Abbas Chokor
email: abbas.j.chokor@gmail.com

Reine Talj

Moustapha Doumiati

Ali Hamdan

Ali Charara

A comparison between a centralized multilayer LPV /H ∞ and a decentralized multilayer sliding mode control architectures for vehicle's global chassis control

Keywords: LPV /H ∞ centralized control, sliding mode decentralized control, global chassis control

This paper investigates new achievements in global chassis control, involving Active Front steering (AFS) and Direct Yaw Control (DYC), to improve the overall vehicle performance, i.e. the vehicle maneuverability, lateral stability and rollover avoidance, in different driving situations. Two multilayer control architectures, each formed by three hierarchical layers, are developed, validated and compared. The lower layer represents the actuators implemented into the vehicle which generate their control inputs based on the orders sent from the middle layer. The middle layer is the control layer which is responsible to generate the control inputs that minimize the errors between the desired and actual vehicle yaw rate, side-slip angle, and roll angle, regardless of the driving situation. The control layer is the main difference of the proposed architectures, where a centralized and a decentralized controllers are developed. In the centralized architecture, the novelty with respect to other works in the field of chassis control is that one single Multi-Input-Multi-Output MIMO optimal controller generates the optimal additive steering angle provided by the AFS and the optimal differential braking provided by the DYC to minimize -at once-all the vehicle state errors (yaw rate, side-slip angle, and roll angle). The optimal H ∞ control technique based on offline Linear Matrix Inequality (LMI) optimal solutions, in the framework of Linear-Parameter-Varying (LPV ) systems, is applied to synthesize the controller. In the decentralized architecture, a heuristic solution is proposed by decoupling the control problem where the Super-Twisting Sliding Mode (STSM) technique is applied to derive the AFS control input which minimizes only the errors on the yaw rate, and the roll angle. Similarly, the DYC control input is privileged to minimize only the error on the side-slip angle. The higher layer of both architectures is the decision making layer which instantly monitors two criteria laying on lateral stability and rollover risks. Then, it generates two weighted parameters which adapt the controller(s) dynamics and performance(s) according to the driving conditions in order to improve the vehicle's maneuverability, lateral stability and rollover avoidance. Both control architectures are tested and validated on the professional simulator "SCANeR Studio". Simulation shows that both architectures are relevant to the global chassis control. The centralized one is optimal, complex and overall closed-loop stability is guaranteed, while the decentralized one does not guarantee the overall closed-loop stability, but it is intuitive, simple, and robust.

Introduction

Motivation

Driving safety is a major challenge for our society. According to the "National Highway Traffic Safety Administration (NHTSA)" statistics, human errors commit almost 90% of road accidents as explained in [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF]. The integration of an Advanced Driving Assistance System (ADAS) in the vehicle permits to act in an appropriate way to avoid accidents, skidding and rollover. ADAS systems are formed by several single-actuator approaches, such as: Electronic Stability Program (ESP) or Direct Yaw Control (DYC) to enhance the vehicle lateral stability; Active Front Steering (AFS) to mainly improve the vehicle maneuverability or lane keeping; and (Semi-)Active Suspensions (AS) to improve comfort (roll, pitch and heave motions attenuation), road holding and rollover avoidance. In the objective of improving the global performance of the vehicle in different driving situations, more sophisticated chassis control systems are developed in literature to create synergies between several ADAS systems. These sophisticated controllers, known as Global Chassis Control (GCC) systems have to deal with the control of complex problems for Multi-Input-Multi-Output (MIMO) systems.

Related Works

The coordination between the AFS and the DYC to improve the vehicle maneuverability and lateral stability depending on the driving situation is one of the main tasks in GCC field. Several advanced control methods have been developed for this issue. In a decentralized approach, authors in [START_REF] He | Coordination of active steering, driveline, and braking for integrated vehicle dynamics control[END_REF] have developed a DYC controller for lateral stability purpose and an AFS controller for maneuverability purpose, based on sliding mode technique, and then a monitor switches between both stand-alone controllers according to the driving situations. Similarly, based on the fuzzy-logic technique, a coordination approach between AFS and DYC has been developed in [START_REF] Karbalaei | A new intelligent strategy to integrated control of afs/dyc based on fuzzy logic[END_REF]. However, the decentralized strategy does not guarantee the overall closed-loop stability of the system when switching between controllers or when both controllers are actuated simultaneously. Authors of [START_REF] Poussot-Vassal | Robust vehicle dynamic stability controller involving steering and braking systems[END_REF], [START_REF] Doumiati | Integrated vehicle dynamics control via coordination of active front steering and rear braking[END_REF], and [START_REF] Doumiati | Robust lpv control for vehicle steerability and lateral stability[END_REF] have developed several robust and optimal MIMO centralized controllers based on LPV/H ∞ control technique, where the LPV /H ∞ controller penalizes or relaxes the steering and braking to enhance maneuverability and lateral stability. The overall closed-loop stability of the system is thus guaranteed, since the controllers' actuation is automated based on the polytopic approach. However, these controllers do not directly involve the vehicle roll motion and rollover problem in the controller synthesis. Some of them state the advantage on the rollover problem as a consequence of the controller without guaranteeing the rollover avoidance. From the other side, many recent research (centralized and decentralized) such as T. [START_REF] Vu | Enhancing roll stability of heavy vehicle by lqr active anti-roll bar control using electronic servo-valve hydraulic actuators[END_REF], [START_REF] Yao | Lateral stability control based on the roll moment distribution using a semiactive suspension[END_REF], and [START_REF] Mirzaei | Fuzzy scheduled optimal control of integrated vehicle braking and steering systems[END_REF] propose to control the vertical load transfer, as a function of the roll angle and its angular velocity to avoid rollover. They also conclude the enhancements on lateral stability as a consequence. Nevertheless, these research require the integration of new actuators into the chassis like (semi-)active suspensions or active anti-roll bar. Authors of Sename (2013), [START_REF] Chen | A new lpv/h global chassis control through load transfer distribution and vehicle stability monitoring[END_REF] and [START_REF] Chen | A new lpv/h global chassis control through load transfer distribution and vehicle stability monitoring[END_REF] have developed several powerful centralized LPV /H ∞ controllers, where the decoupled lateral and vertical vehicle dynamics are respectively controlled by the AFS+DYC and by the (semi-)active suspensions. Some other relevant research such as Ackermann and D. [START_REF] Ackermann | Robust steering control for active rollover avoidance of vehicles with elevated center of gravity[END_REF], [START_REF] Odenthal | Nonlinear steering and braking control for vehicle rollover avoidance[END_REF]), andSolmaz (2007) propose to control the roll motion by the steering and/or braking to avoid the rollover, regardless of the maneuverability and the vehicle trajectory.

All these interesting research have motivated us to study the control of the vehicle yaw rate, the side-slip angle and the roll angle to improve vehicle's maneuverability, lateral stability, and also rollover avoidance using only steering and braking actuators, by adopting both control architectures (centralized and decentralized).

Contribution

This work exposes the design and validation of two new (centralized and decentralized) multilayer control architectures to improve the vehicle maneuverability, the lateral stability and the rollover avoidance, using only steering and braking actuators. This work contributions with respect to literature are:

• a new centralized control structure, which combines the yaw rate control, the sideslip angle control, and the roll control, in one single centralized controller, ensuring internal stability when switching between maneuverability, lateral stability and rollover avoidance objectives; • a new decentralized control structure, which facilitates the global chassis control, by decoupling the control problem, into two sub-control problems, such as: AFS is responsible on the control of the yaw rate, and the roll angle; DYC is responsible on the control of the side-slip angle. Despite of the decoupling procedure, high maneuverability, lateral stability and rollover avoidance performances are guaranteed; • a comparison between both approaches using SCNeR Studio simulator.

The paper structure is as the following: Section 2 exposes the extended bicycle model which combines the coupled lateral (yaw and side-slip) and roll motions. The lateral stability and rollover criteria, as the performance criteria used in this work, are also defined in the same section. Section 3 details the centralized and the decentralized control architectures. We present first an overview of both architectures applied to the vehicle dynamics and a general methodology of comparison between both architectures. Then, we present the detailed MIMO LPV /H ∞ centralized controller structure, as a model-based controller, we also detail the control objectives represented as variable-weighted filters, to finally find the LPV /H ∞ controller which guarantees H ∞ performances between the exogenous inputs and the controlled variables, based on offline LMI optimization, in the framework of the polytopic approach. Later on, we detail both AFS and DYC ST SM controllers, as model-based controllers, where the control inputs are saturated and filtered a-posteriori. In the same section, the decision and the actuator layers of both the centralized and decentralized control architectures are presented. In Section 3.4, we test and validate both architectures on a simulation vehicle thanks to the co-simulation between Simulink and SCANeR Studio Simulator. Finally, in Section 5, we conclude on the achievements of this work, and provide a glance of our future work.

Vehicle dynamics

vehicle modeling

The vehicle is a complex system, whose behavior is strongly nonlinear. It contains several dynamical components such as wheels (acceleration, braking), steering system, suspensions, etc... which define the vehicle dynamics such as, lateral (yaw and side-slip), roll, pitch, heave and wheels motions, etc...(see Figure 1). A complete nonlinear vehicle model has already been developed and validated using the professional simulator "SCANeR Studio" Chokor 2016) and [START_REF] Chokor | Rollover prevention using active suspension system[END_REF]. This model serves to understand the details of the vehicle dynamics by performing accurate simulations. However, this complex model does not suit the control problem formulation because of several real-time problems like unknown parameters, unavailable variables, and computational issues. Thus, a linear simplified vehicle model, which only combines the coupled lateral and roll motions is inspired from literature T. [START_REF] Vu | Enhancing roll stability of heavy vehicle by lqr active anti-roll bar control using electronic servo-valve hydraulic actuators[END_REF]. This model suits the control problem of this work and it is given by the "Plant P" of the following system:

θ x ϕ y ψ z δ l f l r t f t r t f l f l r t r β ψ ⃗ V = ⃗ ẋ + ⃗ ẏ
Plant P :            I z ψ = F y f l f + F yr l r + I xz θ + M z + M d, ψ , MV β + ψ = F y f + F yr + M s h θ θ + F d,y , I x + M s h 2 θ θ = M s h θ V β + ψ + (M s gh θ -K θ )θ -C θ θ + M d,θ , (1)
where ψ, β , and θ are respectively the vehicle yaw rate, the vehicle side-slip angle, and the suspended mass roll angle. M d, ψ , F d,y , M d,θ represent the external disturbances and modeling errors respectively on the vehicle yaw rate, the lateral motion (side-slip angle) and the roll motion.M z is the active yaw moment to be generated. F y f represents the lateral force of the front left and right tires merged together at the center of the front axle. Similarly, F yr is noted for the rear axle. The front and rear tires lateral forces F y f and F yr are supposed to be linear respectively to the front and rear tires side-slip angles α f and α r , such that:

F y f = µC f α f , F yr = µC r α r , (2)
where C f and C r are the cornering stiffness of the front and rear tires. The tires slip angles α f and α r are given in the following equations:

α f = -β - l f ψ V + δ t , α r = -β + l r ψ V , (3)
where δ t = δ d + δ c is the total front wheel steering respectively composed by the driver steering and the active front steering angles. The remaining notations of these equations and the vehicle parameters used for simulation are given in Table 1. Even though these equations are valid when the vehicle operates in the stable region (no rollover or lateral stability risks), they are sufficient and recommended to synthesize a robust controller.

Ẋ =     ψ β θ θ     =    
a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 0 0 0 1 a 41 a 42 a 43 a 44

    A     ψ β θ θ     X +     b u,11 b u,12 b u,21 b u,22 0 0 b u,41 b u,42     Bu δ d + δ c M z U +     b d,11 0 0 0 b d,22 0 0 0 0 0 0 b d,43     B d   M d, ψ F d,y M d,θ   D ; y = X. (4) Ẋbic =     ψbic βbic θbic θbic     =    
a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 0 0 0 1 a 41 a 42 a 43 a 44

    A     ψbic β bic θ bic θbic     X bic +     b u,11 b u,21 0 b u,41     B u,bic δ d U bic ; y = X.
(5) By substituting (3) in (2), and then in (1), the state space representation of the Plant P can be formalized as in (4) (see next page), where

X = [ ψ, β , θ , θ ] T is the state vector, U = [δ c , Mz] T is the vector of control inputs, D = [M d, ψ , F d,y , M d,θ ]
T is the vector of exogenous inputs. The elements of the state matrix A ∈ IR 4×4 , and the input matrices B u ∈ IR 4×2 and B d ∈ IR 4×3 are formalized in Appendix A. The output variables ψ and θ are supposed to be measured and given at the CG of the vehicle, in real time control, by gyro-meters; θ is integrated from θ (θ could be directly taken from the Inertial Measurement Unit IMU if available). The side-slip angle β (and its velocity β ) could be estimated. Several observer approaches that suit the real time constraints implementation and vehicle dynamics have been proposed in literature to estimate β , e.g. an Extended Kalman Filter EKF based observer as done in [START_REF] Chen | A new lpv/h global chassis control through load transfer distribution and vehicle stability monitoring[END_REF] and [START_REF] Doumiati | Vehicle dynamics estimation using Kalman filtering: experimental validation[END_REF]. The extended linear bicycle model (5) (see next page) is similar to (4), while no control and exogenous inputs are considered (δ c = M z = M d, ψ = F d,y = M d,θ = 0), except the driver's one δ d . Based on the vehicle speed V and the driver's steering angle δ d , the extended bicycle model generates the "bicycle trajectory vector" X bic = [ ψbic , β bic , θ bic , θbic ] T . These trajectories are coherent to each other, feasible and represent the ideal stable states of the vehicle. Authors of [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF] propose to saturate ψbic , and β bic , as described in (6), to maintain the adherence between the tires and the road. This issue is done here.

| ψbic | ≤ 0.85µ g/V x , |β bic | ≤ arctan(µ g). (6)
Without loss of similarity, the trajectories provided by the extended bicycle model are used as the reference trajectories in the centralized approach, while they are slightly modified in the decentralized approach in order to introduce the weighting parameters. This issue is more developed later.

Performance criteria

The criterion by which the lateral stability can be quantified is called "lateral stability index" SI. SI reflects the orientation of the vehicle w.r.t its speed vector at the CG, and its rate of change. The lateral stability index (SI) used in Chen ( 2016) is expressed in (7) as: where q 1 and q 2 are identified depending on the vehicle parameters and road adherence. SI varies between 0 and 1. For SI ≤ SI (a predefined lower threshold depending on the vehicle and road parameters), the vehicle is in normal driving situations. When the vehicle reaches critical lateral stability, The lateral stability index SI becomes bigger than a predefined higher threshold SI (SI ≥ SI).

SI = q 1 β + q 2 β , (7)
The criterion by which the rollover risk is evaluated is called "Load Transfer Ratio" LT R. LT R reflects vertical load transfer from the inside to the outside wheels w.r.t the corner (turn). An estimation of LT R is given in (8) as a function of the roll angle and its rate of change [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF]:

LT R = r 1 θ + r 2 θ , (8)
where r 1 and r 2 are identified depending on the vehicle parameters. LT R varies between -1 and 1. When |LT R| > LT R, where LT R a positive constant higher threshold, a rollover risk is detected. Under a lower positive constant threshold LT R, the vehicle is not subjected to a rollover risk.

Centralized vs decentralized control architectures

3.1. Vehicle-related Architectures Overview

Centralized approach

The global centralized multilayer control architecture is shown in Fig. 2. In the control layer, the controlled variables i.e. the vehicle yaw rate ψ, the vehicle side-slip angle β , and the suspended mass roll angle θ are fed-back from "SCANeR Studio" vehicle and are controlled/optimized together by the optimal MIMO LPV /H ∞ centralized controller, to simul-

δ d δ d V x δ c M z θ θ bic =θ ref - + + - ψ , β , β ,θ , θ , V x ... β Decision Layer Control Layer MIMO LPV / H ∞ β bic =β ref ψ ψ bic = ψ ref + - δ c a Tb rr Tb rl Centralized Controller ρ 1 ρ 2 (ρ 1 , ρ 2 ) SI (β , β ) LTR (θ , θ )
Actuator Layer Based on all these information, the MIMO LPV /H ∞ centralized controller generates the control steering angle δ c and the corrective yaw moment M z as the control inputs, while considering actuators constraints (saturation and cut-off frequencies), to maintain the overall closed-loop stability.

Disturbances( M d , ψ , F d , y , M d ,θ ) + + δ t

Decentralized approach

The global decentralized multilayer control architecture is shown in Fig. 3. The main difference w.r.t the centralized one is in the control layer, where each control input is generated by neglecting the other. Intuitively, AFS control input δ c is devoted to control the yaw rate ψ and the roll angle θ , while DYC control input M z is privileged to control side-slip angle β , to restrain the braking actuation. Later in the paper, it is proven that each standalone controller is stable by itself, while the overall closed-loop stability problem arises because of the decoupling of control problem. However, this procedure represents a heuristic solution to facilitate the controller development complexity, by benefiting from the robustness of the super-twisting algorithm. For more simplicity, actuators constraints are not considered in the controller structure, while a posterior filter is implemented to make the control inputs feasible. The decision layer is similar in structure to the one of the centralized approach. Based on (SI) and (LT R), it generates three weighting gains λ ψ , λ β and λ θ . These gains are reduced to two because, as explained later, λ β and λ ψ are linearly dependent variables. The goal of these gains is to promote/attenuate the ST SM controllers depending on the driving situation. These gains are introduced into the reference trajectories, by modifying the extended bicycle model trajectories, to maintain each standalone controller stability (more details on the way these gains are injected in the controllers are given later).

Stability and rollover Monitor

Vehicle (SCANeR)

AFS DYC Driver Pedals δ d δ d V x δ c M z θ , θ ψ , β , β ,θ , θ , V x ...

Decision Layer

Control Layer

ψ ψ bic ,θ bic , θ bic δ c a Tb rr Tb rl λ ψ λ β SI (β , β ) LTR (θ , θ )
Actuator Layer

Disturbances ( M d , ψ , F d , y , M d ,θ ) + + δ t λ θ β ref β STSM SISO STSM Extended Bicycle Model

. Architectures comparison criteria

A general methodology to compare the centralized and decentralized architectures could contain several criteria like for example design complexity, performance, robustness, scalability, computational time, closed-loop stability, sensors and actuators fault tolerance, failure diagnosability, functional safety, etc. . . Table 2 provides some general comparison criteria between both control architectures Chen (2016), [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF]. However, this paper mainly focuses on the design of both centralized and decentralized control architectures, with global performances evaluation and robustness observations with respect to some parameter variations (vehicle speed, road adherence).

Control layers

In this sub-section, a detailed description of the control layer of both the centralized and decentralized global chassis control architectures is presented.

Centralized control layer synthesis: LPV /H ∞ controller

The control layer architecture is drawn in Fig. 4. As a standard H ∞ structure, it contains the controller K LPV /H ∞ (ρ 1 , ρ 2 ) to be synthesized, and the generalized plant ∑ g , where ρ 1 (SI) and ρ 2 (LT R) are two endogenous weighted parameters calculated by the decision making monitor to adapt the controller dynamics and performances according to the driving conditions. The controller K LPV /H ∞ (ρ 1 , ρ 2 ) has as inputs the errors between the desired trajectories and the actual ones of the yaw rate e ψ , the side-slip angle e β , and the roll angle e θ . Since the H ∞ approach is a model-based robust control technique, the actual yaw rate, side-slip angle, and roll angle are calculated based on a LTI vehicle model of (4) (Plant P). Plant P of the generalized plant ∑ g is expressed in (4). It has δ c and M z as control inputs; M d, ψ , F d,y , and M d,θ as disturbances (exogenous inputs); and the actual yaw rate ψ, side-slip angle β , and roll angle θ as outputs to be controlled. The remaining subsystems of ∑ g i.e. the weighting functions W ψ (ρ 1 ), W β (ρ 1 ), W θ (ρ 2 ), W δ (ρ 1 , ρ 2 ), and W M z (ρ 1 ) of Fig. 4 are defined to characterize the performance objectives Z 1 , Z 2 , and Z 3 and the actuators' constraints Z 4 , and Z 5 (Dynamics of the actuators, given in Subsection 3.4, are neglected during the controller design process). The general form of these weights Doumiati ( 2014) is given by the following (numerical values are given in Section 3.4, since they depend on the simulated vehicle and and checked) -W ψ (ρ 1 ) weights the yaw rate control objective:

+ - + - + - W ψ (ρ 1 ) W β (ρ 1 ) W θ ( ρ 2 ) Wδ (ρ 1 , ρ 2 ) W M z (ρ 1 ) δ c M z K LPV / H ∞ (ρ 1 , ρ 2 ) z 1 z 2 z 3 z 4 z 5 Vehicle LTI model (Plant P) ψ β θ ψ β θ ψ ref β ref θ ref e ψ e β e θ M d , ψ F d , y M d ,θ δ c M z θ Σ g
W ψ (ρ 1 ) = ρ 1 s/M 1 + 2π f 1 s + 2π f 1 A 1 , (9)
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 shaped to reduce the yaw rate error in the range of frequencies below a roll-off frequency f 1 where the vehicle operates [START_REF] Heissing | Chassis handbook: fundamentals, driving dynamics, components, mechatronics, perspectives[END_REF]. W ψ (ρ 1 ) is linearly parametrized by the varying parameter ρ 1 , where ρ 1 ∈ ρ 1 ≤ ρ ≤ 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 (maneuverability is enhanced), on the contrary, when ρ 1 = ρ 1 , e ψ is relaxed (lateral stability becomes a priority).

-W β (ρ 1 ) weights the side-slip angle control objective:

W β (ρ 1 ) = 1 ρ 1 s/M 2 + 2π f 2 s + 2π f 2 A 2 . ( 10
)
M 2 , A 2 and f 2 have similar meanings as M 1 , A 1 and f 1 . W β (ρ 1 ) is designed similarly to W ψ (ρ 1 ). The main difference is that W β (ρ 1 ) is inversely dependent on the varying parameter ρ 1 . This is because the lateral stability is more prioritized than maneuverability in critical situations. This issue is explained later in the decision layer.

-W θ (ρ 2 ) weights the roll angle control objective according to a scheduling parameter ρ 2 :

W θ (ρ 2 ) = ρ 2 s/M 3 + 2π f 3 s + 2π f 3 A 3 . (11)
M 3 , A 3 and f 3 have similar meanings as M 1 , A 1 and f 1 . W θ (ρ 2 ) is linearly parametrized by the varying parameter ρ 2 , where ρ 2 ∈ ρ 2 ≤ ρ 2 ≤ ρ 2 (ρ 2 and ρ 2 are constants representing the lower and higher values of ρ 2 ). When ρ 2 = ρ 2 , the performance objective e θ is prioritized (rollover avoidance is a priority). On the contrary, when ρ 2 = ρ 2 , e θ is relaxed (rollover is not a risk).

-W δ (ρ 1 , ρ 2 ) weights the steering control input, δ c :

W δ (ρ 1 , ρ 2 ) = ( 1 ρ 1 + 1 ρ 2 )G 0 δ (s/2π f 4 + 1)(s/2π f 5 + 1) (s/α2π f 5 + 1) 2 , G 0 δ = (∆ f /α2π f 5 + 1) 2 (∆ f /2π f 4 + 1)(∆ f /2π f 5 + 1) , ∆ f = 2π( f 4 + f 5 )/2, (12)
where [ f 4 , f 5 ] is the filter bandwidth. This filter forces the steering system to act at frequencies higher than the driver ones ( f 4 ), to avoid driver annoyance, and lower than the actuator cutoff frequency ( f 5 ). This filter design is inspired from [START_REF] Doumiati | Robust lpv control for vehicle steerability and lateral stability[END_REF]. The novelty here is the dependency of W δ (ρ 1 , ρ 2 ) on ρ 1 and ρ 2 , which allows to relax (promote) or penalize the steering depending on all possible situations. For instance, when rollover stability risk occurs, active steering is relaxed/promoted to maintain vertical stability.

-W M z (ρ 1 ) weights the braking control input, M z :

W M z (ρ 1 ) = ρ 1 10 -5 s/(2π f 6 ) + 1 s/(κ2π f 6 ) + 1 , (13)
where f 6 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 1 = ρ 1 , the braking input is penalized, on the contrary, when ρ 1 = ρ 1 , the braking control signal is relaxed. This design will be related to the vehicle lateral stability.

The controlled outputs Z 1 , Z 2 , Z 3 , Z 4 , and Z 5 have to be minimized for any exogenous input.

To do so, the powerful H ∞ control technique is applied here. See Sename (2013) and W. [START_REF] Gu | Robust control design with MATLAB R[END_REF] for more information about the robust (LPV )H ∞ theory. Interconnection between ∑ g subsystems is done using "sysic" Matlab function (Robust Control Toolbox). Since the generalized plant ∑ g is LPV Apkarian (1995), it can be 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   , (14)
where ρ = {ρ 1 , ρ 2 }, x includes the state variables of Plant P and of the weighting functions,

w = [ ψre f , β re f , θ re f , M d, ψ , F d,y , M d,θ ] T is the exogenous input vector, U = [δ c , M z ] T repre- sents the control inputs, y = [ ψ, β
, θ ] T is the measurement vector fed-back to the controller, y e = [ θ ] T is the exogenous output, and

z = [Z 1 , Z 2 , Z 3 , Z 4 , Z 5 ]
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 issue is relaxed using some filter on the control input Apkarian (1995).

Problem resolution: LMI based LPV /H ∞ : The LPV /H ∞ problem consists in finding the controller K LPV /H ∞ (ρ 1 , ρ 2 ), scheduled by the parameters ρ 1 and ρ 2 , such that:

K LPV /H ∞ (ρ) : ẋc u = A c (ρ) B c (ρ) C c (ρ) 0 x c y , (15)
which minimizes the H ∞ norm of the closed-loop LPV system formed by the interconnection of equations ( 14) and ( 15). Thanks to the Bounded Real Lemma (BRL) extended to LPV systems, this controller can be found. According to system (14) and via the change of basis expressed in [START_REF] Scherer | Multiobjective output-feedback control via lmi optimization[END_REF], a non conservative LMI that expresses the same problem as the BRL is formulated in ( 16) (see next page) and solved by a Semi-Definite Program (SDP), while minimizing γ for ρ ∈

Ω = [ρ 1 , ρ 1 ] × [ρ 2 , ρ 2 ].
The polytopic approach aims at finding A, B and C at each vertex of the polytope described by ρ ∈ Ω, by using a common Lyapunov function, i.e common X > 0 and Y > 0. Thus, the solution can be obtained by solving the system (17) at each vertex

     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; X(ρ) I I Y (ρ) > 0.
(16)

ρ 1 ρ 1 ρ 1 ρ 2 ρ 2 ρ 2 k H ∞ (ω 2 ) k H ∞ (ω 1 ) k H ∞ (ω 3 ) k H ∞ (ω 4 ) k LPV / H ∞ Critical
ω 1 = (ρ 1 , ρ 2 ), ω 2 = (ρ 1 , ρ 2 ), ω 3 = (ρ 1 , ρ 2 ), ω 4 = (ρ 1 , ρ 2 ) of the convex hull Ω:        C c (ρ) = C(ρ)M -T B c (ρ) = N -1 B(ρ) A c (ρ) = N -1 ( A(ρ) -YA(ρ)X -NB c (ρ)C 2 X -Y B 2 (ρ)C c (ρ)M -T )M -T , (17)
where M(ρ) and N(ρ) are defined by the user so that M(ρ)N(ρ) T = I -X(ρ)Y (ρ). See [START_REF] Scherer | Multiobjective output-feedback control via lmi optimization[END_REF] for more details on the computation solution.

According to the polytopic approach, the final controller, K LPV /H ∞ (ρ 1 , ρ 2 ), is a convex combination of the controllers synthesized at the vertices of the polytope Apkarian (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 ), (18)
where

∑ i=4 i=1 α i (ρ 1 , ρ 2 ) = 1; α i (ρ 1 , ρ 2 ) > 0.
The polytopic coordinates α i (ρ 1 , ρ 2 ) weight the controllers on the vertices to construct the final controller, depending on the driving situation, as shown in Fig. 5. An instant evaluation of α i (ρ 1 , ρ 2 ) is given by the following equations (the Matlab function"polydec" (Robust Control Toolbox) is also useful to evaluate polytopes with more vertices):

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 . ( 19
) Ẋ =     ψ β θ θ     =    
a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 0 0 0 1 a 41 a 42 a 43 a 44

    A     ψ β θ θ     X +     b u,11 b u,21 0 b u,41     B u,1 δ c U1 +     b d,11 0 0 0 b d,22 0 0 0 0 0 0 b d,43     B d   M d, ψ F d,y M d,θ   D ; y = X. (20) Ẋ =     ψ β θ θ     =    
a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 0 0 0 1 a 41 a 42 a 43 a 44

    A     ψ β θ θ     X +     b u,12 b u,22 0 b u,42     B u,2 M z U2 +     b d,11 0 0 0 b d,22 0 0 0 0 0 0 b d,43     B d   M d, ψ F d,y M d,θ   D ; y = X.
(21)

Decentralized control layer synthesis: Super-Twisting Sliding Mode ST SM controllers

In an intuitive way, the decentralized approach decouples the control problem into two subproblems: AFS is responsible on the control of the vehicle yaw rate ψ and the roll angle θ by neglecting the effect of the DYC on these dynamics when developing the controller; DYC which is effective to control both the vehicle yaw rate and the side-slip angle, is privileged to control only the side-slip angle β to limit its intervention, in order to prevent long braking durations which decelerate the vehicle, annoy the driver and cause long term tires wearing.

To be noted also, as a consequence of the AFS controller, the side-slip angle is enhanced in the low-to-mid range of lateral stability, while it becomes ineffective at high critical lateral dynamics.

The ST SM AFS control synthesis model ( 20) is similar to (4), while considering M z = 0, and thus, reducing B u to its first column. The driver steering input δ d is neglected in the synthesis model, and then it is considered as a feed-forward of the entire system. The ST SM DYC control synthesis model ( 21) is similar to (4), while considering δ c = 0, and thus, reducing B u to its second column. The driver steering input δ d is also neglected in the synthesis model, and then it is considered as a feed-forward of the entire system. Under these assumptions, a robust control technique, which deals with modeling uncertainties and decoupling phenomenon, is needed. Thus, the super-twisting sliding mode ST SM control technique, which is one of the most powerful robust control techniques that suit this control problem is chosen. The super-Twisting algorithm is a second order sliding mode control that handles a relative degree equal to one. It generates the continuous control function that drives the sliding variable and its derivative to zero in finite time in the presence of smooth matched disturbances. Consider the affine system form written as:

Ẍ = f (X,t) + g(X,t)u(t), (22)
where X = [ ψ, β , θ , θ ] T , and f (X,t) = AX. In the case of the AFS controller synthesis g(X,t) = B u,1 and u = δ c as can be seen from ( 20). In the case of the DYC controller synthesis g(X,t) = B u,2 and u = M z as can be seen from ( 21).

Let define E = [e ψ , e β , e θ , ėθ ] T = [ ψψre f , ββ re f , θθ re f , θθre f ] T the error vector between the actual and the desired states. ψre f , β re f , θ re f , and θre f depend on the extended bicycle model and the weighting gains sent from the decision layer. Their expressions are given later in this section.

Let define three sliding variables as the following:

s ψ = e ψ = ψ -ψre f , s β = e β = β -β re f , s θ = ėθ + k θ e θ = ( θ -θre f ) + k θ (θ -θ re f ), (23)
where s ψ (resp. s θ ) has a relative degree of 1 w.r.t the control input δ c since b u,11 (resp. b u,41 ) is not zero as can be seen in the AFS synthesis model of ( 20). Similarly, s β has a relative degree of 1 w.r.t the control input M z since b u,11 is not zero as can be seen in the DYC synthesis model of ( 21). k θ is a positive constant gain which determines the time convergence of the state errors ėθ and e θ . Since the AFS is responsible of the control of both state variables ψ and θ , let define a new sliding variable s ψ,θ such that:

s ψ,θ = c 1 s ψ + c 2 s θ , (24)
where c 1 and c 2 are positive constant weights relatively scaling the sliding variables s ψ and s θ . s ψ,θ , and s β to be controlled respectively by the AFS and the DYC, have their control inputs appear in their first derivatives (relative degree 1). This means their second derivative can be written as:

s(s,t) = Φ(s,t) + ξ (s,t) u(t) (25)
where Φ(s,t) and ξ (s,t) are unknown bounded signals.

The control objective is to achieve the convergence to the sliding surface defined by s = 0.

Only the knowledge of s is required in real time.

Suppose that there exist positive constants S 0 , b min , b max , C 0 , U max such that ∀x ∈ ℜ n and |s(x,t)| < S 0 , the system satisfies the following conditions:

   |u(t)| ≤ U max |Φ(s,t)| < C 0 0 < b min ≤ |ξ (s,t)| ≤ b max (26)
The sliding mode control law, based on the Super-Twisting algorithm, is given by:

u(t) = u 1 + u 2 u 1 = -α 1 |s| τ sign(s), τ ∈]0, 0.5] u2 = -α 2 sign(s), ( 27
)
where α 1 and α 2 are positive gains. The finite time convergence is guaranteed by the following conditions:

   α 1 ≥ 4C 0 (b max α 2 +C 0 ) b 2 min (b min α 2 -C 0 ) α 2 > C 0 b min (28)
The convergence analysis is shown in [START_REF] Utkin | On convergence time and disturbance rejection of super-twisting control[END_REF]. The function sign is smoothed by the approximation sign(s) = s |s|+ε , where ε is a positive small value.

Based on the above discussion, the STSM control inputs of the AFS and the DYC are respectively given by: (29) where α δ ,1 and α δ ,2 (resp. α M z ,1 and α M z ,2 ) are positive gains satisfying conditions of (28). τ δ and τ M z are constants in the interval ]0, 0.5]. The STSM control inputs guarantee the convergence of s ψ,θ and s β in a finite time to zero. Once s β = 0, this means that the state β is converged to β re f . Once s ψ,θ = 0, this means that s ψ = 0 and s θ = 0 because the state matrix A is Hurwitz (all eigenvalues in the left half plane). Thus, ψ converges to ψre f and ėθ + k θ e θ → 0, which means that θ (resp. θ ) exponentially converges to θ re f (resp. θre f ) if k θ > 0. As mentioned before, ψre f , β re f , θ re f , and θre f depend on the extended bicycle model and the weighting gains sent from the decision layer. Thus, let define

δ 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,
ψre f = λ ψ ψbic + (1 -λ ψ ) ψ, β re f = λ β β bic + (1 -λ β )β , θ re f = λ θ θ bic + (1 -λ θ )θ , θre f = λ θ θbic + (1 -λ θ ) θ , ( 30
)
where λ ψ (SI), λ β (SI) and λ θ (LT R) are the scheduling gains which vary between 0 and 1. Their instant values are sent from the decision layer depending on the vehicle situation. When one of these gains approaches to 1, this means that the corresponding reference trajectory is equal to the one of the extended bicycle model, thus, the corresponding controller is promoted to control the corresponding variable. When it approaches to 0, this means that the reference trajectory is equal to the actual vehicle one, thus, the control of the corresponding variable is attenuated since the corresponding sliding variable is vanished. Consequently, the sliding variables of (23) become equivalent to:

s ψ = ψ -ψre f = λ ψ ( ψ -ψbic ), s β = β -β re f = λ β (β -β bic ), s θ = ( θ -θre f ) + k θ (θ -θ re f ) = λ θ [( θ -θbic ) + k θ (θ -θ bic )], (31)
These new forms of sliding variables mean that the actual state variables are forced to converge to the extended bicycle reference model only if the scheduling gains are high (close to 1). In this case, we say that the control objective is promoted. Otherwise, the control objective is attenuated (relaxed) and the state variables remain without control. Note: The modification of the sliding variables by the multiplication with the scheduling gains maintains the closed-loop stability of the individual AFS and DYC since these gains are introduced in the reference trajectories.

Decision Layers

Centralized approach: ρ 1 and ρ 2 calculations

Once the control layer is developed, the decision layer is responsible to monitor the driving situations.

SI ρ

1 SI SI ρ 1 ρ 1 SI ρ 1 ρ 1 ρ 1 |LTR| ρ 2 LTR LTR ρ 2 ρ 2 1 1 a) b) 0 0 Figure 6.: Scheduling parameters
For SI ≤ SI, the vehicle is in normal driving situations, thus, the AFS is promoted for maneuverability purpose. It also enhances the lateral stability up to a moderate level. In this range, DYC is penalized. When the vehicle reaches critical lateral stability SI ≥ SI, then the DYC is promoted to enhance the lateral stability. Based on this analysis, the scheduled gain ρ 1 is designed to feed the LPV /H ∞ controller sufficient knowledge about the weights to be promoted or attenuated. A "sigmoid" function (32) (see Fig. 6.a) governs the relation between ρ 1 and SI, to ensure a continuous and a relatively smooth variation of ρ 1 .

ρ 1 = ρ 1 - ρ 1 -ρ 1 1 + e -8 SI-SI (SI-SI+SI 2 ) . (32)
When |LT R| > LT R, where LT R a positive constant threshold, a rollover risk is detected, and thus, the controller is informed by the scheduling parameter ρ 2 , to handle this risk. To ensure a smooth transition of ρ 2 , a lower positive constant threshold LT R is defined. A "sigmoid" function (33) (see Fig. 6.b) governs the relation between ρ 2 and |LT R|.

ρ 2 = ρ 2 + ρ 2 -ρ 2 1 + e - 8 LT R-LT R (|LT R|-LT R+LT R 2 ) . (33)
3.3.2. Decentralized approach: λ ψ , λ β and λ θ calculations Similar to the decision layer of the centralized approach, the decision layer of the decentralized approach monitors all the control objectives based on monitoring criteria (SI and LTR), then, it calculates and sends instantly the values of λ ψ , λ β and λ θ to attenuate/promote the corresponding control objective depending on the vehicle situation. λ ψ approaches to 1 when the vehicle maneuverability is the control objective. This means when SI ≤ SI. In this case, λ β approaches to 0 since no lateral stability risk is detected.

When SI ≥ SI, λ ψ approaches to 0 because the vehicle maneuverability is not a priority, while λ β approaches to 1 since the lateral stability risk is high. A "sigmoid" function (34) (Fig. 7.a) governs the relation between λ ψ (resp. λ β ) and SI, to ensure a continuous and a relatively smooth variation of λ ψ and λ β .

λ β = 1 1+e -8 SI-SI (SI- SI+SI 2 )
, By the same reasoning λ θ is related to LT R. λ θ approaches to 0 when no rollover risk is detected (LT R ≤ LT R) and approaches to 1 when rollover risk is detected (LT R ≥ LT R).

λ ψ = 1 -λ β . (34
A "sigmoid" function (35) (Fig. 7.b) governs the relation between λ θ and LT R, to ensure a continuous and a relatively smooth variation of λ θ .

λ θ = 1 1 + e - 8 LT R-LT R (LT R-LT R+LT R 2 ) , (35)

Actuator layer

In the actuator layer, the Active Front Steering AFS system is formed by an electrical motor that generates the physical additive steering called "actuator control steering" δ a c that tracks δ c . AFS provides also the mechanical link between δ a c and δ d , the driver steering angle, where the total steering δ t = δ a c + δ d (for more information refer to [START_REF] Klier | Concept and functionality of the active front steering system[END_REF]). In order to ensure that controller demand is achievable by the actuator, a simple actuator model is implemented into the control loop of the actuator layer. AFS actuator is modeled as:

δ a c = 2π f 5 (δ c -δ a c ), (36)
where f 5 is the actuator cut-off frequency. This actuator is bounded between -δ a c,max , +δ a c,max , where δ a c,max is the saturation of the AFS actuator. In order to prevent direct interference with the active steering on the front tires, the corrective yaw moment M z is allocated by the DYC to rear Electro-Mechanical Brakes EMB (right T b rr and left T b rl ) [START_REF] Doumiati | Integrated vehicle dynamics control via coordination of active front steering and rear braking[END_REF]. The DYC moment M z is generated as a braking torque T b,r j = 2r t r M z at one of the rear wheels of radius r (at the same instant), depending on the direction of M z [START_REF] Doumiati | Robust lpv control for vehicle steerability and lateral stability[END_REF]. The Electro Mechanical Braking (EMB) actuators providing T a b,r j (that tracks T b,r j ) model is given by: Ṫ a b,r j = 2π f 6 (T b,r j -T a b,r j ),

where f 6 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 actuators.

Simulation-based testing

This section is dedicated to test the proposed controllers. Testing is done on "SCANeR studio" simulator, by analyzing several vehicle variables in two scenarios: a double lane change test and a fishhook test; both at an initial speed 110 km/h. Both scenarios are considered as hard tests which solicit the vehicle lateral stability, yet the fishhook test influences more the rollover risk phenomenon since a long duration constant high steering amplitude is applied on the vehicle. In both scenarios, the comparison is done between an uncontrolled vehicle, where the controller is not implemented ("OL" as Open Loop), and by integrating the proposed controllers i.e. the centralized controller ("LPV /H ∞ ") and the decentralized controller ("SM" as Sliding Mode) into the vehicle. Several simulation tests for different scenarios have been done to select the best controller gains for both architectures. Chosen numerical values are provided in Table 3. Meanwhile, the roll angle has tracked its reference only in some regions (especially around 1 s and 3 s). These tracking results can be further explained by observing the monitoring criteria (SI and LTR) and the scheduling gains of each architecture. Therefrom, Fig. 11 and Fig. 12 respectively show the lateral stability index and the load transfer ratio with their corresponding scheduling gains of both control architectures. The lateral stability index SI of the uncontrolled vehicles (Fig. 11) 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 on almost all the time. When SI ≤ SI (almost all the time except around 1 s and 3 s), the scheduling gain ρ 1 (resp. λ ψ ) of the centralized (resp. decentralized) architecture is set to ρ 1 = ρ 1 (resp. λ ψ = 1) which accounts for the maneuver-

= M 2 = M 3 ; A 1 = A 2 = A 3 ; κ 2; 0.1 = 10%; 100 f 1 = f 2 = f 3 ; f 4 ; f 5 ; f 6
11.15 Hz; 1 Hz; 10 Hz; 10 Hz c 1 ; c 2 ; k θ 1;1;1 α δ ,1 ; τ δ ; α δ ,2 0.5;0.5;0.01 α Mz,1 ; τ Mz ; α Mz,2 500;0.5;0.1 ρ 1 ; ρ 1 ; ρ 2 ; ρ 2 70; 85; 75; 85 q 1 ; q 2 ; r 1 ; r 2 9.55; 2.49; 12; 1 SI; SI; LT R; LT R 0.6; 0.7; 0.6; 0. shows the driver steering angle δ d , the AFS steering angle of both controllers δ c , and the total steering δ t . One can notice, that both controllers provide similar steering control angles. Fig. 14 shows the braking of the EMB at the left and right rear wheels. The centralized controller activates a little bit more the braking since it is somehow useful to control the roll motion and the yaw rate, on contrary to the decentralized controller which activates the braking only to control the side-slip angle when necessary. The vehicle speed, which drops due to frictions, is improved by both controllers compared to the uncontrolled vehicle since less lost in the lateral motion is achieved as shown in Fig. 15.

Fishhook scenario

The fishhook test at high speed is a hard test used to perform extremely critical behaviors. It solicits both the lateral stability and the rollover more than in the double lane change. The test consists of turning the steering angle to one direction, keeping a high steer constant angle, then doing the same in the opposite direction. The driver steering angle is shown in Fig. 21. The uncontrolled vehicle yaw rate of Fig. 16 shows that the vehicle turns in one side, and thus, it could not well perform the scenario as the ideal motion should be (reference vehicle). Only the vehicle implemented with the centralized controller could achieve an accurate yaw rate, while the decentralized controller was able to improve the motion without high accuracy. Similar results can be observed on the side-slip angle and the roll angle respectively in Fig. 17 These figures show that the extreme behavior of the uncontrolled vehicle could be reduced but remain unstable using the decentralized controller, while the centralized controller enhances more these behavior providing a marginal stability (the lateral stability and rollover avoidance are achieved but still risky). The fluctuations of the scheduling gains show that this time the controllers are more solicited to ensure the rollover avoidance objective in the presence of the lateral stability risk. The optimality of the centralized solution was more aware about these coupled dynamics which is reflected by the AFS steering angle as can be seen from Fig. 21, where the centralized steering angle was more stable than the decentralized one. Fig. 22 shows the braking of the EMB at the left and right rear wheels. The centralized controller less activates the braking with an overall enhancement of the root mean square by 48% at the left braking, and 38% at the right braking. The peak amount of the braking is also reduced by 33% at the left wheel, and 14% at the right wheel. The vehicle speed is less dropped in the centralized approach since less braking is applied as can be seen from Fig. 23.

Simulation-based robustness evaluation

In order to evaluate the robustness of the proposed control architectures with respect to the vehicle speed and road conditions variations, the double lane change test used in Section 4.1 Figure 24 shows the vehicle yaw rate of the uncontrolled vehicle and both control architectures under three different speeds (V = 110; 140; 180 km/h), where the adherence coefficient is µ = 1. This figure shows that both control architectures are robust to the speed variation, since the vehicle yaw rate is remained lower than the uncontrolled vehicle. Furthermore, both architectures have almost identical properties in terms of speed variation robustness since curves are almost superposed. This observation is confirmed by Figure 25, where both architectures maintain the vehicle stability at the different speeds.

Figure 26 shows the vehicle yaw rate of the uncontrolled vehicle and both control architectures under three different adherence coefficients (µ = 1; 0.8; 0.5), where the vehicle speed is V = 110km/h. This figure shows that both control architectures are robust to a mid-range variation of the adherence coefficient (from µ = 1 to µ = 0.8). However, for a low adherence coefficient (µ = 0.5), only the centralized architecture has succeeded to remain the vehicle under control. This observation is confirmed by Figure 27, where only the centralized architecture has succeeded to maintain the vehicle stability at a low adherence coefficient.

CONCLUSION AND PERSPECTIVES

In this paper, decision-based centralized and decentralized global chassis control architectures have been developed to improve the vehicle maneuverability, the lateral stability and the rollover prevention, by acting on the active front steering and the differential braking. The effectiveness of the proposed architectures has been validated on SCANeR Studio simulator compared to an uncontrolled vehicle. Both the centralized and the decentralized architectures are relevant to control this complex system. It has been shown that the decentralized architecture is more simple to synthesize, while the centralized architecture handles more complex situations. From a research point of view, it seems that the centralized approach is stronger than the decentralized one especially because the global system closed-loop stability is theoretically provable. However, from an engineering point of view, the decentralized approach is more acceptable since it is more practical to develop sub-systems with local objectives, rather than thinking on the whole system where a lot of complexities arise. Besides, in general, more than one engineer work on the architecture, so that each of them can work on a part and then share blocks (model-based design approach). In other words, the engineering point of view is more realistic while the research point of view is more idealistic. Future work will focus more on the other comparison criteria and a real-time implementation and validation of the proposed architectures will be performed on the platform being developed at Heudiasyc laboratory. Moreover, a Lyapunov-based theoretical study will be investigated to prove the stability of the decentralized architecture, and, the limits and conditions of its convergence. Investigations in chassis control will continue to tolerate possible faults on the ADAS actuators and to introduce more actuators. Learning based intelligent controllers should also be developed to improve the decision making process.

  Figure 1.: Full vehicle model

Figure 2 .

 2 Figure 2.: Centralized global chassis control architecture

Figure 3 .

 3 Figure 3.: Decentralized global chassis control architecture

Figure 4 .

 4 Figure 4.: Control layer architecture

Figure 5 .

 5 Figure 5.: Controller -Polytopic approach

Figure 7 .

 7 Figure 7.: Scheduling gains λ ψ and λ β

4. 1 .

 1 Double lane change scenarioIn the double lane change test, the driver is intended to change the lane then go back to the same lane in a short duration. At high speed the vehicle is subjected to lateral stability risk. This simulated test shows the advantage of having such ADAS systems implemented into the vehicle. The driver steering angle applied on the vehicle is represented by the variable δ d on Fig.13. The remaining curves of the same figure are discussed later. The proposed control architectures monitor the vehicle situation and control its dynamics to follow, when necessary, the reference trajectories of the extended bicycle model in order to enhance the vehicle maneuverability, lateral stability and rollover avoidance. Figures8, 9, and 10 respectively show the yaw rate, the side-slip angle and the roll angle which are the controlled variables. Both control architectures (centralized and decentralized) have achieved a high accuracy of tracking the yaw rate and the side-slip angle references, compared to the uncontrolled vehicle.

Figure 12

 12 Figure 8.: Yaw rate comparison

  Figure 17.: Side-slip angle comparison

Figure 25

 25 Figure 19.: Lateral stability and scheduled gains

Figure 26

 26 Figure 26.: Robustness to adherence (µ) variation -Yaw rate comparison

Figure 27 .

 27 Figure 27.: Robustness to adherence (µ) variation -Stability Index comparison

Table 1 .

 1 : Notations and Parameters Values for Simulation

	Symbols	Description	Parameters values
	i	i = { f : f ront, r : rear}
	j	j = {r : right, l : le f t}
	θ	Sprung mass roll angle	[rad]
	φ	Sprung mass pitch angle	[rad]
	ψ	Vehicle yaw rate	[rad/s]
	β	Vehicle side-slip angle at CG	[rad]
	F y i	Lateral forces at the i axle	[N]
	δ d	Driver steering angle	[rad]
	V	Vehicle speed	[m/s]
	I x	Roll moment of inertia of sprung mass	534 [kg.m 2 ]
	I z	Vehicle yaw moment of inertia	1970 [kg.m 2 ]
	I xz	Vehicle yaw-roll product of inertia	743 [kg.m 2 ]
	h θ	Sprung mass roll arm	0.27 [m]
	M s	Sprung mass weight	1126.4 [kg]
	M	Vehicle weight	1286 [kg]
	t r	Half rear track	0.773 [m]
	l f	Wheelbase to the front	1.0385 [m]
	l r	Wheelbase to the rear	1.6015 [m]
	g	Gravity constant	9.81 [m/s 2 ]
	µ	Road adherence coefficient	dry surface= 1 [-]
	C f ,C r	Front, rear tire cornering stiffness	76776 [N/rad]
	K θ	Roll suspension angular stiffness	30000 [N.m/s]
	C θ	Roll suspension angular damper	10000 [N.m/s]

Table 2

 2

	.: Centralized Vs Decentralized Architectures
	Criteria	Centralized	Decentralized
	Performance	Very good	Good
	(including robustness)	quality	quality
	Design	Complex	Simple
	Complexity
	Scalability	Not scalable	scalable (add
		(Re-design)	sub-system)
	Computational	Slow (central	Quick (several
	time	calculator)	calculators)
	Sensors fault	Could damage	Damage only
	effect	the overall system	corresponding
			sub-system
	Actuator fault	Passive/adaptive	Active fault
	effect	fault tolerance	tolerance needed
		possible	(new configuration)
	Closed-loop	Global system	Local sub-systems
	stability
	Failure	Complex	Feasible
	diagnosability	(complexity of isolating (elementary sub-systems
		elementary sub-systems	can be monitored
		with SISO)

Table 3 .

 3 : Controllers' Parameters for Simulation

	Parameters	Values
	M 1

ACKNOWLEDGMENT

The authors would like to thank the Hauts-de-France Region and the European Regional Development Fund (ERDF) 2014/2020 for the funding of this work, through the SYSCOVI project. This work was also carried out in the framework of the Labex MS2T, (Reference ANR-11-IDEX-0004-02) and the Equipex ROBOTEX (Reference ANR-10-EQPX-44-01) which were funded by the French Government, through the program "Investments for the future" managed by the National Agency for Research.

where, (A4)