H T Sa

Anti-lock Braking System ADAS Advanced Driving Assistance Systems ADB Active Dierential Braking

Keywords: Active Braking for Collision Avoidance ABS Global Chassis Control, Hierarchical control, Centralized control, Decentralized control, Super-twisting sliding mode, LP V /H ∞

Longitudinal tire stiness

Introduction

List of Tables 1 Notations .

Passive suspension vertical force (corner ij)

[N ] F s,ij Suspension total force (corner ij)

[N ] U ij Active suspension force (corner ij) [N ] F z,ij Tire ij vertical force [N ] F x,ij Tire ij longitudinal force [N ] F y,ij
Tire ij lateral force

Publications

The contributions presented in this thesis were the subject of some published papers and other submitted ones.

Journal articles (submitted)

• A.Chokor, R. Talj, M. Doumiati, A. Hamdan, and A. Charara. A comparison between a centralized multilayer LP V /H ∞ and a decentralized multilayer sliding mode control architectures for vehicle's global chassis control. Submitted to International Journal of Control (IJC).

• A.Chokor, M. Doumiati, R. Talj, and A. Charara. Eects of roll motion control on vehicle dynamics behavior in the context of global chassis control.

Submitted to European Journal of Control (EJC).

Conference articles (accepted/published) Conference articles (submitted) ordres envoyés depuis la couche intermédiaire. La couche intermédiaire est la couche de contrôle qui est chargée de générer les entrées de contrôle qui minimisent les erreurs entre les variables d'état souhaitées et réelles du véhicule, à savoir les mouvements de lacet, de dérapage, de roulis, de tangage et de soulèvement, quelle que soit la situation de conduite. La couche supérieure est la couche de prise de décision.

Elle surveille instantanément la dynamique du véhicule selon diérents critères, puis génère des paramètres de pondération pour adapter les performances des contrôleurs en fonction des conditions de conduite, c'est-à-dire pour améliorer la manoeuvrabilité, la stabilité latérale, l'évitement du renversement et le confort de conduite du véhicule.

Les architectures proposées se dièrent dans les couches de contrôle et de décision en fonction des actionneurs intégrés proposés. Par exemple, les couches de décisions se dièrent par les critères qui surveillent la dynamique du véhicule et la manière dont la décision est prise (logique oue ou relations explicites). Les couches de con-

Abstract

Global Chassis Control (GC C) is crucial task in intelligent vehicles. It consists of assisting the driver by several automated functionalities especially for active safety and comfort purposes. Due to the fact that the dynamics of these functionalities are interconnected, thus the awaited performances are sometimes contradictory. Hence, the main task in GCC eld is to coordinate the dierent Advanced Driving Assistance Systems (ADAS) to create synergies between the interconnected dynamics in order to improve the overall vehicle performance. Several powerful coordination strategies have already been developed either in the academic world or in the industrial one to manage these interconnections. Because the active safety needs are increasing from one side, and the technology that can be embedded into vehicles is evolving, an intense research and development is still involved in the eld of global chassis control.

This thesis analyzes dierent dynamics interconnections and develops new several GCC strategies where the Active Front Steering, Active Dierential Braking, and the Active Suspensions are coordinated -all together or partially -to improve the vehicle overall performance i.e. the rollover avoidance, the lateral stability, the driving comfort (maneuverability), and the ride comfort. Several multilayer architectures formed by three hierarchical layers are proposed. 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 state variables i.e. the yaw, side-slip, roll, pitch, and heave motions, regardless of the driving situation. The higher layer is the decision making layer. It instantly monitors the vehicle dynamics by dierent criteria, then, it generates weighting parameters to adapt the controllers performances according to the driving conditions i.e. to improve the vehicle's maneuverability, lateral stability, rollover avoidance, and ride comfort.

The proposed architectures dier in the control and decision layers depending on the proposed embedded actuators. For instance, the decision layers dier in the monitored criteria and the way the decision is taken (fuzzy logic or explicit relations).

The control layers dier in structure, where centralized and decentralized controllers are developed. In the centralized architecture, one single Multi-Input-Multi-Output optimal controller generates the optimal control inputs based on the Linear Parameter Varying (LP V )/H ∞ control technique. In the decentralized architecture, the controllers are decoupled, where the Super-Twisting Sliding Mode (ST SM ) technique is applied to derive each control input apart.

The proposed architectures are tested and validated on the professional simulator xxi SCANeR Studio and on a Full vehicle nonlinear complex model. Simulation shows that all architectures are relevant to the global chassis control. The centralized one is optimal, complex and overall stability is guaranteed, while the decentralized one does not guarantee the overall stability, but it is intuitive, simple, and robust.

Introduction

Ground vehicles comprise several mechanical, hydraulic, electronic and electrical components and systems that have dierent objectives/goals. Each of these systems is involved and eects the vehicle dynamics and performance. Indeed, the suspension system, the steering system, the braking system, the power transmission system and even the tires have been evolved for several decades in order to improve the vehicle performance. In this thesis, we treat one of several manners to improve the behavior of these systems which is the development of automatically controlled systems where the electronic devices and algorithms enhance the performance of the mechanical components. These enhancements in systems behaviors are reected to the entire vehicle performance. Due to the fact that these systems are inter-connected by acting on the vehicle/chassis dynamics, creating synergies between them removes the conicts among the control objectives and retains the maximum benece of each of these systems. These synergies consist of making some or all the sub-systems cooperate together to achieve new global goals like comfort enhancement, stability enhancement, energy consumption reduction, etc...

Motivation

Driving safety is a major challenge for our society. According to the US Department of Transportation National Highway Trac Safety Administration (N HT SA)

statistics, human errors commit almost 90% of road accidents as explained in [Rajamani, 2012]. The integration of an Advanced Driving Assistance System (ADAS) or active safety system in the vehicle permits to act in an appropriate way to avoid accidents, skidding, and rollover. Moreover, nowadays ADAS systems cross over vehicle active safety to include ride and driving comfort [START_REF] Chen | Integrated vehicle dynamics and control[END_REF], [START_REF] Sename | Robust control and linear parameter varying approaches: application to vehicle dynamics[END_REF], [START_REF] Chokor | Active suspension control to improve passengers comfort and vehicle's stability[END_REF]. to improve the vehicle maneuverability or lane keeping; and Active Suspensions (ASus), Semi-Active Suspensions (Semi-ASus), and Active anti-Roll Bar (ARB) to improve comfort and road holding. There are also several other active controllers concerned with the vehicle dynamics like Active Braking for Collision Avoidance (ABC A), (adaptive-)cruise controller for speed control, etc...However, these systems 2 are not involved in this thesis.

ADAS systems, when equipped into vehicles, often operate independently, while inevitably, it occurs interactions among their local control objectives. Indeed, the vehicle motions in the vertical, lateral, and longitudinal directions are coupled together. Consequently, creating synergies among these dierent systems can exploit the existence of the interactions between the dynamics to achieve new objectives using the available ADAS.

Current and future intelligent vehicles are incorporating more sophisticated chassis control systems, known by Global Chassis Control (GC C) systems. GCC is an integrated vehicle chassis control system that coordinates several ADAS systems to improve the overall vehicle performance, including handling stability (lateral and longitudinal stability), vertical stability (rollover avoidance), and ride and driving comfort.

The noticed enhancements in the GCC approach have motivated us to study the ways these synergies can be created, either by using dierent actuators technology or by creating dierent control architectures.

This thesis is a part of the SYSCOVI (approche SYstèmes de Systèmes pour la COmmande de la dynamique de Véhicules Intelligents ) project, co-funded by the Hauts-de-France region and the European Regional Development Fund (ERDF)

2014/2020.

State of the Art 0.2.1 Global Chassis Control

In the GCC eld, there is a distinguish between two types of controllers coordination, one type is concerned with the nature of the equipped technology and the controlled objectives, and the other type concerns the form of the designed GCC architecture. In this context, several coordination strategies can be developed, either by coordinating two or more actuators to achieve one, two or more objective(s), or by designing dierent coordination architectures such as centralized, decentralized, hierarchical, and multilayer architectures, etc...

In this section, we will briey introduce the actuators, control objectives, and the dierent control architectures that are involved in the GCC. Dierent actuator's technologies can be embedded into vehicles to achieve the GCC objectives. We cite some of the active actuators that can be added and controlled:

AF S using a controlled electric motor, ADB using electro-hydraulic independent braking, ASus using electro-hydraulic cylinder, semi-ASus using controlled damper, ARB, 4-Wheel Motorized and (4W S) systems.

In this thesis, we focus on the coordination of the AF S, DY C through ADB, and the ASus. More discussions on the actuators are developed later throughout this manuscript.

The control objectives -in interest of this thesis-are the lateral stability, rollover avoidance, driving comfort (maneuverability) and ride comfort. These objectives are to be achieved by the active actuators, knowing that each of the actuator acts 0.2. STATE OF THE ART 3 either by enhancing or deteriorating one or more objective(s).

The lateral stability is the feature of handling the road without lateral skidding.

This latter can happen due to several vehicle's excitations such as: sudden steering (high steering frequency/rate) at high speed when avoiding accident, high steering amplitude at high or low speed when cornering or drifting, change of the road adherence or µ-split...

The rollover is the phenomenon when the vehicle turns around the virtual axis joining the vehicle wheels of the same side (left of right). Rollover happens either by external eect (accident, wind, bump...), or at high speed when cornering.

Rollover and lateral skidding commit the major fatal injuries [START_REF] Dot | Trac safety facts research note: summary of statistical ndings[END_REF], [Deutermann, 2002]. For that reason, rollover avoidance and lateral stability are essential objectives in lot of research concerned by GCC. These objectives were usually achieved by coordinating the AF S and the DY C, while the ASus are lately introduced for road holding and passengers comfort.

The driving comfort, the maneuverability or the steer-ability aims at enhancing the response of the vehicle when the driver steers, in the normal driving range. The steerability is the feature of making the vehicle yaw rate linear to the driver steering angle.

This linear relation makes the driver feels comfortable since the vehicle orientation becomes more predictable by the driver when steering.

The ride comfort includes the roll, pitch and heave motions of the vehicle (sprung mass). These solicitations should be controlled to enhance the passenger's comfort because the natural/induced motions due to the vehicle dynamics (lateral acceleration, longitudinal acceleration...) or external eects (bump, road irregularities...)

are not awaited despite the fact that passengers are used to be subjected to these solicitations. These solicitations have to be attenuated or controlled to predened reference trajectories.

The coordination between the AF S and the DY C to simultaneously improve the vehicle maneuverability and lateral stability depending on the driving situation is one of the main tasks in GCC eld. Indeed, an intense research is involved by the coordination of the AF S with the DY C, where 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] and [START_REF] Bardawil | Integrated vehicle lateral stability control with dierent coordination strategies between active steering and dierential braking[END_REF] have developed a DY C controller for lateral stability purpose and an AF S 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 AF S and DY C 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 stability of the system when switching between controllers or when both controllers are actuated simultaneously. [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], [START_REF] Fergani | Integrated vehicle control through the coordination of longitudinal/lateral and vertical dynamics controllers: Flatness and lpv/-based design[END_REF], [START_REF] Fergani | A new lpv/h∞ global chassis control through load transfer distribution and vehicle stability monitoring[END_REF], [Fergani et al., 2016a] and [START_REF] Doumiati | Robust lpv control for vehicle steerability and lateral stability[END_REF] have developed several robust and optimal Multi-Input-Multi-Output (M IM O) centralized controllers based on Linear Parameter Varying (LP V )/H ∞ control technique, where the LP V /H ∞ controller penalizes or relaxes the steering and braking to enhance maneuverability and lateral stability. The overall stability of the system is thus guaranteed, since the controllers' actuation is automated based on the polytopic approach. However, these controllers does 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.

The vehicle rollover is treated, by some authors, as a dierent ADAS system either by braking (dierential or normal) or by steering [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], [START_REF] Solmaz | A methodology for the design of robust rollover prevention controllers for automotive vehicles with active steering[END_REF], [START_REF] Gáspár | Recongurable control structure to prevent the rollover of heavy vehicles[END_REF]. These approaches had to be included in the GCC architecture, thus, the focus has started on creating a unied chassis controller, centralized or hierarchical, which merges all the control systems in a one single controller [START_REF] Yoon | Design of an unied chassis controller for rollover prevention, manoeuvrability and lateral stability[END_REF], [START_REF] Akhmetov | Global Chassis Control for active safety of heavy vehicles[END_REF], [START_REF] Alberding | Nonlinear hierarchical control allocation for vehicle yaw stabilization and rollover prevention[END_REF], [START_REF] Chou | Global vehicle control using dierential braking torques and active suspension forces[END_REF]. In [START_REF] Yoon | Design of an unied chassis controller for rollover prevention, manoeuvrability and lateral stability[END_REF], the authors propose a DYC controller that has the desired yaw rate to switch between two expressions, one is to enhance the lateral stability and the other one is to avoid rollover. At low level control the desired yaw rate is achieved through simple braking, which has the disadvantage of decelerating the vehicle. Authors in [START_REF] Akhmetov | Global Chassis Control for active safety of heavy vehicles[END_REF] have shown a better performance by achieving the same DY C objective through ADB.

Some other relevant research such as [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], and [START_REF] Solmaz | A methodology for the design of robust rollover prevention controllers for automotive vehicles with active steering[END_REF] propose to control the roll motion by the steering and/or braking to avoid the rollover, regardless of the maneuverability and the vehicle trajectory.

When the (semi)-ASus started to appear, the intention was to improve the ride comfort and road holding [START_REF] Chamseddine | Control of linear full vehicle active suspension system using sliding mode techniques[END_REF], [START_REF] Savaresi | Semi-active suspension control design for vehicles[END_REF], which indirectly improves the stability handling. ASus is thus the main tool to achieve passengers comfort, where the goal of this type of controllers is to isolate the chassis from any road perturbation by ensuring less roll, pitch and vertical displacement of the sprung mass [START_REF] Chokor | Active suspension control to improve passengers comfort and vehicle's stability[END_REF]. ASus also aim to improve road-holding by minimizing tires' strokes and suspensions' deections. One type of the ASus is the CRONE suspension, where the traditional suspension components i.e. the spring and the damper are replaced by a mechanical and hydro-pneumatic system dened by a fractional (so-called non-integer) order force-displacement transfer function [START_REF] Oustaloup | From fractal robustness to non-integer approach in vibration insulation: the crone suspension[END_REF]. The Crone Suspension is a powerful tool to isolate the chassis since the analysis and the design of the controller are done in the frequencydomain, where robust performance and internal stability are guaranteed [START_REF] Moreau | The crone suspension: Modelling and stability analysis[END_REF].

Later on, authors in [START_REF] Alberding | Nonlinear hierarchical control allocation for vehicle yaw stabilization and rollover prevention[END_REF] have solved the yaw stabilizing optimal control problem after introducing the rollover as a constraint on the control allocation process. Authors in [START_REF] Chou | Global vehicle control using dierential braking torques and active suspension forces[END_REF] have completely decoupled the problem of GCC stabilization, such that the yaw rate is controlled through the braking while the roll and pitch rates are controlled through the suspensions. From the other side, many recent research (centralized and decentralized)

such as [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, through the integration of the (semi-)ASus or ARB into the chassis to avoid rollover. They also conclude the enhancements on lateral stability as a consequence. [START_REF] Sename | Robust control and linear parameter varying approaches: application to vehicle dynamics[END_REF], [START_REF] Chen | Integrated vehicle dynamics and control[END_REF] and [START_REF] Fergani | A new lpv/h∞ global chassis control through load transfer distribution and vehicle stability monitoring[END_REF] have developed several powerful centralized LP V /H ∞ controllers, where the decoupled lateral and vertical vehicle dynamics are respectively controlled by the AF S+DY C and by the (semi-)ASus.

Performance Criteria -Stability and Comfort

Several criteria are dened in literature to quantify the vehicle performance, especially the rollover risk, the lateral stability, the maneuverability, and the driving comfort, while a few criteria are involved in the quantication of the riding comfort, since the ride comfort is a subjective feeling. These criteria are used either to posteriorly evaluate the performance objective or to schedule the dierent controllers to adapt their performance to the driving situation. This section introduces dierent criteria developed in literature for each of the cited objectives.

Rollover Criteria

Despite the fact that the rollover is a rare phenomenon, but its consequence are very dangerous. This is why there is a serious work either in industry or academic institutions to avoid the rollover. The vehicle rollover could happen due to external perturbation (lateral wind, bump...), in this case it is called tripped rollover, or to an internal dynamical eect such as an excess of the lateral acceleration, high frequency steering, high amplitude and rate of the roll motion, in this case it is called un-tripped rollover. In this thesis, we are interested in the un-tripped rollover whose study is given in Chapter 1. In [Dahlberg, 2000], the rollover criterion is calculated based on the maximal lateral acceleration that the vehicle could handle before starting rollover in the steady state regime. Thus, the rollover criterion is called SRT for Steady-state Rollover Threshold. In the same paper, another criterion to detect the rollover risk in the dierent dynamical regimes is provided. It is called DRT as Dynamic Rollover Threshold.

In [Dahlberg, 1999], an energetic criterion is addressed to detect the rollover. Mainly, the kinetic and potential energies of the roll motion expressed as functions of the roll rate, the roll stiness, and the heave rate are used to calculate the Dynamic Rollover Energy Margin DREM criterion.

In [START_REF] Chen | Rollover warning of articulated vehicles based on a time-to-rollover metric[END_REF], a criterion called Time To Rollover T T R predicts in the horizon of 3 seconds the roll angle of the vehicle that corresponds to the steering angle at the current time, thus, 3 seconds before rollover the driver is alerted. roll angle, roll rate, and lateral accelerations, where weighting coecients have to be identied for each vehicle. The estimated criterion is still called LT R [START_REF] Doumiati | Vehicle dynamics estimation using Kalman ltering: experimental validation[END_REF] or Rollover Index RI as in [START_REF] Yoon | Design of a rollover indexbased vehicle stability control scheme[END_REF] or Rollover Warning RW as in [START_REF] Peng | Vehicle dynamics control with rollover prevention for articulated heavy trucks[END_REF]. The LT R criterion is adopted in this thesis, its detailed formulation is provided in Chapter 1.

Lateral Stability Criteria

Vehicle lateral stability is a crucial objective in chassis control. When performing a lateral motion, the road/tires lateral forces might be saturated and thus the vehicle laterally skids causing dangerous consequences. Therefrom, researchers have done an intensive work to estimate the tires lateral forces, we cite [Hsu, 2009], [Doumiati et al., 2010a], [START_REF] Anderson | Estimation of tire cornering stiness using gps to improve model based estimation of vehicle states[END_REF], and [START_REF] Wang | Tireroad friction coefcient and tire cornering stiness estimation based on longitudinal tire force dierence generation[END_REF].

The saturation of the tires lateral forces could be caused due to an excess of the lateral acceleration beyond the maximal value that could be handled by the vehicle to maintain adherent to the road. In this context, authors of [Rajamani, 2012] propose to saturate the vehicle yaw rate at a dynamic threshold dependent on the road adherence coecient and the vehicle speed, where the lateral acceleration of the vehicle remains 15% lower than the maximal one calculated based on the adherence and the gravitational acceleration constant. In [START_REF] Yu | Potential energy analysis and limit cycle control for dynamics stability of in-wheel driving electric vehicle[END_REF],

authors propose to control the kinetic energy of the vehicle yaw rate based on the yaw-rate/side-slip phase plan and limit cycle analysis. However, the most famous criterion to evaluate the lateral stability is called Stability Index (SI ). SI is designed in several works based on the side-slip-angle/side-slip-velocity (β -β) phase plane [START_REF] Inagaki | Analysis on vehicle stability in critical cornering using phase-plane method[END_REF], [Rajamani, 2012]. The use of SI as a monitoring criterion needs the real-time estimation of the side-slip angle and velocity at the vehicle center of gravity. For this, several observers are developed in literature such as: [START_REF] Baet | Estimation of vehicle sideslip, tire force and wheel cornering stiness[END_REF], [START_REF] Chung | Design and evaluation of side slip angle-based vehicle stability control scheme on a virtual test track[END_REF], [START_REF] Doumiati | Onboard real-time estimation of vehicle lateral tireroad forces and sideslip angle[END_REF] and [START_REF] Baet | Experimental evaluation of observers for tireroad forces, sideslip angle and wheel cornering stiness[END_REF]. The SI is the criterion chosen in this thesis to evaluate and monitor the lateral stability of the vehicle, a detailed formulation of SI is provided in Chapter 1.

Driving Comfort/Maneuverability Criteria

The driving comfort, the maneuverability or the steer-ability is the feature of making the vehicle yaw rate linear to the driver steering angle. This linear relation makes the driver feels comfortable since the vehicle orientation becomes more predictable by the driver when steering, especially in the stable driving situations. Static and dynamic relations are derived in literature to enhance the maneuverability. In [START_REF] Aripin | A review of active yaw control system for 114 Bibliography vehicle handling and stability enhancement[END_REF], and [START_REF] Xiao | Integrated vehicle dynamics control through coordinating electronic stability program and active suspension system[END_REF] a static linear relation between the ideal yaw rate and the driver steering angle is given, for the steady state regime, depending on the vehicle parameters, speed, and stability factors. While the dynamic relation known as the bicycle model, provides the ideal yaw acceleration and side slip angle for a given steering angle at a given speed, then the desired yaw rate is integrated from the ideal yaw acceleration [START_REF] Sierra | Cornering stiness estimation based on vehicle lateral dynamics[END_REF], [START_REF] Smith | Eects of model complexity on the performance of automated vehicle steering controllers: Model development, validation and comparison[END_REF], [START_REF] Yoon | Design of an unied chassis controller for rollover prevention, manoeuvrability and lateral stability[END_REF], and [START_REF] Oh | The development of an advanced control method for the steerby-wire system to improve the vehicle maneuverability and stability[END_REF]. Another denition of the driving comfort is the requirement of making the active safety actuators operate in the ranges where the driver does not feel their existence. This requirement can be fullled through ltering the actuation signals as done in [START_REF] Ackermann | Yaw disturbance attenuation by robust decoupling of car steering[END_REF], [START_REF] Poussot-Vassal | Robust vehicle dynamic stability controller involving steering and braking systems[END_REF], and [START_REF] Doumiati | Integrated vehicle dynamics control via coordination of active front steering and rear braking[END_REF].

0.3. THESIS CONTRIBUTION 0.2.2.4 Ride Comfort Criteria
Ride comfort is a subjective passengers feeling. It is known as attenuating any chassis body motion i.e the sprung mass roll, pitch and heave motions at some or all frequencies -if possible- [Gillespie, 1992]. To do this, many developments have been done to isolate the vehicle body from external disturbances (like bumps) and internal disturbances caused by the vehicle dynamics interactions (like acceleration, braking, cornering...). These developments are not limited to the design of the passive suspension parameters, but also to the control of the (semi-)ASus or the ARB. Mainly, two methods are widely developed in literature for both ASus and semi-ASus: the rst one is to directly control the roll, pitch and heave motionswith their velocities-at the body center of gravity, through virtual inputs, then an allocation procedure is applied to nd the actuator forces at each vehicle quarter suspension as done in [START_REF] Chou | Global vehicle control using dierential braking torques and active suspension forces[END_REF], [START_REF] Yoon | Design of an unied chassis controller for rollover prevention, manoeuvrability and lateral stability[END_REF], and [START_REF] Vu | Enhancing roll stability of heavy vehicle by lqr active anti-roll bar control using electronic servo-valve hydraulic actuators[END_REF]; the second one is to control each quarter vehicle suspension alone, where the objective is to minimize the quarter sprung mass vibrations as done in [START_REF] Moreau | Application of the crone control-design method to a low-frequency active suspension system[END_REF], [START_REF] Savaresi | Semi-active suspension control design for vehicles[END_REF] and [START_REF] Chamseddine | Control of linear full vehicle active suspension system using sliding mode techniques[END_REF]. To evaluate the comfort, research focus on the peak values and root mean square reductions by doing time-domain tests, or in the frequency-domain by the power-spectral density measure [START_REF] Chen | Integrated vehicle dynamics and control[END_REF], [Rajamani, 2012], and [START_REF] Savaresi | Semi-active suspension control design for vehicles[END_REF].

Ride comfort is a conict objective with the road holding, this is why some research are interested in orienting the suspension control depending the suspension frequency and vehicle stability, we cite [START_REF] Moreau | Application of the crone control-design method to a low-frequency active suspension system[END_REF], and [START_REF] Savaresi | Semi-active suspension control design for vehicles[END_REF].

Thesis Contribution

The main contributions of this thesis are:

• a new ASus controller which attenuates the roll, pitch, and heave motions based on virtual inputs allocation;

• a frequency and time domain analysis of the eect (enhancement) of the roll control on the lateral stability and rollover avoidance. Then, a development of a Linear Quadratic Regulator (LQR), a Lyapunov-based, and a robust Super-Twisting Sliding Mode ST SM controllers to control the roll motion. A

comparison between the roll angle control towards zero, which can be achieved using either ASus, or semi-ASus or ARB, and its control to the new designed roll angle which can be only achieved using the ASus is also done;

• a new multilayer GCC controller which coordinates the AF S, DY C and ASus to enhance the maneuverability, lateral stability, rollover avoidance, and

ride comfort (roll, pitch, and heave motions) at once. The sub-controllers are designed based on the ST SM . The decision layer monitors the dynamics of the vehicle (driving situation), based on fuzzy logic rules and a stability criterion, to promote/attenuate the control objectives;

• a new multilayer decentralized control architecture which facilitates the GCC, by decoupling the control problem into two sub-control problems, and adopting the ST SM control technique, where AF S is responsible on the control of the yaw rate, and the roll angle, and DY C is responsible on the control of the side-slip angle, while maintaining a high maneuverability, lateral stability and rollover avoidance performances;

• a new multilayer centralized control architecture which coordinates the AF S and DY C, and combines the yaw rate control, the side-slip angle control, and the roll control, in one single M IM O LP V /H ∞ centralized controller, ensuring internal stability when switching between maneuverability, lateral stability and rollover avoidance objectives.

Thesis Outline

The thesis is organized in 5 chapters:

• Chapter 1: presents a full-vehicle nonlinear model validated on the professional simulator SCANeR Studio, which can serve as a validation model, and dierent simplied vehicle models used for control purposes. Then, rollover and lateral stability performance criteria are evoked in the same chapter.

• Chapter 2: provides a brief introduction to the control techniques that are used in this thesis i.e. the immersion and invariance as a robust Lyapunovbased technique, the ST SM (robust) technique, and the M IM O LP V /H ∞ (robust and optimal) technique.

• Chapter 3: exposes the eect of roll control through the ASus on the ride comfort, rollover avoidance, and the lateral stability based on time and frequency domains analysis. Roll reference generator is then proposed. Finally, several control laws are developed and compared together to control the roll motion through the ASus.

• Chapter 4: develops a decentralized multilayer GCC controller involving AF S, DY C, and ASus. The ST SM technique is applied to develop the control layer, while some logic and fuzzy-logic rules are developed in the decision layer to coordinate the dierent controllers.

• Chapter 5: develops and compares a centralized and a decentralized multilayer architectures for GCC, involving only AF S and DY C. showing the interaction between both dynamics.

Beside the vehicle models, the lateral stability (Stability Index SI) and the rollover avoidance (Load Transfer Ratio LT R) performance criteria dened in the Introduction Chapter are exposed.

Finally, the active actuators that will be controlled throughout this thesis i.e., the AF S, ADB, and ASus are briey introduced.

Full Vehicle Model

The literature is rich in vehicle models, more or less complex, depending on the use objective. Authors interested in vehicle stability have developed vehicle dynamics models in the horizontal plane called longitudinal and lateral models [START_REF] Villagra | A realistic vehicle model for esp-like control laws synthesis[END_REF], [Rajamani, 2012], [START_REF] Guldner | Analysis of automatic steering control for highway vehicles with look-down lateral reference systems[END_REF], [START_REF] Doumiati | Vehicle dynamics estimation using Kalman ltering: experimental validation[END_REF], and [Ray, 1997]. Others motivated by the passengers' comfort and road holding, have developed quarter, semi and/or full vertical models to describe sprung mass roll, pitch and vibrations [START_REF] Milliken | Race car vehicle dynamics[END_REF], [START_REF] Kiencke | Automotive control systems: for engine, driveline, and vehicle[END_REF], [Rajamani, 2012], and [START_REF] Chou | Global vehicle control using dierential braking torques and active suspension forces[END_REF]. Based on these models, several full vehicle model have been developed in literature which consider the interaction eect between the dierent dynamics. In this section, a full vehicle model is presented, including a description of the sub-systems (components) that inuence the most the vehicle dynamics i.e the tires, the suspension system, the steering system, the braking system, etc... These sub-models are:

• the vehicle vertical model which describes the suspension deection, the tire deection, and the sprung mass roll, pitch and heave motions;

• the vehicle longitudinal and lateral model (in the horizontal plan) which describes the longitudinal acceleration, the lateral acceleration and the vehicle yaw rate;

• the tire/road contact model;

• the wheels dynamics model.

Vertical Model

The vehicle vertical model describes a part of the vehicle dynamics, i.e. the suspension deection, the wheels bounce, and the sprung mass roll, pitch and heave motions.

1.1. FULL VEHICLE MODEL 13

The suspension system relies the sprung and unsprung masses (chassis to wheel link). The classical vehicle suspension system is shown in Figure 1.1. It is formed by a spring and a damper with constant coecients. The suspension system has two main goals: the rst one is to isolate the vehicle chassis from an uneven ground in order to improve passenger's comfort; the second one is to provide good road-holding properties, in order to ensure passenger's safety. Indeed, the suspensions help the wheels to maintain a sucient contact with the road in presence of irregularities and load transfer.

In the recent decades, automotive societies and researchers have had interest to improve passenger's comfort and the road-holding properties, either through the road prole estimation or through evolving the performance of the suspension system.

Indeed, the road prole estimation and prediction represents a useful information that can be used in the feedback signals to adapt the behavior of the new controlled types of suspensions [START_REF] Doumiati | Estimation of road prole for vehicle dynamics motion: experimental validation[END_REF], [START_REF] Doumiati | Road prole estimation using an adaptive youlaku£era parametric observer: Comparison to real prolers[END_REF], and [START_REF] Imine | Road prole inputs for evaluation of the loads on the wheels[END_REF]. From the other hand, there has been a development of new types of suspension systems, the semi-ASus and the ASus systems [Rajamani, 2012].

Figure 1.2.a shows the passive suspension system model formed by a spring and a damper which has a passive behavior due to its constant parameters (stiness and damping coecient). Generally, these parameters are selected by analyzing several evaluation criteria [Rajamani, 2012]. This type of suspensions lacks the capability of responding accurately to some frequency excitations and to adjust its behavior depending on the driving situation. Figure 1.2.b shows the semi-ASus system model which has a variable damping coecient making the suspension properties more exible to handle both comfort and road holding objectives. Mainly, there are two types of semi-ASus: the variable-orice damper and the magneto(electro)rheological (MR) damper. In the the variable-orice damper, the damping force of can be changed through controlling the diameter of the orice in the piston. In the magneto(electro)-rheological (MR) damper, the uid has rheological properties that can be changed with a controlled magnetic (electric) eld, making the damping coecient controllable. For more details about the semi-ASus refer to [START_REF] Guglielmino | Semi-active suspension control: improved vehicle ride and road friendliness[END_REF], and [START_REF] Savaresi | Semi-active suspension control design for vehicles[END_REF]. Figure 1.2.c shows the ASus system model which is formed -in addition to the spring and the damper-by a controlled hydraulic or pneumatic actuator which adds to the system a new controlled force called the active force. This type of suspension system is the most reactive with the variation of the vehicle's dynamics, but the most expensive due to its cost and energy consumption.

F s,ij = -K s,ij (z s,ij -z us,ij ) -C s,ij ( żs,ij -żus,ij ) + U ij , (1.1) zus,ij = 1 m us,ij (F z,ij -F s,ij ), (1.2) F z,ij = -K t,ij (z us,ij -z r,ij ) -C t,ij ( żus,ij -żr,ij ), (1.3) where K t,ij , C t,ij , K s,ij , C s,ij , U ij , F s,ij
, and F z,ij , are respectively, the tire stiness coecient, the tire damping coecient, the suspension stiness coecient, the suspension damping coecient, the actuator force of the ASus system (active force), the total (passive+active) suspension force, and the vertical force on the tire of the ij corner. z s,ij and z us,ij are respectively the vertical displacement of the sprung mass and the unsprung mass (wheel bounce). z r,ij and g are respectively the vertical prole of the road and the gravitational constant. z s,ij and z us,ij , with their time derivatives, are exogenous variables to this model, they are provided by the full vehicle vertical model developed in the next sub-section.

Full Vehicle Vertical Model

In order to express the vertical displacement of each corner, and the roll, pitch and heave motions of the sprung mass at the center of gravity, the full vehicle vertical model is developed (see Figure 1.4).

Let θ, φ and z s be respectively the roll, pitch and heave of the sprung mass. Geometrically, the vertical displacements of the vehicle corners z s,ij can be approximated by:

z s,f r = z s -t f sinθ -l f sinφ, (1.4) z s,f l = z s + t f sinθ -l f sinφ, (1.5) z s,rr = z s -t r sinθ + l r sinφ, (1.6) z s,rl = z s + t r sinθ + l r sinφ, (1.7)
where t f , t r , l f and l r are respectively half front track, half rear track, wheelbase to the front and wheelbase to the rear. żs,rr = żs -t r θcosθ + l r φcosφ, (1.10) żs,rl = żs + t r θcosθ + l r φcosφ.

(1.11)

The dynamic equations of the sprung mass i.e. the roll, pitch and heave (angular) accelerations ( θ, φ, and zs ) can be modeled as:

θ = 1 I x + M s h 2 θ [(-F s,f r + F s,f l ) t f + (-F s,rr + F s,rl ) t r + M s (h θ cos (θ) + z s ) a y + M s (h θ sin (θ) + z s ) g],
(1.12)

φ = 1 I y + M s h 2 φ [-(F s,f r + F s,f l ) l f + (F s,rr + F s,rl ) l r + M s (h φ cos (φ) + z s ) a x + M s (h φ sin (φ) + z s ) g], (1.13) zs = 1 M s (F s,f r + F s,f l + F s,rr + F s,rl ), (1.14)
where, M s , h θ and h φ are respectively the sprung mass mass, the distance between the center of gravity of the sprung mass and the roll rotation center, and the distance between the center of gravity of the sprung mass and the pitch rotation center. I x and I y are respectively the moment of inertia of the sprung mass around x axis and y axis. a x and a y are considered as exogenous inputs to this model, they are calculated in the lateral and longitudinal vehicle model.

CHAPTER 1. VEHICLE DYNAMICS

Longitudinal-Lateral Model

The longitudinal and lateral vehicle models can be derived by applying the second Newton law to the horizontal vehicle scheme shown in Figure 1.5. The governed equations are given by:

M a x =(F x,f l cos δ f l + F x,f r cos δ f r -F y,f l sin δ f l -F y,f r sin δ f r + F x,rl + F x,rr ), (1.15) M a y =(F x,f l sin δ f l + F x,f r sin δ f r -F y,f l cos δ f l + F y,f r cos δ f r + F y,rl + F y,rr ),
(1.16)

I z ψ = -t f (F x,f l cos δ f l -F x,f r cos δ f r -F y-f l sin δ f l + F y,f r sin δ f r ) + l f (F x,f l sin δ f l + F x,f r sin δ f r + F y,f l cos δ f l + F y,f r cos δ f r ) -l r (F y,rl + F y,rr )) -t r (F x,rl + F x,rr )),
(1.17)

where a x , a y , and ψ are respectively the longitudinal acceleration, the lateral acceleration, and the yaw acceleration. M is the total vehicle mass and I z is the moment of inertia of the vehicle around z s axis. F x,ij and F y,ij are respectively the longitudinal and lateral forces applied to the tire ij as shown in Figure 1.7.

These forces are determined (in the next sub-section) based on the tire/road contact properties and the vertical force F z,ij applied to the tire.

A virtual variable called side-slip angle β is an important variable used to determine the vehicle stability. The side-slip angle is the angle between the longitudinal direction of the vehicle and the speed vector V of the vehicle at its center of gravity as shown in Figure 1.6. The side-slip velocity can be found by projecting the tires' lateral forces to the speed vector V (Figure 1.5). The side-slip dynamics is given by the following equation:

β = 1 M V (F x,f r sin(δ f r -β) + F x,f l sin(δ f l -β) + F y,f r cos(δ f r -β) + F y,f l cos(δ f l -β) -(F x,rr + F x,rl ) sin(β) + (F y,rr + F y,rl ) cos(β)) -ψ.
(1.18)

When the vehicle operates in the linear region (lateral tire forces not saturated), the side-slip dynamics can be approximated by projecting the lateral acceleration a y to the longitudinal speed vector V x such as:

β = a y M V x -ψ.
( In literature, the tire has dierent vertical models, some researchers model the tire as a spring and a damper with constant stiness and damper coecients, while others neglect the damping coecients. Automotive societies and vehicle dynamics simulators as SCANeR Studio (OKTAL) model the tire as a spring and a damper with passive variable stiness and damping coecients, depending on the suspension deection and its velocity. In this thesis, the tire is supposed a punctual mass at its center of gravity and modeled as a spring and a damper with constant stiness and damping coecients (see Figure 1.8).

On the same time, the literature is rich in tire/road contact models which aim at describing the lateral and longitudinal tire forces as nonlinear functions of the vertical load, tire properties, and road properties. Here, we cite the well-known models:

• analytical models which are developed from the basic theory of sliding and adherence constraints, taking into account tire parameters (material, pressure, rigidity) and the environment (temperature, road nature);

• Pacejka-Baker model known by its magic formula [START_REF] Pacejka | Magic formula tyre model with transient properties[END_REF]];

• DUGOFF model [START_REF] Dugo | An analysis of tire traction properties and their inuence on vehicle dynamic performance[END_REF];

• Burckhardt/Kiencke model [Kiencke, 1993];

• LuGrue model or Carlos Canudas-de-Wit model [Canudas- [START_REF] Wit | Dynamic friction models for road/tire longitudinal interaction[END_REF].

In this thesis, the DUGOFF model is adopted because of its simplicity and computational implementation where the tire longitudinal and lateral forces can

F x,ij = C σ σ x,ij 1 -σ x,ij f (λ ij ), (1.20) F y,ij = C α tan(α ij ) 1 -σ x,ij f (λ ij ), (1.21) f (λ ij ) = (2 -λ ij )λ ij f or λ ij < 1 1 f or λ ij > 1, (1.22) λ ij = µF z,ij (1 -σ x,ij ) 2 × (C σ σ x,ij ) 2 + (C α tan(α ij )) 2
.

(1.23)

As these equations show, the lateral and longitudinal tire forces (F y,ij and F x,ij ) only depend on three parameters i.e. the longitudinal tire stiness C σ , the lateral tire stiness (cornering nesses) C α , and the road adherence µ. From DUGOFF model, F y,ij and F x,ij are nonlinear functions of the tire variables i.e. the longitudinal tire slipping σ x,ij , the side slip angle of the tire α ij , and the vertical load applied on the tire F z,ij . σ x,ij and α ij are calculated in the wheels dynamics model.

Wheels Dynamics Model

The wheel side slip angle α ij represents the deviated angle between the wheel speed vector and the wheel orientation as shown in Figure 1.9. α ij of each wheel can be obtained based on the speed vector of the vehicle at its center of gravity and the

α f r = δ f r -arctan V y + l f ψ V x + t f ψ , (1.24) α f l = δ f l -arctan V y + l f ψ V x -t f ψ , (1.25) α rr = -arctan V y -l r ψ (V x + t r ψ , (1.26) α rl = -arctan V y -l r ψ V x -t r ψ .
(1.27)

The longitudinal tire slipping represents the dierence between the actual linear wheel speed in the longitudinal direction V x,ij at its center of gravity and the expected one r ij Ω ij due to its rotation (see Figure 1.10). The longitudinal tire slipping of the tire ij is given by the following equation:

σ x,ij = r ij Ω ij -V x,ij rΩ ij acceleration r ij Ω ij -V x,ij V x,ij braking (1.28)
where r ij is the eective tire radius and Ω ij is its angular velocity. Ω ij has the dynamics given in the following equation: C m,ij is transmitted from the motor to the wheels through the power transmission system, and C f,ij is generated by the braking system.

I r Ωij = -r ij F x,ij + C m,ij -C f,ij , (1.29) 1.2. VEHICLE DYNAMICS PERFORMANCE CRITERIA 21 Ω F x, ij C m,ij C f ,ij ij
In this thesis, the motor and braking torques are considered as the inputs to the vehicle, neither the power transmission system nor the braking system are modeled, since we limit the study to the chassis dynamics. Similar for steering system, we directly consider the steering angle on the front wheels.

Before validating the full vehicle model, let develop the vehicle performance criteria based on the vehicle dynamics. Then the full vehicle model and the performance criteria will be validated using SCANeR Studio Simulator.

Vehicle Dynamics Performance Criteria

Several performance criteria are already dened in sub-section 0.2.2. This sub-section provides analytical equations of the LT R, and the lateral SI which evaluate the most critical features discussed in this thesis i.e. the rollover avoidance and the lateral stability.

LT R reects the vertical load transfer from the inside to the outside wheels w.r.t the corner (turn). the LT R is dened in [Rajamani, 2012] and described in (1.30) by:

LT R = F zr -F zl F zr + F zl , (1.30)
where F zr and F zl are respectively the vertical forces on the right and left side wheels. The rollover is supposed to start when the vehicle inner wheels lift o from ground (even if in some cases the wheels can lift o and then come back to the ground, but the full nonlinear models cannot recover these special cases which create discontinuity in the tire/road contact model). Thus, the rollover starts when F zl or F zr becomes zero, which means all the load is on the outer wheels. Hence, it occurs when the LT R = ±1. 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.

The LT R dened in (1.30) can be used only to evaluate the rollover risk a posterori by simulation, since the vertical tire forces are not measured in real time application.

An estimation of LT R, when the roll dynamics is moderate, is given in (1.31) as a linear combination of the roll angle, its rate of change and the lateral acceleration [START_REF] Yoon | Design of a rollover indexbased vehicle stability control scheme[END_REF], [Rajamani, 2012], [START_REF] Peng | Vehicle dynamics control with rollover prevention for articulated heavy trucks[END_REF]:

LT R = r 1 θ + r 2 θ + r 3 a y , (1.31)
where r 1 , r 2 and r 3 are identied depending on the vehicle parameters. The identied value of r 1 and r 2 are given later in Table 5.1, while r 3 = 0.

The estimated LT R of (1.31) is used in this thesis to monitor the rollover risk. To do so, θ is supposed to be measured by a gyrometer and given at the CG of the vehicle, in real time control; θ is integrated from θ (θ could be directly taken from the Inertial Measurement Unit IMU if available).

The lateral SI reects the orientation of the vehicle w.r.t its speed vector at the CG, and its rate of change. SI is expressed in (1.32) as [START_REF] Chen | Integrated vehicle dynamics and control[END_REF]:

SI = q 1 β + q 2 β , (1.32)
where q 1 and q 2 are identied depending on the vehicle parameters and road adherence µ to characterize the stable boundary of the β -β phase plane. SI is normalized and varies between 0 and 1. For SI ≤ SI (a predened lower threshold depending on the vehicle and road parameters), the vehicle is in normal driving situations (stable region). Up to a predened higher threshold SI, the vehicle is considered in the critical lateral stability region, where active safety controllers have to be triggered to cover back the lateral stability of the vehicle. Beyond SI the vehicle operates in the unstable region.

In order to use the SI criterion for monitoring, the side-slip angle β (and its velocity β) have to 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 | Integrated vehicle dynamics and control[END_REF] and [START_REF] Doumiati | Vehicle dynamics estimation using Kalman ltering: experimental validation[END_REF].

Full Vehicle Model Validation

The full vehicle model serves to perform several driving scenarios in order to analyze the vehicle dynamics. This model is recommended when the professional simulators or real experimentation are not available. Indeed, in this thesis, some of the vehicle controllers are validated on the full vehicle model and others on the professional simulator SCANeR Studio (this issue is discussed more later). However, this section aims at validating the full vehicle model in order to show its accuracy. Thus, the ISO 2.

I z ψ = F yf l f + F yr l r , M V β + ψ = F yf + F yr .
(1.33) F yf and F yr respectively represent the lateral force of the tire on the front axle and on the rear axle. F yf and F yr are supposed to be linear to the wheels side-slip angle such that:

F yf = µC f α f , F yr = µC r α r , (1.34)
where C f and C r are respectively the double of the front and rear tires cornering stiness. The wheels side-slip angles are found using the following equations:

α f = -β - l f ψ V + δ f , α r = -β + lr ψ V , (1.35)
where δ f is the front steering angle. Throughout of this thesis, δ f could represent δ d or δ t (where δ t = δ d + δ c ) depending on the context. To be noted also, the vehicle speed V and the longitudinal vehicle speed V x are used equivalently when the vehicle

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     +     b u,11 b u,21 0 b u,41     B 1 δ f (1.38) motion is linear.
The state space representation of the bicycle model can be written as:

ψbic βbic = -µ l f 2 c f +lr 2 cr IzVx µ lrcr-l f c f Iz -1 + µ lrcr-l f c f M Vx 2 -µ c f +cr M Vx ψbic β bic + µ l f c f Iz µ c f M Vx δ f .
(1.36)

To be mention that the bicycle model does not consider any of the vehicle vertical motion i.e. wheels bounce, sprung mass motion, load transfer, etc...

The next section presents an extended bicycle model including the sprung mass roll dynamics due to its particular eect on the lateral motion behavior as will be shown in Chapter 5.

Extended Bicycle Model

The literature [START_REF] Vu | Enhancing roll stability of heavy vehicle by lqr active anti-roll bar control using electronic servo-valve hydraulic actuators[END_REF] as the following:

I z ψ = F yf l f + F yr l r + I xz θ, M V β + ψ = F yf + F yr + M s h θ θ, (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.
(1.37)

As can be seen, the extended bicycle model is an improved version of the bicycle model where rstly the roll dynamics is described by a linear dierential equation and secondly it is included into the lateral motion equations. By substituting (1.35) in (1.34), and then in (1.37), the state space representation of the extended bicycle model can be formalized as in (1.38), where X = [ ψ, β, θ, θ] T is the state vector.

The elements of the state matrix A ∈ IR 4×4 , and the input matrix B 1 ∈ IR 4×1 are formalized in Appendix .1.

Vehicle Active Actuators

This section briey introduces the active actuators of the AF S, ADB, and ASus systems. This section serves later to generate the control actions demanded by the CHAPTER 1. VEHICLE DYNAMICS controllers.

The AF S system is formed by an electrical motor that generates the physical additive steering called actuator control steering δ a c that tracks the control steering angle δ c provided by the controller. AF S also provides 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 . However, the actuator detailed model is not in the scope of this thesis (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 loops. AF S actuator is modeled as:

δa c = 2πf 5 (δ c -δ a c ), (1.39)
where f 5 is the actuator cut-o frequency. This actuator is bounded between -δ a c,max , +δ a c,max , where δ a c,max is the saturation of the AF S actuator.

The ADB can provide the corrective yaw moment M z of the DY C. In order to prevent direct interference with the AF S on the front tires, the ADB is generated by the rear Electro-Mechanical Brakes (EM B) (right T b rr and left T b rl braking torques) [START_REF] Doumiati | Integrated vehicle dynamics control via coordination of active front steering and rear braking[END_REF]. Thus, The DY C torque M z is generated as a braking torque T b,rj = 2r

tr 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 EM B actuators providing T a b,rj (that tracks T b,rj ) model is given by: Ṫ a b,rj = 2πf 6 (T b,rj -T a b,rj ),

(1.40)

where f 6 is the actuator cut-o frequency. This actuator control is bounded between [0, T a b,max ], where T a b,max is the saturation of the EM B actuators.

The ASus actuator is a spool valve controlled electro-hydraulic component. The spool valve control makes the ASus providing at its output a force U * ij that tracks U ij sent from the ASus controller. Several works in literature have focused on the control of the electro-hydraulic actuators in order to track a desired force specied by an ASus controller [Rajamani, 2012]. However, the control of the actuator valve is not in the scope of this thesis, for the details of the dynamics and the control of the actuator please refer to [Rajamani, 2012]. In this thesis, in order to have a feasible active suspension control input, a simple actuator model is used to deal with the actuator constraints. These constraints are mainly the response time (cut-o frequency f ) and the saturation (maximal achievable force U ij,max ) of the actuator.

The actuator model is given in (1.41).

U * ij = 2πf * (min(U ij , U ij,max ) -U * ij ).
(1.41)

The actuators should be designed to be faster enough and capable to provide sucient force, to control the vehicle dynamics (cut-o frequency f , and maximal active force U ij,max ). These characteristics are generally time-varying, while a proper selection of their mean values is done in [START_REF] Chamseddine | Control of linear full vehicle active suspension system using sliding mode techniques[END_REF] where f = 10 Hz and U ij,max = 9800 N . Chapter 2

Review on Control Architectures and Techniques

This chapter briey reviews some of the control architectures and the robust and/or optimal control techniques existing in literature that are applied in GCC led. We

Introduction

A crucial task in control system engineering is the choice of the most suitable control technique. One should carefully study the characteristics of the system to be controlled, i.e., the representativeness of the model (how much precise or uncertain), the complexity of this model (number of inputs/outputs, monovariable or multivariable, types of parameters(constant or variant), interconnection between dynamics (linear, nonlinear, ordinary and partial dierential equations)), and the computational time for real world experiments.

As can be seen from the full vehicle model of Section 1.1, the vehicle is a complex multivariable nonlinear system, whose control is relatively dicult compared to other dynamical systems. The literature is rich in chassis control systems, either to control one, a part, or all the chassis variables through one or several control inputs.

Therefrom comes the notion of decentralized and centralized control [START_REF] Chen | Integrated vehicle dynamics and control[END_REF]. Indeed, in the decentralized control, the control researcher/engineer selects, for each set of control variables (one variable or more), one control input (generally, the most inuencing one) and neglects the eect of the other inputs on the variables of interest. Then, for each combination (variables-input), a specic control technique is applied in order to achieve a precise reference/equilibrium tracking in presence of model uncertainties, neglected inputs, and eventual external perturbations. For

CHAPTER 2. REVIEW ON CONTROL ARCHITECTURES AND TECHNIQUES

the case where only one variable is controlled through one input, this part of the control system is said SISO or monovariable. The cooperation between the control inputs is generally achieved in a higher layer, the decision layer, which is responsible to determine the actual/future vehicle operation situation. Thus, the control architecture is said multilayer. On the opposite side, in the centralized control, the control inputs cooperate all together to control all the variables of interest. The controlled part of the control system is said M IM O or multivariable.

Meanwhile, a higher layer, the decision layer, monitors the vehicle situation to adjust the controller dynamics to the encountered situation.

Multilayer Centralized vs Multilayer Decentralized Control Architectures

The multilayer control architecture consists of decomposing the controller design into several layers as shown in Therefore, both the advantages and disadvantages are obvious. The centralized architecture has the advantages of controlling and observing all the subsystems in an integrated manner. This means, control actions are more precise, optimal, and less conicting. However, the disadvantages cannot be ignored: the curse of dimensionality caused by the increasing number of subsystems results in tremendous design diculties. Moreover, the failure of the centralized controller inevitably leads to a total failure of the whole chassis control system. Finally, when the centralized architecture needs to include more required subsystems, the entire centralized architecture has to be redesigned since the architecture lacks exibility. [START_REF] Canale | [END_REF]Fagiano, 2008] [Falcone et al., 2007b] [ Falcone et al., 2007a], random sub-optimal control [START_REF] Chen | Integrated control of automotive electrical power steering and active suspension systems based on random sub-optimal control[END_REF], robust and optimal H ∞ and LP V /H ∞ control [START_REF] Fergani | An lpv/H ∞ integrated vehicle dynamic controller[END_REF]] [Doumiati et al., 2013] [ [START_REF] Poussot-Vassal | Robust vehicle dynamic stability controller involving steering and braking systems[END_REF]] [Gaspar et al., 2007], sliding mode [START_REF] Li | Integrated vehicle chassis control based on direct yaw moment, active steering and active stabiliser[END_REF] [ [START_REF] Chamseddine | Control of linear full vehicle active suspension system using sliding mode techniques[END_REF]] [Chen et al., 2016] [Yoon et al., 2010], Immersion and Invariance control (Lyapunov-based) [START_REF] Tagne | Design and validation of a robust immersion and invariance controller for the lateral dynamics of intelligent vehicles[END_REF], (adaptive) Fuzzy logic control [START_REF] Wei | Integrated chassis control system for improving vehicle stability[END_REF]] [Xiao et al., 2009], nonlinear (adaptive or fuzzy) PID [Pedro et al., 2013] [Moradi andFekih, 2013], Inverse model control [START_REF] Andreasson | Global chassis control based on inverse vehicle dynamics models[END_REF], and articial neural networks [START_REF] Nwagboso | Development of neural-network control of steer-by-wire system for intelligent vehicles[END_REF].

Among these many control techniques, in this thesis, we have chosen to apply three advanced control techniques to the GCC: a Lyapunov-based control technique in the framework of Immersion and Invariance, the ST SM control technique, and the H ∞ control technique in the framework of the LP V system. Both the Lyapunov-based and the ST SM are nonlinear robust control techniques applied in the decentralized architectures for SISO systems, while the H ∞ is a linear robust and optimal control technique applied in the centralized architectures for the M IM O systems. These control techniques are introduced in the following sections.

Lyapunov-Based Control Technique

Consider the second order system written as:

ẍ = f (X, t) + g(X, t)u(t), (2.1)
where u is the control input, X = [x, ẋ] T ∈ 2 is the state vector, f and g are continuous functions, and g is invertible.

Let the desired trajectory of the state vector

X is X des = [x des , ẋdes ] T ∈ 2 . Let E = [e, ė] T = X -X des = [x -x des , ẋ -ẋdes ] T ∈ 2 is the state error vector.
The control objective is to converge the error vector E to zero.

As the system has a relative degree 2 w.r.t the control input u, let dene a new variable z such as:

z = ė + k 1 e + k 2 t 0 edτ. (2.2) ż z ė e ∫e e ė ∫e V V V 0 V (z )
Figure 2.2 Immersion and Invariance (modied from [START_REF] Astol | Nonlinear and adaptive control with applications[END_REF])

Based on Immersion and Invariance approach [START_REF] Astol | Nonlinear and adaptive control with applications[END_REF], the o-themanifold variable z has to converge to the target dynamics corresponding to z = 0 in the manifold as shown in Figure 2.2. In order to render the manifold attractive, i.e. to ensure the convergence of z to zero, given that z has a relative degree of 1 w.r.t the control input u, let dene a positive denite Lyapunov candidate function as follows:

V (z) = 1 2 z 2 , (2.3)
V should be negative (Lyapunov stability conditions), thus,

V = z ż ≤ 0, (2.4)
then, let: ż = -αz,

(2.5) which makes (2.4) always negative if α > 0. Thus:

ë + k 1 ė + k 2 e = -α( ė + k 1 e + k 2 t 0 edτ ), (2.6)
where

ë = ẍ -ẍdes = f (X, t) + g(X, t)u(t) -ẍdes , (2.7)
then, the control input u can be found such as:

u = g -1 (X, t)[-f (X, t) + ẍdes -(α + k 1 )( ẋ -ẋdes ) -(αk 1 + k 2 )(x -x des ) -αk 2 t 0 (x -x des )dτ ].
(2.8)

g -1 (X, t) and -f (X, t) compensate all the dynamics of the system as a feed-forward command, beside the robust terms.

In the manifold, once z = 0, the error dynamics obey to the following equation:

ė + k 1 e + k 2 t 0 edτ = 0.
(2.9)

A sucient condition to guarantee the convergence of ė, e, and t 0 edτ to zero, is to have k 1 > 0 and k 2 > 0 (Routh-Hurwitz stability condition for a second order characteristic polynomial). Despite the fact that the convergence of t 0 edτ to zero is not a necessary condition, its addition in z helps to reduce the permanent steadystate error.

SUPER-TWISTING SLIDING MODE CONTROL TECHNIQUE

Super-Twisting Sliding Mode Control Technique

The Super-Twisting algorithm is a second order sliding mode control that handles a relative degree equal to one [START_REF] Shtessel | Sliding mode control and observation[END_REF]. It generates the continuous control function that drives the sliding variable and its derivative to zero in nite time in the presence of smooth matched disturbances [START_REF] Shtessel | Sliding mode control and observation[END_REF].

Consider the system written as:

ẍ = f (X, t) + g(X, t)u(t), (2.10)
where u is the control input, X = [x, ẋ] T ∈ 2 is the state vector, and f , g are continuous functions. Let the desired trajectory of the state vector

X is X des = [x des , ẋdes ] T ∈ 2 , and let E = [e, ė] T = X -X des = [x -x des , ẋ -ẋdes ] T ∈ 2
s = ė + k e,
(2.11) with a second derivative written as:

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

The control objective is to achieve the convergence to the sliding surface dened as 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 ∈ 2 and |s(x, t)| < S 0 , the system satises the following conditions:

   |u(t)| ≤ U max , |Φ(s, t)| < C 0 , 0 < b min ≤ |ξ(s, t)| ≤ b max .
(2.13)

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| ∇ sign(s); ∇ ∈]0, 0.5], u2 = -α 2 sign(s).
(2.14) α 1 and α 2 are positive gains. The nite time convergence is guaranteed by the following conditions:

α 1 ≥ 4C 0 (bmaxα 2 +C 0 ) b 2 min (b min α 2 -C 0 ) , α 2 > C 0 b min .
(2.15)

The convergence is shown in Figure 2.3. The convergence analysis is given in [Utkin, 2013].

Note :

-The ST SM controller is known for its robustness against parameters uncertainties and disturbances. It converges to the sliding surface in nite time. However, it could -An equivalent input can be added as a feedforward control input to the ST SM control input (feedback) given in (2.14). This equivalent input compensates the dynamics of the system (when f and g are suciently well estimated) to make the sliding variable converges faster as in the Lyapunov based approach. The nal control input becomes:

u(t) = g -1 (X, t) -f (X, t) + ẍdes -k( ẋ -ẋdes ) -α 1 |s| ∇ s |s| + ε -α 2 t 0 s |s| + ε dτ .
(2.16)

Once s = 0, the error dynamics obey to the following equation:

ė + ke = 0.

(2.17)

ė and e exponentially converge to zero if k > 0.

LP V /H ∞ Control Technique

For any system, the H ∞ control synthesis is a disturbance attenuation problem. It consists in nding a stabilizing controller that minimizes the impact of an input disturbances w(t) on a (weighted) controlled output z(t). Generally, the H ∞ control technique is applied to LT I and LP V systems. Figure 2.4 shows the plant P (S) to be controlled, the input W i (S) and output W o (S) weighting dynamic functions, and the H ∞ controller K H∞ (S) to be synthesized (S is the Laplace transformation). The open-loop system formed by the interconnection of P (S), W i (S), and W o (S) is called the generalized plant Σ g (S). In this thesis, P (S) represents an LT I vehicle model, and W i (S) and W o (S) are scheduled LP V lters. Thus, Σ g (S) is a LP V system. The closed-loop system formed by the interconnection of Σ g (S) and K H∞ (S) is called C l (S).

w (t) z (t ) W i (S) W o (S) P (S) Σ g (S) K H ∞ (S) ∞ y(t ) u(t) C l (S)
Figure 2.4 H ∞ control architecture (modied from [START_REF] Sename | Robust control and linear parameter varying approaches: application to vehicle dynamics[END_REF])

To understand the H ∞ control technique, let consider in a rst time that the generalized system g and the controller K H∞ are LT I. The H ∞ control design consists in nding the controller K H∞ that minimizes the H ∞ norm of the closedloop system C l (S) as given in the following equation:

||C l (S)|| ∞ = sup ω∈ σ(C l (jω)) < γ, (2.18)
where σ is the largest singular value, and γ is the attenuation level. The H ∞ norm represents the maximal gain of the frequency response of the system. It is also called the worst case attenuation level in the sense that it measures the maximum amplication that the system can deliver on the whole frequency set. For SISO (respectively M IM O) systems, it represents the maximal peak value on the Bode magnitude (respectively singular value) plot of C l (jω); in other words, it is the largest gain if the system is fed by harmonic input signal [START_REF] Sename | Robust control and linear parameter varying approaches: application to vehicle dynamics[END_REF]. For more information on the H ∞ control refer to [START_REF] Zhou | Robust and optimal control[END_REF], [Apkarian et al., 1995], and [Apkarian and Gahinet, 1995]. A useful summary is given in chapter 2 of the dissertation of [Poussot-Vassal, 2008]. Now, let consider a LP V system dened by its generalized form g , such as:

Σ g (ρ) :   ẋ z y   =   A(ρ) B 1 (ρ) B 2 C 1 (ρ) D 11 (ρ) D 12 C 2 D 21 0     x w u   , (2.19)
where x includes all the state variables of the system, w is the exogenous input vector, u represents the control inputs, y is the measurement vector fed-back to the controller, y e is the exogenous output, and z is the weighted controlled output vector.

A, B 1 , B 2 , C 1 , C 2 , D 11 , D 12 , and D 22 are known matrices with nite dimensions. ρ is the vector of the varying parameters, it is known and bounded. Without loss of generality, we treat the case where ρ = {ρ 1 , ρ 2 }, since it will serve for the next chapters.

The LP V /H ∞ problem consists in nding the controller K LP V /H∞ (ρ 1 , ρ 2 ), scheduled by the parameters ρ 1 and ρ 2 , such that:

K LP V /H∞ (ρ) : ẋc u = A c (ρ) B c (ρ) C c (ρ) 0 x c y , (
ρ ∈ Ω = [ρ 1 , ρ 1 ] X [ρ 2 , ρ 2 ].
The polytopic approach aims at nding 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 (2.21) at each vertex

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) -Y A(ρ)X -N B c (ρ)C 2 X -Y B 2 (ρ)C c (ρ)M -T )M -T , (2.21)
where M (ρ) and N (ρ) are dened 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 nal controller, K LP V /H∞ (ρ 1 , ρ 2 ), is a convex combination of the controllers synthesized at the vertices of the polytope [Apkarian et al., 1995] such as:

K LP V /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 ), (2.22)
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 nal controller (see Figure 2.5). α i (ρ 1 , ρ 2 ) are instantly evaluated 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 .
(2.23)

Conclusion

In this chapter, a dierentiation between the multilayer centralized and decentralized architectures is provided. Some advanced control techniques are also evoked.

The power of the super-twisting algorithm, compared to the Lyapunov-based approach (Immersion and Invariance), is its robustness to modeling uncertainties (when the dynamics f and g are not well estimated and measured). Indeed, the

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 ∞ Figure 2.5 Polytopic LP V /H ∞ controller     A(ρ)X + XA(ρ) T + B 2 C(ρ) + C(ρ) T B T 2 ( * ) T ( * ) T ( * ) T A(ρ) + A(ρ) T Y A(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.
(2.24) feedforward equivalent input is not mandatory to proove the closed-loop stability of the super-twisting algorithm. This feature is essential in GCC because some of the vehicle and road parameters are time-varying. The comparison between the ST SM and the LP V /H ∞ is more dicult since they are both robust to uncertainties and disturbances. However, each of these two techniques has its pros and cons. The ST SM is simple to be applied since the controller gains can be easily adjusted for SISO system, while the dynamics interconnection eects are neglected. In contrary, the LP V /H ∞ considers the dynamics interconnection eects through the generation of simultaneous control inputs, while the control objectives are not scalable. Later on, both architectures are compared together through performance simulation.

Next chapter discusses the eect of the roll motion control on the vehicle performance.

Chapter 3

Eect of Roll Motion Control on Vehicle Performance

This chapter discusses the eects of the roll control on the vehicle performance.

Roll motion control is achieved through the generation of a virtual roll moment that can be allocated to the ASus, the semi-ASus, or the ARB. GCC architecture) to achieve their objectives. The idea of this chapter is to make the (semi-)ASus, usually developed for ride comfort and road holding [START_REF] Chamseddine | Control of linear full vehicle active suspension system using sliding mode techniques[END_REF], [START_REF] Savaresi | Semi-active suspension control design for vehicles[END_REF], intervene to achieve new functionalities supporting the other vehicle stability controllers aiming at avoiding rollover and maintaining lateral stability. This coordination makes the vehicle operates in a larger stable range, since for the same driving conditions, the dierent vehicle stability controllers will be less solicited.

The eect of the roll motion control on the rollover phenomenon is obvious, since the rollover problem is related to the roll motion and the lateral acceleration. This eect is detailed later in this chapter.

The eect of the roll motion control on the lateral stability is more complicated. The dynamics coupling between the vertical and lateral tire forces [START_REF] Li | Eect of vertical and lateral coupling between tyre and road on vehicle rollover[END_REF] is an essential key to enhance the lateral stability. Several studies on the (semi-)ASus are conducted to explicitly try to enhance the lateral stability [START_REF] Jin | Improving vehicle handling stability performance via integrated control of active front steering and suspension systems[END_REF], [START_REF] Vivas-Lopez | Global chassis control system using suspension, steering, and braking subsystems[END_REF], [START_REF] Zhao | Vehicle chassis integrated control based on multimodel and multilevel hierarchical control[END_REF], [START_REF] Zulkarnain | Application of an active anti-roll bar system for enhancing vehicle ride and handling[END_REF]. The basic idea is to prevent the saturation of tires lateral forces, which causes the vehicle to laterally skid when cornering, through controlling the vertical tires forces. That means, an inverse tire/road contact model should be evaluated, which is somehow dicult, especially in the nonlinear saturated regions. For that, researchers as [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], [Fergani et al., 2016a] propose to control the vertical load transfer when cornering, or minimize the vertical displacements of the unsprung masses, as they are the most inuencing factors on the lateral forces. However, this method may ensure a posterior enhancement on the lateral stability but not a prior demonstrated guaranty. One of this chapter objectives is to demonstrate, a priori, in the frequency domain, that the roll angle control can always enhance the lateral stability.

Roll Control Eect on Rollover Problem

Rollover Problem Formulation

Vehicle rollover is dened as 90 • or more rotation of the vehicle around its longitudinal axis [Gillespie, 1992]. This phenomenon starts when two vehicle wheels of the same side lift o from ground. Equivalently, rollover risk can be also evaluated around the axis joining the other two wheels remaining on the ground. In spite of the fact that the rollover crash constitutes a small percentage of all road accidents (3%), it commits fatal injuries, nearly 33% of all deaths from passenger's vehicle crashes, according to the N HT SA, 2011 statistics. For this reason, rollover avoidance has become an important safety issue for many researchers and automotive societies.

According to [Gillespie, 1992], there are two types of rollover: tripped and untripped rollover. Tripped rollover occurs due to an external force on the vehicle, like wheels impact with a curb or a pot hole, an accident with another vehicle, or even a violent wind. Untripped rollover occurs due to an excess in the lateral acceleration or due to the roll dynamics in a vehicle equipped with passive suspensions. For instance, untripped rollover occurs when performing a curved road with a sharp From the full vehicle model, the roll motion is expressed as:

θ = 1 I x + M s h 2 θ [(-F s,f r + F s,f l ) t f + (-F s,rr + F s,rl ) t r + M s (h θ cos (θ) + z s ) a y + M s (h θ sin (θ) + z s ) g]. (3.1)
The roll dynamics is equivalent to:

θ = 1 I x + M s h 2 θ [(-F f r + F f l ) t f + (-F rr + F rl ) t r + M s (h θ cos (θ) + z s ) a y + M s (h θ sin (θ) + z s ) g + M θ ],
(3.2) as shown in Figure 3.1, where F ij is the passive suspension force on the vehicle corner ij, and M θ is the active roll moment to be generated through the ASus forces U ij such as:

M θ = (-U f r + U f l ) t f + (-U rr + U rl ) t r . (3.3)
From the roll dynamics (3.2), the major variables that aect the induced roll angle are the passive suspension forces F ij and the lateral acceleration a y . As the lateral acceleration becomes high when cornering, as the induced roll angle turns to the outside of the corner and the vehicle becomes subjected to a rollover risk. In order to avoid the rollover, the active roll moment M θ should be generated to act on the roll motion.

Static Untripped Rollover

To better understand the rollover problem, let consider, in a rst time, that the vehicle is a rigid body (without suspension system), for the reason to show the eect of the lateral acceleration alone on the vertical load transfer. In a second time, the eect of the induced roll motion in committing rollover, in a vehicle equipped The lateral acceleration a y applied on the vehicle center of gravity creates the d'Alembert's force (centrifugal force) M.a y which acts to turn the vehicle towards the outside of the turn, with the torque M a y h acting on the axis formed by the outside wheels contact points with the ground. However, an opposite torque takes place to counterbalance the torque M a y h. It is performed by two vertical equal forces (in opposiste direction) distanced by t f forming a load transfer from inner to outer wheels, such as:

F zo -F zi = M a y h t f , (3.4)
where F zo and F zi are respectively the outer and inner vertical forces formed by adding each two tire vertical forces in the same vehicle side. Taking the moments of all forces around the same axis (Newton's law), the equilibrium is maintained (before starting rollover) if:

M a y h -M gt f + F zi 2t f = 0.

(3.5)

If the vehicle speed V increases, or the radius of the turn R decreases (the curvature increases), the lateral acceleration increases (for any rigid body: a y = V 2 /R in a non-mobile frame). Increasing a y happens while guarantying equilibrium in (3.5), which is done by a natural decreasing of the other single variable F zi , up to a certain amount of a y where F zi becomes 0, which represents inner wheels lift-o. Equation (3.5) becomes:

M a y,lif t-of f h -M gt f = 0, (3.6)
where a y,lif t-of f is the minimal lateral acceleration that causes wheels lift-o. Hence:

a y,lif t-of f = t f h g.
(3.7)

The factor t f h is called Static Stability Factor (SSF ). It is a constant value that depends on the vehicle geometry (around 1.2 for passengers cars, 1 for light trucks and 0.45 for heavy trucks [Gillespie, 1992]), that means heavy trucks rollover occurs at lower lateral accelerations.

This analysis to quantify the rollover risk is equivalent to the vertical LT R dened in [Rajamani, 2012] and described in (1.30).

3.2. ROLL CONTROL EFFECT ON ROLLOVER PROBLEM 47

Dynamic Untripped Rollover

In real situations, due to the suspension system, the vehicle is not a simple rigid body.

To study the motion roll eect on rollover, suppose that the vehicle has one degree of freedom represented by the roll angle θ between the suspended and unsuspended masses (see Figure 3.1). This means that the suspended mass center of gravity deviates by a positive angle θ (toward the outside) around the roll axis. Hereby, the moment of all forces around the axis joining outer wheels becomes:

M s a y h -M s g(t f -(h -h r )sinθ) + F zi 2t f = 0. (3.8)
By analogy with previous analysis, under the assumption of small angles sinθ ≈ θ, a y,lif t-of f becomes:

a y,lif t-of f = t f -(h -h r )θ h g, (3.9)
which means that a y,lif t-of f is reduced when considering the passive suspension system by approximately 5% [Gillespie, 1992]. Hence, comparing to a rigid body vehicle, the vehicle equipped with a passive suspension system starts rollover at lower lateral accelerations.

The transient response of θ is ignored in this analysis, while a roll overshoot may occur due to the developed roll dynamics. This overshoot adds to θ a small amount, which consequently, reduce a y,lif t-of f another 5%, and may cause vehicle to rollover when moving at a lower lateral acceleration. This analysis also ignores the lateral acceleration measurement error (accelerometer subjected to a roll motion), and other physical uncertainties causing rollover, like a vertical shift of the center of gravity, changes in tires and ground contact surfaces. In order to stay in a safe driving region, a safety factor of 0.7 of the total a y,lif t-of f expressed in (3.9) is proposed. A new variable is declared and denoted a y,saf e , where:

a y,saf e = 0.7a y,lif t-of f = 0.7

t f -(h -h r )θ h g (3.10)
A lateral acceleration a y above a y saf e risks the vehicle inner wheels to lift o when cornering. Consequently, to avoid the rollover, the lateral acceleration a y should be maintained below this threshold.

Roll Reference Generation

To avoid the rollover, the AF S and ADB aim to reduce a y , to maintain a y < a y saf e , while the ASus, semi-ASus and ARB aim, by controlling the roll angle, to elevate the maximal safe lateral acceleration a y,saf e . For instance, stiening the suspensions using the semi-ASus or the ARB can reduce the vehicle roll angle towards zero. This procedure arises a y,saf e as exhibits equation (3.10), which prevents the rollover to happen without the need to reduce a y . The contribution that adds the ASus system is the ability to continue turning the roll angle in the negative direction (to the inner side of the corner), that means a y,saf e will be more shifted to a higher value. The choice of the desired roll angle θ des is done as follows:

-At zero lateral acceleration (straight road), the desired roll angle is 0 -The map between θ des and a y is supposed to be linear to make a smooth comfortable roll change rate. Thus, the desired roll angle θ des is given as:

θ des = - 10 π 180 0.7 t f h g a y .
(3.11)

Figure 3.3 illustrates the safety range dened as the remoteness between the vehicle lateral acceleration and the maximal safe lateral acceleration which is a function

of the vehicle roll angle. It also shows the eect of turning the vehicle roll angle θ towards θ des of (3.11) in enlarging the safety range more than minimizing the roll angle towards zero (θ des = 0). However, a comparison is done in this chapter to compare both references.

Roll Control Eect on Lateral Stability

Frequency Analysis Setup

The analysis of the enhancement that the roll control brings on the lateral stability is not that much easy as for rollover avoidance, because the relation between the roll angle θ and the lateral stability quantied by the SI criterion (function of the sideslip dynamics β, β) is not static. This relation is rather governed by the dynamical dierential equations of the vehicle. Therefore, this section analyzes, in the frequency and time domains, the eect of the roll control on the lateral stability.

The frequency analysis requires, in general, a linear model. Therefore, the extended bicycle model of (1.37), as a linear model describing the relation between the lateral and the roll dynamics, is augmented to include the active roll moment M θ as the roll motion control input, and then it is used to make the study. The state-space representation of this model is given in (3.12). It is similar to that of the extended

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     +     b u,11 b u,13 b u,21 b u,23 0 0 b u,41 b u,43     B 1,3 δ d M θ (3.12)
bicycle model given in (1.38), with the simple addition of the control input M θ . The elements of the matrices A and B 1,3 , given in Appendix .1, depend on several time-varying parameters such as: V , µ, C f , C r , and M s ... However, the study is done for nominal/constant values of these parameters. Then, the study is enlarged to dierent vehicle speeds V , as it is the most varying one. Consequently, the statespace representation given in (3.12) is considered as a Linear Time-Invariant LT I system.

The LT I model gives an approximation of the real vehicle behavior in stable to midcritical regions. Hence, as the vehicle is being controlled before loosing stability, this LT I model is sucient to study the eect of the roll control on the lateral stability.

In fact, the risk of the vehicle to operate in the unstable region is minor, since other controllers like the DY C in the GCC structure are supposed to maintain the lateral stability. However, to analyze the eect of the roll control on the lateral stability when the vehicle operates in the unstable region, an extension of this study can be done using frequency response techniques for nonlinear systems like Pseudo-Bode computation [START_REF] Nassirharand | Input/output characterization of highly nonlinear multivariable systems[END_REF].

The LT I model (3.12) can be written as:

Ẋ = AX + B 1 δ d + B 3 M θ , (3.13)
where B 1 and B 3 are respectively the rst and second colomun of B1, 3.

The objective is to compare the frequency response of the vehicle with a controlled roll motion as in (3.13), and a the vehicle without a roll controller (M θ = 0) as in (3.14):

Ẋ = AX + B 1 δ d .
(3.14) Thus, let consider the state feedback LQR control law:

M θ = -KX. (3.15)
The optimal closed loop system becomes:

Ẋ = (A -B 3 K)X + B 1 δ d , (3.16)
which has the same form as (3.14), but with a controlled roll motion.

The optimization procedure consists in nding the control input U = M θ which minimizes the performance index J:

J = ∞ 0 (X T QX + U T RU )dt, (3.17)
where Q and R are the weighting matrices.

The control purpose is to minimize the roll angle and roll velocity, by controlling M θ . Thus, the performance index J becomes:

J = ∞ 0 (ρ 1 θ 2 + ρ 2 θ2 + M 2 θ )dt, (3.18)
where the weighting coecients ρ 1 and ρ 2 are adjusted to promote the weight on the roll angle, while minimizing the energy of the control input.

The matrix gain K has the form:

K = R -1 B T 3 P, (3.19)
where the matrix P is the solution of the Algebric Riccati Equation :

A T P + P A -P B 3 R -1 B T 3 P + Q = 0.

Note:

The state feedback LQR controller only serves to analyze the response of the vehicle.

However, more ecient controllers are designed in Section 3.4.

Lateral Stability Frequency Analysis

Steering Input

The steering input can be approximated (within a sample time) by a sinusoidal form with the amplitude A and the frequency ω as in (3.21):

δ d = A sin (ωt).
(3.21)

The driver eective steering frequency ω is between [0, 12.5] rad/s as explained in [START_REF] Heiÿing | Chassis handbook: fundamentals, driving dynamics, components, mechatronics, perspectives[END_REF]. This range of frequencies is considered as the useful range needed to perform a maneuver. Beyond this range, the vehicle becomes non-sensitive to the steering angle [START_REF] Heiÿing | Chassis handbook: fundamentals, driving dynamics, components, mechatronics, perspectives[END_REF]. The reason of this limitation is that the vehicle yaw rate cannot respond to the demand of the steering angle amplitude when the steering rate (frequency) is too high. The driver, in the normal driving situations, can provide a steering in the frequency range [0, 3] rad/s. Hence, the critical range of frequencies is when the driver perform a sudden steering with a frequency between [3, 12.5] rad/s which will be the range of the frequency response to be studied.

The eective steering amplitude A depends on the vehicle speed and steering frequency. For the same steering frequency, as the vehicle speed becomes higher as the steering amplitude has to be maintained small, which is a natural tendency of the driver, this fact will be validated in the next section. In the frequency range

[3, 12.5] rad/s and a speed range between [80,130] km/h, a steering amplitude between [0, 0.1] rad (approximately [0 • , 5 • ]) is sucient to make the study cover both normal and critical driving situations. Later, a discussion will be done to generalize for higher amplitudes. (3.23) Equation (3.23) shows that for the same frequency ω, the lateral SI becomes greater as the steering amplitude A increases. Thus, let A be set at the maximum of the eective steering amplitude A = 0.1 rad. Equation (3.23) also shows that SI becomes higher when M (ω) is higher. Consequently, the frequency response curve of the vehicle without roll control in Figure 3.4 shows that in the frequency range [0, 3] rad/s, the magnitude M (ω

) of G SI δ d
is between [15.8, 17] db.This means that for A = 0.1 rad, the SI can reach SI = 0.1 * 10 (17/20) = 0.7, which demonstrates that in this frequency range, the driving situation is normal.

In the [3, 12.5] rad/s frequency range, the magnitude response of G SI In the frequency ranges [3,5] rad/s and [11, 12.5] rad/s, the curves are close to each other, meaning that the lateral stability enhancement is not as signicant as in the frequency range [5,11] Beyond 12.5 rad/s, the magnitudes of both curves start decreasing, which means that the vehicle response to the steering becomes limited. Even if the roll control enhances well the lateral stability, however, as mentioned before, the vehicle will not be driven in this region.

In order to generalize for any speed, the frequency responses G SI δ d of the controlled and uncontrolled vehicles are evaluated at dierent speeds V = 70, 85, 100, 115 km/h. The results are illustrated in Figure 3.5. This gure shows that at the speed V = 70 km/h, both magnitude curves of G SI δ d are low and approximately confounded in all the frequency range [0, 12.5] rad/s, that means even if the roll control can slightly reduce the SI, there is no need to control the roll motion, because SI is already low. As the speed becomes higher, as the magnitude curves start at higher values, and the dierence between both curves of each couple (at the same speed) starts to appear and becomes bigger (for V ≥ 85 km/h), especially in the critical frequency range [5,11] rad/s.

Time-Domain Test

Roll Motion Controllers Design

This section is dedicated to design robust controllers to control the roll motion.

The Lyapunov-based control technique described in Section 2.3 and the ST SM control technique described in Section 2.4 are used to develop the roll motion controllers. These robust control techniques are chosen to deal with the nonlinear behavior/dynamics of the vehicle. A performance comparison between the Lyapunovbased, the ST SM , and the LQR controllers is also performed.

As discussed before, the objectives of these controllers are either to converge the nonlinear roll motion θ given in (3.2) to zero or to θ des given in (3.11). The controllers will be developed for a general reference θ des of θ, then, they will be tested for both references.

Let rst dene: (3.24) the error between the actual and desired roll angles.

e θ = θ -θ des ,

Lyapunov-Based Controller

The control objective is to converge the roll error variable e θ (of relative degree 2 w.r.t the control input M θ ) to zero. Let dene the o-the-manifold variable z θ , such as:

z θ = ėθ + k 1θ e θ + k 2θ t 0 e θ dτ.
(3.25)

Based on Immersion and Invariance approach [START_REF] Astol | Nonlinear and adaptive control with applications[END_REF], the o-themanifold variable z θ has to converge to the target dynamics corresponding to z θ = 0 where k 1θ > 0 and k 2θ > 0 (Routh-Hurwitz stability condition for a second order characteristic polynomial).

In order to render the manifold attractive, i.e. to ensure the convergence of z θ to zero, let dene a positive denite Lyapunov candidate function as follows:

V θ = 1 2 z θ 2 , (3.27)
Vθ should be negative (Lyapunov stability conditions), thus, Vθ = z θ żθ ≤ 0,

M θ = (I x + M s h 2 θ )[-M θeq + θdes -(α θ + k 1θ )( θ -θdes ) -(α θ k 1θ + k 2θ )(θ -θ des ) -α θ k 2θ t 0 (θ -θ des )dτ ], (3.32)
where

M θeq = 1 I x + M s h 2 θ [(-F f r + F f l )t f + (-F rr + F rl )t r + M s (h θ cos(θ) + z s )a y + M s (h θ sin(θ) + z s )g]. (3.33)
This control input means that the vehicle parameters should be well estimated and several variables need to be measured or estimated. Indeed, -M θeq and (I

x + M s h 2 θ )
compensate all the dynamics of the roll angle expressed in (3.2) as a feed-forward command, beside the robust terms of the feedback on e θ , its time derivative and integral of equation (3.32). θ, θ, and a y , z s are measured by the Inertial Measurement Unit IMU, the suspension forces F ij could be estimated from suspensions' deections.

For the case where θ des is zero, thus, θdes and θdes are also zeros. For the case where θ des is as in (3.11), thus, θdes and θdes are the rst and second derivatives of (3.11).

θdes and θdes are bounded to avoid sending high demand control to the actuators and to consider the driver feeling when turning.

Once the control input M θ is generated, an allocation procedure on the ASus forces U ij is proposed in Section 3.5.

s θ = ėθ + k θ e θ , (3.34)
where k θ > 0. The variable s θ has a relative degree of 1 w.r.t the control input M θ , thus the sliding mode super-twisting algorithm can be applied. Unlike the preceding Lyapunov controller, the integral term of e θ has not been considered inside the variable s θ , because the super-twisting algorithm is a second order sliding mode, it contains an integral term on the sign of s θ [Utkin, 2013]. s θ has a relative degree equal to one, thus, based on the discussion given in Section 2.4, the control input can be given by:

M θ = -α θ |s| 0.5 s |s| + ε -β θ t 0 s |s| + ε dτ, (3.35)
where α θ and β θ are positive gains verifying conditions (2.15). Only the knowledge of s θ is required in real time. This means, there is no need to compensate the roll motion dynamics.

Once s θ = 0, the states θ and θ exponentially converge to θ des and θdes if k θ > 0.

To be noted that the term M θeq of (3.33) can be added to the control input M θ of (3.35), as a feed-forward, to achieve a faster convergence towards the sliding surface.

However, the performance shown later by simulations is performed while omitting this term.

Closed-Loop Control Architecture

The general closed loop control scheme is presented in Figure 3

U f l = 0.5 lr l f +lr M θ t f , U f r = -0.5 lr l f +lr M θ t f , U rl = 0.5 l f l f +lr M θ tr , U rr = -0.5 l f l f +lr M θ tr . (3.36)
The choice of this distribution is done in order to avoid any inuence on the pitch angle and the bounce displacement as shown in Figure 4.3.

The four forces in (4.14) have to be generated by the ASus real physical actuators described in Section 1.6. To be noted, the controller gains are chosen to respect the

Controllers Validation and Performance Comparison

In this section, the proposed controllers will be validated on the simulation model (full vehicle model) because the ASus are not available on SCANeR Studio simulator. Indeed, the simulation model is ecient enough to validate the controllers, since it provides sucient knowledge on the dynamical behavior of the vehicle.

To do so, the shhook maneuver is the appropriate test to evaluate both the vehicle rollover risk and the lateral stability, because for the passengers cars, the lateral skidding happens around 0.8g lateral acceleration, while the rollover happens around 1.1g [Gillespie, 1992]. The shhook maneuver represents a sharp steering in one direction, in a very short duration, then a similar steering in the opposite direction as shown in Figure 3.9.

The vehicle initial speed is V = 130 km/h, while the throttle and the braking pedals are dropped.

As seen before, the roll controller is more useful above SI = 0.7 for lateral stability enhancement, and at high lateral acceleration for rollover avoidance. However, for now, as there is no decision layer to guaranty a stable switching, it will be activated This section is divided into three parts: rstly, the static roll reference(θ des = 0) will be considered, and then the dynamic one based on equation (3.11); nally, a performance comparison will be elaborated.

Controllers Validation (Static θ des )

Lyapunov-based, ST SM and LQR controllers are compared in this section to evaluate their performances when minimizing the roll angle to θ des = 0. This issue gives the sliding mode controller an advantage over the Lyapunovbased one especially in real-time application, where the estimated components could be imprecise, the vehicle parameters are uncertain, the model is not exact or perturbations occur on the system. The performance of the LQR controller is also acceptable to control the roll angle towards zero. Obviously, the roll control is not sucient at higher lateral accelerations (higher speed and/or higher road curvature).

Figure 3.12 shows the lateral SI of the uncontrolled roll vehicle which exceeds 0.7, while the controlled ones (by all the proposed controllers) enhance the lateral stability. Even if the enhancement is not sucient, because the SI remains above 0.7, the roll control to zero can be used to help the other controllers (AF S and DY C) to avoid the lateral skidding in a GCC strategy. Note that SI will be more improved when controlling the roll motion towards the dynamic reference in the opposite direction.

Controllers Validation (Dynamic θ des )

In this section, the controllers performances will be evaluated when controlling θ to θ des in the opposite direction expressed in (3.11). Figure 3.15 shows the LT R for the uncontrolled roll vehicle which increases up to 1 (respectively -1), that means, the vehicle is risked to rollover, while the LT R for the controlled roll vehicles (both Lyapunov-based and sliding mode) is enhanced and reduced to 0.85 (respectively -0.85). Figure 3.16 shows the lateral SI of the uncontrolled roll vehicle that exceeds the value 0.7 up to 1.2 which leads the vehicle to loose its lateral stability. This behavior can be explained by the high lateral acceleration which exceeds the lateral acceleration handling limit µ.g [Rajamani, 2012]. It can be also explained by the insuciency of the inner tires lateral forces due to low vertical loads on these tires. Controlling the roll angle toward the inside wheels elevates these vertical forces, thus, the tire lateral forces become higher, Figure 3.17 shows the vehicle longitudinal speed, it drops during all tests due to frictions and the aggressive steering at the same rate, that means the roll control dedicated to avoid rollover and lateral skidding has no eect on the vehicle speed, in contrary to other known stabilizing controllers like ADB or simply normal braking [Rajamani, 2012].

Figure 3.18 shows the advantage of the roll control for lateral stability enhancement and rollover avoidance on the performed trajectories, where they are very close to each other comparing to other controllers like AF S or ADB dedicated to the same objectives as explained in [Rajamani, 2012].

Roll Reference Performance Comparison

Turning the roll angle in the opposite direction requires more energy than just minimizing it to zero, because it can be only achieved by the ASus which consume more energy comparing to the semi-ASus or the ARB. This fact has made from the ASus a non preferable system to be integrated into series vehicles. However, the results of this chapter show the ability of the ASus in maintaining the lateral stability and avoiding the rollover when turning the roll angle in the opposite direction.

Conclusion

This chapter has shown the enhancement that the vehicle roll control could bring on its lateral stability and rollover avoidance, either by turning it to zero or to a In the context of GCC, the roll control can be activated for two objectives: rst, when the lateral SI becomes higher than SI = 0.7 (basing on the lateral stability enhancement), and second, when the LT R becomes higher than LT R = 0.8 (basing on rollover avoidance enhancement). However, the controllers can be used all the time without these conditions because they do not deteriorate in any way neither the lateral stability nor the rollover avoidance. The advantages of these criteria could be to minimize the consumed energy, when a higher decision layer is proposed to coordinate the suspensions with other stabilizing systems, like the AF S and the DY C, in the context of GCC.

Next chapter develops a decentralized architecture for the global chassis control, involving AF S, DY C, and ASus.

Introduction

The main goal of the GCC system developed in this chapter is to enhance the overall vehicle performance i.e. maneuverability, lateral stability, rollover avoidance, and ride comfort. Figure 4.1 depicts the general architecture of the proposed GCC.

It consists of a multilayer control architecture, formed by three hierarchical layers.

The lower layer is the actuator layer which represents the actuators implemented into the vehicle that generate their control inputs based on the orders sent from the middle layer (see Section 1.6). The middle layer is the control layer which is responsible to generate the control inputs that minimize the errors between the reference and actual vehicle state variables, i.e., yaw, side-slip, roll, pitch, and heave motions, regardless of the driving situation. Since the control architecture is decentralized, a heuristic solution is proposed by decoupling the control problem.

The ST SM robust control technique is applied to derive the control inputs, where the AF S control input minimizes only the error on the yaw rate, mainly to improve the maneuverability and enhance the lateral stability; the DY C control input is privileged to minimize only the error on the side-slip dynamics, mainly to guarantee the lateral stability of the vehicle; and the ASus control inputs minimize only the errors on the roll, pitch, and heave dynamics, mainly to improve the ride comfort, lateral stability, and rollover avoidance. The contributions of this chapter are as follows:

ψ ref δ d δ d V x δ c C z U ij ×4 θ θ ref ϕ z - + + - Roll Ref Generator Data ( ψ , β , β ,θ ,ϕ , z , V x ...)
• single-input single-output ST SM controllers are developed to control the ASus, the DY C, and the AF S in the presence of modeling errors, external disturbances and exogenous inputs.

• new objectives are achieved by the ASus controller, usually developed for ride comfort. It is exploited to improve the vertical stability (rollover avoidance) and lateral stability (lateral skidding avoidance). General improvements are also observed, e.g., the DY C will be less solicited, the vehicle speed will less drop, and others...

• development of a decision layer that promotes/attenuates the local subcontrollers objectives. This layer monitors the dynamics of the vehicle, calculates and sends scheduled gains to the sub-controllers, based on fuzzy logic rules and a stability criterion to enhance the overall vehicle performance according to the driving conditions.

Global Chassis Control Controller Design

Control Synthesis Model

The full vehicle model has been already developed in Section 1.1. It is a complex nonlinear model which combines the vertical, lateral, longitudinal, and tire/road contact (Dugo model) sub-models, in addition to four wheels angular dynamics, with a 26 state variables gathered in the state vector X of (4.1).

X = [θ, θ, φ, φ, z s , żs , z us,ij , żus,ij , ω ij , x, ẋ, y, ẏ, ψ, ψ, β, β] T . (4.1)
In order to develop a GCC controller, some dynamics of the full model (see Figure 4.2) are rewritten in the ane form to serve as a control synthesis model, such as:

θ = g θ (X) + f θ (X)M θ , (4.2) 4.2. GLOBAL CHASSIS CONTROL CONTROLLER DESIGN 67 Figure 4.2 Full vehicle model φ = g φ (X) + f φ (X)M φ , (4.3) zs = g z (X) + f z (X)M z , (4.4) ψ = g ψ(X ) + f ψ,δ (X)δ t + f ψ,Cz (X)C z , (4.5) β = g β (X) + f β,δ (X)δ t + f β,Cz (X)C z , (4.6)
where M θ , M φ , and M z represent respectively the active roll torque, active pitch torque, and active heave force, as intermediate control inputs. These inputs have to be generated at a lower level control by physical actuators, e.g. the ASus forces U ij integrated on four corners. δ t = δ d + δ c is the total steering angle at the front wheels, where δ d is the one provided by the driver and δ c is the one provided by the AF S controller. C z is the active yaw torque provided by the DY C controller. C z has to be generated at a lower level control as an ADB on the rear wheels. g q (X), f v (X), f l,δ (X), and f l,Cz (X) where q = {v, l}, v = {θ, φ, z s } and l = { ψ, β} are the nonlinear functions of the full vehicle model detailed in Appendix .2.

Since the control synthesis model is nonlinear, where some functions are hard to be estimated in real time, and in presence of modeling errors and external disturbances, the ST SM robust control technique is relevant to design the GCC controller, without compensating the vehicle dynamics.

Control Layer

Active Suspensions Controller

The common objectives of the ASus widely developed in literature are improving the ride comfort and road holding [START_REF] Savaresi | Semi-active suspension control design for vehicles[END_REF], [START_REF] Yoon | Design of an unied chassis controller for rollover prevention, manoeuvrability and lateral stability[END_REF] and [START_REF] Akhmetov | Global Chassis Control for active safety of heavy vehicles[END_REF]. One contribution of this chapter is to emphasize new achievable enhancements on the global chassis performance through the coordinated integration of the ASus. These enhancements concern directly the rollover and the lateral stability. Hence, the other sub-controllers in the GCC structure become less solicited, and consequently, the vertical and lateral stability ranges of the vehicle manipulation can be enlarged to more hard maneuvers.

Let rst develop an

v = g v (X) + f v (X)M v . (4.7)
Each of these equations has a unique control input M v , that acts only on the corresponding variable v. Thus, similar controllers with particular gains can be developed for all of these dynamics.

To control these dynamics, one can choose the ST SM control law known by its robustness to modeling errors and external disturbances.

Let v ref = {θ ref , φ ref , z ref }
be the general reference trajectories to be tracked, and vref = θref , φref , żs,ref be their time derivatives.

The reference trajectories are determined based on several factors. The objective of ride comfort in literature is to minimize the roll, pitch and heave angles and velocities/accelerations [START_REF] Savaresi | Semi-active suspension control design for vehicles[END_REF], [START_REF] Yoon | Design of an unied chassis controller for rollover prevention, manoeuvrability and lateral stability[END_REF] and [START_REF] Akhmetov | Global Chassis Control for active safety of heavy vehicles[END_REF], thus, v des = vdes = {0, 0, 0}. As seen in Chapter 3, the roll control towards zero contributes to the enhancement of the lateral stability and rollover avoidance. To provide more enhancement on both objectives, the dynamic roll reference of Section 3.2.4 can be followed. Moreover, scheduling parameters are lately introduced, in the decision making layer, inside these reference trajectories to promote/attenuate the control objectives.

Let consider now the control towards general references v ref . Thus, let: (4.8) be the error between the actual and reference state. Let:

e v = v -v ref ,
s v = ėv + λ v e v ,
(4.9) be the sliding variable, chosen with a relative degree of 1 w.r.t the control input M v (the control input appears in the rst time derivative of the sliding variable) to meet the super-twisting constraints. This means that the discontinuous function appears in the second derivative of the sliding variable such that:

sv (s v , t) = Φ v (s v , t) + ξ v (s v , t) Ṁv (t), (4.10)
where Φ v (s v , t) and ξ v (s v , t) are unknown bounded functions satisfying conditions of (2.13).

The sliding mode control input, based on the Super-Twisting algorithm, is given by: The optimal allocation has the advantage of tolerating the faults of the ASus.

M v (t) = -α v,1 |s v | τv sign(s v ) -α v,2 t 0 sign(s v )dτ,
However, this is not treated in this thesis. Thus, we propose two simple allocation methods as the following:

Pseudo-Inverse Matrix:

From Figure 4.3, the virtual control inputs M θ , M φ , and M z can be written in function of the ASus forces such as:

  M θ M φ M z   =   -t f t f -t r t r -l f -l f l r l r 1 1 1 1   L     U f r U f l U rr U rl     .
(4.12)

Thus, the ASus forces U ij can be found by applying the pseudo-inverse matrix method, such as:

    U f r U f l U rr U rl     = L T .L -1 L T   M θ M φ M z   (4.13)
The pseudo-inverse matrix method may hold singularities in the solution, thus, another constraint could be added to make a square matrix L, or a geometrical distribution method can be applied.

Geometrical distribution:

This methods consists of generating for each dynamics the needed control input, by doing a geometrical distribution between the four suspensions, while maintaining no eect on the other two dynamics, as given in (4.14) (see Figure 4.3).

U f l = 0.5 lr l f +lr

M θ t f -0.5 M φ l f +lr + 0.5 lr l f +lr M z , U f r = -0.5 lr l f +lr M θ t f -0.5 M φ l f +lr + 0.5 lr l f +lr M z , U rl = 0.5 l f l f +lr M θ tr + 0.5 M φ l f +lr + 0.5 l f l f +lr M z , U rr = -0.5 l f l f +lr M θ tr + 0.5 M φ l f +lr + 0.5 l f l f +lr M z .

Active Front Steering Controller

The maneuverability or steer-ability means having a linear relation between the steering provided by the driver and the achieved vehicle yaw rate. The objective of the AF S controller is to enhance the steer-ability, thus, converging the real vehicle yaw rate to a reference one linear to the steer angle provided by the driver. The linear relation can be derived from the bicycle model exposed in Section 1.4, which represents a stable and ideal motion of the vehicle, where the tires lateral forces are supposed to be linear to the tires side-slip angles. From the other side, the AF S controller will be promoted/attenuated based on the decision layer. Thus, the yaw rate reference trajectory ψref is related to the bicycle model given in Section 1.4 and to the scheduling parameters of the decision layer. The explicit relation will be given later in Section 4.2.3.

The AF S can also enhance the lateral stability of the vehicle. As the lateral stability is related to the lateral acceleration a y , the authors in [Rajamani, 2012] propose to maintain a y below a threshold depending on the maximal possible adherence (4.15), by saturating ψref , as described in (4.16).

a y V x ( ψ + β) ≤ µ g, (4.15) ψref,max = 0.85µ g/V x , (4.16)
where µ is the road adherence coecient and g is the gravity constant.

Let consider for now that the objective of the AF S is to converge the vehicle yaw rate ψ, whose dynamics is described in (4.5), to a general reference one ψref .

Since the yaw equation (4.5) does not reect the vehicle real yaw dynamics, because it is a simplied linear representation, and since some vehicle parameters are hard to be instantly estimated like µ, the ST SM robust control technique is applied here without compensating the yaw dynamics. For simplicity, the corrective yaw torque control input is omitted in the AF S controller design (C z = 0). This assumption makes from the ST SM controller a heuristic solution. This issue is resolved in the next chapter when adopting the M IM O H ∞ optimal control technique.

Thus, let dene the sliding variable as follows:

s ψ = e ψ = ψψref .

(4.17)

The variable s ψ has a relative degree of 1 w.r.t the control input δ c . Thus, s ψ(s ψ, t) = Φ ψ(s ψ, t) + ξ ψ(s ψ, t) δc (t).

(4.18) Φ ψ(s ψ, t) and ξ ψ(s ψ, t) are unknown bounded functions satisfying conditions of (2.13). By the same reasoning applied above, the ST SM control input δ c can be formulated as (4.19):

δ c = -α ψ,1 s ψ τ ψ sign(s ψ) -α ψ,2 t 0 sign(s ψ)dτ. (4.19)
This algorithm guarantees the convergence of s ψ to zero in a nite time, if the gains α ψ,1 and α ψ,2 satisfy the same convergence conditions of (2.15), and τ ψ ∈]0, 0.5].

Since s ψ is formed only by the error on the yaw rate, thus, ψ converges in nite time towards ψref .

This control strategy enhances the maneuverability of the vehicle and maintain the lateral acceleration below the lateral-skidding threshold µ.g. However, the lateral stability also depends on the vehicle side-slip angle β and its rate of change β. Thus, this control strategy enhances without guarantee the lateral stability, especially when these variables are solicited enough to destabilize the vehicle. To resolve the problem, the rst intuitive solution is to introduce the control of β and β in the objectives of the AF S controller. However, the yaw torque provided by the steering in the critical range of lateral stability is not enough to stabilize the vehicle since the lateral tires forces will be saturated. Alternatively, the DY C controller using dierential rear braking is known to be eective to control β and β, while it has the disadvantages of: decelerating the vehicle, long-term wheels wear, and driver discomfort. Thus, it is recommended to actuate the DY C controller only under critical situations.

Direct Yaw Control Controller:

The objective of the DY C controller is to control the side-slip angle β and its rate of change β when the vehicle is under critical driving situations by generating a corrective yaw torque. The physical actuators to create the yaw torque are selected to be the rear EM B. This choice prevents direct interference with the AF S on the front tires [START_REF] Doumiati | Integrated vehicle dynamics control via coordination of active front steering and rear braking[END_REF].

In this subsection, the DY C controller will be developed as a decentralized controller, i.e. to control β and β whatever the driving situation is. Later, in the decision layer, the DY C controller will be activated based on the decision rules. Similar to the case of the AF S, the ST SM control law is adopted to control β and β of (4.6) respectively towards the reference trajectories β ref and βref . These general reference trajectories are related to the bicycle model given in Section 1.4 and to the scheduling parameters of the decision layer. The explicit relation will be given later in Section 4.2.3.This control is done through generating the corrective yaw torque C z as the control input while omitting δ c . Thus, let:

e β = β -β ref , (4.20)
be the error between the actual and reference side-slip angles. Let the corresponding sliding variable be:

s β = ėβ + λ β e β .
(4.21)

The sliding variable has a relative degree of 1 w.r.t the control input C z . By the same reasoning as before, the super-twisting algorithm guarantees the convergence of s β to zero in a nite time. Thus, β and β exponentially converge to β ref and βref if λ β > 0. Finally, the ST SM control input C z is given in (4.22) by:

C z = -α β,1 |s β | τ β sign(s β ) -α β,2 t 0 sign(s β )dτ, (4.22)
where α β,1 and α β,2 satisfy conditions of (2.15), and τ β ∈]0, 0.5].

CHASSIS CONTROL INVOLVING ACTIVE FRONT STEERING, DIRECT YAW CONTROL AND ACTIVE SUSPENSIONS -A DECENTRALIZED ARCHITECTURE

+ - Sliding Mode Controller + + S q q des λ obj λ q d dt e q Monitor Control input ėq Data ( ψ , β , β ,θ ,ϕ , z , V x ...) q Figure 4.4 Scheduled controller

Decision Layer

A decision layer will be developed in this section to coordinate the GCC subcontrollers. The main idea is to promote/attenuate the control objectives by multiplying each sliding variable by a scheduling gain {λ obj } = {λ ψ, λ β , λ θ , λ φ , λ z } as depicted in Figure 4.4. Each of λ obj varies between 0 and 1. As much λ obj approaches to 1, as the control objective is promoted, vice-versa, the control objective is completely attenuated when λ obj approaches to 0.

In order to maintain the stability of each ST SM controller, these scheduling parameters have to be endogenous in the sliding variables. This can be done through introducing these scheduling parameters inside the reference trajectories.

As mentioned before, ψref , and β ref depend on the bicycle model and the weighting gains sent from the decision layer, while θ ref , θref , φ ref , φref , z ref , żref are to be minimized and depend on the weighting gains sent from the decision layer. Thus, let dene the dierent trajectory references such as:

ψref = λ ψ ψbic + (1 -λ ψ) ψ, β ref = λ β β bic + (1 -λ β )β, θ ref = λ θ 0 + (1 -λ θ )θ, θref = λ θ 0 + (1 -λ θ ) θ, φ ref = λ φ 0 + (1 -λ φ )φ, φref = λ φ 0 + (1 -λ φ ) φ, z s,ref = λ z 0 + (1 -λ z )z s , żs,ref = λ z 0 + (1 -λ z ) żs . (4.23)
When λ obj approaches to 1, the corresponding reference trajectory becomes equal to the bicycle trajectory (for ψ and β), and equal to 0 (for θ, φ, and z s ). Thus, the corresponding controller is promoted to control the corresponding variable. When λ obj approaches to 0, the reference trajectory becomes equal to the actual vehicle one, thus, the control of the corresponding variable is attenuated since the corresponding sliding variable is vanished.

Consequently, the dierent sliding variables become equivalent to: These new forms of sliding variables mean that the actual state variables are forced to converge to the extended bicycle reference model (for ψ and β), and to 0 (for θ, φ, and z s ), 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.

s ψ = ψ -ψref = λ ψ ( ψ -ψbic ), s β = β -β ref = λ β (β -β bic ), s θ = ( θ -θref ) + k θ (θ -θ ref ) = λ θ [ θ + k θ θ], s φ = ( φ -φref ) + k φ (φ -φ ref ) = λ φ [ φ + k φ φ], s z = ( żs -żs,ref ) + k z (z s -z s,ref ) = λ z [ żs + k z z s ].
Note: The modication of the sliding variables by the multiplication with the scheduling gains maintains the closed-loop stability of the individual AF S and DY C since these gains are introduced in the reference trajectories.

The decision layer monitors all controllers objectives based on monitoring criteria (data) and a set of coordination rules dened in the following, then, it calculates and sends instantly the exact value of λ obj to attenuate/promote the corresponding objective.

Active Front Steering and Direct Yaw Control Coordination Rules:

The criteria by which the AF S and DY C are coordinated is the lateral SI. The coordination rules are as follows:

* If the vehicle situation is normal SI ≤ SI, then the AF S controller should be promoted to improve the maneuverability. In this range, the DY C controller is disabled.

* If the vehicle operates in the unstable region SI ≥ SI, then the DY C controller should be promoted to enhance the lateral stability. In this range, the AF S controller has a poor eect to enhance the lateral stability, thus, it may not be actuated. * In the critical region SI ≤ SI ≤ SI, the AF S and DY C should be attenuated/promoted smoothly and continuously in this range which can be done by a sigmoid function.

Based on these rules, the scheduling gains λ ψ and λ β can be given in function of SI as:

λ β = 1 1+e - 8 SI-SI (SI- SI+SI 2 ) , λ ψ = 1 -λ β .
(4.25)

These functions are plotted in Figure 4.5.

Active Suspensions Coordination/Actuation Rules:

The proposed ASus controller will participate in the GCC by achieving three main objectives: roll control, pitch control and heave control. To reduce the excessive actuation of the ASus, while maintaining good ride and stability (vertical and lateral) qualities, the following supervision rules will promote/attenuate these objectives:

* Since the natural tendency of the driver is to turn his body towards the * The pitch control objective has to be attenuated/promoted depending on the severity of braking/acceleration, to reduce the use of the ASus while maintaining good ride quality (in terms of pitch motion). Thus, only the harsh and considerable pitch motion has to be minimized by making λ φ approaches to 1 to promote the pitch control objective. As much the pitch motion becomes soft, as λ φ approaches to 0 to attenuate the pitch control objective.

One suggests treating λ φ as a fuzzy-scheduling gain which attenuates/promotes the pitch control objective, based on the pitch angle error of (4.8) and its rate of change.

The pitch angle error and its rate of change are applied to the Fuzzy Logic Controller (F LC) as inputs, and the fuzzy-scheduling gain λ φ is the output. The reason of choosing the F LC for the decision-making process is due to its simplicity to make the relation between the needed control input and the controlled variables in an intuitive way. Five fuzzy sets are dened for each input, and three for the output such as:

e φ , ėφ ∈ {N B (N egative Big), N S (N egative Small), ZE (Zero), P S (P ositive Small), P B (P ositive Big)}; and λ φ ∈ {P S, P M (P ositive M edium), P B}. The normalized Membership Functions (MFs) of fuzzication of the controller inputs and defuzzication of the controller output are respectively given in Figures 4.6, 4.7 and 4.8. To determine the fuzzy controller output λ φ for the given fuzzy controller inputs e φ and ėφ , the decision matrix of the linguistic control rules is designed and presented in Table 4.1. These fuzzy sets, membership functions, and the linguistic rules are usually determined based on an expert knowledge of the system by performing several simulations. Finally, to defuzzify the result/output, the Mamdani centroid fuzzy inference method is used [ [Reznik, 1997]].

* The heave control objective attenuation/promotion can be done in a similar manner to the pitch control objective. That means, a fuzzy-scheduling gain λ z can be obtained to regulate the degree of achievement of the heave control objective depending on the harshness of this motion. However, this is not done in this thesis since the literature is rich in controlling the heave motion as mentionned before.

Global Chassis Control Validation and Simulation

In this section, the proposed GCC system will be validated through two simulation tests using Matlab/Simulink. The simulation model of the full vehicle is developed and validated on the professional vehicle simulator SCANeR Studio [START_REF] Chokor | Active suspension control to improve passengers comfort and vehicle's stability[END_REF] and [START_REF] Chokor | Rollover prevention using active suspension system[END_REF].

The rst test is a sine steer (Figure 4.9) at 100 km/h initial speed. This test solicits the vehicle yaw and lateral motions, as well as the roll motion. The uncontrolled (induced), desired and controlled roll angles are shown in Figure 4.10. The lateral SI, without controlling the vehicle, increases to reach more than SI = 1 as shown in Figure 4.11. That means, the vehicle has lost its lateral stability. The AF S controller alone (dedicated to the maneuverability) can reduce the SI to 1 as shown in the same gure. However, this improvement is not sucient. The coordinated AF S and DY C controller can maintain the lateral stability under SI = 0.8. This improvement is obtained thanks to the torque C z generated by the DY C that stabilizes the side slip angle β and its rate of change β. The addition of the roll control to the GCC structure (by the ASus) can enhance more the lateral stability by reducing the peak value of SI (SI = 0.8) to less than 0.7. The lateral stability can be alternatively studied in the β -β phase plane shown in Figure 4.12. The boundaries are for SI = 1. As much β -β relation is near the ideal one -calculated from the bicycle reference model-, as the lateral stability is more enhanced. It can be noticed that the uncontrolled vehicle exceeds the boundaries, while the GCC controller is the nearest one to β -β reference. The vehicle yaw rate is shown in Figure 4.13. In the ranges below the saturation of the yaw rate reference, the uncontrolled vehicle is somehow far away from the yaw rate reference. Meanwhile, all the adopted strategies (AF S, AF S + DY C, and the GCC) converge to the desired yaw rate. This means that the maneuverability is enhanced regardless of the adopted strategy. When the vehicle yaw rate becomes too much high, the control objective attempts to saturate the yaw rate in order to simultaneously enhance the lateral stability and avoid the nonlinear relation between the yaw rate and the driver steering. The AF S is shown When adding the roll control (by means of ASus) to the GCC system to enhance the stability, the DY C controller becomes less solicited, and thus, the yaw rate reapproaches to the desired one.

Figure 4.14 shows the LT R of the uncontrolled vehicle and the dierent control strategies. The results show that the LT R is the best when the GCC strategy is adopted compared to other strategies. It can also be noticed that activating the AF S alone has a drawback on the LT R. The fact is because the vertical stability is not considered in the development of the AF S controller. Divers comfort enhancements are noticed when adding the roll control to the GCC system. One observes that:

-The ADB on the rear wheels, provided by the AF S + DY C controller to stabilize the vehicle, are reduced as shown in Figure 4.15. The justication is that the ASus contributes to the stabilization process. According to the same gure, the Root Mean Square (RM S) of the braking torques are reduced by 47% on the rear left wheel and by 36% on the rear right wheel. The RM S reects the dissipated energy by the braking actuator, which has an impact on its life time. The peak values of both braking are also considerably reduced by 53% and 30% respectively.

-The vehicle speed drop caused by the braking is less reduced as shown in Figure 4.16.

-The critical longitudinal slipping of the rear tires caused by the ADB are limited as shown in Figure 4.17. Consequently, the ABS control system (supposed to be integrated into the chassis) will be less solicited. In the centralized architecture, the novelty with respect to other works in the eld of chassis control is that one single M IM O optimal controller generates the optimal additive steering angle provided by the AF S and the optimal ADB provided by the DY C to minimize -at once-all the vehicle state errors (yaw rate, side-slip angle, and roll angle). The optimal H ∞ control technique based on oine Linear Matrix Inequality (LM I) optimal solutions, in the framework of LP V systems, is applied CHAPTER 5.

CENTRALIZED AND DECENTRALIZED GLOBAL CHASSIS CONTROL ARCHITECTURES,

INVOLVING ACTIVE FRONT STEERING AND DIRECT YAW CONTROL

to synthesize the controller.

In the decentralized architecture, a heuristic solution is proposed by decoupling the control problem where the ST SM robust control technique is applied to derive the AF S control input which minimizes only the errors on the yaw rate, and the roll angle. Similarly, the DY C 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 GCC. 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.

Chapter's contributions with respect to literature are:

• a new centralized control structure, which combines the yaw rate control, the side-slip 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 GCC, by decoupling the control problem, into two sub-control problems, such as: AF S is responsible on the control of the yaw rate, and the roll angle; DY C 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 SCANeR Studio simulator.

To do so, the extended bicycle model of (1.37) is modied to include the control and exogenous inputs, then, it is used as the control synthesis model.

    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     B 1,2 δ d + δc Mz 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.
(5.2)

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     B1 δ d U bic ; y bic = X bic .
(5.3)

Control Synthesis Model

The control synthesis model, denoted by Plant P, is a linear vehicle model, which combines the coupled lateral and roll motions, while considering the control and exogenous inputs. This model suits the control problem of this work and it is given by the following system: done here on ψbic , to maintain the adherence between the tires and the road.

P lant P :            I z ψ = F yf l f + F yr l r + I xz θ + M z + M d, ψ, M V β + ψ = F yf + 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 , (
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 modied in the decentralized approach in order to introduce the weighting parameters. This issue is more developed later.

Centralized vs

Decentralized Approach

The global decentralized multilayer control architecture is shown in Figure 5.2. The main dierence w.r.t the centralized one is in the control layer, where each control input is generated by neglecting the other. Intuitively, AF S control input δ c is devoted to control the yaw rate ψ and the roll angle θ, while DY C control input M z is privileged to control side-slip angle β, to restrain the braking actuation. It will be proven later, 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 beneting from the robustness of the supertwisting algorithm. For more simplicity, actuators constraints are not considered in the controller structure, while a posterior lter is implemented to make the control inputs feasible. The decision layer is similar in structure to the one of the centralized approach. Based on the vehicle dynamics monitoring criteria, this layer generates three weighting gains λ ψ, λ β and λ θ . The goal of these gains is to promote/attenuate the ST SM controllers depending on the driving situation. These gains will be introduced into the reference trajectories, by modifying the extended bicycle model trajectories, to maintain each standalone controller stability.

Control Layers

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

CHAPTER 5.

CENTRALIZED AND DECENTRALIZED GLOBAL CHASSIS CONTROL ARCHITECTURES, The control layer architecture is drawn in Figure 5.3. As a standard H ∞ structure, it contains the controller K LP V /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.

INVOLVING ACTIVE FRONT STEERING AND DIRECT YAW CONTROL

+ - + - + - 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
The controller K LP V /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 LT I vehicle model of (5.2) (Plant P ).

Plant P of the generalized plant g is expressed in (5.2). 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 Mz (ρ 1 ) of Figure 5.3 are dened 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 1.6, are neglected during the controller design process). The general form of these weights [START_REF] Doumiati | Robust lpv control for vehicle steerability and lateral stability[END_REF] is given by the following (numerical values are given in Section 5.4, since they depend on the simulated vehicle and integrated actuators):

-W ψ(ρ 1 ) weights the yaw rate control objective:

W ψ(ρ 1 ) = ρ 1 s/M 1 + 2πf 1 s + 2πf 1 A 1 , (5.4)
where M 1 is suciently 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-o frequency f 1 where the vehicle operates [START_REF] Heiÿing | 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 + 2πf 2 s + 2πf 2 A 2 .
(5.5)

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 dierence 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 + 2πf 3 s + 2πf 3 A 3 .
(5.6) 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/2πf 4 + 1)(s/2πf 5 + 1) (s/α2πf 5 + 1) 2 , G 0 δ = (∆ f /α2πf 5 + 1) 2 (∆ f /2πf 4 + 1)(∆ f /2πf 5 + 1) , ∆ f = 2π(f 4 + f 5 )/2, (5.7)
where [f 4 , f 5 ] is the lter bandwidth. This lter forces the steering system to act at frequencies higher than the driver ones (f 4 ), to avoid driver annoyance, and lower than the actuator cut-o frequency (f 5 ). This lter 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, AF S is relaxed/promoted to maintain vertical stability.

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

W Mz (ρ 1 ) = ρ 1 10 -5 s/(2πf 6 ) + 1 s/(κ2πf 6 ) + 1 , (5.8)
where f 6 is the braking actuator cut-o 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 [START_REF] Sename | Robust control and linear parameter varying approaches: application to vehicle dynamics[END_REF] and [START_REF] Gu | Robust control design with MATLAB R[END_REF] for more information about the robust LP V /H ∞ theory.

CENTRALIZED AND DECENTRALIZED GLOBAL CHASSIS CONTROL ARCHITECTURES, et al., 1995], it can be formulated as: is the weighted controlled output vector.

INVOLVING ACTIVE FRONT STEERING AND DIRECT YAW

CONTROL 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
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   , ( 5
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 lter on the control input [Apkarian and Gahinet, 1995].

Problem resolution: LM I based LP V /H ∞ : The LM I based LP V /H ∞ problem consists in nding the controller K LP V /H∞ (ρ 1 , ρ 2 ), scheduled by the parameters ρ 1 and ρ 2 , such that:

K LP V /H∞ (ρ) : ẋc u = A c (ρ) B c (ρ) C c (ρ) 0
x c y ,

(5.10) which minimizes the H ∞ norm of the closed-loop LP V system formed by the interconnection of equations (5.9) and (5.10). The LP V /H ∞ controller of (5.10) can be found using the development given in Section 2.5.

According to the polytopic approach, the nal controller, K LP V /H∞ (ρ 1 , ρ 2 ), is a convex combination of the controllers synthesized at the vertices of the polytope [Apkarian et al., 1995] such as:

K LP V /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 ), (
    A     ψ β θ θ     X +     b u,11 b u,21 0 b u,41     B1 δ c U 1 +     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.
(5.12)

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     B2 M z U 2 +     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.
(5.13) 5.3.2.2 Decentralized Control Layer Synthesis: Super-Twisting Sliding Mode Controllers

In an intuitive way, the decentralized approach decouples the control problem into two sub-problems: AF S is responsible on the control of the vehicle yaw rate ψ and the roll angle θ by neglecting the eect of the DY C on these dynamics when developing the controller; DY C which is eective 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 duration which decelerates the vehicle, annoys the driver and causes long term tires wearing. To be noted also, as a consequence of the AF S controller, the side-slip angle is enhanced in the lowto-mid range of lateral stability, while it becomes ineective at high critical lateral dynamics.

The ST SM -based AF S control synthesis model (5.12) is similar to (5.2), while considering M z = 0, and thus, reducing B 1,2 to its rst column. The driver steering input δ d is neglected in the synthesis model, and then it is considered as a feedforward of the entire system.

The ST SM -based DY C control synthesis model (5.13) is similar to (5.2), while considering δ c = 0, and thus, reducing B 1,2 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 ST SM control technique, which is one of the most powerful robust control techniques that suit this control problem is chosen.

Consider the ane system form written as: Ẍ = f (X, t) + g(X, t)u(t),

(5.14) CHAPTER 5.

CENTRALIZED AND DECENTRALIZED GLOBAL CHASSIS CONTROL ARCHITECTURES, INVOLVING ACTIVE FRONT STEERING AND DIRECT YAW CONTROL

where X = [ ψ, β, θ, θ] T , and f (X, t) = AX. In the case of the AF S controller synthesis g(X, t) = B 1 and u = δ c as can be seen from (5.12). In the case of the DY C controller synthesis g(X, t) = B 2 and u = M z as can be seen from (5.13).

Let Their expressions are given later in this section.

Let dene three sliding variables as the following:

s ψ = e ψ = ψ -ψref , s β = e β = β -β ref , s θ = ėθ + k θ e θ = ( θ -θref ) + k θ (θ -θ ref ), (5.15)
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 AF S synthesis model of (5.12). 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 DY C synthesis model of (5.13). k θ is a positive constant gain which determines the time convergence of the state errors ėθ and e θ . Since the AF S is responsible of the control of both state variables ψ and θ, let dene a new sliding variable s ψ,θ such that:

s ψ,θ = c 1 s ψ + c 2 s θ , (5.16)
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 AF S and the DY C, have their control inputs appear in their rst derivatives (relative degree 1). This means their second derivative can be written as:

s(s, t) = Φ(s, t) + ξ(s, t) u(t)

(5.17) where Φ(s, t) and ξ(s, t) are unknown bounded signals.

The control objective is to achieve the convergence to the sliding surface dened by s = 0. Only the knowledge of s is required in real time.

Based on Section 2.4, the ST SM control inputs of the AF S and the DY C are respectively given by: (5.18) where α δ,1 and α δ,2 (resp. α Mz,1 and α Mz,2 ) are positive gains satisfying conditions of (2.15). τ δ and τ Mz are constants in the interval ]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 ψ,θ and s β in a nite time to zero. Once s β = 0, this means that the state β is converged to β ref . 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 ψref and ėθ + k θ e θ → 0, As mentioned before, ψref , β ref , θ ref , and θref depend on the extended bicycle model and the weighting gains sent from the decision layer. Thus, let dene ψref = λ ψ ψbic + (1 -λ ψ) ψ, (5.19) 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.

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 Mz,1 |s β | τ Mz sign(s β ) -α Mz,2 t 0 sign(s β )dτ,
β ref = λ β β bic + (1 -λ β )β, θ ref = λ θ θ bic + (1 -λ θ )θ, θref = λ θ θbic + (1 -λ θ ) θ,
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 (5.15) become equivalent to:

s ψ = ψ -ψref = λ ψ ( ψ -ψbic ), s β = β -β ref = λ β (β -β bic ), s θ = ( θ -θref ) + k θ (θ -θ ref ) = λ θ [( θ -θbic ) + k θ (θ -θ bic )],
(5.20) 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 modication of the sliding variables by the multiplication with the scheduling gains maintains the closed-loop stability of the individual AF S and DY C 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.

For SI ≤ SI, the vehicle is in normal driving situations, thus, the AF S is promoted for maneuverability purpose. It also enhances the lateral stability up to a moderate level. In this range, DY C is penalized. When the vehicle reaches critical lateral stability SI ≥ SI, then the DY C is promoted to enhance the lateral stability. Based on this analysis, the scheduled gain ρ 1 is designed to feed the LP V /H ∞ controller sucient knowledge about the weights to be promoted or attenuated. A sigmoid function (5.21) (see Figure 5.5.a) governs the relation between ρ 1 and SI, to ensure a continuous and a relatively smooth variation of ρ 1 . CHAPTER 5.

CENTRALIZED AND DECENTRALIZED GLOBAL CHASSIS CONTROL ARCHITECTURES, INVOLVING ACTIVE FRONT STEERING AND DIRECT YAW CONTROL

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 5.5 Scheduling parameters ρ 1 = ρ 1 - ρ 1 -ρ 1 1 + e -8 SI-SI (SI-SI+SI 2
) .

(5.21)

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 dened. A sigmoid function (5.22) (see Figure 5.5.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 ) .
(5.22) 5.3.3.2 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 LT R), 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 (5.23) (Figure 5.6.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 ) , λ ψ = 1 -λ β .
(5.23)

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 θ = 1 1 + e - 8 LT R-LT R (LT R-LT R+LT R 2 ) ,
(5.24)

Controllers Validation

This section is dedicated to validate the proposed controllers. It is composed of two sub-sections. The rst one is to show the eect of controlling the roll motion in a GCC architecture on the vehicle dynamics. To do so, the proposed LP V /H ∞ controller (centralized architecture of this chapter) is compared to the LP V /H ∞ controller of [START_REF] Doumiati | Robust lpv control for vehicle steerability and lateral stability[END_REF] (denoted by [6] in the simulations), where the roll angle is not introduced in the controller structure (as many powerful controllers developed in literature and cited in Section 0.2.1). The second sub-section is to validate and compare the proposed centralized and decentralized architectures.

Parameters numerical values of the proposed controllers used in simulations are provided in Table 5.1.

M 1 = 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; 0.7 δ a c,max ; T a b,max proposed controller the ability to handle more combinations of complex situations like maneuverability and rollover at the same time by using only AF S, and lateral stability and maneuverability at the same time, by using AF S+DY C. This summary is illustrated by the weights α i of Figure 5.12, which correspond to the controllers vertices of Figure 5.4. The controller of [6] has only two vertices, which oblige to switch between maneuverability and lateral stability objectives, while, the proposed controller is able to cover more complex combinations of situations thanks to four vertices controllers.

Figure 5.13 shows the uctuations of the scheduling parameters ρ 1 and ρ 2 , based on SI and LT R criteria. To be noted, ρ 2 remains at ρ 2 the most of the time, this means the rollover risk is rarely detected, and thus, the proposed controller is not totally stimulated, to prove its eectiveness. This issue is due to the fact that lateral stability handling risk appears in passengers cars before rollover risk. Thus, enhancing the provide more ecient results than the one in [6] for vehicles with higher center of gravity, where rollover risk can be detected at lower values than the lateral stability risk. Figure 5.14 shows the driver steering angle δ d , the AF S steering angle of both controllers δ c , and the total steering δ t . One can notice, that both controllers provide similar steering control angles, except at peaks and troughs, where the proposed controller, actuates more the AF S in order to handle all objectives (the combined complex objectives discussed before). Figure 5.15 shows the braking of the EM B at the left and right rear wheels. The proposed controller less activates the braking with an overall enhancement of the root mean square by 59% at the left braking, and 22% at the right braking. The peak amount of the braking is also reduced by 72% at the left wheel, and 16% at the right wheel. The vehicle speed, which drops due to frictions, is slightly improved as shown in Figure 5.16.

Centralized and Decentralized Architectures Validation and Comparison

This section is dedicated to validate and compare the proposed centralized and

Double Lane Change Scenario

The same DLC test done before is performed here. This simulation test shows the advantage of having such ADAS systems implemented into the vehicle. 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.

Figures 5.17 Meanwhile, the roll angle has tracked its reference only in some regions (especially around 1 s and 3 s). 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 maneuverability objective. When SI ≤ SI (specically around 1 s and 3 s), the scheduling gain ρ 1 (resp. λ ψ) of the centralized (resp. decentralized) architecture deviates toward ρ 1 = ρ 1 (resp. λ ψ = 0 or λ β = 1) which accounts for the lateral stability objective.

Based on this discussion, one can conclude on the yaw rate tracking of Figure 5.17 almost all the time except around 1 s and 3 s, and on the side-slip angle tracking of Figure 5.18 only around 1 s and 3 s. However, the side-slip angle tracks its reference all the time as a consequence of the yaw rate tracking, since both dynamics are correlated, and so their references. The LT R of the uncontrolled vehicle (Figure 5.21) exceeds LT R = LT R, which means that the vehicle is risky to roll-over, while both control architectures have covered back the LT R under LT R = LT R almost all the time except around 1 s and 3 s, and thus, they have succeeded to remove the rollover risk. The scheduling gain ρ 2 (resp. λ θ ) of the centralized (resp. decentralized) architecture was ρ 2 = ρ 2 (resp. λ θ = 0) almost all the time except around 1 s and 3 s where it deviates to a higher value. This means, only around 1 s and 3 s the proposed control architectures have switched the control objective to rollover avoidance, while at the remaining time of the simulation the LT R is enhanced because the roll angle of Figure 5.19 is diminished compared to the uncontrolled one as a consequence of the yaw rate enhancement since both dynamics are correlated. Figure 5.22 shows the driver steering angle δ d , the AF S steering angle of both controllers δ c , and the total steering δ t . One can notice, that both controllers provide similar steering control angles. 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 uctuations 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 reected by the AF S steering angle as can be seen from Figure 5.30, where the centralized steering angle was more stable than the decentralized one.

Figure 5.31 shows the braking of the EM B 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 Figure 5.32.

Conclusion

In this chapter, decision-based centralized and decentralized GCC architectures have been developed to improve the vehicle maneuverability, the lateral stability and the rollover prevention, by acting on the AF S and the ADB.

Conclusion and Perspectives

In this nal section, we rst summarize the contributions concerning the developments of GCC architectures. We then suggest possible improvements upon our work and topics for future studies.

Conclusion

After reviewing the vehicle dynamics and control developed in literature, we have contributed, in this thesis, some achievements in GCC that have led to several publications. We have proposed several original GCC architectures, in complementary with the existing works in literature, through immersing in the analysis of every interconnection between the dynamics and thanks to the robust control techniques dealing with M IM O complex and uncertain systems.

More precisely, in this thesis, we have contributed in the:

• exposition of the eect of the roll motion control through the ASus on the ride comfort, rollover avoidance, and the lateral stability based on time and frequency domains analysis. To do so, several control laws and roll reference trajectories are developed and compared.

• development of a decentralized multilayer GCC controller involving AF S, DY C, and ASus. Several control objectives are achieved, i.e, the maneuverability, lateral stability, rollover avoidance, and ride comfort. The ST SM technique is applied to develop the control layer, while some logic and fuzzylogic rules are developed in the decision layer to coordinate the dierent controllers.

• development and comparison of a centralized and a decentralized multilayer architectures for GCC, involving only AF S and DY C. To conclude, these are new features that can be embedded into on-road vehicles which enhance road safety and comfort. Of course, we have limited the studies to the scientic GCC aspect, without studying the feasibility of these developments which depends on several other factors, like actuator costs (especially for ASus),

vehicle size and mechanical assembly, sensors precision, computational complexity for calculators... All these factors have to be studied to nd the best architecture among others.

Perspectives

In the following, we present some potential research improvements in GCC, and future works beyond GCC:

• Test and validate the proposed architectures in real-time experiments.

• Deepen the study in the vehicle dynamics modeling to include more interconnections in the full vehicle model. This can help to better understand the real vehicle behavior.

• Extend the cooperation between the active safety actuators to include more coordination in the GCC architecture.

• Consider the driver in the loop, to develop more intelligent and interactive GCC controllers. This task enhances the human-machine cooperation for assisted driving.

• Develop learning-based decision makers. This is a very interesting task to cover more vehicle situations and to take more delicate and precise decisions. The learning process can be done by simulations and experiments which is more precise than researcher tuned parameters.

• Proceed to fault-tolerant control via actuators tasks reconguration and controllers' adaptation. This task has been started at Heudiasyc Laboratory in SYSCOVI project within two masters internships.

• Adapt the proposed GCC architectures to autonomous vehicle. This task can provide safe autonomous driving.

Appendices

.1 Extended Bicycle Model

where, g ψ(X ) = 1 Iz [-t f (F x,f l -F x,f r ) + l f (F y,f l + F y,f r ) -l r (F y,rl + F y,rr ) -t r (F x,rl -F x,rr )],

(33) f ψ,δ (X) = 1 I z [t f (F y,f l -F y,f r ) + l f (F x,f l + F x,f r )],

(34)

f ψ,Cz = 1 I z , (35)
g β (X) = (-1 +

lrcr-l f c f M Vx 2 )(- l f 2 c f +lr 2 cr IzVx ) ψ +(-1 + lrcr-l f c f M Vx 2 )( lrcr-l f c f Iz )β + (- c f +cr M Vx ) β, ( 36
) f β,δ (X) = (-1 + l r c r -l f c f M V x 2 ) l f c f I z , ( 37
) f β,Cz (X) = 1 I z (-1 + l r c r -l f c f M V x 2
).

(38)

F ij , F x,ij , and F y,ij are nonlinear functions of X detailed in Section 1.1.

  i i = {f : f ront, r : rear} [-] j j = {r : right, l : lef t} [-] z s,ij Sprung mass bounce at the corner ij [m] z us,ijUnsprung mass bounce at the corner ij[m] z r,ijRoad vertical prole at the corner ij [m] z s CG vertical displacement (bounce/heave) [m] z s,ij Sprung mass bounce at the corner ij [m] z us,ij Unsprung mass bounce at the corner ij [m] F ij

  angle of the tire ij[rad] C m,ij , C f,ij Motor, braking torques at the wheel ij [N.m] α f , α r Front, rear tire side-slip angle (bicycle model) [rad] σ x,ijLongitudinal tire slipping[-]

  Unsprung mass mass at the corner ij 40 [kg] K s,f r , K s,f l Suspension stiness coecient (front tires) 20000 [N/m] K s,rr , K s,rl Suspension stiness coecient (rear tires) 13000 [N/m] C s,f r , C s,f l Suspension damping coecient (front tires) 9830 [N.s/m] C s,rr , C s,rl Suspension damping coecient (rear tires) Roll moment of inertia of sprung mass 534 [kg.m 2 ] I y Pitch moment of inertia of sprung mass 1860 [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 ] g Gravity constant 9.81 [m/s 2 ] C σ,ij

•

  A.Chokor, R. Talj, M. Doumiati, and A. Charara. Vehicle roll control for lateral stability enhancement and rollover avoidance. Submitted to IEEE 2020 American Control Conference (ACC 2020). xvii Résumé Le Contrôle Global du Châssis (C GC) est une tâche cruciale dans les véhicules intelligents. Il consiste à assister le conducteur par l'intermédiaire de plusieurs fonctionnalités automatisées, notamment à des ns de sécurité active et de confort. Etant donné que les dynamiques de ces fonctionnalités sont interconnectées, les performances attendues sont parfois contradictoires. Par conséquent, le CGC consiste à coordonner les diérents systèmes ADAS Advanced Driving Assistance Systems an de créer des synergies entre les dynamiques interconnectées pour améliorer les performances globales du véhicule. Plusieurs stratégies de coordination puissantes ont déjà été développées, soit dans le monde académique, soit dans le monde industriel pour gérer ces interconnexions. Du fait que les besoins en matière de sécurité active augmentent d'un côté et que la technologie pouvant être intégrée dans les véhicules évolue, une intense activité de recherche et développement est toujours en cours dans le domaine du contrôle global du châssis. Cette thèse analyse diérentes interconnexions dynamiques et développe des nouvelles stratégies CGC dans lesquelles le braquage actif avant, le freinage diérentiel actif et les suspensions actives sont coordonnés -tous ensemble ou partiellement -an d'améliorer les performances globales du véhicule, à savoir l'évitement du renversement, la stabilité latérale, le confort de conduite (manoeuvrabilité) et confort des passagers. Plusieurs architectures multicouches formées par trois couches hiérarchiques sont proposées. La couche inférieure représente les actionneurs implémentés dans le véhicule qui génèrent leurs entrées de commande en fonction des

  ADAS systems are formed by several single-actuator approaches that have been proposed and marketed, such as: Vehicle Stability Control (V SC) or Electronic Stability Program (ESP ) including Direct yaw Control (DY C), Anti-lock Braking System (ABS), Active Dierential Braking (ADB) and others, to enhance the vehicle handling and stability; Active Front Steering (AF S), 4 Wheel Steering (4W S), and Electronic Power Steering (EP S)

  The most commonly used criterion in literature is based on the load transfer between the right and left tires' vertical forces. It is called the Load Transfer Ratio (LT R) which normalizes the load transfer by the total vertical forces on the tires. Because tires' vertical forces are hard to be measured, the LT R criterion is used only for performance evaluation by simulation, while to schedule the controllers' performances in real-time application, several estimations of LT R are developed in literature. The estimated criterion is based on a liner relation between the vehicle

From

  Figure 1.1 Passive suspension system

Figure 1

 1 Figure1.3 shows the quarter vehicle vertical model. When a vehicle moves, the suspension system and the tires undergo dynamical motions/vibrations. The equations that describe the system dynamics at the ij (i = {f : f ront, r : rear} and

  Figure 1.4 Full vehicle vertical model

F

  Figure 1.5 Vehicle lateral/longitudinal model

Figure 1 . 8

 18 Figure 1.8 Tire vertical model

  Figure 1.9 Tire side-slip

Figure 1 .

 1 Figure 1.11 Wheel dynamics

Figure 1 .

 1 Figure 1.12 Steering angle

Figure 1 .

 1 Figure1.12 shows the front wheels steering angle applied to both simulations, where at 100 km/h, it solicits the lateral and roll motions of the vehicle, making the vehicle operates in a moderate region in terms of lateral stability and load transfer. Figures 1.13, 1.14, and 1.15 respectively show and validate the solicitations of the vehicle yaw rate, the lateral acceleration, and the roll angle. The pitch angle and the vehicle heave are less solicited by this test as shown in Figures 1.16, and 1.17. Figures 1.18, 1.19, and 1.20 respectively show and validate the vertical, lateral, and longitudinalforces of all the vehicle tires. Figure1.21 shows the longitudinal speed of the vehicle which remains constant 100 km/h. The longitudinal speed had to drop because of the friction of tires' with the road, but to perform the ISO test, the built-in longitudinal speed controller of the simulator adjusts the speed by introducing a motor torque on the wheels which is also applied to the full vehicle model. Figures 1.22, 1.23, and 1.24 respectively show and validate the side-slip angle, side-slip rate, and the SI of the vehicle which operates in the moderate lateral stability region (SI ≈ 0.5). Figure1.25 shows and validates the LT R which is also in the moderate range.

Figure 1 .

 1 25 shows and validates the LT R which is also in the moderate range.1.4 Bicycle ModelThe vehicle bicycle model is a simplied version of the full vehicle model. It is usually used as the desired reference model when controlling/stabilizing the lateral vehicle motion. The bicycle model consists of merging each two wheels of the same axle together at the center of the axle as shown in Figure 1.26. The bicycle expresses the yaw rate and the side-slip angle of the vehicle at its center of gravity as the following:

Figure 1

 1 Figure 1.13 Yaw rate

Figure

  Figure 1.26 Bicycle model

  extended bicycle model is a coupled lateral-roll vehicle model. It is a linear simplied vehicle model which combines the vehicle yaw and side-slip to the roll motion. This model is suitable to analyze the eect of the roll motion control on the lateral stability of the vehicle. A frequency and time domain study is conducted in Chapter 3 to show this coupled-eect. The extended bicycle model is inspired from

  , the vehicle dynamics is discussed. A full vehicle nonlinear model is presented and validated, and some other linearized models are exposed. The performance criteria are also quantied. Finally, the active actuators that are involved in this thesis are modeled. Next chapter reviews the control architectures and techniques used in the GCC eld.

  can identify two types of control architectures: multilayer centralized and multilayer decentralized. The reviewed control techniques are the essential keys for controllers' design in next chapters. Mainly, three advanced control techniques are used in this thesis: a Lyapunov-based control technique in the framework of Immersion and Invariance, the Super-Twisting Sliding Mode ST SM control technique, and the H ∞ control technique in the framework of the LP V system. Both the Lyapunov-based and the ST SM are nonlinear robust control techniques applied in the decentralized architectures for Single-Input-Single-Output (SI SO) systems, while the H ∞ is a linear robust and optimal control technique applied in the centralized architectures for M IM O systems.

  Figure 2.1. After collecting all the vehicle operation information, including data from the sensors and the state estimators, the upper layer (decision layer) monitors the driver's intentions and the current vehicle state, then, it informs, through scheduling parameters, the middle layer about the encountered situation to adjust its behavior. By the way, the upper layer is designed to coordinate interactions amongst all the subsystem controllers. Thereafter, the control inputs are generated, by the middle layer (control layer), in order to achieve the desired vehicle states, depending on the architecture of this layer (centralized or decentralized). If the middle layer architecture is centralized (Figure 2.1.a), that means a single central controller generates the control inputs which cooperate all together to control all the variables of interest. If the middle layer architecture is decentralized (Figure 2.1.b), that means several controllers exist in the middle layer, where each controller generates only one control input which controls a part of the variables of interest. Finally, each of the individual lower layer (actuator layer) executes its local control objectives to control the vehicle dynamics.

Figure 2 . 1

 21 Figure 2.1 Multilayer centralized and decentralized control architectures

Figure 2

 2 Figure2.3 Super-twisting sliding mode (modied from[START_REF] Shtessel | Sliding mode control and observation[END_REF]

  Figure 3.2 Static rollover

  of the vehicle with roll controller (3.16) and the vehicle without roll controller (3.14) can be now compared. More specically, the transfer functions of the SI (as function of the side-slip dynamics) w.r.t the exogenous steering input are evaluated for both vehicles in the next sub-section.

Figure 3 . 4

 34 Figure 3.4 Bode plot SI/δ d ; V = 100 km/h

  increases signicantly, while controlling the roll angle remarkably reduces the response of G SI δ . In this frequency range, for a high amplitude (A = 0.1 rad), the SI of the uncontrolled roll vehicle exceeds SI = 1, while the controlled one establishes acceptable behavior.

AFigure 3 . 6

 36 Figure 3.6 Roll angle and SI for δ d = 0.1sin(6t); V = 100 km/h

  54 CHAPTER 3. EFFECT OF ROLL MOTION CONTROL ON VEHICLE PERFORMANCE in the manifold, where the roll error dynamics obey to the following equation: ėθ + k 1θ e θ + k 2θ

  żθ = -α θ z θ , (3.29) which makes (3.28) always negative if α θ > 0. Thus: ëθ + k 1θ ėθ + k 2θ e θ = -α θ ( ėθ + k 1θ e θ + k 2θ θ from (3.2) in (3.30), the control input M θ can be found as in (3.32):

  .7. The roll reference generator block represents the development given in (3.11). The switch block is manually actuated upon simulation to choose the desired roll reference. The roll controller block represents the three dierent control techniques: Linear QuadraticRegulation LQR, Lyapunov-based, and ST SM . This block has the roll angle error and its rst derivative as inputs, and M θ as controller output. The control input M θ has to be generated by the actuators forces or torques depending on the integrated technology. An example (but not restricted to) is the distribution of M θ between the four active forces U ij of the ASus. This is done in the control allocation unit as described in (4.14):

  Figure 3.7 Control scheme

Figure 3 .

 3 Figure 3.9 Fishhook steering

Figure 3 .

 3 Figure 3.10 shows the roll angle for a vehicle with passive suspensions (uncontrolled roll) which turns up to 7 • in both directions. All the three controllers are ecient to control the roll angle toward zero. As a comparison, beside the robust terms of the feedback (3.32), the Lyapunov-based controller compensates all the dynamics of the roll angle as expressed in (3.33) to have this performance. That means, in real application, a strong knowledge (estimation, measurement, parameters exactitude, ideal modeling) on the components of M θeq is needed. On the other side, the ST SM controller is ecient in controlling the roll angle by only the generation of the feedback from the roll angle and velocity (3.35), with no need to compensate the roll dynamics. This fact is due to the robustness of the ST SM control law.

Figure 3 .

 3 Figure 3.11 ay comparison ; θ des = 0

Figure 3 .

 3 Figure 3.11 shows the lateral acceleration of the uncontrolled roll vehicle which is approximately the same for all roll controllers. This means that the roll control does not aect the lateral acceleration to avoid rollover. In fact, the controller aims to elevate the maximal safe lateral acceleration expressed in (3.10), which depends on the roll angle. As the roll angle is minimized to zero, thus, the maximal safe lateral acceleration increases as shown in the same gure. Obviously, the roll control

Figure 3 .Figure 3 .

 33 Figure 3.13 shows the uncontrolled roll angle (same as in Figure 3.10), the desired one, and the controlled ones. The Lyapunov-based controlled one, and the ST SM controlled one accurately track the desired roll trajectory with a small error especially for the ST SM controller between 4 and 5 seconds, where the roll angle turns quickly. In fact, it is because the sliding mode controller has no knowledge on the roll dynamics. This transient behavior can be enhanced when considering the equivalent control input. The LQR controller, as is synthesized to minimize the

Figure 3 .

 3 Figure 3.14 shows the approximately confounded lateral accelerations of the uncontrolled roll vehicle, and the controlled ones. It also shows the maximal safe lateral acceleration which increases more comparing to the case where the roll angle is minimized to zero. This issue drives away the rollover risk at this range of lateral acceleration.

Figure 3 .

 3 Figure 3.14 ay comparison ; θ des opposite direction

Figure 3 .

 3 Figure 3.19 shows the control inputs for both Lyapunov-based and sliding mode controllers which are in fact the ASus forces provided by the actuators (after saturating and ltering). This gure shows that their maximal value is around 4000 N which is feasible by the ASus actuators without any saturation. This fact makes these developed forces realistic and can be implemented to the vehicle after controlling the ASus actuators.

  Figure 3.20 compares the SI for the vehicle when controlling the roll angle toward zero and toward θ des of eq. (3.11). This Figure also shows that the control of the roll angle in the opposite direction brings more enhancement on the lateral stability than just minimizing it to zero. Hence, it could contribute to reduce the energy consumption of other actuators used for lateral stability (AF S, ADB...).

  in the opposite direction. For rollover avoidance, the proof of this improvement is done in time domain, by means of dening the safe lateral acceleration threshold and analyzing it when controlling the roll angle. The proof of the improvement on the lateral stability is done in the frequency domain, by means of analyzing the frequency response of the lateral SI w.r.t the steering input, in both cases, when the roll angle is controlled or not. Lyapunov-based, ST SM , and LQR controllers have been developed to generate the roll moment which controls the roll angle. Controlling the roll in the opposite direction has demonstrated more advantages on the lateral stability and rollover avoidance compared to only minimizing the roll angle to zero.

  Figure 4.1 Proposed GCC scheme

  α v,1 and α v,2 are positive gains satisfying conditions of (2.15), and τ v ∈]0, 0.5]. The super-twisting algorithm guaranties the convergence of s v in a nite time to zero. Once s v = 0, the states v and v exponentially converge to v ref and vref respectively if λ v > 0. The function sign is smoothed by the approximation sign(s v ) = sv |sv|+εv , where ε v > 0.

Figure 4 . 3

 43 Figure 4.3 Active forces distribution

  (4.14) CHASSIS CONTROL INVOLVING ACTIVE FRONT STEERING, DIRECT YAW CONTROL AND ACTIVE SUSPENSIONS -A DECENTRALIZED ARCHITECTURE

Figure 4

 4 Figure 4.5 AF S and DY C coordination functions

Figure 4 . 8

 48 Figure 4.6 Fuzzy sets of the input e φ

Front

  Figure 4.9 Front wheels steering

Figure 4 .

 4 Figure 4.11 Lateral Stability Index SI

CHASSIS

  Figure 4.15 Active Brake torques

  Figure 5.1 Centralized GCC architecture

  multilayer control architecture is shown in Figure 5.1. 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 M IM O LP V /H ∞ centralized controller, to simultaneously enhance the vehicle maneuverability, the lateral stability and the rollover avoidance. Trajectories' references ψref , β ref , and θ ref of the controlled variables are exactly the ones generated by the Extended Bicycle Model 5.3. Two endogenous time-varying scheduling gains/parameters ρ 1 and ρ 2 schedule the M IM O LP V /H ∞ controller objectives. A decision maker (in the higher layer) monitors the vehicle situation and instantly sends the values of the scheduling parameters, based on lateral stability (SI ) and rollover (LT R) criteria. Based on all these information, the M IM O LP V /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-o , β ,θ , θ , V x ...

Figure 5 . 2

 52 Figure 5.2 Decentralized GCC architecture

  Figure 5.3 Control layer architecture

  .9) where ρ = {ρ 1 , ρ 2 }, x includes the state variables of Plant P and of the weighting functions, w = [ ψref , β ref , θ ref , M d, ψ, F d,y , M d,θ ] T is the exogenous input vector, U = [δ c , M z ] T represents the control inputs, y = [ ψ, β,θ] T is the measurement vector fedback to the controller, y e = [ θ] T is the exogenous output, and z = [Z 1 , Z 2 , Z 3 , Z 4 , Z 5 ] T

  5.11) as shown in Figure 5.4, where each vertex represents an objective (more discussion is given in Section 5.3.3.1).

  dene E = [e ψ, e β , e θ , ėθ ] T = [ ψψref , β -β ref , θ -θ ref , θθref ] T the error vector between the actual and the desired states. ψref , β ref , θ ref , and θref depend on the extended bicycle model and the weighting gains sent from the decision layer.

  θ (resp. θ) exponentially converges to θ ref (resp. θref ) if k θ > 0.

Figure 5 . 6

 56 Figure 5.6 Scheduling gains λ ψ and λ β

5Figure 5 . 7 Figure 5 .

 575 Figure 5.7 Yaw rate comparison

Figure 5 .

 5 Figure 5.13 Decision layer -Inputs vs Outputs

Figure 5 .

 5 Figure 5.14 Steering angle comparison

  Figures 5.17, 5.18, and 5.19 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 5 .

 5 Figure 5.19 Roll angle comparison (DLC test)

  Figure 5.23 shows the braking of the EM B 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 5.5. CONCLUSION 101 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 Figure 5.24. 5.4.2.2 Fishhook Scenario The shhook 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 DLC. 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 Figure 5.30. The uncontrolled vehicle yaw rate of Figure 5.25 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 Figure 5.26 and Figure 5.27. Figure 5.28 and Figure 5.29 respectively show the SI and the LT R with their corresponding scheduling gains. These gures

Figure 5 .Figure 5 .Figure 5 .Figure 5 .

 5555 Figure 5.20 Lateral stability and scheduling gains (DLC test)

  The novelty w.r.t literature is the introduction of the roll control into the GCC strategy without the need to include the suspensions. We have achieved several control objectives, i.e, the maneuverability, lateral stability, rollover avoidance, and ride comfort (only roll motion). The M IM O LP V /H ∞ (respectively ST SM ) robust control technique is applied to develop the control layer of the centralized (respectively decentralized) architecture, while endogenous weighting parameters are developed in the decision layer for both architectures to coordinate the dierent controllers and objectives. CHAPTER 5. CENTRALIZED AND DECENTRALIZED GLOBAL CHASSIS CONTROL ARCHITECTURES, INVOLVING ACTIVE FRONT STEERING AND DIRECT YAW CONTROL

a

  11 = (c 1 + I xz * d 1 * (1 + b 1 ))/(d 4 * I z ), a 12 = (c 2 + I xz * d 1 * b 2 )/(d 4 * I z ), a 13 = I xz * d 2 /(d 4 * I z ), a 14 = I xz * d 3 /(d 4 * I z ), a 21 = b 1 + M s * h θ * a 41 /(M * V ), a 22 = b 2 + M s * h θ * a 42 /(M * V ), a 23 = M s * h θ * a 43 /(M * V ), a 24 = M s * h θ * a 44 /(M * V ), a 41 = d 1 * (1 + b 1 )/d 4 , a 42 = d 1 * b 2 /d 4 , a 43 = d 2 /d 4 + d 5 * a 13 , a 44 = d 3 /d 4 + d 5 * a 14 , (25) b u,11 = (c δ + I xz * a δ /I z )/c 4 , b u,21 = b δ + M s * h θ * b u,41 /(M * V ), b u,41 = d 1 * b δ /d 4 + d 5 * b u,11 , b u,12 = 1/(I z * c 4 ), b u,22 = M s * h θ * b u,42 /(M * V ), b u,42 = d 5 * b u,12 , b u,13 = d 5 /I z , b u,23 = M s * h θ * d 5 /(I xz * M * V ), b u,43 = d 5 /I xz ,(26) b d,11 = 1/I z , b d,22 = 1/M * V, b d,22 = 1/I z .

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 2 Vehicle parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 4.1 Fuzzy rules decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1 Controllers' Parameters for Simulation . . . . . . . . . . . . . . . . . . 93

		Notations
	Table 1 Notations
	Symbol Description	Unit
	ix

Table 2

 2 Vehicle parameters

	Symbol	Description

  Entrées-Multi-Sorties génère les entrées de commande optimales basées sur la technique de commande LP V /H ∞ . Dans l'architecture décentralisée, les contrôleurs sont découplés. La technique ST SM (Super-Twisting Sliding Mode) est appliquée pour déduire chaque entrée de commande. Les architectures proposées sont testées et validées sur le simulateur professionnel SCANeR Studio et sur un modèle complexe non linéaire du véhicule. La simulation montre que toutes les architectures sont pertinentes pour le contrôle global du châssis. Celle centralisée est optimale, complexe et garantit la stabilité globale, tandis que celle décentralisée ne garantit pas la stabilité globale, mais elle est intuitive, simple et robuste. Mots Clés : Contrôle Global du Châssis, Contrôle hiérarchique, Contrôle centralisé, Contrôle décentralisé, Mode glissant Super-twisting, LP V /H ∞ .

trôle se dièrent par leurs structures, où des contrôleurs centralisés et décentralisés sont développés. Dans l'architecture centralisée, un seul contrôleur optimal MEMS xix Multi-

  The novelty w.r.t literature is the introduction of the roll control into the GCC strategy without the need to include the suspensions. The M IM O LP V /H ∞ (respectively ST SM ) robust control technique is applied to develop the control layer of

	0.5. CONCLUSION	9
	0.5 Conclusion	Chapter 1
	In this chapter, the motivation of this work and a state of the art on GCC
	are exposed. Then, the vehicle dynamics performance criteria, including rollover avoidance, lateral stability, driving comfort, and ride comfort, are presented. Next chapter reviews the vehicle dynamics. Vehicle Dynamics
	This chapter presents and analyzes several vehicle models aiming at understanding
	the vehicle dynamics. A complex nonlinear model called full vehicle model which is
	composed of several connected sub-models i.e. the vertical model, the lateral model,
	the longitudinal model, the road/tire contact model and the wheels dynamics model,
	is developed and validated on the professional simulator SCANeR Studio. The full
	vehicle model serves to validate the active controllers when real experimentation and
	access to a professional simulator are not available. Another vehicle model which is
	called bicycle model is also presented in this chapter. The bicycle model is a simple
	linear model which describes the lateral vehicle motion when the vehicle operates in
	a stable region (no lateral skidding). This model can be used as a nominal (synthesis)
	model for controllers development. An extended bicycle model is also presented in
	this chapter. It a simple linear model which describes the coupled roll-lateral motion,

the centralized (respectively decentralized) architecture, while endogenous weighting parameters are developed in the decision layer for both architectures to coordinate the dierent controllers and objectives.

Finally, the thesis concludes with a summary about the obtained results and an outlook about potential improvements of our work.

  2.20) CHAPTER 2. REVIEW ON CONTROL ARCHITECTURES AND TECHNIQUES which minimizes the H ∞ norm of the closed-loop LP V system formed by the interconnection of equations (2.19) and (2.20).Problem resolution: LM I based LP V /H ∞ : Thanks to the Bounded Real Lemma (BRL) extended to LP V systems, this controller can be found. According to system (2.19) and via the change of basis expressed in[START_REF] Scherer | Multiobjective output-feedback control via lmi optimization[END_REF], a non conservative LM I that expresses the same problem as the BRL is formulated in (2.24) and solved by a Semi-Denite Program (SDP ), while minimizing γ for

  • .

			a y
		Rollover zone
		Uncertain zone
			t f / h
		Safety zone	0.7 t f / h		a y lift -off
		a y -10°θ θ min θ des	10°θ	a y safe max	θ
		Safety range :
		Figure 3.3 Rollover risk evaluation
	-At a lateral acceleration equal to the maximal static safe lateral acceleration
	threshold 0.7	t f

h g, the desired roll angle is equal to the maximal achievable roll angle 10 • (vehicle design constraints).

  ASus controller dedicated to control the roll, pitch and heave

	CHASSIS CONTROL INVOLVING ACTIVE FRONT STEERING, DIRECT YAW CONTROL
	AND ACTIVE SUSPENSIONS -A DECENTRALIZED ARCHITECTURE
	motions of the sprung mass, respectively described by equations (4.2), (4.3) and
	(4.4). A general form of these dynamics can be written as:

Table 4

 4

		.1 Fuzzy rules decision matrix
	λ φ e φ	ėφ NB NS ZE PS PB NB PB PB PM PM PS NS PB PM PS PS PM ZE PM PS PS PS PM PS PM PS PS PM PB PB PS PM PM PB PB

  Based on this model, the M IM O LP V /H ∞ centralized controller structure is formalized, while detailing the control objectives represented as variable-weighted lters, to nd the LP V /H ∞ controller which guarantees H ∞ performances between the exogenous inputs and the controlled variables, based on oine LM I optimization, in the framework of the polytopic approach. Then, we present, in a decentralized architecture, both the ST SM based AF S and DY C controllers, as model-based controllers, where the control inputs are saturated and ltered a-posteriori. The decision layer of both the centralized and decentralized control architectures are also presented. The SI (1.32) and the estimated LT R (1.31) are used as the performance criteria to evaluate the lateral stability and rollover risks. Finally, we test and validate both architectures thanks to the co-simulation between Simulink

	5.2. CONTROL SYNTHESIS MODEL	83
	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
	and SCANeR Studio simulator.

Table 5 .

 5 1 Controllers' Parameters for Simulation

	Parameters	Values

Acknowledgments