The relation between ρ 1 and λ is presented by a "sigmoid" function (15) (see Fig. 4).

ρ 1 = ρ 1 + ρ 1 -ρ 1 1 + e -8 λ -λ (λ -λ +λ 2 ) (15)
For the ρ 2 calculation, ρ 2 depends on the lateral stability indicator SI.

Lateral stability index SI:

The Lateral stability index SI reflects the orientation of the vehicle depending on the speed vector at the center of gravity CG, and its rate of change. SI is given as (see [START_REF] Hamdan | A centralized multilayer lpv/h-infinity control architecture for vehicle's global chassis control, and comparison with a decentralized architecture[END_REF]):

SI = c 1 β + c 2 β , (16)
where c 1 and c 2 are estimated w.r.t the vehicle parameters and the shape of the road. SI is between 0 and 1 in stable driving. SI determines the driving situations. SI has two rules: * When SI ≤ SI (a predefined lower threshold depending on the road's parameters and vehicle), the vehicle is in normal driving situation (stable region) and ADAS system for stability (DYC) is penalized (not needed). * when SI ≥ SI (a predefined higher threshold), the vehicle is in critical lateral stability region (unstable region) and a stability ADAS system DYC is needed to be triggered to reestablish the lateral stability of the vehicle.

According to this discussion, ρ 2 is calculated based on the SI in order to provide the LPV /H ∞ controller information about the weights to be pushed or attenuated. The relation between ρ 2 and SI is given through a "sigmoid" function (17) (see Fig. 4) that guarantees a continuous and smooth variation of ρ 2 .

ρ 2 = ρ 2 - ρ 2 -ρ 2 1 + e -8 SI-SI (SI-SI+SI 2 ) ; ( 17
)

Actuator Layer:

The Actuator Layer contains the two actuators AFS and the DYC used to generate the physical control inputs of the system. The AFS is an electrical motor which provides the added steering angle δ a sw-c . In order to ensure that the AFS actuator is able to provide the added steering angle demanded by the controller δ sw-c , the AFS is modeled as follows: δ a sw-c = 2π f 3 (δ sw-cδ a sw-c ) (18) where δ a sw-c follows δ sw-c , f 3 is the actuator cut-off frequency. This actuator is bounded between -δ a sw-c,max , +δ a sw-c,max , with δ a sw-c,max the maximum amount of steering angle that can be added by the AFS actuator for assistance purpose. Note that the total steering wheel angle is given as:

δ total = δ sw-h + δ a sw-c
, where ρ 1 is implicitly expressed in δ a sw-c . δ sw-h is the human's steering angle, representing the human driver in-theloop, delivered by the developed Driver Model in [START_REF] Hamdan | A fuzzy logic shared steering control approach for semi-autonomous vehicle[END_REF]. Concerning the DYC actuator, the DYC moment M z can be realized by applying on the low-level a differential braking torque on one rear wheel (left or right) of radius r [START_REF] Hamdan | A centralized multilayer lpv/h-infinity control architecture for vehicle's global chassis control, and comparison with a decentralized architecture[END_REF]]. The applied braking torque is given as follows:

               T b rr = -2 * M z * r t r , T b lr = 0, i f M z ≤ 0, T b lr = 2 * M z * r t r , T b rr = 0, i f M z > 0, (19)
where T b lr and T b rr are the left and right differential braking torques respectively. A simple model for the electro mechanical braking (EMB) actuator is used. The EMB actuator is modeled as: Ṫ a b,r j = 2π f 4 (T b,r j -T a b,r j ), (20) where T a b,r j follows T b,r j , f 4 is the actuator cut-off frequency. This actuator control is bounded between [0, T a b,max ], where T a b,max is the saturation of the EMB actuator.

DECENTRALIZED ARCHITECTURE

In the decentralized approach, we decoupled the two objectives (trajectory tracking and vehicle's stability improvement) into two sub-control problems. The ST SM control technique is applied to provide the control inputs, such as: AFS is responsible on the control of the lateral error (e y ) in order to follow the trajectory; DYC is responsible on the control of side-slip angle (β ) to enhance stability. The global decentralized multilayer control architecture is shown in Fig. 5. The main difference w.r.t the centralized one is in the control layer, where each controller generates his input by neglecting the other. The inputs are: AFS control input δ sw-c dedicated to minimize the lateral error (e y ) in order to follow the trajectory, and DYC control input M z devoted to control the side-slip angle β to improve vehicle's stability. However, the decision and actuator layers are similar to the ones of the centralized approach, where the decision layer generates the two weights, called α and γ in this case; function of λ and SI respectively. The aim of these gains is to promote/attenuate both ST SM controllers depending on driver's behavior (λ ) and the lateral stability indicator (SI). Fig. 5. Decentralized ST SM ADAS system architecture.

Decentralized Control Layer synthesis: Super-twisting sliding mode controllers

The control layer consists of two Single-Input-Single-Output SISO ST SM controllers generating the control inputs. For that, the ST SM control technique detailed in [START_REF] Hamdan | A fuzzy logic shared steering control approach for semi-autonomous vehicle[END_REF] is used here to find the control inputs: δ sw-c and M z . The AFS provides the δ sw-c to minimize the lateral error e y (defined later) and realize the path following. The DYC generates the yaw moment M z to control the side-slip angle β and enhance the vehicle's lateral stability. The longitudinal movement for both architectures is realized also using a STSM controller developed in [START_REF] Hamdan | A fuzzy logic shared steering control approach for semi-autonomous vehicle[END_REF].

The ST SM-based AFS control synthesis model is similar to (2), while considering M z = 0, and thus, reducing B to its first column B 1 . The driver steering input δ sw-h is neglected in the synthesis model and it is fed-forward to the system. The ST SM-based DYC control synthesis model is similar to (2), while considering δ sw-c = 0, and thus, reducing B to its second column B 2 . The driver steering input δ sw-h is also neglected in the synthesis model. Let's recall the second order system given as: Ẋ = f (X,t) + g(X,t)u(t) (21) where X = [e y , β ] T , and f (X,t) = AX. In the case of the AFS controller synthesis g(X,t) = B 1 and u = δ sw-c . In the case of the DYC controller synthesis g(X,t) = B 2 and u = M z . Let's define E = [e y , e β ] = [yy * , β re fβ ] T the error vector between the actual and the desired states. y * and β re f = 0 are the desired lateral coordinate of the road and the desired side-slip angle respectively. e y is calculated by a Map Matching block at a look-ahead distance l s , that localizes the vehicle on the reference map. Let's define the two sliding variables for the two controllers as follows:

s e y = ėy + λ y e y , λ y > 0

s β = γ(e β + λ β ėβ ) λ β > 0 (22)
where λ y and λ β are positive constants. The sliding variables s e y and s β have a relative degree equal to one w.r.t the control inputs δ sw-c (for the lateral dynamics) and M z respectively, since B 1 and B 2 are not zero, as can be seen in the control synthesis model (2). γ (SI) is a scheduling gain which varies between 0 and 1, provided by the decision layer discussed later. When γ (SI) is equal to 1, the DYC controller is promoted. When γ (SI) is equal to 0, the DYC controller is attenuated.

Thus, in order to achieve the convergence of the sliding variables to the sliding surface defined by s = 0 and based on the discussion in [START_REF] Hamdan | A fuzzy logic shared steering control approach for semi-autonomous vehicle[END_REF], the ST SM control inputs of the AFS and the DYC are respectively given by:

δ sw-c = -α δ ,1 |s ψ,θ | τ δ sign(s ψ,θ ) -α δ ,2 t 0 sign(s ψ,θ )dτ, M z = -α M z ,1 |s β | τ Mz sign(s β ) -α M z ,2 t 0 sign(s β )dτ, (23)
where α δ ,1 and α δ ,2 (resp. α M z ,1 and α M z ,2 ) are positive gains. τ δ and τ M z are constants in ]0, 0.5]. The function sign is smoothed by the approximation sign(s) = s |s|+ε , where ε is a positive small value. The ST SM control inputs guarantee the convergence of s e y and s β in a finite time to zero. Once s e y = 0 and s β = 0 , this means that y will converge to y * and the sideslip angle β will converge to β re f respectively. Similar to the centralized approach, a decision layer of the decentralized approach monitors all the control objectives based on some monitoring criteria: λ and SI. Then, it calculates and sends instantly the values of α and γ to relax/promote the corresponding control objective depending on the vehicle situation. α depends on the driver's behavior λ . * When λ ≤ λ , that means |e y | ≤ e y and DA=1, α approaches to 0 and the ADAS system AFS is attenuated. * When λ ≥ λ , that means |e y | ≥ e y and/or DA=0, α approaches to 1 so the ADAS system AFS should be promoted to assist the driver. By the same way, γ depends on the lateral stability index SI. * When SI ≤ SI, γ approaches to 0 since no lateral stability risk and the ADAS system DYC is penalized . * When SI ≥ SI, γ approaches to 1 since the lateral stability risk is high and the ADAS system DYC is activated. "Sigmoid" functions (24) (see Fig. 6) govern the relation between α and λ and between γ and SI respectively, to ensure a continuous and smooth variation of α and γ. Concerning the actuator layer, it is the same as the centralized architecture. Note that the final steering wheel angle is given as: δ total = δ sw-h + α * δ a sw-c (see Fig. 5). Parameters Values M 1 = M 2 ; M 3 ; A 1 = A 2 ; ε u ; κ 2; 1; 0.1 = 10%; 0.01; 100 f 1 = f 2 ; f 3 = f 4 11.15 Hz; 10 Hz c 1 ; c 2 ; l s ; r;t r ; λ y ; λ β ; ε 9.55; 2.49; 3; 0.3; 0.75; 8; 0.1; 1 e y ; e y ; λ ; λ ; SI; SI; ρ 1 ; ρ 1 ; ρ 2 ; ρ 2 0.5; 0.7; 0.5; 0.7; 0.6; 0.7; 0.01; 1; 0.45; 10 α δ ,1 ; α Mz ,1 ; τ δ = τ Mz ; α δ ,2 ; α Mz ,2 0.1;1000;0.5;0.01;0.1 δ a sw-c,max ; T a b,max 5 • ; 1200 N.m

α = 1 1 + e -8 λ -λ (λ -λ +λ 2 ) ; γ = 1 1 + e -8 SI-SI (SI-SI+SI 2 ) (24) 1 0 λ λ λ 1 α 1 0 𝑆𝐼 𝑆𝐼

Simulation results

As mentioned before, this section is dedicated to validate and compare the proposed centralized MIMO LPV /H ∞ and the decentralized ST SM architectures. A scenario is defined using the track given in the Fig. 7 to test and validate the proposed controllers which aim to assist the driver in the lane keeping maneuver at speed 60 Km/h (see Fig. 13), and improve the vehicle's stability, when needed, based on the decision monitoring. To do that, two errors are injected between 35s and 50s and between 70s and 85s on the driver's behavior, where he is no more available. The vehicle is deviated from the lane and the ADAS system (represented by the centralized or decentralized controller) is activated to diminish the driver's error (through AFS) and retain the lateral stability of the vehicle (through DYC). The simulation results show the importance of having an ADAS system in the vehicle, compensating driver's error and enhancing vehicle's stability. During this scenario, a comparison is done between the vehicle without ADAS (where driver is alone), and the proposed architectures i.e the driving with ADAS LPV /H ∞ centralized and the driving with ADAS ST SM decentralized controllers. The Fig. 8 shows the lateral error of: driver without ADAS, driving with ADAS-LPV /H ∞ -centralized and driving with ADAS-ST SM-decentralized. As we can see, the errors are injected on the driver's behavior between 35s and 50s and between 70s and 85s. Both ADAS systems (LPV /H ∞ and ST SM controllers) diminish this error (initially 1m) caused by the driver on the curvy road (see Fig. 13). Thus, the two controllers have achieved the assistance goal with acceptable accuracy of lane keeping (e y between -60cm and 60cm, Fig. 8). However, Fig. 8 shows that the ADAS-ST SM-decentralized controller is capable to diminish more the lateral error e y to zero compared to the ADAS-LPV /H ∞ -centralized controller that is less performant, because the SISO SM-AFS controller realizes the lane keeping objective independently of SISO ST SM-DYC controller (see Fig. 5).

The two ADAS controllers are switched-on until t=50s, where

λ i ≥ λ (i = {1 : ADAS -LPV /H ∞ -centralized, 2 : ADAS - ST SM -decentralized})
because the driver's availability DA is equal to 0 and the lateral error is more than e y (see Fig. 9). Remember that DA reflects the driver's status (1 for available or 0 for not available). Again at t=70s, an ADAS controller is still needed because there is a second error caused by the driver's behavior and DA=0. The ADAS controllers assist the driver until t=85s. At t=85s, the driver's behavior returns normal and DA=1, then the two ADAS are switched-off.

The switching-on of the two ADAS -AFS controllers can be explained by observing the decision layer of each ADAS system architecture, in other words, the monitoring criterion λ 1 and λ 2 . Fig. 9 shows the two parameters λ 1 and λ 2 , with the corresponding scheduling parameter ρ 1 of the LPV /H ∞ -centralized, and the percentage of assistance α of the ST SM-decentralized ADAS system architecture, and, the driver's availability DA.

For λ 1 ≥ λ (resp. λ 2 ≥ λ ), this means that the driver lost control and the two ADAS -AFS controllers have switchedon to assist and help him. When λ 1 ≥ λ (resp. λ 2 ≥ λ ), that means DA = 0 and/or e y is more than e y , especially between 35s and 50s and between 70s and 85s, the scheduling gain ρ 1 of the LPV /H ∞ -centralized controller (resp. the percentage of assistance α of the ST SM-decentralized controller) is set to ρ 1 = ρ 1 (resp. α = 1), which activates the assistance. For the region, when λ 1 ≤ λ (resp. λ 2 ≤ λ ), the scheduling gain ρ 1 (resp. the percentage of assistance α) is set to ρ 1 = ρ 1 (resp. α = 0), which means the driver acts correctly and the ADAS -AFS controllers are switched-off. Thus, both ADAS -AFS controllers have almost the same behavior and they are able to help the driver by compensating his errors. On the other hand, both control architectures have similar influence on the lateral stability index SI as shown in the Fig. 10. The vehicle's lateral stability objective is achieved by the two controllers in order to prevent an undesirable driving situation (SI > 1). Fig. 10 shows the lateral SI with the corresponding scheduling parameter ρ 2 and the scheduling gain γ of both control architectures. The lateral index SI of the vehicle without ADAS (driver without ADAS) exceeds SI = 1, which means that the vehicle has lost its stability, while both control architectures have covered back the SI under SI = SI, and thus, they have succeeded to remain the vehicle stable during the γ is chosen as γ = 1) when SI ≥ SI for the lateral stability enhancement through the activation of the differential braking actuator (DYC). When SI ≤ SI, ρ 2 (resp. γ) deviates to ρ 2 = ρ 2 (resp γ = 0) which means the deactivation of the differential braking actuator (DYC), and there is no risk of lateral stability.

Concerning the centralized LPV /H ∞ ADAS system control architecture, the choice and the tuning of the parameters ρ 1 and ρ 2 (Fig. 1) is not obvious since the LPV /H ∞ controller aims to compromise between the different control objectives in order to give an optimal result. Many simulations with tuning of these parameters are done to get the optimal one. Fig. 11 shows the human driver steering angle δ sw-h , the AFS steering angle δ sw-c and the different total steering angles of both controllers. One can notice, both controllers provide almost the similar steering angle. The two controllers are in conflict with the driver between 35s and 50s and between 70s and 85s in order to compensate his errors. As we can notice, oscillations appear more with the ST SM-decentralized than the LPV /H ∞ -centralized controller. Moreover, the advantage of using ST SM control technique is that it is simple and easy to implement with a low cost. However, the point of weakness of this technique is the oscillation and chattering. Fig. 12 shows the braking of the EMB at the left and right rear wheels. The LPV /H ∞ -centralized controller activates a little bit more the braking to compromise between the different objectives, on contrary to the ST SM-decentralized controller which activates the braking only to cover back the lateral stability when necessary. Finally, Fig. 13 shows the longitudinal speed tracking to the desired one through a ST SM longitudinal controller, the road curvature of the desired trajectory and finally the lateral and longitudinal accelerations. The lateral acceleration does not exceed the ±5m/s 2 , and the actual longitudinal acceleration is pertinent (< ±2m/s 2 ) which demonstrate a comfortable and

Fig. 4 .

 4 Fig. 4. Scheduling parameters ρ 1 and ρ 2

Fig. 9 .

 9 Fig. 7. Map of the tracked trajectory

Fig. 11 .

 11 Fig. 10. SI vs ρ 2 and γ

  • A comparison between both ADAS architectures in terms of effectiveness and performance of each approach.

					λ(𝑒 𝑦 , 𝐷𝐴)	𝑆𝐼(β, ሶ β)
							Driver's behavior
							ρ 1	and stability Monitor	ρ 2	Decision Layer
	Yaw rate generator	ሶ Ψ 𝑟𝑒𝑓 ሶ Ψ	+ -		MIM𝑂 𝐿𝑃𝑉/𝐻 ∞ Centralized Controller	Control Layer
							(ρ 1 , ρ 2 )
						δ 𝑠𝑤-𝑐
					𝛅 𝑠𝑤-𝑐 𝑎	AFS	Actuator Layer
	Driver	𝛅 𝑠𝑤-ℎ	+ +	δ 𝑡𝑜𝑡𝑎𝑙	Nonlinear Vehicle Model	ሶ Ψ, β, ሶ β, 𝑉 𝑥 …
							Disturbances

Table 1 .

 1 Controllers' Parameters for Simulation

	γ
	𝑆𝐼	1
	Fig. 6. Scheduling gains α and γ
	5. SYSTEM'S VALIDATION
	The proposed centralized MIMO LPV /H ∞ and the decentral-
	ized ST SM ADAS system architectures are validated in this sec-
	tion. Validation is done by simulation using Matlab/Simulink
	with a complete nonlinear model of the vehicle, validated on
	"SCANeR Studio" (OKtal) simulator. Then, a comparison is
	done by integrating the proposed both architectures into the
	vehicle, and comparing it to a vehicle without ADAS, where
	the controllers are not implemented (driving without ADAS)
	in order to show the difference in terms of performance and
	effectiveness of each control technique. The different param-
	eters numerical values of the two controllers used during the
	simulation are given in Table 1.

CONCLUSION AND PERSPECTIVES

To conclude, in this paper two ADAS system architectures have been developed involving the Active Front Steering (AFS) and the Direct Yaw Control (DYC) to assist the driver in the Lane Keeping maneuver and to enhance the lateral stability. The development of both ADAS is done using the LPV /H ∞ and the Super-Twisting Sliding Mode (ST SM) control techniques. The different layers including the decision layer for the decision making process are detailed. The proposed (ADAS) system architectures are validated in Matlab/Simulink for a given scenario with a complete nonlinear model of the vehicle, validated on "SCANeR Studio" (OKtal) simulator. In addition, a comparison between both control strategies is done. The results show the effectiveness and the performance both approaches in terms of driver's error and undesirable driving situation prevention. In the we will consider other criteria to prove performance of the decision and validate the approaches "SCANeR Studio" simulator, connected to hardware-in-the-loop steering system. Apkarian, P. and P. (1995). A convex characterization of gain-scheduled h∞ controllers. IEEE Transactions on Automatic Control, 40(5), 853-864. Apkarian, P., Gahinet, P., and Becker, G. (1995). Self-scheduled h∞ control of linear parameter-varying systems: a design example. Automatica, 31(9), 1251-1261. Atoui, H., Milanés, V., Sename, O., and Martinez, J.J. (2021).

Real-time look-ahead distance optimization for smooth and robust steering control of autonomous vehicles. In 2021


Michel Nabaa
email: michel.nabaa@univ-lehavre.fr

Cyrille Bertelle
email: cyrille.bertelle@univ-lehavre.fr

Antoine Dutot
email: antoine.dutot@univ-lehavre.fr

Damien Olivier
email: damien.olivier@univ-lehavre.fr

Pascal Mallet
email: pascal.mallet@agglo-havraise.fr

A DYNAMIC VULNERABILITY MAP TO ASSESS THE RISK OF ROAD NETWORK TRAFFIC UTILIZATION

Keywords: Dynamic Vulnerability Map, Decision Making, Hazard, Dynamic Graph, Communities Detection, Self organization, GIS, Traffic Flow

Le Havre agglomeration (CODAH) includes 16 establishments classified Seveso 3 with high threshold. In the literature, we construct vulnerability maps to help decision makers assess the risk. Such approaches remain static and do take into account the population displacement in the estimation of the vulnerability. We propose a decision making tool based on a dynamic vulnerability map to evaluate the difficulty of evacuation in the different sectors of CODAH. We use a Geographic Information system (GIS) to visualize the map which evolves with the road traffic state through a detection of communities in large graphs algorithm.

INTRODUCTION

The population of the Seine estuary is exposed to several types of natural and industrial hazards. It is included in the drainage basin of the "Lézarde" and is also exposed to significant technological risks. The modeling and assessment of the danger is useful when it intersects with the exposed stakes. The most important factor is people. Recent events have shown that our agglomerations are vulnerable in front of emergency situations. The examination of impacted populations remains a difficult exercise. In this context, the Major Risk Management Direction team (DIRM) of Le Havre Agglomeration (CODAH) has developed a model of spatial and temporal population exposed allocation PRET-RESSE; the scale is the building [START_REF] Bourcier | Allocation spatio-temporelle de la population exposée aux risques majeurs. Contribution à l'expologie sur le bassin de risques majeurs de l'estuaire de la seine: modèle PRET-RESSE[END_REF]; thus by distinguishing their day and night occupation. The model was able to locate people during the day both in their workplace and their residence (the unemployed and retirees). Although the model was able to locate the diurnal and nocturnal population, it remains static because it does not take into account the daily movement of people and the road network utilization. For a better evacuation of people in the case of a major risk, we need to know the state of road traffic to determine how to allocate the vehicles on the road network and model the movement of these vehicles. In fact, the panic effect of some people can lead to accidents and traffic jams, which may be too grievous with a danger that spreads quickly. The panic generally results from the lack of coordination and dialogue between individuals [START_REF] Provitolo | A proposition for a classification of the catastrophe systems based on complexity criteria[END_REF]. In the literature, several models were developed to calculate a score of the vulnerability related to the road network utilization. This score may depend on social, biophysical, demographical or other aspects. Most of these models adopt a pessimistic approach to calculate this vulnerability: this case is met when a group of people in a hazardous area decide all to take the same route to evacuate this area, which unfortunately happens quite often in the real world evacuation situations. Although it helps decision-makers to estimate the risk by a census vulnerability map, this approach remains static and does not take into account the evolution of the road network traffic. In this paper, we have to simplify the representation of the population displacement, which is a complex phenomena. We also propose a dynamic and pessimistic approach related to the access to the road network. To this end, we model the road network by a dynamic graph (the dynamics is due to the traffic evolution). A simple model based on traffic flow will also be proposed. Then, we apply a self-organization algorithm to detect communities on the graph belonging to the collective intelligence algorithms. The algorithm allows us to define the different vulnerable neighborhoods of the agglomeration in the case of an evacuation due to a potential danger, while taking into account the evolution of the road network traffic. The result of this algorithm will be visualized into a GIS on a dynamic vulnerability map which categorizes various sectors depending on the difficulty of access to the road network. The map will help decisionmakers in a better estimation of risk in the communes of the CODAH. It will enrich PRET-RESS static model developed at the CODAH, taking into account the mobility of the population.

VULNERABILITY ASSESSMENT APPROACHES

Traditional methods of conception and evaluation of the population at risk do not sometimes treat the behavioral evacuee's response (e.g. initial response to an evacuation, travel speed, family interactions / group, and so on.); they describe prescriptive rules as the travel distance. These traditional methods are not very sensitive to human behavior for different emergency scenarios. The computerized models offer the potential to evaluate the evacuation of a neighborhood in emergency situations and overcome these limitations [START_REF] Castel | Developing a prototype agent-based pedestrian evacuation model to explore the evacuation of King's Cross St Pancras underground station, centre for advanced spatial analysis[END_REF]. Recently, some interesting applications have been developed by including the population dynamics, the models of urban growth patterns and land use. For computer modelers, this integration provides the ability to have computing entities as agents that are linked to real geographical locations. For GIS users, it provides the ability to model the emergence of phenomena by various interactions of agents in time and space by using a GIS [START_REF] Najlis | Repast for GIS[END_REF]). Many researchers have emphasized the need to create models of based vector Multi Agent Systems (MAS), which may require topological data structure provided in GIS [START_REF] Parker | Integration of geographic information systems and agent-based models of land use : Challenges and prospects[END_REF]). In this way, one can represent an ABM in which may coexist n levels of organizations with several classes of agents (e.g. Level 1: individuals or companies, Level 2 and 3: agents, cities, communities. . .) (Daude 2005). In [START_REF] Cutter | Revealing the vulnerability of people and places: a case study of Georgetown County, South Carolina[END_REF], the author presents a method to spatially estimate the vulnerability and treats the biophysical and social aspects (access to resources, people with evacuation special needs, people with reduced mobility ...). Several layers are created into the GIS (a layer by a danger), and all these layers are combined into one composed of intersecting polygons to build a generic vulnerability map. To complete, it was necessary to take into account the infrastructure and various possible emergency exits. So, a new map has been constructed and a new layer has been incorporated. This work has been applied to the George Town canton in which we find various natural and industrial risks, and where there are different types of people. In the neighborhood evacuation cases on a micro scale, a number of studies based on micro simulation have been developed. In their paper [START_REF] Church | Mapping evacuation risk on transportation networks using a spatial optimization model[END_REF], the authors presented a model to estimate the necessary time to evacuate a neighborhood according to the effective of the population, number of vehicles, roads capacity and number of vehicles per minute. The model is based on the optimization in order to find the critical area around a point at a potential danger in a pessimistic way. This model has been coupled with a GIS (ArcInfo) to visualize the results (identify evacuation plans) and construct an evacuation vulnerability map for the city (Santa Barbara). [START_REF] Cova | Modelling community evacuation vulnerability using GIS[END_REF] opened the way on the study based on geographic information systems to evacuate people. Their study identified the communities that may face transport difficulties during an evacuation. Research has modeled the population by lane occupation during an evacuation emergency using the city of Santa Barbara. An optimization based model (graph partitioning problem) was realized to find the neighborhood that causes the highest vulnerability around each node in the graph and a vulnerability map around nodes in the city was constructed. A constructive heuristic has been used to calculate the best cluster around each node. This heuristic was developed in C and the result was displayed on a map (with ARCINFO). Nevertheless, in this approach, we predefine the maximum number of nodes in a neighborhood, which may not always be realistic and does not take into account the traffic evolution during the calculation of critical neighborhoods. So, the vulnerable neighborhoods don't evolve according to traffic state. In our work, we try to build a dynamic vulnerability map evolving with the traffic dynamics, in which the nodes number of in a critical neighborhood, is not predetermined and can change depending on traffic state.

PROBLEMATIC

In this paper, the vulnerability is related to the access to the road network. We are persuaded that an accident on the network may cause traffic jams and therefore serious problems. So, it is important to have a pessimistic approach which takes into account the worst behavior of evacuees during a disaster (e.g. all individuals evacuate by taking the same exit route). To address this vulnerability, we have to finely represent the population and the dynamic state of road traffic. In PRET-RESSE model developed within the major risks management team of CODAH, we have ventilated the day / night population at the buildings. The model was able to locate people during the day both in their workplace and their residence (the unemployed and retirees). It has been estimated that people will be in their residence during the night. A displacement survey was also realized in CODAH agglomeration and will be included in our work. PRET-RESS will be enriched by our model to dynamically assess the vulnerability related to the road traffic evolution.

System architecture

The system consists of two related modules. The first one is dedicated to simulate the flow and apply the communities detection algorithm on the graph. The second one allows visualizing the result into a GIS. The architecture is illustrated in the following figure.

Environment modeling

The road network is integrated as a layer in the Geographic Information System (GIS). From this layer, we extract the data by using the open source java GIS toolkit Geotools. This toolkit provides several methods to manipulate geospatial data and implements Open Geospatial Consortium (OGC) specifications, so we can read and write ESRI shapefile format. Once the necessary road network data extracted, we use the GraphStream tool developed within LITIS laboratory of Le Havre to construct a graph. This tool is designed for modeling; processing and visualizing graphs. The data extracted from network layer contains the roads circulation direction, roads id, roads type, their lengths and geometry. The extracted multigraph G (t) = (V (t), E (t)) represents the road network where V (t) is the set of nodes and E (t) the set of arcs. We deal with a multigraph because there was sometimes more than one oriented arc in the same direction between two adjacent nodes due to multiple routes between two points in Le Havre road network. GraphStream facilitates this task because it is adapted to model and visualize multigraphs. In the constructed multigraph: -The nodes represent roads intersections; -The arcs represent the roads taken by vehicles; -The weight on each arc represents the needed time to cross this arc, depending on the current load of the traffic; -Dynamic aspect relates to the weights of the arcs, which can evolve in time, according to the evolution of the fluidity of circulation; we have also constructed a Voronoi tessellation (Thiessen polygon) around nodes and projected the population in buildings on these nodes. The population in buildings is extracted from PRET-RESS model.

Traffic management

For a better evacuation of people in a major risk situation, we need to know the state of the road network to determine how to allocate the vehicles on this network and to model the movement of these vehicles. Different types of models can be adopted:

-The microscopic model details the behavior of each individual vehicle by representing interactions (modeled by a car following model) with other vehicles and in general by using a spatialization. We can extend the model by adding a regulation model with priorities rules, traffic lights… Microscopic models may be applied on crowds movement as in boids collective approaches [START_REF] Reynolds | Flocks, Herds, and Schools: A Distributed Behavioral Model[END_REF]). In our problem, a microscopic model is used on the scale of a sector or a neighborhood. It has the advantage to model vehicle behavior in an evacuation of a neighborhood, people panic, interactions between vehicles, accidents… -The macroscopic model is based on the analogy between vehicular traffic and the fluid flow within a canal. It allows us to visualize the flow on the roads rather than individual vehicles. It is used at many sectors or the entire city scale. The hybrid model allows coupling the two types of dynamics flow models within the same simulation. Several works have already borrowed this direction [START_REF] Hennecke | Macroscopic simulation of open systems and micro-macro link[END_REF][START_REF] Bourrel | Mixing micro and macro representation of traffic flow: a first theoretical step[END_REF][START_REF] Magne | Toward an hybrid macro and micro traffic flow simulation model[END_REF]. However, this approach is relatively new and very few have adopted it to our knowledge [START_REF] El Hman | Simulation hybride de flux de trafic basée sur les systèmes multi-agents[END_REF]). In risk context, the use of a hybrid model is very important especially when dealing with large volume of data: changing the scale from micro to macro in a region where we haven't a crisis situation (everything is normal) allows to economize the computation and the change from macro to micro in a critical situation allows to zoom and detect the behaviors and interactions between entities in danger. In this paper, we used a simple model of macroscopic flow: -A set of flow of cars moving from one arc to another adjacent one.

-The arcs are limited in capacity of vehicles.

-The flow can be broken and two or more flows can gather on a node.

-Traffic jams may appear in certain places of the road network; those places will be more vulnerable than others. We have adopted a macroscopic model in which flows circulate normally (no accidents) because the goal now is to establish a dynamic pessimistic vulnerability map which is not always the case in the real world (90% of people takes an exit route and the rest takes another route for example). Hence, it is important to have in the near future a micro approach with a change of scale (from micro to macro and vice versa during the simulation) to simulate scenarios of danger in real time (accidents, behavior of drivers, vehicles interactions ...), a study on which we are working actually .

Complex system

A complex system is characterized by numerous entities of the same or different nature that interact in a non-trivial way (non-linear, feedback loop ...); the global emergence of new properties not seen in these entities: Properties or evolution cannot be predicted by simple calculations. CODAH can be seen as a complex system in which the environment may influence on evacuation by imposing some rules which may reduce the flow fluidity (existence or not of safe refuge and emergency exits, routes traffic direction, priorities, traffic lights...) and vice versa (a fire or an accident may cause damages and change the environment). This system is in perpetual evolution; it is far from equilibrium dynamics with an absence of any global control. Some organizations may appear or disappear according to different interactions. Entities as vehicles and pedestrians interact between them and with environment.

COMUNITIES DETECTION ALGORITHM

Our aim is to identify communities in graphs, i.e. dense areas strongly linked to each other and more weakly linked to the outside world. The concept of communities in a graph is difficult to explain formally. It can be seen as a set of nodes whose internal connections density is higher than the outside density without defining formal threshold [START_REF] Pons | Détection de structures de communautés dans les grands réseaux d'interactions[END_REF]). The goal is to find a partition of nodes in communities according to a certain predefined criteria without fixing the number of such communities or the number of nodes in a community. Radicchi [START_REF] Radicchi | Defining and identifying communities in networks[END_REF] proposes two possible definitions to quantify a community definition:

• Community C in a Strong Sense:

C i i d i d out C out C ∈ ∀ >
), ( ) ( . A node belongs to a strong community if it has more connections within the community than outside.

• Community C in a Weak Sense:

∑ ∑ ∈ ∈ ∈ ∀ > C i out C C i out C C i i d i d
), ( ) ( . A community is qualified as weak if the sum of all degrees inside is more important than the sum of degrees towards the rest of the graph. In our problem, we look for a self-organization in networks with an evolutionary algorithm close to the detection of communities in large graphs and belonging to collective intelligence algorithms. This algorithm is based on the spread of forces in graphs.

Figure 2: Communities detection in a graph

The algorithm principle is to color the graph using pheromones and it uses several colonies of ants, each of a distinct color. Each colony will collaborate to colonize zones, whereas colonies compete to maintain their own colored zone (see figure 2). Solutions will therefore emerge and be maintained by the ant behavior. The solutions will be the color of each vertex in the graph. Indeed, colored pheromones are deposited by ants on edges. The color of a vertex is obtained from the color having the largest proportion of pheromones on all incident edges. We have an interaction between each two local adjacent nodes according to the attraction force that exists between them. This force depends in our case on N/C, where N is the number of vehicles on the arc between 2 nodes neighbors and C represents vehicles capacity of the arc. This report was chosen because, in each community, we will have a large number of vehicles which all decide to exit through the same route in the case of a potential danger; this responds well to one of the purposes listed in beginning to have a pessimistic approach in the calculation of vulnerability. The algorithm has the advantage of not allowing the breaking of a link between 2 adjacent nodes to maintain the structure of the road network. When the traffic evolves, the algorithm detects that and communities can change or disappear as a result of local forces that change between the nodes locally. At each simulation time step, the flow on the arcs change following traffic conditions and the attraction forces may change also. Once communities are formed on the graph, the result will be transmitted to the road network layer into the GIS to be visualized.

CONCLUSION

In this paper, we have proposed a decision making tool to assess the danger depending on the road network use by the vehicles. This tool enables decision makers to visualize, on a geographic information system, a dynamic vulnerability map linked to the difficulty of evacuating the various streets in the metropolitan area of Le Havre agglomeration. We simulated the road network traffic by using a simple model of vehicles flow. A communities detection algorithm in the large graphs was adopted. It enabled us to form communities in a graph thanks to local force propagation rules between adjacent nodes. The communities evolve according to the current state of road network traffic. The result of the evolution of communities is visualized by using a GIS. The adopted approach allowed us to estimate the risk due to the use of the road network by vehicles and categorize Le Havre agglomeration areas by their vulnerability. We will complete our work by using a micro model of traffic with a possibility of change from micro to macro and vice versa when necessary, and this depending on the situation.

Figure 1 :

 1 Figure 1: System architecture

  edges number from a node i belonging to the community C towards the nodes of the same community. out C d is the exiting edges number from a node i belonging to the community C towards the nodes of other communities. Finding organizations is a new field of research (Albert and Barabasi, 2002; Newman, 2004a). Interesting works were developed in the literature on the detection of structure in large communities in graphs (Clauset et al. 2004, Newman 2004a; b, Pons and Latapy 2006).

Directive SEVESO is an European directive, it lays down to the states to identify potential dangerous site. It intends to prevent major accidents involving dangerous substances and limit their consequences for man and the environment, with a view to ensuring high levels of protection throughout the Community.