Mahdi, Kasem Mohammad Kasem

Ali

Ali Hassan

Maarouf Hussein

Ahmad Moham- Mad Mahdi

Title Hierarchical Approach for Global Chassis Control of an Electric Vehicle

Keywords: ABS control, Active suspension system, Automotive, CRONE Controller, Global Chassis Control, Lateral Stability, Vehicle dynamics

Road transportation is shifting significantly toward electrification around the globe. A market for light electric mobility solutions had emerged where all-inwheel devices are expected to play an important role in this new trend. This technology offers new opportunities and raises new challenges in Global Chassis Control (GCC) that rises, recently, to remarkable levels. This study is based on a supervision control approach for vertical, longitudinal, and lateral control in light electric vehicles. The developed control system designs rely on the CRONE method which can ensure the robustness of the stability degree against the system parametric variations. For vertical dynamics, various control solutions are developed for automotive suspensions to improve passenger comfort and road holding. For longitudinal dynamics, a study for the ABS function is done for braking system enhancement while considering the effect of the vertical dynamics. Then, a combination of ABS control and suspension control is presented in the sense of reducing the deterioration effect of vertical dynamics. Finally, the work is concerned by the development of vehicle lateral stability control, where the effect of the vehicle vertical dynamics was analyzed. The obtained results verify the effectiveness of the designed control strategies in enhancing the vehicle comfort, handling, and safety. Moreover, the well understanding of the influence of the vertical dynamics, as well as the key role of the controlled suspension on other vehicle dynamics, will open up new prospects to the development of new strategies for global chassis control of light electric vehicle.

Titre Approche Hiérarchique pour le Contrôle Global du

Châssis d'un Véhicule Électrique

Résumé

Le transport routier évolue considérablement vers l'électrification dans le monde entier. Un marché pour les solutions de mobilité légère électrifiée a émergé. Les dispositifs à transmission intégrale joue un rôle important dans cette nouvelle tendance. Cette technologie offre de nouvelles opportunités et pose de nouveaux défis dans le contrôle de châssis globale « Global Chassis Control (GCC) » qui a récemment atteint des niveaux remarquables. Cette étude présente une approche de supervision pour le contrôle vertical, longitudinal et latéral dans les véhicules électriques légers. Le système de contrôle développé repose sur la méthode CRONE qui peut assurer la robustesse du degré de stabilité contre les variations paramétriques du système. Pour la dynamique verticale, diverses solutions de contrôle sont développées pour les suspensions automobiles afin d'améliorer le confort des passagers et la tenue de route. Pour la dynamique longitudinale, une étude de la fonction ABS est réalisée pour l'amélioration du système de freinage, tenant en compte l'effet de la dynamique verticale. Ensuite, une combinaison de contrôle ABS et de contrôle de suspension est présentée dans le but de réduire l'effet de détérioration de la dynamique vertical. Enfin, les travaux portent sur le développement du contrôle latéral de la stabilité du véhicule, où l'effet de la dynamique verticale du véhicule est analysé. Les résultats obtenus permettent de vérifier l'efficacité des stratégies de contrôle conçues pour améliorer le confort, la maniabilité et la sécurité du véhicule. En outre, la bonne compréhension de l'influence de la dynamique verticale, ainsi que le rôle clé de la suspension contrôlée sur les autres dynamiques du véhicule, ouvrira de nouveaux horizons pour le développement de nouvelles stratégies pour le contrôle global du châssis des véhicules électriques légers. First and foremost, I would like to thank God Almighty for giving me the strength, knowledge, ability and opportunity to undertake this research study. Without His blessings, this achievement would not have been possible. In the belief that: "The most thankful among you to God is he who is most thankful towards people"-Ali Zayn Al-Abidin [AS], I would like to express some words to the people who have guided me, shared time with me, and supported me during the past three years.

Mots-clés

First, I would like to gratefully and sincerely thank my advisors, Professors Xavier Moreau and Clovis Francis, for their scientific understanding and for their precious support during my PhD studies. I have been so lucky to work with them, who were always available and gave me the best research conditions and professional opportunities. I look forward to further collaboration in the future. I would like to thank Professor Hassan Noura for being the president of my doctoral committee; Professors Lydie Nouveliere, Reine Talj, and Hassan Shraim for their time to review my PhD thesis. Their detailed and valuable comments have contributed to the quality of this manuscript.

I have great pleasure in acknowledging my gratitude to my colleague in IMS Lab, Emna Hamrouni, for having interesting scientific and non-scientific discussions and sharing beautiful moments with me. I would like also to thank André Benine-Neto for his time, advices, and sharing knowledge which were really helpful to me.

General Introduction

Curiosity leads researchers from different academic disciplines to look backward in the past for the origins of their subjects. They constantly ask questions like how it was started and where was the beginning? Who invent or discover and when? Indeed, reviewing the history is essential, it helps us to understand the linkages between past and present, and analyze the legacies of the past in the present to contribute to our future. In technology, history reveals a story of great inventors with brilliant ideas and inspired events. In those days, inventions were still attributed to "inventors" and met the societal needs, however today, they can no longer compete where inventions are now developed by giant and powerful corporations.

Back to History of Automobile

To 'automobile', we're at least talking about 540 years of car history. That definition is confined to vehicles that are capable to propel itself [START_REF] Dunn | Understanding the Impact of Technology : Do Advanced Driver Assistance and Semi-Automated Vehicle Systems Lead to Improper Driving Behavior[END_REF], otherwise, humans have possessed knowledge of the wheel for several thousand years, and we've been using animals as a source of transportation for nearly that long. The automobile as we know it nowadays was not invented in a single day by a single inventor. The history of the automobile reflects an evolution that took place worldwide involving many different innovators.

Sometime around the year 1478, Leonardo da Vinci drew out his plans for the world's first self-propelled vehicle powered by coiled springs (Fig. 1) [START_REF] Mcshane | The Automobile: A Chronology of Its Antecedents, Development, and Impact[END_REF]. The vehicle was actually designed as a special attraction for Renaissance festivals, and like many of Leonardo's sketches, the car remained on paper throughout his lifetime. Amusingly, the first built world's car might be meant to be a toy for the Chinese Kangxi Emperor and invented by Ferdinand Verbiest (Iannaccone, 2004). At that time, steam technology was still in its infancy, but Verbiest was able to build a rudimentary, ball-shaped boiler, which then forced steam towards a turbine that could turn the back wheels (Fig. 2). The 1700s were dominated by various inventors working to perfect the steam engine -Thomas Newcomen and James Watt are probably the most famous of these, but there were many more. The first person to take a steam engine and place it on a full-sized vehicle was probably a Frenchman named Nicolas-Joseph Cugnot, who between 1769 and 1771 built a steam-powered automobile called "fardier à vapeur" [START_REF] Lucendo | Cars of Legend: First Cars of History[END_REF]. The machine could reputedly move at a little over 2 miles per hour, but it needed to be refueled with wood every 15 minutes, for which the fardier had to come to a complete stop. One story says that the second of Cugnot's two vehicles crashed into a wall in 1771 (Fig. 3) due to poor weight distribution and so was unable to handle even moderately rough terrain, which might make it the first ever automobile accident. iii

The development of automobile continues where American Oliver Evans receives the first US patent for a steam-powered land vehicle in 1789. In 1807 the Switzerland inventor Francois Isaac de Rivaz invented the first automobile powered by internal combustion, however, his engine turns to be unsuccessful. It takes until 1867, where significant improvement is done on the internal combustion engine by the German Nikolaus August Otto. His engine was the first to efficiently burn fuel directly in a piston chamber, then, in 1877 the four-cycle internal combustion engine considered as the first prototype of modern car engines (E. C.S, 1932). Thereafter, the internal combustion engine provided the pathway to the first modern automobiles, with Karl Benz generally getting the credit for his successful invention in 1886, the "Motorwagen", and thus dominates the automobile market until the 1920s (Fig. 4). Later on, Henry Ford introduced the Model T and innovated mass-production techniques manufacturing over 15 million Model Ts (Fig. 5) by 1927 [START_REF] Parker | Henry Ford and the Automobile Industry[END_REF], and thus, Ford, in addition to General Motors and Chrysler emerged as the "Big Three" auto companies by the 1920s.

Advancements in Vehicular Technology

The evolution of the automobile was not limited to just improving the engines. Indeed, research and innovations had increased considerably in order to improve the performance, safety, and comfort of vehicles. Among these inventions are, for example:

the tire: invention by Robert William Thomson (1845) then first patent filings by John Boyd Dunlop (1888) and the Michelin brothers (1891).

braking systems: drum brakes (Louis Renault, 1902), internal shoe brakes, Hydraulic brakes (Malcolm Loughead,1918), disc brakes (William Lanchester, 1902).

transmission systems: manual or automatic gearboxes;

optical systems: electric headlights, mirrors;

suspensions: mechanical suspensions, pneumatic suspensions… safety systems: deformable materials, seat belt by Nils Bohlin, Volvo's first safety engineer, in 1959.

In line with the increasing demands on high comfort and safety vehicles, a variety of advancements continued to appear in a wide spectrum of scientific and engineering fields. This was associated with the evolution of enabling technologies, (electrical, electronics and software technologies) such as microprocessors and wireless networks. The rapid development of electronics and software tools during the last four decades coupled with the increase in computational speeds and a reduction in the cost of these tools has touched every aspect of life, industry and, of course, the automobile. Some of the developed systems that provided a significant improvement regarding safety are:

-Global Positioning System (GPS) Navigation by the U.S. Department of Defense (DoD) in the early 1970s [START_REF] Rahiman | An overview of development GPS navigation for autonomous car[END_REF], -Proximity sensors, -Anti-Lock Braking System (ABS) [START_REF] Bosch | Automotive handbook / Bosch[END_REF], v -Electronic Stability Control (ESC) or Program (ESP) [START_REF] Linder | Methods for the evaluation of traffic safety effects of Antilock Braking System (ABS) and Electronic Stability Control (ESC): A literature review[END_REF] -Adaptive Cruise Control (ACC) -Adaptive headlamps using Optoelectronic Technologies, in addition to other systems that developed and are implemented in most nearly manufactured vehicles. Notably, the number of electronic control units (ECUs) has grown from just one or two per vehicle in the 1970s to over 70 in today's high-end automobiles, which offer about 500 features [START_REF] Gharavi | Scanning advanced automobile technology[END_REF].

Road to "Autonomous" automobile

Today, the evolution of automobile directed to the development of socalled autonomous vehicles (AV), and also known as, automated car, selfdriving car or driverless car. The basic concept of road vehicle automation refers to the replacement of some or all of the human labor of driving by electronic and/or mechanical devices [START_REF] Shladover | Connected and automated vehicle systems: Introduction and overview[END_REF]. Origins of the automated driving technology can be traced back to the early 20th century. At that time, the technology was concentrated on autonomous speed, brake, lane control, and other basic cruise control aspects [START_REF] Pendleton | Perception, planning, control, and coordination for autonomous vehicles[END_REF].

In fact, global automobile production is increasing rapidly, and almost certain to rise far higher, where annual production has nearly doubled since 1997-the year the world's governments signed the Kyoto climate change agreement (Fig. 6). In a parallel manner, automotive companies have developed many technologies that can facilitate and improve vehicular systems to assist drivers for safe and better driving conditions [START_REF] Anderson | Autonomous Vehicle Technology: A Guide for Policymakers, Autonomous Vehicle Technology: A Guide for Policymakers[END_REF], paving the way for autonomy, and are referred to as Advanced Driver Assistance Systems (ADAS).

The ADAS systems range on the spectrum of passive and active systems. A passive system relies on the driver to take corrective action by providing alerts of a potentially dangerous situation. For example:

-Lane Departure Warning (LDW) alerts the driver of unintended/unindicated lane departure; vi -Forward Collision Warning (FCW) indicates that under the current dynamics relative to the vehicle ahead, a collision is imminent.

The driver then needs to brake in order to avoid the collision.

-Blindspot warning (BSW) systems that typically use cameras, radar, or LiDAR to detect vehicles traveling in the equipped vehicle's blind spot and provide some form of warning to the driver.

-Fatigue warning systems alert the driver if signs of drowsiness are detected based on steering input, eye blinks, and other driver behaviors. In contrast, active safety systems take action to help avoid a crash, like:

-The Automatic Emergency Braking (AEB) that identifies the imminent collision using typically radars, LiDAR, or cameras and then apply the vehicle's brakes without any driver intervention to avoid the crash or at least reduce its severity.

vii -Adaptive Cruise Control (ACC) maintains a set speed and following distance from the vehicle ahead using information from a camera or radar sensor. Frequently, it is present on vehicles equipped with FCW or front AEB.

-Lane Keeping Assist (LKA) that use cameras to determine the position of the vehicle in relation to lane markings. When the system detects that the vehicle begins to leave its travel lane it automatically steers the vehicle to prevent it from leaving the lane.

-Lane Centering.

-Traffic Jam Assist (TJA) that supports the driver in slow-moving traffic, combining the LKA and ACC.

Several police-reported vehicle crash data and insurance claim data have attempted to quantify the actual effects of these systems by comparing selected vehicles of the same model with and without the systems of interest. For example, the Highway Loss Data Institute (HLDI) found that Subaru's EyeSight system (which includes FCW, AEB, LDW, and ACC) reduced the frequency of bodily injury liability and property damage liability claims by 35% and 15%, respectively. Moreover, early statistics found that Subaru's EyeSight system reduced the frequency of bodily injury claims for crashes involving pedestrians by 35% [START_REF] Wakeman | Effect of Subaru EyeSight on pedestrian-related bodily injury liability claim frequencies[END_REF]. Additionally, a European study concluded that Volvo's LDW and LKA systems reduced the systems' target crashes by 53%, which equated to 30% of all single-vehicle and head-on crashes that resulted in injuries [START_REF] Sternlund | The effectiveness of lane departure warning systems-A reduction in realworld passenger car injury crashes[END_REF]. Furthermore, a study by Insurance Institute for Highway Safety of police-reported crashes of similar vehicles with and without BSW systems estimated that the systems reduced all lane-change crashes of equipped vehicles by 14% [START_REF] Cicchino | Effects of blind spot monitoring systems on policereported lane-change crashes[END_REF].

Table 1 lists the potential help of some ADAS technologies in reducing crashes, injuries, and Deaths. It can be concluded that the current and future vehicle safety systems have the potential to dramatically reduce the number of crashes, injuries, and fatalities on our roadways. It is anticipated that the increasing market penetration of these systems and improvements in their functionality and performance will contribute to overall improvements in traffic safety [START_REF] Benson | Potential Reductions in Crashes , Injuries , and Deaths from Large-Scale Deployment of Advanced Driver Assistance Systems[END_REF]. ADAS. Up to 43.4 % of all car accidents in the database then become preventable [START_REF] Hummel | Advanced Driver Assistance Systems: An investigation of their potential safety benefits based on an analysis of insurance claims in Germany[END_REF]. The emergency braking system becomes even more effective if it is also able to address accidents with pedestrians and cyclists [START_REF] Kühn | Accident statistics and the potential of driver assistance systems[END_REF]. For this reason, automobile manufacturers are introducing AEB technology in their new vehicle fleets (Fig. 7). Fig. 8 shows the predicted percent of vehicles equipped with some ADAS technologies. In (Highway Loss Data Institute, 2018), significant statistics were presented that evaluates the collision avoidance technologies since 2009, e.g. for 2019 as shown in Fig. 9. On the other hand, annual forums (AAA Foundation for Traffic Safety, 2019) and various studies were done to investigate the impact of vehicle x technologies and automation on vulnerable road users and driver behavior and performance [START_REF] Blanco | Automated vehicle crash rate comparison using naturalistic data[END_REF][START_REF] Schoettle | A preliminary analysis of real-world crashes involving self-driving vehicles[END_REF]. A recent study by the AAA Foundation for Traffic Safety found a substantial lack of awareness of certain safety critical limitations of ADAS technologies among owners of vehicles equipped with the technologies [START_REF] Mcdonald | Vehicle Owners' Experiences with and Reactions to Advanced Driver Assistance Systems[END_REF]. One main objective of their study was to investigate driver behavior and the associated risks of ADAS use. The results indicated that the use of lateral and longitudinal ADAS culminated in an increased occurrence of distracted driving behaviors [START_REF] Dunn | Understanding the Impact of Technology : Do Advanced Driver Assistance and Semi-Automated Vehicle Systems Lead to Improper Driving Behavior[END_REF]. Drivers with activated level 2 ADAS took more frequent and longer duration non-driving-related task glances, subsequently spending less time with their eyes on driving-related tasks. The authors, therefore, propose a three-phase model of ADAS operation, comprising a novelty phase (learning and testing in real time), a postnovelty operational phase (attain driver awareness of the limitations), and an experienced user phase. Moreover, comprehensive training for drivers purchasing ADAS-equipped vehicles may have a positive outcome. Today, every ADAS-equipped vehicle currently operates differently across functionality, ADAS activation, capabilities, and operating speeds, resulting in a learning curve each time a driver uses a different system. As such, standardization to a greater degree may also be beneficial in that it will limit the impact of novelty symptoms. The researchers claim that these propositions could help to mitigate the drawbacks of the use of ADAS [START_REF] Dunn | Understanding the Impact of Technology : Do Advanced Driver Assistance and Semi-Automated Vehicle Systems Lead to Improper Driving Behavior[END_REF]. In the Horizon of Self-driving cars At first, a 4-level taxonomy of vehicle automation was developed by the National Highway Traffic Safety Administration (NHTSA) in 2013 [START_REF] Wadud | Help or hindrance? The travel, energy and carbon impacts of highly automated vehicles[END_REF], and then 5-level automation was introduced by the Society of Automotive Engineers International (SAE) in 2014-later on updated in 2016. In 2016, NHTSA adopted SAE's taxonomy and automation levels (NHTSA, 2016), and thus, they have become an industry standard, and also frequently referred in the academic literature (SAE International, 2018). Fig. 10 defines the different levels of automation possible for a vehicle. Just during the last decade or so, incubating conditions of the Digital and 4th Industrial Revolutions gave birth to rapid technological advancements in the field. A recent development consists of networking vehicles and allowing data sharing. For example, wireless communication techniques such as 5G or WIFI are adapted to form so-called VANets (Vehicle Ad-hoc Networks) networks specifically developed for road traffic. These networks can thus allow vehicles and infrastructures to communicate with each other in order to transmit information without the need for central base stations or any controller [START_REF] Malik | A comparative study on routing protocols for VANETs[END_REF]. Vehicular communication in VANETs can be achieved by exchanging information using vehicle-to-vehicle(V2V) and vehicle to infrastructure (V2I) message flows. Figure 11 [START_REF] Sheikh | A comprehensive survey on VANET security services in traffic management system[END_REF].

Autonomous vehicles are largely motivated by passenger safety. Every year numerous people die or are seriously injured in road accidents that are largely caused by human errors in perception and decision-making, or by lack of attention and sometimes by late reactions. The autonomous vehicles can use the active percept of a number of redundant sensors to timely make correct driving decisions and hence eliminate accidents. Autonomous vehicles further result in more efficient travel. This means one has to spend much less on insurance, while the insurances will naturally become less expensive. The vehicles benefit from efficient onboard algorithms, communication with the vehicles around, transportation management centers and information from different sources. This facilitates making better informed and optimal decisions. Further, these vehicles can park themselves close to each other and operate in high-speed platoons, which significantly increase the operational efficiency. The implications of a single vehicle may seem limited, but a network of such intelligent vehicles has the potential to make the overall transportation system operate efficiently. As an example, the vehicles can collectively operate to avoid congestions [START_REF] Kala | On-Road Intelligent Vehicles Motion Planning for Intelligent Transportation Systems British Library Cataloguing-in-Publication Data[END_REF].

Here are some of the benefits we can reap from incorporation autonomous vehicles into our everyday lives: Consequently, nowadays, numerous prototype AVs are being trailed on the roads. It can be identified over 40 companies developing road-going selfdriving vehicles from the automotive industry stalwarts to leading technology brands and telecommunications companies. Beyond trendy names like Tesla, Uber, Amazon and Google chasing self-driving cars (Fig. 12), a host of auto brands are also investing in autonomous R&D. For instance, the German auto supplier Continental AG and the France-based PSA Group (including Peugeot, Citroën, and DS. The PSA Group announced that the "eyes off" mode is slated to arrive by 2021, while semi-autonomous "hands-off" modes will be available by 2020. These features, along with electric vehicles and new models, form the core of PSA's broader "Push to Pass" growth strategy for the period 2016 -2021 [START_REF] Holtz | Décryptage-nouveau plan" push to pass" de psa peugeot-citroen: l'ambition raisonnable de carlos tavares, son patron[END_REF]PSA Group, 2016), in addition to the Autonomous Vehicle for All (AVA) program that brings together a range of driving assistance functions that will lead to the autonomous car (PSA Group, 2017). Fig. 13 shows three examples for PSA contribution in the field: AVA program collaborated with VINCI Autoroutes; a Level 4 AI-based autonomy based on AImotive sensor, computing hardware, and autonomous software, installed on a Citroen C4 Picasso test car; a collaboration between PSA and the Spain's xiv Automotive Technology Centre of Galicia (CTAG) to test IoT and vehicle to vehicle/Infrastructure technologies on autonomous driving that aims to reinforce the creation of smart and sustainable cities. However, high levels of vehicle automation pose regulatory challenges for the AV manufacturing countries [START_REF] Nowakowski | Determining the readiness of automated driving systems for public operation: Development of behavioral competency requirements[END_REF]. The uptake of new technology like AV should be regulated through federal and state governments' pre-deployment policy. Major regulating policies are revolving around testing and deployment, cybersecurity and privacy, liabilities and insurance, ethics, and repair/maintenance and calibration. As an example, AV legislation and policies in the US, the Netherlands, the UK, and Sweden are paving the way for other countries [START_REF] Vellinga | From the testing to the deployment of self-driving cars: Legal challenges to policymakers on the road ahead[END_REF]. However, the first fatal crash by a self-driving UBER involving pedestrian in the US proves that more research, development, legislation and planning are needed for a safer and wider AV uptake. The Victoria Transport Policy Institute (VTPI) mentioned in its report, published in October 2019, different overlooks for additional risks that these technologies can introduce, in contrary to the claims of optimists that AVs will reduce crash rates and insurance costs by 90% [START_REF] Litman | Autonomous Vehicle Implementation Predictions Implications for Transport Planning[END_REF]. Some of these risks are: Hardware and software failures (false sensor, distorted signal, or software error); Malicious hacking; Increased risk-taking ("over-trusting" technology); Additional risks to nonauto travelers. These new risks will probably cause crashes, so net safety impacts are likely to be smaller than the 90% crash reductions that xv advocates claim. Thus, VTPI concluded that autonomous vehicles may be no safer per mile than an average driver, and may increase total crashes when self-and human-driven vehicles mix [START_REF] Penmetsa | Perceptions and expectations of autonomous vehicles -A snapshot of vulnerable road user opinion[END_REF].

Zero crashes, Zero congestion, and Zero emissions! It becomes obvious that AVs are moving from science fiction to a very near-term reality, thus, the time to consider the potential climate impacts of autonomous vehicles is now. (2020-25) (Till [START_REF] Bunsen | Global EV Outlook 2019 to electric mobility[END_REF].

Road transportation is shifting significantly toward electrification around the globe (Fig. 14). The potential to reduce energy consumption and greenhouse emissions are the most important driver for electric mobility. Besides the growing awareness of environmental and population pressures, the innovations market acceptance [START_REF] Li | New entrants versus establishers in China and US electric vehicle marketplace: a comparative analysis[END_REF] and government support (ER-TRAC et al., 2017;European Commission, 2017) results in the advancement of Electric Vehicle (EV) technology. This technology starts to challenge its internal combustion counterparts with large-scale investments and highly coordinated research and development [START_REF] Ahmad | A Review of the Electric Vehicle Charging Techniques, Standards, Progression and Evolution of EV Technologies in Germany[END_REF]. It becomes essential for meeting the goals of decarbonization and energy security (Paris agreement at COP21 2015 -C40 Committee (COP21, 2015) / Madrid COP 25 (Sustainable Innovation Forum, 2019)).

xvi Between 1990 and 2007, the biggest increase in CO2 emissions could be attributed to road transport with an increase of 200.7% in that period. Moreover, 40% of nitrogen oxide emissions could be attributed to road transport (EEA, 2015(EEA, , 2012)). According to the Lancet Commission on Pollution and Health, around 6.5 million fatalities in 2015 caused by conventional cars using internal combustion engines (ICE's) (Fernández, 2019) linked to that kind of pollution. This, therefore, has forced the implementation of new urgent measures intended to protect citizens, applying new regulations, driving restrictions and even banning most pollutant internal combustion engine (ICE) vehicles when air quality reaches unacceptable levels of toxicity [START_REF] Held | On the road to electrification -A qualitative comparative analysis of urban e-mobility policies in 15 European cities[END_REF]. The results show that CO2 emission of the vehicle decreases up to 50.6% according to an urban driving cycle due to applying the electrification strategy for hybrid electric vehicles [START_REF] Mourad | Electrification Strategy of Hybrid Electric Vehicle as a Solution to Decrease CO2 Emission in Cities[END_REF].

Many countries have already made a bold announcement regarding electrification of mobility that opt to ban diesel vehicles from their cities' center by the next decades [START_REF] Pereirinha | Main Trends and Challenges in Road Transportation Electrification[END_REF]. Germany aims to reduce its Green House Gasses (GHG) emissions by 80-95%, where today, around 85% of these emissions are energy-related, in which 22% of them caused by road transportation [START_REF] Ruhnau | Direct or indirect electrification? A review of heat generation and road transport decarbonisation scenarios for Germany[END_REF]. Given the reported high energy efficiency and zero tailpipe emission, the EV has been given focus as a potential solution for enhancing the sustainability in our road transportation sector [START_REF] Chen | Electrification of roads : Opportunities and challenges[END_REF]. Other countries that have introduced a ban include Norway (by 2025), France, and The United Kingdom (both by 2040) [START_REF] Dugdale | European countries banning fossil fuel cars and switching to electric[END_REF]. More recently, Sweden and Denmark have announced a ban on sales for the year 2030. The EV market is taking up on this and an adequate selection of vehicles will be available [START_REF] Bischoff | Impacts of vehicle fleet electrification in Sweden-a simulation-based assessment of longdistance trips[END_REF].

For this purpose, in the long term, building all-electric vehicles with autonomous capabilities integrated from the beginning is the most efficient way to unlock the tremendous potential societal benefits of self-driving cars, rather than through retrofits. Electric vehicles allow for simpler integration of the advanced technologies required for the cleanest and safest operation of autonomous vehicles. Researchers assign the main advantages of All-electric autonomous vehicles over their gasoline -or hybrid powered vehiclescounterparts to: xvii -Cleaner Environment: It is estimated that there will be 40 cities with populations exceeding 10 million by 2040, then All-electric shared autonomous vehicles will be ideal for dense cities that need solutions for congestion and noise pollution. Thanks to zero tailpipe emissions, EVs are well suited to address air pollution issues, especially in urban areas and along road networks, where a large number of people are exposed to harmful pollutants from road transport vehicles. Moreover, increasing electric mobility in association with a progressive increase in low-carbon electricity generation can deliver significant reductions in GHG emissions from road transport relative to ICE vehicles.

-Stable Power: The advanced sensing and computing hardware on an autonomous vehicle needs a lot of electric power. Compared to an internal combustion engine, an all-electric battery pack acts as a more stable power source that can enable higher-powered AV components.

Manufacturers see that EV/AV is a very complementary, technology-wise, equation. The existence of various EVs manufacturers evidence that the world's established and new automakers are increasingly focusing on electrified vehicles with increasing levels of connectivity and autonomy. For example, one eagerly awaited new car on the horizon is Porsche's Taycan (Fig. 15). The production Taycan is presented in September (at the 2019 IAA) and hit the market by the end of 2019. There are already more than 20,000 prospective buyers already making a down payment. A primary differentiator of the car is fast charging. Fast charging is enabled by the car's high- xviii voltage (around double what is EV-typical) 800-V architecture, guaranteeing its around 90-kW•h lithium-ion battery pack can be recharged in just 4 min for 100 km of range, where the total vehicle range is over 500 km [START_REF] Jost | EVs and AVs for 2020 and beyond[END_REF]. (2020-25) (Till [START_REF] Bunsen | Global EV Outlook 2019 to electric mobility[END_REF].

In 2018, the global electric car fleet exceeded 5.1 million, up 2 million from the previous year and almost doubling the number of new electric car registrations. The People's Republic of China (hereafter "China") remained the world's largest electric car market, followed by Europe and the United States. Norway was the global leader in terms of electric car market share (46%) (Till [START_REF] Bunsen | Global EV Outlook 2019 to electric mobility[END_REF]. In the European Union, several significant policy instruments were approved. In 2030, global EV sales estimated to reach 23 million and the stock will exceed 130 million vehicles in the New Policies Scenario (Fig. 16). In the EV30@30 Scenario, EV sales and stock nearly double by 2030: sales reach 43 million and the stock is larger than 250 million. Many countries, including France, have committed to the EV30@30 Campaign, where they have set a common vision to strive to make all passenger vehicle sales electric by 2050 [START_REF] Slowik | EXPANDING ZERO-EMISSION MOBILITY EQUITY AND ACCESS[END_REF].

New Dimension in the sense of Autonomous vehicles Suspensions' Control

Nowadays, comfort driving and road safety are highly enhanced thanks to a variety of advanced technologies where inventive active suspension systems show promise in electric/autonomous fleets.

xix

In line with the process of launching All-electric autonomous vehicles, allin-wheel devices are expected to play an important role in this new trend. Such vehicles are able to adapt to urban lifestyles, mainly, to contribute to the air quality, noise reduction in urban cities as well as fulfilling ride quality requirements. Since a lighter vehicle means also a sharp reduction in energy consumption, a modern solution can contain integrated powertrain suspension located directly inside the wheel.

Besides the ProteanDrive project [START_REF] Tran | ProteanDRIVE® in-wheel motors on show at auto Shanghai[END_REF], the eCorner wheel by Siemens [START_REF] Gombert | eCorner: Propulsion by Hub Motors[END_REF], and a variety of similar approaches, the Active Wheel (AW) developed by Michelin serves as one propitious technology in this field. This advanced technology initiates a new era in road transportation where all essential components have been integrated within the wheel itself: the pneumatic part, a compact traction electric motor with regenerative braking, an electric active suspension system, and the braking system.

In fact, the ride comfort in vehicles with driving automation at level 4 or higher will need to provide a superior experience. As mentioned earlier, autonomous vehicles will offer increased productivity and convenience by freeing occupants from the responsibility of driving. However, studies indicate that the occurrence of motion sickness in autonomous vehicles will be substantially higher than in conventionally driven vehicles. Occupants of autonomous vehicles are more likely to be involved in performing tasks and activities, such as reading, writing and using a computer or tablet, that typically increase the occurrence of motion sickness [START_REF] Ekchian | A High-Bandwidth Active Suspension for Motion Sickness Mitigation in Autonomous Vehicles[END_REF]. Therefore, new and different approaches needed rather than conventional solutions.

Here, the structure of the Active Wheel offers new opportunities in Global Chassis Control (GCC) that rises, recently, to remarkable levels. In AW, the conventional mechanical suspension has been replaced by a passive spring and a DC electrical motor. In comparison with other force generating actuators (hydraulic, Hydractive, hydro-pneumatic...), the DC linear characteristics of the AW's suspension system provides an extremely fast response time, low contact friction, high precision and efficiency, good controllability, and less energy consumption [START_REF] Nguyen | Direct current linear actuator using in the active suspension system of automobile[END_REF].

xx Along with active suspension systems, the innovative technologies in the frame of sensors, electronics, and advanced software, in addition to the networking and data sharing between vehicles and infrastructures, all these coordinated could inevitably provide better performance in the horizon of ride comfort and road safety. In this sense, the mechanical aspects of the systems could be linked to forward-facing sensors -cameras and LiDAR, for example -to bring a predictive aspect to the technology. Thus, in the future, what is certain is that these technologies will be needed if autonomous vehicles are to reach their full potential. This, adequately, must be accompanied by new or updated control strategies and theories. The existence of diverse dynamic control systems and a wide range of actuators integrated into the All-electric autonomous vehicle, will place greater emphasis and give more prominence to the coupling effect between the embedded active systems, and in particular, the effect of the active suspension system on other vehicle dynamics and functionality.

xxi xxii

Thesis framework and content

This dissertation synthesizes the results of the three years PhD work (from January 2017 to April 2020), performed in collaboration between the IMS laboratory under the supervision of Xavier Moreau, professor at the University of Bordeaux, and the CRSI laboratory under the supervision of Clovis Francis, professor at the Lebanese University. This thesis is concerned on the Hierarchical approach for global chassis control of an electric vehicle in the sense of improving the vehicle comfort, stability, and safety through the control of active suspension system, braking systems, and active steering.

For this purpose, the thesis is subdivided into five chapters:

-Chapter 1 provides a general introduction about vehicle dynamics, and mainly on the automotive suspension systems. It is devoted also to recall modeling concepts and the CRONE control tools in addition to some recent developments in vehicle control. Moreover, it introduces the role of the suspension in vertical dynamics and the defined the different areas of study and operating, which the suspension must meet different objectives, and sometimes even conflicting. The problem formulation and the assigned objectives to be achieved are then presented.

-In Chapter 2, entitled "Vertical Dynamic Control", deals with different solutions for the inherent problems in traditional suspensions.

Two main controllers were designed for two different objectives. While the first one is assigned to regulate the height of the center of gravity of the vehicle, in addition to its roll and pitch angle, the second controller offers two fractional order damping controllers. Using the xxiii CRONE methodology, a comfort-oriented and road holding oriented controller were developed and have achieved remarkable performances.

-In Chapter 3, entitled "Hierarchical Approach for Driver Disturbance Rejection", presents a hierarchical control approach where two-level controllers are proposed for the AW suspension system. The systems are controlled using the first-generation CRONE controllers while taking into account the vehicles' parametric uncertainties. The main objective of the proposed approach is to reject the driver disturbances in light electric vehicles to enhance ride comfort. The proposed strategy permits to maximize the benefit of the embedded active suspension actuator in the Active Wheel to regulate the main chassis dynamics, namely, the heave, roll and pitch. It is worth noting that the road disturbances have been considered in this work and the strategy was announced applicable for specific road classes and profiles.

-In Chapter 4, entitled "Study of the ABS function of an electromechanical braking system", studies have been done on improving the braking performance of the electric vehicle by an ABS control on even and rough roads. Using CRONE methodology, the controller is designed to maximize the efficiency of the braking system of Active Wheel. The study takes into consideration the road adhesion variations, vehicle mass variation, and vehicle velocity variations. The final objective of this system is to reduce the braking distances while taking into consideration system limitations and noise measurement criterion. Then, the half-vehicle model is used to study the effect of the load transfer disturbance on the performance of the ABS controller. In the end, the road disturbances were taken into account, and some testing scenarios made to find out its effect on the ABS performance. In order to minimize the performance deterioration, the fractional-order damping controllers introduced in Chapter 2, are also implemented to test for its influence on the overall system, in its different orientation, i.e. comfort-oriented and road holding oriented controllers. xxiv -In Chapter 5, entitled "Vehicle Stability Control", the study was concerned on the lateral dynamics control. The objectives were assigned to enhance the performance of the lateral dynamics of the vehicle against disturbances and parametric variations. For this reason, three 3 rd generation CRONE controllers were designed: two for yaw rate regulation using braking torques and active steering, and the third one designed for the lateral position control of the vehicle. Finally, the effect of the vertical dynamics of the chassis was analyzed on the performance of the designed lateral controllers.

Introduction

In the past years, increasing demand for high comfort and safety vehicles had motivated the academics and industrials to intensify the studies and massively investigate vehicle dynamic control. Recently, and in line with the newly emerged markets for autonomous or/and electric vehicles, new challenges are added to the old numerous problems facing vehicular control. For safety-oriented cases, the efforts are made on stabilizing the vehicle in critical situations using the antilock-braking system (ABS) [START_REF] Brugger | Anti-lock brake system[END_REF][START_REF] Soliman | An investigation of anti-lock braking system for automobiles[END_REF] and the electronic stabilizing program (ESP) [START_REF] Ferguson | The effectiveness of electronic stability control in reducing real-world crashes: a literature review[END_REF], for example. While for comfort-oriented problems, the use of controllable suspension systems shows very promising potential in improving ride comfort [START_REF] Tseng | State of the art survey: Active and semiactive suspension control[END_REF]. Several real implementation and testing prototypes for active and semi-active suspension systems had proved their ability to fulfill comfort specifications [START_REF] Casciati | Active and semi-active control of structures-theory and applications: A review of recent advances[END_REF]. However, dealing with these systems needs a good knowledge of the vehicle dynamics, forces and moments acting on the vehicle, and coupling effects between the embedded subsystems in both parts: the passive and the active ones.

In fact, disturbances caused by engines, gravity, aerodynamics, and road excitations are highly influencing the accelerating and braking performance, as well as steering and ride performance. Thus, control-wise, to improve the overall performance of any vehicle, convenient coordination and strategies should be applied between active systems and appropriate control rules should be adopted [START_REF] Vivas-López | Review on global chassis control[END_REF].

Besides the power train and engine systems, three actuators related to braking, steering, and suspension subsystem are emphasized to enhance the vehicle's performance in this thesis. Many concerns are related to these essential systems that are very interesting in both vehicle dynamics and automatic control. A brief overview of each subsystem's characteristics reveals an important common factor between them. Braking performance defines the vehicle safety and determined by a braking coefficient that, in turn, is derived from an adhesive and hysteresis friction [START_REF] Gillespie | Fundamentals of Vehicle Dynamics[END_REF]. For better braking performance, this coefficient should have high values. The ABS technology is based on this fact, where the designed strategy should keep the braking coefficient around the highest possible values. On the other side, the handling of the vehicle is related to steering performance. One main factor in steering performance is the sideslip angle, i.e. the angle between the direction in which a wheel is pointing and the direction in which it is actually traveling. The value of the slip angle is negligible at low speeds in which cornering objectives are easy to achieve. However, at high speeds, the slip angle is significant at each wheel. Thus, the steering tasks are no longer simple and could lead to under-steering or over-steering scenarios. In fact, this is directly related to the available grip in tires. In critical steering situations and at high speeds, the available grip is exceeded and steering is no longer efficient for vehicle rotation.

The idea is that the performances of both systems, braking, and steering, are related to the tire, more precisely, it's characteristics. Actually, the tire dynamics play a key role in vehicle performance. The tire characteristics have been strongly studied in the literature [START_REF] Gillespie | Fundamentals of Vehicle Dynamics[END_REF][START_REF] Kiencke | Automotive control systems: For engine, driveline, and vehicle: Second edition[END_REF][START_REF] Pacejka | Basic Tire Modeling Considerations[END_REF]Rajamani, 2012a). Here, the suspension systems, in its all kinds, turn out to be critical for the vehicle dynamics, since they provide the link between the tires and the vehicle's body. At first glance, it seems that the suspension systems influence only the tire vertical forces, i.e. the wheel load forces, but it was shown that they indirectly affect the lateral and longitudinal dynamics. Indeed, the lateral and longitudinal wheel forces are only the results of those vertical tire forces [START_REF] Pacejka | Basic Tire Modeling Considerations[END_REF]. In that sense, the braking, steering, and suspension systems are related to each other, thus, a well understanding for the overall vehicle dynamics and in particular, the suspension system, along with a good control design strategy, will inevitably improve the performance not only the ride comfort but also the ride safety.

Vehicle Dynamics 1.2.1 Introduction to the Global Vehicle Dynamics

Based on the laws of physics, the vehicle model can be built analytically and represents the different system dynamics. In the past, the main difficulty was in the computation process of finding mathematical solutions. Indeed, the study of global vehicle dynamics is a study of complex systems. The complexity of the vehicle dynamics relies on the form of nonlinearities, a large number of components and systems, variations in parameters, and in particular mass and load distributions [START_REF] Rajamani | Vehicle Dynamics and Control, Mechanical Engineering Series[END_REF].

The global dynamics of a vehicle can be defined into 14 degrees of freedom (DoF) distributed into three main sections, as illustrated in Figure 1.1: longitudinal, lateral, and vertical dynamics [START_REF] Gillespie | Fundamentals of Vehicle Dynamics[END_REF]. More precisely, these 14DoF are divided as follow:

-5DoF for longitudinal dynamics These dynamics are mainly excited by means of specific systems that are the linkage between the chassis of the vehicle and the road. These systems, the so-called "Liaisons Au Sol (LAS)" systems, are:

-The steering: 1 actuator (or two in case there exist rear steering); -The traction: 2 or 4 actuators in case of 2 or 4 driven wheels; -The braking: 4 actuators; -The suspension: 4 actuators, one for each vehicle corner.

The action of these systems might affect the dynamics of each of the 14DoF directly, indirectly, or by the coupling effect. Table 1.1 illustrates the type of influence between the different subsystems in the vehicle. Obviously, the input action of the steering system has a direct influence on the transversal dynamics by generation lateral translational forces as well as yaw torques. But, at the same time, it indirectly generates a rolling torque on the chassis, and has a coupling effect, that affects the vertical dynamics of the vehicle. On the other hand, the traction and braking actions have a direct effect on the longitudinal and lateral dynamics, but also, they generate a pitching torque that influences the suspension (vertical dynamics). And in return, the suspension system has an indirect effect on both the longitudinal and lateral dynamics via the variation of the wheel contact forces, in addition to the direct influence on the vertical dynamics.

By coupling By coupling

One main coupling effect is the one between the longitudinal and lateral dynamics on one side and the tires on the other side. The nonlinear characteristic of the tire contact forces is the critical factor that mostly influences the overall dynamics of the vehicle [START_REF] Pacejka | Basic Tire Modeling Considerations[END_REF].

For best illustration, the study of these dynamics and characteristics, one can divide the vehicle operating region into different fields that are bounded by the available grip at each tire, assuming driving on dry roads. As shown in Figure 1.2a, the three domains represent the following:

-The green region, the comfort region, is defined by low longitudinal and lateral accelerations that are less than 0.4g in absolute value. In this zone, the vertical dynamics has no significant influence on the other dynamics.

-The blue region represents the limits of the linear zone of the tire model. This zone corresponds to the normal driving conditions where it represents the majority of driving time in the vehicle.

-The security region in grey represents the nonlinear zone of the tire model where safety is the main concern, e.g. obstacle avoidance, emergency braking, ... Furthermore, due to the aforementioned coupling effect between the vertical dynamics and other dynamics, the 2-dimensional diagram (g-g) related to longitudinal and lateral accelerations is extended to a 3-dimensional representation while adding a third axis related to vertical acceleration. Figure 1.2b illustrates the 3D diagram that includes the vertical acceleration, mainly the sprung mass acceleration, that is generally generated by the driver disturbance in addition to the road perturbations.

Vertical dynamics of the vehicle

In the context of vertical dynamics, the study is focusing on the suspension systems that are the main contributors in this framework. Those dynamics are divided into 7DoF as mentioned before.

Two main kinds of disturbances affect the vertical vehicle dynamics (Figure 1.3):

-Road disturbances, and are classified as shocks (bump, potholes, ...) or vibrations. Shocks are discrete events of relatively short duration and high intensity that causes large suspension deflections, where vibrations are typically specified as a random process that consists of high-frequency components. Thus, the road perturbations are the result of vertical variation of the road profile and are a combination of the vehicle forward speed, spectral power density, and its geometry.

For this reason, dealing with these kinds of disturbances means dealing with a wide range of frequencies of interest [0-50Hz], mainly.

-Driver disturbances that are caused by the driver inputs upon braking, accelerating, and maneuvering, for example. However, the dynamics of these perturbations are related to the driver, thus, it is related to a small range of frequencies of interest and particularly at low frequen-

cies [0-5Hz].
Road disturbances, in its all kind, directly influence the vertical dynamics and concern the ride comfort of the passengers. While the driver disturbances are affecting the same DoF but by coupling effect. Actions on longitudinal dynamics (traction, braking, …) and on lateral dynamics (maneuvering), in the form of pitch, roll, and even yaw motions, are exciting the vertical dynamics by coupling. For this reason, the effect of the vertical dynamics should be emphasized on the other dynamics (longitudinal and transversal). This study can be held by a single quarter vehicle model that is excited by road disturbances and the resulting load transfers due to driver inputs during acceleration and deceleration phases, and denoted by 𝑓𝑓 0 (𝑇𝑇). The detailed modeling will be presented in the next sections.

Range of Vehicle models

As mentioned before, dealing with a full real vehicle model is dealing with highly complex systems that include high nonlinearities, varying parameters, etc. However, there exist three main types of vehicle modeling, each one has its own degree of complexity and role, and are introduced in the decreasing order of complexity as follow [START_REF] Bouvin | Towards an alternative version of the Hydractive CRONE car suspension[END_REF][START_REF] Létévé | Study of the influence of passenger vehicle suspension on vibrational comfort, driving behaviour and operating limits : CRONE approach to formalization, analysis and synthesis[END_REF]:

-Validation Model: used mainly for 'validation', in a simulation framework, for any new high or low-level strategy or controller design.

One example of these models is the vehicle simulators like SCANeR™ The resulted vehicle model leads to an imperfect representation of the real system; however, this degree of deficiency is related to the settled objectives, e.g. validate, understand/analyze, or synthesize.

One common approach adopted by researchers upon collaboration with industrials, is the V-Shaped approach, as illustrated in Figure 1.4. It is also known as the Verification and Validation model. It is based on the association of a testing phase for each corresponding development stage. The development of each step is directly associated with the testing phase. The next phase starts only after completion of the previous phase i.e. for each development activity, there is a testing activity corresponding to it. This approach is comprised of:

-The industrial side provides the "reference" vehicle model, that undergoes identifications and re-adjusting vehicular parameters to attain similar responses compared to that of real bench tests, either on track or on road. Here, the domain of validity is then determined and the degree of correlation between the reference model and the real vehicle is then identified.

-Extract from the "reference" model the needed specification related to the new prescribed implemented systems, e.g. steering, braking, traction, suspension, etc.

-Implementing and applying the newly designed strategy, part, or control that aims to improve the existing one.

-The last step is to establish a comparison of the performance of the new strategy designed with that of the "reference" model.

Full Vehicle Validation Model

The simulations for strategy validation are done using the full non-linear car model with 14 degrees of freedom taking into consideration most of the nonlinearities existing in the vehicle dynamics. This model has been widely studied in the literature [START_REF] Ahmadian | Ride Performance Analysis of Semiactive Suspension Systems Based on a Full-Car Model[END_REF][START_REF] Berger | From a Competition for Self-Driving Miniature Cars to a Standardized Experimental Platform: Concept[END_REF][START_REF] Connair | Development of a common vehicle model for chassis control design[END_REF][START_REF] Giangiulio | 14 dof VERTEC vehicle model VDSIM--Vehicle Dynamic SIMulator Implementation and validation[END_REF][START_REF] Gillespie | Fundamentals of Vehicle Dynamics[END_REF]Rajamani, 2012a). This model is developed in a MATLAB/Simulink environment as illustrated in Figure 1.5, accompanied by several file scripts that define the parametric configuration of the vehicle including tire characteristics.

Figure 1.4. V -cycle approach for designing of vehicles' elements [START_REF] Létévé | Study of the influence of passenger vehicle suspension on vibrational comfort, driving behaviour and operating limits : CRONE approach to formalization, analysis and synthesis[END_REF].

Quarter Vehicle model: a compromise between complexity and dynamics

The quarter vehicle system is modeled using the well-known quarter-model that exists in the literature [START_REF] Jazar | Suspension Mechanisms, in: Vehicle Dynamics: Theory and Application[END_REF]Rajamani, 2012a;[START_REF] Savaresi | Semi-Active Suspension Control Design for Vehicles, Semi-Active Suspension Control Design for Vehicles[END_REF]. This approach is facilitating the analysis where the degree of complexity is reduced from 14Dof in a full vehicle to the only 2DoF related to each corner, taking advantage of the symmetry of the vehicle properties. Here, the four wheels are assumed decoupled and the suspension dynamics are considered in the frequency range of interest for the vertical dynamics [0 -30 Hz]. Although of the nonlinear characteristics of the suspension components, in reality, it is pointed out that such a model can be used for optimal suspension control design and analysis [START_REF] Hrovat | Survey of Advanced Suspension Developments and Related Optimal Control Applications[END_REF], where it represents a convenient modeling framework to evaluate the performance and robustness of a controller. [START_REF] Létévé | Study of the influence of passenger vehicle suspension on vibrational comfort, driving behaviour and operating limits : CRONE approach to formalization, analysis and synthesis[END_REF], where 𝑚𝑚 2 represents the sprung mass (in kg), 𝑚𝑚 1 represents the unsprung mass (in kg), 𝑓𝑓 0 (𝑇𝑇) is the load transfer forces mainly caused by driver disturbances (in N), and 𝑣𝑣 2 (𝑇𝑇), 𝑣𝑣 1 (𝑇𝑇), and 𝑣𝑣 0 (t) are the velocity of sprung mass, unsprung mass, and road variation at the contact point pneumatic-ground (in m/s), respectively.

By applying the fundamental laws of physics, two equations are defined representing the dynamics of the vertical velocity of the sprung mass 𝑣𝑣 2 (𝑇𝑇) and the unsprung mass 𝑣𝑣 1 (𝑇𝑇) in (m/s) as follow:

𝑣𝑣 2 (𝑇𝑇) = 1 𝑚𝑚 2 � 𝐹𝐹 Σ 1 (𝜏𝜏 )𝐺𝐺𝜏𝜏 + 𝑣𝑣 2 (0) 𝑡𝑡 0 , (1.1)
and

𝑣𝑣 1 (𝑇𝑇) = 1 𝑚𝑚 1 � 𝐹𝐹 Σ 2 (𝜏𝜏 )𝐺𝐺𝜏𝜏 + 𝑣𝑣 1 (0) 𝑡𝑡 0 , (1.2)
with 𝐹𝐹 Σ 1 (𝑇𝑇) and 𝐹𝐹 Σ 2 (𝑇𝑇) represent the algebraic sum of all forces (in N) applied on the unsprung mass and the sprung mass, respectively, such that: [START_REF] Bouvin | Towards an alternative version of the Hydractive CRONE car suspension[END_REF].

� 𝐹𝐹 Σ 1 (𝑇𝑇) = 𝐹𝐹 1 (𝑇𝑇) -𝐹𝐹 𝑠𝑠 (𝑇𝑇) -𝑚𝑚 1 𝑔𝑔 𝐹𝐹 Σ 2 (𝑇𝑇) = 𝑓𝑓 0 (𝑇𝑇) + 𝐹𝐹 𝑠𝑠 (𝑇𝑇) -𝑚𝑚 2 𝑔𝑔 , (1.3)
where 𝐹𝐹 1 (𝑇𝑇) = 𝐹𝐹 1 0 + 𝑓𝑓 1 (𝑇𝑇) and 𝐹𝐹 𝑠𝑠 (𝑇𝑇) = 𝐹𝐹 𝑠𝑠 0 + 𝑓𝑓 𝑠𝑠 (𝑇𝑇), are respectively, the vertical forces developed by the pneumatic of the wheel on the unsprung mass 𝑚𝑚 1 (in N) and by the suspension system on the sprung mass 𝑚𝑚 2 (in N), while 𝑓𝑓 1 (𝑇𝑇) and 𝑓𝑓 𝑠𝑠 (𝑇𝑇) represent the variations of these forces around the characterized equilibrium point, where 𝐹𝐹 1 0 and 𝐹𝐹 𝑠𝑠 0 are the static components defined at the equilibrium state.

For a given suspension configuration, the force developed by the suspension system 𝐹𝐹 𝑠𝑠 (𝑇𝑇) depends, in a non-linear manner, on the suspension stroke and its velocity. Similarly, for a given wheel pressure, size, and structure, the vertical force generated by the wheel 𝐹𝐹 1 (𝑇𝑇) depends, in a nonlinear manner, on the wheel deflection and its velocity.

The study of quarter vehicle dynamics is carried out incongruent with the vehicle's domain of operation, as illustrated in Figure 1.2.

Domains of operation and domains of study

The domains of study considered in this thesis represent the different zones of operation of a light electric vehicle. As mentioned before, the study of the dynamic vehicle behavior can be divided mainly into two regions as a function of lateral and longitudinal accelerations, and independent of the road solicitations, that is:

-Linear zone that corresponds to the normal driving situations; -The nonlinear zone that corresponds to critical and emergency situations where the grip limits of the tires could be exceeded.

In fact, by adding the effect of road solicitations, three fields of study can be defined on the (g-g) diagram as shown in Figure 1.7. These fields are as follow:

-The vibratory comfort; -Road holding;

-Operating limits or active safety.

In the context of suspension system control and analysis, the field of studies is divided into the three defined zones, in which every zone is related to the amplitude of the variations in the suspension stroke (suspension deflection) around its static equilibrium point [START_REF] Bouvin | Towards an alternative version of the Hydractive CRONE car suspension[END_REF][START_REF] Létévé | Study of the influence of passenger vehicle suspension on vibrational comfort, driving behaviour and operating limits : CRONE approach to formalization, analysis and synthesis[END_REF].

Domain of operations

The three domains of operation shown in Figure 1.7 are then related to three types of variations:

At Small variations

This domain of operation corresponds to a small variation of the suspension deflection around the static equilibrium positions. The limits of this domain are determined generally by a predefined value of the amplitude of the variations. It is worth noted that the vehicle behavior in this zone is considered linear, thus, it permits the establishment of a linearized model around a given static equilibrium position, as well as the use of time-frequency duality. This gives the advantage for the use of the linear model for the analysis of the vehicle's behavior and to the synthesis/design of the suspension architecture/control.

Limit of Adherence

Limit of Linear Zone The domain of "small variation" is then associated with the vertical dynamics in the field of study of "vibrational comfort", where it won't have a significant influence in the vehicle dynamic behavior of the "road holding" zone [START_REF] Rizzo | L'approche CRONE dans le domaine des architectures complexes des suspensions de véhicules automobiles : la suspension CRONE Hydractive[END_REF]. In this framework, only a small variation of road profile variation 𝑣𝑣 0 (𝑇𝑇) are taken into account, while neglecting the small variation could be generated from load transfers 𝑓𝑓 0 (𝑇𝑇) applied on the vertical vehicle's dynamics.

Using the model presented in Figure 1.6, and after linearization around the position of static equilibrium, the Laplace transform is applied with the assumption of zero initial conditions, and a linear model can then be deduced [START_REF] Létévé | Study of the influence of passenger vehicle suspension on vibrational comfort, driving behaviour and operating limits : CRONE approach to formalization, analysis and synthesis[END_REF]. The diagram and the configuration of this linear model are shown in Figure 1.8a. The vertical behavior of the tire is defined by the Kelvin -Voigt model [START_REF] Gillespie | Fundamentals of Vehicle Dynamics[END_REF], where 𝑘𝑘 1 and 𝑏𝑏 1 represent, respectively, the equivalent tire stiffness (in N/m) and the damping coefficient (in Ns/m). The advantage is that this generic model is independent of the suspension architecture, where the mechanical impedance is dented by 𝑃𝑃 𝑠𝑠 (𝑠𝑠) and defined by: 𝑃𝑃 𝑠𝑠 (𝑠𝑠) = 𝐹𝐹 𝑠𝑠 (𝑠𝑠) 𝑉𝑉 12 (𝑠𝑠) , (1.4) where 𝐹𝐹 𝑠𝑠 (𝑠𝑠) = ℒ{𝑓𝑓 𝑠𝑠 (𝑇𝑇)} and 𝑉𝑉 21 (𝑠𝑠) = -𝑉𝑉 12 (𝑠𝑠) = ℒ{𝑣𝑣 21 (𝑇𝑇)} , where ℒ denotes the Laplace Transform operator. Furthermore, 𝑣𝑣 21 (𝑇𝑇) = 𝑣𝑣 2 (𝑇𝑇) -𝑣𝑣 1 (𝑇𝑇) represents the velocity of the suspension deflection (in m/s).

Finally, the causal diagram, according to the Bond Graph approach [START_REF] Margolis | Bond graph model incorporating sensors, actuators, and vehicle dynamics for developing controllers for vehicle safety[END_REF], is presented in Figure 1.8b with the representation of the admittance term 𝐶𝐶 1 (𝑠𝑠) and 𝐶𝐶 2 (𝑠𝑠) of the masses 𝑚𝑚 1 and 𝑚𝑚 2 , respectively, such that:

𝐶𝐶 1 (𝑠𝑠) = 𝑉𝑉 1 (𝑠𝑠) 𝐹𝐹 Σ 1 (𝑠𝑠) = 1 𝑚𝑚 1 𝑠𝑠 , (1.5)
and

𝐶𝐶 2 (𝑠𝑠) = 𝑉𝑉 2 (𝑠𝑠) 𝐹𝐹 Σ 2 (𝑠𝑠) = 1 𝑚𝑚 2 𝑠𝑠 , (1.6)
where the 𝑃𝑃 1 (𝑠𝑠) the impedance of the tire is defined by:

𝑃𝑃 1 (𝑠𝑠) = 𝐹𝐹 1 (𝑠𝑠) 𝑉𝑉 01 (𝑠𝑠) = 𝑘𝑘 1 𝑠𝑠 + 𝑏𝑏 1 . (1.7)
The variable 𝑉𝑉 01 (𝑠𝑠) = ℒ{𝑣𝑣 01 (𝑇𝑇)}, where 𝑣𝑣 10 (𝑇𝑇) = -𝑣𝑣 01 (𝑇𝑇) = 𝑣𝑣 1 (𝑇𝑇) -𝑣𝑣 0 (𝑇𝑇) is the velocity of wheel deflection in (m/s). The remaining terms that appear in the causal diagram are discussed later in detail in the next chapters.

At Medium variations

The second operating domain of the suspension is defined for a greater value of the suspension deflection corresponding to medium variation around the static equilibrium position. Here, these variations result from both: the road disturbances 𝑣𝑣 0 (𝑇𝑇), and the driver disturbances 𝑓𝑓 0 (𝑇𝑇) (load transfer). These variations affect the vehicle's overall dynamic behavior, except these that results from singular events that could lead to reaching the critical limits of suspension travel. This domain of operation is closely linked to the area of study of road holding, and consequently, it requires a more complex model than the quarter vehicle model to analyze all the dynamics involved (at least a four-wheel model with suspension). [START_REF] Létévé | Study of the influence of passenger vehicle suspension on vibrational comfort, driving behaviour and operating limits : CRONE approach to formalization, analysis and synthesis[END_REF].

At high variations

The domain of operation, in this case, is defined by the set of large variations around the static equilibrium position. These variations result mainly from road disturbances (speed bump, potholes, …) as well as the driver disturbances (emergency braking, …), which could lead to reaching the limits of suspension deflection.

This area of operation is closely linked to the active security region, where it is necessary to consider the limit of the suspension deflections. This is critical for comfort and safety, as well as road holding. Moreover, it requires a full vehicle model to analyze the overall vehicle's behavior.

Adequation of the different domains

Table 1.2 summarizes the adequacy between the areas of operating and the areas of study [START_REF] Létévé | Study of the influence of passenger vehicle suspension on vibrational comfort, driving behaviour and operating limits : CRONE approach to formalization, analysis and synthesis[END_REF]. Thus, vibrational comfort can be analyzed within the framework of the three operating domains (green-fill), but the synthesis of a new suspension architecture could be done with small variations operations regions, while it couldn't be done within medium and / or large variations (red-fill). Then, road behavior can be analyzed, too, within the framework of the three operating domains, knowing that in this work no synthesis method road behavior-oriented is proposed. Finally, obviously, the analysis of the operating limits in small/medium variations are not necessary (grey-fill), only the operating domain with large variations could be studied, not only in terms of analysis but also in terms of synthesis.

Conclusion

The action of the LAS systems generates torques and forces that influence the overall dynamic behavior of a vehicle, i.e. longitudinal, lateral, and vertical dynamics. On one hand, the study of the global dynamics of the vehicle, including high complex systems: nonlinearities, variable parameters, and coupling effects between various embedded systems, requires convenient and suitable modeling. In particular, concerning the suspension systems studies, the models range from the full complete vehicle model for validation to the quarter vehicle mode for synthesis. These models are based on the predefined domain of operation of the vehicle, and the objective of the study in order to facilitate the synthesis and the validation of the control/strategy designed.

Suspension system 1.4.1 Introduction

The suspension system used in the automobile is made up mainly of two components, the spring and the damping element, so-called the shock absorber (Figure 1.10). This simple system passes through one hundred years of development, from the appearance of passive dampers in Roll Royce in 1913 to the highly complex controllable Magneto-Rheological (MR) dampers used in the most recent cars, for example. Today, it plays an important role in the automotive industry due to its increased potential in improving the safety and handling of the vehicle, besides the vibration isolation and ride comfort. More detailed historical facts on suspension systems are presented in [START_REF] Aubouet | Semi-active SOBEN suspensions modeling and control[END_REF].

The suspension system is the system that is responsible to link the unsprung mass (e.g. wheels) to the vehicle chassis or body. As illustrated in Figure 1.9, three main roles fall on the automobile suspension system: -Support the static load of the vehicle; -Ensuring passenger comfort through vibration isolation; -Ensuring road-wheel contact by road holding.

The two main components of the suspension system should work adequately to fulfill these roles. The spring should reduce the effect of traveling over rough roads, leading to improved ride quality. While shock absorbers serve the purpose of limiting excessive suspension movement and damp spring oscillations. For this reason, without shock absorbers, the vehicle is able to absorb bumps, but the undamped suspension means that the vehicle will continue to bounce which might cause the tires to lose contact with the ground. Thus, adding shock absorber in parallel to springs, means that the vehicle could dampen the motion to prevent the vehicle from bouncing, in addition to bump's absorption. In this sense, it guarantees the contact between the wheel and the ground that is necessary for maintaining road holding and therefore for passenger safety. Indeed, the roads on which vehicles are driven are highly heterogeneous (Figure 1.11). The needed suspension characteristics depend on the degree of roughness of the road considered. Generally, road roughness has varying dimensions and shapes and consists of random distribution spatially and temporally spectrum [START_REF] Bouvin | Towards an alternative version of the Hydractive CRONE car suspension[END_REF]. Thus, these solicitations generate continuous vertical displacements whose direction and intensity depend on the profile of the obstacle as well as on the speed of crossing. To further understand the suspension role, it should be worthwhile to assume a vehicle chassis that is rigidly connected to the wheel without the suspension system. Then these accelerations are directly transmitted to the body of the vehicle, thus, transferred to the passengers. Consequently, this implies a discomfort driving experience and could be harmful or even dangerous for passengers. In this sense, the suspension system fulfills its critical role by acting on the comfort of the vehicle.

Moreover, vibrations affect our ability to work in both physical and mental sense. It may impair concentration or sensing important information or signals. In the physical sense, vibrations are a risk for the body tissues and organs since they absorb the energy of such vibrations. Particularly, the muscles compensate for the small forces that vibrations expose the body to, and if the body is exposed to vibration for a long-time duration this could pose a risk to the joints and could tire the muscles [START_REF] Berlin | Production Ergonomics: Designing Work Systems to Support Optimal Human Performance, Production Ergonomics: Designing Work Systems to Support Optimal Human Performance[END_REF]. Besides that, in vehicles where the body is sitting on a vibrating base, an additional risk factor is that different body tissues have different resonance [START_REF] Bouvin | Towards an alternative version of the Hydractive CRONE car suspension[END_REF].

frequencies, meaning that there is a range of vibrations at which somebody tissue will experience local discomfort (Figure 1.12). These resonance frequencies will vary from individual to another but tend to lie within a certain range, e.g. the resonance frequency of the shoulder typically lies between 4 to 6Hz, according to [START_REF] Bohgard | Work and Technology on Human Terms[END_REF][START_REF] Rasmussen | HUMAN BODY VIBRATION EXPOSURE AND ITS MEASUREMENT[END_REF].

However, not only joints and muscles are at risk, even the internal organs like the brain and eyes are sensitive to vibration at different frequencies. For example, the eyes at resonance frequency cannot function properly and lead to impaired visual work, while low-frequency vertical vibration (<1Hz) has a particular tendency to cause nausea or drowsiness. This explains why some people can experience motion sickness in the vehicle, while others are lulled to sleep. For this reason, it's very critical to ensure passenger comfort by filtering vibrations at a different range of frequencies. Although the tires and the seats partially absorb high-frequency irregularities, however, it is not enough to fulfill the global objectives.

In order to ensure passenger safety, the influence of the suspension on the road holding behavior of the vehicle must also be taken into account. Indeed, during emergency driving situations, the priority then goes to safety rather than comfort. Although this is mainly dependent on the tire because it's in direct contact with the road, the suspension also plays a large role in maintaining this contact. In some cases, road irregularities could lead the wheel to lift off the ground and lose contact with the road, thus, losing the ability of handling, braking, and traction functionality.

Therefore, as the suspension system could affect the overall dynamics of the vehicle, the work and studies should be oriented to optimize the performance of the suspension system in the sense of improving the overall dynamics of the vehicle.

Classification of Suspension systems

Along with the strong development of the automotive industry during past decades, the suspension system has evolved continuously to be able to meet the requirements mentioned above. The evolution was based on the mechanics, hydraulics, and electrical technological advances, as well as the control strategies applied. The suspension systems could be divided into two main sets:

-Passive suspension systems -Controllable suspension systems.

The suspension system is passive if the characteristics of the components are fixed. These characteristics are determined by the designer of the suspension, according to the design goals and the intended application.

On the other hand, the controllable suspension may be carried out according to the energy input and the bandwidth of the actuator. Here, three features may be observed: the controllability range, in other words, the range of forces that the actuators may deliver; the control bandwidth which is a measure of how fast the actuator action is; the power request that is mainly a result of the combination of controllability range and control bandwidth. The resulting classification is concisely shown in Figure 1.13, which highlight five families of controllable suspensions [START_REF] Savaresi | Semi-Active Suspension Control Design for Vehicles, Semi-Active Suspension Control Design for Vehicles[END_REF]:

-Adaptive suspension: the control action is applied as slow damping coefficient modulation. Thus, the control range is limited by the passivity constraint. The adaptive damping control is characterized by a small bandwidth and no energy is introduced in the system; therefore, the power demand is relatively very low (few Watts).

-Semi-active suspension: the control action is applied as fast damping coefficient modulation, in which it features an electronic shock absorber with a relatively large bandwidth up to 40Hz. However, the generated forces follow the passivity limits of the damper, thus no energy may be introduced into the system. Therefore, the demanded power is also relatively low (tens of Watts).

-Load-leveling suspension: it is considered as the first attempt towards full active suspensions, where they are capable of introducing energy in the system to change the steady-state condition. This kind of suspension acts on the parameter of the springs usually, e.g. air spring suspension [START_REF] Allen | Air Suspension Systems[END_REF]. The bandwidth generally is within 0.1-1 Hz, but the power request is around hundreds of Watts.

-Slow-active suspensions: in active suspension systems, the passivity limits are fully overcome and energy is introduced into the system.

The control input is represented by a suspension force delivered by an actuator placed parallel/series to passive components or replace them.

Generally, the bandwidth of such systems is relatively limited to a few Hertz, where the power consumption is in the range of kilo-Watts.

-Fully-active suspensions: the main difference here is the bandwidth relative to the slow active suspension. The fully active actuator is able to generate force at high frequencies and could reach the limit of 30Hz for some systems. This advantage comes at the expense of high-power requirements where it ranges in the region of tens of kilo-Watts.

Some Suspension system realization

Various attempts have been done in finding solutions for the inherent tradeoff of passive suspension systems since the 1980s. One example is the BOSE systems (Figure 1.14) [START_REF] Rakshith | Bose Automotive Suspension[END_REF]. This suspension uses [START_REF] Savaresi | Semi-Active Suspension Control Design for Vehicles, Semi-Active Suspension Control Design for Vehicles[END_REF].

an electromagnetic actuator that serves as driver and road disturbance rejection. The heart of the system is based upon a linear electromagnetic motor that is installed at each wheel. Inside the linear electromagnetic motor are magnets and coils of wire. When electrical power is applied to the coils, the motor retracts and extends, creating motion between the wheel and the car body. One of the key advantages of an electromagnetic approach is its fast response. The linear electromagnetic motor responds quickly enough to counter the effects of bumps and potholes, maintaining a comfortable ride. Additionally, the motor has been designed for maximum strength in a small package, allowing it to put out enough force to prevent the car from rolling and pitching during aggressive driving maneuvers.

Another example is the Hydractive suspension system developed by Citroën. Recent developments in electronics and computers made it possible to delegate many functions previously solved by mechanical-hydraulically components to electronic units. The latest version of this technology was the third generation, the so-called Hydractive 3+. Just like its predecessor, this system also has two modes, firm and soft. A stiffness regulator, an additional sphere and a hydraulic control block per axle, isolates or connects the corner and center spheres. The Hydractive suspension system also includes an automatic attitude correction device illustrated in Figure 1.15. This system maintains the ground clearance of the vehicle by modifying it at the request of the driver or, dynamically, depending on the speed and the road condition [START_REF] Bouvin | Towards an alternative version of the Hydractive CRONE car suspension[END_REF]. Furthermore, the CREUAT suspension, another active form of the suspension system, is a very close solution to Hydractive one (Figure 1.16). The big difference in the CREUAT system is although it has four gas spring chambers, however, they are not allocated one to each corner of the car.

Finally, the CRONE suspension system that had been intensively studied for years and could be found in details in [START_REF] Bouvin | Towards an alternative version of the Hydractive CRONE car suspension[END_REF][START_REF] Daou | Effect of hydropneumatic components nonlinearities on the CRONE suspension[END_REF][START_REF] Létévé | Study of the influence of passenger vehicle suspension on vibrational comfort, driving behaviour and operating limits : CRONE approach to formalization, analysis and synthesis[END_REF][START_REF] Moreau | The CRONE suspension: Management of the dilemma comfort-road holding[END_REF][START_REF] Moreau | The CRONE Suspension[END_REF][START_REF] Oustaloup | The crone suspension[END_REF][START_REF] Rizzo | L'approche CRONE dans le domaine des architectures complexes des suspensions de véhicules automobiles : la suspension CRONE Hydractive[END_REF].

Active Wheel

Automobile manufacturers are in the process of launching EVs and a market for electric mobility solutions will emerge where all-in-wheel devices are expected to play an important role in this new trend. Such vehicles are able to adapt to urban lifestyles, mainly, to contribute to the air quality and noise reduction in urban cities. Since a lighter vehicle means a sharp reduction in energy consumption, a modern solution can contain integrated powertrain suspension located directly inside the wheel. One example of such technology is the Active Wheel (AW) developed by Michelin (Figure 1.17). This advanced technology initiates a new era in road transportation where all essential components have been integrated within the wheel itself: the pneumatic part, a compact traction electric motor with regenerative braking, an electric active suspension system, and the braking system. The structure of the Active Wheel offers new opportunities in Global Chassis Control (GCC) that rises, recently, to remarkable levels. At first, the conventional mechanical suspension has been replaced by a passive spring and a DC electrical motor. In comparison with other force-generating actuators (hydraulic, Hydractive, hydro-pneumatic...), the DC linear characteristics of the AW's suspension system provides an extremely fast response time (just 3/1000 th of a second), low contact friction, high precision and efficiency, good controllability, and less energy consumption [START_REF] Nguyen | Direct current linear actuator using in the active suspension system of automobile[END_REF].

Depending on the amount of power or type of usage desired, a given vehicle may integrate four motors (one in each wheel) or two motors (in the front wheels, for example). In this way, the Michelin Active Wheel allows automobile manufacturers to continue designing both two and four-wheel drive cars.

With the Michelin Active Wheel, the energy that powers the integrated motor is always electric whether it comes from a lithium-ion or other type of battery, a fuel cell, and /or supercapacitors. In all cases, these power sources offer two important benefits -zero pollution and optimal comfortsince vehicles equipped with the Michelin Active Wheel emit no greenhouse gases. What's more, electrical power trains are extremely quiet, which is an advantage not only to passengers but above all to people outside the vehicle. In short, a car powered by the Michelin Active wheel solution is like a breath of fresh air in urban environments.

The Michelin Active Wheel greatly simplifies the vehicle design process by making the mechanical components on a conventional car superfluous. Vehicles powered by the Michelin Active Wheel have no need for a gearbox, clutch, transmission shaft, differential or shock absorbers. This, in turn, makes cars lighter and thus more energy efficient. As a result, the vehicle can achieve an operating range in line with motorist expectations. This technology is clearly a game-changer, providing an effective and elegant response to today's critical road transportation issues: energy (quantity, diversity, and cost), greenhouse gas emissions, urban congestions, and urban pollution. Figure 1.18 shows two examples of light electric vehicles equipped with Active Wheel technology. For example, the Heuliez WILL electric vehicle, which has a maximum mass of 1000kg, and a maximum speed of 140km/h, with 2-wheel motor drive. The other example, the Venturi Volage, which has two seats only, with four Active wheels, with a maximum speed of 150km/h. The structure of the AW is adopted in this thesis, where it considers a light vehicle equipped with Active Wheel technology. The study and the control of the suspension systems are based on the structure of the electrical active suspension systems integrated into the Active wheel.

Conclusion

The technology of suspension systems had been developed for around a hundred years. Much innovative passive and active versions are revealed to deal with the increasing demand for comfort and road holding requirements. They are distinct in terms of energy consumption and control bandwidth, mainly. The final objective of any designed suspension system is to full fill the specifications of the comfort, handling, and safety of the vehicle. In this work, the Active Wheel structure is adopted, taking advantage of its electrical and linear characteristics. The work is then oriented to investigate the contribution of such structure in the global chassis control of a light electric vehicle. In the framework of global chassis control of a light electric vehicle, the focus is oriented to control the critical active systems embedded in the vehicle. Pointing towards the well-known and common objective s of all global chassis control strategies, i.e. comfort, handling, road holding, and safety.

Strategy, Approach, and Objectives

Firstly, in this study, a frequential synthesis of robust controllers are taken into account, e.g. CRONE, 𝐻𝐻 ∞ , QFT, … However, the CRONE controller, in its all generations, is the most used controller along with this work in order to get the most advantage of collaborations with the CRONE group at the IMS Lab. Furthermore, the comparison between these types of controllers with temporal synthesized controllers could be as a perspective.,. e.g. LQ, LQG, sliding mode controller, backstepping controller, ….

Two-level Approach

The "Top -Down" Approach In this approach the designer is looking forward for the maximum performance could be achieved, while assuming the following:

-All the state variables are considered measurable, and no limitations are existing on the number of sensors needed.

-All the sensors used are working perfectly, that is their dynamics are neglected, and no noise measurements are introduced.

-All the actuators are assumed to work perfectly, where their dynamics are neglected with no saturation limits, that is all desired control inputs are supposed to be equal to the generated one.

The "Down -Top" Approach

In this approach, a variety of constraints are taken into account and introduced to the study, as a result, a more realistic performance is achieved. Some of these realistic constraints are as follows:

-The number of sensors is limited, then the use of observers and estimators is necessary.

-The limitation of the saturation of the control input of actuators is considered as well as the dynamics of the actuator.

-The dynamics of sensors and noise measurements and the range of operating are no longer neglected.

Vertical Control Objectives

In the framework of Global Chassis control, the primary work done was regarding the vertical dynamics control of the light electric vehicle. The objectives are listed as follows:

1. Varying the height of the vehicle at a stop, e.g. loading and unloading phases.

2. Holding chassis against driver solicitations.

3. Holding chassis against road perturbations. 4. Attain good vibratory isolation against road perturbations. 5. Holding the wheel (unsprung mass) under road disturbances by:  Level 1: limiting the vibrations from the wheel to the chassis  Leve 2: limiting the variations of the wheel deflections in the sense of control of road holding.

The global problem of vertical dynamics control of the vehicle is then decomposed into different local issues.

-At the scale of the quarter vehicle model: using the 2DoF model to deal with chassis dynamics (sprung mass) and wheel dynamics (unsprung mass).

-At the scale of the complete vehicle model, using the 7DoF model that includes: Heave, pitch, and roll motions, and the four movements related to wheel displacements.

These objectives are to be discussed in detail in the next chapters.

Longitudinal Control Objective

In the context of analyzing the longitudinal dynamics of the light electric vehicle, the study is focused on the effect of the vertical dynamics of the vehicle on the longitudinal dynamics. Here, the work done is related to braking behavior only.

Along with the many advances in vehicular technology development, the safety requirements for automobiles are of great importance. The techniques applied to various vehicles have already improved the system's stability and passenger's safety with the use of several significant control systems, such as anti-lock braking systems (ABS) [START_REF] Dix | ABS Control Using Optimum Search via Sliding Modes[END_REF][START_REF] Peeie | Skid control of a small electric vehicle with two in-wheel motors: simulation model of ABS and regenerative brake control[END_REF][START_REF] Schinkel | Anti-lock braking control using a sliding mode like approach[END_REF] Thus, an ABS CRONE controller is introduced and then a coordination strategy between the longitudinal control and the designed vertical controllers is done in the sense of optimizing the braking performance of the vehicle.

Lateral Control Objective

For lateral dynamics, the study is divided into different parts and can be listed as follows:

-Improve vehicle's stability by yaw rate regulation using a microbraking system controller.

-Improve vehicle's stability by yaw rate regulation using a microsteering system controller.

-Improve the tracking trajectory using a controlled active steering system.

Using the CRONE controller, the following objectives are to be achieved, while taking into account the effect of vertical dynamics on each system. Then, the final objective is to establish the effect of the controlled vertical dynamics system on each of the lateral controlled dynamic systems and implement collaboration between these systems for optimizing vehicle lateral stability and tracking trajectory.

Conclusion

In the framework of GCC, there exists three main architecture in the literature, namely, the centralized control, decentralized control, and control by supervision [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF]. This study adopts a supervision control strategy, that represents an intermediate structure between the other two ones, by a proposed hierarchical approach for vertical dynamic control of a light electric vehicle coordinated with longitudinal and lateral dynamics. The supervisor represents the highest level of this hierarchical approach. It receives all available measurements coming from the vehicle itself (sensors, pedals, ...), environment (exteroceptive sensors, camera, LiDAR, ...), and information that could be provided by road infrastructure or by other vehicles (like in autonomous vehicles). All these data are used in the perception of the environment, observation, path planning, motion generation, and last but not least, diagnosis.

The CRONE Control Approach 1.6.1 Introduction to the CRONE Control System Design Methodology

The CRONE (the French acronym for Commande Robuste d'Ordre Non Entier, which means fractional-order robust control) CSD methodology is a frequency domain approach that has been developed since the 1980s (Oustaloup, 1995(Oustaloup, , 1991a[START_REF] Oustaloup | Linear feedback control systems of fractional order between 1 and 2[END_REF]. It is based on the common unity-feedback configuration presented in Figure 1.19. Three CRONE CSD methods have been developed, successively extending the field of application. In these three methods, the controller or open-loop transfer function is defined using fractional order integro-differentiation. In the frequency domain, they enable linear robust control systems to be simply and methodologically designed.

The major advantage over the 𝐻𝐻 ∞ design is that the plant uncertainties (or perturbations) are taken into account with no distinction of their nature, whether they are structured (parametric) or unstructured.

The fractional-order, which is either real or complex depending on the generation of the control design, enables parameterization of the open-loop transfer function with a small number of high-level parameters. The optimization of the control is thus reduced to only the search for the optimal values of these parameters. In the absence of supplementary information, mainly the one related to noise measurements, it could be difficult to determine the frequency at the unity gain of the open-loop transfer function.

Control diagram and data for synthesis

First-generation CRONE Control

The first-generation CRONE control proposes to use a controller without phase variation around the cross over frequency 𝜔𝜔 𝑐𝑐𝑐𝑐 . This strategy has to be used when the frequency 𝜔𝜔 𝑐𝑐𝑐𝑐 is within a frequency range where the plant phase is constant (with respect to the frequency). In this range where the plant frequency response is asymptotic (this frequency band is called a plant asymptotic-behavior band), the plant variations are only gain-like.

Within a frequency range [𝜔𝜔 𝐴𝐴 , 𝜔𝜔 𝐵𝐵 ] around the desired frequency 𝜔𝜔 𝑐𝑐𝑐𝑐 , the CRONE controller is defined from the fractional transfer function of an order 𝑚𝑚 integro-differentiator, such as:

𝐶𝐶(𝑠𝑠) = 𝐶𝐶 0 � 1 + 𝑠𝑠 𝜔𝜔 𝑙𝑙 1 + 𝑠𝑠 𝜔𝜔 ℎ � 𝑚𝑚 , (1.12)
where 𝐶𝐶 0 , 𝜔𝜔 𝑙𝑙 , 𝜔𝜔 ℎ , and 𝑚𝑚 ∈ ℝ .

In order to satisfy the predefined specifications, the cascade form of the first generation CRONE controller is extended to include: a proportional constant 𝐶𝐶 0 , a band-limited fractional-order derivative (𝑚𝑚) within two corner frequencies 𝜔𝜔 𝑙𝑙 and 𝜔𝜔 ℎ , an integer band-limited integrator of order 𝑚𝑚 𝐼𝐼 upper bounded by the frequency 𝜔𝜔 𝐼𝐼 to manage steady-state error and a lowpass filter of order 𝑚𝑚 𝑓𝑓 with a cutoff frequency 𝜔𝜔 𝑓𝑓 to manage the control sensitivity level and reduce noise amplification at high frequencies. Therefore, the general transfer function is formulated as follows:

𝐶𝐶(𝑠𝑠) = 𝐶𝐶 0 �1 + 𝜔𝜔 𝐼𝐼 𝑠𝑠 � 𝑚𝑚 𝐼𝐼 � 1 + 𝑠𝑠 𝜔𝜔 𝑙𝑙 ⁄ 1 + 𝑠𝑠 𝜔𝜔 ℎ ⁄ � 𝑚𝑚 1 �1 + 𝑠𝑠 𝜔𝜔 𝑓𝑓 ⁄ � 𝑚𝑚 𝑓𝑓 . (1.13)
The transitional frequencies defined are calculated so that a constant phase is achieved on a frequency range [𝜔𝜔 𝐴𝐴 , 𝜔𝜔 𝐵𝐵 ] which covers the frequency range where 𝜔𝜔 𝑐𝑐 could vary due to the plant parametric variation (Sabatier et al., 2015a). Then, the fractional-order 𝑚𝑚 and the gain 𝐶𝐶 0 are calculated to ensure the required nominal phase margin 𝑀𝑀 𝜙𝜙 and open-loop gain crossover frequency 𝜔𝜔 𝑐𝑐 .

However, for time-domain simulations, the fractional part of the cascade form should be replaced by a rational transfer function. Thanks to Oustaloup's method (Oustaloup, 1995(Oustaloup, , 1991a)), using N recursive zeros and N recursive poles, this point could be achieved. Then the final cascade form of the first-generation CRONE controller is written as follows: .14) where 𝑚𝑚 𝐼𝐼 ,𝑚𝑚 𝑓𝑓, 𝑁𝑁 ∈ ℕ and 𝜔𝜔 𝑖𝑖 ′ , 𝜔𝜔 𝑖𝑖 ∈ ℝ.

𝐶𝐶(𝑠𝑠) = 𝐶𝐶 0 �1 + 𝜔𝜔 𝐼𝐼 𝑠𝑠 � 𝑚𝑚 𝐼𝐼 � 1 + 𝑠𝑠 𝜔𝜔 𝑖𝑖 ′ ⁄ 1 + 𝑠𝑠 𝜔𝜔 𝑖𝑖 ⁄ 𝑁𝑁 𝑖𝑖=1 1 �1 + 𝑠𝑠 𝜔𝜔 𝑓𝑓 ⁄ � 𝑚𝑚 𝑓𝑓 . ( 1
𝜔𝜔 𝑖𝑖+1 ′ 𝜔𝜔 𝑖𝑖 ′ = 𝜔𝜔 𝑖𝑖 𝜔𝜔 𝑖𝑖 = 𝛼𝛼𝛼𝛼 > 1, (1.15
)

𝜔𝜔 𝑖𝑖 𝜔𝜔 𝑖𝑖 = 𝛼𝛼 and 𝜔𝜔 𝑖𝑖+1 ′ 𝜔𝜔 𝑖𝑖 ′ = 𝛼𝛼, (1.16
) Generally, the parameters 𝜔𝜔 𝑖𝑖 and 𝜔𝜔 𝑖𝑖 ′ can be calculated by fixing a value for 𝑁𝑁 which ensures 𝛼𝛼𝛼𝛼 ≈ 5.

𝛼𝛼𝛼𝛼 = � 𝜔𝜔 ℎ 𝜔𝜔 𝑙𝑙 � � 1/𝑁𝑁 , ( 1
The robustness of the phase margin 𝑀𝑀 𝜙𝜙 is ensured only if 𝜔𝜔 𝑐𝑐𝑐𝑐 is in the frequency range where the system phase is constant. When this condition is not met, use the second-generation CRONE command.

Second-generation CRONE controller

For problems of control effort level, it is sometimes impossible to choose a 𝜔𝜔 𝑐𝑐𝑐𝑐 within an asymptotic behavior frequency band of the plant. Nevertheless, as Bode first state for the 'design of single loop absolutely stable amplifiers' whose tube gains vary, the robust controller is the one which permits the open-loop transfer function to be defined by a constant phase in a useful band. So, the CRONE approach defines the open-loop transfer function 6m m around 𝜔𝜔 𝑐𝑐𝑐𝑐 from the open-loop transfer function of a fractional integrator [START_REF] Lanusse | CRONE control system design toolbox for the control engineering community: Tutorial and case study[END_REF]:

𝛽𝛽(𝑠𝑠) = � 𝜔𝜔 𝑐𝑐𝑐𝑐 𝑠𝑠 � 𝜈𝜈 (1.20)
with 𝜔𝜔 𝑐𝑐𝑐𝑐 ∈ ℝ and 𝜈𝜈 ∈ [1,2].

The complementary sensitivity function 𝑇𝑇 (𝑠𝑠), and sensitivity function 𝐶𝐶(𝑠𝑠), can be expressed as:

𝑇𝑇 (𝑠𝑠) = 𝛽𝛽(𝑠𝑠) 1 + 𝛽𝛽(𝑠𝑠) = 1 1 + � 𝑠𝑠 𝜔𝜔 𝑐𝑐𝑐𝑐 � 𝜈𝜈 (1.21)
and

𝐶𝐶(𝑠𝑠) = 1 1 + 𝛽𝛽(𝑠𝑠) = � 𝑠𝑠 𝜔𝜔 𝑐𝑐𝑐𝑐 � 𝜈𝜈 1 + � 𝑠𝑠 𝜔𝜔 𝑐𝑐𝑐𝑐 � 𝜈𝜈 . (1.22)
Around 𝜔𝜔 𝑐𝑐𝑐𝑐 , the Nichols plot of 𝛽𝛽(𝑠𝑠) is a vertical straight line whose phase location is determined by order 𝜈𝜈 only (Figure 1.21). This vertical straight line being the desired shape of the open-loop frequency response Nichols plot, we define it as a frequency template or simply template (with a different meaning from that given by the QFT approach where the template is a frequency uncertainty domain). The vertical template thus defined slides on its own axis as the plant parameters vary (i.e. as 𝜔𝜔 𝑐𝑐𝑐𝑐 varies).

The gray area of Figure 1.21 represents the uncertainty (or perturbation) of the open-loop frequency response.

This vertical displacement of the template ensures:

-that the robust phase margin 𝑀𝑀 𝜙𝜙 equals (2 -𝜈𝜈)𝜋𝜋/2, -a robust resonant peak 𝑀𝑀 𝑟𝑟 expressed by:

𝑀𝑀 𝑟𝑟 = sup ω |𝑇𝑇 (𝑗𝑗𝜔𝜔)| |𝑇𝑇 (𝑗𝑗0)| = 1 sin (𝜈𝜈𝜋𝜋/2)
.

(1.23) -a robust modulus margin 𝑀𝑀 𝑚𝑚 expressed by:

𝑀𝑀 𝑚𝑚 = inf ω |𝛽𝛽(𝑗𝑗𝜔𝜔) + 1| = �sup ω |𝐶𝐶(𝑗𝑗𝜔𝜔)|� -1 = sin (𝜈𝜈𝜋𝜋/2). (1.24)
-a robust damping ratio 𝜁𝜁 directly deduced from the central half-angle 𝜃𝜃 formed by the two closed-loop conjugate complex poles of T(s) and of S(s).

𝜁𝜁 = cos θ = cos �π - π 𝜈𝜈 � -cos � 𝜋𝜋 𝜈𝜈 �, (1.25)
thus, introducing the notion of robust oscillatory mode.

To manage the control effort level and the steady-state errors provided by the CRONE controller, the fractional open-loop transfer function (1.14) has to be bandlimited and to include integral and low-pass effects. The nominal fractional open-loop transfer function is defined by:

𝛽𝛽 0 (𝑠𝑠) = 𝐾𝐾 � 𝜔𝜔 1 ′ 𝑠𝑠 + 1� 𝑛𝑛 1 � 1 + 𝑠𝑠 𝜔𝜔 ℎ 1 + 𝑠𝑠 𝜔𝜔 1 � 𝜈𝜈 ⎝ ⎜ ⎛ 1 1 + 𝑠𝑠 𝜔𝜔 ℎ ′ ⎠ ⎟ ⎞ 𝑛𝑛 ℎ , (1.26)
Figure 1.21. Vertical sliding of a frequency template (bold solid line) ensuring the robustness of phase margin 𝑀𝑀 𝜙𝜙 and of resonant peak 𝑀𝑀 𝑟𝑟 (Sabatier et al., 2015a) .

where 𝐾𝐾, 𝜔𝜔 1 ′ , 𝜔𝜔 ℎ , 𝜔𝜔 1 , and 𝜔𝜔 ℎ ′ ∈ ℝ + , and 𝑛𝑛 1 and 𝑛𝑛 ℎ ∈ ℕ + . Figure 1.22 presents the shape of the frequency response of the perturbed open-loop 𝛽𝛽(𝑠𝑠) defined from its nominal value. Integer orders 𝑛𝑛 𝑙𝑙 and 𝑛𝑛 ℎ are set by taking into account the performance specification and the plant magnitude asymptotic behavior at low and high frequencies.

The simplest way to obtain a rational controller from the fractional openloop transfer function is to use Oustaloup's method to approximate the band-limited fractional order integrator of (1.26). Using N recursive zeros and N recursive poles, the nominal open-loop transfer function becomes rational: .27) where ⌊𝜈𝜈⌋ is the integer part of 𝜈𝜈 and with:

𝛽𝛽 0𝑅𝑅 (𝑠𝑠) = 𝐾𝐾 � 𝜔𝜔 1 ′ 𝑠𝑠 + 1� 𝑛𝑛 1 � 1 + 𝑠𝑠 𝜔𝜔 ℎ 1 + 𝑠𝑠 𝜔𝜔 1 � ⌊𝜈𝜈⌋ � 1 + 𝑠𝑠 𝜔𝜔 𝑖𝑖 1 + 𝑠𝑠 𝜔𝜔 𝑖𝑖 ′ 𝑁𝑁 1 ⎝ ⎜ ⎛ 1 1 + 𝑠𝑠 𝜔𝜔 ℎ ′ ⎠ ⎟ ⎞ 𝑛𝑛 ℎ , ( 1
𝜔𝜔 1 = �𝛼𝛼 𝜔𝜔 1 , 𝜔𝜔 1 ′ = 𝛼𝛼𝜔𝜔 1 , 𝜔𝜔 𝑖𝑖+1 = 𝛼𝛼𝛼𝛼𝜔𝜔 𝑖𝑖 and 𝜔𝜔 𝑖𝑖+1 ′ = 𝛼𝛼𝛼𝛼𝜔𝜔 𝑖𝑖 ′ ,
(1.28) and

𝛼𝛼 = � 𝜔𝜔 ℎ 𝜔𝜔 1 � 𝜈𝜈-⌊𝜈𝜈⌋ 𝑁𝑁 and 𝛼𝛼 = � 𝜔𝜔 ℎ 𝜔𝜔 1 � 1-𝜈𝜈+⌊𝜈𝜈⌋ 𝑁𝑁 . (1.29)
Then, the rational controller 𝐶𝐶 𝑅𝑅 (𝑠𝑠) is defined by:

𝐶𝐶 𝑅𝑅 (𝑠𝑠) = 𝐺𝐺 0 -1 (𝑠𝑠)𝛽𝛽 0𝑅𝑅 (𝑠𝑠).
(1.30)

The third-generation CRONE Controller

Just as the second-generation CRONE control permits the choice of frequency 𝜔𝜔 𝑐𝑐𝑐𝑐 outside an asymptotic behavior band of the plant, and widens the application domain of the first generation of CRONE control, the third generation widens the second by permitting the handling of more general uncertainties than just gain-like perturbations.

Indeed, the vertical sliding of the previous vertical template is a requirement which is only achieved on perturbation of:

-all the parameters of the plant in the particular case when the frequency 𝜔𝜔 𝑐𝑐𝑐𝑐 is within a frequency band where the plant frequency response is close to its asymptote; -a gain, and corner frequencies which are greatly different from 𝜔𝜔 𝑐𝑐𝑐𝑐 in the general case when the frequency 𝜔𝜔 𝑐𝑐𝑐𝑐 is not within an asymptotic behavior band.

Except for these cases, the vertical template does not slide on itself when the plant transfer function is perturbed and this template does not necessarily ensure the best robustness of the control-loop system.

The open-loop transfer function is now defined as the real part of a complex fractional order integrator, that is:

𝛽𝛽(𝑠𝑠) = Re /i � 𝜔𝜔 𝑐𝑐𝑐𝑐 𝑠𝑠 � 𝑛𝑛 , (1.31)
with 𝑛𝑛 = 𝑎𝑎 + 𝑃𝑃𝑏𝑏 ∈ ℂ.

The real order determines the phase placement of this Nichols locus, socalled generalized template, and then the imaginary order 𝑏𝑏 determines its angle to the vertical. From an infinity of these templates which tangent the magnitude contour related to a required magnitude peak 𝑄𝑄 𝑟𝑟 for the nominal parametric state of the plane, the optimal template can be defined as the template which permits minimization of the variations of 𝑄𝑄 stemming from the uncertainty domain which in turn stems from the various parametric states of the plant. The principle of the third-generation CRONE control methodology is to optimize the parameters of the nominal open-loop transfer function 𝛽𝛽 0 (𝑠𝑠) that includes a band-limited complex fractional-order integration, that is:

𝛽𝛽 0 (𝑠𝑠) = 𝛽𝛽 1 (𝑠𝑠)𝛽𝛽 𝑚𝑚 (𝑠𝑠)𝛽𝛽 ℎ (𝑠𝑠), (1.32)
where 𝛽𝛽 𝑚𝑚 (𝑠𝑠) is a set of band-limited generalized templates:

𝛽𝛽 𝑚𝑚 (𝑠𝑠) = � 𝐶𝐶 𝑘𝑘 𝑠𝑠𝑖𝑖𝑐𝑐𝑛𝑛(𝑏𝑏 𝑘𝑘 ) 𝑁𝑁+ 𝑘𝑘=-𝑁𝑁 - �𝛼𝛼 𝑘𝑘 1 + 𝑠𝑠 𝜔𝜔 𝑘𝑘+1 1 + 𝑠𝑠 𝜔𝜔 𝑘𝑘 � 𝑎𝑎 𝑘𝑘 ⎝ ⎜ ⎛ Re /i ⎩ � ⎨ � ⎧ �𝛼𝛼 𝑘𝑘 1 + 𝑠𝑠 𝜔𝜔 𝑘𝑘+1 1 + 𝑠𝑠 𝜔𝜔 𝑘𝑘 � 𝑖𝑖𝑏𝑏 𝑘𝑘 ⎭ � ⎬ � ⎫ ⎠ ⎟ ⎞ -𝑞𝑞 𝑘𝑘 𝑠𝑠𝑖𝑖𝑐𝑐𝑛𝑛(𝑏𝑏 𝑘𝑘 ) (1.33) with, 𝛼𝛼 𝑘𝑘 = � 𝜔𝜔 𝑘𝑘+1 𝜔𝜔 𝑘𝑘 � 1/2 for 𝑘𝑘 ≠ 0 and 𝛼𝛼 0 = ⎝ ⎜ ⎜ ⎛ 1 + � 𝜔𝜔 𝑟𝑟 𝜔𝜔 0 � 2 1 + � 𝜔𝜔 𝑟𝑟 𝜔𝜔 1 � 2 ⎠ ⎟ ⎟ ⎞ 1 2 , (1.34)
where, .35) where 𝑁𝑁 + , 𝑁𝑁 -, and 𝑞𝑞 𝑘𝑘 ∈ ℕ + ; and 𝜔𝜔 𝑟𝑟 , 𝜔𝜔 𝑘𝑘 , 𝜔𝜔 𝑘𝑘+1 , 𝜔𝜔 𝑘𝑘 , 𝛼𝛼 𝑘𝑘 , 𝐶𝐶 𝑘𝑘 , 𝐶𝐶 1 , 𝐶𝐶 ℎ , 𝑎𝑎 𝑘𝑘 and 𝑏𝑏 𝑘𝑘 ∈ ℝ.

𝛽𝛽 1 (𝑠𝑠) = 𝐶𝐶 1 � 𝜔𝜔 -𝑁𝑁 - 𝑠𝑠 + 1� 𝑛𝑛 1 and 𝛽𝛽 ℎ (𝑠𝑠) = 𝐶𝐶 ℎ � 𝑠𝑠 𝜔𝜔 𝑁𝑁 + + 1� -𝑛𝑛 ℎ , ( 1
The fractional controller 𝐶𝐶 𝐹𝐹 (𝑠𝑠) is defined then by its frequency response as:

𝐶𝐶 𝐹𝐹 (𝑗𝑗𝜔𝜔) = 𝛽𝛽 0 (𝑗𝑗𝜔𝜔)/𝐺𝐺 0 (𝑗𝑗𝜔𝜔).

(1.36)

The parameters of a rational transfer function 𝐶𝐶 𝑅𝑅 (𝑠𝑠) with a predefined low-order structure are tuned to fit the ideal frequency response 𝐶𝐶 𝐹𝐹 (𝑠𝑠) using any frequency-domain system identification technique. An advantage of this design method is that, whatever the complexity of the control problem, the reduced value of the controller order (usually around 6) can be used. More details are explained intensively in [START_REF] Lanusse | CRONE control system design toolbox for the control engineering community: Tutorial and case study[END_REF] and (Sabatier et al., 2015b).

CRONE Control System Design Toolbox

The CRONE toolbox, developed gradually since the nineties [START_REF] Oustaloup | The CRONE toolbox for Matlab[END_REF] by the CRONE research group, and granted by CNRS and PSA car company, is a MatLAB and Simulink toolbox dedicated to applications of fractional (or non-integer) derivatives in engineering and science.

The toolbox is not only intended to researchers, including Ph.D. students, but also to industrials whose growing interest in fractional systems and whose readiness to invest in the development of its applications are manifest. Original theoretical mathematical concepts developed in the group were integrated in the first version of the CRONE toolbox as four main modules each of which deals with a specific application domain of fractional derivatives: the "Mathematical module" that contains several algorithms of fractional calculus, the "System identification module" that contains frequency and time-domain system identification algorithms; the "CRONE control module" that uses fractional derivatives as high-level design parameters in order to simplify the design of robust control systems; and the "Path planning module" that contains algorithms to optimize paths in dynamic environments During the last ten years, the CRONE toolbox was only available for some privileged researchers and industrials who granted its development. On the other hand, the last two decades have witnessed a growing interest in fractional derivatives and their applications. From now on, the CRONE toolbox is freely available for the international scientific community for research and pedagogical purposes. The aim of the CRONE group is to share its developments and its knowledge with scientists, researchers, and engineers worldwide.

Conclusion

The Control System Design of a controller is represented using the frequency-domain approach. It provides an overview of the three generations of the CRONE controllers. Fractional-order controllers provide more tuning parameters that are used to improve robustness by the CRONE methodology. The first-generation CRONE controllers can be used for plants with magnitude variation and constant phase. When the plant phase varies with respect to the frequency, the second generation of CRONE controllers can be used. As described before, this robust controller is designed through a fractional-order open-loop transfer function. From real, the order will become complex for both gain and phase perturbed plants. This defines the third generation CRONE methodology.

For more details concerning the design methodology of the CRONE controller, the reader could refer to the following reference (Sabatier et al., 2015b).

Conclusion

A brief overview of the overall vehicle's dynamics had confirmed the role of the suspension systems in the overall performance of the vehicle. As the suspension system influences the vertical dynamics directly, it affects the longitudinal and lateral dynamics indirectly. Three domains of operations and three domains of studies were demonstrated. Therefore, different models are defined each one is dedicated to the domain of operation and study. They range from validation to analysis, and synthesis models. Depending on the region of study considered, suspension performance must meet multiple objectives, and mostly they are in contradiction with each other. Here the use of the active suspension system gives the ability to find solutions for the main dilemma inherent in passive systems. This had motivated many studies to investigate the optimal performance of suspension systems. Among all types of suspension, the electrical active suspension system integrated with the Active wheel is studied. The interests and the challenges in fully active suspension systems control has been presented by summarizing the results of several already existing studies in the literature related to modeling and controlling methodology. This thesis deals with global vertical dynamics effects on the global chassis control's perspectives. Thus, controllers fulfilling suspension objectives are implemented, and furthermore, different controllers are designed for lateral and longitudinal dynamics, aiming to improve vehicle's performance in terms of lateral stability, tracking trajectory, and braking optimality. The CRONE CSD methodology is considered in almost every controller to examine the relative benefits of fractional order controllers in the vehicle's dynamics control.

Strategies and approaches had been indicated to pave the way for achieving the assigned objectives and provide a solid scientific reference on the context of global chassis control for the new trend technology, the so-called, all-in-wheel electric/autonomous vehicles.

Introduction

Investigations on improving suspension systems for vehicles have drawn much interest in recent years for better ride quality. Briefly, the purposes of the vehicle suspension system are to (1) isolate the vehicle body from road disturbances in order to provide good ride quality; (2) keep good road holding; (3) provide good handling; and ( 4) support the vehicle static weight (Rajamani, 2012a). This chapter is focusing on these global vertical control objectives that were previously listed in detail in Chapter 1.

At first, an intensive study is done on the quarter vehicle model to analyze the different modes of dynamics in the frequency domain. For analysis purposes, various transfer functions are introduced representing the relations between the input and outputs of the system, as well as the effect of the active forces, generated by the active suspension system, on each degree of freedom.

This chapter focuses on two main concepts, the chassis's height control and the fractional-order damping control, in order to handle various suspension objectives. However, other solutions are introduced in the next chapters regarding reject driver solicitations, for example. For this reason, a CRONE controller is designed in order to regulate the height of the chassis of the vehicle. Then, a comfort-oriented controller is designed to regulate the vertical displacement of the sprung mass against road perturbations using a fractional damping proposed strategy. The designed fractional-order damping controllers are tested for road disturbance rejection in the sense of comfort, road holding, and vibratory isolation.

Frequency domain study for quarter vehicle model

Following the brief introductory section related to the quarter vehicle vertical dynamics in section 1.2.3, the present study will get deeper into the frequency analysis of those dynamics. Figure 2.1 shows the quarter vehicle model with the integrated active suspension system in the Active Wheel (AW). The vehicle's vertical dynamics are modeled using the well-known quarter-model that exists in the literature [START_REF] Jazar | Suspension Mechanisms, in: Vehicle Dynamics: Theory and Application[END_REF][START_REF] Savaresi | Semi-Active Suspension Control Design for Vehicles, Semi-Active Suspension Control Design for Vehicles[END_REF].

Here, the four quarter vehicle systems are assumed decoupled and the suspension dynamics are considered in the frequency range of interest [ 0 -30 Hz]. Despite the nonlinear characteristics of the suspension components, it is pointed out that such a model can be used for optimal suspension control design and analysis [START_REF] Hrovat | Survey of Advanced Suspension Developments and Related Optimal Control Applications[END_REF], where it represents a convenient modeling framework to evaluate the performance and robustness of a controller.

The suspension configuration of the AW has been considered as a high bandwidth active suspension system due to the presence of the force-generating actuator mounted in parallel to the passive spring-damper components [START_REF] Sun | Vibration isolation for active suspensions with performance constraints and actuator saturation[END_REF]. Assuming that the actuator is ideal, that is the desired force is equal to the generated force, the vertical motion of this system is described by two differential equations as follows:

𝑚𝑚 2 𝑎𝑎 ̈2(t) = -𝑓𝑓 𝑠𝑠𝑝𝑝 (t) + 𝑓𝑓 0 (t) + 𝑢𝑢 𝑎𝑎 (𝑇𝑇), (2.1
)

𝑚𝑚 1 𝑎𝑎 ̈1(t) = 𝑓𝑓 𝑠𝑠𝑝𝑝 (t) -𝑓𝑓 1 (t) -𝑢𝑢 𝑎𝑎 (𝑇𝑇), (2.2)
with,

z 2 z 1 z 0 m 2 m 1 k 2 k 1 b 1 b 2 u a f 0 Figure 2.1. Active Wheel Quarter vehicle model representation. 𝑓𝑓 𝑠𝑠𝑝𝑝 (𝑇𝑇) = 𝑘𝑘 2 (𝑎𝑎 2 (𝑇𝑇) -𝑎𝑎 1 (𝑇𝑇)) + 𝑏𝑏 2 (𝑎𝑎 ̇2(𝑇𝑇) -𝑎𝑎 ̇1(𝑇𝑇)), (2.3
)

𝑓𝑓 1 (𝑇𝑇) = 𝑘𝑘 1 (𝑎𝑎 1 (𝑇𝑇) -𝑎𝑎 0 (𝑇𝑇)) + 𝑏𝑏 1 (𝑎𝑎 ̇1(𝑇𝑇) -𝑎𝑎 ̇0(𝑇𝑇)).
(2.4)

The force 𝑓𝑓 0 (𝑇𝑇) describes the external disturbances to denote the variations of the static payload, aerodynamical loads, and inertial forces induced by driver operations, e.g. cornering, braking, and accelerating. The variables 𝑎𝑎 2 (𝑇𝑇) and 𝑎𝑎 1 (𝑇𝑇) represents the sprung mass and unsprung mass displacements, respectively. The term 𝑢𝑢 𝑎𝑎 (𝑇𝑇) represents the value of the desired control input that assumed to be equal to the generated force by the actuator.

Under the assumption of zero initial conditions, the Laplace transform of the aforementioned equations result in the following formulas:

𝑚𝑚 2 𝑠𝑠 2 𝑍𝑍 2 (𝑠𝑠) = (𝑘𝑘 2 + 𝑏𝑏 2 𝑠𝑠)�𝑍𝑍 1 (𝑠𝑠) -𝑍𝑍 2 (𝑠𝑠)� + 𝑈𝑈 𝑎𝑎 (𝑠𝑠), (2.5)
and According to the diagrams of Figure 2.4 and 2.5, the following transfer functions can be obtained, while assuming 𝐹𝐹 0 (𝑠𝑠) = 𝑉𝑉 0 (𝑠𝑠) = 0:

𝑚𝑚 1 𝑠𝑠 2 𝑍𝑍 1 (𝑠𝑠) = (𝑘𝑘 1 + 𝑏𝑏 1 𝑠𝑠)�𝑍𝑍 0 (𝑠𝑠) -𝑍𝑍 1 (𝑠𝑠)� -(𝑘𝑘 2 + 𝑏𝑏 2 𝑠𝑠)�𝑍𝑍 1 (𝑠𝑠) -𝑍𝑍 2 (𝑠𝑠)� -𝑈𝑈 𝑎𝑎 (𝑠𝑠). (2.6)
𝑉𝑉 2 (𝑠𝑠) = 𝑠𝑠 𝑚𝑚 2 𝑠𝑠 2 + 𝑏𝑏 2 𝑠𝑠 + 𝑘𝑘 2 𝑈𝑈 𝑎𝑎 (𝑠𝑠) + 𝑏𝑏 2 𝑠𝑠 + 𝑘𝑘 2 𝑚𝑚 2 𝑠𝑠 2 + 𝑏𝑏 2 𝑠𝑠 + 𝑘𝑘 2 𝑉𝑉 1 (𝑠𝑠), (2.7)
and

𝑉𝑉 1 (𝑠𝑠) = - 𝑠𝑠 𝑚𝑚 1 𝑠𝑠 2 + (𝑏𝑏 1 + 𝑏𝑏 2 )𝑠𝑠 + (𝑘𝑘 2 + 𝑘𝑘 1 )
𝑈𝑈 𝑎𝑎 (𝑠𝑠)

+ 𝑏𝑏 2 𝑠𝑠 + 𝑘𝑘 2 𝑚𝑚 1 𝑠𝑠 2 + (𝑏𝑏 1 + 𝑏𝑏 2 )𝑠𝑠 + (𝑘𝑘 2 + 𝑘𝑘 1 )
𝑉𝑉 2 (𝑠𝑠).

(2.8)

From (2.7) and (2.8), the effect of the introduced control effort on the unsprung mass dynamics and sprung mass dynamics could be analyzed using the following two transfer functions respectively:

𝑉𝑉 1 (𝑠𝑠) 𝑈𝑈 𝑎𝑎 (𝑠𝑠) = - 𝑚𝑚 2 𝑠𝑠 3 𝐺𝐺𝑟𝑟𝑛𝑛(𝑠𝑠) , (2.9) 𝑉𝑉 2 (𝑠𝑠) 𝑈𝑈 𝑎𝑎 (𝑠𝑠) = 𝑠𝑠(𝑚𝑚 1 𝑠𝑠 2 + 𝑏𝑏 1 𝑠𝑠 + 𝑘𝑘 1 ) 𝐺𝐺𝑟𝑟𝑛𝑛(𝑠𝑠) , ( 2
𝐺𝐺𝑟𝑟𝑛𝑛(𝑠𝑠) = 𝑚𝑚 1 𝑚𝑚 2 𝑠𝑠 4 + (𝑚𝑚 2 (𝑏𝑏 1 + 𝑏𝑏 2 ) + 𝑚𝑚 1 𝑏𝑏 2 )𝑠𝑠 3 +(𝑚𝑚 2 (𝑘𝑘 1 + 𝑘𝑘 2 ) + 𝑚𝑚 1 𝑘𝑘 2 + 𝑏𝑏 1 𝑏𝑏 2 )𝑠𝑠 2 + (𝑏𝑏 1 𝑘𝑘 2 + 𝑘𝑘 1 𝑏𝑏 2 )𝑠𝑠 + 𝑘𝑘 1 𝑘𝑘 2 .
(2.11)

The influence of the road and driver disturbances on the quarter vehicle system dynamics could also be analyzed using the following transfer functions:

𝑍𝑍 2 (𝑠𝑠) 𝑍𝑍 0 (𝑠𝑠) = (𝑏𝑏 1 𝑠𝑠 + 𝑘𝑘 1 )(𝑏𝑏 2 𝑠𝑠 + 𝑘𝑘 2 ) 𝐺𝐺𝑟𝑟𝑛𝑛(𝑠𝑠) , (2.12
) .15) In order to demonstrate the frequency responses of the transfer functions that appeared in Figure 2.5, a nominal parametric configuration has been considered. Table 2.1 lists the nominal parameters of the quarter vehicle model used for plotting the transfer functions in the frequency domain. It illustrates the two modes of functionality, namely, the slow mode corresponding to chassis motions or sprung mass motion, and the rapid mode corresponding to the unsprung dynamics or wheel mode. Numerically, for the given parametric configuration, the chassis mode is around 1.22Hz where the wheel mode is approximately 16Hz. These values will change a little bit with the parametric uncertainties of the quarter vehicle model, for front and rear axles as well. However, this characteristic (two modes) usually exists in all touristic vehicles.

𝑍𝑍 1 (𝑠𝑠) 𝑍𝑍 0 (𝑠𝑠) = (𝑏𝑏 1 𝑠𝑠 + 𝑘𝑘 1 )(𝑚𝑚 2 𝑠𝑠 2 + 𝑏𝑏 2 𝑠𝑠 + 𝑘𝑘 2 ) 𝐺𝐺𝑟𝑟𝑛𝑛(𝑠𝑠) , (2.13) 𝑍𝑍 2 (𝑠𝑠) 𝐹𝐹 0 (𝑠𝑠) = 𝑚𝑚 1 𝑠𝑠 2 + (𝑏𝑏 1 + 𝑏𝑏 2 )𝑠𝑠 + (𝑘𝑘 1 + 𝑘𝑘 2 ), 𝐺𝐺𝑟𝑟𝑛𝑛(𝑠𝑠) (2.14) 𝑍𝑍 1 (𝑠𝑠) 𝐹𝐹 0 (𝑠𝑠) = 𝑏𝑏 2 𝑠𝑠 + 𝑘𝑘 2 𝐺𝐺𝑟𝑟𝑛𝑛(𝑠𝑠) . ( 2

Slow mode

Rapid mode

Exploitation of the dynamic decoupling for synthesis models

One of the particularities of touristic vehicles is the ability to decouple the dynamics between the chassis mode (around 1 Hz) and the wheel mode (around 15Hz) [START_REF] Létévé | Study of the influence of passenger vehicle suspension on vibrational comfort, driving behaviour and operating limits : CRONE approach to formalization, analysis and synthesis[END_REF].

From a mathematical point of view, the dynamics decoupling conditions are as follows:

if 𝑘𝑘 2 ≪ 𝑘𝑘 1 and if 𝑏𝑏 2 = �𝑘𝑘 2 𝑚𝑚 2 , (2.16)

then, the polynomial 𝐺𝐺𝑟𝑟𝑛𝑛(𝑠𝑠) of fourth-order in 𝑠𝑠 could be factorized into a product of two polynomial of order 2, such as:

𝐺𝐺𝑟𝑟𝑛𝑛(𝑠𝑠) = (𝑚𝑚 1 𝑠𝑠 2 + (𝑏𝑏 1 + 𝑏𝑏 2 )𝑠𝑠 + 𝑘𝑘 1 )(𝑚𝑚 2 𝑠𝑠 2 + 𝑏𝑏 2 𝑠𝑠 + 𝑘𝑘 2 ).
(2.17)

Thus, the search for the roots of this polynomial becomes easier, such as:

𝐺𝐺𝑟𝑟𝑛𝑛(𝑠𝑠) = 0, ⟹ (𝑠𝑠 2 + 2𝜁𝜁 1 𝜔𝜔 𝑛𝑛1 𝑠𝑠 + 𝜔𝜔 𝑛𝑛1 2 )(𝑠𝑠 2 + 2𝜁𝜁 1 𝜔𝜔 𝑛𝑛2 𝑠𝑠 + 𝜔𝜔 𝑛𝑛2 2 ) = 0, (2.18)
where,

𝜔𝜔 𝑛𝑛1 = � 𝑘𝑘 1 𝑚𝑚 1
: the natural frequency of wheel mode;

𝜁𝜁 1 = (𝑏𝑏 1 +𝑏𝑏 2 ) 2�𝑘𝑘 1 𝑚𝑚 1
: the damping factor of wheel mode;

𝜔𝜔 𝑛𝑛2 = � 𝑘𝑘 2 𝑚𝑚 2
: the natural frequency of chassis mode;

𝜁𝜁 2 = 𝑏𝑏 2 2�𝑘𝑘 2 𝑚𝑚 2
: the damping factor of chassis mode.

(2.19)

It should be noted that according to the condition of (2.16), then (2.14) with 𝑏𝑏 2 = �𝑘𝑘 2 𝑚𝑚 2 , leads to 𝜁𝜁 2 = 0.5.

On the other hand, from a practical point of view, for touristic vehicles, the condition 𝑘𝑘 2 ≪ 𝑘𝑘 1 is always satisfied. Especially, for the light electric vehicle adopted in this study, where the front stiffness of the suspension system is 12000N/m, and the wheel stiffness is 300000N/m. This condition is also met at the rear axle, where the suspension stiffness is 10000N/m, and the wheel stiffness is 200000N/m. However, concerning the equality related to the value of suspension damping coefficient 𝑏𝑏 2 , there is no need to be rigorously satisfied. Indeed, for values of 𝜁𝜁 2 that is close to 0.5, the dynamic decoupling could always be applied.

Consequently, for the purpose of the analysis and synthesis, various transfer functions mentioned lately could be simplified, and therefore reduce the complexity of the study. This approach depends on the domain of study as well as on the domain of operation. One example is the sensitivity of the wheel displacement relative to the road perturbations, namely, 𝑇𝑇 10 (𝑠𝑠) = 𝑍𝑍 1 (𝑠𝑠)/𝑍𝑍 0 (𝑠𝑠), such that:

𝑇𝑇 10 (𝑠𝑠) = 𝑍𝑍 1 (𝑠𝑠) 𝑍𝑍 0 (𝑠𝑠) ≈ 1 � 𝑠𝑠 𝜔𝜔 𝑛𝑛1 � 2 + 2𝜁𝜁 1 𝜔𝜔 𝑛𝑛1 𝑠𝑠 + 1 .
(2.20)

The second example is the transfer function 𝑇𝑇 20 (𝑠𝑠), that can be simplified in a range of frequencies of interest (low frequencies) as follows:

∀𝜔𝜔 ≪ 𝜔𝜔 𝑛𝑛1 , 𝑇𝑇 20 (𝑠𝑠) = 𝑍𝑍 2 (𝑠𝑠) 𝑍𝑍 0 (𝑠𝑠) ≈ 𝑘𝑘 2 + 𝑏𝑏 2 𝑠𝑠 𝑘𝑘 2 + 𝑏𝑏 2 𝑠𝑠 + 𝑚𝑚 2 (𝑠𝑠) 2 .
(2.21) Figure 2.7a shows the bode response of the sensitivity transfer function of the chassis dynamics relative to the road disturbances as formulated in (2.12) and its simplified form due to the decoupling as in (2.21). It shows that the simplification form 𝑇𝑇 20 (𝑠𝑠) is approximately valid in the frequency range of [0-5Hz]. On the other side, Figure 2.7b shows that among a wide range of frequencies, the road disturbances are transmitted without any amplification or attenuation to the vertical movement of the unsprung mass (𝑍𝑍 1 (𝑠𝑠)). Therefore, the dynamics of the unsprung mass (wheel) is negligible compared to that of the chassis dynamics (𝑍𝑍 2 (𝑠𝑠)). Similarly, it could be concluded form the bode response in Figure 2.7b that the simplification form of 𝑇𝑇 10 (𝑠𝑠) extends over the entire range of frequency of operation of the suspension, which is approximately [0-30Hz].

Thus, for analysis and synthesis, the 2Dof quarter vehicle model can then be reduced to two linear 1Dof models. One model is relative to the zone of frequencies [0 -5Hz] and is associated with the expression (2.21). The other one is associated with the expression (2.20) and related to the range of frequencies.

Moreover, in the context of vibrational comfort, the following two points could be considered:

-In the framework of holding chassis and wheel against road perturbations, the performance is to be characterized in the time-domain using the damping factors 𝜁𝜁 1 and 𝜁𝜁 2 . In the frequency domain, the performances are characterized by the values of the resonance factors 𝑄𝑄 1 and 𝑄𝑄 2 of the transfer functions 𝑇𝑇 10 (𝑠𝑠) and 𝑇𝑇 20 (𝑠𝑠), respectively, in which a lower values 𝑄𝑄 ′ 𝑠𝑠 reflects better performance.

-The vibration isolation is a filtering process, where it could be represented by the decrease in the gain of the Bode response in the zone of interest [2 -8] Hz, in which the human body is greatly sensitive to vertical vibrations, in addition to the range of frequencies beyond the wheel mode (rapid mode), that is around 15 to 30 Hz.

Taking into account the aforementioned remarks, the establishment of new controllers should be pointed towards one of these ranges. Based on these formalizations, different objectives for vertical dynamics control are assigned, each one related to one region of interest where it should deal with one inherent problematic dilemma related to passive suspension systems. Figure 2.8 shows the analysis model corresponding to each range of interest and domain of study. For road disturbances of low frequencies, the model presented in Figure 2.8a can be used as an analysis model to study the effect of road perturbations on chassis dynamics. However, to analyze the effect of road perturbations on unsprung mass displacement, the model presented in Figure 2.8b based on 𝑇𝑇 10 (𝑠𝑠) is sufficient. These models should facilitate the study upon reducing the complexity into 1DoF modeling.

In the following upcoming sections, and based on the range of interest, one of these two models should be applied for synthesis where needed. Vehicle's height control

Introduction

Vehicle height adjustment is one of the main features of active suspension systems. The primary objective of this control system is to facilitate the loading and unloading process on the vehicle by adjusting the height of the car body. When the car is at a standstill, raising the body by 5 to 10cm makes the entry and exit easier for the occupants, which provides a comfortable treatment for passengers.

Furthermore, the application of this feature was extended to prove its advantage in reducing death rate fatalities. This function could help to lessen the impending side impact. This is done in combination with vehicles' safety systems and the central driver assistance systems, which use the merged sensor data to identify hazardous situations around the car. At a specific speed of impending side impact, the active suspension system could raise the body up to a few centimeters on the side of impact. This brings the sill into a better position to absorb the impact energy. Deformation of the cabin and the loads acting on the occupants, above all in the chest and abdominal areas, can thus be reduced by up to 50% compared with a lateral collision in which the suspension is not raised [START_REF] Millikin | Audi's predictive active suspension in the A8[END_REF].

In this section, a 2 nd generation CRONE controller is designed in order to fulfill this objective. The robustly designed controller is synthesized using the simplified quarter vehicle model and then tested on the 2DoF quarter vehicle model, and later on, is applied on the complete 14DoF vehicle model.

Synthesis model

In this section, the study takes into consideration the decoupling analysis discussed in the previous section. Indeed, the assigned objective doesn't need a rapid closed-loop system; thus, the study is focused on low band frequencies. Accordingly, the 1DoF quarter vehicle model could be used as a synthesis model. Besides, under the assumption of a standstill situation, the road profile is considered perfectly even, thus, 𝑎𝑎 1 (𝑇𝑇) = 𝑎𝑎 0 (𝑇𝑇) = 0. Therefore, the 2DoF quarter vehicle model is reduced to the 1DoF model, while considering only the suspension dynamics and the chassis dynamics. Figure 2.9 represents the synthesis model, where the following equation is then formulated to represents the plant's transfer function:

𝑃𝑃 (𝑠𝑠) = 𝑍𝑍 2 (𝑠𝑠) 𝑈𝑈 𝑎𝑎 (𝑠𝑠) = 1 𝑚𝑚 2 𝑠𝑠 2 + 𝑏𝑏 2 𝑠𝑠 + 𝑘𝑘 2 (2.22)
Moreover, this transfer function can be written in the following canonical form:

𝑃𝑃 (𝑠𝑠) = 𝑍𝑍 2 (𝑠𝑠) 𝑈𝑈 𝑎𝑎 (𝑠𝑠) = 𝑃𝑃 0 � 𝑠𝑠 𝜔𝜔 𝑛𝑛2 � 2 + 2𝜁𝜁 2 𝜔𝜔 𝑛𝑛2 𝑠𝑠 + 1 (2.23)
where,

𝑃𝑃 0 = 1 𝑘𝑘 2 , 𝜔𝜔 𝑛𝑛2 = � 𝑘𝑘 2 𝑚𝑚 2 , 𝜁𝜁 2 = 𝑏𝑏 2 2�𝑘𝑘 2 𝑚𝑚 2 .
( 2.24) This final form of the transfer function is based on (2.10) while taking into account the decoupling characteristics.

Control design methodology

Defining parametric uncertainty

Four controllers are implemented, one for each quarter vehicle. However, due to the geometrical symmetry of the chassis about the x-axis, one controller is designed for every axle. According to the assigned objective, each controller is assigned to track the reference height of the chassis 𝑎𝑎 2,𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) provided by the supervisor.

In the quarter vehicle synthesis model, three main suspension parameters are considered uncertain. Table 2.2 shows the range of uncertainty of each parameter. These uncertainties caused by the variation of the total mass of the electric vehicle when it is empty (only the driver) or full load (four passengers). For the other suspension parameters, it was assumed that their values have an uncertainty of ±10%. Thus, three transfer functions are defined: 𝑃𝑃 𝑖𝑖,𝑚𝑚𝑖𝑖𝑛𝑛 (𝑠𝑠), 𝑃𝑃 𝑖𝑖,𝑚𝑚𝑟𝑟𝑑𝑑 (𝑠𝑠), and 𝑃𝑃 𝑖𝑖,𝑚𝑚𝑎𝑎𝑥𝑥 (𝑠𝑠), in which for front axle 𝑃𝑃 = 1, and for rear axle, 𝑃𝑃 = 2. Figure 2.10 illustrates the range of variations of the parameters of the canonical form, more specifically, the natural frequency of the chassis mode 𝜔𝜔 𝑛𝑛2 , the damping factor 𝜁𝜁 2 , and the proportional parameter 𝑃𝑃 0 , for front and rear axles. For this purpose, 10000 samples were considered, where each sample represents a transfer function of random parametric configuration. Using this method, the range of variation (minimum and maximum values) of each term of the canonical transfer function can be extracted from the range of variation listed in Table 2.2. Figure 2.11 and 2.12 show the bode responses of the synthesis models and the effect of the parametric uncertainties, for front and rear axles, respectively.

Specifications

In this framework, different points are specified for the assigned control strategy and will be listed as follows:

-Rapidity: the crossover frequency of the closed-loop system is to be much less than the natural frequency 𝜔𝜔 𝑛𝑛2 . -Degree of stability: the resonant peak of the complementary sensitivity function is 𝑄𝑄 𝑇𝑇 ≤ 3 𝐺𝐺𝐶𝐶 that is equivalent to 𝑚𝑚𝑃𝑃𝑛𝑛(𝑀𝑀 𝜙𝜙 ) = 40°.

-Precision at steady state: the error less than 1%.

-The AW actuator control limit: 𝑚𝑚𝑎𝑎𝑎𝑎 |𝑢𝑢 𝑎𝑎 | ≤ 4000𝑁𝑁.

Particularly here, the "med" plants are used as nominal plants in the design process of the two controllers (front and rear axles). Respecting the aforementioned specifications, the following parameters are considered:

𝜔𝜔 𝑐𝑐1 = 1.1 𝑀𝑀𝑎𝑎𝐺𝐺/𝑠𝑠 ≤ 𝜔𝜔 𝑛𝑛2,𝑛𝑛𝑛𝑛𝑚𝑚1 5 ⁄ , (2.25
)

𝜔𝜔 𝑐𝑐2 = 1.1 𝑀𝑀𝑎𝑎𝐺𝐺/𝑠𝑠 ≤ 𝜔𝜔 𝑛𝑛2,𝑛𝑛𝑛𝑛𝑚𝑚2 5 ⁄ , (2.26)
𝑀𝑀 𝜙𝜙 = 45°.

(2.27)

Second Generation CRONE design controller

A second-generation CRONE controller can be designed to enforce the following form for the open-loop transfer function:

𝛽𝛽(𝑠𝑠) = 𝐾𝐾 � 𝜔𝜔 𝑙𝑙 𝑠𝑠 + 1� 𝑛𝑛 𝑙𝑙 � 1 + 𝑠𝑠 𝜔𝜔 ℎ 1 + 𝑠𝑠 𝜔𝜔 𝑙𝑙 � 𝑛𝑛 � 1 1 + 𝑠𝑠 𝜔𝜔 ℎ � 𝑛𝑛 ℎ , (2.28)
where 𝐾𝐾, 𝜔𝜔 𝑙𝑙 , and 𝜔𝜔 ℎ ∈ ℝ + , and 𝑛𝑛 𝑙𝑙 and 𝑛𝑛 ℎ ∈ ℕ + . Integer orders 𝑛𝑛 𝑙𝑙 and 𝑛𝑛 ℎ are set by taking into account the performance specification and the plant magnitude asymptotic behavior at low and high frequencies. This form includes a derivative behavior of order '𝑛𝑛' around the crossover frequency enabling a phase margin constancy, therefore, a robust performance.

Then, particularly for this study, the transfer function of the CRONE controller (𝐶𝐶(𝑠𝑠)) can be formulated as the following:

𝐶𝐶(𝑠𝑠) = 𝛽𝛽(𝑠𝑠)𝑃𝑃 -1 (𝑠𝑠) = 𝐾𝐾 � 𝜔𝜔 𝑙𝑙 𝑠𝑠 + 1� 𝑛𝑛 𝑙𝑙 � 1 + 𝑠𝑠 𝜔𝜔 ℎ 1 + 𝑠𝑠 𝜔𝜔 𝑙𝑙 � 𝑛𝑛 � 1 1 + 𝑠𝑠 𝜔𝜔 ℎ � 𝑛𝑛 ℎ � 𝑠𝑠 𝜔𝜔 𝑛𝑛2 � 2 + 2𝜁𝜁 2 𝜔𝜔 𝑛𝑛2 𝑠𝑠 + 1 𝑃𝑃 0 .
(2.29) Thus, the controller transfer function can be written in the following form:

𝐶𝐶(𝑠𝑠) = 𝐶𝐶 0 �� 𝑠𝑠 𝜔𝜔 𝑛𝑛2 � 2 + 2𝜁𝜁 2 𝜔𝜔 𝑛𝑛2 𝑠𝑠 + 1� � 𝜔𝜔 𝑙𝑙 𝑠𝑠 + 1� 𝑛𝑛 𝑙𝑙 � 1 + 𝑠𝑠 𝜔𝜔 ℎ 1 + 𝑠𝑠 𝜔𝜔 𝑙𝑙 � 2-𝑚𝑚 � 1 1 + 𝑠𝑠 𝜔𝜔 ℎ � 𝑛𝑛 ℎ , (2.30)
where 𝑛𝑛 = 2 -𝑚𝑚 and 𝐶𝐶 0 = 𝛽𝛽 0 /𝑃𝑃 0 , where 𝛽𝛽 0 is the gain magnitude of the open-loop transfer function. Figure 2.14 illustrates the effect of the non-integer order on the bode diagram.

Finally, the 2 nd generation CRONE controller is expressed as the product of two terms, an integer order expression and non-integer order expression, as follows:

𝐶𝐶(𝑠𝑠) = 𝐶𝐶 𝐸𝐸 (𝑠𝑠)𝐶𝐶 𝑁𝑁𝐸𝐸 (𝑠𝑠) (2.31) where, 𝐶𝐶 𝐸𝐸 (𝑠𝑠) = 𝐶𝐶 0 ⎝ ⎜ ⎜ ⎛ � 𝑠𝑠 𝜔𝜔 𝑛𝑛2 � 2 + 2𝜁𝜁 2 𝜔𝜔 𝑛𝑛2 𝑠𝑠 + 1 � 𝑠𝑠 𝜔𝜔 𝑙𝑙 � 𝑛𝑛 𝑙𝑙 ⎠ ⎟ ⎟ ⎞ �1 + 𝑠𝑠 𝜔𝜔 𝑙𝑙 � 𝑛𝑛 𝑙𝑙 -2 �1 + 𝑠𝑠 𝜔𝜔 ℎ � 𝑛𝑛 ℎ -2
(2.32)

𝐶𝐶 𝑁𝑁𝐸𝐸 (𝑠𝑠) = � 1 + 𝑠𝑠 𝜔𝜔 ℎ 1 + 𝑠𝑠 𝜔𝜔 𝑙𝑙 � -𝑚𝑚
, where 0 < 𝑚𝑚 < 1.

(2.33) The second step after the selection of the crossover frequency and the phase margin needed is to define the value of 𝑛𝑛 𝑙𝑙 , 𝑛𝑛 ℎ , and 𝑛𝑛. At low frequencies, and in order to achieve the specified steady-state error, then 𝑛𝑛 𝑙𝑙 = 2 is adopted. At high frequencies, the objective is to limit the sensibility of control effort to noise measurements, that is, to minimize the gain of the regulator at high frequencies, therefore, we have chosen 𝑛𝑛 ℎ = 𝑛𝑛 𝑝𝑝 + 1 = 2 + 1 = 3, where 𝑛𝑛 𝑝𝑝 is the order of the controlled plant at high frequencies as shown previously in Figure 2.11 and 2.12.

At medium frequencies, the value of 𝑛𝑛 is computed to attain the specified degree of stability for the closed-loop system. In fact, the calculation of the value of the non-integer order could be done using selected value for phase margin 𝑀𝑀 𝜙𝜙 , the factor of the resonance 𝑄𝑄 𝑇𝑇 , module margin 𝑀𝑀 𝑚𝑚 , or the damping factor 𝜁𝜁 of the system. Here, the value of 𝑛𝑛 is calculated using the assigned value for the phase margin 𝑀𝑀 𝜙𝜙 by the following formula:

𝑀𝑀 𝜙𝜙 = (2 -𝑛𝑛)𝜋𝜋/2, (2.34)
then, for 𝑀𝑀 𝜙𝜙 = 45°= 𝜋𝜋 4 ⟹ 𝑛𝑛 = 1.5 ⟹ 𝑚𝑚 = 0.5.

The third step is to determine the values for the transitional frequencies at low and high frequencies, 𝜔𝜔 𝑙𝑙 and 𝜔𝜔 ℎ , respectively.

For the front axle, according to Figure 2.11, at 𝜔𝜔 𝑐𝑐1 = 1.1 rad/s the uncertainty is approximately gain-like variation. From the rate of decrease of the open-loop magnitude gain '𝑀𝑀' and from the plant magnitude uncertainty, in order to ensure the robustness of the phase margin, the frequency range [𝜔𝜔 𝐴𝐴 , 𝜔𝜔 𝐵𝐵 ] needs to cover 0.65 decade, where: The values of 𝜔𝜔 𝑙𝑙 and 𝜔𝜔 ℎ , the corner frequencies, are set using the following formulations:

r =
𝜔𝜔 𝑙𝑙 = 𝜔𝜔 𝐴𝐴 /𝑎𝑎 and 𝜔𝜔 ℎ = 𝜔𝜔 𝐵𝐵 × 𝑏𝑏.

(2.38)

The selection of the values of (𝑎𝑎, 𝑏𝑏) depends on an optimizing methodology of the length of the frequency template as presented in Figure 2

𝑛𝑛 = -𝜋𝜋 + 𝑀𝑀 𝜙𝜙 + 𝑛𝑛 ℎ arctan 𝜔𝜔 𝑐𝑐1 𝜔𝜔 ℎ + 𝑛𝑛 𝑙𝑙 ( 𝜋𝜋 2 -arctan 𝜔𝜔 𝑐𝑐1 𝜔𝜔 𝑙𝑙 ) arctan 𝜔𝜔 𝑐𝑐1 𝜔𝜔 ℎ -arctan 𝜔𝜔 𝑐𝑐1 𝜔𝜔 𝑙𝑙 , (2.40)
and (2.42)

𝐾𝐾 = �1 + 𝜔𝜔 𝑐𝑐1 2 𝜔𝜔 ℎ 2 � 𝑛𝑛 ℎ 2 � 𝜔𝜔 𝑙𝑙 𝜔𝜔 ℎ � 𝑛𝑛 2 �1 + 𝜔𝜔 𝑙𝑙 2 𝜔𝜔 𝑐𝑐1 2 � 𝑛𝑛 𝑙𝑙 2 . ( 2
Then, the final open-loop transfer function for the front axle is formulated as follows:

𝛽𝛽 1 (𝑠𝑠) = 46.18 � 0.1038 𝑠𝑠 + 1� 2 � 1 + 𝑠𝑠 23.3 1 + 𝑠𝑠 0.1038 � 1.42 � 1 1 + 𝑠𝑠 23.3 � 3 .
(2.43)

In the same manner, the open-loop transfer function's parameters are computed for the rear axle plant. According to Figure 2.12, at 𝜔𝜔 𝑐𝑐2 = 1.1 rad/s the uncertainty is approximately gain-like variation. From the rate of decrease of the open-loop magnitude gain and from the plant magnitude uncertainty, in order to ensure the robustness of the phase margin, the frequency range [𝜔𝜔 𝐴𝐴 , 𝜔𝜔 𝐵𝐵 ] needs to cover 0.68 decade: 2.18 and 2.19, show that the phase margins only vary from 45.3° to 45.1° as the crossover frequency ranges from 1.03 to 1.18 rad/s, for front and rear axle as well. Furthermore, the Nichols chart contours indicate that the resonance peak of 𝑇𝑇 (𝑠𝑠) are approximatley equal. The band-limitation achieved by 𝜔𝜔 𝑙𝑙 and 𝜔𝜔 ℎ permits a good management of the performance/robustness tradeoff. -The variation on the resonant peak of 𝑇𝑇 (𝑠𝑠) is approximately zero.

-The resonant peaks of 𝐶𝐶(𝑠𝑠) are approximately equal.

-At low frequencies, 𝑇𝑇 (𝑠𝑠) is close to 0dB, which will result in inaccurate tracking of the reference signal, and 𝐶𝐶(𝑠𝑠) is small, which means desensitization of 𝑇𝑇 (𝑠𝑠) versus the plant uncertainty.

-𝐶𝐶𝐺𝐺(s) decreases for low frequencies (efficient rejection of input disturbances) and 𝐶𝐶𝐶𝐶(𝑠𝑠) reachers about 85dB and then decreases (low amplification of noise measurements).

For time-domain simulations, a rational form of the controllers' transfer functions is necessary. The simplest way is to use Oustaloup's method (Oustaloup, 1991b) to replace the band-limited fractional-order integrator of (2.33), using 𝑁𝑁 recursive zeors and poles. As the frequency range [𝜔𝜔 𝑙𝑙 , 𝜔𝜔 ℎ ] cover three decades, then 𝑁𝑁 = 4 is enough to approximate well the fractional-order part '𝑚𝑚'. Thus, the resulted rational form of the 2 nd generation CRONE controllers'' transfer functions can be written in the following form: (2.51)

𝐶𝐶 1 (𝑠𝑠) = 665 � 193𝑠𝑠 2 + 200𝑠𝑠 + 12𝑟𝑟3 (𝑠𝑠 0.1038 ⁄ ) 2 � (1 + 𝑠𝑠 23.3 ⁄ ) -1 × (𝑠𝑠 +
Figure 2.22 shows the bode response of the rational transfer functions of the two controllers defined by (2.50) and (2.51), for front and rear plants, respectively. These transfer functions are then used in time-domain simulations and applied to the validation model for performance and robustness analysis.

Time-domain simulations and analysis

The two designed controllers are tested for step responses, at first, for the synthesis models. Figure 2.23 and 2.24 are presented respectively the front and rear step responses of minimal, medium, and maximal systems. They show that for tolerable durations, the tracking of 𝑎𝑎 2𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) is efficient. Moreover, the CRONE controllers provide a percentage of overshoot for the response of 𝑎𝑎 2 (sprung mass displacement) to 𝑎𝑎 2𝑟𝑟𝑟𝑟𝑓𝑓 that remains almost constant with respect to the plant uncertainties. Besides, all plants have the same damping behavior which reflects the robustness of the damping factor of the closed-loop systems obtained thanks to the CRONE controllers.

In order to validate the designed controller, two test simulations were done on the quarter vehicle model (Figure 2.1), which includes the unsprung mass dynamics as well as the sprung mass dynamics. The two closed-loop systems have given a reference value of 50mm elevation, and the controller should work on tracking the reference robustly versus parametric variations. Each controller is tested on the three plant configurations to examine the robustness of the controllers as well as their performances.

Figure 2.25 shows the simulation results on the validation models for the front and rear axles' suspensions, respectively. It obviously shows the robustness, in both scenarios, of the designed controllers against parametric variation of the plant. All time-domain responses have the same overshoot, steady-state error, and damping behavior. The graphs also plot the effect of the introduced control effort on wheel deformation, where it represents the behavior of the unsprung mass displacements 𝑎𝑎 1 (𝑇𝑇).

To extend the benefit of the control strategy, three different scenarios are tested on the full vehicle model. First, the objective is to elevate the center of gravity of the chassis by 5cm. The second scenario is assigned to obtain a reference pitch angle. The final scenario is to elevate the chassis from one side, and thus, obtain an adequate roll angle. In fact, all these scenarios are achieved by means of translational relationship between the car-body reference values and the four vertical positions at each corner (𝑎𝑎 2 (𝑇𝑇)).

Each scenario was tested for three different vehicles' parametric configuration, i.e. empty vehicle (P1), half-loaded vehicle (P2), and fully-loaded vehicle (P3).

The objective is to examine the robustness of the controllers with carbody dynamics versus masses variations.

Figure 2.26 shows the results of the 3 scenarios. Although of the variation of the total mass of the vehicle, the designed controller provides the same response while varying the height of the chassis, the pitch angle around the y-axis, and the roll angle around the x-axis. The only difference appeared at the height of the center of gravity of the vehicle in the 2 nd scenario, which is normal and expected due asymmetry distribution of the mass of the vehicle in the longitudinal direction. Therefore, the CG is initially not at the middle of the chassis length '𝐿𝐿', then any variation in the pitch angle will vary the value of the height CG.

Conclusion

The first designated objective was to regulate the height of the chassis. For this purpose, two 2 nd generation CRONE controllers are designed for the front axle and rear axle of the vehicle. The frequency and time-domain analysis show the robustness of the designed controller versus parametric variations. Besides, the controllers were able to achieve good tracking reference performance. These controllers proved their ability to regulate the car-body motions when needed. Mainly, these controllers are dedicated to standstill situations for the purpose of exit/enter the vehicle. Thus, it provides more feasibility for occupants, especially for children and passengers with physical disabilities.

Moreover, the control strategy can be considered as an adequate solution for reducing the influence of emergency impending side impact on the vehicle. In fact, it depends mainly on the ability of early detection of such alert and the time response of the controllers, in which it proved that the system could reach a good elevation in less than 1s, where the objective is no longer tracking a reference at all.

Fractional-order damping control

Introduction

Active suspension systems are proving their importance in the field of vehicle dynamics. Many studies in the literature show their ability to enhance comfort for passengers [START_REF] Moreau | Application of the CRONE control-design method to a low-frequency active suspension system[END_REF], vehicle stability [START_REF] Altet | Stability of Hydractive CRONE Suspension[END_REF] and driving performance (Termous et al., 2018b) compared to passive systems. However, due to the coupling effects between the suspension dynamics and longitudinal and lateral dynamics [START_REF] Hamersma | Improving the braking performance of a vehicle with ABS and a semi-active suspension system on a rough road[END_REF], active suspension control laws, where the strategy is comfortoriented, may degrade the performance of braking on rough roads, for example.

Indeed, the active suspension forces introduced by the electric actuator affect the wheel normal forces, which in turn impact the wheel longitudinal contact forces that are directly responsible for the braking, acceleration, and cornering behavior of a vehicle. For this purpose, searching for new design control strategies using suspension control becomes a topic of research nowadays in order to enhance lateral and longitudinal dynamics [START_REF] Sammier | Skyhook and H8 Control of Semi-active Suspensions: Some Practical Aspects[END_REF] [START_REF] Gáspár | Towards Global Chassis Control By Integrating the Brake and Suspension Systems[END_REF]. In this domain, fractional-order control laws have gained their prevalence where they show their ability in solving different control problems [START_REF] Moreau | Effect of the Crone Suspension Control System on Braking[END_REF].

In this section, the impact of fractional-order control methods has been studied, where we derive optimized different fractional-order (FO) damping control strategies, a comfort-oriented and a road-holding oriented strategy, based on the non-integer derivation of the suspension deflection using the active suspension system introduced in the Active Wheel [START_REF] Laurent | Assembly comprising a wheel and a suspension integrated into the wheel[END_REF].

The following study represents one of the proposed solutions for the objectives assigned for vertical dynamics control for light electric vehicles. Besides a comfort-oriented control strategy, a road-holding oriented strategy is also proposed where the main purpose is to reduce the variation of the wheel deflections, thus, improves significantly the lateral and longitudinal dynamics of the vehicle.

Here, the study presents the control strategy and design methodologies. Moreover, frequency-time-domain analysis is done on the level of quarter vehicle models. However, the simulations that reflect real-time scenarios are presented in later chapters while dealing with lateral and longitudinal dynamics on full vehicle validation models.

Suspension model

In general, the vehicle suspension system is required to fulfill a number of functions. These can be stated as follows:

isolate the vehicle from road surface irregularities; -control variations in vertical load at the road/tire interface; -control body motion caused by road inputs, inertia forces, and aerodynamic forces; -maintain vehicle directional stability during maneuvers.

In fact, the main problem is the conflict between these various aspects of vehicle behavior.

Traditional Suspension

A passive vibration control unit consists of a spring and an energy dissipator. As detailed in previous sections, the suspension system can be modeled as a linear spring and a damper. The expression of the force 𝑓𝑓 𝑠𝑠 (𝑇𝑇) developed by the suspension is formulated as (2.3) for passive systems:

𝑓𝑓 𝑠𝑠 (𝑇𝑇) = 𝑘𝑘 2 (𝑎𝑎 2 (𝑇𝑇) -𝑎𝑎 1 (𝑇𝑇)) + 𝑏𝑏 2 (𝑎𝑎 ̇2(𝑇𝑇) -𝑎𝑎 ̇1(𝑇𝑇)),
(2.52)

The Active Wheel suspension system The active suspension system embedded in the Active Wheels can be modeled by a spring, an electric actuator that acts as a force generator and a small frictional damping ratio that results from the mechanical links of the system. Thus, the expression of the force developed by the suspension system is then formulated by: 𝑓𝑓 𝑠𝑠 (𝑇𝑇) = 𝑘𝑘 2 �𝑎𝑎 2 (𝑇𝑇) -𝑎𝑎 1 (𝑇𝑇)� + 𝑏𝑏 2 �𝑎𝑎 ̇2(𝑇𝑇) -𝑎𝑎 ̇1(𝑇𝑇)� + 𝑢𝑢 𝑎𝑎 (𝑇𝑇).

(2.53)

The idea is to use the force generator as a continuously controllable damper that develops a force 𝑢𝑢 𝑎𝑎 (𝑇𝑇) that is proportional to the fractional derivative of the suspension deflection. This force is then written as:

𝑢𝑢 𝑎𝑎 (𝑇𝑇) = 𝑏𝑏 𝑎𝑎 � 𝐺𝐺 𝐺𝐺𝑇𝑇 � 𝑛𝑛 (𝑎𝑎 2𝑖𝑖 (𝑇𝑇) -𝑎𝑎 1𝑖𝑖 (𝑇𝑇)), (2.54)
With 0 < 𝑛𝑛 < 2 is the non-integer derivative order, and 𝑏𝑏 𝑎𝑎 is a constant gain factor. The Laplace transform of relation (2.54) leads to the following transfer function:

𝑈𝑈 𝑎𝑎 (𝑠𝑠) = -𝑏𝑏 𝑎𝑎 1 𝑠𝑠 1-𝑛𝑛 (𝑉𝑉 2 (𝑠𝑠) -𝑉𝑉 1 (𝑠𝑠)).
(2.55)

If we neglect the small friction coefficient of the mechanical links in the AW' suspension system, the force 𝑓𝑓 𝑠𝑠 (𝑇𝑇) developed by the active suspension is then given by:

𝑓𝑓 𝑠𝑠 (𝑇𝑇) = 𝑘𝑘 2 (𝑎𝑎 2 -𝑎𝑎 1 ) + 𝑏𝑏 𝑎𝑎 � 𝐺𝐺 𝐺𝐺𝑇𝑇 � 𝑛𝑛 (𝑎𝑎 2 (𝑇𝑇) -𝑎𝑎 1 (𝑇𝑇)).
(2.56)

The Laplace transform of the vertical dynamics of the quarter vehicle system with fractional-order damping control leads to the following transfer functions: It should be noticed that for 𝑏𝑏 𝑎𝑎 = 0 is the case of passive suspension, and when 𝑏𝑏 2 = 0 is the case of active suspension (assumption). Moreover, for the particular case, 𝑛𝑛 = 1 and 𝑏𝑏 𝑎𝑎 = 𝑏𝑏 2 , the active suspension system acts as the passive system.

𝑉𝑉 1 (𝑠𝑠) = (𝑚𝑚 2 𝑠𝑠

Control structure

The proposed strategy for the active suspension is shown in Figure 2.27. It consists of a feedback control law used to regulate the suspension deflection versus road input disturbances. Supposing "ba" is constant, the only controlled parameter is the fractional-order "n". The value of n should be chosen according to the selected objectives.

For this purpose, three sensitivity transfer functions were formulated in order to find the optimized value of '𝑛𝑛', that are: where 𝐻𝐻 𝑎𝑎 (𝑠𝑠) represents the sensitivity of the vehicle's body vertical acceleration to road disturbances. It evaluates the vibration isolation, in which small gains results in better ride comfort.

𝐻𝐻 12 (𝑠𝑠) = 𝑍𝑍 1 (𝑠𝑠) -𝑍𝑍 2 (𝑠𝑠) 𝑉𝑉 0 (𝑠𝑠) � 𝐹𝐹 0 =0 , (2.61)
where 𝐻𝐻 12 (𝑠𝑠) is the sensitivity of suspension deflection to road disturbances.

The final sensitivity function is the one relative to the wheel deflection, 𝐻𝐻 01 (𝑠𝑠), which plays the role of an indicator for road-holding and a constraint for wheel normal forces variations. This transfer function is formulated as follows:

𝐻𝐻 01 (𝑠𝑠) = 𝑍𝑍 1 (𝑠𝑠) -𝑍𝑍 0 (𝑠𝑠) 𝑉𝑉 0 (𝑠𝑠) � 𝐹𝐹 0 =0
.

(2.62)

The transfer functions of these three sensitivity functions are easily formulated using the expressions detailed in section 2. Using these sensitivity functions, three criteria expressions are formulated for optimizations purposes and are given as follows:

⎩ � � � � ⎨ � � � � ⎧ 𝐽𝐽 1 (𝑛𝑛) = 𝐽𝐽 𝑎𝑎 (𝑛𝑛) = 1 𝛼𝛼 1 � |𝐻𝐻 𝑎𝑎 (𝑗𝑗𝜔𝜔)| 2 𝐺𝐺𝜔𝜔 𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚 𝐽𝐽 2 (𝑛𝑛) = 𝐽𝐽 12 (𝑛𝑛) = 1 𝛼𝛼 2 � |𝐻𝐻 12 (𝑗𝑗𝜔𝜔)| 2 𝐺𝐺𝜔𝜔 𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚 𝐽𝐽 3 (𝑛𝑛) = 𝐽𝐽 01 (𝑛𝑛) = 1 𝛼𝛼 3 � |𝐻𝐻 01 (𝑗𝑗𝜔𝜔)| 2 𝐺𝐺𝜔𝜔 𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚 𝜔𝜔 𝑚𝑚𝑚𝑚𝑚𝑚 , (2.66)
where 𝛼𝛼 1 , 𝛼𝛼 2 , and 𝛼𝛼 3 represent the values of the calculated criteria for the passive suspension system, i.e. 𝑏𝑏 𝑎𝑎 = 0. The range of integration is considered between 𝜔𝜔 𝑚𝑚𝑖𝑖𝑛𝑛 = 2 𝜋𝜋 0.1 rad/s and 𝜔𝜔 𝑚𝑚𝑎𝑎𝑥𝑥 = 2 𝜋𝜋 30 rad/s.

According to these criteria, when the value is less than 1, it indicates an improvement over the passive system. The optimal control problem can thus be formulated as finding the minimal values of 𝑛𝑛 for each criterion. show the graph plots of the criterion values normalized relative to passive value as a function of the fractional-order 𝑛𝑛. Minimum values are obtained for front and rear axles, for empty load and full-loaded parametric configurations. Table 2.3 lists the values of 𝑛𝑛 while considering 𝑏𝑏 𝑎𝑎 = 1200 𝑁𝑁𝑠𝑠/𝑚𝑚. The suspension control strategy must be changed according to the road state of the surface. In case of a smooth road, the road-holding strategy is then preferred. It is the same case also when a critical braking situation revealed where comfort-oriented strategy is no longer a priority, here the suspension must be switched to road-holding oriented strategy in order to minimize the variation of the wheels' normal forces to reduce the braking distance. These considerations are assigned to the "supervisor", the high-level control unit in the global chassis control, that takes decisions based on all available measurements and estimated variables.

Rationalized approach of FO damping control

In order to show the influence of the fractional damping control, several scenarios were done using the quarter vehicle validation model. The scenario was chosen to be a quarter vehicle test rig driven on a rough surface, in which it includes a range of low and high frequencies components so that it affects ride comfort, road holding, and handling.

In this study, two values for 𝑛𝑛 are chosen, and then a comparative study is done. Then, for time-domain simulations, the fractional-order damping control in (2.55) is formalized as a 2 nd generation CRONE controller to assure a band-limited derivation of order 'n' around the crossover frequency enabling a phase margin constancy, and therefore, a robust performance. The controller is expressed as the following:

𝐶𝐶(𝑠𝑠) = 𝐶𝐶 0 � 1 + 𝑠𝑠 𝜔𝜔 ℎ 1 + 𝑠𝑠 𝜔𝜔 𝑙𝑙 � 1-𝑛𝑛 , (2.67)
where

𝐶𝐶 0 = 𝑏𝑏 𝑎𝑎 ⎝ ⎜ ⎜ ⎛ 1 + � 𝜔𝜔 𝑚𝑚 𝜔𝜔 𝑙𝑙 � 2 1 + � 𝜔𝜔 𝑚𝑚 𝜔𝜔 𝑙𝑙 � 2 ⎠ ⎟ ⎟ ⎞ 1-𝑛𝑛 2 .
(2.68)

Particularly for this study, the central frequency 𝜔𝜔 𝑚𝑚 of the band-limited integrator is selected 1 rad/s. The rate of decrease of the open-loop magnitude gain '𝑀𝑀' in the frequency range [𝜔𝜔 𝐴𝐴 , 𝜔𝜔 𝐵𝐵 ] needs to cover 5 decades, where the range of interest is between 0.1 and 30 Hz. (2.69)

The values of 𝜔𝜔 𝑙𝑙 and 𝜔𝜔 ℎ , the corner frequencies, are set using the following formulations:

𝜔𝜔 1 = 0.0032/10 = 0.00032 rad/s, and 𝜔𝜔 ℎ = 316.2 × 10 = 3162 rad/s.

(2.70)

Then fractional part of the 2 nd generation CRONE controller should be replaced by a rational transfer function. Thanks to the Oustaloup method, the rational form of the transfer function is obtained for each value of 𝑛𝑛.

The two values of 𝑛𝑛 are chosen as follows:

𝑛𝑛 1 = 0.65 and 𝑛𝑛 2 = 1.1.

(2.71) show the bode response of the fractional-order damping control 𝑈𝑈 𝑎𝑎 (𝑠𝑠), compared to the fractional-order 2 nd generation CRONE control, and the rational form of the controller used in time-domain simulations, for the two values of 𝑛𝑛.

It is important to consider the identical responses for all transfer functions in the range of 0.1 -30 Hz, i.e. the range of interest of frequency for road disturbances. The rational transfer functions were obtained using 𝑁𝑁 recursive zeros and 𝑁𝑁 recursive poles by Oustaloup's method. The 2 nd generation CRONE control proved (in the frequency domain) to be an adequate approach for representing the fractional damping control proposed in this section.

Figure 2.34 shows the resulted bode responses of the two sensitivity functions, 𝐻𝐻 𝑎𝑎 (𝑠𝑠) and 𝐻𝐻 12 (𝑠𝑠), with the two fractional damping controllers with 𝑛𝑛 = 0.65 and 𝑛𝑛 = 1.1. In Figure 2.34a, it can be deduced that for 𝑛𝑛 = 0.65 the damping control is comfort-oriented strategy, where it attenuates the acceleration response to road disturbances on frequencies greater than 2Hz, especially in the range of interests . On the other hand, Figure 2.34b the fractional damping control with order 𝑛𝑛 = 1.1 shows a significant attenuation in suspension deflection versus road disturbances on a wide range of frequencies, which is featured prominently at the resonance frequencies of chassis mode and wheel mode. This behavior is eventually granting a roadholding improvement in the time-domain simulations. For this reason, the fractional damping control with order 1.1 could be adopted as a road-holding oriented control strategy.

Time-domain simulations

Using the 2DoF quarter vehicle validation model, two scenarios were done to examine the performance of the fractional-order damping controllers. Each scenario adopts one kind of road disturbances that includes a range of frequencies and specific amplitudes. One parametric configuration is used in all simulations, in which the controlled systems are compared to traditional suspension passive systems of the damping coefficient 𝑏𝑏 2 = 1200 𝑁𝑁𝑠𝑠/𝑚𝑚.

The first scenario uses a sinusoidal profile road disturbance with a sweeping frequency starting from 0.1 to 40 Hz. The amplitude of the sinusoidal form is 1mm. Generally, this scenario is made for testing vibratory isolation of a designed controller. Figure 2.35 represents the results obtained for the two FO damping controllers compared to the passive system.

It is clearly showing that FO damping control with comfort-oriented control have a lower amplitude in terms of vertical accelerations 𝑎𝑎 ̈2(𝑇𝑇) compared to passive and road-holding oriented one. Significantly, the comfort-oriented controller has also much lower control effort values. These results reveal a good performance in vibratory isolations where the main goal is to attenuate the vertical accelerations and thus provide comfort for passengers. This improvement comes at the expense of higher suspension deflections and wheel deflections, which degrades the road-holding performance of the vehicle and may affect the handling process as well. The results of the second scenario are plotted in Figure 2.36. Here, we increase the amplitude of the sinusoidal profile to 2mm. This test is generally done to test the road-holding performance of control design. Although of high vertical accelerations, the road-holding oriented FO damping controller was able to reduce the suspension deflection amplitudes compared to passive and comfort-oriented systems. This behavior reflects an enhancement in terms of road-holding of the vehicle. Moreover, for n=1.1, the system has lower wheel deflections that are an important indication for good handling.

In the third scenario, the sinusoidal profile is now swept between 0.1 to 5.5Hz with an amplitude of 15mm. This scenario is done to examine the comfort performances of a given designed controller. Figure 2.37 shows the results on the dynamics of different variables. At first, the three systems show approximately the same behavior in terms of suspension deflection. However, the comfort-oriented controller shows a significant attenuation for vertical accelerations compared to passive and road-holding oriented controlled systems. At the same time, the FO damping control of order n=0.65 needs lower control efforts. It is worthwhile to mention that in this range of frequencies, the comfort-oriented controller has a better performance of handling compared to other systems where it has a lower variation in wheel deflections as well.

Conclusion

In order to examine the benefits of fractional damping control in rejecting road disturbances, two fractional orders are chosen. According to an optimized strategy, two values have resulted in a specific parametric configuration. The study resulted in two control strategies: a comfort-oriented and a road-holding oriented system. Each control system is related to one value of the fractional-order damping control.

It was shown, in frequency and time-domain simulations, the effectiveness of the comfort-oriented controller in improving comfort performance while it reduces the vertical accelerations versus different ranges of amplitude and frequencies of road disturbances. It proves its ability to isolated vibrations as well compared to other systems, especially the passive systems.

On the other hand, the road-holding FO damping control proves its ability to minimize the variation of suspension deflection against road disturbances compared to the passive systems, and thus, fulfilling the road-holding criterion.

The practical use of these control designs is then to integrates them among other control strategies to improve, for example, braking performance, cornering, lateral stability, etc. These studies will be presented in detail in the next chapters.

Conclusion

This chapter addresses two main objectives of the active suspension system of an electric vehicle. Using the CRONE CSD methodology, fractional damping control is implemented in two manners: a comfort-oriented controller and a road-holding oriented controller. The proposed control law could achieve adequate performance in rejecting road disturbances. The experimental results reveal that the presented strategy could provide an important performance as far as a consistent damping non-integer order control is selected. For this purpose, an optimization methodology is adopted for finding the optimized value of the fractional-order of damping control. Hence, the presented method could effectively enhance ride comfort, together with the road-holding and handling of the vehicle.

Moreover, a 2 nd generation CRONE control is developed for the control of the vehicle's active suspension system for chassis' height regulation. It was clearly demonstrated that the closed-loop system performance is pretty good enough in varying the car-body mass. The CRONE designs feature significant robustness versus vehicle's parametric variations. This control law is able to vary the height of the chassis successfully for facilitates the exit and entry process to the vehicle. In addition, it could be used for safe manner for reducing the impact of side accidents on the vehicle by elevating the side of impact before the car collision

Introduction

Road transportation is shifting significantly toward electrification around the globe. The potential to reduce energy consumption and greenhouse emissions are the most important driver for electric mobility. Besides the growing awareness of environmental and population pressures, the innovations market acceptance [START_REF] Li | New entrants versus establishers in China and US electric vehicle marketplace: a comparative analysis[END_REF] and government support [START_REF] Ertrac | European Roadmap Electrification of Road Transport Status: final for publication[END_REF](European Commission, 2017) results in the advancement of Electric Vehicle (EV) technology. This technology starts to challenge its internal combustion counterparts with large-scale investments and highly coordinated research and development [START_REF] Ahmad | A Review of the Electric Vehicle Charging Techniques, Standards, Progression and Evolution of EV Technologies in Germany[END_REF]. It becomes essential for meeting the goals of decarbonization and energy security (Paris agreement at COP21 2015 -C40 Committee (COP21, 2015), Madrid COP 25 (Sustainable Innovation Forum, 2019)). Some countries have already made a bold announcement regarding the electrification of mobility that opts to ban diesel vehicles from their cities' center by the next decades [START_REF] Pereirinha | Main Trends and Challenges in Road Transportation Electrification[END_REF].

Consequently, automobile manufacturers are in the process of launching EVs and a market for electric mobility solutions had been emerged where all-in-wheel devices are expected to play an important role in this new trend. Such vehicles are able to adapt to urban lifestyles, mainly, to contribute to the air quality and noise reduction in urban cities. Since a lighter vehicle means a sharp reduction in energy consumption, a modern solution can contain integrated powertrain suspension located directly inside the wheel. One example of such technology, adopted in this work, is the Active Wheel (AW) developed by Michelin (Fig. 3.1). This advanced technology initiates a new era in road transportation where all essential components have been integrated within the wheel itself: the pneumatic part, a compact traction electric motor with regenerative braking, an electric active suspension system, and the braking system.

The structure of the Active Wheel offers new opportunities in Global Chassis Control (GCC) that rises, recently, to remarkable levels. GCC is the design of cooperative control between the different mechatronic subsystems to enhance vehicle performance, generally: safety, stability, and comfort. While, for example, the (Anti-lock Braking System) ABS optimizes the braking distance [START_REF] Benine-Neto | Robust control for an electro-mechanical anti-lock braking system: the CRONE approach[END_REF][START_REF] Reif | Antilock braking system (ABS)[END_REF], (Electronic Stability Program) ESP stabilizes the vehicle along the desired trajectory (Rajamani, 2012b), engine control optimizes both traction torque and emissions [START_REF] Morand | Robust cruise control using CRONE approach[END_REF]Rajamani, 2012c), the active suspension systems had proved their ability to control the vertical dynamics of the vehicle in the sense of improving ride quality. Indeed, they can ease the inherent tradeoff in passive systems between comfort and road holding by the integration of actuators between the chassis and wheel mass [START_REF] Hrovat | Survey of Advanced Suspension Developments and Related Optimal Control Applications[END_REF][START_REF] Karnopp | Active Damping in Road Vehicle Suspension Systems[END_REF]. Although of the high energy consumption of these systems in real applications compared to passive and semi-active systems, it was found that the potential to develop a vibration energy recovery system could achieve an energy balance requirement [START_REF] Wang | An investigation on energy recovery analysis of active suspension system[END_REF]. Moreover, the recoverable energy-dependent not only on the driving and road conditions but also on the type of control algorithms [START_REF] Huang | Active controller design for an electromagnetic energy-regenerative suspension[END_REF]. This fact drives the academic and industrial stakeholders to investigate also control concepts.

In congruence with the global trend and due to its compatibility with the long pursued full electric vehicle, considerable attempts have been done in the past few decades on the development of the control algorithms of active suspension systems [START_REF] Tseng | State of the art survey: Active and semiactive suspension control[END_REF]. Starting from PID [START_REF] Li | Motion-based active disturbance rejection control for a non-linear full-car suspension system[END_REF] and intelligent PD controllers [START_REF] Haddar | Intelligent PD controller design for active suspension system based on robust model-free control strategy[END_REF], to linear Quadratic Regulators LQR [START_REF] Wu | A Simultaneous Mixed LQR/H∞ Control Approach to the Design of Reliable Active Suspension Controllers[END_REF], and finite-time output feedback controller [START_REF] Pan | Nonlinear Output Feedback Finite-Time Control for Vehicle Active Suspension Systems[END_REF]. However, robust controllers have been also proved their effectiveness to fulfill the suspension control objectives, like 𝐻𝐻 ∞ controllers [START_REF] Akbari | Output feedback H∞/GH2 preview control of active vehicle suspensions: A comparison study of LQG preview[END_REF][START_REF] Jing | Robust finite-frequency H∞ control of full-car active suspension[END_REF][START_REF] Yu | Parallel Active Link Suspension: A Quarter-Car Experimental Study[END_REF], fuzzy 𝐻𝐻 ∞ [START_REF] Li | Reliable fuzzy control for active suspension systems with actuator delay and fault[END_REF], particularly, for in-wheel motors in electric vehicles [START_REF] Shao | Output feedback H∞ control for active suspension of in-wheel motor driven electric vehicle with control faults and input delay[END_REF]. In addition, nonlinear controllers are also applied in this context, such as backstepping controllers [START_REF] Lin | Nonlinear Design of Active Suspensions[END_REF][START_REF] Sun | Filter-Based Adaptive Vibration Control for Active Vehicle Suspensions with Electrohydraulic Actuators[END_REF], adaptive backstepping [START_REF] Sun | Adaptive backstepping control for active suspension systems with hard constraints[END_REF], sliding mode controller [START_REF] Chamseddine | Sliding Mode Control Applied to Active Suspension Using Nonlinear Full Vehicle and Actuator Dynamics[END_REF][START_REF] Li | Adaptive Sliding-Mode Control for Nonlinear Active Suspension Vehicle Systems Using T -S Fuzzy Approach[END_REF], model-free fractionalorder sliding mode control [START_REF] Wang | Model-free fractional-order sliding mode control for an active vehicle suspension system[END_REF], model predictive controllers [START_REF] Gohrle | Design and vehicle implementation of preview active suspension controllers[END_REF], and fuzzy control with wheelbase preview [START_REF] Zhao | Design of an integrated controller for active suspension systems based on wheelbase preview and wavelet noise filter[END_REF].

In this chapter, a hierarchical control strategy, based on two-level controllers (global and local), is proposed for the active suspension system embedded in the AW, where the conventional mechanical suspension has been replaced by a passive spring and a DC electrical motor. In comparison with other force-generating actuators (hydraulic, Hydractive, hydro-pneumatic...), the DC linear characteristics of the AW's suspension system provides an extremely fast response time, low contact friction, high precision and efficiency, good controllability, and less energy consumption [START_REF] Nguyen | Direct current linear actuator using in the active suspension system of automobile[END_REF]. This approach is, in fact, a global solution for the 2 nd objective listed in the introduction in the frame of vertical dynamic control of the light electric vehicle. Recalling the main objectives: (1) regulating the height of the center of gravity of the vehicle; (2) holding chassis under driver solicitations; (3) holding chassis against road disturbances; (4) obtain tolerable vibration isolation under road perturbations; finally (5) holding the unsprung mass against road disturbances by reducing the variation of the wheel deflection to enhance vehicle handling.

The present study focuses particularly on the second objective, and then evaluates the ability of the designed controller on satisfying objective ( 3) and ( 4). Accordingly, the proposed strategy assumes that the vehicle operates in the "comfort" region which represents the majority of the time of use of the vehicle, especially in urban cities, in which the chassis accelerations are within 0.4g. Then, a comfort-oriented approach has been developed using a robust non-integer order controller, namely, the CRONE controller (Sabatier et al., 2015a), to examine the relevant benefits of the fractionalorder techniques in the context of suspension control.

We aim from this method to obtain the maximum possible improvement of ride comfort while respecting the road holding and handling criteria. Generally, the ride quality is influenced by two main sets of disturbances. One is caused by the driver upon braking, accelerating, maneuvering, and wind gusts, for example. The other one is the ground input disturbances caused by road roughness that could be classified as shocks (bumps, …) or vibrations. In fact, improving the performance of such a system can decrease the number of traffic fatalities by preventing physical fatigue of the driver as well as positively influence the driver's subjective impression of the vehicle [START_REF] Koch | Driving state adaptive control of an active vehicle suspension system[END_REF]. Furthermore, in line with the innovative advancement in autonomous vehicles, a comfort driving style is highly demanded to overcome motion sickness [START_REF] Ekchian | A High-Bandwidth Active Suspension for Motion Sickness Mitigation in Autonomous Vehicles[END_REF].

Different test simulations were done to evaluate the performance and robustness of the proposed control approach on a realistic driving scenario using a full nonlinear 14DOF vehicle model. Usually, the strategies implemented to deal with the second objective assumes that the road is perfectly even, which is not the case in reality. Therefore, the simulations will examine the effectiveness of the hierarchical approach in compensating chassis dynamics under driver disturbances, and additionally inspect the ability of the comfort-oriented local controller to reject road disturbances. Evaluations were done using International Standardization (ISO) and well-known indexes regularly used in the context of suspension performance.

Hierarchical Approach

As mentioned in the general introduction, Advanced Driver-Assistance Systems (ADAS) are implemented nowadays in nearly most of the newly manufactured vehicles (e.g. ABS, ESP, Adaptive Cruise Control (ACC), ...). These systems were designed to operate individually to fulfill its own goal, however, some of these systems are directly or indirectly may affect each other's performance due to the complex coupling between the various vehicle dynamics. Therefore, the absence of any coordination between the actuators may lead to degrading the overall performance of the vehicle regarding handling and safety.

In the frame of GCC, there exists three main architecture in the literature, namely, the centralized control, decentralized control, and control by supervision [START_REF] Gordon | Integrated control methodologies for road vehicles[END_REF]. This study adopts a supervision control strategy, that represents an intermediate structure between the other two ones, by a proposed hierarchical approach for vertical dynamic control of a light electric vehicle. Figure 3.2 illustrates the main four levels of the proposed strategy: Supervisor, Global Control, Control allocation, and Local Control. The supervisor represents the highest level of this hierarchical approach. It receives all available measurements coming from the vehicle itself (sensors, pedals, ...), environment (exteroceptive sensors, camera, Li-DAR, ...), and information that could be provided by road infrastructure or by other vehicles (like in autonomous vehicles). All these data are used in the perception of the environment, observation, path planning, motion generation, and last but not least, diagnosis.

One main role of the supervisor is to estimate the domain in which the vehicle is operating. Using the observed or measured values of the lateral (𝑎𝑎 𝑦𝑦 ) and longitudinal (𝑎𝑎 𝑥𝑥 ) accelerations [START_REF] Regolin | A Sliding-Mode Virtual Sensor for Wheel Forces Estimation with Accuracy Enhancement via EKF[END_REF], as well as the vertical acceleration (𝑎𝑎 𝑧𝑧 ), the supervisor thoughtfully allocates the control efforts and references of all active systems. Figure 3.3 illustrates, in an (𝑎𝑎 𝑥𝑥 , 𝑎𝑎 𝑦𝑦 ) diagram, the main three range of operations. The first one in green, the comfort region, where it is defined by low longitudinal and lateral accelerations that are less than 0.4g in absolute value. In this zone, the vertical dynamics of the vehicle has no significant influence on the other dynamics, assuming driving on dry and good roads. The second region (blue), represents the limits of the linear zone of tire model [START_REF] Pacejka | Basic Tire Modeling Considerations[END_REF], and here the suspension system starts to influence the lateral dynamics so that comfort -road holding dilemma should be managed using adapted control strategy. Finally, the active security region (red), which represents the nonlinear zone of the tires where safety is then the priority.

Previously, the hierarchical approach has been introduced and tested in braking phases [START_REF] Hamrouni | Hierarchical approach for Global Chassis Control[END_REF], and cornering driving scenarios at moderate constant speeds [START_REF] Bouvin | Hierarchical approach for Global Chassis Control[END_REF]Termous et al., 2018a). This study extends the application to control the three main chassis dynamics to compensate for the heave, pitch, and roll motion simultaneously by a global controller. Moreover, the study adopted the 2DOF quarter vehicle model that includes both the sprung and unsprung mass modes, controlled by a local controller, in order to investigate the level of disturbance rejection of road unevenness and vibration isolation. The hierarchical approach is then stated as a validated approach for certain road structures and classes.

Vehicle model

Validation Model

The simulations for strategy validation are done using a full non-linear car model with 14 degrees of freedom, widely studied in the literature, taking into consideration most of the nonlinearities existing in the vehicle dynamics [START_REF] Ahmadian | Ride Performance Analysis of Semiactive Suspension Systems Based on a Full-Car Model[END_REF][START_REF] Connair | Development of a common vehicle model for chassis control design[END_REF][START_REF] Giangiulio | 14 dof VERTEC vehicle model VDSIM--Vehicle Dynamic SIMulator Implementation and validation[END_REF].

Synthesis model

Two main models are synthesized, the first one is related to chassis dynamics for the global controller, and the second one is the quarter vehicle model for the local controllers. Considering the proposed domain of operation (comfort-zone), a simplified model can be used where neglecting some nonlinearities, while taking into account the main fundamental laws of dynamics. The following two subsections describe the synthesis model used for each control design.

Chassis Model

Regarding the synthesis model of the chassis of the vehicle, two planes are introduced. The First plane 𝑃𝑃 1 , is the one that passes through the center of the four unsprung masses (wheels), where the second plane 𝑃𝑃 0 is the one that passes throughout the four points of contact between the wheel and the road. Thus, in addition to the reference related to the car-body, two references are introduced relative to the unsuspended masses and to the road, 𝑅𝑅 1 and 𝑅𝑅 0 , respectively. Generally, for the car-body, the application of the fundamental law of dynamics leads to the following three equations of motion:

𝑣𝑣 𝑧𝑧 (𝑇𝑇) = 1 𝑀𝑀 2 � 𝑓𝑓 ∑ 𝑧𝑧 (𝜏𝜏 ) 𝑡𝑡 0 𝐺𝐺𝜏𝜏 + 𝑣𝑣 𝑧𝑧 (0), (3.2)
𝜔𝜔 𝜑𝜑 (𝑇𝑇) = 1 𝑃𝑃 𝑦𝑦𝑦𝑦 � 𝐶𝐶 ∑𝜑𝜑 (𝜏𝜏 ) 𝑡𝑡 0 𝐺𝐺𝜏𝜏 + 𝜔𝜔 𝜑𝜑 (0), (3.3
)

𝜔𝜔 𝜃𝜃 (𝑇𝑇) = 1 𝑃𝑃 𝑥𝑥𝑥𝑥 � 𝐶𝐶 ∑𝜃𝜃 (𝜏𝜏 ) 𝑡𝑡 0 𝐺𝐺𝜏𝜏 + 𝜔𝜔 𝜃𝜃 (0). (3.4)
The three variables 𝑣𝑣 𝑧𝑧 (𝑇𝑇), 𝜔𝜔 𝜑𝜑 (𝑇𝑇), and 𝜔𝜔 𝜃𝜃 (𝑇𝑇) represent the velocity of heave displacement of the center of gravity, the angular pitch rate about the yaxis, and the angular roll rate about the x-axis, respectively. The term 𝑓𝑓 ∑ 𝑧𝑧 (𝜏𝜏 ) (𝑁𝑁 ) represents the summation of all forces applied in the vertical direction on the chassis, that is:

𝑓𝑓 ∑ 𝑧𝑧 (𝜏𝜏 ) = � 𝑓𝑓 𝑠𝑠𝑖𝑖𝑠𝑠 (𝑇𝑇) 2 𝑖𝑖=1,𝑠𝑠=1
+ 𝑢𝑢 𝑧𝑧 (𝑇𝑇), (3.5) where,

𝑓𝑓 𝑠𝑠 𝑖𝑖𝑠𝑠 (𝑇𝑇) = 𝑓𝑓 𝑠𝑠𝑝𝑝,𝑖𝑖𝑠𝑠 (𝑇𝑇) + 𝑓𝑓 0,𝑖𝑖𝑠𝑠 (𝑇𝑇),

with 𝑓𝑓 𝑠𝑠𝑝𝑝,𝑖𝑖𝑠𝑠 (𝑇𝑇) and 𝑓𝑓 0,𝑖𝑖𝑠𝑠 (𝑇𝑇) represent the passive suspension forces and the load transfer forces at each corner respectively. The term 𝑢𝑢 𝑧𝑧 (𝑇𝑇) is the desired force for heave velocity (𝑉𝑉 𝑧𝑧 (𝑠𝑠)) regulation, such that:

𝑈𝑈 𝑧𝑧 (𝑠𝑠) = -𝐶𝐶 𝑧𝑧 (𝑠𝑠) 𝑉𝑉 𝑧𝑧 (𝑠𝑠), (3.7)
where 𝐶𝐶 𝑧𝑧 (𝑠𝑠) represents the transfer function of the heave global controller. 𝐶𝐶 ∑𝜑𝜑 (𝑇𝑇) (N. m) is the algebraic sum of all external torques applied around the y-axis, and is defined as:

𝐶𝐶 ∑𝜑𝜑 (𝑇𝑇) =
with 𝑊𝑊 0𝜃𝜃 (𝑇𝑇) being the rolling torque induced due to maneuvering, 𝑊𝑊 𝑠𝑠𝜃𝜃 (𝑇𝑇) represents the vehicle passive anti-roll torque, and 𝑢𝑢 𝜃𝜃 (𝑇𝑇) is the active anti-roll torque provided by the active suspension control efforts, and defined by: 𝑢𝑢 𝜃𝜃 (𝑇𝑇) = 𝑢𝑢 𝐹𝐹𝐹𝐹 ,𝜃𝜃 (𝑇𝑇) + 𝑢𝑢 𝐹𝐹𝐵𝐵,𝜃𝜃 (𝑇𝑇), (3.12) where 𝑢𝑢 𝐹𝐹𝐹𝐹 ,𝜃𝜃 (𝑇𝑇) is the feed-forward torque and the 𝑢𝑢 𝐹𝐹𝐵𝐵,𝜃𝜃 (𝑇𝑇) is the feedback torque that regulates the roll rate Ω 𝜃𝜃 (𝑠𝑠) using 𝐶𝐶 𝜃𝜃 (𝑠𝑠) transfer function, such that:

𝑈𝑈 𝐹𝐹𝐵𝐵,𝜃𝜃 (𝑠𝑠) = -𝐶𝐶 𝜃𝜃 (𝑠𝑠)𝛺𝛺 𝜃𝜃 (𝑠𝑠) (3.13)
The feedforward control efforts are estimated based on the following two formulas:

𝑢𝑢 𝐹𝐹𝐹𝐹 ,𝜑𝜑 (𝑇𝑇) = -ℎ 𝜑𝜑 𝑀𝑀 2 𝜆𝜆 𝜑𝜑 𝑎𝑎̃𝑝𝑝 𝜑𝜑 (𝑇𝑇), (3.14)
𝑢𝑢 𝐹𝐹𝐹𝐹 ,𝜃𝜃 (𝑇𝑇) = -ℎ 𝜃𝜃 𝑀𝑀 2 𝜆𝜆 𝜃𝜃 𝑎𝑎̃𝑝𝑝 𝜃𝜃 (𝑇𝑇), (3.15) where 𝑎𝑎̃𝑝𝑝 𝜑𝜑 (𝑇𝑇) and 𝑎𝑎̃𝑝𝑝 𝜃𝜃 (𝑇𝑇) are the measured displacement of traction/braking pedal and driver steering angle, respectively. The constant values of 𝜆𝜆 𝜑𝜑 and 𝜆𝜆 𝜃𝜃 depend on the braking and traction system response, steering system behavior, vehicle geometry, and inertial parameters. The main objective of these feedforward torques is to oppose the driver disturbances, thus, provide fast compensation for chassis dynamics.

Figure 3.5 shows a simplified schematic that illustrate the car body dynamics. In fact, this modeling is used for analysis and control synthesis, where its range of validity is defined by two main assumptions: the frequency range is limited to [0 5]Hz, i.e. the study of low frequency comfort, and the inertia matrix of the body is assumed diagonal, in which the three axes of heave, pitch, and roll are passing through the center of gravity of the chassis.

Accordingly, passive anti-torques and forces can be extracted as follows:

𝐹𝐹 𝑠𝑠𝑧𝑧 (𝑠𝑠) = -� 𝑘𝑘 𝑧𝑧 𝑠𝑠 + 𝑏𝑏 𝑧𝑧 � �𝑉𝑉 𝑧𝑧 (𝑠𝑠) -𝑉𝑉 𝑧𝑧1 (𝑠𝑠)�, (3.16
)

𝐶𝐶 𝑠𝑠𝜑𝜑 (𝑠𝑠) = -� 𝑘𝑘 𝜑𝜑 𝑠𝑠 + 𝑏𝑏 𝜑𝜑 � �Ω 𝜑𝜑 (𝑠𝑠) -Ω 𝜑𝜑1 (𝑠𝑠)�, (3.17
)

𝐶𝐶 𝑠𝑠𝜃𝜃 (𝑠𝑠) = -� 𝑘𝑘 𝜃𝜃 𝑠𝑠 + 𝑏𝑏 𝜃𝜃 � �Ω 𝜃𝜃 (𝑠𝑠) -Ω 𝜃𝜃1 (𝑠𝑠)�. (3.18)
At equilibrium, with zero initial conditions, three transfer functions can be deduced using (3.2 -3.4) and (3.16 -3.18) representing the global chassis dynamics, and are formulated as follows:

Quarter vehicle model

The vehicle suspension system is modeled using the well-known quartermodel that exists in the literature [START_REF] Jazar | Suspension Mechanisms, in: Vehicle Dynamics: Theory and Application[END_REF][START_REF] Savaresi | Semi-Active Suspension Control Design for Vehicles, Semi-Active Suspension Control Design for Vehicles[END_REF]. Here, the four wheels are assumed decoupled and the suspension dynamics are considered in the frequency range of interest for the vertical dynamics [ 0 -30 𝐻𝐻𝐻𝐻]. The suspension configuration of the AW is considered as a high bandwidth active suspension system due to the presence of the force generating actuator mounted parallel to the passive spring-damper components [START_REF] Sun | Vibration isolation for active suspensions with performance constraints and actuator saturation[END_REF]. Assuming that the actuator is ideal, the vertical motion of this system is described by the same equations presented in section 2.2.

At this level, four local regulators are implemented, one for each quarter vehicle. However, due to the geometrical symmetry of the chassis about the x-axis, one controller is designed for every axle. According to the proposed strategy, the objective of each local controller is assigned to regulate the vertical displacement of the chassis 𝑎𝑎 2,𝑖𝑖𝑠𝑠 (𝑇𝑇). Denote by 𝐺𝐺 𝑖𝑖 (𝑠𝑠) the transfer function of the controlled plant, 𝐾𝐾 𝑖𝑖 (𝑠𝑠) the transfer function of the designed local controller, and the control command law 𝑈𝑈 𝑖𝑖𝑠𝑠 (𝑠𝑠) the feedback control input, then: Control system design

𝑈𝑈 𝑖𝑖𝑠𝑠 (𝑠𝑠) = -𝐾𝐾 𝑖𝑖 (𝑠𝑠) 𝑍𝑍 ̃2,𝑖𝑖𝑠𝑠 (𝑠𝑠), ( 3

CRONE control approach

The CRONE (the French acronym for Commande Robuste d'Ordre Non-Entier, which means fractional-order robust control) design methodology is a frequency domain approach that has been developed since the 1980s (Oustaloup, 1991a;Sabatier et al., 2015a). It is based on the common unityfeedback configurations where three CRONE control generations have been developed. In these three methods, the open-loop transfer function is defined using fractional order integro-differentiation. One major advantage of the CRONE controller is that the plant uncertainties are taken into account. A direct consequence of these uncertainties is the variation of the stability degree of the controlled system, in other words, the variation of the phase margin of the closed-loop systems, namely, 𝑀𝑀 𝜙𝜙 . This implies a different performance of the plant for each set of parameters.

The selection of the CRONE generation depends on the degree of variations of gain or/and phase of the plant due to parametric uncertainties at the desired open-loop gain crossover frequency 𝜔𝜔 𝑐𝑐 [START_REF] Lanusse | CRONE control system design toolbox for the control engineering community: Tutorial and case study[END_REF]. In this contribution, the first-generation CRONE controller is used for both global and local controllers.

Optimization approach of control parameters' selection

The CRONE controller enables the parameterization of the open-loop transfer function with a small number of high-level parameters. Then, the optimization is thus reduced to only the search for the optimal values of these parameters.

The cascade form of the first-generation CRONE controller consists of a proportional constant 𝐶𝐶 0 , a band-limited fractional-order derivative (𝑚𝑚) within two corner frequencies 𝜔𝜔 𝑙𝑙 and 𝜔𝜔 ℎ , an integer band-limited integrator of order 𝑚𝑚 𝐼𝐼 upper bounded by the frequency 𝜔𝜔 𝐼𝐼 to manage steady-state error, and a low-pass filter of order 𝑚𝑚 𝑓𝑓 with a cutoff frequency 𝜔𝜔 𝑓𝑓 to manage the control sensitivity level and reduce noise amplification at high frequencies. Therefore, the general transfer function is formulated as follows:

𝐶𝐶(𝑠𝑠) = 𝐶𝐶 0 �1 + 𝜔𝜔 𝐼𝐼 𝑠𝑠 � 𝑚𝑚 𝐼𝐼 � 1 + 𝑠𝑠 𝜔𝜔 𝑙𝑙 ⁄ 1 + 𝑠𝑠 𝜔𝜔 ℎ ⁄ � 𝑚𝑚 1 �1 + 𝑠𝑠 𝜔𝜔 𝑓𝑓 ⁄ � 𝑚𝑚 𝑓𝑓 . (3.25)
The transitional frequencies defined are calculated so that a constant phase is achieved on a frequency range [𝜔𝜔 𝐴𝐴 , 𝜔𝜔 𝐵𝐵 ] which covers the frequency range where 𝜔𝜔 𝑐𝑐 could vary due to the plant parametric variation (Sabatier et al., 2015a). Then, the fractional-order 𝑚𝑚 and the gain 𝐶𝐶 0 are calculated to ensure the required nominal phase margin 𝑀𝑀 𝜙𝜙 and open-loop gain crossover frequency 𝜔𝜔 𝑐𝑐 . In this study, particularly, it was considered that 𝜔𝜔 𝐼𝐼 = 𝜔𝜔 𝑙𝑙 and 𝜔𝜔 𝑓𝑓 = 𝜔𝜔 ℎ .

The selection process of the controlled parameters is based on obtaining the highest possible control performance while attaining a reasonable degree of robustness against uncertain parameters. The robustly designed controller should produce a constant open-loop phase in [𝜔𝜔 𝐴𝐴 , 𝜔𝜔 𝐵𝐵 ] whose Nichols locus defined by a vertical straight line, so-called the frequency template (Figure 3.6). Ideally, this template ensures the robustness by minimizing the variations of phase margins 𝑀𝑀 𝜙𝜙 , modulus margins 𝑀𝑀 𝑟𝑟 , and the resonant peaks (𝑄𝑄 𝑠𝑠 and 𝑄𝑄 𝑇𝑇 ) of the sensitivity and the complementary sensitivity functions 𝐶𝐶(𝑠𝑠) and 𝑇𝑇 (𝑠𝑠), respectively, where: In the frequency domain, the robustness of the designed controller can be assessed by the maximum variation of phase margins ∆𝑀𝑀 𝜙𝜙 or the maximum difference of resonant peaks (∆𝑄𝑄 𝑇𝑇 ,𝑆𝑆 ) with plant uncertainties. On the other side, besides the value of 𝜔𝜔 𝑐𝑐 , the performance of the controller is directly affected by the value 𝜔𝜔 𝑙𝑙 , such that as 𝜔𝜔 𝑙𝑙 increases the steady-state error decreases, whereas the robustness is deteriorated. Same for 𝜔𝜔 ℎ , whereas its value increases it provides higher robustness to the system, but it amplifies more the noise effect of the measured signals.

𝐶𝐶(𝑠𝑠) = 1 �1 + 𝛽𝛽(𝑠𝑠)� , ( 3
To deal with this tradeoff between performance and robustness, the optimization methodology is then translated to optimize the 'length' of the asymptotic phase behavior (or the frequency template) in the Nichols chart. Consequently, reduce the variation of 𝑀𝑀 𝜙𝜙 (or (∆𝑄𝑄 𝑇𝑇 ,𝑆𝑆 ) ) of the different plants as much as possible.

Performance specifications

The specifications of the global controller and the local controllers are as follows:

-For the stability degree, the resonant peak of the complementary sensitivity function is 𝑄𝑄 𝑇𝑇 ≤ 3 𝐺𝐺𝐶𝐶 that is equivalent to min(𝑀𝑀 𝜙𝜙 ) = 40°.

-For the rapidity of the closed-loop system, the widest possible bandwidth while taking into consideration the actuator limits, noise amplification, and plant frequency response.

-For the precision at steady state, the relative error lower than 1%.

-The AW actuator control limit: max|𝑢𝑢 𝑎𝑎 | ≤ 4000𝑁𝑁.

Global controller design

All parameters of the three controlled plants related to chassis dynamics presented in (3.16 -3.18) are considered uncertain. The range of uncertainty of each parameter is listed in Table 3.1. Figure 3.7 shows the plants bode responses of the extremum plants. It can be easily deduced that the plants' variations are mostly gain-like only at high frequencies (𝜔𝜔 ≥ 30 rad/s). Therefore, in this range, the first-generation CRONE controller can be used to control the three plants. The open-loop transfer functions for the three systems are defined as follows:

𝛽𝛽 𝑧𝑧 (𝑠𝑠) = 𝐶𝐶 𝑧𝑧 (𝑠𝑠)𝐺𝐺 𝑧𝑧,𝑛𝑛𝑛𝑛𝑚𝑚 (𝑠𝑠), 𝛽𝛽 𝜑𝜑 (𝑠𝑠) = 𝐶𝐶 𝜑𝜑 (𝑠𝑠)𝐺𝐺 𝜑𝜑,𝑛𝑛𝑛𝑛𝑚𝑚 (𝑠𝑠), 𝛽𝛽 𝜃𝜃 (𝑠𝑠) = 𝐶𝐶 𝜃𝜃 (𝑠𝑠)𝐺𝐺 𝜃𝜃,𝑛𝑛𝑛𝑛𝑚𝑚 (𝑠𝑠).

(3.31)

According to (Sabatier et al., 2015a), the control algorithm starts at the selection of the desired crossover frequencies of the open-loop transfer functions, namely, 𝜔𝜔 𝑧𝑧,𝑐𝑐 , 𝜔𝜔 𝜑𝜑,𝑐𝑐 , and 𝜔𝜔 𝜃𝜃,𝑐𝑐 . Respecting the aforementioned specifications, the upper limits of crossover frequencies are indicated taking into account the active wheel suspension actuator limit in addition to the noise measurement specifications, that is: -Control input related to measurement noise is less than 10%.

-The noise level for heave motion is max(𝑁𝑁 𝑧𝑧 ) ≤ 2𝑊𝑊𝑚𝑚/𝑠𝑠, and for pitch and roll motion max�𝑁𝑁 𝜑𝜑 , 𝑁𝑁 𝜃𝜃 � ≤ 1.5 °/𝑠𝑠.

To translate these specifications to the frequency domain criterion, the following input sensitivity function 𝐶𝐶𝐶𝐶(𝑠𝑠) is considered, where: 𝐶𝐶𝐶𝐶(𝑠𝑠) = 𝐶𝐶(𝑠𝑠)/�1 + 𝛽𝛽(𝑠𝑠)�.

(3.32)

The magnitude |𝐶𝐶𝐶𝐶(𝑗𝑗𝜔𝜔)| is computed using the following: To reduce complexity, one value is chosen for the three crossover frequencies satisfying (3.28 -3.30), then 𝜔𝜔 𝑐𝑐 = 80 rad/s.

|𝐶𝐶𝐶𝐶 𝑧𝑧 (𝑗𝑗𝜔𝜔)| = � 𝑈𝑈 𝑧𝑧,𝐹𝐹𝐵𝐵 (𝑗𝑗𝜔𝜔) 𝑁𝑁 𝑧𝑧 (𝑗𝑗𝜔𝜔) � ≤ � 10% × (4 × 4000) 0.02 � = 80𝑟𝑟3𝑁𝑁 . 𝑠𝑠/𝑚𝑚, (3.33
In fact, the controlled plants have very similar behavior in the frequency domain (Figure 3.7). For this reason, one control parametric configuration can be used for the three controllers, except the value of 𝐶𝐶 0 that is adapted according to the controlled plant to obtain the desired crossover frequency of each open-loop transfer function. The value of the fractional-order derivation 𝑚𝑚 is chosen so that 𝑀𝑀 𝜙𝜙 = 45°, where 𝑚𝑚 𝐼𝐼 = 2 and 𝑚𝑚 𝑓𝑓 = 1. Then, the first-generation CRONE control parameters are computed and the resulted fractional-order controllers are as follow: Finally, for time-domain simulations, the fractional part of the controller (𝐶𝐶 𝑁𝑁𝐸𝐸 (𝑠𝑠)) should be replaced by another rational transfer function. Thanks to Oustaloup's Method [START_REF] Oustaloup | Frequencyband complex noninteger differentiator: Characterization and synthesis[END_REF], the rational form of the controller transfer function is obtained. Using N recursive zeros and N recursive poles, the band-limited fractional-order derivation is replaced as follows:

𝐶𝐶 𝑧𝑧 (𝑠𝑠) = 357120 × 𝐶𝐶 𝐸𝐸 (𝑠𝑠) 𝐶𝐶 𝑁𝑁𝐸𝐸 (𝑠𝑠), ( 3
� 1 + 𝑠𝑠 2 ⁄ 1 + 𝑠𝑠/1702 � -0.47 → � � 1 + 𝑠𝑠/𝜔𝜔 𝑧𝑧𝑟𝑟 𝑁𝑁 1 + 𝑠𝑠/𝜔𝜔 𝑝𝑝𝑛𝑛 𝑁𝑁 � , 5 𝑁𝑁=1 (3.44)
where, 5, 20, 78, 304, and 1187 rad/s are the values of zeros, and 3, 10, 41, 160, and 625 rad/s are the values of poles.

Figure 3.8 shows the bode responses of the designed global controllers. Figure 3.9 shows the open-loop bode plots for the nominal plants of three global systems. The consistency between the three plants is expected due to the adopted design strategy in which all global controllers used to have the same cross-over frequency, stability margins, and phase behavior.

Local control design

In the quarter vehicle synthesis model, three main suspension parameters are considered uncertain. Table 3.2 shows the range of uncertainty of each parameter, thus, three transfer functions are defined: 𝐺𝐺 𝑖𝑖,𝑚𝑚𝑖𝑖𝑛𝑛 (𝑠𝑠), 𝐺𝐺 𝑖𝑖,𝑚𝑚𝑟𝑟𝑑𝑑 (𝑠𝑠), and 𝐺𝐺 𝑖𝑖,𝑚𝑚𝑎𝑎𝑥𝑥 (𝑠𝑠).

Figure 3.10 shows the bode responses of the local synthesis models and the effect of the parametric uncertainties. It is clear that the uncertainty leads to higher variation for the rear local plant in comparison to the front one. Also, it is shown that both plants have an asymptotic phase behavior in the frequency range between the resonance and anti-resonance frequency. This behavior imposes a lower and upper bound for the rapidity (𝜔𝜔 𝑐𝑐 ) of the closed-loop system. Similar to global controller design, and considering the specifications, a first-generation CRONE controller is designed for every axle. Here, the "𝑚𝑚𝑚𝑚𝑚𝑚" plants are used as nominal plants in the design process. The crossover frequencies of the open-loop transfer functions are selected, namely, 𝜔𝜔 𝑐𝑐1 = 50 𝑀𝑀𝑎𝑎𝐺𝐺/𝑠𝑠 and 𝜔𝜔 𝑐𝑐2 = 40 𝑀𝑀𝑎𝑎𝐺𝐺/𝑠𝑠, for front and rear controllers respectively, where the variation is approximately gain-like only. Taking into account the required phase margin (𝑀𝑀 𝜙𝜙 = 45°) and the value of the crossover frequencies, then the obtained control parameters for front and rear axle controllers are listed in Table 3.3. Thus, the fractional-order local CRONE controllers are:

𝐾𝐾 1 (𝑠𝑠) = 83028 �1 + 2.79 𝑠𝑠 � � 1 + 𝑠𝑠 2.79 ⁄ 1 + 𝑠𝑠 897 ⁄ � 0.62 1 (1 + 𝑠𝑠 897 ⁄ ) , (3.45) 𝐾𝐾 2 (𝑠𝑠) = 33591 �1 + 2 𝑠𝑠 � � 1 + 𝑠𝑠 2 ⁄ 1 + 𝑠𝑠 1610 ⁄ � 0.58 1 (1 + 𝑠𝑠 1610 ⁄ ) . (3.46)
Again, the band-limited fractional-order derivation is replaced by 5 recursive zeros and 5 recursive poles for each as follow: where the values of zeros are: 3.5, 11, 35, 110, and

� 1 + 𝑠𝑠 2.79 ⁄ 1 + 𝑠𝑠 897 ⁄ � 0.62 → � � 1 + 𝑠𝑠/𝜔𝜔 𝑧𝑧𝑟𝑟 𝑁𝑁 1 + 𝑠𝑠/𝜔𝜔 𝑝𝑝𝑛𝑛 𝑁𝑁 � , 5

Control Allocation

Once the desired global control efforts are determined, a new repartition is applied to assure the passage of the different efforts to the four suspensions. The relations that enable the transition of 𝑢𝑢 𝑧𝑧 (𝑇𝑇), 𝑢𝑢 𝜑𝜑 (𝑇𝑇), and 𝑢𝑢 𝜃𝜃 (𝑠𝑠) to the AW actuators are described in this section.

Recalling the global objective of the hierarchical approach: to compensate chassis dynamics under driver disturbances, then the input variable references are given by:

𝑣𝑣 𝑧𝑧 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) = 𝜔𝜔 𝜑𝜑 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) = 𝜔𝜔 𝜃𝜃 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) = 0. (3.49)
Assuming that the pitch and roll angles have small values and the chassis is infinitely rigid, the relation between the global control efforts and the four suspension forces can be formulated as follows: However, in order to establish a direct relationship between the forces 𝑓𝑓 𝑠𝑠𝑖𝑖𝑠𝑠 (𝑇𝑇) and the chassis dynamics, it is necessary to add a fourth relation.

𝑓𝑓 𝑠𝑠11 (𝑇𝑇) + 𝑓𝑓 𝑠𝑠12 (𝑇𝑇) + 𝑓𝑓 𝑠𝑠21 (𝑇𝑇) + 𝑓𝑓 𝑠𝑠22 (𝑇𝑇) = 𝑢𝑢 𝑧𝑧 (𝑇𝑇), (3.50
Therefore, the Roll Stiffness Distribution (RSD) factor 𝛼𝛼 𝑅𝑅𝑆𝑆𝑅𝑅 (𝑇𝑇) [START_REF] Abe | Study on effects of roll moment distribution control in active suspension on improvement of limit performance of vehicle handling[END_REF] was used. This factor is important for the stability of the vehicles during maneuvers. Since the track width of the front and rear axles are equal, then 𝛼𝛼 𝑅𝑅𝑆𝑆𝑅𝑅 is given by: Therefore, the relations (3.50)-(3.54) can be written under the following matrix form:

𝛼𝛼 𝑅𝑅𝑆𝑆𝑅𝑅 (𝑇𝑇) = 𝑊𝑊�𝑓𝑓 𝑠𝑠 12 (𝑇𝑇) -𝑓𝑓 𝑠𝑠 11 (𝑇𝑇)� 𝑊𝑊�𝑓𝑓 𝑠𝑠21 (𝑇𝑇) -𝑓𝑓 𝑠𝑠22 (𝑇𝑇)� -𝑊𝑊(𝑓𝑓 𝑠𝑠11 (𝑇𝑇) -𝑓𝑓 𝑠𝑠12 (𝑇𝑇)) . ( 3
⎣ ⎢ ⎡ 1 -𝑎𝑎 -𝑊𝑊 -𝑊𝑊 1 -𝑎𝑎 𝑊𝑊 𝑊𝑊 1 𝑏𝑏 𝑊𝑊 0 1 𝑏𝑏 -𝑊𝑊 0 ⎦ ⎥ ⎤ ⎝ ⎜ ⎜ ⎜ ⎛ 𝑓𝑓 𝑠𝑠11 𝑓𝑓 𝑠𝑠12 𝑓𝑓 𝑠𝑠21 𝑓𝑓 𝑠𝑠22 ⎠ ⎟ ⎟ ⎟ ⎞ = ⎝ ⎜ ⎜ ⎛ 𝑢𝑢 𝑧𝑧 𝑢𝑢 𝜑𝜑 𝑢𝑢 𝜃𝜃 𝛼𝛼 𝑅𝑅𝑆𝑆𝑅𝑅 𝑢𝑢 𝜃𝜃 ⎠ ⎟ ⎟ ⎞ .
(3.55) Thus, using the inverse of (3.52), the desired values of the four suspensions 𝑓𝑓 ̅ 𝑠𝑠,𝑖𝑖𝑠𝑠 (𝑇𝑇) can be computed as follows:

⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 𝑓𝑓 ̅ 𝑠𝑠11 𝑓𝑓 ̅ 𝑠𝑠12 𝑓𝑓 ̅ 𝑠𝑠21 𝑓𝑓 ̅ 𝑠𝑠22 ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 𝑏𝑏 2𝐿𝐿 -1 2𝐿𝐿 0 -1 2𝑊𝑊 𝑏𝑏 2𝐿𝐿 - 1 2𝐿𝐿 0 1 2𝑊𝑊 𝑎𝑎 2𝐿𝐿 1 2𝐿𝐿 1 2𝑊𝑊 -1 2𝑊𝑊 𝑎𝑎 2𝐿𝐿 1 2𝐿𝐿 - 1 2𝑊𝑊 1 2𝑊𝑊 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ ⎝ ⎜ ⎜ ⎛ 𝑢𝑢 𝑧𝑧 𝑢𝑢 𝜑𝜑 𝑢𝑢 𝜃𝜃 𝛼𝛼 𝑅𝑅𝑆𝑆𝑅𝑅 𝑢𝑢 𝜃𝜃 ⎠ ⎟ ⎟ ⎞ .
(3.56)

Finally, the overall control input of the active suspension system is then expressed by the algebraic sum of the local control efforts of local controllers (𝑢𝑢 𝑖𝑖𝑠𝑠 (𝑇𝑇)), the desired force of the global controller (𝑓𝑓 ̅ 𝑠𝑠 𝑖𝑖𝑠𝑠 (𝑇𝑇)), and the estimated force of the passive suspension elements (𝑓𝑓 ̂𝑠𝑠𝑝𝑝, 𝑖𝑖𝑠𝑠 ), that is: Figure 3.13 illustrates the overall schematics of the hierarchical approach. Finally, the development of the proposed control algorithm requires various feedback values of the vehicle states variables, including the sprung mass displacements and the main chassis dynamics. Plenty of methods have been proposed for this purpose and experimental validation is reported in [START_REF] Hashemi | Real-time estimation of the road bank and grade angles with unknown input observers[END_REF][START_REF] Viehweger | Vehicle state and tyre force estimation : demonstrations and guidelines[END_REF]. In this study, it is assumed that all these variables can be measured or estimated accurately.

In fact, after the global references of the control efforts are perfectly known, then three strategies could be conceived while taking into consideration the technology of the used actuator that is the DC electric motor. These strategies can be listed as follows: -Current regulation (closed-loop control): same for the first strategy but with a closed-loop control system to regulate the current provided to the electric actuator to track the reference value of the computed current.

-Current and/or sprung mass regulation: like the previous strategy but with the ability for adding an additional closed-loop system for sprung mass regulation combined with regulation of the current electric suspension actuator.

In this study, it is assumed that the actuator is ideal, that is perfectly providing the reference values obtained from the hierarchical approach combined with the part for sprung mass regulations.

Study requirements

Disturbance model

In order to evaluate the performance of the designed suspension local controller, the quarter models are subjected to different road disturbances. In the context of vehicle ride comfort and handling, the suspension system is disturbed by two main road inputs that are mainly classified as shock and vibrations [START_REF] Hrovat | Survey of Advanced Suspension Developments and Related Optimal Control Applications[END_REF]. Shocks are discrete events of relatively short duration and high intensity that caused large suspension deflections, where vibrations are typically specified as random processes that consist of highfrequency components. Both types have a major influence on ride comfort and safety of the vehicle. In general, a well-designed controlled suspension must perform adequately in a wide range of shock and vibration environments.

Random road profiles

Beside driver disturbances, if the vibration induced by power and transmission systems is ignored, the unevenness of the road surface is the most important factor affecting the passenger ride comfort. In simulations, the random road profile is generated according to the International Standard Organization ISO 8608 (ISO-8608, 2011) where eight road roughness categories labeled from A to H are defined by the single-sided power spectral density PSD in the space-frequency (cycles/m) domain. The roughness profile of the road surface can be defined using the following equation:

𝐺𝐺 𝑑𝑑 (𝛺𝛺) = 𝐺𝐺 𝑑𝑑 (𝛺𝛺 0 ) ⋅ � 𝛺𝛺 𝛺𝛺 0 � -𝑤𝑤 , (3.58)
where Ω 0 = 1 𝑀𝑀𝑎𝑎𝐺𝐺/𝑚𝑚 is the conventional value of angular spatial frequency, and 𝐺𝐺 𝑑𝑑 (Ω 0 ) is a constant derived from consistent tables found on the basis of the considered road class [START_REF] Agostinacchio | The vibrations induced by surface irregularities in road pavements -a Matlab® approach[END_REF][START_REF] Andrén | Power spectral density approximations of longitudinal road profiles[END_REF]ISO-8608, 2011;[START_REF] Pan | Adaptive tracking control for active suspension systems with non-ideal actuators[END_REF][START_REF] Tyan | Generation of random road profiles[END_REF]. In this study, an artificial road profile is generated using a stochastic representation as a function of PSD of vertical displacements. Then, the road profile can be described by the following expression [START_REF] Agostinacchio | The vibrations induced by surface irregularities in road pavements -a Matlab® approach[END_REF]:

𝑎𝑎 0 (𝑎𝑎) = � 𝐶𝐶 𝜄𝜄 𝑁𝑁 𝜄𝜄 𝜄𝜄=0 sin(𝛺𝛺 𝜄𝜄 ⋅ 𝑎𝑎 + 𝑀𝑀 𝜄𝜄 ) = � �2 ⋅ 𝛥𝛥𝛺𝛺 ⋅ 𝐺𝐺 𝑑𝑑 (𝛺𝛺 𝜄𝜄 ) 𝑁𝑁 𝜄𝜄 𝜄𝜄=0
⋅ sin(𝛺𝛺 𝜄𝜄 ⋅ 𝑎𝑎 + 𝑀𝑀 𝜄𝜄 ), (3.59) where 𝐶𝐶 𝜄𝜄 is the amplitude, 𝑀𝑀 𝜄𝜄 is the random phase angle that follows a uniform probabilistic distribution within [0 -2𝜋𝜋], 𝑎𝑎 is the abscissa variable within the range 0 → 𝐿𝐿 0 , and 𝐿𝐿 0 is the length of the road profile. ΔΩ is the step size of the PSD bounded by two lower and upper angular spatial frequencies Ω 1 = 0.04 𝑀𝑀𝑎𝑎𝐺𝐺/𝑚𝑚 and Ω 𝑁𝑁 𝜄𝜄 = 10 𝑀𝑀𝑎𝑎𝐺𝐺/𝑚𝑚 respectively, in order to cover an appropriate band of speeds of the passing vehicle. Table 3.4 lists the values of 𝐺𝐺 𝑑𝑑 (Ω 0 ) of some road classes while adopting 𝑤𝑤 = 2 in (3.58). 𝛀𝛀 𝟎𝟎 = 𝟏𝟏 𝒓𝒓𝒓𝒓𝒓𝒓/𝒎𝒎

Singular disturbance: Bump

In order to provide a realistic framework regarding shock road disturbances, a pronounced bump on smooth road is introduced. A widely used simple model for the vertical road displacement of a singular disturbance event is expressed by the following: 3.60) where ℎ 𝑏𝑏 represents half of the bump height, 𝐿𝐿 𝑏𝑏 represents the bump length and 𝑣𝑣 𝑥𝑥 represents the passing-vehicle's velocity.

𝑎𝑎 0 (𝑇𝑇) = ⎩ � ⎨ � ⎧ ℎ 𝑏𝑏 �1 -cos � 2𝜋𝜋 𝜐𝜐 𝑥𝑥 𝐿𝐿 𝑏𝑏 𝑇𝑇�� , for 0 ≤ 𝑇𝑇 ≤ 𝐿𝐿 𝜐𝜐 𝑥𝑥 0 , else (

Benchmark systems

Different suspension systems are used as performance benchmarks for the designed local active suspension control system. In this section, one parameter configuration for quarter vehicle model is adopted for all the systems presented, in which 𝑚𝑚 2 = 218 𝑘𝑘𝑔𝑔, 𝑘𝑘 2 = 12𝑟𝑟3 𝑁𝑁/𝑚𝑚, 𝑘𝑘 1 = 3𝑟𝑟5 𝑁𝑁/𝑚𝑚, 𝑏𝑏 1 = 50 𝑁𝑁. 𝑠𝑠/𝑚𝑚, and 𝑚𝑚 1 = 32 𝑘𝑘𝑔𝑔.

Based on the frequency response of 𝑍𝑍 2 (𝑠𝑠)/𝑍𝑍 0 (𝑠𝑠) and 𝑍𝑍 1 (𝑠𝑠)/𝑍𝑍 0 (𝑠𝑠), the sensitivity transfer functions of the sprung mass and unsprung mass motion relative to road disturbance, respectively, the value of 𝑏𝑏 2 (damping factor) is adapted to get: a) A passive suspension system (PSc) acting as a baseline for the minimal behavior needed to ride comfort where the body factor resonance 𝑄𝑄 2 of |𝑍𝑍 2 (𝑠𝑠)/𝑍𝑍 0 (𝑠𝑠)| should not exceed 2, (𝑄𝑄 2 ≤ 2). Then 𝑏𝑏 2𝑐𝑐 = 1040 𝑁𝑁. 𝑠𝑠/𝑚𝑚.

b) A passive suspension system (PSr) that represents the lower-limit performance needed for the unsprung mass dynamic, where the unsprung factor resonance 𝑄𝑄 1 of |𝑍𝑍 1 (𝑠𝑠)/𝑍𝑍 0 (𝑠𝑠)| should not exceeds 3, (𝑄𝑄 1 ≤ 3 ). Then, for 𝑏𝑏 2𝑟𝑟 = 975 𝑁𝑁𝑠𝑠/𝑚𝑚, minimal security is ensured.

In fact, the use of these systems is a rule of thumb methodology adopted by the PSA (Peugeot Citroën Associates) group to evaluate any newly designed strategies at the very first stage [START_REF] Hamrouni | Skyhook and CRONE active suspensions: A comparative study[END_REF].

c) The third benchmark system is a system based on (PSr) with an optimized Skyhook controller (Sh), such that: 3.61) where 𝑉𝑉 2 (𝑠𝑠) is the Laplace transform of the velocity of the vertical displacement of the chassis (𝑎𝑎 ̇2(𝑇𝑇)). The transfer function 𝐶𝐶 𝑆𝑆ℎ (𝑠𝑠) consists of a simple damping coefficient '𝑏𝑏 𝑠𝑠ℎ ' multiplied by a low pass filter with a well-selected cut-off frequency to limit unsprung damping factor. The selection of 𝑏𝑏 𝑠𝑠ℎ is based on an optimized strategy used in [START_REF] Hamrouni | Skyhook and CRONE active suspensions: A comparative study[END_REF]. Then, obtained transfer function of 𝐶𝐶 𝑆𝑆ℎ (𝑠𝑠) is as follows:

𝑈𝑈 𝑆𝑆ℎ (𝑠𝑠) = -𝐶𝐶 𝑆𝑆ℎ (𝑠𝑠) 𝑉𝑉 2 (𝑠𝑠), (
𝐶𝐶 𝑠𝑠ℎ (𝑠𝑠) = 𝑏𝑏 𝑆𝑆𝐻𝐻 1 + 𝑠𝑠 𝜔𝜔 𝑆𝑆ℎ = 2225 1 + 𝑠𝑠 63 .
(3.62) Figure 3.14 shows the frequency response of the benchmark systems. We observe that, for the assumed parameters, the considered PSc system has similar dynamic behavior of PSr system at the wheel-mode frequency, (close 𝑄𝑄 1 values). Therefore, in the evaluation section, only the PSc passive system will be considered for comparison. Finally, for the sake of comparing the performance, a PID controller is designed. In order to get a fair comparison, two main criteria are taken into account: the PID controller should have the same rapidity and stability margin as the designed CRONE controller. Since the chosen parametric configuration is for the front axle, therefore, the selected crossover frequency of the open-loop transfer function with the PID controller is 𝜔𝜔 𝑐𝑐,𝑃𝑃𝐼𝐼𝑅𝑅 = 50 𝑀𝑀𝑎𝑎𝐺𝐺/𝑠𝑠.

Moreover, the integration and the low pass filter transitional frequencies are supposed to be equal to that of the 𝐾𝐾 1 (𝑠𝑠). Then, the transfer function of the PID controller is written as follows: The rest control parameters of the PID are chosen in order to have the open-loop transfer function tangent to 3dB contour in the Nichols chart, the same as the CRONE controller. In the same manner, a PID controller is also designed for the rear axle, and the resulting transfer function is formulated as follows:

𝐾𝐾 1,𝑃𝑃𝐼𝐼𝑅𝑅 (𝑠𝑠) = 167690 � 1 + 𝑠𝑠/2.

Evaluation Indexes

Various methods are used to evaluate the performance of the proposed control law with respect to the nominal benchmark systems. In general, in academic and industrial applications, there are two main aspects of interest: The comfort and road-holding characteristics.

-Ride comfort is primarily associated with low vertical accelerations 𝐻𝐻2(𝑡𝑡). For this reason, the root means square (RMS) value of the vertical chassis acceleration is considered as a measure for ride comfort. However, human comfort depends also on the frequency of the vertical acceleration, as well as its direction, location, and duration, especially, in the frequency range [4][5][6][START_REF]Bode diagram of the transfer functions (a) 𝑇𝑇20(𝑠𝑠) and (b) 𝑇𝑇10[END_REF][START_REF]Nichols chart for open-loop transfer functions with CRONE controller 𝐶𝐶1[END_REF] where the human sensitivity for vertical vibrations is distinctive (Standard, 1997). For this reason, ISO 2631 specifies a method to asses ride comfort through weighting the value of the vertical acceleration by a shaping filter 𝑊𝑊 𝑐𝑐 (𝑠𝑠) that emphasizes the comfort relevant frequency range [START_REF] Zuo | Low order continuous-time filters for approximation of the ISO 2631-1 human vibration sensitivity weightings[END_REF]. A good approximation for the frequency weighting curve of ISO 2631-1 can be described as follows [START_REF] Koch | Driving state adaptive control of an active vehicle suspension system[END_REF]: (3.65) Figure 3.17 shows the bode response of 𝑊𝑊 𝑐𝑐 (𝑠𝑠), in which it presents a bandlimited filter for frequencies in the range of interest [4 -8] Hz.

𝑊𝑊 𝑐𝑐 (𝑠𝑠) = 10
Furthermore, the maximum value of the vertical acceleration is also calculated which is indicative in singular disturbance events.

-The suspension deflection |𝑎𝑎 2 (𝑇𝑇) -𝑎𝑎 1 (𝑇𝑇)| is limited to 9𝑊𝑊𝑚𝑚 in the AW suspension system. Moreover, the RMS value of suspension deflection is also considered to analyze road holding.

Study requirements

133

-The dynamic wheel load's RMS value represents a good index to evaluate ride safety to ensure the contact between the tire and the road. Thus, the transfer of longitudinal and lateral forces is guaranteed so that the driver can control the vehicle by steering, braking, etc... For this reason, the following formula is adopted: 3.66) where 𝐹𝐹 𝑑𝑑𝑦𝑦𝑛𝑛 is the wheel deflection, and 𝐹𝐹 𝑠𝑠𝑡𝑡𝑎𝑎𝑡𝑡 is the static wheel load. This is derived from the 3σ -rule where 𝐹𝐹 𝑑𝑑𝑦𝑦𝑛𝑛 would remain within ± 𝐹𝐹 𝑠𝑠𝑡𝑡𝑎𝑎𝑡𝑡 for 99.7% of the time [START_REF] Mitschke | Beurteilungsmaßstäbe und ihre Berechnung (Assessment standards and their calculation)[END_REF].

𝑚𝑚𝑎𝑎𝑎𝑎��𝐹𝐹 𝑑𝑑𝑦𝑦𝑛𝑛 �𝑀𝑀𝑚𝑚𝑠𝑠� ≤ 𝐹𝐹 𝑠𝑠𝑡𝑡𝑎𝑎𝑡𝑡 3 ≤ (𝑚𝑚 2 + 𝑚𝑚 1 )𝑔𝑔 3 , (
-The peak and the RMS value of the actuator's forces is also considered. Additionally, power consumption is also taken into consideration to compare the active systems, and the following relation has been selected [START_REF] Marzbanrad | Stochastic optimal preview control of a vehicle suspension[END_REF]):

Frequency-domain analysis

In order to assess the robustness of the designed controllers, the Nichols charts of the open-loop transfer functions are plotted for global controllers with extremum plants as shown in Figure 3.18. It clearly shows the frequency template that is tangent to 3dB contour, which reflects the robustness of each controller against the parametric variation. Thus, the variation in the stability margins is approximately zero. the peak response of the complementary sensitivity functions 𝑇𝑇 (𝑠𝑠) are as follows: for front axle 𝑚𝑚𝑎𝑎𝑎𝑎(∆𝑄𝑄 𝑇𝑇 ) = 0.02𝐺𝐺𝐶𝐶 with CRONE controller where for PID is 0.27𝐺𝐺𝐶𝐶, and for the rear axle, 𝑚𝑚𝑎𝑎𝑎𝑎(∆𝑄𝑄 𝑇𝑇 ) = 0.1𝐺𝐺𝐶𝐶 with CRONE controller while with PID is 0.43𝐺𝐺𝐶𝐶. Therefore, although of high uncertainty in the suspension parameters, the designed CRONE controller shows its robustness in general, and particularly compared to PID.

Regarding the performance of the local controllers, Figure 3.21 and Figure 3.22 show respectively the frequency responses of the sensitivity of the sprung mass acceleration and suspension deflection to road disturbance for nominal plants. It is obvious that the controlled plants are able to attain high attenuation for the sprung mass acceleration for a wide range of frequencies compared to the PSc system, especially in the range of interest (≤ 8𝐻𝐻𝑎𝑎). However, this improvement comes at the expense of a degradation at the resonance wheel-mode frequency. It can be observed in Figure 3.22 the high gain at 𝑓𝑓 ≈ 15𝐻𝐻𝑎𝑎 for the controlled systems compared to the passive and Skyhook systems. This fact will be translated to high variation for wheel deflection in case of high frequency road disturbances, and consequently less able to handle the vehicle by the driver. Moreover, at low frequencies (≤ 1𝐻𝐻𝑎𝑎), the controlled systems are mostly not attenuating the suspension deflection comparing to other systems, which will reflect a higher values suspension deflection in the time-domain. In contrast, the Skyhook system keeps the damping behavior of the PSc system at wheel-mode frequencies, but it has less vertical acceleration attenuation. However, it effectively eliminates the effect of the resonance behavior at the chassis-mode frequency (around 1.5 Hz).

It should be noted that the difference in magnitudes at high frequencies appeared in Figure 3.21 is due to the difference in the damping coefficient of the suspension systems. For Active Wheel, as mentioned earlier, the damping coefficient has a very low value that results only from the mechanical parts used in such a system, where no shock absorber is used in this structure. Therefore, the high gain at wheel mode frequency shown in Figure 3.22 is not only due to the closed-loop system but also because of the small value of the damping coefficient of the suspension system.

Although this analysis is done on one parametric configuration of the quarter vehicle, it could be applied to other configurations in which they share the same characteristics and no significant changes could be detected.

Time-domain analysis

The simulations are divided into two subsections: one for testing the limits of the designed local controller in rejection of road disturbances, where the quarter vehicle used is for the front axle with the parametric configuration of section 3.

Rejection of road disturbances

Singular event disturbance

An isolated bump is employed in this simulation as described in (3.5.1.2). The vehicle is passing over a bump, of a height of 10cm and length 1m, at a speed of 10km/h. Figure 3.23 shows the performance of the controlled system compared to the passive and Skyhook system. The CRONE and PID control show a high reduction in vertical acceleration which reveals an important improvement of ride comfort, however, they exceed the suspension deflection limits. Numerically, Table 3.5 lists the RMS values and maximums that evaluate the performance of the controlled systems relative to the passive system in percentage. It is obvious that the CRONE controller gains more comfort improvement compared to Skyhook and PID controlled systems. It can be concluded from the results obtained that the Skyhook controlled system could be an adequate solution for this kind of singular event road disturbances unless the height of the bump doesn't exceed the limits of the suspension deflection (≤ 9𝑊𝑊𝑚𝑚), where the designed controller provides more ride comfort for passengers, especially that the wheel deflection is within the admissible limits.

Practically, this simulation is not enough for analyzing the performance of the proposed strategy on singular disturbances. A holistic study should be done on the effect of all characteristics of the road profile. A comfortoriented active suspension control strategy could be an adequate solution for road profiles like in Highways, national roads, or secondary roads. Moreover, it will be adopted also for singular obstacles that lead to acceptable suspension deflections, where a threshold value should be predefined. The decision of what control algorithm should be applied is based on the detected obstacles' characteristics like its geometry, speed of passing vehicle, and its power spectral density (PSD), for example.

For this reason, the study of the effect of such characteristics should be done and strategies could be implemented based on a holistic analysis. This work is probably one of the author's concerns for future studies. road profiles used and their PSD graph. The road class A (green) is generated by 𝐺𝐺 𝑑𝑑,𝐴𝐴 (Ω 0 ) = 1 and 𝐺𝐺 𝑑𝑑,𝐴𝐴-𝐵𝐵 (Ω 0 ) = 2 for road class A-B. Figure 3.25 and Figure 3.26 show the performance of the CRONE controller compared to PSc system with road class A and A-B, respectively, where the variable 𝐶𝐶 2 (𝑇𝑇) represents the vertical acceleration of the sprung mass. Table 3.6 and Table 3.7 lists the numerical results of the controlled quarter vehicle systems driven in road class A and A-B, respectively. The numerical results showed that the designed comfort-oriented local controller can improve ride comfort significantly by reducing 𝐶𝐶 2 (𝑇𝑇) up to 47 -50 % relative to PSc system, while respecting road holding critical limits formulated in section 3.5.3. The degradations in the performance of the dynamic wheel load 𝐹𝐹 𝑑𝑑𝑦𝑦𝑛𝑛 are acceptable in these scenarios where the RMS values are still Figure 3.24. ISO road profiles and their PSD graphs. less than the limit (𝐹𝐹 𝑠𝑠𝑡𝑡𝑎𝑎𝑡𝑡 3 ⁄ = 250 * 𝑔𝑔/3 = 817.5). However, it is clear that for rougher road profiles (class ≥ 𝐶𝐶), the RMS value of the wheel load dynamics could increase and exceed the limits, and as a result, the comfortoriented controller could then affect the handling performance of the vehicle.

The Skyhook system provides more safety margins but at the expense of low comfort improvement. These results are consistent with the frequency domain analysis where in frequencies range [4][5][6][START_REF]Bode diagram of the transfer functions (a) 𝑇𝑇20(𝑠𝑠) and (b) 𝑇𝑇10[END_REF][START_REF]Nichols chart for open-loop transfer functions with CRONE controller 𝐶𝐶1[END_REF], the controlled systems have a significant attenuation in contrast to the Skyhook controller. In terms of power consumption, PID and CRONE have very close values with an advantage for the latter in the desired actuator forces. The obtained results give the preference for the CRONE controller in terms of robustness and performance for random road rejection testing scenarios.

Driver Disturbance Rejection Scenario

The adopted driving scenario consists of two phases, the vehicle starts at an initial speed at 90𝑘𝑘𝑚𝑚/ℎ then the driver starts to slow down the vehicle until it reaches a speed of around 55𝑘𝑘𝑚𝑚/ℎ. In the second phase, at 𝑇𝑇 = 5.5𝑠𝑠, the driver introduces a positive steering input to turn left the vehicle. Figure 3.27 shows the steering input profile, the introduced braking/motor torques, the values of 𝑎𝑎 𝑦𝑦 and 𝑎𝑎 𝑥𝑥 , and the vehicle trajectory. It should be noted that the resulted values of the longitudinal and lateral accelerations are within the limits (0.4g), which means that the vehicle is operating in the comfort zone, which is the region of interest in this study.

Figure 3.28-30 display the chassis dynamics of the four systems: Passive system, system with only CRONE local controllers (LC), the system with CRONE global and local controllers without feedforward (GC+LC), system with CRONE global controller with feedforward control efforts only (GC+FF), and finally system with all components (All, i.e. GC+LC+FF). Primarily, the controlled systems prove their ability to highly compensate for the chassis dynamics. However, the plots reveal the advantage of the global controller and the feedforward control efforts. Obviously, the controlled system, where all control efforts elements are included, has better performance in compensating chassis dynamics. Table 3.8 shows the RMS values of different variables for all possible control configurations. The complete hierarchical approach proves its advantage in reducing pitch and roll accelerations, as well as sprung mass acceleration. On the other hand, in Figure 3.31, systems without local controller achieved better suspension deflection values that reflect more road-holding safety margins. However, this improvement achieved at the expense of less vibration isolation, where higher acceleration values are obtained.

Conclusion

Based on the hierarchical control approach, two-level controllers are presented for the AW suspension system. The systems are controlled using the first-generation CRONE controllers while taking into account the vehicles' parametric uncertainties. The main objective of the proposed approach is to reject the driver disturbances in light electric vehicles. In the range of interest of the study, the simulation results show the effectiveness of the strategy to enhance ride comfort and respecting the road holding/handling specifications. The combination of two-level controller permits to maximize the benefit of the embedded active suspension actuator in the AW. The controller shows its ability to fulfill the needed performance and robustness against the parametric variation of the vehicle. Consequently, it is shown that the local controller is able to deal with road disturbances of the A-B class. It is able to operate in a safe region without any significant effect on handling performances.

The extension of this work, in the framework of Global Chassis Control, is to implement other robust controllers that deal with the remaining objectives listed in the introduction, e.g. rejecting singular events and uneven road profiles (higher than B classes), and design road-holding oriented suspension controllers. The switching algorithm between the different designed controllers should be based on various sensing data given to the supervisor. Moreover, another control strategy should be implemented to deal with critical situations where the vehicle will operate outside the comfort region. The supervisor will then switch to the controller (e.g. road holding oriented controller, …) that is most convenient to the vehicle and to the environment/road situation.

Introduction

The safety of modern vehicles can be attributed to a wide range of complex systems that aim to prevent loss of lives, whether it assists the driver in an active sense or simply provides passive protection. However, the effectiveness of the braking system of the vehicle has always been one of the primary factors that can save occupant's lives.

The anti-lock brake system (ABS) is considered as a landmark invention of braking systems. As shown in the general introduction, the yaw stability control (YSC), electronic stability program (ESP), and automatic emergency brake (AEB) are gradually invented and integrated into various braking systems. The continuous integration of these new features and improvement of the braking system had been greatly improved safety performance and ride comfort [START_REF] Ahn | Analysis of a regenerative braking system for Hybrid Electric Vehicles using an Electro-Mechanical Brake[END_REF].

Several ABS strategies and control methodologies are available today. State-of-the-art control strategies, such as sliding mode, gain scheduling and fuzzy logic control, finds its use in ABS research as software implemented simulation models [START_REF] Aly | An Antilock-Braking Systems[END_REF]. However, the operation of ABS on rough road is a relatively new research area that is yet to be fully understood. The ability of ABS to perform on rough terrains is deteriorated by fluctuations in wheel speeds and by the fluctuation of vertical loading in the tire. It was found that the variation in the wheel speeds, due to terrain excitations, could partially explain why ABS does not perform as well on rough terrains, as on smooth terrains [START_REF] Penny | The test and simulation of ABS on rough, non-deformable terrains[END_REF]. Braking on rough terrains causes the tires to lose contact with the terrain momentarily and the contact patch of the tires also varies, reducing a vehicle's ability to perform a stable emergency maneuver [START_REF] Van Der Merwe | ABS braking on rough terrain[END_REF]. A scientific explanation as to why braking distance on rough terrains is much larger than on smooth terrains, is identified by [START_REF] Hamersma | Improving the braking performance of a vehicle with ABS and a semi-active suspension system on a rough road[END_REF] that state that the rough surface of the terrain causes body roll and pitch motions in the vehicle which in-turn causes large fluctuations in wheel loads. This decreases the longitudinal forces on the tires because of the vertical force variations, thus, increasing the braking distance.

In the literature, investigations have been done on improving the braking performance of ABS on rough terrain, through altering the stiffness and damping settings of a vehicle's suspension [START_REF] Hamersma | Improving the braking performance of a vehicle with ABS and a semi-active suspension system on a rough road[END_REF]. Numerous tests with different suspension characteristic combinations were performed and it was found that the braking distance enhancement could be significant.

Since longitudinal braking force generation is heavily dependent on vertical load, the rough terrain braking problem can essentially be improved with adequate suspension control [START_REF] Niemz | Improving Braking Performance By Control of Semi-Active Suspension[END_REF][START_REF] Reul | Enhanced braking performance by integrated ABS and semi-active damping control[END_REF]. Further advances in this control strategy can be carried out using continuous damping along with an accommodating ABS algorithm. This study aims to develop a simulation model and a test platform to conduct further research into the rough road ABS braking problem.

At first, a 3 rd generation CRONE controller is designed to maximize the efficiency of the braking system on AW. The study takes into consideration the road adhesion variations, vehicle mass variation, and vehicle velocity variations. Various testing scenarios were implemented on a quarter vehicle model while comparing the designed controller with the uncontrolled system in addition to a PID and 1 st generation CRONE controllers. The final objective of this system is to reduce the braking distances while taking into consideration system limitations and noise measurement criterion.

Then, the half-vehicle model is depicted to study the effect of the load transfer disturbance on the performance of the ABS controller. By the end, the road disturbances were taken into account, and some testing scenarios made to find out its effect on the ABS performance. In order to minimize the performance deterioration, the fractional-order damping controllers introduced in Chapter 2, are also implemented to test for its influence on the overall system, in its different scenarios, i.e. comfort-oriented and road holding oriented controllers.

The work done in the frame of the ABS controller was based on the previous study presented in [START_REF] Benine-Neto | Robust control for an electro-mechanical anti-lock braking system: the CRONE approach[END_REF], where it didn't consider the vertical dynamics. The contribution here is the adaptation of the designed controller and the analysis of the effect of the vertical dynamics on the closed-loop system's performance .

Modeling

Throughout this chapter, the following notation convention of the different physical variables is adopted as follows:

𝑋𝑋(𝑇𝑇) = 𝑋𝑋 𝑟𝑟 + 𝑎𝑎(𝑇𝑇), (4.1) where:

-𝑋𝑋(𝑇𝑇) represents the physical magnitude of the variable in the absolute reference;

-𝑋𝑋 𝑟𝑟 defines the reference value (constant) fixed by the stationary regime (equilibrium) around which the study of the dynamic behavior is carried out;

-𝑎𝑎(𝑇𝑇) is the small variations in the physical quantity 𝑋𝑋(𝑇𝑇) around its equilibrium value 𝑋𝑋 𝑟𝑟 .

Thus, all the notations in lower case are considered as variations and all physical variables without capital letter notations are variables whose reference are zero. Finally, the Laplace transform of the variation 𝑎𝑎(𝑇𝑇) is noted as 𝑋𝑋(𝑠𝑠) = 𝑇𝑇𝐹𝐹{𝑎𝑎(𝑇𝑇)}.

Schematization and configuration

As the vehicle is considered moving in a straight line, particularly in this chapter, the transverse and yaw dynamics can be neglected. In addition, the vehicle is assumed to be perfectly symmetrical along the longitudinal axis, thus, a bicycle model could be considered in order to represent the longitudinal dynamics (Figure 4.1a) where needed. At first, the subject focuses only on half of the front axle, that is to say, that we consider a quarter model of the front vehicle (Figure 4.1b).

As the Electro-mechanical braking system (EMB) is considered in this study, Figure 4.2 shows an overview and the block diagram of the EMB actuator. As a brand-new form of braking, it completely abandons the traditional hydraulic braking system. Instead, it is powered by electric energy. It uses wires to transfer energy and data cables to transfer signals. It drives brake pads to clamp the friction plate by motor and related enforce mechanism, ultimately stopping the wheel. EMB has become the developing direction of the braking system due to its' high-efficiency, high braking performance, being environmentally friendly and easily integrated. The EMB system is composed of a controller and an actuator. As one of the key parts of the EMB system, the performance of its actuator influences the brake of the vehicle directly [START_REF] Zhuo | The Design and Evaluation of EMB Actuator Scheme The Establishment of Parametric 3D Model[END_REF].

Although there are numerous EMB actuator schemes, they are composed of some basic models: drive module, enforce module, motion conversion module and sensor module. The Drive module, as the core of EMB, means the drive motor. Recently, Brush DC Motor (BDC), Brushless DC Motor (BLDC), Torque Motor and Switched Reluctance Motor (SR) are widely used. Enforce module includes reducer and other enforcement mechanisms. The commonly used reducers are cylindrical gear reducer, bevel gear reducer, planetary gear reducer, and worm reducer. The main enforce mechanisms are wedge mechanism, hinge mechanism, lever mechanism, cam mechanism, and crank slider mechanism. Motion conversion module plays the role of converting the rotational motion of drive motor to transitional motion, its main implementations are spiral drive mechanism, rack and pinion mechanism, crank slider mechanism and cam mechanism. The sensor module is mainly used in the control of the EMB actuator and it is required to provide higher sensitivity, as well as a lower price. In the classic three closed-loop control system, clamping force, motor speed, and motor current signal are essential. Correspondingly, a pressure sensor, a speed sensor, and a current sensor are required [START_REF] Zhuo | The Design and Evaluation of EMB Actuator Scheme The Establishment of Parametric 3D Model[END_REF].

Dynamic equations

The equation describing the longitudinal dynamics of the quarter vehicle model using Bond-Graph approach is given by:

𝑉𝑉 𝑥𝑥 (𝑇𝑇) = 1 𝑚𝑚 � 𝐹𝐹 𝛴𝛴 (𝜏𝜏 ) 𝐺𝐺𝜏𝜏 𝑡𝑡 0 + 𝑉𝑉 𝑥𝑥 (0), (4.2)
where the algebraic sum of the external forces 𝐹𝐹 𝛴𝛴 (𝑇𝑇) applied to the mass 𝑚𝑚 = 𝑚𝑚 1 + 𝑚𝑚 2 (𝑚𝑚 total mass of the quarter of the vehicle considered, 𝑚𝑚 1 unsprung mass and 𝑚𝑚 2 sprung mass) is given by:

𝐹𝐹 𝛴𝛴 (𝑇𝑇) = 𝐹𝐹 𝑥𝑥 (𝑇𝑇) -𝑓𝑓 0𝑥𝑥 (𝑇𝑇) -𝐹𝐹 𝑎𝑎 (𝑉𝑉 𝑥𝑥 ) -𝐹𝐹 𝑟𝑟𝑟𝑟 (𝑉𝑉 𝑥𝑥 ), (4.3)
with 𝐹𝐹 𝑥𝑥 (𝑇𝑇) is the longitudinal force developed by the tire, 𝑓𝑓 0𝑥𝑥 (𝑇𝑇) is the longitudinal gust of wind (variation) modeled by a force applied to the center of gravity 𝐶𝐶𝐺𝐺, and 𝐹𝐹 𝑎𝑎 (𝑉𝑉 𝑥𝑥 ) is the resistance aerodynamics whose expression is given by: with 𝜌𝜌 is the air density, 𝐶𝐶 𝑥𝑥 represents the projected area of a quarter of the vehicle in the longitudinal direction, and 𝐶𝐶 𝑥𝑥 corresponds to the aerodynamic drag coefficient.

𝐹𝐹 𝑎𝑎 (𝑉𝑉 𝑥𝑥 ) = 1 2 𝜌𝜌 𝐶𝐶 𝑥𝑥 𝐶𝐶 𝑥𝑥 𝑉𝑉 𝑥𝑥 2 (𝑇𝑇), (4.4)
Moreover, 𝐹𝐹 𝑟𝑟𝑟𝑟 (𝑉𝑉 𝑥𝑥 ) is the result of the rolling resistance expressed at the center of gravity 𝐶𝐶𝐺𝐺 of the quarter vehicle, and is expressed by:

𝐹𝐹 𝑟𝑟𝑟𝑟 (𝑉𝑉 𝑥𝑥 ) = �𝜆𝜆 0 + 𝜆𝜆 2 𝑉𝑉 𝑥𝑥 2 (𝑇𝑇)� 𝑚𝑚 𝑔𝑔, (4.5)
where 𝜆𝜆 0 and 𝜆𝜆 2 are two positive constants.

In the same manner, the dynamics of rotation of the wheel is described by:

𝛺𝛺(𝑇𝑇) = 1 𝐽𝐽 𝑟𝑟 � 𝐶𝐶 𝛴𝛴 (𝜏𝜏 ) 𝐺𝐺𝜏𝜏 + 𝛺𝛺(0) 𝑡𝑡 0 , (4.6)
where the algebraic sum of the external couples 𝐶𝐶 𝛴𝛴 (𝑇𝑇) applied to the inertia 𝐽𝐽 𝑟𝑟 is given by: 𝐶𝐶 𝛴𝛴 (𝑇𝑇) = 𝐶𝐶 𝑟𝑟 (𝑇𝑇) -𝑀𝑀 0 𝐹𝐹 𝑥𝑥 (𝑇𝑇) -𝐶𝐶 𝑓𝑓𝑓𝑓 (𝑇𝑇),

with 𝐽𝐽 𝑟𝑟 is the inertia of the wheel, 𝛺𝛺(𝑇𝑇) represents the absolute rotation speed of the wheel, 𝑀𝑀 0 is its radius under load, 𝐶𝐶 𝑟𝑟 (𝑇𝑇) represents the torque applied to the wheel by the traction motor (positive torque) or by the braking actuator (torque negative), and 𝐶𝐶 𝑓𝑓𝑓𝑓 (𝑇𝑇) is the resulting viscous friction torque given by:

𝐶𝐶 𝑓𝑓𝑓𝑓 (𝑇𝑇) = 𝑏𝑏 𝑟𝑟 𝛺𝛺(𝑇𝑇), (4.8)
where 𝑏𝑏 𝑟𝑟 is a viscous coefficient of friction.

In this study, the tire model used is the Pacejka model [START_REF] Pacejka | Basic Tire Modeling Considerations[END_REF]. The inputs and outputs for this model are described in Figure 4.3, in which the inputs are:

-The wheel vertical normal force 𝐹𝐹 𝑧𝑧 (𝑇𝑇), -the sideslip wheel angle 𝛿𝛿 𝑝𝑝 (𝑇𝑇), which is considered null in this study (straight-line motion), -the road adhesion 𝜇𝜇, -the wheel slip rate 𝑇𝑇�(𝑉𝑉𝑎𝑎(𝑇𝑇), �Ω(𝑇𝑇)).

More precisely, the relationship between the wheel slip rate 𝑇𝑇 (𝑇𝑇) and the longitudinal wheel forces (also called the magic formula in literature), is given by: 𝐹𝐹 𝑥𝑥 (𝑇𝑇) = 𝜇𝜇𝑃𝑃 𝑥𝑥 sin �𝐶𝐶 𝑥𝑥 arctan �𝐶𝐶 𝑥𝑥 �(1 -𝐸𝐸 𝑥𝑥 ) 𝛵𝛵 (𝑇𝑇) + 𝐸𝐸 𝑥𝑥 𝐶𝐶 𝑥𝑥 arctan�𝐶𝐶 𝑥𝑥 𝛵𝛵 (𝑇𝑇)����, (4.9)

where the parameters 𝐶𝐶 𝑥𝑥 , 𝐶𝐶 𝑥𝑥 , 𝑃𝑃 𝑥𝑥 , and 𝐸𝐸 𝑥𝑥 depend on the characteristics of the tire used and on the value of the normal force 𝐹𝐹 𝑧𝑧 (𝑇𝑇).

In addition, the road adhesion 𝜇𝜇 is considered to be an uncertain parameter that could vary between 0.1 (icy-ground) and 1 (dry-ground). In the same way, the static component of the normal force is uncertain because it is equal to the weight distributed on each wheel, the uncertainty is linked to the mass 𝑚𝑚 = 𝑚𝑚 1 + 𝑚𝑚 2 varying between unloaded and loaded vehicle, i.e.:

𝐹𝐹 𝑧𝑧 𝑟𝑟 = 𝑚𝑚 𝑔𝑔, (4.10)

The expression of the slip rate 𝑇𝑇�(𝑉𝑉𝑎𝑎(𝑇𝑇), �Ω(𝑇𝑇)) depends on the acceleration (𝑇𝑇 𝑎𝑎𝑐𝑐𝑐𝑐 ��𝑉𝑉𝑎𝑎(𝑇𝑇), �Ω(𝑇𝑇)�) or braking (𝑇𝑇 𝑓𝑓𝑟𝑟𝑟𝑟 ��𝑉𝑉𝑎𝑎(𝑇𝑇), �Ω(𝑇𝑇)�), as:

𝑇𝑇 (𝑇𝑇) = ⎩ � � ⎨ � � ⎧ 𝛺𝛺(𝑇𝑇)𝑀𝑀 0 -𝑉𝑉 𝑥𝑥 (𝑇𝑇) ≥ 0, then 𝑇𝑇 𝑎𝑎𝑐𝑐𝑐𝑐 (t) = 𝛺𝛺(𝑇𝑇) 𝑀𝑀 0 -𝑉𝑉 𝑥𝑥 (𝑇𝑇) 𝛺𝛺(𝑇𝑇) 𝑀𝑀 0 ≥ 0 𝛺𝛺(𝑇𝑇)𝑀𝑀 0 -𝑉𝑉 𝑥𝑥 (𝑇𝑇) < 0, then 𝑇𝑇 𝑓𝑓𝑟𝑟𝑟𝑟 (t) = 𝛺𝛺(𝑇𝑇) 𝑀𝑀 0 -𝑉𝑉 𝑥𝑥 (𝑇𝑇) 𝑉𝑉 𝑥𝑥 (𝑇𝑇) < 0 . (4.11)
Finally, in the framework of this study, the dynamics of the EMB actuator is considered negligible, then the relationship between the control voltage input 𝑢𝑢 𝑐𝑐 (𝑇𝑇) and the braking control torque 𝑊𝑊 𝑢𝑢 (t) applied to the wheel is formulated as follows:

𝑊𝑊 𝑢𝑢 (𝑇𝑇) = 𝐾𝐾 𝑢𝑢 𝑐𝑐 (𝑇𝑇) , (4.12)
where 𝐾𝐾 = 125 Nm/V and max|𝑈𝑈 𝑐𝑐 | = 12V.

Model Linearization

In accordance with the remark made at the beginning of section 4.2, the following notations are used:

⎩ � � � � � � � ⎨ � � � � � � � ⎧ 𝑉𝑉

Tire linearization

The first hypothesis of the linearization process supposed that the equilibrium point leads to a slip rate 𝑇𝑇 𝑟𝑟 belonging to the interval [ -10%, +10%], that is related to the linear operating zone of the tire (see Figure 4.4). In this case, the longitudinal force 𝐹𝐹 𝑥𝑥 (𝑇𝑇) is written in the form:

𝐹𝐹 𝑥𝑥 (𝑇𝑇) = 𝐹𝐹 𝑥𝑥 𝑟𝑟 + 𝑓𝑓 𝑥𝑥 (𝑇𝑇), (4.14)
with

𝐹𝐹 𝑥𝑥 𝑟𝑟 = 𝑝𝑝 𝑥𝑥 𝑇𝑇 𝑟𝑟 , (4.15)
and 𝑓𝑓 𝑥𝑥 (𝑇𝑇) = 𝑝𝑝 𝑥𝑥 𝜏𝜏(𝑇𝑇), (4.16) where, 𝑝𝑝 𝑥𝑥 is the slope of the tangents to the curve (𝑇𝑇 (𝑇𝑇), 𝐹𝐹 𝑥𝑥 (𝑇𝑇)), such that: (4.17)

𝑝𝑝 𝑥𝑥 = 𝜕𝜕𝐹𝐹 𝑥𝑥 𝜕𝜕𝛵𝛵 � 𝑇𝑇 =𝑇𝑇 𝑒𝑒 = 𝜇𝜇 [𝐶𝐶𝐶𝐶𝑃𝑃] 𝑥𝑥 �1 -𝐸𝐸 𝑥𝑥 + 𝐸𝐸 𝑥𝑥 1 + 𝐶𝐶 𝑥𝑥 2 𝑇𝑇 𝑟𝑟2 � 1 + 𝐶𝐶 𝑥𝑥 2 ⋅
However, in the indicated linear region of operation, the values of 𝑝𝑝 𝑥𝑥 is formulated as follows:

𝑝𝑝 𝑥𝑥 = 𝜇𝜇 [𝐶𝐶𝐶𝐶𝑃𝑃] 𝑥𝑥 . (4.18)
The value of [𝐶𝐶𝐶𝐶𝑃𝑃] 𝑥𝑥 depends on different parameters and characteristics of the tires and is studied in detail in later sections.

Regarding the linearization for the slip rate variable, it is important to distinguish between the two phases, acceleration and deceleration. For acceleration, we have: 𝑇𝑇 𝑎𝑎𝑐𝑐𝑐𝑐 �𝑉𝑉 𝑥𝑥 (𝑇𝑇), 𝛺𝛺(𝑇𝑇)� = 𝑇𝑇 𝑎𝑎𝑐𝑐𝑐𝑐 𝑟𝑟 + 𝛼𝛼 𝑎𝑎𝑐𝑐𝑐𝑐 𝜔𝜔(𝑇𝑇) + 𝜇𝜇 𝑎𝑎𝑐𝑐𝑐𝑐 𝑣𝑣 𝑥𝑥 (𝑇𝑇)

(4.19) with, ⎩ � � � � ⎨ � � � � ⎧ 𝑇𝑇 𝑎𝑎𝑐𝑐𝑐𝑐 𝑟𝑟 = 1 - 𝑉𝑉 𝑥𝑥 𝑟𝑟 𝑀𝑀 0 Ω 𝑟𝑟 ≥ 0 𝛼𝛼 𝑎𝑎𝑐𝑐𝑐𝑐 = 𝜕𝜕𝑇𝑇 𝑎𝑎𝑐𝑐𝑐𝑐 𝜕𝜕𝛺𝛺 � 𝑉𝑉 𝑚𝑚 =𝑉𝑉 𝑚𝑚 𝑒𝑒 𝛺𝛺=Ω 𝑒𝑒 = 𝑉𝑉 𝑥𝑥 𝑟𝑟 𝑀𝑀 0 Ω 𝑟𝑟 2 = 𝑊𝑊𝑠𝑠𝑇𝑇𝑟𝑟 > 0 𝜇𝜇 𝑎𝑎𝑐𝑐𝑐𝑐 = 𝜕𝜕𝑇𝑇 𝑎𝑎𝑐𝑐𝑐𝑐 𝜕𝜕𝑉𝑉 𝑥𝑥 � 𝑉𝑉 𝑚𝑚 =𝑉𝑉 𝑚𝑚 𝑒𝑒 𝛺𝛺=Ω 𝑒𝑒 = - 1 𝑀𝑀 0 Ω 𝑟𝑟 = 𝑊𝑊𝑠𝑠𝑇𝑇𝑟𝑟 < 0 , (4.20)
where for the braking phases, we have:

𝑇𝑇 𝑓𝑓𝑟𝑟𝑟𝑟 �𝑉𝑉 𝑥𝑥 (𝑇𝑇), 𝛺𝛺(𝑇𝑇)� = 𝑇𝑇 𝑓𝑓𝑟𝑟𝑟𝑟 𝑟𝑟 + 𝛼𝛼 𝑓𝑓𝑟𝑟𝑟𝑟 𝜔𝜔(𝑇𝑇) + 𝜇𝜇 𝑓𝑓𝑟𝑟𝑟𝑟 𝑣𝑣 𝑥𝑥 (𝑇𝑇), (4.21)
with,

⎩ � � � � ⎨ � � � � ⎧ 𝑇𝑇 𝑓𝑓𝑟𝑟𝑟𝑟 𝑟𝑟 = 𝑀𝑀 0 Ω 𝑟𝑟 𝑉𝑉 𝑥𝑥 𝑟𝑟 -1 < 0 𝛼𝛼 𝑓𝑓𝑟𝑟𝑟𝑟 = 𝜕𝜕𝑇𝑇 𝑓𝑓𝑟𝑟𝑟𝑟 𝜕𝜕𝛺𝛺 � 𝑉𝑉 𝑚𝑚 =𝑉𝑉 𝑚𝑚 𝑒𝑒 𝛺𝛺=Ω 𝑒𝑒 = 𝑀𝑀 0 𝑉𝑉 𝑥𝑥 𝑟𝑟 = 𝑊𝑊𝑠𝑠𝑇𝑇𝑟𝑟 > 0 𝜇𝜇 𝑓𝑓𝑟𝑟𝑟𝑟 = 𝜕𝜕𝑇𝑇 𝑓𝑓𝑟𝑟𝑟𝑟 𝜕𝜕𝑉𝑉 𝑥𝑥 � 𝑉𝑉 𝑚𝑚 =𝑉𝑉 𝑚𝑚 𝑒𝑒 𝛺𝛺=Ω 𝑒𝑒 = - 𝑀𝑀 0 Ω 𝑟𝑟 𝑉𝑉 𝑥𝑥 𝑟𝑟 2 = 𝑊𝑊𝑠𝑠𝑇𝑇𝑟𝑟 < 0 . (4.22)
In resume, after linearization, the longitudinal force 𝐹𝐹 𝑥𝑥 (𝑇𝑇) is written as follows:

-in acceleration:

⎩ � ⎨ � ⎧ 𝐹𝐹 𝑥𝑥 (𝑇𝑇) = 𝐹𝐹 𝑥𝑥 𝑟𝑟 + 𝑓𝑓 𝑥𝑥 (𝑇𝑇) 𝐹𝐹 𝑥𝑥 𝑟𝑟 = 𝑝𝑝 𝑥𝑥 𝑇𝑇 𝑟𝑟 𝑓𝑓 𝑥𝑥 (𝑇𝑇) = 𝑝𝑝 𝑥𝑥 𝛼𝛼 𝑎𝑎𝑐𝑐𝑐𝑐 𝜔𝜔(𝑇𝑇) + 𝑝𝑝 𝑥𝑥 𝜇𝜇 𝑎𝑎𝑐𝑐𝑐𝑐 𝑣𝑣 𝑥𝑥 (𝑇𝑇) , (4.23)
-in braking:

⎩ � ⎨ � ⎧ 𝐹𝐹 𝑥𝑥 (𝑇𝑇) = 𝐹𝐹 𝑥𝑥 𝑟𝑟 + 𝑓𝑓 𝑥𝑥 (𝑇𝑇) 𝐹𝐹 𝑥𝑥 𝑟𝑟 = 𝑝𝑝 𝑥𝑥 𝑇𝑇 𝑟𝑟 𝑓𝑓 𝑥𝑥 (𝑇𝑇) = 𝑝𝑝 𝑥𝑥 𝛼𝛼 𝑓𝑓𝑟𝑟𝑟𝑟 𝜔𝜔(𝑇𝑇) + 𝑝𝑝 𝑥𝑥 𝜇𝜇 𝑓𝑓𝑟𝑟𝑟𝑟 𝑣𝑣 𝑥𝑥 (𝑇𝑇) . (4.24)

Linearization of the aerodynamic resistance

The linearization of the expression (4.4) related to the aerodynamic resistance force 𝐹𝐹 𝑎𝑎 (𝑉𝑉 𝑥𝑥 ) can be formulated as follows:

𝐹𝐹 𝑎𝑎 �𝑉𝑉 𝑥𝑥 (𝑇𝑇)� = 𝐹𝐹 𝑎𝑎 𝑟𝑟 + 𝑓𝑓 𝑎𝑎 �𝑣𝑣 𝑥𝑥 (𝑇𝑇)�, (4.25)
where, 𝐹𝐹 𝑎𝑎 𝑟𝑟 = 𝐹𝐹 𝑎𝑎 (𝑉𝑉 𝑥𝑥 𝑟𝑟 ), (4.26) and, 𝑓𝑓 𝑎𝑎 �𝑣𝑣 𝑥𝑥 (𝑇𝑇)� = 𝑏𝑏 𝑎𝑎 𝑣𝑣 𝑥𝑥 (𝑇𝑇), (4.27) with,

⎩ � � � ⎨ � � � ⎧ 𝐹𝐹 𝑎𝑎 𝑟𝑟 = 1 2 𝜌𝜌 𝐶𝐶 𝑥𝑥 𝐶𝐶 𝑥𝑥 𝑉𝑉 𝑥𝑥 𝑟𝑟2 = 𝑊𝑊𝑠𝑠𝑇𝑇𝑟𝑟 𝑏𝑏 𝑎𝑎 = 𝜕𝜕𝐹𝐹 𝑎𝑎 𝜕𝜕𝑉𝑉 𝑥𝑥 � 𝑉𝑉 𝑚𝑚 =𝑉𝑉 𝑚𝑚 𝑒𝑒 = 𝜌𝜌 𝐶𝐶 𝑥𝑥 𝐶𝐶 𝑥𝑥 𝑉𝑉 𝑥𝑥 𝑟𝑟 = 𝑊𝑊𝑠𝑠𝑇𝑇𝑟𝑟 . (4.28)
The parameter 𝑏𝑏 𝑎𝑎 here is related to the viscous aerodynamic friction coefficient (in Ns/m) that depends mainly on the speed of the vehicle 𝑉𝑉 𝑥𝑥 𝑟𝑟 = 𝑉𝑉 𝑥𝑥0

(in m/s).

Linearization of rolling resistance force

In the same manner, the linearization of the expression (4.5) related to the rolling resistance force 𝐹𝐹 𝑟𝑟𝑟𝑟 (𝑉𝑉 𝑥𝑥 ) can be formulated as follows:

𝐹𝐹 𝑟𝑟𝑟𝑟 �𝑉𝑉 𝑥𝑥 (𝑇𝑇)� = 𝐹𝐹 𝑟𝑟𝑟𝑟 𝑟𝑟 + 𝑓𝑓 𝑟𝑟𝑟𝑟 �𝑣𝑣 𝑥𝑥 (𝑇𝑇)�, (4.29)
where, 𝐹𝐹 𝑟𝑟𝑟𝑟 𝑟𝑟 = 𝐹𝐹 𝑟𝑟𝑟𝑟 (𝑉𝑉 𝑥𝑥 𝑟𝑟 ), (4.30) and, 𝑓𝑓 𝑟𝑟𝑟𝑟 �𝑣𝑣 𝑥𝑥 (𝑇𝑇)� = 𝑏𝑏 𝑟𝑟𝑟𝑟 𝑣𝑣 𝑥𝑥 (𝑇𝑇), (4.31) with,

⎩ � � ⎨ � � ⎧𝐹𝐹 𝑟𝑟𝑟𝑟 𝑟𝑟 = �𝜆𝜆 0 + 𝜆𝜆 2 𝑉𝑉 𝑥𝑥 𝑟𝑟2 � 𝑚𝑚 𝑔𝑔 = 𝑊𝑊𝑠𝑠𝑇𝑇𝑟𝑟 𝑏𝑏 𝑟𝑟𝑟𝑟 = 𝜕𝜕𝐹𝐹 𝑟𝑟 𝜕𝜕𝑉𝑉 𝑥𝑥 � 𝑉𝑉 𝑚𝑚 =𝑉𝑉 𝑚𝑚 𝑒𝑒 = 2 𝜆𝜆 2 𝑚𝑚 𝑔𝑔 𝑉𝑉 𝑥𝑥 𝑟𝑟 = 𝑊𝑊𝑠𝑠𝑇𝑇𝑟𝑟 , (4.32)
where, 𝑏𝑏 𝑟𝑟𝑟𝑟 is coherent with a viscous coefficient of friction (Ns/m) depending on the speed of the vehicle 𝑉𝑉 𝑥𝑥 𝑟𝑟 = 𝑉𝑉 𝑥𝑥0 (m/s) and the mass 𝑚𝑚 of the quarter vehicle.

Equilibrium values and initial conditions

Considering the assumption of small variations and taking into account the previous equations development, (4.2) and (4.6) are rewritten in the following form, respectively:

𝑚𝑚 𝑣𝑣̇𝑥𝑥(𝑇𝑇) = 𝐹𝐹 𝑥𝑥 𝑟𝑟 +𝑓𝑓 𝑥𝑥 (𝑇𝑇) -𝑓𝑓 0𝑥𝑥 (𝑇𝑇) -𝐹𝐹 𝑎𝑎 𝑟𝑟 -𝑓𝑓 𝑎𝑎 (𝑇𝑇) -𝐹𝐹 𝑟𝑟𝑟𝑟 𝑟𝑟 -𝑓𝑓 𝑟𝑟𝑟𝑟 (𝑇𝑇), (4.33)
𝐽𝐽 𝑟𝑟 𝜔𝜔(𝑇𝑇) = 𝐶𝐶 𝑚𝑚 𝑟𝑟 + 𝑊𝑊 𝑢𝑢 (𝑇𝑇) -𝑀𝑀 0 𝐹𝐹 𝑥𝑥 𝑟𝑟 -𝑀𝑀 0 𝑓𝑓 𝑥𝑥 (𝑇𝑇) -𝐶𝐶 𝑓𝑓𝑓𝑓 𝑟𝑟 -𝑊𝑊 𝑓𝑓𝑓𝑓 (𝑇𝑇). The numerical expression of each variable at the equilibrium point is done in accordance with the different steps presented using the block diagram of Figure 4.5 where the starting point is from the selection of the speed 𝑉𝑉 𝑥𝑥0 . Thus, the magnitude of which the expression is determined in Step N-1 is given to a step N to determine the following expression, etc. Therefore, we get the following ordered steps:

⎩ � � � � � � � � � � ⎨ � � � � � � � � � � ⎧ 0 -𝑉𝑉 𝑥𝑥 𝑟𝑟 = 𝑉𝑉 𝑥𝑥0 [ 𝑚𝑚/𝑠𝑠] 1 -𝐹𝐹 𝑎𝑎 𝑟𝑟 = 1 2 𝜌𝜌 𝐶𝐶 𝑥𝑥 𝐶𝐶 𝑥𝑥 𝑉𝑉 𝑥𝑥 𝑟𝑟2 , and 𝐹𝐹 𝑟𝑟𝑟𝑟 𝑟𝑟 = �𝜆𝜆 0 + 𝜆𝜆 2 𝑉𝑉 𝑥𝑥 𝑟𝑟2 � 𝑚𝑚𝑔𝑔 [𝑁𝑁] 2 -𝐹𝐹 𝑥𝑥 𝑟𝑟 = 𝐹𝐹 𝑎𝑎 𝑟𝑟 + 𝐹𝐹 𝑟𝑟𝑟𝑟 𝑟𝑟 [ 𝑁𝑁] 3 -𝑇𝑇 𝑎𝑎𝑐𝑐𝑐𝑐 𝑟𝑟 = 1 [𝐶𝐶𝐶𝐶𝑃𝑃] 𝑥𝑥 𝐹𝐹 𝑥𝑥 𝑟𝑟 [ %] 4 -𝛺𝛺 𝑟𝑟 = 1 𝑀𝑀 0 � 𝑉𝑉 𝑥𝑥 𝑟𝑟 1 -(𝑇𝑇 𝑎𝑎𝑐𝑐𝑐𝑐 𝑟𝑟 100 ⁄ ) � [𝑀𝑀𝑎𝑎𝐺𝐺/𝑠𝑠] 5 -𝐶𝐶 𝑚𝑚 𝑟𝑟 = 𝑀𝑀 0 𝐹𝐹 𝑥𝑥 𝑟𝑟 + 𝑏𝑏 𝑟𝑟 𝛺𝛺 𝑟𝑟 [ 𝑁𝑁𝑚𝑚]
.

(4.36)

Synthesis model

The designed controller is based on the frequency response of the plant; therefore, a linearized model of the plant dynamics is required. The linearization process during the braking phase around an operating point is adopted and it could be defined by the following state-space representation:

� 𝑎𝑎 = 𝐶𝐶 𝑎𝑎 ¯ + 𝐶𝐶 𝑢𝑢 𝑎𝑎 = 𝐶𝐶 𝑎𝑎 ¯ + 𝑃𝑃 𝑢𝑢 , (4.37)
with,

𝑎𝑎 = � 𝑎𝑎 1 = 𝑣𝑣 𝑥𝑥 (𝑇𝑇) 𝑎𝑎 2 = 𝜔𝜔(𝑇𝑇) � , 𝑢𝑢 = � 𝑓𝑓 0𝑥𝑥 𝑢𝑢 𝑐𝑐 (𝑇𝑇)
� and 𝑎𝑎 = 𝜔𝜔(𝑇𝑇), (4.38) which leads to the transfer function 𝐺𝐺(𝑠𝑠) that is given by:

𝐺𝐺(𝑠𝑠) = 𝛺𝛺(𝑠𝑠) 𝑈𝑈 𝑐𝑐 (𝑠𝑠) � 𝑓𝑓 0𝑚𝑚 =0 𝑏𝑏 𝑚𝑚 =0 , (4.39)
where Ω(𝑠𝑠) = ℒ{𝜔𝜔(𝑇𝑇)} and 𝑈𝑈 𝑐𝑐 (𝑠𝑠) = ℒ{𝑢𝑢 𝑐𝑐 (𝑇𝑇)}, and taking into consideration the parametric uncertainty of the plant, and in the absence of the noise measurements, 𝑏𝑏 𝑚𝑚 = 0, and neglecting the gust wind forces, 𝑓𝑓 𝑥𝑥0 = 0.

The obtained linearized longitudinal dynamics is described by: 𝑚𝑚 𝑣𝑣̇𝑥𝑥(𝑇𝑇) = 𝑓𝑓 𝑥𝑥 (𝑇𝑇) -𝑓𝑓 𝑎𝑎 (𝑇𝑇) -𝑓𝑓 𝑟𝑟𝑟𝑟 (𝑇𝑇) -𝑓𝑓 𝑥𝑥0 .

(4.40)

Concerning the wheel dynamics, the angular acceleration Ω 𝑟𝑟 ̇ can be computed for a constant slip rate 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 and a given longitudinal velocity 𝑉𝑉 𝑥𝑥 𝑟𝑟 , by the following formula:

Ω ̇(𝑇𝑇) = 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 + 1 𝑀𝑀 0 𝑉𝑉 ̇𝑥𝑥(𝑇𝑇). (4.41)
For braking phases, the wheel traction torque is considered null, 𝐶𝐶 𝑚𝑚 = 0, then the corresponding control effort is computed as:

𝑈𝑈 𝑐𝑐 𝑟𝑟 = - 𝐽𝐽 𝑟𝑟 Ω e -𝑀𝑀 0 𝐹𝐹 𝑥𝑥 𝑟𝑟 -𝑏𝑏 𝑟𝑟 Ω e 𝐾𝐾 . (4.42)
Finally, linearizing (4.6) around a given optimal slip rate 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 , we get:

𝐽𝐽 𝑟𝑟 𝜔𝜔(𝑇𝑇) = -𝐾𝐾 𝑢𝑢 𝑐𝑐 (𝑇𝑇) -𝑀𝑀 0 𝑝𝑝 𝑥𝑥 𝜏𝜏(𝑇𝑇) -𝑏𝑏 𝑟𝑟 𝜔𝜔(𝑇𝑇). (4.43)
Based on (4.41) and (4.43), the state-space matrix's parameters are obtained as follows:

𝐶𝐶 = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ - 𝛺𝛺 𝑟𝑟 𝑀𝑀 0 𝑝𝑝 𝑥𝑥 𝑉𝑉 𝑥𝑥 𝑟𝑟2 + 𝜌𝜌𝐶𝐶 𝑥𝑥 𝐶𝐶 𝑥𝑥 𝑉𝑉 𝑥𝑥 𝑟𝑟 2 + 2𝑉𝑉 𝑥𝑥 𝑟𝑟 𝑔𝑔𝜆𝜆 2 𝑚𝑚 𝑚𝑚 𝑀𝑀 0 𝑝𝑝 𝑥𝑥 𝑉𝑉 𝑥𝑥 𝑟𝑟 𝑚𝑚 𝛺𝛺 𝑟𝑟 𝑀𝑀 0 2 𝑝𝑝 𝑥𝑥 𝐽𝐽 𝑟𝑟 𝑉𝑉 𝑥𝑥 𝑟𝑟2 - 𝑏𝑏 𝑟𝑟 + 𝑀𝑀 0 2 𝑝𝑝 𝑥𝑥 𝑉𝑉 𝑥𝑥 𝑟𝑟 𝐽𝐽 𝑟𝑟 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ , (4.44)
𝐶𝐶 = ⎣ ⎢ ⎢ ⎡ - 1 𝑚𝑚 0 0 - 𝐾𝐾 𝐽𝐽 𝑟𝑟 ⎦ ⎥ ⎥ ⎤ , 𝐶𝐶 = [0 1],
and 𝑃𝑃 = 0.

The transfer function 𝐺𝐺(𝑠𝑠) can be thus obtained as follows:

𝐺𝐺(𝑠𝑠) = -𝐾𝐾 � 𝑠𝑠 𝐽𝐽 𝑟𝑟 + 2𝛺𝛺 𝑟𝑟 𝑀𝑀 0 𝑝𝑝 𝑥𝑥 + 𝜌𝜌𝐶𝐶 𝑥𝑥 𝐶𝐶 𝑥𝑥 𝑉𝑉 𝑥𝑥 𝑟𝑟3 + 4𝑉𝑉 𝑥𝑥 𝑟𝑟3 𝑔𝑔𝜆𝜆 2 𝑚𝑚 2𝐽𝐽 𝑟𝑟 𝑉𝑉 𝑥𝑥 𝑟𝑟2 𝑚𝑚 � 𝑠𝑠 2 + 𝑝𝑝𝑠𝑠 + 𝑞𝑞 , ( 4

Modeling of the reference

The generation of the reference inputs to the control process is composed of a pre-compensation part which is defined by the voltage control input reference 𝑢𝑢 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) and the generation of the reference value of the angular velocities of the wheel 𝜔𝜔 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇).

In fact, the variation of the values of 𝑢𝑢 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) is the image of the driver's desired braking torques. Thus, from the measured values 𝑎𝑎 𝑝𝑝 (𝑇𝑇) of the position of the braking pedal (from 0 to 12cm), the reference voltage can be calculated (from 0 to 12V), as follows:

𝑢𝑢 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) = 𝐾𝐾 𝑝𝑝 𝑎𝑎 𝑝𝑝 (𝑇𝑇), (4.48)
where 𝐾𝐾 𝑝𝑝 = 100V/m.

On the other hand, the values of the reference of the wheel desired angular velocity 𝜔𝜔 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) could be computed in the following manner: the estimated value of the wheel slip rate can be defined as the following formula depending on the estimated or a measured value of the wheel angular and longitudinal velocity, as follows:

𝑇𝑇 ̂𝑓𝑓𝑟𝑟𝑟𝑟 (𝑇𝑇) = 𝛺𝛺 ̂(𝑇𝑇) 𝑀𝑀 0 𝑉𝑉 ̂𝑥𝑥(𝑇𝑇) -1 < 0 ⇔ 𝛺𝛺 ̂(𝑇𝑇) 𝑀𝑀 0 < 𝑉𝑉 ̂𝑥𝑥(𝑇𝑇). (4.49)
Here, two cases are defined in order to obtain the reference value of the angular wheel velocity:

-If 𝑇𝑇 ̂𝑓𝑓𝑟𝑟𝑟𝑟 (𝑇𝑇) > 𝑇𝑇 ̂𝑛𝑛𝑝𝑝𝑡𝑡 , then 𝜔𝜔 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) is taken as:

𝛺𝛺 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) = 𝛺𝛺 ̂(𝑇𝑇). (4.50)
where 𝑇𝑇 ̂𝑛𝑛𝑝𝑝𝑡𝑡 is selected according to the tire response curve, and represents the optimal slip rate, e.g. 8%, 10%. This value generally depends on the tire characteristics and the values of the available grip/adhesion.

-Otherwise, if 𝑇𝑇 ̂𝑓𝑓𝑟𝑟𝑟𝑟 (𝑇𝑇) ≤ 𝑇𝑇 ̂𝑛𝑛𝑝𝑝𝑡𝑡 , then:

𝛺𝛺 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) = �1 + 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 � 𝑉𝑉 ̂𝑥𝑥(𝑇𝑇) 𝑀𝑀 0 . (4.51)
Control system design

Strategy

The control scheme for the ABS is presented in Figure 4.6. The block Regulator indicates the designed controller (CRONE). The signal 𝑏𝑏 𝑚𝑚 (𝑇𝑇) represents the noise on the measurement of the wheel angular velocity, Ω(𝑇𝑇). The term 𝑈𝑈 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑠𝑠) represents the reference voltage that is proportional to the displacement of the braking pedal as illustrated in (4.48). The values Ω 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑠𝑠) are generated in the Generation of references block as explained in (4.50) and (4.51). The block Vehicle, particularly, in this section represents the quarter vehicle validation model.

The ABS strategy is consisting of two main phases. At regular braking conditions, in which no critical situation or sliding process may occur, the ABS is deactivated (inactive), and therefore Ω 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) = Ω 𝑚𝑚𝑟𝑟𝑠𝑠 (𝑇𝑇). In this case, the error is null and the feedback controller has no effect at all, that is 𝑈𝑈 𝑓𝑓𝑏𝑏 = 0. Thus, the only input to the plant, in this case, is the feedforward part, 𝑈𝑈 𝑓𝑓𝑓𝑓 (𝑇𝑇).

The ABS system becomes active when there is a risk of wheel locking. This situation, practically, is detected when the estimated slip rate 𝑇𝑇 (𝑇𝑇) < 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 (braking phase). Then, the reference signal is given as:

Ω 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇) = �1 + 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 �𝑉𝑉 𝑥𝑥 (𝑇𝑇) 𝑀𝑀 0 , (4.52)
where it differs from the Ω 𝑚𝑚𝑟𝑟𝑠𝑠 (𝑇𝑇), and therefore the error is no longer zero, and the feedback loop starts to counteracts the feedforward signal provided by the driver.

In order to deactivate the ABS, two conditions must be met simultaneously:

-𝑇𝑇 (𝑇𝑇) ≥ 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 , which ensures a safe situation in which the tire forces are not saturated anymore.

-A decrease in the value of 𝑎𝑎 𝑝𝑝 (𝑇𝑇), which indicates the driver's intention to reduce or bring to an end of the braking action (braking pedal release).

It should be noted that some previous works found in the literature [START_REF] Nyandoro | Linear slip control formulation for vehicular anti-lock braking system with suspension effects[END_REF][START_REF] Tang | Fractional order sliding mode controller design for antilock braking systems[END_REF] prefer to formulate the control strategy as a regulation of the slip rate dynamics. However, one major drawback of such strategies is in switching in the controller structure derived from the ABS active or inactive modes. Hence, it requires treating the problem with hybrid-system tools to ensure stability and performance under these commutations. The presented strategy proposed by (Benine-Neto et al., 2017) consists of tracking a reference wheel speed. The activation of the ABS changes the reference-generator model, thus avoiding commutations on the feedback loop, and providing the same results as a slip rate regulation. The overall strategy for ABS is illustrated in Figure 4.7.

Control specifications

Several specifications should be taken into account before launching the design process. Regarding the controller, the following specifications are required:

-Control input related to noise measurement is less than 15% of the greatest measurement voltage.

-A Nominal overshoot of step response related to reference signal: about 30%.

As mentioned before, the range of the control input is 𝑈𝑈 𝑐𝑐 (𝑇𝑇) ∈ [0 -12]𝑉𝑉 . The noise level of Ω(𝑇𝑇) corresponds to 2% sensor range, that is max (𝑏𝑏 𝑚𝑚 (𝑇𝑇))=2 rad/s.

In order to translate these specifications into frequency-domain terms, following equations associated with Figure 4.6 are introduced: According to the maximum control effort available and the indicated measurement noise percentage related to it, an upper boundary limit is calculated for the closed-loop cutoff frequency, so-called 𝜔𝜔 𝑐𝑐𝑐𝑐 . Depending on the formulation of the control sensitivity function 𝐶𝐶𝐶𝐶(𝑠𝑠), we get the following:

� 𝑈𝑈 𝐹𝐹𝐵𝐵 (𝑗𝑗𝜔𝜔) 𝑏𝑏 𝑚𝑚 (𝑗𝑗𝜔𝜔) � = |𝐶𝐶𝐶𝐶(𝑗𝑗𝜔𝜔)| ≤ � 0.15 × 12𝑉𝑉 2𝑀𝑀𝑎𝑎𝐺𝐺/𝑠𝑠 � = 0.89𝑉𝑉𝑠𝑠/𝑀𝑀𝑎𝑎𝐺𝐺. (4.56)
Taking advantage of the fact that at high frequencies 𝜔𝜔 𝐻𝐻𝐹𝐹 , the frequency response of the control sensitivity functions is very close to that of the controller's transfer function, that is:

|𝐶𝐶𝐶𝐶(𝑗𝑗𝜔𝜔 𝐻𝐻𝐹𝐹 )| ≈ |𝐶𝐶(𝑗𝑗𝜔𝜔 𝐻𝐻𝐹𝐹 )| ≤ 0.89𝑉𝑉𝑠𝑠 𝑀𝑀𝑎𝑎𝐺𝐺 , (4.57)
and, at the open-loop gain crossover frequency 𝜔𝜔 𝑐𝑐𝑐𝑐 , we have:

𝛽𝛽(𝑠𝑠) = �𝐶𝐶�𝑗𝑗𝜔𝜔 𝑐𝑐𝑐𝑐 �𝐺𝐺�𝑗𝑗𝜔𝜔 𝑐𝑐𝑐𝑐 �� = 1, (4.58)
then, one can consider a proportional controller such that 𝐶𝐶(𝑠𝑠) = 𝐶𝐶 𝑝𝑝 ≤ 0.89𝑉𝑉𝑠𝑠/𝑀𝑀𝑎𝑎𝐺𝐺, and can compute the upper bound of 𝜔𝜔 𝑐𝑐𝑐𝑐 using (4.58) by:

�𝐺𝐺 𝑛𝑛𝑛𝑛𝑚𝑚 �𝑗𝑗𝜔𝜔 𝑐𝑐𝑐𝑐 �� ≤ 1 𝐶𝐶 𝑝𝑝 = 1.12 𝑀𝑀𝑎𝑎𝐺𝐺/𝑠𝑠𝑉𝑉 . (4.59)
Therefore, depending on the frequency response of the nominal plant proposed, the upper bound limit is computed so that the inequality (4.59) is satisfied.

Finally, regarding the allowed overshoot, it can be represented as the peak gain of the complementary sensitivity function 𝑇𝑇 (𝑠𝑠), which will be denoted as resonant gain 𝑀𝑀 𝑟𝑟0 for the nominal plant. The 𝑀𝑀 𝑟𝑟0 = 3𝐺𝐺𝐶𝐶 corresponds to an overshoot of about 30%.

Plant definition

Three main parameters have the influence on the frequency response of the system transfer function 𝐺𝐺(𝑠𝑠), that are more explicitly: 𝑉𝑉 𝑥𝑥 𝑟𝑟 , that determines the values of Ω 𝑟𝑟 through (4.51); 𝑚𝑚, that appears explicitly on the coefficient of the plant transfer function (4.45); and finally 𝜇𝜇 𝑟𝑟 that affects the value of 𝑝𝑝 𝑥𝑥 according to (4.17) that in turn also appears in (4.45). It is worth noting that these three parameters are obtained after considering the linearized model about a trajectory of constant slip rate, namely 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 . The range of variations of these parameters as assumed in this study are listed in Table 4.1.

In order to analyze the effect of the parametric uncertainty on the frequency response of the controlled plant 𝐺𝐺(𝑠𝑠), a study should be made to find how the variation of road adhesion and vehicle's mass affects the variation of the value of 𝑝𝑝 𝑥𝑥 .

Two main approaches are applicable to calculate the variation of 𝑝𝑝 𝑥𝑥 . The first approach is to assume that 𝑝𝑝 𝑥𝑥 corresponds to the slope of the tangent to the curve (𝑇𝑇 , 𝐹𝐹 𝑥𝑥 ) at the origin. In this case, the values can be computed by the following algorithm for each value of 𝑚𝑚 and 𝜇𝜇 :

𝑝𝑝 𝑥𝑥 = 𝜇𝜇 [𝐶𝐶𝐶𝐶𝑃𝑃] , (4.60)
where [𝐶𝐶𝐶𝐶𝑃𝑃] is an expression related to the Pacejka tire model, and it depends on the micro parameters of the model, and is formulated as:

𝝁𝝁 𝒎𝒎 0.5 -1 - 𝒎𝒎 200 -250 kg [𝐶𝐶𝐶𝐶𝑃𝑃] = (𝑏𝑏3 × 𝑍𝑍 2 + 𝑏𝑏4 × 𝑍𝑍)𝑟𝑟 -𝑏𝑏5×𝑍𝑍 , (4.61)
where b3, b4, and b5 are three models' parameters and are equal to 23.9, 242, and 0, respectively. Here, 𝑍𝑍 represents the vertical wheel contact force 𝐹𝐹 𝑧𝑧 in kN, that is:

𝑍𝑍 = 𝐹𝐹 𝑧𝑧 /1000 = 𝑚𝑚 × 𝑔𝑔/1000, (4.62)
where 𝑔𝑔 is the gravity constant. It should be noted that the value of 𝐹𝐹 𝑧𝑧 here is assumed constant in which it represents the static load of the vehicle.

The second approach to calculate the variation of 𝑝𝑝 𝑥𝑥 is what adopted here in this study. A numerical function is implemented to calculate the slope of the tangent at a predefined value of slip rate, 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 . The function, based on two values of slip rates nearly around 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 , calculates the slope of the tangent drawn between these two points as illustrated in Figure 4.8.

In fact, the range of variation of 𝑝𝑝 𝑥𝑥 is highly depending on the value of 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 . According to the used Pacejka tire model, and after analyzing the different response of 𝐹𝐹 𝑥𝑥 as a function of mass variation and road adhesion variations, the range of uncertainty of 𝑝𝑝 𝑥𝑥 is obtained. Figure 4.9 plots the variation of 𝑝𝑝 𝑥𝑥 as a function of 5000 samples of bounded random values of wheel contact static force 𝐹𝐹 𝑧𝑧 , and road adhesion 𝜇𝜇. The green points represent the calculated values of 𝑝𝑝 𝑥𝑥 at each given sample (𝐹𝐹 𝑧𝑧 , 𝜇𝜇). A random process is used to select random numbers from a predefined range of variations for each parameter. The red contour represents the sorted samples (in terms of magnitude) in which it reveals the overall behavior of the variation of this parameter and indicates the extremums of such variations. The minimum and the maximum value are described by two dashed black lines as shown in Figure 4.9. It should be noted that these results are obtained at the wheel slip rate value of 7%.

However, each given value of slip rate results in a different range of variations of 𝑝𝑝 𝑥𝑥 . For this reason, for study and control design, it is critical to indicate the value of 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 precisely. Table 4.2 lists the variations of the tireforce slope coefficient 𝑝𝑝 𝑥𝑥 as a function of specified values of slip rates. In this study, although the maximum values at which tire-force slope is in the range of 8.5-9%, the optimal value is considered 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 = 8% in order to avoid fast switching between positive and negative values of 𝑝𝑝 𝑥𝑥 . For this reason, the values are chosen just behind the maximum value. As a result, in addition to the specified range of variations for 𝑚𝑚, 𝑉𝑉 𝑥𝑥 𝑟𝑟 , and 𝜇𝜇, the range of variation of 𝑝𝑝 𝑥𝑥 is then [ 1355 -2942].

Knowing that all the parameters of the plant transfer function 𝐺𝐺(𝑠𝑠) are now defined, then the frequency responses of the plant against uncertainties can be studied.

In order to analyze the influence of the parametric uncertainties on the frequency responses of the plant 𝐺𝐺(𝑠𝑠), 50 random plants were computed and plotted as shown in Figure 4.10 (yellow). As a result, three main plants were considered: 𝐺𝐺 𝑛𝑛𝑛𝑛𝑚𝑚 (𝑠𝑠) , 𝐺𝐺 𝑚𝑚𝑖𝑖𝑛𝑛 (𝑠𝑠) , and 𝐺𝐺 𝑚𝑚𝑎𝑎𝑥𝑥 (𝑠𝑠) . Figure 4.10 shows the responses of these plants, where their parametric configurations are detailed in Table 4.3.

CRONE controller

The selection of the generation of CRONE controller, as illustrated in Chapter 1, depends mainly on the magnitude and phase behavior of the controlled nominal plant around the selected crossover frequency. Therefore, the cross over frequency should be assigned.

According to the previous section, the nominal plant's transfer function thus could be obtained and formulated as follows:

𝐺𝐺 𝑛𝑛𝑛𝑛𝑚𝑚 (𝑠𝑠) = 31250𝑠𝑠 + 16610 250𝑠𝑠 2 + 2567𝑠𝑠 + 149.2 . (4.63)
Taking into consideration the specifications of section 4.3.2, then the crossover frequency 𝜔𝜔 𝑐𝑐𝑐𝑐 can be easily now calculated. Based on the frequency response and the control sensitivity specifications, thus the upper bound for the crossover frequency is:

𝜔𝜔 𝑐𝑐𝑐𝑐 ≤ 113 rad/s. (4.64)
Consequently, the third generation must be used since there are high variations of gain and phase of the plant due to parametric uncertainties at the desired open-loop gain crossover frequency 𝜔𝜔 𝑐𝑐𝑐𝑐 , as shown in Figure 4.10. The usage of this generation should provide more robust performance for the controller versus parametric variations. However, it is impossible to attain an ideal robustly controller with all these variations, therefore, the process is directed to find the optimized controller for the robustness/performance tradeoff.

The open-loop transfer function with the 3 rd generation CRONE controller is defined by an integrator of complex fractional-order 𝑛𝑛 = 𝑎𝑎 + 𝑏𝑏𝑃𝑃 ∈ ℂ 𝑖𝑖 , such that: Hence, the 3 rd generation CRONE controller design consists of an optimization to minimize the 𝑀𝑀 𝑟𝑟 variations computed by a robustness-oriented cost function defined by: (4.66) while taking into consideration the constraints based on the sensitivity functions, where 𝑇𝑇 (𝑗𝑗𝜔𝜔) here represents the complementary transfer function.

𝛽𝛽(𝑠𝑠) = ℜ /𝑖𝑖 � 𝜔𝜔 𝑢𝑢 𝑠𝑠 � 𝑛𝑛 . ( 4
𝐽𝐽 = sup 𝜔𝜔,𝐺𝐺 |𝑇𝑇 (𝑗𝑗𝜔𝜔)| -𝑀𝑀 𝑟𝑟0 ,
Four independent high-level parameters are optimized: 𝜔𝜔 𝑙𝑙 , 𝜔𝜔 ℎ , resonant frequency 𝜔𝜔 𝑟𝑟 , and open-loop gain at 𝜔𝜔 𝑟𝑟 , namely 𝑌𝑌 𝑟𝑟 which corresponds to the gain at the tangent point to the Nichols contour grid 𝑀𝑀 𝑟𝑟0 . The real and imaginary parts of the complex non-integer order 𝑎𝑎 and 𝑏𝑏, depending on 𝑌𝑌 𝑟𝑟 . Considering the bandwidth limitation in (4.56) and including integral and low pass effects, the open-loop becomes as follows:

𝛽𝛽(𝑠𝑠) = 𝐾𝐾 � 𝜔𝜔 𝑙𝑙 𝑠𝑠 + 1� 𝑛𝑛𝑙𝑙 � 1 + 𝑠𝑠 𝜔𝜔 ℎ 1 + 𝑠𝑠 𝜔𝜔 ℎ � 𝑎𝑎 ⎝ ⎜ ⎛ ℜ ∕𝑖𝑖 ⎣ ⎢ ⎡ �𝐶𝐶 0 1 + 𝑠𝑠 𝜔𝜔 ℎ 1 + 𝑠𝑠 𝜔𝜔 𝑙𝑙 � 𝑖𝑖𝑏𝑏 ⎦ ⎥ ⎤ ⎠ ⎟ ⎞ -𝑠𝑠𝑖𝑖𝑐𝑐𝑛𝑛(𝑏𝑏) �1 + 𝑠𝑠 𝜔𝜔 ℎ � -𝑛𝑛ℎ , ( 4
.67) where,

𝐶𝐶 0 = ⎝ ⎜ ⎜ ⎛ 1 + � 𝜔𝜔 𝑟𝑟 𝜔𝜔 𝑙𝑙 � 2 1 + � 𝜔𝜔 𝑟𝑟 𝜔𝜔 ℎ � 2 ⎠ ⎟ ⎟ ⎞ 1 2
, with 𝑛𝑛 𝑙𝑙 and 𝑛𝑛 ℎ ∈ ℕ + .

(4.68)

The values of 𝑛𝑛 𝑙𝑙 and 𝑛𝑛 ℎ are selected considering the assigned specification and the frequency response of the controlled plant. Thus, for the proposed electro-mechanical ABS controller, it is chosen 𝑛𝑛 𝑙𝑙 = 2 in order to track ramp inputs (linear variation of Ω 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇)). Then, in order to obtain better filtering of measurement noise, and considering the specified overshoot for a step response (30%), i.e. |𝑇𝑇 (𝑗𝑗𝜔𝜔 𝑟𝑟 ) 𝑛𝑛𝑛𝑛𝑚𝑚 | ≈ 3𝐺𝐺𝐶𝐶, the order of low pass filter is chosen as 𝑛𝑛 ℎ = 2.

The tuning parameters 𝑎𝑎, 𝑏𝑏, 𝜔𝜔 𝑙𝑙 , and 𝜔𝜔 ℎ are used to shape the nominal open-loop frequency response. One aim of this loop shaping is to move the frequency uncertainty domains away from the critical point (-𝜋𝜋, 0). According to (Sabatier et al., 2015a), each parameter acts mainly on one shape feature of the nominal open-loop transfer function and thus can be optimized easily.

As the plant uncertainty is taken into account by the least conservative method, a nonlinear optimization method must be used to find the optimal values of the independent parameters. As the performance of the closed-loop system is as important as its stability degree of robustness, in order to manage precisely the performance related to tracking, regulation, and control effort level, while minimizing the cost function (4.66), the 3 rd generation CRONE controller considers also 5 frequency-domain inequality constraints that have to be fulfill for all the plants (with all parametric variations) and for every 𝜔𝜔 ∈ ℝ + : Briefly from (Sabatier et al., 2015b), the bounds 𝑇𝑇 1 and 𝑇𝑇 𝑢𝑢 can be used to avoid the sluggishness of the response to a step variation, and could limit the lowers value of the crossover frequency and the highest value of the resonant peak 𝑀𝑀 𝑟𝑟 . Then, 𝐶𝐶 𝑢𝑢 permits the limitation of the effect of the output disturbances by limiting the resonant peak of |𝐶𝐶(𝑗𝑗𝜔𝜔)|, that could be addressed also by the lowers value of the modulus margin 𝑀𝑀 𝑚𝑚 . Moreover, the bound 𝐶𝐶𝐶𝐶 𝑢𝑢 is a boundary on the control effort sensitivity function and can be used to limit the control effort 𝑢𝑢(𝑇𝑇) with respect to measurement noise 𝑏𝑏 𝑚𝑚 (𝑇𝑇) and output disturbances. It should be noted that usually the closed-loop bandwidth is usually limited by taking into account this constraint as in this study. Finally, bound 𝐺𝐺𝐶𝐶 𝑢𝑢 is also useful in the case of the control of low-damped modes of the plant by limiting the effect of the disturbance 𝐺𝐺 𝑢𝑢 (𝑇𝑇), and thus ensuring the accuracy of the control loop.

inf 𝐺𝐺 |𝑇𝑇 (𝑗𝑗𝜔𝜔) ≥ 𝑇𝑇 1 (𝜔𝜔), (4.69
The optimization is achieved by MATLAB using the CRONE toolbox. The main purpose of this toolbox is to calculate the CRONE controller transfer function based on the plant transfer function as well as the parametric uncertainties of the system. The toolbox can specify which generation is the most suitable to answer the user specifications and it can deliver the controller transfer function in a rational form after applying the Oustaloup's approximation. For interested authors, they can refer to the following references for more information concerning the toolbox and its characteristics [START_REF] Lanusse | Toward a CRONE toolbox for the design of full MIMO controllers[END_REF][START_REF] Lanusse | CRONE control system design toolbox for the control engineering community: Tutorial and case study[END_REF][START_REF] Malti | Object-oriented CRONE toolbox for fractional differential signal processing[END_REF][START_REF] Malti | Towards an object oriented CRONE Toolbox for fractional differential systems[END_REF][START_REF] Malti | CRONE Toolbox for system identification using fractional differentiation models[END_REF][START_REF] Oustaloup | The CRONE toolbox for Matlab[END_REF].

The resulting open-loop parameters of the CRONE controller, 𝐶𝐶 𝐴𝐴𝐵𝐵𝑆𝑆 (𝑠𝑠), are listed in Table 4.4. The resulting open-loop crossover frequency 𝜔𝜔 𝑐𝑐𝑐𝑐 = 90 rad/s, satisfying the specifications of (4.64). Figure 4.12 shows a comparison between the fractional and the rational form of the controller's transfer function 𝐶𝐶 𝐴𝐴𝐵𝐵𝑆𝑆 (𝑠𝑠). The constraints imposed on the sensitivity function for the design of the controllers are represented with the dashed black lines in Figure 4.15 which also shows the magnitude of each sensitivity function considering the uncertain plants with the designed controller. Table 4.5 shows the maximum gain of each sensitivity function considering the perturbed plants. Since all of them satisfy the assigned constraints, then robustness is ensured.

Time domain analysis

In order to evaluate the performance of the designed ABS controller, several testing scenarios were done. The quarter vehicle validation model used for testing is the one described in 4.2.2 while considering a different parametric configuration from that of nominal one to examine the robustness of the controller versus mass variations.

For comparison reason, two more controllers are designed, a first-generation crone controller,𝐶𝐶 1𝐴𝐴𝐵𝐵𝑆𝑆 (𝑠𝑠) and a PID controller, 𝐶𝐶 𝑃𝑃𝐼𝐼𝑅𝑅 (𝑠𝑠). Both controllers are designed to have the same crossover frequency (Iso-rapidity) and the same stability margin with the nominal plant 𝐺𝐺 𝑛𝑛𝑛𝑛𝑚𝑚 (𝑠𝑠). The first scenario (A) is a vehicle moving with an initial velocity of 𝑉𝑉 𝑥𝑥 𝑟𝑟 (𝑇𝑇 = 0) = 30𝑚𝑚/𝑠𝑠, driven on a dry and even road (𝜇𝜇 = 1). The driver starts to slow down the vehicle by pressing the braking pedal slowly. The driver input has a steady-state position that corresponds to 𝑈𝑈 𝑐𝑐 = 10𝑉𝑉 .

Figure 4.18 shows the vehicle longitudinal and angular velocities of the different systems. The uncontrolled vehicle shows the worst braking performance, as it presents the highest braking distance (distance till 𝑉𝑉 𝑥𝑥 = 0.5𝑚𝑚/𝑠𝑠), which confirms the importance of ABS control. In fact, it can be seen that the ABS controllers prevent the wheels from locking. At 𝑇𝑇 = 2.3s, the wheel slip rate, shown in Figure 4.19, becomes smaller than the 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 , and therefore the ABS is activated. All controllers manage to satisfactorily track the wheel reference angular velocity. The corresponding control inputs are plotted in Figure 4.20. Up to 2.3s, only the feed-forward term is present, but once the ABS is activated, the feedback term counteracts in order to track Ω 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑇𝑇), avoiding wheel locking-up.

It is also interesting to consider the influence of the measurement noise on the control input. For this reason, the power spectrum density is plotted for each control effort for every controller, as shown in Figure 4.21. It is clear that the 3 rd generation CRONE controller designed has a lower PSD magnitude compared to other controllers. Such behavior can be explained by the high gains on the open-loop Bode response of other controller compared to the 𝐶𝐶 𝐴𝐴𝐵𝐵𝑆𝑆 (𝑠𝑠) (Figure 4.16). This filtering properties in high frequencies for the 3 rd generation controller causes less agitation on the control input.

In this sense, such an advantage does not seem significant in the braking performance, where the braking distances are almost very close to each other, i.e. 50.69, 50.67, and 50.66m for PID, 𝐶𝐶 1𝐴𝐴𝐵𝐵𝑆𝑆 , and 𝐶𝐶 𝐴𝐴𝐵𝐵𝑆𝑆 , respectively. Nevertheless, it is very important mainly for two reasons [START_REF] Benine-Neto | Robust control for an electro-mechanical anti-lock braking system: the CRONE approach[END_REF]: Firstly, it reduces the energy consumption of the actuator and increases its life-time; Secondly, if such control input agitation level, as for other PID controller, for example, is acceptable, then the CRONE 3 rd generation design allows the use of lower quality (nosier) wheel speed sensors, which may be cheaper. In order to examine the robustness of the controllers in the time domain against road adhesion variations, a second scenario is simulated. The second scenario (B) presents an emergency braking, where the driver input has a steady-state position corresponding to 𝑈𝑈 𝑐𝑐 = 12𝑉𝑉 (full pedal displacement). The road adhesion is degraded from 1(dry) to 0.5 (wet) at 𝑇𝑇 = 2.5𝑠𝑠, i.e. 𝜇𝜇 = 0.5, ∀𝑇𝑇 > 2.5𝑠𝑠. All other parameters are the same as in scenario (A).

The braking distance are respectively 45.86, 45.79, and 45.65m for PID, 𝐶𝐶 1𝐴𝐴𝐵𝐵𝑆𝑆 , and 𝐶𝐶 𝐴𝐴𝐵𝐵𝑆𝑆 , respectively. The wheel is locked due to high braking pressure at 1.5s for the uncontrolled system. As shown in Figure 4.22, the controlled systems show a good performance in preventing wheel blocking. Moreover, at 𝑇𝑇 = 2.5𝑠𝑠 the controlled systems prove their ability to adapt to the variations on the road adhesion. As shown in Figure 4.23, all controllers responded to the adhesion variation at 𝑇𝑇 = 2.5𝑠𝑠, and in few milliseconds, they could track the references angular wheel velocities again. Figure 4.24 plot the control inputs for the different controllers. As for scenario (A), in this scenario, the PSD graph for the control efforts are plotted in Figure 4.25. It clearly shows the advantage of the designed 3 rd generation CRONE controller over the other controllers in the frame of power consumption.

The simulations done in this section takes into consideration only the longitudinal dynamics of the quarter vehicle model while neglecting the vertical dynamics. However, it is highly critical to consider the suspension dynamics and its effect on the performance of the ABS controller in cases where the load transfer is introduced, and more importantly, when the road is no longer even. Thus, the effect of load transfer is taken into consideration on the half-vehicle model, and road disturbances are then introduced on the quarter vehicle model. As a result, the ABS controller faces a new challenge of robustness and performance against high variations on the wheel contact vertical forces, denoted by 𝐹𝐹 𝑧𝑧 [N]. Effect of Vertical dynamics on the ABS control system

Introduction

Many theories and design methods for ABS and active suspension systems have been proposed individually over the years. As introduced in the last sections, the ABS system manipulates the braking of automobiles in order to prevent uncontrollable skidding in accidents as a result of locked-up wheels. It can provide directional stability and shortens the braking distance [START_REF] Lin | Nonlinear control design of anti-lock braking systems with assistance of active suspension[END_REF].

On the other hand, active suspension systems must be able to guarantee the ride comfort of passengers. Hence, in the design of active suspensions, the improvement of ride quality is a major control objective to be emphasized. In this framework, many control designs schemes were developed for both a quarter-car and a half-car active suspension system [START_REF] Fialho | Road adaptive active suspension design using linear parameter-varying gain-scheduling[END_REF][START_REF] Huang | Road-adaptive algorithm design of half-car active suspension system[END_REF][START_REF] Lin | Nonlinear backstepping active suspension design applied to a half-car model[END_REF], which aimed to improve the tradeoff between the ride quality of passenger comfort and the utilization of suspension travel.

The purpose of this section is to take advantage of the designed ABS controller combined with the active suspension system using the fractional damping control to further reduce vehicle braking time and stopping distance. During the braking process, if the vertical normal force is increased (this situation is equivalent to an increase of the weight of a car, driver disturbances, …), the friction force between the car and road surface is increased to help the vehicle to stop in a shorter distance. As normal force cannot be increased naturally, active suspensions should, therefore, be the tool employed to achieve that adjustment of the normal force. With the choice of an appropriate integrated algorithm, the vertical normal force can be adjusted with the braking torque in order to achieve a reduction of braking time and stopping distance.

However, one main important cases in which the performance of the ABS control is degraded is when driving over rough terrain [START_REF] Hamersma | Improving the braking performance of a vehicle with ABS and a semi-active suspension system on a rough road[END_REF][START_REF] Kaldas | Improvement of Heavy Vehicles Ride and Braking Performance via Combined Suspension and Braking Systems Control[END_REF][START_REF] Sanchez-Torres | ABS design and active suspension control based on HOSM[END_REF]. Not only is the effective rolling radius is changing due to undulations in the road surface, but the vertical and torsional dynamics of the wheel may be excited to such an extent that contact between the tire and the road is lost or that the tire starts oscillating. The tire oscillation has a significant effect on the performance of an ABS system [START_REF] Adcox | Interaction of anti-lock braking systems with tire torsional dynamics[END_REF].

As stated in [START_REF] Hamersma | Improving the braking performance of a vehicle with ABS and a semi-active suspension system on a rough road[END_REF], the situation is further complicated by the so-called run-in effect of tire force generation. Gillespie [START_REF] Gillespie | Fundamentals of Vehicle Dynamics[END_REF] describes the run-in effect stating that a tire's response to a step input is not instantaneous. The lag is related to the rotation of the tire, typically taking between one-half and one full revolution of the wheel to reach the steady-state force condition. A varying vertical load on the tire, as is the case when driving over a rough road, thus further complicates the optimum application of ABS. When the wheel load increases, the size of the contact patch is increased with rubber that is not deformed. The 'new' rubber has to deform before it generates a significant longitudinal force, which lags the increase in vertical load. In contrast, when the wheel load decreases, the contact patch area decreases and the force generated is decreased instantaneously. Furthermore, due to the low damping present in the tire, natural frequencies may be excited due to the terrain input and the periodic braking input from the ABS. The net effect of this process is a reduction in the brake force and increased stopping distances [START_REF] Breuer | Brake Technology Handbook[END_REF].

Therefore, due to this coupling effect between the suspension dynamics and the braking system [START_REF] Hamersma | Improving the braking performance of a vehicle with ABS and a semi-active suspension system on a rough road[END_REF], active suspension control laws, where the strategy is comfort-oriented, may degrade the performance of braking on rough roads. Indeed, the active suspension forces introduced by the electric actuator affect the wheel normal forces, which in turn impact the wheel longitudinal contact forces that are directly responsible for the braking behavior of a vehicle. For this reason, a convenient combination between ABS control and Fractional-order damping control is proposed to minimize the effect of road roughness on the performance of the braking system of the vehicle.

Dynamic model

In this study, the half vehicle model is adopted, representing the motion of the two-wheeled car model, as shown in Figure 4.26. There are four equations to derive a linear model for the vertical dynamics and three equations to derive a nonlinear model for the longitudinal dynamics.

The only difference here in this study is the addition of the vertical dynamics of the quarter vehicle model, and the chassis dynamics, i.e. heave and pitch motion. Moreover, the vertical wheel normal forces 𝐹𝐹 𝑧𝑧 are no longer static, and it includes wheel dynamics. Thus, it is formulated as follows:

𝐹𝐹 𝑧𝑧𝑖𝑖 (𝑇𝑇) = 𝑚𝑚 𝑖𝑖 𝑔𝑔 + 𝑓𝑓 1𝑖𝑖 (𝑇𝑇), (4.74) where 𝑃𝑃 = 1 for front axle 𝑃𝑃 = 2 for the real axle. The dynamic force 𝑓𝑓 1𝑖𝑖 (𝑇𝑇) is formulated as in (2.2) The heave and pitch motion of the chassis could be calculated using the dynamics of the front and rear sprung masses, by: 𝑎𝑎 21 = 𝑎𝑎 𝐺𝐺 -𝑎𝑎sin(𝜃𝜃 𝐺𝐺 ), 𝑎𝑎 22 = 𝑎𝑎 𝐺𝐺 + 𝑏𝑏sin(𝜃𝜃 𝐺𝐺 ).

(4.77)

Supposing that 𝜃𝜃 𝐺𝐺 has a small value, and using the inverse of (4.77), we get the following relation:

� 𝑎𝑎 𝐺𝐺 𝜃𝜃 𝐺𝐺 � = � 1 -𝑎𝑎 1 𝑏𝑏 � -1 � 𝑎𝑎 21 𝑎𝑎 22 �.
(4.78)

Control structure

For purpose of comparing the performance and analyzing the different effects of the vertical dynamics characteristics, the fractional-order damping control discussed in Chapter 2 is introduced and for ABS control the 3 rd generation CRONE controller designed in the previous section is adopted here as it demonstrated the best performances among the three controllers studied.

Testing simulations

In order to test the influence of the vertical dynamics of the vehicle on the ABS control process, different scenarios should be done, each one to analyze one variable at a time. All simulations are done using the same scenario made in the previous sections where a vehicle starting at an initial speed of 𝑉𝑉 𝑥𝑥 (𝑇𝑇 = 0) = 30𝑚𝑚/𝑠𝑠. Two kinds of disturbances are taken into account to excite the vertical dynamics, the driver disturbance and road irregularities, each one using its convenient vehicle model.

Driver disturbance

Passive suspension

At first, two simulations were done to compare the system's performance with and without vertical dynamics. At this stage, the road is taken perfectly even so that we study the effect of the load shift only. Here, the ABS system is deactivated during all the simulation time. where it shows that the two systems have identical dynamics except for a small duration around 𝑇𝑇 = 3𝑠𝑠. In Figure 4.28a and b, the wheel normal forces and the longitudinal forces are plotted respectively. Due to the variation of the load transfer at each axle, it influences the wheel contact forces, and as a result, we could observe the difference between the two systems in the values of the longitudinal forces at each wheel. As a result, the braking difference of these systems is recorded as 64.41 and 64.74m, for a system without and with the suspension system, respectively.

Passive suspension with ABS Control

We consider the previous scenario, with an ABS controller implemented at each wheel. The braking distance is respectively, 54.92 and 55.44m for the system without and with the suspension system. Figure 4.29 shows the wheel slip rate response and the longitudinal velocities of the vehicle, in which the latter shows a little bit difference between the two systems that explain the difference found in the braking distances. Figure 4.30 shows the wheel normal forces and the longitudinal forces, in which it describes the effect of the variation of the load transfer on the vehicle.

In both testing scenarios, it was concluded that the introduction of the vertical dynamics degraded the overall performance of the braking system. Although of the increase in the longitudinal forces on the front axle for the systems with a suspension due to load transfer, but we can see the high reduction on the rear longitudinal forces on both cases. These variations lead to obtaining a degradation on the value of the stopping distances. It should be noted that these responses may highly depend on the tire characteristics and the testing scenario. However, this conclusion could be generalized due to its consistency with several studies found in the literature.

Controlled suspension With ABS system

Two more simulations were done, one with a passive suspension system and controlled ABS and another one with a controlled suspension system and ABS controller. The second system was tested for three fractional-order damping configurations as described in Chapter 2, i.e. 𝑛𝑛 = 0.65, 𝑛𝑛 = 1.1, and 𝑛𝑛 = 1.4. The braking distances of these simulations gave very close values that are around 55.44m.

Road disturbances

Many testing simulations were done taking into consideration road roughness. The results of these simulations are listed in 6 different tables. Tables 4.6 4.7,and 4.8 present the results on the front axle quarter vehicle model, while Tables 4.9,4.10,and 4.11 correspond to the rear axle quarter vehicle model. Tables 4.6 and4.9 are for the braking distance without an ABS control system, Tables 4.7 and4.10 are for systems with ABS control system with 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 = 8%, and finally Tables 4.8 and4.11 are for systems with ABS control system but with 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 = 7%, where (A) means a normal braking scenario and (B) is an emergency braking scenario.

The overall performance leads to the following conclusions:

-The results confirm the deteriorative effect of the road unevenness on the braking performance where higher stopping distances were observed in all tested configurations.

-They also confirm the ability of the designed ABS control strategy in reducing braking time and distances, and thus attain more safety for passengers and pedestrians, in different configurations.

-The passive suspension system effect becomes more significant in emergency braking scenarios on systems with ABS controllers, thus, necessitating more future attention. It should be noted that this effect is more indicative on heavier vehicles, in which for front axle it recorded up to 55cm difference compared to the rear axle quarter vehicle model in which the maximum difference was few centimeters.

-The convenient selection of the optimal value of the wheel slip rate plays an important role where all the braking distances obtained with 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 = 8% were less than those obtained with of 𝑇𝑇 𝑛𝑛𝑝𝑝𝑡𝑡 = 7%, more specifically, they were 0.6m smaller on average.

-The FO damping control shows an advantage in reducing the braking distance where 𝑛𝑛 = 1.1, compared to other fractional-orders and even compared to the passive suspension system.

For further analysis of the key role of FO damping control for this enhancement in braking distance, three selected scenarios were plotted, where Figure 4.31 and 4.32 show the wheel contact forces and longitudinal forces respectively. The results were shown for a quarter vehicle model controlled with the designed ABS control while considering a fractional damping control with 𝑛𝑛 = 0.65 (comfort-oriented), 𝑛𝑛 = 1.1 (road holding oriented), and a system without vertical dynamics (no suspension).

The road-holding oriented FO damping control was able to reduce the variation of the wheel normal contact forces, thus, the variation of the longitudinal forces 𝐹𝐹 𝑥𝑥 , as illustrated in Figure 4. 31 and 4.32, respectively. The RMS values of these forces relative to the system without vertical dynamics are listed in Table 4.12. It shows a reduction of 8.5% and 7% for 𝑛𝑛 = 1.1 compared to passive system, and 38% and 34% compared to 𝑛𝑛 = 0.65, in the RMS values of 𝐹𝐹 𝑧𝑧 and 𝐹𝐹 𝑥𝑥 respectively.

Conclusion

The test simulations on the quarter vehicle model show that rough terrain excitation further contributes to dynamic tire effects such as loss of vertical contact and poor contact patch generation that leads to reduced longitudinal force generation.

It is concluded that the combined use of the ABS control and a convenient active suspension control can be a useful approach in the research of ABS braking on rough terrains. Moreover, the effect of load shifting is studied on the ABS control on the half-vehicle model. The results show a degradation in the performance of the ABS control due to the variation of wheel normal contact forces at the front and rear axles. In this sense, the fractional-order damping control seems not efficient in improving braking performance in even road scenarios.

Conclusion

Many factors could affect vehicle's safety in road traffic situations. One main factor is the wet or slippery road surfaces and the abrupt reaction of the driver. In such hazardous driving conditions, it is possible for the wheels of a vehicle to lock up under high braking torque. As a result, the vehicle can become uncontrollable or may go into a slip or leave the road. Here, the antilock braking control system detects the tendency of the wheels to lock up under braking and regulate the braking pressure conveniently to prevent the wheels from locking up and the vehicle, thus, remains steerable. As a consequence, the vehicle can be braked or stopped quickly and safely.

For this reason, a 3 rd generation CRONE controller was designed in order to track a reference value for the wheel angular velocity that attain enhanced braking performance. In the framework of the quarter vehicle model, the controller designed proves its ability in tracking the generated reference, and thus, reducing the braking distance. Besides model parametric variations, the designed controller shows its robustness against high road adhesion variations. Moreover, the simulation results show the 3 rd generation CRONE controller was less sensitive to noise measurement compared to a 1 st generation CRONE and PID controller.

However, due to the coupling effect between the vertical and longitudinal dynamics, the influence of load shifting and wheel fluctuations due to road excitations were studied. Road input excitation deteriorates braking performance which results in longer stopping distances due to a number of suspensions and tire contributing factors, in addition to the vehicle's body motions such as heave and pitch that are excited by rough road or driver input. It was shown that the rough terrain braking problem can essentially be improved with suitable fractional-order damping suspension control.

This study can be extended to include another suspension control algorithm that might improve the braking performance. Moreover, investigation of the role of ABS in stabilizing the vehicle during maneuvering is to be considered later on. In fact, the system should also be able to ensure that handling stability and steerability are maintained.

Introduction

Many active systems have been widely used in automotive vehicles in order to increase safety and avoid accidents. For light electric vehicles, it is essential to give importance for lateral stability during a maneuver. The two most famous active systems that are used to provide lateral stability are the anti-lock braking systems (ABS) [START_REF] Benine-Neto | Robust control for an electro-mechanical anti-lock braking system: the CRONE approach[END_REF] and the Electronic Stability Program (ESP).

Several studies exist in the literature dealing with yaw rate regulation, and could be divided into three categories: regulation with an active steering system that provides additional steering angle [START_REF] Benine-Neto | Vehicle yaw rate control based on piecewise affine regions[END_REF], regulation with differential braking to create the needed moment around the z-axis of the chassis [START_REF] Wang | Independent wheel torque control of 4WD electric vehicle for differential drive assisted steering[END_REF], and finally a coordination control strategy that combine both active systems to improve stability and safety for vehicles (Termous et al., 2018b). These objectives were achieved using a different kind of controllers that used to regulate the yaw motion. Starting from the very simple methods using PID controller [START_REF] Deur | Parameter optimization of PID class of yaw rate controllers[END_REF], to the sliding mode controller [START_REF] Tchamna | Yaw rate and side-slip control considering vehicle longitudinal dynamics[END_REF], the H∞ controller [START_REF] Lian | Direct Yawmoment Robust Control for Electric Vehicles Based on Simplified Lateral Tire Dynamic Models and Vehicle Model[END_REF], the backstepping [START_REF] Zhou | Vehicle yaw stability-control system design based on sliding mode and backstepping control approach[END_REF], model predictive control (MPC) [START_REF] Xu | Design and Experiments of Safeguard Protected Preview Lane Keeping Control for Autonomous Vehicles[END_REF], and many others.

In this chapter, the study is focused on yaw motion control, and the objective is to enhance the performance of the lateral dynamics of the vehicle. This can be achieved by introducing a well-designed controller that could help the driver, through additional yaw moment, to avoid skidding out of the trajectory when a sudden change of road adhesion occurs, the side wind force is applied to the vehicle, or upon emergency maneuvering. In fact, the yaw moment can be provided by two main actuators: the braking actuator and the steering actuator. For this purpose, two controllers were designed individually, in the sense of regulating yaw rate dynamics. The study was aimed at examining the performance of each controller at a time, the future work was assigned to implement adequate coordination between these active systems for an optimized performance of the vehicle's stability. However, since enhancing the vehicle's stability doesn't lead always to the best lateral position for a vehicle, another controller was designed to track a generated reference for the lateral position.

The efficiency of each controller can be analyzed by its effectiveness in regulating the yaw rate or lateral position of the vehicle to its reference against disturbances, and its robustness against uncertain parameters of the vehicle. Starting from the velocity of the vehicle to the total mass, and last but not least the road adhesion that has the biggest influence on steering performance [START_REF] Monot | Dynamic Stability Control system: the CRONE approach[END_REF].

Finally, since most studies on lateral stability don't take into consideration the effect of the vertical dynamics on the lateral control, our purpose in this study is to highlight the effect of the vertical dynamics of the chassis on the performance of the designed lateral controllers.

Vehicle model

In this section, all the notations in lower case are considered as variations and all physical variables without capital letter notations are variables whose reference are zero. Finally, the Laplace transform of the variation 𝑎𝑎(𝑇𝑇) is noted as 𝑋𝑋(𝑠𝑠) = 𝑇𝑇𝐹𝐹{𝑎𝑎(𝑇𝑇)}.

Schematization and configuration

In this study, the Four-wheel vehicle model (FWVM) was used, commonly called the two tracks model as seen in Figure 5.1, and it is widely addressed in the literature to study the transversal vehicle behavior (Rajamani, 2012b).

The first remark concerning this model is that the steering input angle introduced by the driver, 𝛽𝛽 𝑓𝑓 , is considered less than 10°. Moreover, the width of both axles is considered equal, that is the vehicle is symmetric with respect to the x-axis. Regarding the active systems, all used actuators are assumed ideal, that is all control efforts designed are introduced directly to the vehicle model where the actuator dynamics were neglected.

The chosen vehicle velocities are considered constant in this study, assuming the vehicle is equipped with active cruise control. The value of this speed is selected around 50km/h. Some test scenarios could consider a speed higher than 50km/h to analyze the effect of high velocities on the stability of the designed controller.

Dynamic equations

The main equation describing the yaw motion of the lateral model is given by: 𝑃𝑃 𝑧𝑧 𝜓𝜓 ̈= 𝑤𝑤 �cos𝛽𝛽 𝜐𝜐 (𝐹𝐹 𝑥𝑥12 -𝐹𝐹 𝑥𝑥11 ) + sin𝛽𝛽 𝜐𝜐 �𝐹𝐹 𝑦𝑦11 -𝐹𝐹 𝑦𝑦12 �� + 𝐿𝐿 𝑓𝑓 �sin𝛽𝛽 𝜐𝜐 (𝐹𝐹 𝑥𝑥11 + 𝐹𝐹 𝑥𝑥12 ) + cos𝛽𝛽 𝜐𝜐 �𝐹𝐹 𝑦𝑦12 + 𝐹𝐹 𝑦𝑦11 �� -𝐿𝐿 𝑟𝑟 �𝐹𝐹 𝑦𝑦22 + 𝐹𝐹 𝑦𝑦21 � -𝑤𝑤(𝐹𝐹 𝑥𝑥21 -𝐹𝐹 𝑥𝑥22 ) + 𝑀𝑀 𝑟𝑟𝑟𝑟𝑠𝑠 , (5.1) where 𝑃𝑃 𝑧𝑧 is the moment of inertia, 𝑀𝑀 𝑟𝑟𝑟𝑟𝑠𝑠 represents the resistance moment about the z-axis, and 𝐹𝐹 𝑥𝑥 and 𝐹𝐹 𝑦𝑦 are respectively the longitudinal and the transversal forces applied on the tires. The variable 𝛽𝛽 𝜐𝜐 represents the steering angle introduced by the driver, Δ 𝐺𝐺 is the sideslip angle at the CG, and 𝑤𝑤, 𝐿𝐿 𝑓𝑓 and 𝐿𝐿 𝑟𝑟 are the chassis' dimensions as illustrated in Figure 5.1.

The equation representing the lateral acceleration 𝑎𝑎 𝑦𝑦 , in the vehicle frame of reference, is given by: 𝑎𝑎 𝑦𝑦 = 𝑉𝑉 ̇𝑦𝑦 + 𝑉𝑉 𝑥𝑥 𝜓𝜓 ̇, (5.2) where Vx represents the longitudinal speed of the vehicle, ψ represents the yaw rate, and Vy is the lateral speed. The variable y V  can be expressed by the following equation:

𝑀𝑀 𝑉𝑉 ̇𝑦𝑦 = -𝑀𝑀 𝜓𝜓 ̇ 𝑉𝑉 ̇𝑥𝑥 -𝑓𝑓 𝑦𝑦0 + sin𝛽𝛽 𝜐𝜐 (𝐹𝐹 𝑥𝑥12 + 𝐹𝐹 𝑥𝑥11 ) + 𝐹𝐹 𝑦𝑦22 + 𝐹𝐹 𝑦𝑦21 + cos𝛽𝛽 𝜐𝜐 �𝐹𝐹 𝑦𝑦12 + 𝐹𝐹 𝑦𝑦11 �, (5.3) where fy0 represents the side wind gust applied on the center of gravity of the vehicle. The tire forces are generated according to the well-known Pacejka Model [START_REF] Pacejka | Basic Tire Modeling Considerations[END_REF], where more than 30 empirical parameters are needed to fully describe the experimental measurements. The calculation of these forces depends mainly on the value of the road adherence µ, the tire ground contact forces 𝐹𝐹 𝑧𝑧 , and 𝛥𝛥 𝑝𝑝 the tire's sideslip angle. The sideslip angle at the center of gravity of the vehicle can be described by:

𝛥𝛥 𝐺𝐺 = 𝑉𝑉 𝑦𝑦 𝑉𝑉 𝑥𝑥 . (5.4)
The inputs and outputs of the tire model are described in Figure 5.2, in which the inputs are:

-The wheel vertical normal force 𝐹𝐹 𝑧𝑧 (𝑇𝑇), -The tire's sideslip wheel angle Δ 𝑝𝑝 (𝑇𝑇), -the road adhesion 𝜇𝜇, -the wheel slip rate 𝑇𝑇��𝑉𝑉𝑎𝑎(𝑇𝑇), �Ω(𝑇𝑇)�.

More precisely, the relationship between Δ 𝑝𝑝 (𝑇𝑇) and the lateral wheel forces are given, according to the magic formula, by: 𝐹𝐹 𝑦𝑦 (𝑇𝑇) = 𝜇𝜇𝑃𝑃 𝑦𝑦 sin �𝐶𝐶 𝑦𝑦 arctan �𝐶𝐶 𝑦𝑦 ��1 -𝐸𝐸 𝑦𝑦 �Δ 𝑝𝑝 (𝑇𝑇) + 𝐸𝐸 𝑦𝑦 𝐶𝐶 𝑦𝑦 arctan �𝐶𝐶 𝑦𝑦 Δ 𝑝𝑝 (𝑇𝑇)����, (5.5) where the parameters 𝐶𝐶 𝑦𝑦 , 𝐶𝐶 𝑦𝑦 , 𝑃𝑃 𝑦𝑦 , and 𝐸𝐸 𝑦𝑦 depend mainly on the characteristics of the tire used and on the value of the normal force 𝐹𝐹 𝑧𝑧 (𝑇𝑇).

The wheel sideslip angle dynamic is formulated for each wheel as follows:

Δ 𝑝𝑝12 = 𝛽𝛽 𝑓𝑓 -arctan( 𝑎𝑎̇+ 𝐿𝐿 𝑓𝑓 𝜓𝜓 ȧ𝑎̇+ 𝑤𝑤𝜓𝜓 ̇ ), Δ 𝑝𝑝11 = 𝛽𝛽 𝑓𝑓 -arctan( 𝑎𝑎̇+ 𝐿𝐿 𝑓𝑓 𝜓𝜓 ȧ𝑎̇- 𝑤𝑤𝜓𝜓 ̇ ) , Δ 𝑝𝑝22 = -arctan( 𝑎𝑎̇-𝐿𝐿 𝑟𝑟 𝜓𝜓 ȧ𝑎̇+ 𝑤𝑤𝜓𝜓 ̇ ) , Δ 𝑝𝑝21 = -arctan( 𝑎𝑎̇-𝐿𝐿 𝑟𝑟 𝜓𝜓 ȧ𝑎̇- 𝑤𝑤𝜓𝜓 ̇ ) .
(5.6)

Equation of the trajectory

The trajectory of the vehicle is defined by the coordinates 𝑋𝑋 𝐺𝐺 (𝑇𝑇) and 𝑌𝑌 𝐺𝐺 (𝑇𝑇) of the center of gravity CG in the absolute reference frame of the vehicle (𝑅𝑅 0 ).

The coordinates 𝑋𝑋 𝐺𝐺 (𝑇𝑇) and 𝑌𝑌 𝐺𝐺 (𝑇𝑇) are obtained by integrating the components 𝑎𝑎̇𝐺𝐺(𝑇𝑇) and 𝑎𝑎̇𝐺𝐺(𝑇𝑇), the speeds of the center of gravity, defined in the absolute frame of reference 𝑅𝑅 0 , that is:

𝑉𝑉 ⃗ 𝐺𝐺 𝑅𝑅 0 ⁄ = 𝑉𝑉 𝑥𝑥0 𝑎𝑎⃗ + 𝑉𝑉 𝑦𝑦 (𝑇𝑇) 𝑎𝑎⃗ = � 𝐺𝐺𝑂𝑂𝐺𝐺 ����� ����� �� 𝐺𝐺𝑇𝑇 � 𝑅𝑅 0 = � 𝐺𝐺 𝐺𝐺𝑇𝑇 (𝑋𝑋 𝐺𝐺 (𝑇𝑇) 𝑎𝑎⃗ 0 + 𝑌𝑌 𝐺𝐺 (𝑇𝑇) 𝑎𝑎⃗ 0 )� 𝑅𝑅 0 , (5.7)
or, after derivation, where 𝑋𝑋 𝐺𝐺0 and 𝑌𝑌 𝐺𝐺0 are the initial values for the position of the center of gravity of the vehicle.

Model Linearization

The design of the frequency-based controllers is based on the frequency response of the plant; therefore, the vehicle model should be linearized around some operating points. Starting from the major nonlinear variable, the lateral forces 𝐹𝐹 𝑦𝑦 represented in Figure 5.3 shows that the values of the lateral forces have linear behavior when 𝛥𝛥 𝑝𝑝 is in the range of [-1.5°,1.5°]. Thus, at this operating area, the lateral forces can be calculated by: 𝐹𝐹 𝑦𝑦 (𝑇𝑇) = 𝑊𝑊 𝑦𝑦 𝛥𝛥 𝑝𝑝 (𝑇𝑇), (5.12) where 𝑊𝑊 𝑦𝑦 represents the cornering stiffness that supposed to be constant for a given adhesion coefficient and a total vehicle mass. The computation process of this parameter is exactly the same as for the longitudinal stiffness 𝑝𝑝 𝑥𝑥 that was obtained and illustrated in Chapter 4.

Moreover, the following assumptions were taken into account:

• The value of the steering angle (at the vehicle's wheels level) is less 5°, then we consider cos • The vehicle is equipped with a cruise control system and the ESP is short and rapid in order not to influence the longitudinal speed, therefore the value of the longitudinal speed is considered constant and denoted by 𝑉𝑉 𝑥𝑥0 . • The vehicle is considered to be symmetric with respect to the x-axis of the chassis, thus at equilibrium, the longitudinal forces Fx that are at the same axle, are considered to be equal (𝐹𝐹 𝑥𝑥11 = 𝐹𝐹 𝑥𝑥12 and 𝐹𝐹 𝑥𝑥21 = 𝐹𝐹 𝑥𝑥22 ). • The tire sideslip angles at the same axle are equal. Therefore, at equilibrium, the variables representing the lateral dynamics could be simplified and formulated by the following equations: (5.15)

The variable 𝑢𝑢 𝑧𝑧 (𝑇𝑇) represents the variation of the yaw torque control introduced by the designed controllers that would be generated by additive braking torques applied on the tires.

State-space representation

The state-space representation of the linearized lateral model is then established and can be described by: � 𝑎𝑎(𝑇𝑇) = 𝐶𝐶𝑎𝑎(𝑇𝑇) + 𝐶𝐶𝑢𝑢(𝑇𝑇) 𝑎𝑎(𝑇𝑇) = 𝐶𝐶𝑎𝑎(𝑇𝑇) + 𝑃𝑃𝑢𝑢(𝑇𝑇) , (5.16) where, (5.20)

𝑎𝑎(𝑇𝑇) = � 𝑎𝑎 1 = 𝜓𝜓 ̇(𝑇𝑇) 𝑎𝑎 2 = 𝛥𝛥 𝐺𝐺 (𝑇𝑇) = 𝑣𝑣 𝑦𝑦 /𝑉𝑉 𝑥𝑥0 � , 𝑢𝑢(𝑇𝑇) = ⎝ ⎜ ⎜ ⎛ 𝑓𝑓 0𝑦𝑦 (𝑇𝑇) 𝑢𝑢 𝑧𝑧 (𝑇𝑇) 𝛽𝛽(𝑇𝑇) ⎠ ⎟ ⎟ ⎞ , (5.17
) 𝐶𝐶 = ⎣ ⎢ ⎢ ⎡ -2 𝐿𝐿 𝑓𝑓 2 𝑊𝑊 𝑦𝑦𝑓𝑓 + 𝐿𝐿 𝑟𝑟 2 𝑊𝑊 𝑦𝑦𝑟𝑟 𝑃𝑃 𝑧𝑧 𝑉𝑉 𝑥𝑥0 -2𝑃𝑃𝑃𝑃𝐸𝐸 𝑃𝑃 𝑧𝑧 -2 𝑀𝑀 𝑉𝑉 𝑥𝑥0 2 𝑃𝑃𝑃𝑃𝐸𝐸 -1 -2 𝑀𝑀𝑉𝑉 𝑥𝑥0 �𝑊𝑊 𝑦𝑦𝑓𝑓 + 𝑊𝑊 𝑦𝑦𝑟𝑟 � ⎦ ⎥ ⎥ ⎤ , 𝐶𝐶 = ⎣ ⎢ ⎢ ⎡ 0 1 𝑃𝑃 𝑧𝑧 2𝐿𝐿 𝑓𝑓 𝑊𝑊 𝑦𝑦𝑓𝑓 𝑃𝑃 𝑧𝑧 1 𝑀𝑀𝑉𝑉 𝑥𝑥0 0 2𝑊𝑊 𝑦𝑦𝑓𝑓 𝑀𝑀 𝑉𝑉 𝑥𝑥0 ⎦ ⎥ ⎥ ⎤ , 𝐶𝐶 = [1 0], 𝑃𝑃 = 0, ( 5
The transfer function 𝐺𝐺 1 (𝑠𝑠) is dependent on three main parameters: the longitudinal speed 𝑉𝑉 𝑥𝑥0 , the road adhesion µ that is included in the 𝑊𝑊 𝑦𝑦𝑓𝑓 and 𝑊𝑊 𝑦𝑦𝑟𝑟 coefficients, and the total mass of the vehicle M. The value of the front and rear wheelbase 𝐿𝐿 𝑓𝑓 and 𝐿𝐿 𝑟𝑟 , and the yaw inertia 𝑃𝑃 𝑧𝑧 are directly related to the value of M, where they are changing according to its value.

Uncertain Plants

The range of variation of each uncertain parameter is shown in Table 5.1. According to these variations, three plants were selected and named as: the nominal plant, minimum plant, and maximum plant, where the nominal plant corresponds to the normal driving conditions. Figure 5.4 shows the influence of the uncertainty in the frequency domain. It should be noted that these variations correspond to a light electric vehicle parametric configuration in two conditions: an empty vehicle (only driver) and a fully-loaded vehicle (five passengers). The study considers a wide range of road adhesion variation, where 𝜇𝜇 = 0.1 corresponds to an icy road surface and 𝜇𝜇 = 1 to a dry road surface.

Yaw rate regulation by corrective yaw moment

In this section, the design of the 3 rd generation CRONE controller and the PID controller are detailed. In order to have a fair comparison between the two controllers, they should be designed so that the rapidity and stability margin are identical for both controllers for the nominal plant chosen. Both controllers are designed to track the yaw rate reference generated by a supervisor based on the steering profile introduced by the driver. The evaluation is done in terms of robustness and performance against disturbance rejection and parametric variations.

Specifications

Several specifications are required for the controllers:

• The control input related to the measurement noise is less than 10% of the greatest value of the control input. • The nominal overshoot of the step response related to the reference signal is about 30%.

Denote by 𝑀𝑀 𝑟𝑟0 the value of the closed-loop resonant gain of the contour that is tangent to the nominal open-loop transfer function on Nichols chart. Thus, the stability could be granted by fixing the value of 𝑀𝑀 𝑟𝑟0 that is chosen to specify a value of 3dB for the nominal plant. Therefore, it ensures a phase margin greater than 40° and therefore, satisfying stability. Figure 5.5 shows the control scheme for the yaw rate regulation using braking torques. In order to export these specifications in the frequency domain, the following functional equations associated with The open-loop crossover frequency, denoted by 𝜔𝜔 𝑢𝑢 , is chosen so that the sensitivity of the input control to the noise measurement (Nm) is less than 10%. Meaning that: 5.22) where 𝑁𝑁 𝑚𝑚0 is the maximum magnitude of the sensor noise measurements that is usually specified in the sensor's datasheet. Knowing that the Maximum control input is 3000Nm, and the noise level of yaw rate is considered 1% of the sensor range (1 rad/s), then: )| = 1, and considering the controller as a proportional controller 𝐶𝐶 0 , then it is easy to calculate an upper bound for the crossover frequency 𝜔𝜔 𝑢𝑢 . In this study and for the nominal conditions, 𝜔𝜔 𝑢𝑢,𝑚𝑚𝑎𝑎𝑥𝑥 = 82𝑀𝑀𝑎𝑎𝐺𝐺/𝑠𝑠. To manage the performances related to tracking, regulation, and control effort, the sensitivity functions must fulfill the following inequality constraints for all parametric states of the plant.

� 𝑈𝑈 𝑧𝑧 (𝑗𝑗𝜔𝜔) 𝑁𝑁 𝑚𝑚 (𝑗𝑗𝜔𝜔) � = 𝐶𝐶𝐶𝐶(𝑗𝑗𝜔𝜔) ≤ � 10%𝑈𝑈 𝑧𝑧max 𝑁𝑁 𝑚𝑚0 �, (
|𝐶𝐶𝐶𝐶(𝑗𝑗𝜔𝜔)| ≤ � 0.
To limit the highest resonant peak: ∀𝜔𝜔 ∈ ℝ + : |𝑇𝑇 (𝑗𝑗𝜔𝜔)| < 𝑇𝑇 𝑚𝑚𝑎𝑎𝑥𝑥 = 6 dB.

(5.24)

To limit the lowest modulus margin: (5.27)

∀𝜔𝜔 ∈ ℝ + : |𝐶𝐶(𝑗𝑗𝜔𝜔)| < 𝐶𝐶 𝑚𝑚𝑎𝑎𝑥𝑥 = 6dB. ( 5

CRONE Controller

In this study, the third generation CRONE controller must be used because of the presence of high variation of gain and phase about the desired open-loop crossover frequency 𝜔𝜔 𝑢𝑢 . The same methodology used in Chapter 4 is applied here to design the 3 rd generation CRONE controller 𝐶𝐶 1 (𝑠𝑠).

For good tracking for ramp inputs references and better filtering of measurements noise, it is chosen 𝑛𝑛 𝑙𝑙 = 2 and 𝑛𝑛 ℎ = 2, respectively. Using the CRONE Toolbox, the remaining parameters are calculated as follows: 𝜔𝜔 𝑟𝑟 = 𝜔𝜔 0 = 10rad/s, 𝑌𝑌 𝑟𝑟 = 7, 𝜔𝜔 1 =25rad/s, 𝑎𝑎 0 = 0.345, and 𝑏𝑏 0 = 2.4. The resulting open-loop crossover frequency is 𝜔𝜔 𝑢𝑢 = 19.2 rad/s for the nominal plant.

One of the main seeking objectives while designing the parametric configuration of the CRONE controller is to choose the smallest possible crossover frequency that satisfies all the aforementioned criteria. This concept explains the small value of 𝜔𝜔 𝑢𝑢 obtained compared to the maximum allowable value of 𝜔𝜔 𝑢𝑢 . In fact, this option has several advantages related to noise measurement rejection at high frequencies, sensors selection, and facilitates the work of the actuators that inevitably have inherent limits on its dynamics.

PID Controller

The PID controller was designed so that we get identical 𝜔𝜔 𝑢𝑢 as for the CRONE controller. The obtained PID controller has the following cascade form:

𝐶𝐶 𝑃𝑃𝐼𝐼𝑅𝑅 (𝑠𝑠) = 𝐶𝐶 0𝑃𝑃𝐼𝐼𝑅𝑅 � 1 + 𝑠𝑠 𝜔𝜔 𝑖𝑖 ⁄ 𝑠𝑠 𝜔𝜔 𝑖𝑖 ⁄ � 2 1 + 𝑠𝑠 𝜔𝜔 𝑏𝑏 ⁄ 1 + 𝑠𝑠 𝜔𝜔 ℎ ⁄ �1 + 𝑠𝑠 𝜔𝜔 𝑓𝑓 ⁄ � -1 , (5.28)
where 𝐶𝐶 0𝑃𝑃𝐼𝐼𝑅𝑅 is a proportional gain and 𝜔𝜔 𝑖𝑖 is the integral transitional frequency. The parameters of the phase-lead compensator are chosen in order to have the nominal open-loop transfer function tangent to the 3dB contour on Nichols chart, and finally, a low pass filter is added with a cutoff frequency 𝜔𝜔 𝑓𝑓 . The resulting PID controller parameters are: 𝐶𝐶 0𝑃𝑃𝐼𝐼𝑅𝑅 =3.6e5, 𝜔𝜔 𝑖𝑖 = 2.4rad/s, 𝜔𝜔 𝑓𝑓 = 192𝑀𝑀𝑎𝑎𝐺𝐺/s, 𝜔𝜔 𝑏𝑏 = 75rad/s, and 𝜔𝜔 ℎ = 0.5rad/s. The fact that the 3 rd generation CRONE controller can bring more phaselead after 𝜔𝜔 𝑢𝑢 than the PID controller, provides more robustness against uncertain parameters. It is obvious that the variation of 𝑀𝑀 𝑟𝑟 is smaller for the CRONE controller compared to the PID controller. Moreover, it can be noticed that the variation of the phase margin 𝑀𝑀 𝜙𝜙 is also small compared to that of the PID controller. Numerically, with the CRONE controller, the cross over frequency varies between 12.8 and 26.6rad/s versus plants' parametric variations, while with the PID it varies between 14.2 and 23rad/s. On the other hand, the phase margins vary in the range 48.4 -72.2° with the CRONE controller, while with PID it varies between 3.92 -53.4°.

Frequency-domain analysis

The constraints imposed on the sensitivity functions for the design of the controllers are represented with the dashed black lines in Figures 5.10 and 5.11,for CRONE and PID controller,respectively. They also show the magnitude of each sensitivity function considering the uncertain plants with the designed controller. Table 5.2 shows the maximum gain of each sensitivity function considering the perturbed plants. Since all of them satisfy the assigned constraints with CRONE controller, then robustness is ensured, in contrast to the case of PID controller.

Test simulations

In order to evaluate the performance of the designed controller in stabilizing the vehicle, several testing scenarios were done. Particularly here, only the lateral and longitudinal dynamics were considered while neglecting the vertical dynamics of the vehicle. Later on, the effect of the vertical dynamics on the lateral stability is studied separately in the next sections.

The first scenario (A) consists of a vehicle driven at an initial speed of 55km/h, on dry and even road. The vehicle parameters are considered different from their nominal values to examine the robustness of the controllers against vehicle's parametric variations. The electric vehicle goes through a sudden change lane maneuver whose steering profile plotted in Figure 5.12. The lane change starts at t=0.5s, during the maneuver and about t=0.7s, the vehicle goes through an ice patch (𝜇𝜇 = 0.1) and lasts for 1.5s. The maneuver finishes at t=2s. Figure 5.13 shows a comparison of the different yaw motions. Although of low vehicle speed adopted in this scenario, the uncontrolled system shows a high deviation from the generated reference of yaw rate. The CRONE controller shows a more stable performance where we can see smaller peak oscillations around the reference (t=1.25s to t=2.5s) compared to the performance of the PID controller that reveals a tendency to lose stability. Although both controllers have consistent trajectory response, as shown in Figure 5.14, the CRONE controller demanded less control effort for yaw rate reference tracking as shown in Figure 5.15. In fact, the steady-state error in Figure 5.14 between the reference trajectory and the controlled system trajectories is understandable and expected since the control objective is to track the yaw rate reference and not the lateral position of the reference model. For this purpose, another controller is needed in which it uses steering control to track the reference values of the lateral position of the vehicle, and this point will be discussed in later sections. Numerically, the maximum values of 𝑈𝑈 𝑐𝑐 (𝑇𝑇) are 1771 and 2565Nm for CRONE and PID, respectively. Moreover, the RMS values of 𝑈𝑈 𝑐𝑐 (𝑇𝑇) were recorded 445 and 880Nm for CRONE and PID respectively. These results reflect the robustness of the designed 3 rd generation CRONE controller against parametric variation, mainly, the road adhesion, compared to the PID controller.

Yaw rate regulation by active steering

Introduction

In some scenarios, the drivers may not be capable to maintain vehicle's stability when they want to turn the vehicle, where might easily skid and spin out, possibly resulting in a road accident. The main reason is that the nominal yaw rate motion during turning is difficult to be achieved in the presence of external disturbances and parametric variations.

As shown in the previous sections that the vehicle's stability is highly enhanced with braking torques, for modern vehicles, the yaw rate can also be controlled by adjusting the steering angle with an active front steering (AFS) system. In addition to the steering wheel mechanism, an electromechanical actuator (usually a geared DC motor) is installed on the steering column, which can provide a superposition angle to the steering wheel angle to adjust the actual steering angle. The yaw rate can then be corrected via proper control of the active front steering (AFS) actuator. In order to guarantee the vehicle's yaw stability with the AFS system, the objective of the vehicle yaw rate controller is to determine an appropriate front steering angle for the AFS-actuator to adjust yaw rate dynamics [START_REF] Wong | Adaptive Control of Vehicle Yaw Rate with Active Steering System and Extreme Learning Machine -A Pilot Study[END_REF].

Synthesis model

Taking advantage of the previous illustration and models' equations, and using the state-space representation of the lateral dynamics of the electric vehicle, the synthesis model for the active steering control can be formulated while taking into consideration the third input of vector 'B' in (5.18), that is the steering input 𝑢𝑢 𝛽𝛽 (𝑇𝑇).

Therefore, the model used for control design in the frequency-domain is represented by the transfer function 𝐺𝐺 2 (𝑠𝑠) and can be written as follows: (5.31)

𝐺𝐺 2 (𝑠𝑠) = 𝛹𝛹 ̇(𝑠𝑠) 𝑈𝑈 𝛽𝛽 (𝑠𝑠) � 𝑓𝑓 𝑦𝑦0
The range of variation of each uncertain parameter is shown in Table 5.3. According to these variations, three plants were chosen and named: the medium plant 𝐺𝐺 2𝑚𝑚𝑟𝑟𝑑𝑑 , minimum plant 𝐺𝐺 2𝑚𝑚𝑖𝑖𝑛𝑛 , and maximum plant 𝐺𝐺 2𝑚𝑚𝑎𝑎𝑥𝑥 . Figure 5.16 shows the influence of the uncertainty in the frequency domain, in addition to the frequency response of the three perturbed plants.

Control Design

Specifications

Several specifications are required for the controllers:

• Control input related to measurement noise less than 10% of the greatest value of the control input. • The nominal overshoot of step response related to the reference signal is about 30%. • Use the smallest possible value 𝜔𝜔 𝑐𝑐 .

• Maximum limit of introduced steering effort is max�𝑢𝑢 𝛽𝛽 � < 10°. (5.32)

The maximum control input considered in this study is max (|𝑢𝑢 𝛽𝛽| ) =10°,

and the maximum noise level of yaw rate ( ) t ψ is considered around 𝑁𝑁 𝑚𝑚0 = 0.01rad/s, that corresponds to 1% of sensor range (1 rad/s). Thus, the control input sensitivity can be formulated as the following: In this study and for the nominal conditions, the upper bound for the crossover frequency is obtained as 𝜔𝜔 𝑢𝑢,𝑚𝑚𝑎𝑎𝑥𝑥 = 164𝑀𝑀𝑎𝑎𝐺𝐺/𝑠𝑠. To manage the performances related to tracking, regulation and control effort, the sensitivity functions must fulfill the following inequality constraints for all the parametric states of the plant.

To limit the highest resonant peak: ∀𝜔𝜔 ∈ ℝ + : |𝑇𝑇 (𝑗𝑗𝜔𝜔)| < 𝑇𝑇 𝑚𝑚𝑎𝑎𝑥𝑥 = 3.5dB.

(5.34)

To limit the lowest modulus margin: (5.37)

CRONE Controller

According to the frequency response of the plant model, the third generation CRONE controller must be used because of the presence of high variation of gain and phase about the desired open-loop crossover frequency 𝜔𝜔 𝑢𝑢 . It should be noted that the 'med' plant is used as a nominal plant for control synthesis. Using the same methodology described in Chapter 4, the 3 rd generation CRONE controller 𝐶𝐶 2 (𝑠𝑠) was designed.

In order to obtain a good tracking for ramp inputs references and better filtering of measurements noise, it was chosen 𝑛𝑛 𝑙𝑙 = 2 and 𝑛𝑛 ℎ = 2 respectively. Using the CRONE Toolbox, the rest parameters are calculated as follows: 𝜔𝜔 𝑟𝑟 = 80 rad/s, 𝜔𝜔 0 = 5 rad/s, 𝑌𝑌𝑀𝑀 = 1.2 , 𝜔𝜔 1 =500rad/s, 𝑎𝑎0 = 1.43 , and 𝑏𝑏0 = -0.48. The resulting open-loop crossover frequency is 𝜔𝜔 𝑢𝑢 = 88.6 rad/s for the nominal plant.

PID Controller

The PID controller was designed so we get identical 𝜔𝜔 𝑢𝑢 , and in the same methodology illustrated in section 5.3.3. The obtained PID control parameters are: 𝐶𝐶 0𝑃𝑃𝐼𝐼𝑅𝑅 =1.51, 𝜔𝜔 𝑖𝑖 = 7.38rad/s, 𝜔𝜔 𝑓𝑓 = 177𝑀𝑀𝑎𝑎𝐺𝐺/s, 𝜔𝜔 𝑏𝑏 = 126.7rad/s, and 𝜔𝜔 ℎ = 62rad/s. The Nichols chart of the 3 rd generation CRONE controller shows more robustness against uncertain parameters compared to the PID controller. It is obvious that the variation of the 𝑀𝑀 𝑟𝑟 is smaller for the CRONE controller compared to the PID controller. Moreover, it can be noticed that the variation of the phase margin 𝑀𝑀 𝜙𝜙 is also small compared to that of the PID controller. Numerically, with the CRONE controller, the crossover frequency varies between 52.7 and 127 rad/s versus plants' parametric variations, while with PID it varies between 53 and 125 rad/s. On the other hand, the phase margins vary in the range 40 -42.8° with the CRONE controller, while with PID it varies between 41.2 -46° °.

Frequency analysis

The constraints imposed on the sensitivity function for the design of the controllers are represented with the dashed black lines in Figure 5.22 and 5.23,with CRONE and PID respectively,in addition to the magnitude of each sensitivity function considering the uncertain plants with the designed controller. Table 5.4 shows the maximum gain of each sensitivity function considering the perturbed plants.

Simulation results

The scenario (A) simulated in section 5.4 is also adopted here. The only difference is that the road adhesion dropped down from 𝜇𝜇 = 1 (dry road) to 𝜇𝜇 = 0.5 (wet road). The simulation compares three systems: an uncontrolled system, a system controlled by 𝐶𝐶 2𝑃𝑃𝐼𝐼𝑅𝑅 (𝑠𝑠), and a system controlled by the 3 rd generation CRONE controller 𝐶𝐶 2 (𝑠𝑠).

Figure 5.24 shows the yaw motion response of the different systems. It shows that both designed controllers were able to track the generated yaw rate reference, unlike the uncontrolled system that performs far from the reference. The main reason for this deviation for the passive system caused by the low road adhesion introduced. The additive steering angle 𝑢𝑢 𝛽𝛽 (𝑇𝑇) provided by the controllers, as shown in Figure 5.25, allows the vehicle to perform around the generated reference. Figure 5.26 plots the vehicle trajectories where it reflects the enhancement of the controlled system compared to the passive one. In fact, this controller is applicable for a reasonable road adhesion. This means that for lower road adhesion (𝜇𝜇 < 0.35), this control strategy is no longer effective. In order for the steering dynamics to influence the vehicle dynamics needs a tolerable amount of grip. This is not the case when the road adhesion is very low, in which the grip is no longer enough for traction and steering at the same time. For this reason, for future studies when control coordination will be implemented, this kind of controllers (yaw rate regulation using active steering) should be deactivated when a specified threshold is exceeded. In this situation, the yaw rate (or sideslip) regulation by differential braking torques is more efficient.

Lateral position Control

Introduction

The objective of this study is to control the lateral position 𝑌𝑌 𝐺𝐺 (𝑇𝑇) of the center of gravity of the vehicle to track a reference lateral position 𝑌𝑌 𝐺𝐺𝑟𝑟𝑟𝑟𝑓𝑓 generated by a nominal model whose parametric state does not vary, unlike the validation model which present parametric uncertainties.

The previous stability control shows a good performance in tracking yaw rate reference, and therefore, an important enhancement in stability and safety using introduced braking torques or active steering angle. However, it was shown that the trajectories of the active systems, although they address vehicle stability, they can't perfectly attain the best trajectory. In some situations, it went 1m far from the generated reference trajectory. This result could be critical in real scenarios and a controller for the lateral position is then needed to ensure the vehicle's safety. The designed controller is addressed individually, in which the study neglects the other controllers introduced in previous sections.

Synthesis model

Following the same modeling process detailed in section 5.2, the synthesis model for lateral position regulation can be formulated with the help of the previous synthesis models and state-space representation in section 5.2.5.

Assuming that the longitudinal velocity is constant and that the equilibrium of the lateral position is 0, the lateral dynamics of the vehicle can be formulated as follows: 𝑎𝑎̇𝐺𝐺(𝑇𝑇) = 𝑉𝑉 𝑥𝑥0 𝜓𝜓(𝑇𝑇) + 𝑣𝑣 𝑦𝑦 (𝑇𝑇) .

( 5.38) Then, the synthesis model or the plant transfer function 𝐺𝐺 3 (𝑠𝑠) related to lateral position control can be obtained using the following:

𝐺𝐺

Uncertain Plants of 𝑮𝑮 𝑮𝑮 (𝒔𝒔)

The range of variation of each uncertain parameter is similar to what shown in Table 5.3 since it uses the same actuator (steering) for regulation. According to these variations, three plants were chosen and named as: the medium plant, minimum plant, and maximum plant. Figure 5.27 shows the influence of the uncertainty in the frequency domain.

Control design

Figure 5.28 shows the control scheme for the lateral position regulation using the active steering control input 𝑢𝑢 𝑦𝑦 (𝑇𝑇). The open-loop crossover frequency is chosen to be less then 10rad/s. For this reason, the third generation CRONE controller must be used. It should be noted that the 'med' plant is used as a nominal plant for control synthesis. Following the same process of the previous methodology, the parameters of the 3 rd generation CRONE controller 𝐶𝐶 3 (𝑠𝑠) are calculated. For 𝑛𝑛 𝑙𝑙 = 3 and 𝑛𝑛 ℎ = 2, the control parameters obtained are: 𝜔𝜔 𝑟𝑟 = 8.21rad/s, 𝑌𝑌 𝑟𝑟 = 0.01, 𝜔𝜔 -1 =2.3rad/s, 𝑎𝑎 -1 = 3.7 , 𝑏𝑏 -1 = 3.8 , 𝜔𝜔 0 =2.57rad/s, 𝑎𝑎 0 = -1.33 , 𝑏𝑏 0 = -1.75 , 𝜔𝜔 1 =6rad/s, 𝑎𝑎 1 = 2.18 , 𝑏𝑏 1 = 0.55 , and 𝜔𝜔 2 = 12 rad/s. The resulting open-loop crossover frequency is 𝜔𝜔 𝑢𝑢 = 8.3 rad/s for the nominal plant. Then the PID control parameters are computed as follows: 𝜔𝜔 𝑖𝑖 = 0.28 rad/s, 𝜔𝜔 𝑏𝑏 = 1.43 rad/s, 𝜔𝜔 ℎ = 48 rad/s, and 𝐶𝐶 0 = 0.246.

Test simulations

In order to assess the performance of the lateral position control, a vehicle driven with an initial speed of 55km/h is simulated with a steering profile similar to that used in scenarios (A). At t=1s, a lateral wind gust force of 400N hit the vehicle and last for 1.5s. In this simulation, the road is assumed perfectly dry and even.

Figure 5.33 plots the yaw rate performance of the controlled system and the passive system compared to the generated reference yaw rate. In this study, the objective is not to track the yaw rate, so it is anticipated that the controlled systems are not tracking the yaw rate reference. However, it can be seen that they have less deviation compared to the uncontrolled system. Figure 5.34 shows the active steering control angle of both controllers. Figure 5.35 shows the lateral acceleration of the controlled system compared to the passive one, and Figure 5.36 plots the vehicle trajectories. It can be deduced that the control systems are perfectly tracking the reference trajectory but at the expense of higher lateral acceleration.

In some situations, higher values of the lateral acceleration mean degradation in vehicle's stability. For this reason, a compromise must be done between the tracking trajectory and vehicle stability based on the sensing data from the environment and the situation in which the vehicle is operating.

Effect of the Vertical dynamics on the lateral stability control

The objective of the following section is to analyze the effect of the vertical dynamics of the vehicle on the designed lateral stability control systems. For this purpose, a simulation was done on a complete 14DoF vehicle model that includes a passive suspension configuration and compared to the results obtained with a model neglecting the vertical dynamics.

Scenario (A) is replicated here and only the CRONE controllers are considered. Then, a comparison is done between a system with the vertical dynamics, denoted by (with VD) and another one neglecting the vertical dynamics (without VD). It should be noted that the performance of the passive system that appears in the following figures includes the effect of the vertical dynamics.

In the first simulation, the controlled system is regulating the yaw rate dynamics by braking torques, that is 𝐶𝐶 1 (𝑠𝑠) is used. Figure 5.37 shows the yaw rate response that compares the controlled systems with and without vertical dynamics with the passive system. It is shown that the two controlled systems differ at the end of the maneuvering, approximately around 2 < 𝑇𝑇 < 3𝑠𝑠. It can be noticed that the yaw motions are higher for systems with vertical dynamics. Figure 5.38 shows the control effort of the controllers in both cases, in which it is clear that with vertical dynamics, higher control effort torques are needed to track the generated reference. The reason for increasing the needed control effort for tracking the yaw rate reference is well illustrated in Figures 5.39 and 5.40. In fact, the significant variation of normal wheel forces (Fz) leads to the difference between the lateral forces at each tire between the two compared system. Figure 5.39 shows the lateral forces generated on each tire for the CRONE controlled system, comparing the systems with and without vertical dynamics.

The big difference occurred at instances where the lateral acceleration is higher than 0.4g, namely the emergency operating region, as shown in Figure 5.40. In this region, the high value of lateral acceleration increases the transverse load transfer and this leads to a higher variation of 𝐹𝐹 𝑧𝑧 . Consequently, the system will go far from the nominal behavior of the reference model (bicycle model), which requires a higher control effort to be able to track the generated reference.

These results translated in Figure 5.41: the uncontrolled system went out of the trajectory, where the vehicle lost its maneuverability. On the other hand, the controlled systems attain stability with a small difference in the obtained trajectories between the systems with and without the vertical dynamics. The second simulation will examine the limitations of the designed controller in stabilizing the vehicle. In this scenario, a vehicle with an initial speed of 90km/h will undergo a double lane maneuver (DLM) whose steering profile plotted in Figure 5.42. The road surface is considered perfectly dry, which is 𝜇𝜇 = 1. In fact, this kind of scenario is considered as critical and emergent maneuvering in which the driver is trying to avoid an obstacle. Figure 5.43,5.44,and 5.45 show the yaw rate response, active steering control angle, and the lateral accelerations in the double lane maneuver scenario, respectively. The controlled systems show a good tracking of yaw rate references whether with vertical dynamic or not, with a little advantage to the system without vertical dynamics. However, as shown in Figure 5.44, the control effort of the system without vertical dynamics was saturated for around 0.25s, in addition to the higher mean value of the active steering angle demanded to regulate the yaw rate dynamics. Figure 5.45 shows the high lateral accelerations, that admits the nonlinear behavior of the vehicle (outside the linear zone).

Figure 5.46 shows the difference in the lateral forces between the two systems, where it is obvious that the transverse load transfer is highly influencing the forces distribution. Finally, Figure 5.47 plots the vehicle trajectories in which the uncontrolled system deviates from the reference path and the controlled systems have stabilized the vehicle around the generated reference.

Conclusion

Vehicle lateral instability may be caused by many factors such as a sudden side wind or sudden braking and maneuvering on different road conditions. It is necessary to generate a corrective yaw moment to compensate for the lateral instability under such situations. There are two main methods to achieve a corrective yaw moment: first, by active steering and, second, by active braking. In addition, the control of the lateral position of the vehicle is of high interest nowadays, particularly incongruent with the new trend of autonomous vehicles.

Three control strategies were designed and tested in this chapter in order to enhance the vehicle's stability and lateral position of the vehicle. Each controller was tested individually against external disturbances and parametric variations. The designed controllers proved their ability to attain good performance in stabilizing the vehicle in different testing conditions. The study adopted the assumption of a vehicle with a constant speed and low steering profiles. For this reason, these studies can be extended to consider higher speeds and higher steering profiles as well, where different controllers are expected to be implemented.

Moreover, the effect of vertical dynamics on the lateral stability control was studied in this chapter. It was shown that the influence of the load transfer leads to a significant effect on the stability control in scenarios where the vehicle is no longer operating in the comfort zone, in which the lateral acceleration is higher than 0.4g. In fact, these active control systems were designed independently for improving specific vehicle performance, and so their functions are limited because, mainly, of the non-linearity of tires forces and the essential dynamic coupling between these systems.

Finally, the coordination between these systems is needed to optimize further the overall performance of vehicle dynamics. Nowadays, the integration of these active control systems has become an interesting topic in vehicle dynamics and control fields. For this reason, this study is assigned as future work where a coordinating strategy should be done for yaw rate regulation, that takes into account the actuator dynamics (braking and active steering), combined with the lateral position control, in the sense of improving the overall performance of the electric vehicle's safety and stability.

General Conclusion and Perspectives

General Conclusion

This thesis has been devoted to the problem of improving the vehicle dynamics in terms of comfort, road holding, and safety. Being aware of the new rising trend of autonomous vehicles, and the fact that the world is shifting towards mobile electrification, the study adopted the light electric vehicle as a study platform. Moreover, as all-in-wheels considered one of the promising technologies in this new trend, the Active Wheel structure was introduced.

The thesis has been addressed various studies aiming at enhancing the vertical, longitudinal, and lateral dynamics of the vehicle using the different active systems. For this purpose, several control problems have been solved through the use of the CRONE methodology CSD. Emphasis was placed on using the CRONE controller in order to take advantage of its robustness against plant uncertainties (or perturbations) that are taken into account with no distinction of their nature in control design, whether they are structured (parametric) or unstructured. More importantly, the CRONE studies paved the way for further studies related to fractional-order controllers, which have become of high interest to the author as future concerns.

In summary, the works of the thesis are presented in 3 main parts and 5 chapters as follows:

-In the first chapter, a general introduction to the automotive suspension systems have been presented. Then, some well-known vehicle models, used for design and simulation and focusing on the vertical dynamics, are given. A general introduction of the suspension role and the three generations of CRONE controllers are presented. In addition, it introduced the strategies adopted, the used approach, and the assigned objectives.

-The second Chapter recalls some background regarding the vertical dynamics of the vehicle, where it focuses on the vertical dynamics of the quarter vehicle model. Then, it addressed in detail the frequencydomain studies of this model. The first investigation has been made in designing a low bandwidth control that adjusts the height of the vehicle, and the chassis dynamics as well, that is the roll and pitch angle.

Then, for the purpose of ride comfort and road holding enhancement, two oriented fractional-order damping controllers were designed using the 2 nd generation CRONE control approach.

-In the third chapter, a hierarchical control strategy, based on two-level controllers (global and local), is proposed for the active suspension system embedded in the Active Wheel, for driver disturbance rejection.

The study aims to control the three main chassis dynamics to compensate for the heave, pitch, and roll motion simultaneously by a global controller. Moreover, the study adopted the 2DOF quarter vehicle model that had been controlled by a local controller, in order to investigate the level of disturbance rejection of road unevenness and vibration isolation. The hierarchical approach is then stated as a validated approach for certain road structures and classes.

-In the fourth chapter, investigations have been done on improving the braking performance of ABS control on even and rough terrains. At first, a 3 rd generation CRONE controller is designed to maximize the efficiency of the braking system of Active Wheel. The study takes into consideration the road adhesion variations, vehicle mass variation, and vehicle velocity variations. The final objective of this system is to reduce the braking distances while taking into consideration system limitations and noise measurement criterion. Then, the half-vehicle model is used to study the effect of the load transfer disturbance on the performance of the ABS controller. In the end, the road disturbances were taken into account, and some testing scenarios made to find out its effect on the ABS performance.

In order to minimize the performance deterioration, the fractional-order damping controllers introduced in Chapter 2, are also implemented to test for its influence on the overall system, in its different orientation, i.e. comfort-oriented and road holding oriented controllers.

-In the last chapter, the study is shifted towards lateral dynamics control.

The study was focused on yaw motion control, and the objective is to enhance the performance of the lateral dynamics of the vehicle. Three 3 rd generation CRONE controllers were designed: two for yaw rate regulation using either braking torques or by additive active steering angle, and the third one addresses the lateral position control of the vehicle. Finally, the effect of the vertical dynamics of the chassis was analyzed on the performance of the designed lateral controllers.

Perspectives

During the thesis, several developments were made and some results have been obtained. Besides, regarding the presented works, various studies seem to be of great interest to be continued and developed in the future.

In terms of control methodologies, more rigorous investigations on comparing the designed controllers with other robust controllers are of high value. Mainly in the short term, the priority is given for fractional-order controllers. In this sense, different optimization algorithms are found in the literature, related to control parameters selection, could be used.

For suspension control design and analysis, more strategies have to be designed to fulfill the objectives related to high road classes and singular road disturbances. The study can consider various road characteristics to place a more convenient controller for each case. In fact, due to the continuous improvement on the sensing technologies, the road structure and characteristics can be now extracted and then managed by the vehicle's supervisor. Thus, it facilitates control switching to achieve an optimized suspension performance.

Due to the fact that most controllers consider that the vehicle is operating in the comfort zone, other control strategies should be implemented so that it deals with critical driving scenarios, either in the level of suspension control or lateral stability controls.

Moreover, as mentioned in Chapter 4, the study can be extended to consider the performance of the ABS function in cornering critical scenarios. In fact, the system should also be able to ensure that handling stability and steerability are maintained. The combination of ABS control and suspension control is an area of improvement, where more suspension control strategies might be designed to enhance the overall performance of maneuvering.

Up to now, the suspension systems, steering systems, and braking systems have been studied, mostly, in a separate manner. The well understanding of the overall vehicle dynamics, motivate us for focusing on deep global chassis control studies in the near future. In fact, well-coordinated strategies between the different active systems should inevitably optimize the overall performance of the vehicle in terms of comfort, handling, and safety.

Finally, it is critically important that the study will be extended later on to consider the dynamics of the actuators of different used active systems, and hopefully, a test of the proposed methods on the real platform will be performed someday in the near future…

  : Contrôle ABS, Système de suspension actif, Automobile, Contrôleur CRONE, Contrôle global du châssis, Stabilité latérale, Dynamique du véhicule.

Figure 1 .

 1 Figure 1. The original design of the self-propelled car (Codex Atlanticus, f.812 r) and its replica at museum Clos Lucé, Amboise, France.

Figure 2 .

 2 Figure 2. The steam 'car' designed by Verbiest in 1672.

Figure 3 .

 3 Figure 3. Nicholas-Joseph Cugnot's vehicle, depicted at the famous first road accident at the Arsenal in 1770.

Figure 4 .

 4 Figure 4. Mercedes Benz Patent Motorwagen 1886.

Figure 5 .

 5 Figure 5. 1923 Ford Model T Touring.

Figure 6 .

 6 Figure 6. Global automobile production (cars, trucks, and buses), 1900-2016 (US DOT, 2016).

Figure 7 .

 7 Figure 7. NTHSA -IIHS Percent of MY2017 Fleet Conforming to AEB Voluntary Commitment. (Gotsch, 2018).

Figure 8 .

 8 Figure 8. IIHS/HLDI Predicted Percent of Vehicles Equipped (standard or optionally equipped) with ADAS Technologies.

Figure 9 .

 9 Figure 9. Real World benefits of some ADAS technologies in 2019.

Figure 10 .

 10 Figure 10. SAE j3016 levels of driving automation (SMMT, 2019).

Figure 11 .

 11 Figure 11. VANets mode diagram (Sheikh and Liang, 2019).

Figure 12 .

 12 Figure 12. Waymo's Google self-driving taxis (Google, 2019).

Figure 13 .

 13 Figure 13. Some of PSA Group contributions in Autonomous Vehicles.

Figure 14 .

 14 Figure 14. Projected global electric car stock compared with OEM targets (2020-25) (Till Bunsen et al., 2019).

Figure 15 .

 15 Figure 15. The 2020 Porsche Taycan is Porsche's first-ever electric vehicle.

Figure 16 .

 16 Figure 16. Projected global electric car stock compared with OEM targets (2020-25) (Till Bunsen et al., 2019).

o 1

 1 for vehicle translation o 4 for the rotary motion of the 4 wheels -2DoF for the lateral dynamics o 1 for vehicle translation o 1 for rotational motion around z-axis: yaw motion. -7Dof for vertical dynamics o 1 translation of the chassis center of gravity (CG) in z-direction: heave motion; o 2 rotational motion of the chassis around the y-axis and x-axis: pitch and roll angular rotation. o 4 translations in the z-direction for the 4 unsprung masses (wheel).

Figure 1 . 1 .

 11 Figure 1.1. Schematic of the main vehicle dynamics.

Figure 1 . 2 .

 12 Figure 1.2. Acceleration diagrams: (a) 2D representation (𝑎𝑎 𝑥𝑥 , 𝑎𝑎 𝑦𝑦 ), (b) 3D representation with 𝑎𝑎 𝑧𝑧 .

Figure 1 . 3 .

 13 Figure 1.3. The various type of disturbances applied on the vehicle dynamics.

Figure 1 . 5 .

 15 Figure 1.5. Validation model in MATLAB/Simulink environment.

Figure 1 .

 1 Figure 1.6 shows the general quarter vehicle configuration[START_REF] Létévé | Study of the influence of passenger vehicle suspension on vibrational comfort, driving behaviour and operating limits : CRONE approach to formalization, analysis and synthesis[END_REF], where 𝑚𝑚 2 represents the sprung mass (in kg), 𝑚𝑚 1 represents the unsprung mass (in kg), 𝑓𝑓 0 (𝑇𝑇) is the load transfer forces mainly caused by driver disturbances (in N), and 𝑣𝑣 2 (𝑇𝑇), 𝑣𝑣 1 (𝑇𝑇), and 𝑣𝑣 0 (t) are the velocity of sprung mass, unsprung mass, and road variation at the contact point pneumatic-ground (in m/s), respectively.

Figure 1 . 6 .

 16 Figure 1.6. Schematics of the Quarter vehicle analysis model (Bouvin, 2019).

  Figure 1.7. Representation of the domains of study on a diagram of (𝑎𝑎 𝑥𝑥 , 𝑎𝑎 𝑦𝑦 ).

Figure 1 . 8 .

 18 Figure 1.8. Analysis model of quarter vehicle[START_REF] Létévé | Study of the influence of passenger vehicle suspension on vibrational comfort, driving behaviour and operating limits : CRONE approach to formalization, analysis and synthesis[END_REF].

Figure 1 .

 1 Figure 1.9. Traditional suspension system.

Figure 1 .

 1 Figure 1.10. Main roles of the vehicle suspension system.

Figure 1 . 11 .

 111 Figure 1.11. Representation of different road roughness and obstacles (Bouvin, 2019).

Figure 1 . 12 .

 112 Figure 1.12. Resonance frequencies for different body segments, represented by a simplified mechanical model of a human standing on a vibrating platform (Berlin and Adams, 2017).

Figure 1 . 13 .

 113 Figure 1.13. Classification of electronically controlled suspension[START_REF] Savaresi | Semi-Active Suspension Control Design for Vehicles, Semi-Active Suspension Control Design for Vehicles[END_REF].

Figure 1 .

 1 Figure 1.14. The Architecture of Bose Suspension.

Figure 1 . 15 .

 115 Figure 1.15. Representation of the attitude correction system of the standard Hydractive suspension.

Figure 1 . 16 .

 116 Figure 1.16. Hydraulic implantation of the suspension CREUAT.

Figure 1 .

 1 Figure 1.17. Active Wheel structure.

Figure 1 .

 1 Figure 1.18. Light electric vehicles equipped with Active Wheel: (a) Venturi Volage, (b) Heuliez Will.

Figure 1 .

 1 Figure 1.19. Block-diagram of an output unity negative feedback control system.

Figure 1 .

 1 Figure 1.19 represents the control diagram used for control synthesis. The functional equations associated with this diagram are as follow: -For the output 𝑌𝑌 (𝑠𝑠): 𝑌𝑌 (𝑠𝑠) = 𝐶𝐶(𝑠𝑠)𝑃𝑃 𝑦𝑦 (𝑠𝑠) + 𝐺𝐺𝐶𝐶(𝑠𝑠)𝑃𝑃 𝑢𝑢 (𝑠𝑠) + 𝑇𝑇 (𝑠𝑠)𝑌𝑌 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑠𝑠) -𝑇𝑇 (𝑠𝑠)𝑁𝑁 𝑚𝑚 (𝑠𝑠); (1.8) -For the error 𝜀𝜀(𝑠𝑠): 𝜀𝜀(𝑠𝑠) = -𝐶𝐶(𝑠𝑠)𝑃𝑃 𝑎𝑎 (𝑠𝑠) -𝐺𝐺𝐶𝐶(𝑠𝑠)𝑃𝑃 𝑢𝑢 (𝑠𝑠) + 𝐶𝐶(𝑠𝑠)𝑌𝑌 𝑀𝑀𝑟𝑟𝑓𝑓 (𝑠𝑠) -𝐶𝐶(𝑠𝑠)𝑁𝑁 𝑚𝑚 (𝑠𝑠); (1.9) -For the control input 𝑈𝑈(𝑠𝑠): 𝑈𝑈 (𝑠𝑠) = -𝐶𝐶𝐶𝐶(𝑠𝑠)𝑃𝑃 𝑦𝑦 (𝑠𝑠) -𝑇𝑇 (𝑠𝑠)𝑃𝑃 𝑢𝑢 (𝑠𝑠) + 𝐶𝐶𝐶𝐶(𝑠𝑠)𝑌𝑌 𝑟𝑟𝑟𝑟𝑓𝑓 (𝑠𝑠) -𝐶𝐶𝐶𝐶(𝑠𝑠)𝑁𝑁 𝑚𝑚 (𝑠𝑠) (1.10) where:

Figure 1 .

 1 Figure 1.20 presents the asymptotic Bode diagrams for the non-integer part where:

Figure 1 .

 1 Figure 1.20. Asymptotic Bode diagrams of the (a) fractional and (b) rational CRONE regulator for 𝑚𝑚 ∈ [0, 1].

Figure 1 .

 1 Figure 1.22. Shape of open-loop Nichols plot.

Figure 2 .

 2 Figure 2.2 shows the causal diagram for the different relations between the inputs into the quarter vehicle model, that is: the external forces applied to the sprung mass 𝐹𝐹 0 (𝑠𝑠), the road disturbances 𝑉𝑉 0 (𝑠𝑠), and the control effort applied by the actuator 𝑈𝑈 𝑎𝑎 (𝑠𝑠), and the outputs state variables. Figure 2.3 shows the various sensitivity functions (𝐺𝐺(𝑠𝑠)) that represent the relationships between the input and the outputs of the controlled quarter vehicle model.

Figure 2 . 2 .

 22 Figure 2.2. Block diagram illustrating the controlled QV.

Figure 2 . 3 .

 23 Figure 2.3. Formulation of sensitivity functions between inputs and outputs variables.

Figure 2 . 4 .

 24 Figure 2.4. Block diagram of different sensitive transfer functions (Velocities).

Figure 2 . 5 .

 25 Figure 2.5. Block Diagram of the different sensitive transfer function (Displacements).

Figure 2 .

 2 Figure 2.6 shows the bode responses of the numbered transfer function of Figure2.5. It illustrates the two modes of functionality, namely, the slow mode corresponding to chassis motions or sprung mass motion, and the rapid mode corresponding to the unsprung dynamics or wheel mode. Numerically, for the given parametric configuration, the chassis mode is around 1.22Hz where the wheel mode is approximately 16Hz. These values will change a little bit with the parametric uncertainties of the quarter vehicle model, for front and rear axles as well. However, this characteristic (two modes) usually exists in all touristic vehicles.

Figure 2 . 6 .

 26 Figure 2.6. Bode responses of the numbered transfer functions as described in Figure 2.5.

Figure 2 . 7 .

 27 Figure 2.7. Bode diagram of the transfer functions (a) 𝑇𝑇 20 (𝑠𝑠) and (b) 𝑇𝑇 10 (𝑠𝑠).

Figure 2 . 8 .

 28 Figure 2.8. Analysis model of quarter vehicle model correspondent to transfer functions (a) 𝑇𝑇 20 (𝑠𝑠) and (b) 𝑇𝑇 10 (𝑠𝑠).

Figure 2 .

 2 Figure 2.9. Height regulation synthesis model.

Figure 2 .

 2 Figure 2.11. Bode response of the front axle synthesis model.

Figure 2 .

 2 Figure 2.10. Range of variations of the parameters of the canonical form 𝑃𝑃 (𝑠𝑠).

Figure 2 .

 2 Figure 2.12. Bode response of rear axle synthesis model.

Figure 2 .

 2 Figure 2.13 represents the unity feedback block diagram of the control methodology, where the 𝑍𝑍 2𝑟𝑟𝑟𝑟𝑓𝑓 (𝑠𝑠) is the Laplace transformation of the reference value of the car-body position provided by the supervisor and 𝐶𝐶(𝑠𝑠) is the transfer fucntion of the designed controller.

Figure 2 .

 2 Figure 2.13. Vehicle height regulation's block diagram.

Figure 2 .

 2 Figure 2.14. Illustration of the effect the non-integer order '𝑛𝑛' on bode diagram.

  .15. Starting from 𝑎𝑎 = 5 and 𝑏𝑏 = 10, then: 𝜔𝜔 𝑙𝑙 = 0.52/5 = 0.1038 rad/s, and 𝜔𝜔 ℎ = 2.33 × 10 = 23.3 rad/s. (2.39) Finally, the fractional-order 𝑛𝑛 and gain 𝐾𝐾 of the nominal open-loop transfer function has to ensure the required nominal phase margin 𝑀𝑀 𝜙𝜙 and openloop crossover frequency 𝜔𝜔 𝑐𝑐1 . Thus, they are computed by:

  .41) As a result, the nominal phase margin and open-loop gain crossover frequency are ensured with: 𝑛𝑛 = 1.42 and 𝐾𝐾 = 46.18.

Figure 2 .

 2 Figure 2.15. Vertical sliding of a frequency template (bold red solid line) ensuring the robustness of phase margin 𝑀𝑀 𝜙𝜙 and of resonant peak 𝑀𝑀 𝑟𝑟 .

Figure 2 .

 2 Figure 2.20 and 2.21 present the closed-loop sensitivity functions for front and rear axles, respectively. They show that:

Figure 2 .

 2 Figure 2.16. Open-loop Nichols plot for the 3 front plants.

Figure 2 .

 2 Figure 2.17. Open-loop Nichols plot for the 3 rear plants.

Figure 2 .

 2 Figure 2.18. Open-loop Nichols plot for the 3 front plants.

Figure 2 .

 2 Figure 2.19. Open-loop Nichols plot for the 3 rear plants.

Figure 2 .

 2 Figure 2.20. Closed-loop sensitivity functions for the 3 front plants: 𝑃𝑃 𝑛𝑛𝑛𝑛𝑚𝑚 (-), 𝑃𝑃 𝑚𝑚𝑖𝑖𝑛𝑛 (--), and 𝑃𝑃 𝑚𝑚𝑎𝑎𝑥𝑥 (-•-).

Figure 2 .

 2 Figure 2.21 Closed-loop sensitivity functions for the 3 rear plants: 𝑃𝑃 𝑛𝑛𝑛𝑛𝑚𝑚 (-), 𝑃𝑃 𝑚𝑚𝑖𝑖𝑛𝑛 (--), and 𝑃𝑃 𝑚𝑚𝑎𝑎𝑥𝑥 (-•-)..

Figure 2 .

 2 Figure 2.22. Bode plot of rational controllers' transfer functions

Figure 2 .

 2 Figure 2.24. Closed-loop step responses for the 3 rear plants.

Figure 2 . 26 .

 226 Figure 2.26. Simulation results using the full model of (a) varying height of car-body, (b) attain reference pitch angle, and (c) attain reference roll angle.

Figure 2 .

 2 Figure 2.27. Control diagram of the fractional order damping control.

Figure 2 .

 2 Figure 2.28. Optimized damping order 𝑛𝑛 of front axle for empty vehicle.

Figure 2 .

 2 Figure 2.29. Optimized damping order 𝑛𝑛 of front axle for full loaded vehicle.

Figure 2 .

 2 Figure 2.30. Optimized damping order 𝑛𝑛 of rear axle for empty vehicle.

Figure 2 . 31 .

 231 Figure 2.31. Optimized damping order 𝑛𝑛 of rear axle for full loaded vehicle.

  As the nominal plant leads to the mean magnitude of the open-loop frequency response around 𝜔𝜔 𝑚𝑚 , then:

Figure 2 .

 2 Figure 2.32. FO damping transformation compared to fractional and rational 2 nd generation CRONE controller for 𝑛𝑛 = 0.65.

Figure 2 .

 2 Figure 2.33. FO damping transformation compared to fractional and rational 2 nd generation CRONE controller for 𝑛𝑛 = 1.1.

  Figure 2.34. Bode response of sensitivity functions: (a) 𝐻𝐻 𝑎𝑎 (𝑠𝑠) and (b) 𝐻𝐻 12 (𝑠𝑠), with two fractional damping order 𝑛𝑛 = 0.65 and 𝑛𝑛 = 1.1.

Figure 2 .

 2 Figure 2.35. Simulation results of fractional damping control compared to passive system with sinus road disturbances of frequency 0.1 to 40Hz of amplitude 1mm.

Figure 2 .

 2 Figure 2.36. Simulation results of fractional damping control compared to passive system with sinus road disturbances of frequency 0.1 to 40Hz of amplitude 2mm.

Figure 2 .

 2 Figure 2.37. Simulation results of fractional damping control compared to passive system with sinus road disturbances of frequency 0.1 to 5.5Hz of amplitude 15mm.

Figure 3 . 1 .

 31 Figure 3.1. Active wheel from Michelin.

Figure 3 . 2 .

 32 Figure 3.2. Block Diagram of Global Chassis Control.

Figure 3 . 3 .

 33 Figure 3.3. Comfort operating domain of a vehicle.

Figure 3 .Figure 3 . 4 .

 334 Figure 3.4 illustrates the definition of the three planes/references related to the vertical dynamics of the synthesis model. However, for the range of interest of frequency 0 -5Hz, comfort zone (low frequencies), the three DoF related to heave, pitch, and roll dynamics of the plane 𝑃𝑃 0 is assumed transferred completely to the plane 𝑃𝑃 1 . Therefore:

Figure 3 . 5 .

 35 Simplified schematics illustrating the car body dynamics: (a) Heave, (b) pitch, and (c) roll motions.

  𝑠𝑠) representing the open-loop transfer function such that 𝛽𝛽(𝑠𝑠) = 𝐶𝐶(𝑠𝑠)𝐺𝐺(𝑠𝑠).

Figure 3 . 6 .

 36 Figure 3.6. Frequency template in the open loop Nichols plot.

Figure 3 . 7 .

 37 Figure 3.7. Bode responses of global plants' transfer functions.

  high frequencies 𝜔𝜔 𝐻𝐻𝐹𝐹 , |𝐶𝐶𝐶𝐶(𝑗𝑗𝜔𝜔 𝐻𝐻𝐹𝐹 )| ≈ |𝐶𝐶(𝑗𝑗𝜔𝜔 𝐻𝐻𝐹𝐹 )|, and considering a proportional controller 𝐶𝐶(𝑠𝑠) = 𝐶𝐶 𝑝𝑝 , then it is easy to compute the upper bound of the cross over frequency of each closed-loop system. Given that the open-loop gain at the crossover frequency 𝜔𝜔 𝑐𝑐 is unity, that is: |𝛽𝛽(𝑗𝑗𝜔𝜔 𝑐𝑐 )| = |𝐶𝐶𝐶𝐶(𝑗𝑗𝜔𝜔 𝑐𝑐 ) 𝐺𝐺(𝑗𝑗𝜔𝜔 𝑐𝑐 )| = �𝐶𝐶 𝑝𝑝 𝐺𝐺(𝑗𝑗𝜔𝜔 𝑐𝑐 )� = 1,

Figure 3 . 8 .

 38 Figure 3.8. Bode responses of global controllers' transfer functions.

Figure 3 .

 3 Figure 3.9. Bode responses of global nominal open-loops.

Figure 3 . 10 .

 310 Figure 3.10. Bode responses of local plants.

Figure 3 . 12 .

 312 Figure 3.12. Bode response of open-loop transfer functions of local controllers.

)

  𝑏𝑏(𝑓𝑓 𝑠𝑠21 (𝑇𝑇) + 𝑓𝑓 𝑠𝑠22 (𝑇𝑇)) -𝑎𝑎�𝑓𝑓 𝑠𝑠11 (𝑇𝑇) + 𝑓𝑓 𝑠𝑠12 (𝑇𝑇)� = 𝑢𝑢 𝜑𝜑 (𝑇𝑇), (3.51) 𝑊𝑊�𝑓𝑓 𝑠𝑠21 (𝑇𝑇) -𝑓𝑓 𝑠𝑠22 (𝑇𝑇)� -𝑊𝑊�𝑓𝑓 𝑠𝑠11 (𝑇𝑇) -𝑓𝑓 𝑠𝑠12 (𝑇𝑇)� = 𝑢𝑢 𝜃𝜃 (𝑇𝑇). (3.52)

  then the expression of 𝛼𝛼 𝑅𝑅𝑆𝑆𝑅𝑅 is formulated as follows:𝛼𝛼 𝑅𝑅𝑆𝑆𝑅𝑅 (𝑇𝑇) 𝑢𝑢 𝜃𝜃 (𝑇𝑇) = 𝑊𝑊�𝑓𝑓 𝑠𝑠 12 (𝑇𝑇) -𝑓𝑓 𝑠𝑠 11 (𝑇𝑇)�.(3.54)

  𝑢𝑢 𝑎𝑎,𝑖𝑖𝑠𝑠 (𝑇𝑇) = 𝑢𝑢 𝑖𝑖𝑠𝑠 (𝑇𝑇) + 𝑓𝑓 ̅ 𝑠𝑠 𝑖𝑖𝑠𝑠 (𝑇𝑇) + 𝑓𝑓 ̂𝑠𝑠𝑝𝑝, 𝑖𝑖𝑠𝑠 (𝑇𝑇) (3.57) -Direct control order: apply the inversion law Force -Current of the electric suspension actuator to obtain the value of the reference current. Then introduce the reference current to the electric motor directly.

Figure 3 . 13 .

 313 Figure 3.13. Schematic illustration for the hierarchical approach.

Figure 3 .

 3 Figure 3.14. Frequency response of benchmark systems.

Figure 3 .

 3 Figure 3.15. shows the bode responses of the CRONE controllers compared to the PID controllers for front and rear axles.

Figure 3 .

 3 Figure 3.16 shows the Nichols chart of the open-loop transfer function for the nominal plants with the two controllers. It can be seen that both controllers intersect the 0dB at the same frequency and are tangent to 3dB contour (same phase margins).

Figure 3 . 15 .

 315 Figure 3.15. Bode responses of local CRONE and PID controllers.

Figure 3 . 16 .

 316 Figure 3.16. Nichols chart of open-loop transfer functions of local systems with PID and CRONE controllers.

Figure 3 . 17 .

 317 Figure 3.17. Bode response of 𝑊𝑊 𝑐𝑐 (𝑠𝑠) transfer function.

Figure 3 .

 3 Figure 3.19 and Figure 3.20 show the Nichols charts for the front and rear open-loop transfer functions with CRONE and PID local controllers, respectively. The designed CRONE controller is matching the specification where the open-loop transfer functions are tangent to the 3dB contour ensuring high stability against parametric variations, in contrast to the PID controller, especially for rear plants. Numerically, the resulted maximum variations in

Figure 3 .Figure 3 . 20 .

 3320 19. Nichols chart of the open-loop transfer function of front plants with (a) CRONE and (b) PID controllers. Nichols chart of the open-loop transfer function of front plants with (a) CRONE and (b) PID controllers.

Figure 3 .

 3 Figure 3.21. Sensitivity transfer functions of the sprung mass acceleration relative to road disturbances.

Figure 3 .

 3 Figure 3.22. Sensitivity transfer functions of the suspension deflection relative to road disturbances.

5 . 2 .

 52 The second one is to test the performance of the full vehicle model in rejecting driver disturbance. The parameters of the vehicle used are also here different from the nominal values in order to examine the robustness of the controller as well as its performance (𝑀𝑀 = 820𝑘𝑘𝑔𝑔, 𝑏𝑏 = 6/5𝑚𝑚, 𝑎𝑎 = 4/5𝑚𝑚 , 𝑘𝑘 1,1 = 3𝑚𝑚5 𝑁𝑁/𝑚𝑚, 𝑘𝑘 1,2 = 2𝑚𝑚5 𝑁𝑁/𝑚𝑚, 𝑏𝑏 1 = 50 , 𝑚𝑚 1 = 32 𝑘𝑘𝑘𝑘 , 𝑐𝑐 = 0.71𝑚𝑚 , 𝑘𝑘 2,1 = 12𝑚𝑚3 𝑁𝑁/𝑚𝑚 , 𝑘𝑘 2,2 = 10𝑚𝑚3𝑁𝑁/𝑚𝑚 , ℎ 𝜃𝜃 = ℎ 𝜑𝜑 = 0.5𝑚𝑚 , and 𝑏𝑏 2 = 200 𝑁𝑁𝑠𝑠/𝑚𝑚 for the controlled system).

3. 7 . 1 . 2

 712 Figure 3.23. Simulation results of road bump.

Figure 3 . 25 .

 325 Figure 3.25. Simulation results of CRONE controller with road class A.

Figure 3 . 26 .

 326 Figure 3.26. Simulation results of CRONE controller with road class A-B.

Figure 3 .

 3 Figure 3.27. Driving scenario variables: (a) Steering profile, (b) introduced brake/motor torques, (c) resulted longitudinal and lateral accelerations, and (d) vehicle trajectory.

Figure 3 .

 3 Figure 3.28. Simulation results of the vertical dynamics of CG.

Figure 3 .

 3 Figure 3.29. Simulation results of pitch motion of the chassis.

Figure 3 .

 3 Figure 3.30. Simulation results of roll motion of the chassis.

Figure 3 . 31 .

 331 Figure 3.31. Simulation results of suspension deflections for different controlled systems.

Figure 4 . 1 .

 41 Schematization and configuration: longitudinal bicycle models (a) and front vehicle quarter (b).

Figure 4 . 2 .

 42 Figure 4.2. Functional diagram of the EMB actuator.

Figure 4 . 3 .

 43 Figure 4.3. Black box model of the tire.

  𝑥𝑥 (𝑇𝑇) = 𝑉𝑉 𝑥𝑥 𝑟𝑟 + 𝑣𝑣 𝑥𝑥 (𝑇𝑇) 𝛺𝛺(𝑇𝑇) = 𝛺𝛺 𝑟𝑟 + 𝜔𝜔(𝑇𝑇) 𝑇𝑇 (t) = 𝑇𝑇 𝑟𝑟 + 𝜏𝜏 (𝑇𝑇) 𝐹𝐹 𝑥𝑥 (𝑇𝑇) = 𝐹𝐹 𝑥𝑥 𝑟𝑟 + 𝑓𝑓 𝑥𝑥 (𝑇𝑇) 𝐹𝐹 𝑎𝑎 (𝑇𝑇) = 𝐹𝐹 𝑎𝑎 𝑟𝑟 + 𝑓𝑓 𝑎𝑎 (𝑇𝑇) 𝐹𝐹 𝑟𝑟𝑟𝑟 (𝑇𝑇) = 𝐹𝐹 𝑟𝑟𝑟𝑟 𝑟𝑟 + 𝑓𝑓 𝑟𝑟𝑟𝑟 (𝑇𝑇) 𝐶𝐶 𝑓𝑓𝑓𝑓 (𝑇𝑇) = 𝐶𝐶 𝑓𝑓𝑓𝑓 𝑟𝑟 + 𝑊𝑊 𝑓𝑓𝑓𝑓 (𝑇𝑇)

Figure 4 . 4 .

 44 Figure 4.4. Longitudinal forces as function of tire slip rate according to Pacejka model.

  (4.34) At the point of equilibrium, in steady-state speed 𝑉𝑉 𝑥𝑥 𝑟𝑟 = 𝑉𝑉 𝑥𝑥0 = 50, 70, or 90km/h, all variations are zero. The relationships(4.33) and(4.34) then can be reduced to:

Figure 4 . 5 .

 45 Figure 4.5. Overview of the steps necessary to determine the variables at equilibrium state.

Figure 4 . 6 .

 46 Figure 4.6. Control schema for ABS control loop.

Figure 4 . 7 .

 47 Figure 4.7. Reference generator for wheel angular velocity (flow chart).

Figure 4 . 8 .

 48 Figure 4.8. Second approach of calculating 𝑝𝑝 𝑥𝑥 .

Figure 4 .

 4 Figure 4.9. Variation of 𝑝𝑝 𝑥𝑥 as function of random samples of 𝐹𝐹 𝑧𝑧 and 𝜇𝜇.

Figure 4 . 10 .

 410 Figure 4.10. Bode diagrams of 𝐺𝐺(𝑠𝑠); 𝐺𝐺 𝑛𝑛𝑛𝑛𝑚𝑚 (𝑠𝑠), 𝐺𝐺 𝑚𝑚𝑖𝑖𝑛𝑛 (𝑠𝑠), 𝐺𝐺 𝑚𝑚𝑎𝑎𝑥𝑥 (𝑠𝑠), and the yellow lines representing the variations of magnitude and phase for each frequency considering all parametric variations.

  .65) Its real part determines the phase location at 𝜔𝜔 𝑐𝑐𝑐𝑐 , such that 𝐶𝐶𝑀𝑀𝑔𝑔�𝛽𝛽�𝑗𝑗𝜔𝜔 𝑐𝑐𝑐𝑐 �� = -𝑎𝑎𝜋𝜋/2, and its imaginary part determines the angle with the vertical of the Generalised template on the Nichols chart, as shown in Figure 4.11.

Figure 4 .

 4 Figure 4.11 also shows various Nichols' magnitude contours. The closedloop resonant gain 𝑀𝑀 𝑟𝑟0 corresponds to the value of the contour that is tangent to the nominal open-loop plot. Plant uncertainties are modeled by the ellipses around each frequency 𝜔𝜔, causing an increase on the closed-loop resonant gain 𝑀𝑀 𝑟𝑟𝑚𝑚𝑎𝑎𝑥𝑥1 for the perturbed plants, as illustrated in Figure 4.11a. By choosing an adequate direction of the generalized template it is possible to minimize the effect of the plant uncertainties on the closed-loop resonant gain, as shown by the optimal template in Figure 4.11b.

Figure 4 .

 4 Figure 4.13 shows the Bode response of the open-loop transfer function with the CRONE controller. It is shown that the cross over frequency varies between 88 -90 rad/s versus plants' parametric variations. On the other hand, the phase margins vary in the range of 49 -57 °.Figure 4.14 shows the Nichols diagram of the open-loop plants. It can be shown that the nominal transfer function (in green) is tangent to the 3dB Nichols magnitude contour, according to the design requirement at the open-loop gain 𝑌𝑌 𝑟𝑟 .

  Figure 4.13 shows the Bode response of the open-loop transfer function with the CRONE controller. It is shown that the cross over frequency varies between 88 -90 rad/s versus plants' parametric variations. On the other hand, the phase margins vary in the range of 49 -57 °.Figure 4.14 shows the Nichols diagram of the open-loop plants. It can be shown that the nominal transfer function (in green) is tangent to the 3dB Nichols magnitude contour, according to the design requirement at the open-loop gain 𝑌𝑌 𝑟𝑟 .

Figure 4 . 12 .

 412 Figure 4.12. Bode response of the fractional and rational from of the controller 𝐶𝐶 𝐴𝐴𝐵𝐵𝑆𝑆 (𝑠𝑠) (toolbox).

Figure 4 . 13 .

 413 Figure 4.13. Bode response of open-loop transfer functions with CRONE controller.

Figure 4 . 14 .

 414 Figure 4.14. Nichols diagram for the open-loop transfer functions.

Figure 4 . 15 .

 415 Figure 4.15. Frequency responses of sensitivity functions.

Figure 4 .

 4 [START_REF]Nichols chart of open-loop transfer functions of local systems with PID and CRONE controllers[END_REF] shows the bode response of the controllers compared to the designed third-generation CRONE controller.

Figure 4 .

 4 17 plots the Bode response on the openloop transfer functions of the nominal plants with the three controllers. It can be seen that all of them have the same crossover frequency and stability phase margin.

Figure 4 . 16 .

 416 Figure 4.16. Bode response of the controllers' transfer function.

Figure 4 . 17 .

 417 Figure 4.17. Bode response of the open-loop nominal transfer functions with controllers.

Figure 4 .

 4 Figure 4.18. Longitudinal velocity and wheel angular velocity of the quarter car in braking scenario (A).

Figure 4 .

 4 Figure 4.19. Slip rate of the quarter car in Scenario (A).

Figure 4 . 20 .

 420 Figure 4.20. Control input for the quarter car in braking scenario (A).

Figure 4 .

 4 Figure 4.21. Power spectrum density of control efforts in braking scenario (A).

Figure 4 .

 4 Figure 4.22. Longitudinal velocity and wheel angular velocity of quarter car in braking scenario (B).

Figure 4 .

 4 Figure 4.23. Slip rates of quarter car in braking scenario (B).

Figure 4 . 24 .

 424 Figure 4.24. Control input for the quarter car in braking scenario (B).

Figure 4 . 25 .

 425 Figure 4.25. Power spectrum density of control efforts in braking scenario (B).

Figure 4 .

 4 Figure 4.27a shows the wheel slip rate behaviors of the two systems for front and rear axles. Figure4.27b plots the vehicle longitudinal velocity where it shows that the two systems have identical dynamics except for a small duration around 𝑇𝑇 = 3𝑠𝑠. In Figure4.28a and b, the wheel normal forces and the longitudinal forces are plotted respectively. Due to the variation of the load transfer at each axle, it influences the wheel contact forces, and as a result, we could observe the difference between the two systems in the values of the longitudinal forces at each wheel. As a result, the braking difference of these systems is recorded as 64.41 and 64.74m, for a system without and with the suspension system, respectively.

Figure 4 .

 4 Figure 4.27a shows the wheel slip rate behaviors of the two systems for front and rear axles. Figure4.27b plots the vehicle longitudinal velocity where it shows that the two systems have identical dynamics except for a small duration around 𝑇𝑇 = 3𝑠𝑠. In Figure4.28a and b, the wheel normal forces and the longitudinal forces are plotted respectively. Due to the variation of the load transfer at each axle, it influences the wheel contact forces, and as a result, we could observe the difference between the two systems in the values of the longitudinal forces at each wheel. As a result, the braking difference of these systems is recorded as 64.41 and 64.74m, for a system without and with the suspension system, respectively.

Figure 4 .

 4 27. Braking performance of the system with and without vertical dynamics: (a) wheel slip rate and (b) the vehicle longitudinal velocity, front axle (-) and rear axle (--).

Figure 4 .

 4 28. Braking performance of the system with and without vertical dynamics: front axle (-) and rear axle (--).

Figure 4 .

 4 29. Braking performance of ABS controlled systems with and without vertical dynamics: (a) wheel slip rate and (b) the vehicle longitudinal velocity, front axle (-) and rear axle (--).

Figure 4 .

 4 30. Braking performance of ABS controlled systems with and without vertical dynamics: front axle (-) and rear axle (--).

Figure 4 .

 4 Figure 4.31. Wheel contact forces with ABS control and FO damping control.

Figure 4 .

 4 Figure 4.32. Longitudinal forces with ABS control and FO damping control.

Figure 5 . 1 .

 51 Figure 5.1. Four-wheel model schematization.

Figure 5 . 2 .,

 52 Figure 5.2. Black box model of the tire.

Figure 5 . 3 .

 53 Figure 5.3. Lateral forces as function of wheel sideslip angle.

Figure 5 . 4 .

 54 Figure 5.4. Bode responses of uncertain plants of 𝐺𝐺 1 (𝑠𝑠).

Figure 5

 5 Figure 5.5 shows the control scheme for the yaw rate regulation using braking torques. In order to export these specifications in the frequency domain, the following functional equations associated with Figure5.5 are introduced.

. 25 )

 25 To limit the control effort: ∀𝜔𝜔 ∈ ℝ + : |𝐶𝐶𝐶𝐶(𝑗𝑗𝜔𝜔)| < 𝐶𝐶𝐶𝐶 𝑚𝑚𝑎𝑎𝑥𝑥 = 85.5 dB.(5.26)To improve the input disturbance rejection:∀𝜔𝜔 ∈ ℝ + : |𝐺𝐺𝐶𝐶(𝑗𝑗𝜔𝜔)| < 𝐺𝐺𝐶𝐶 𝑚𝑚𝑎𝑎𝑥𝑥 = -68 dB.

Figure 5 .

 5 Figure 5.6 shows the Bode plot responses of the transfer functions of the two controllers, and Figure 5.7 plots the Bode response of the nominal openloop transfer functions with the two controllers.Figure 5.8 and 5.9 show the Nichols chart of the open-loop transfer functions for the three uncertain plants with CRONE and PID controllers, respectively. It can be seen that the nominal open-loop transfer function for both controllers (in green) is tangent to the 3dB contour.

Figure 5 .

 5 Figure 5.6 shows the Bode plot responses of the transfer functions of the two controllers, and Figure 5.7 plots the Bode response of the nominal openloop transfer functions with the two controllers.Figure 5.8 and 5.9 show the Nichols chart of the open-loop transfer functions for the three uncertain plants with CRONE and PID controllers, respectively. It can be seen that the nominal open-loop transfer function for both controllers (in green) is tangent to the 3dB contour.

Figure 5 . 7 .

 57 Figure 5.7. Bode response of the nominal open-loop transfer function.

Figure 5 . 8 .

 58 Figure 5.8. Nichols chart for open-loop transfer functions with CRONE controller 𝐶𝐶 1 (𝑠𝑠).

Figure 5 .

 5 Figure 5.9. Nichols chart for open-loop transfer functions with PID controller.

Figure 5 . 10 .

 510 Figure 5.10. Frequency responses of sensitivity functions with 𝐶𝐶 1 (𝑠𝑠).

Figure 5 . 11 .

 511 Figure 5.11. Frequency responses of sensitivity functions with 𝐶𝐶 1𝑃𝑃𝐼𝐼𝑅𝑅 (𝑠𝑠).

Figure 5 .

 5 Figure 5.12. Steering profile of scenario (A).

Figure 5 .

 5 Figure 5.13. Yaw rate motions of scenario (A).

Figure 5 .

 5 Figure 5.14. Vehicle trajectories of scenario (A).

Figure 5 . 15 .

 515 Figure 5.15. Yaw moment control efforts of scenario (A).

Figure 5 . 16 .

 516 Figure 5.16. Bode responses of uncertain plants of 𝐺𝐺 2 (𝑠𝑠).

Figure 5 .

 5 Figure 5.17 shows the control scheme for the yaw rate regulation using active steering.

Figure 5 .

 5 Figure 5.17. Functional diagram of yaw rate regulation by active steering.

  input disturbance rejection:∀𝜔𝜔 ∈ ℝ + : |𝐺𝐺𝐶𝐶(𝑗𝑗𝜔𝜔)| < 𝐺𝐺𝐶𝐶 𝑚𝑚𝑎𝑎𝑥𝑥 = 10dB.

Figure 5 .

 5 Figure 5.18 and 5.19 show the Bode plot of the two controllers and of the response of the open-loop transfer functions with the two controllers, respectively. Figure 5.20 and 5.21 show the Nichols chart of the open-loop transfer function for the three uncertain plants for both controllers.

Figure 5 .

 5 Figure 5.18. Bode plot for designed controllers of 𝐺𝐺 2 (𝑠𝑠) plant.

Figure 5 .

 5 Figure 5.19. Bode response of the nominal open-loop transfer function of active steering systems.

Figure 5 . 20 .

 520 Figure 5.20. Nichols chart for open-loop transfer functions with CRONE controller 𝐶𝐶 2 (𝑠𝑠).

Figure 5 .

 5 Figure 5.21. Nichols chart for open-loop transfer functions with 𝐶𝐶 2𝑃𝑃𝐼𝐼𝑅𝑅 (𝑠𝑠).

Figure 5 .

 5 Figure 5.22. Frequency responses of sensitivity functions with 𝐶𝐶 2 (𝑠𝑠).

Figure 5 .

 5 Figure 5.23. Frequency responses of sensitivity functions with 𝐶𝐶 2𝑃𝑃𝐼𝐼𝑅𝑅 (𝑠𝑠).

Figure 5 .

 5 Figure 5.24. Yaw rate motions with active steering control.

Figure 5 . 25 .

 525 Figure 5.25. Control active steering angle.

Figure 5 . 26 .

 526 Figure 5.26. Vehicle trajectories with active systems compared to passive and reference models.

Figure 5 .

 5 Figure 5.27. Bode responses of uncertain plants of G3(s).

Figure 5 .

 5 29 shows the Bode plot of the transfer function of the two controllers and Figure 5.30 presents the Bode response of the open-loop transfer functions with the two designed controllers.

Figures 5 .

 5 Figures 5.18 and 5.19 show the Nichols chart of the open-loop transfer function for the three uncertain plants for both controllers. It can be seen that the open-loop transfer function for the nominal plant for both controllers is tangent to the 3dB contour.

Figure 5 .Figure 5 .

 55 Figure 5.28. Functional diagram of lateral position control by active steering

Figure 5 .

 5 Figure 5.30. Bode plot of the open-loop transfer functions with both controllers.

Figure 5 . 31 .

 531 Figure 5.31. Nichols chart for open-loop transfer functions with CRONE controller 𝐶𝐶 3 (𝑠𝑠).

Figure 5 .

 5 Figure 5.32. Nichols chart for open-loop transfer functions with 𝐶𝐶 3𝑃𝑃𝐼𝐼𝑅𝑅 (𝑠𝑠).

Figure 5 .

 5 Figure 5.33. Yaw motion with lateral position control.

Figure 5 .

 5 Figure 5.34. Active steering control angles.

Figure 5 .

 5 Figure 5.35. Lateral acceleration with lateral position control.

Figure 5 .

 5 Figure 5.36. Vehicle trajectories with lateral position control.

Figure 5 .

 5 Figure 5.37. Yaw rate motions with CRONE controller.

Figure 5 .

 5 Figure 5.38. Yaw moment control efforts.

Figure 5 .

 5 Figure 5.39. Lateral forces of four tires.

Figure 5 .

 5 Figure 5.40. Lateral accelerations comparison.

Figure 5 .

 5 Figure 5.41. Vehicle's trajectory.

Figure 5 .

 5 Figure 5.42. Steering profile of double lane maneuver.

Figure 5 .

 5 Figure 5.43. Yaw rate response in a double lane maneuver scenario.

Figure 5 .

 5 Figure 5.44. Active steering control angle in a double lane maneuver scenario.

Figure 5 .

 5 Figure 5.45. Lateral accelerations in a double lane maneuver scenario.

Figure 5 .

 5 Figure 5.46. Lateral forces at four tires in a double lane maneuver scenario.

Figure 5 . 47 .

 547 Figure 5.47. Vehicle trajectories in a double lane maneuver scenario.

Table of Figures 1 .

 of1 The original design of the self-propelled car (Codex Atlanticus, f.812 r) and its replica at museum Clos Lucé, Amboise, France. .................................................................... ii 2. The steam 'car' designed by Verbiest in 1672. .................................................................. ii 3. Nicholas-Joseph Cugnot's vehicle, depicted at the famous first road accident at the Arsenal in 1770. ........................................................................................................... ii 4. Mercedes Benz Patent Motorwagen 1886. ........................................................................ iii 5. 1923 Ford Model T Touring.............................................................................................. iii 6. Global automobile production (cars, trucks, and buses), 1900-2016 (US DOT, 2016). .. vi 7. NTHSA -IIHS Percent of MY2017 Fleet Conforming to AEB Voluntary Commitment.

	PARAMETERS VARIABLES
	𝑎𝑎 𝑥𝑥 Nomenclature Longitudinal acceleration 𝑀𝑀 Total mas of the vehicle 𝑎𝑎 𝑦𝑦 Lateral acceleration	m/s 2 kg m/s 2
	𝑚𝑚 2 𝑎𝑎 𝐺𝐺	Sprung mass of QV Heave vertical displacement of CG	kg m
	𝑚𝑚 1 𝜑𝜑 𝐺𝐺 ACRONYMS Unsprung mass of QV-Chassis Pitch angle 𝑘𝑘 2 Stiffness coefficient of suspension 𝜃𝜃 𝐺𝐺 Chassis Roll angle	kg rad N/m rad
	ABS ACC 𝑘𝑘 1 𝑎𝑎 2 𝑏𝑏 2 𝑎𝑎 1 ADAS AEB AV 𝑏𝑏 1 𝑎𝑎 0 𝑎𝑎, 𝐿𝐿 𝑓𝑓 𝑉𝑉 𝑥𝑥 𝑏𝑏, 𝐿𝐿 𝑟𝑟 𝑉𝑉 𝑦𝑦 BW 𝑤𝑤, 𝑊𝑊 𝜓𝜓 CG GCC CRONE 𝐿𝐿 Δ 𝐺𝐺 𝑃𝑃 𝑥𝑥𝑥𝑥 Δ 𝑝𝑝 𝑃𝑃 𝑦𝑦𝑦𝑦 𝑇𝑇 ECU 𝑃𝑃 𝑧𝑧 Ω GPS 𝑀𝑀 0 𝐹𝐹 𝑥𝑥 ESC or ESP Stiffness coefficient of the tire Anti-lock Braking System QV sprung mass vertical displacement Damping coefficient of suspension QV unsprung mass vertical displacement Adaptive Cruise Control Advanced Driver-Assistance System Automatic Emergency Braking Damping coefficient of the tire Road elevation Distance from the vehicle's front axle to CG Vehicle longitudinal speed Distance from the vehicle's rear axle to CG Vehicle lateral speed Autonomous vehicles Half width of the Axles Yaw angle of the vehicle Blindspot warning Center of Gravity Global Chassis Control Vehicle length Sideslip angle at CG Moment of inertia around the x-axis Tire sideslip angle Moment of inertia around the y-axis Wheel slip rate Commande Robuste d'Ordre Non Entier Moment of inertia around the z-axis Wheel angular velocity Electronic Control Unit Wheel radius Tire longitudinal forces Global Positioning System 𝐻𝐻 Height of the center of gravity 𝐹𝐹 𝑦𝑦 Tire lateral forces Electronic Stability Control or Electronic Stability N/m m Ns/m m Ns/m m m m/s m m/s m rad/s m rad kg m 2 rad kg m 2 % kg m 2 rad/s m N m N 𝐽𝐽 𝑟𝑟 Wheel inertia kg m 2 𝐹𝐹 𝑧𝑧 Tire vertical forces N Program FCW 𝑔𝑔 Gravity constant m/s 2 𝑋𝑋 𝐺𝐺 Longitudinal displacement of CG m Forward Collision Warning LDW Lane Departure Warning 𝑌𝑌 𝐺𝐺 Lateral displacement of CG m
	LiDAR	Light Detection and Ranging
	LKA	Lane Keeping Assist
	QV	Quarter Vehicle
	TJA	Traffic Jam Assist
	VANets	Vehicular Ad-hoc Network
	V2V	Vehicle-to-Vehicle
	V2I	Vehicle-to-Infrastructure
		xxxix xl xli

[START_REF] Gotsch | Crash Course: Putting the AI in the Automotive Industry[END_REF]

. ............................................................................................................ ix 8. IIHS/HLDI Predicted Percent of Vehicles Equipped (standard or optionally equipped) with ADAS Technologies. ........................................................................................... ix 9. Real World benefits of some ADAS technologies in 2019. ................................................ x 10. SAE j3016 levels of driving automation (SMMT, 2019). ............................................... xi 11. VANets mode diagram (Sheikh and Liang, 2019). ........................................................ xii 12. Waymo's Google self-driving taxis (Google, 2019). ...................................................... xiii 13. Some of PSA Group contributions in Autonomous Vehicles. ...................................... xiv 14. Projected global electric car stock compared with OEM targets (2020-25) (Till Bunsen et al., 2019). ............................................................................................................... xv 15. The 2020 Porsche Taycan is Porsche's first-ever electric vehicle. .............................. xvii 1.1. Schematic of the main vehicle dynamics. ....................................................................... 4 1.2. Acceleration diagrams: (a) 2D representation (𝑎𝑎𝑎𝑎, 𝑎𝑎𝑎𝑎), (b) 3D representation with 𝑎𝑎𝑎𝑎. ................................................................................................................................. 6 1.3. The various type of disturbances applied on the vehicle dynamics. .............................. 8 1.4. V -cycle approach for designing of vehicles' elements (Létévé, 2014). ....................... 10 1.5. Validation model in MATLAB/Simulink environment. ............................................... 11 1.6. Schematics of the Quarter vehicle analysis model (Bouvin, 2019). ............................. 12 1.7. Representation of the domains of study on a diagram of (𝑎𝑎𝑎𝑎, 𝑎𝑎𝑎𝑎). ............................ 14 1.8. Analysis model of quarter vehicle (Létévé, 2014). ........................................................ 16 1.9. Traditional suspension system. ..................................................................................... 18 1.10. Main roles of the vehicle suspension system. .............................................................. 19 1.11. Representation of different road roughness and obstacles (Bouvin, 2019). ............... 20 1.12. Resonance frequencies for different body segments, represented by a simplified mechanical model of a human standing on a vibrating platform (Berlin and Adams, 2017). .......................................................................................................................... 21 1.13. Classification of electronically controlled suspension (Savaresi et al., 2010). ............ 24 1.14. The Architecture of Bose Suspension. ......................................................................... 26

Table 1 .

 1 1. The effect of LAS systems on global vehicle dynamics.

	Dynamic	14DoF	Steering	Traction	Braking	Suspen-sion
	Transversal	1 Translation	Direct	By coupling	By coupling
	2DoF	1 Rotational	Direct	Indirect differential traction	Indirect differential braking	Indirect variation
	Longitudinal	1 Translation		Direct	Direct	in road structure
	5DoF	4 Rotational		Direct	Direct
		1 Heave	By coupling	By coupling	By coupling
	Vertical 7DoF	1 Pitch 1 Roll	By coupling Perturbation rolling couple	Perturbation pitching couple	Perturbation pitching couple	Direct

Table 1 .2. Adequation between domains of study and domains of operation.

 1

				Domains of Operation
		Small Variations	Medium variation	High variations
		Analysis	Synthesis	Analysis	Synthesis	Analysis	Synthesis
		Vibratory
		comfort
	Domains	Road
	of Study	Holding
		Operation
		Limits

Table 2 .

 2 1. Nominal quarter vehicle parameters.

	Parameter	Value	Unit
	𝒎𝒎 𝟐𝟐	200	kg
	𝒌𝒌 𝟐𝟐	12000	N/m
	𝒃𝒃 𝟐𝟐	200	N.s/m
	𝒎𝒎 𝟏𝟏	32	kg
	𝒌𝒌 𝟏𝟏	300000	N/m
	𝒃𝒃 𝟏𝟏	50	N.s/m

Table 2 .

 2 2. The variation range of suspension parameters.

			Front axle (i=1)		Rear axle (i=2)
	Parameter	𝑚𝑚 2 [kg]	𝑘𝑘 2 [N/m]	𝑏𝑏 2 [N.s/m]	𝑚𝑚 2 [kg]	𝑘𝑘 2 [N/m]	𝑏𝑏 2 [N.s/m]
	𝑷𝑷 𝒊𝒊,𝒎𝒎𝒊𝒊𝒎𝒎	168	10800	180	68	9000	180
	𝑷𝑷 𝒊𝒊,𝒎𝒎𝒎𝒎𝒎𝒎	193	12000	200	118	10000	200
	𝑷𝑷 𝒊𝒊,𝒎𝒎𝒎𝒎𝒎𝒎	218	13200	220	168	11000	220

  Within the frequency range [𝜔𝜔 𝐴𝐴 , 𝜔𝜔 𝐵𝐵 ], the nominal plant phase varies by 1.3°, which confirms the reason for selecting the 2 nd generation CRONE controller.

		𝜔𝜔 𝐴𝐴 = 𝜔𝜔 𝑐𝑐1 / √	4.5 = 0.52 rad/s,
		𝜔𝜔 𝐵𝐵 = 𝜔𝜔 𝑐𝑐1 •	√	and 4.5 = 2.33 rad/s.	(2.37)
		log 10	𝜔𝜔 𝐵𝐵 𝜔𝜔 𝐴𝐴	= �	-80.59 -82.32 𝑛𝑛 � = 0.65,	(2.35)
	then,	𝜔𝜔 𝐵𝐵 𝜔𝜔 𝐴𝐴	= 10 0.6526 = 4.5.	(2.36)
	As the nominal plant leads to the mean magnitude of the open-loop fre-
	quency response around 𝜔𝜔 𝑐𝑐1 , then:

  Within the frequency range [𝜔𝜔 𝐴𝐴 , 𝜔𝜔 𝐵𝐵 ], the nominal plant phase varies by 2.4°, which confirms the reason for selecting the 2 nd generation CRONE controller. The values of 𝜔𝜔 𝑙𝑙 and 𝜔𝜔 ℎ , the corner frequencies, are set using the following formulations with same values for (a,b), then:For the three parametric states of the two plants (front, rear), Figure2.16 and 2.17 present respectively the open-loop Nichols plots. They clearly show the frequency template tangent to the 3dB contour. The open-loop frequency responses, as plotted in Figure

	𝑀𝑀 = 𝑙𝑙𝑃𝑃𝑔𝑔 10	𝜔𝜔 𝐵𝐵 𝜔𝜔 𝐴𝐴	= �	-80.668 -78 𝑛𝑛 � = 0.68,	(2.44)
	then,	𝜔𝜔 𝐵𝐵 𝜔𝜔 𝐴𝐴	= 10 0.68 = 4.8.		(2.45)
	As the nominal plant leads to the mean magnitude of the open-loop fre-
	quency response around 𝜔𝜔 𝑐𝑐2 , then:
	𝜔𝜔 𝐴𝐴 = 𝜔𝜔 𝑐𝑐2 / √	4.8 = 0.5 rad/s,
	𝜔𝜔 𝐵𝐵 = 𝜔𝜔 𝑐𝑐2 •	and 4.8 = 2.4 rad/s. √	(2.46)
	𝜔𝜔 𝑙𝑙 = 0.52/5 = 0.1 rad/s,
									and		(2.47)
	𝜔𝜔 ℎ = 2.33 × 10 = 24 rad/s.
	According to (2.40) and (2.41), the nominal phase margin and open-loop
	gain crossover frequency are ensured with:
	𝑛𝑛 = 1.42 and 𝐾𝐾 = 48.74.	(2.48)
	Then, the final open-loop transfer function for the rear axle is formulated
	as follows:
	𝛽𝛽 2 (𝑠𝑠) = 48.74 �	0.1 𝑠𝑠	+ 1�	2	�	1 + 𝑠𝑠 24 1 + 𝑠𝑠 0.1 �	1.42	�	1 1 + 𝑠𝑠 24	�	3	.	(2.49)

  2 + 𝑏𝑏 2 𝑠𝑠 + 𝑏𝑏 𝑎𝑎 𝑠𝑠 𝑛𝑛 + 𝑘𝑘 2 )(𝑏𝑏 1 𝑠𝑠 + 𝑘𝑘 1 ) 𝑃𝑃 𝑛𝑛 𝑟𝑟(𝑠𝑠) = 𝑚𝑚 1 𝑚𝑚 2 𝑠𝑠 4 + (𝑚𝑚 2 (𝑏𝑏 1 + 𝑏𝑏 2 ) + 𝑚𝑚 1 𝑏𝑏 2 )𝑠𝑠 3 +(𝑚𝑚 2 (𝑘𝑘 1 + 𝑘𝑘 2 ) + 𝑚𝑚 1 𝑘𝑘 2 + 𝑏𝑏 1 𝑏𝑏 2 )𝑠𝑠 2 + (𝑘𝑘 1 𝑏𝑏 2 + 𝑏𝑏 1 𝑘𝑘 2 )𝑠𝑠 +𝑘𝑘 1 𝑘𝑘 2 + 𝑏𝑏 𝑎𝑎 (𝑚𝑚 1 + 𝑚𝑚 2 )𝑠𝑠 2+𝑛𝑛 + 𝑏𝑏 𝑎𝑎 𝑏𝑏 1 𝑠𝑠 1+𝑛𝑛 + 𝑏𝑏 𝑎𝑎 𝑘𝑘 1 𝑠𝑠 𝑛𝑛 . (2.59) 𝑉𝑉 2 (𝑠𝑠), 𝑉𝑉 1 (𝑠𝑠), 𝑉𝑉 0 (𝑠𝑠), and 𝐹𝐹 0 (𝑠𝑠) are the Laplace transform of 𝑎𝑎 ̇2(𝑇𝑇), 𝑎𝑎 ̇1(𝑇𝑇), 𝑎𝑎 ̇0(𝑇𝑇), and 𝐹𝐹 0 (𝑇𝑇), respectively.

		+	𝑃𝑃 𝑛𝑛 𝑟𝑟(𝑠𝑠) 𝑃𝑃 𝑛𝑛 𝑟𝑟(𝑠𝑠) 𝑠𝑠(𝑏𝑏 2 + 𝑏𝑏 𝑎𝑎 𝑠𝑠 𝑛𝑛 + 𝑘𝑘 2 )	𝑉𝑉 0 (𝑠𝑠) 𝐹𝐹 0 (𝑠𝑠),	(2.57)
	and
	𝑉𝑉 2 (𝑠𝑠) = +	(𝑏𝑏 2 𝑠𝑠 + 𝑏𝑏 𝑎𝑎 𝑠𝑠 𝑛𝑛 + 𝑘𝑘 2 )(𝑏𝑏 1 𝑠𝑠 + 𝑘𝑘 1 ) 𝑃𝑃 𝑛𝑛 𝑟𝑟(𝑠𝑠) 𝑃𝑃 𝑛𝑛 𝑟𝑟(𝑠𝑠) 𝑠𝑠�𝑚𝑚 1 𝑠𝑠 2 + (𝑏𝑏 2 + 𝑏𝑏 1 )𝑠𝑠 + 𝑏𝑏 𝑎𝑎 𝑠𝑠 𝑛𝑛 + (𝑘𝑘 2 + 𝑘𝑘 1 )� 𝑉𝑉 0 (𝑠𝑠)	𝐹𝐹 0 (𝑠𝑠),	(2.58)
	with,

Table 2 .

 2 3. Optimized values of the damping fractional-order 𝑛𝑛.

	Axle	Load		𝒏𝒏
			𝐽𝐽 1	𝐽𝐽 2	𝐽𝐽 3
	Front	Empty load Full load	0.68 0.7	1.1 1.11	1.69 1.68
	Rear	Empty load Full load	0.64 0.68	1.01 1.06	1.8 1.68

  𝑊𝑊 0𝜑𝜑 (𝑇𝑇) being the pitch torque induced due to braking, accelerating, etc..., 𝑊𝑊 𝑠𝑠𝜑𝜑 (𝑇𝑇) represents the vehicle passive anti-pitch torque, and 𝑢𝑢 𝜑𝜑 (𝑇𝑇) is

	the active anti-pitch torque provided by the active suspension control efforts,
	and defined by:
	𝑢𝑢 𝜑𝜑 (𝑇𝑇) = 𝑢𝑢 𝐹𝐹𝐹𝐹 ,𝜑𝜑 (𝑇𝑇) + 𝑢𝑢 𝐹𝐹𝐵𝐵,𝜑𝜑 (𝑇𝑇),	(3.9)
	where 𝑢𝑢 𝐹𝐹𝐹𝐹 ,𝜑𝜑 (𝑇𝑇) is the feed-forward torque and the 𝑢𝑢 𝐹𝐹𝐵𝐵,𝜑𝜑 (𝑇𝑇) is the feedback
	torque that regulates the pitch rate Ω 𝜑𝜑 (𝑠𝑠) using 𝐶𝐶 𝜑𝜑 (𝑠𝑠) transfer function,
	such that:
	𝑈𝑈 𝐹𝐹𝐵𝐵,𝜑𝜑 (𝑠𝑠) = -𝐶𝐶 𝜑𝜑 (𝑠𝑠)𝛺𝛺 𝜑𝜑 (𝑠𝑠).	(3.10)
	Similarly, 𝐶𝐶

𝑊𝑊 0𝜑𝜑 (𝑇𝑇) + 𝑊𝑊 𝑠𝑠𝜑𝜑 (𝑇𝑇)+𝑢𝑢 𝜑𝜑 (𝑇𝑇) = 𝑓𝑓�ℎ 𝜑𝜑 , 𝑀𝑀 2 , 𝑎𝑎 𝑥𝑥 , . . �+𝑊𝑊 𝑠𝑠𝜑𝜑 (𝑇𝑇) + 𝑢𝑢 𝜑𝜑 (𝑇𝑇),

(3.8)

with ∑𝜃𝜃 (𝑇𝑇) (N. m) is the algebraic sum of all external torques applied around the x-axis, and is defined as:

𝐶𝐶 ∑𝜃𝜃 (𝑇𝑇) = 𝑊𝑊 0𝜃𝜃 (𝑇𝑇) + 𝑊𝑊 𝑠𝑠𝜃𝜃 (𝑇𝑇)+𝑢𝑢 𝜃𝜃 (𝑇𝑇) = 𝑓𝑓�ℎ 𝜃𝜃 , 𝑀𝑀 2 , 𝑎𝑎 𝑦𝑦 , . . �+𝑊𝑊 𝑠𝑠𝜃𝜃 (𝑇𝑇) + 𝑢𝑢 𝜃𝜃 (𝑇𝑇),

  𝑃𝑃𝑟𝑟𝑛𝑛 𝑖𝑖 (𝑠𝑠) = 𝑚𝑚 1 𝑚𝑚 2𝑖𝑖 𝑠𝑠 4 + 𝑘𝑘 1𝑖𝑖 𝑘𝑘 2𝑖𝑖 +(𝑚𝑚 2𝑖𝑖 (𝑏𝑏 1 + 𝑏𝑏 2𝑖𝑖 ) + 𝑚𝑚 1 𝑏𝑏 2𝑖𝑖 ) 𝑠𝑠 3 +(𝑚𝑚 2𝑖𝑖 (𝑘𝑘 1𝑖𝑖 + 𝑘𝑘 2𝑖𝑖 ) + 𝑚𝑚 1 𝑘𝑘 2𝑖𝑖 )𝑠𝑠 2 +𝑏𝑏 1 𝑏𝑏 2𝑖𝑖 𝑠𝑠 2 + (𝑘𝑘 1𝑖𝑖 𝑏𝑏 2𝑖𝑖 + 𝑏𝑏 1 𝑘𝑘 2𝑖𝑖 ) 𝑠𝑠.

	where,
						(3.24)
						.22)
	where 𝑍𝑍 ̃2,𝑖𝑖𝑠𝑠 (𝑠𝑠) is the measured value of the vertical displacement of sprung
	mass at each corner. Then considering the synthesis model 𝐺𝐺 𝑖𝑖 (𝑠𝑠) is formu-
	lated as:
	𝐺𝐺 𝑖𝑖 (𝑠𝑠) =	𝑍𝑍 2,𝑖𝑖𝑠𝑠 (𝑠𝑠) 𝑈𝑈 𝑖𝑖𝑠𝑠 (𝑠𝑠)	=	𝑚𝑚 1 𝑠𝑠 2 + 𝑏𝑏 1 𝑠𝑠 + 𝑘𝑘 1 𝑃𝑃𝑟𝑟𝑛𝑛 𝑖𝑖 (𝑠𝑠)	,	(3.23)

Table 3 .

 3 1. Uncertain parameters of global transfer functions.

	Parameter	Range	Unit
	𝑴𝑴 𝟐𝟐	472-772	𝑘𝑘𝑔𝑔
	𝒌𝒌 𝒛𝒛	39600-48400	𝑁𝑁/𝑚𝑚
	𝒌𝒌 𝝋𝝋	41600-48020	𝑁𝑁/𝑚𝑚
	𝒌𝒌 𝜽𝜽	19960-24400	𝑁𝑁/𝑚𝑚
	𝒃𝒃 𝜽𝜽	363-444	𝑁𝑁. 𝑠𝑠 𝑚𝑚 ⁄
	𝒃𝒃 𝝋𝝋	800-891	𝑁𝑁. 𝑠𝑠 𝑚𝑚 ⁄
	𝒃𝒃 𝒛𝒛	720-880	𝑁𝑁. 𝑠𝑠 𝑚𝑚 ⁄
	𝑰𝑰 𝒎𝒎𝒎𝒎	225-400	𝑘𝑘𝑔𝑔. 𝑚𝑚 2
	𝑰𝑰 𝒚𝒚𝒚𝒚	400-750	𝑘𝑘𝑔𝑔. 𝑚𝑚 2
	For control synthesis, the "max" plants are used as nominal plants in the
	design process, that is:
	𝐺𝐺 𝑧𝑧,𝑛𝑛𝑛𝑛𝑚𝑚 (𝑠𝑠) =	𝑠𝑠 772 𝑠𝑠 2 + 880𝑠𝑠 + 48400	,	(3.28)
	𝐺𝐺 𝜑𝜑,𝑛𝑛𝑛𝑛𝑚𝑚 (𝑠𝑠) =	𝑠𝑠 750 𝑠𝑠 2 + 891𝑠𝑠 + 48020 ,	(3.29)
	𝐺𝐺 𝜃𝜃,𝑛𝑛𝑛𝑛𝑚𝑚 (𝑠𝑠) =	𝑠𝑠 400 𝑠𝑠 2 + 444𝑠𝑠 + 24400	.	(3.30)

Table 3 .

 3 2. Range of variation of suspension parameters.

		Front axle (i=1)		Rear axle (i=2)
	Parameter	𝑚𝑚 2	𝑘𝑘 2	𝑏𝑏 2	𝑚𝑚 2	𝑘𝑘 2	𝑏𝑏 2
		[kg]	[N/m]	[N.s/m]	[kg]	[N/m]	[N.s/m]
	𝑮𝑮 𝒊𝒊,𝒎𝒎𝒊𝒊𝒎𝒎	168	10800	180	68	9000	180
	𝑮𝑮 𝒊𝒊,𝒎𝒎𝒎𝒎𝒎𝒎	193	12000	200	118	10000	200
	𝑮𝑮 𝒊𝒊,𝒎𝒎𝒎𝒎𝒎𝒎	218	13200	220	168	11000	220

Table 3 .

 3 3. Control parameters of the local crone controllers.

	352 rad/s, and the values

Figure 3.11. Bode response of local controllers.

Table 3 .

 3 4. ISO 8608 values of 𝐺𝐺 𝑑𝑑 (𝛺𝛺 0 ) related to some road class.𝑮𝑮 𝒎𝒎 (𝛀𝛀 𝟎𝟎 ) × 𝟏𝟏𝟏𝟏 -𝟔𝟔 (𝒎𝒎 𝟑𝟑 )

	Road class	Lower limit	Geometric Mean Upper limit
	A	-	1	2
	B	2	4	8
	C	8	16	32

Table 3 .

 3 5. Simulation results for the controlled systems with bump road profiles.

			Bump
	Quantity
		SH	CRONE	PID
	|𝒛𝒛̈𝟐𝟐 | 𝒓𝒓𝒎𝒎𝒓𝒓	1.778 +30%	0.3 +88.1%	0.338 +86.76%
	�𝒛𝒛̈𝟐𝟐 ,𝒄𝒄𝒄𝒄𝒎𝒎𝒇𝒇 � 𝒓𝒓𝒎𝒎𝒓𝒓	1.24 +20.56%	0.246 +84.2%	0.277 +82.23%
	𝒎𝒎𝒓𝒓𝒎𝒎(𝒛𝒛̈𝟐𝟐)	5.73 +24.12%	0.97 +87%	1.05 +86%
	|𝒛𝒛 𝟐𝟐 -𝒛𝒛 𝟏𝟏 | 𝒓𝒓𝒎𝒎𝒓𝒓	0.024 +27.7%	0.029 +11.8%	0.0288 +11%
	𝒎𝒎𝒓𝒓𝒎𝒎(|𝒛𝒛 𝟐𝟐 -𝒛𝒛 𝟏𝟏 |)	0.068 +20%	0.0967 -14.2%	0.0973 -14.9%
	�𝑭𝑭 𝒎𝒎𝒚𝒚𝒎𝒎 � 𝒓𝒓𝒎𝒎𝒓𝒓	451 +0.48%	264 +53.4%	271 +51.6%
	𝑷𝑷 𝒘𝒘	141.48	207	208
	‖𝒖𝒖 𝒎𝒎 ‖ 𝒓𝒓𝒎𝒎𝒓𝒓	225	383	392
	𝒎𝒎𝒓𝒓 𝒎𝒎(|𝒖𝒖 𝒎𝒎 |) (𝑵𝑵)	648	1321	1360

Table 3 .

 3 6. Simulation results for the controlled systems with class A roads.

			Profile 1 (Road Class A)
		Sh	CRONE	PID
	|𝒛𝒛̈𝟐𝟐| 𝒓𝒓𝒎𝒎𝒓𝒓	0.5 5.1%	0.262 50.5%	0.27 49.1%
	�𝒛𝒛̈𝟐𝟐 ,𝒄𝒄𝒄𝒄𝒎𝒎𝒇𝒇 � 𝒓𝒓𝒎𝒎𝒓𝒓	0.417 0.9%	0.212 49.5%	0.218 48.2%
	𝒎𝒎𝒓𝒓𝒎𝒎(𝒛𝒛̈𝟐𝟐)	1.5 10.6%	0.855 49.2%	0.72 48.2%
	|𝒛𝒛 𝟐𝟐 -𝒛𝒛 𝟏𝟏 | 𝒓𝒓𝒎𝒎𝒓𝒓	0.0028 11.8%	0.006 -93.9%	0.006 -96.6%
	𝒎𝒎𝒓𝒓𝒎𝒎(|𝒛𝒛 𝟐𝟐 -𝒛𝒛 𝟏𝟏 |)	0.0088 12.5%	0.0157 -55.8%	0.0159 -57.7%
	�𝑭𝑭 𝒎𝒎𝒚𝒚𝒎𝒎 �	328.3 -1.5%	515 -59.3%	503 -55.5%
	𝑷𝑷 𝒘𝒘	6	21.8	21.88
	‖𝒖𝒖 𝒎𝒎 ‖ 𝒓𝒓𝒎𝒎𝒓𝒓	31.3	76.3	78.3
	𝒎𝒎𝒓𝒓𝒎𝒎(|𝒖𝒖 𝒎𝒎 |)	89.2	194.4	202.5
	(% )

relative to the PSc systemTable 3 .

 3 [START_REF]Bode diagram of the transfer functions (a) 𝑇𝑇20(𝑠𝑠) and (b) 𝑇𝑇10[END_REF]. Simulation results for the controlled systems with class A-B roads.

			Profile 2 (Road Class A-B)
		Sh	CRONE	PID
	|𝒛𝒛̈𝟐𝟐| 𝒓𝒓𝒎𝒎𝒓𝒓	0.73 3.7%	0.4 47.2%	0.41 46%
	�𝒛𝒛̈𝟐𝟐 ,𝒄𝒄𝒄𝒄𝒎𝒎𝒇𝒇 � 𝒓𝒓𝒎𝒎𝒓𝒓	0.611 0.35%	0.325 47%	0.332 45.8%
	𝒎𝒎𝒓𝒓𝒎𝒎(𝒛𝒛̈𝟐𝟐)	2.4	1.524	1.544
		4.6%	39.5%	38.7%
	|𝒛𝒛 𝟐𝟐 -𝒛𝒛 𝟏𝟏 | 𝒓𝒓𝒎𝒎𝒓𝒓	0.004 6.3%	0.00892 -112%	0.009 -115%
	𝒎𝒎𝒓𝒓𝒎𝒎(|𝒛𝒛 𝟐𝟐 -𝒛𝒛 𝟏𝟏 |)	0.011 9.56%	0.0219 -82.9%	0.0223 -86.4%
	�𝑭𝑭 𝒎𝒎𝒚𝒚𝒎𝒎 �	470.6 -1.8%	788.261 -70.5%	767.7 -66%
	𝑷𝑷 𝒘𝒘	12.5	57.342	57.124
	‖𝒖𝒖 𝒎𝒎 ‖ 𝒓𝒓𝒎𝒎𝒓𝒓	43.7	109.8	113
	𝒎𝒎𝒓𝒓𝒎𝒎(|𝒖𝒖 𝒎𝒎 |)	116	263	280
		(% ) relative to PSc system

Table 3 .

 3 8. Simulation results of the driving scenario.

	Variable				System
	(RMS)	PSc	LC	GC	GC+LC	GC+FF	All
	𝒛𝒛̈𝑪𝑪 𝑪𝑪 (𝒎𝒎/𝒔𝒔 𝟐𝟐 ) ÷ 𝟏𝟏𝟏𝟏 𝑮𝑮	2.55	0.7	0.26	0.29	0.164	0.156
	𝝋𝝋̈𝑪𝑪 𝑪𝑪 (°/𝒔𝒔 𝟐𝟐 )	1.35	0.17	0.06	0.05	0.036	0.0296
	𝜽𝜽 ̈𝑪𝑪𝑪𝑪 (°/𝒔𝒔 𝟐𝟐 )	4.7	0.375	0.123	0.098	0.039	0.031
	𝑼𝑼 𝒎𝒎 (𝑵𝑵)	0	1050	1007.6	1049.4	1049.3	1060

Table 4 .

 4 1. Range of variation of uncertain parameters of the ABS control plant.

	Parameter	Range	Unit
	𝑽𝑽 𝒎𝒎 𝒎𝒎	15 -30	m/s

Table 4 .

 4 2. Range of variation of 𝑝𝑝 𝑥𝑥 as function the chosen of slip rate.

	Wheel slip rate value 𝑻𝑻 in %		Range of variation of 𝒑𝒑 𝒎𝒎
	7%	→	3200 -6838
	7.5%	→	2260 -4590
	8%	→	1355 -2942
	8.9%	→	95 -810

𝒎𝒎 ∈ [𝟐𝟐𝟏𝟏𝟏𝟏 -𝟐𝟐𝟐𝟐𝟏𝟏]kg and 𝝁𝝁 ∈ [𝟏𝟏.

𝟐𝟐 -𝟏𝟏]

Table 4 .

 4 3. Parametric variation of G(s).

	Plant	𝑽𝑽 𝒎𝒎 𝒎𝒎 [m/s]	𝝁𝝁 𝒎𝒎	𝒎𝒎 [kg]
	𝑮𝑮 𝒎𝒎𝒄𝒄𝒎𝒎	25	1	250
	𝑮𝑮 𝒎𝒎𝒊𝒊𝒎𝒎	30	0.5	250
	𝑮𝑮 𝒎𝒎𝒎𝒎𝒎𝒎	15	1	200

Table 4 .

 4 4. Open-loop parameters of the 3 rd generation CRONE controller.

	Open-loop parameter	value
	𝝎𝝎 𝒍𝒍	55 rad/s
	𝝎𝝎 𝒉𝒉	165 rad/s
	𝝎𝝎 𝒓𝒓	55 rad/s
	𝒀𝒀 𝒓𝒓	5.5
	𝒓𝒓	0.4
	𝒃𝒃	0.8

Performance analysis 4.4.1 Frequency domain

Table 4 .

 4 5. Maximum gain in dB of sensitivity functions for perturbed G(s).

		𝑻𝑻 (𝒔𝒔)	𝑺𝑺(𝒔𝒔)	𝑪𝑪𝑺𝑺(𝒔𝒔)	𝑮𝑮𝑺𝑺(𝒔𝒔)
	CRONE	3.6dB	2.3dB	-1dB	5.6dB

  in chapter 1. The forces 𝑓𝑓 0𝑖𝑖 represented in Figure4.26 are the front and rear load shift due to the longitudinal acceleration or deceleration, formulated as:

	𝑓𝑓 02 = -	ℎ + 𝑎𝑎 𝐺𝐺 (𝑎𝑎 + 𝑏𝑏)	𝑀𝑀 𝑎𝑎 𝑥𝑥 (𝑇𝑇) .	(4.76)
	𝑓𝑓 01 =	ℎ + 𝑎𝑎 𝐺𝐺 (𝑎𝑎 + 𝑏𝑏)	𝑀𝑀 𝑎𝑎 𝑥𝑥 (𝑇𝑇) ,	(4.75)

and Figure 4.26. Half-car model.

Table 4 .

 4 12. RMS values for wheel forces with FO damping controllers.

	RMS	Passive	𝒏𝒏 = 𝟏𝟏. 𝟔𝟔𝟐𝟐	𝒏𝒏 = 𝟏𝟏. 𝟏𝟏
	𝑭𝑭 𝒛𝒛 [N]	305	454	280
	𝑭𝑭 𝒎𝒎 [N]	262	363	240

  𝑃𝑃 𝑧𝑧 𝜓𝜓 ̈= 𝐿𝐿 𝑓𝑓 �𝑓𝑓 𝑦𝑦,11 + 𝑓𝑓 𝑦𝑦,12 � -𝐿𝐿 𝑟𝑟 �𝑓𝑓 𝑦𝑦,21 + 𝑓𝑓 𝑦𝑦,22 � + 𝑢𝑢 𝑧𝑧 ,

					(5.13)
	𝑀𝑀 𝑣𝑣̇𝑦𝑦 = 𝑓𝑓 𝑦𝑦,11 + 𝑓𝑓 𝑦𝑦,12 + 𝑓𝑓 𝑦𝑦,21 + 𝑓𝑓 𝑦𝑦,22 + 𝑓𝑓 𝑦𝑦0 -𝑀𝑀 𝑉𝑉 𝑥𝑥0 𝜓𝜓 ̇,	(5.14)
	𝛥𝛥 𝑝𝑝,11 = 𝛥𝛥 𝑝𝑝,12 = 𝛽𝛽 𝜐𝜐 -𝛥𝛥 𝐺𝐺 -𝐿𝐿 𝑓𝑓	𝜓𝜓	V𝑉𝑥𝑥0	,
	𝛥𝛥 𝑝𝑝,21 = 𝛥𝛥 𝑝𝑝,22 = -𝛥𝛥 𝐺𝐺 + 𝐿𝐿 𝑟𝑟	𝜓𝜓	V𝑉𝑥𝑥0	,
	with 𝛥𝛥 𝐺𝐺 = 𝑣𝑣 𝑦𝑦 𝑉𝑉 𝑥𝑥0 ⁄	.

  .18) with 𝑃𝑃𝑃𝑃𝐸𝐸 = 𝐿𝐿 𝑓𝑓 𝑊𝑊 𝑦𝑦𝑓𝑓 -𝐿𝐿 𝑟𝑟 𝑊𝑊 𝑦𝑦𝑟𝑟 , where 𝑊𝑊 𝑦𝑦𝑓𝑓 and 𝑊𝑊 𝑦𝑦𝑟𝑟 are the front and rear corner stiffness, respectively. The synthesis model used for the design of the controllers in the frequency-domain is represented by the transfer function 𝐺𝐺 1 (𝑠𝑠), and can be written in the following canonical form:

	𝐺𝐺 1 (𝑠𝑠) =	𝛹𝛹 ̇(𝑠𝑠) 𝑈𝑈 𝑧𝑧 (𝑠𝑠)	�	𝑓𝑓 𝑦𝑦0 =𝛽𝛽 𝑣𝑣 =0,	=	𝐾𝐾 0 (1 + 𝑠𝑠 𝜔𝜔 𝑧𝑧 ⁄ ) 1 + 2𝜁𝜁 𝜔𝜔 𝑛𝑛 𝑠𝑠 + � 𝑠𝑠 𝜔𝜔 𝑛𝑛 �	2 ,	(5.19)
	where,
	𝐾𝐾 0 = 𝑉𝑉 𝑥𝑥0 �𝑊𝑊 𝑦𝑦𝑓𝑓 + 𝑊𝑊 𝑦𝑦𝑟𝑟 � �2𝑊𝑊 𝑦𝑦𝑓𝑓 𝑊𝑊 𝑦𝑦𝑟𝑟 𝐿𝐿 2 -𝑀𝑀𝑉𝑉 𝑥𝑥0 𝑃𝑃𝑃𝑃𝐸𝐸� ⁄	,
	𝜔𝜔 𝑧𝑧 = 2�𝑊𝑊 𝑦𝑦𝑓𝑓 + 𝑊𝑊 𝑦𝑦𝑟𝑟 � 𝑀𝑀 ⁄ 𝑉𝑉 𝑥𝑥0 ,
	𝜔𝜔 𝑛𝑛 = ��4𝑊𝑊 𝑦𝑦𝑓𝑓 𝑊𝑊 𝑦𝑦𝑟𝑟 𝐿𝐿 2 -2𝑀𝑀𝑉𝑉 𝑥𝑥0	2 𝑃𝑃𝑃𝑃𝐸𝐸� 𝑃𝑃 𝑧𝑧 𝑀𝑀 𝑉𝑉 𝑥𝑥0 ⁄	2	� 1 2 ⁄ ,
	𝜁𝜁 =	��𝑊𝑊 𝑦𝑦𝑓𝑓 + 𝑊𝑊 𝑦𝑦𝑟𝑟 � 𝑀𝑀 + �𝐿𝐿 𝑓𝑓 ⁄ 𝜔𝜔 𝑛𝑛 𝑉𝑉 𝑥𝑥0 2 𝑊𝑊 𝑦𝑦𝑓𝑓 + 𝐿𝐿 𝑟𝑟	2 𝑊𝑊 𝑦𝑦𝑟𝑟 � 𝑃𝑃 𝑧𝑧 ⁄	�	.

Table 5 .

 5 1. Parametric variations of G1(s).

	Plant	𝑽𝑽 𝒎𝒎𝟎𝟎 km/h	𝝁𝝁	𝑴𝑴 kg	𝑰𝑰 𝒛𝒛 kg.m 2	𝑳𝑳 𝒇𝒇 m	𝑳𝑳 𝒓𝒓 m
	𝑮𝑮 𝟏𝟏𝒎𝒎𝒄𝒄𝒎𝒎	50	1	600	360	2/3	4/3
	𝑮𝑮 𝟏𝟏𝒎𝒎𝒊𝒊𝒎𝒎	45	1	900	420	8/9	10/9
	𝑮𝑮 𝟏𝟏𝒎𝒎𝒎𝒎𝒎𝒎	55	0.1	600	360	2/3	4/3

  Knowing that |𝐶𝐶(𝑗𝑗𝜔𝜔 𝑢𝑢 )𝐺𝐺 1𝑛𝑛𝑛𝑛𝑚𝑚 (𝑗𝑗𝜔𝜔 𝑢𝑢

	1 × 3000 0.01	� = 3𝑟𝑟4 𝑁𝑁 . 𝑚𝑚. 𝑠𝑠/𝑀𝑀𝑎𝑎𝐺𝐺.	(5.23)

Thus, for high frequencies 𝜔𝜔 𝐻𝐻𝐹𝐹 , |𝐶𝐶𝐶𝐶(𝑗𝑗𝜔𝜔)| ≈ |𝐶𝐶(𝑗𝑗𝜔𝜔 𝐻𝐻𝐹𝐹 )| ≤ 3𝑟𝑟4𝑁𝑁𝑚𝑚𝑠𝑠/𝑀𝑀𝑎𝑎𝐺𝐺.

Table 5 .

 5 2. Maximum gain in dB of sensitivity functions for perturbed G1(s).

		𝑻𝑻 (𝒔𝒔)	𝑺𝑺(𝒔𝒔)	𝑪𝑪𝑺𝑺(𝒔𝒔)	𝑮𝑮𝑺𝑺(𝒔𝒔)
	𝑪𝑪 𝟏𝟏 (𝒔𝒔)	5.9dB	2.4dB	85.5dB	-74dB
	𝑪𝑪 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 (𝒔𝒔)	23.4dB	23.3dB	102dB	-55dB

Figure 5.6. Bode plot for designed controllers of 𝐺𝐺 1 (𝑠𝑠) plant.

Table 5 .

 5 3. Parametric uncertainties of G2(s).

		𝑴𝑴 [kg]	𝑽𝑽 𝒎𝒎𝟎𝟎 [m/s 2 ]	𝝁𝝁
	𝑮𝑮 𝟐𝟐𝒎𝒎𝒎𝒎𝒎𝒎	750	50	0.6
	𝑮𝑮 𝟐𝟐𝒎𝒎𝒊𝒊𝒎𝒎	600	55	0.35
	𝑮𝑮 𝟐𝟐𝒎𝒎𝒎𝒎𝒎𝒎	900	45	1

Table 5 .

 5 4. Maximum gain in dB of sensitivity functions for perturbed G2(s).

		𝑻𝑻 (𝒔𝒔)	𝑺𝑺(𝒔𝒔)	𝑪𝑪𝑺𝑺(𝒔𝒔)	𝑮𝑮𝑺𝑺(𝒔𝒔)
	𝑪𝑪 𝟐𝟐 (𝒔𝒔)	3.2 dB	4.8dB	4.5dB	7.4dB
	𝑪𝑪 𝟐𝟐𝟏𝟏𝟏𝟏𝟏𝟏 (𝒔𝒔)	3.05dB	4.52dB	4.28dB	6.84dB

  The transfer function 𝑉𝑉 𝑦𝑦 (𝑠𝑠)/𝑈𝑈 𝛽𝛽(𝑠𝑠) can be easily deduced from the statespace representation in 5.2.5 when considering the third input of 𝑢𝑢(𝑇𝑇), that is 𝑢𝑢 𝛽𝛽 (𝑠𝑠), and the second variable of the state variable 𝑎𝑎(𝑇𝑇) which is the sideslip angle Δ 𝐺𝐺(𝑇𝑇). Knowing that Δ 𝐺𝐺 (𝑠𝑠) = 𝑉𝑉 𝑦𝑦 (𝑠𝑠)/𝑉𝑉 𝑥𝑥0 , then:

		𝑉𝑉 𝑦𝑦 (𝑠𝑠) 𝑈𝑈 𝛽𝛽 (𝑠𝑠)	=	Δ 𝐺𝐺 (𝑠𝑠 ) 𝑈𝑈 𝛽𝛽(𝑠𝑠)	• 𝑉𝑉 𝑥𝑥0 .	(5.40)
	Finally, the transfer function can be expressed as follows:
	𝐺𝐺 3 (𝑠𝑠) =	𝑌𝑌 𝐺𝐺 (𝑠𝑠) 𝑈𝑈 𝛽𝛽 (𝑠𝑠)	= �	𝜔𝜔 3 𝑠𝑠	�	2	⎝ ⎜ ⎜ ⎜ ⎛ 1 + 1 +	2𝜁𝜁 3 ′ 𝜔𝜔 𝑛𝑛3 ′ 𝑠𝑠 + � 𝑠𝑠 𝜔𝜔 𝑛𝑛3 ′ � 2𝜁𝜁 3 𝜔𝜔 𝑛𝑛3 𝑠𝑠 + � 𝑠𝑠 𝜔𝜔 𝑛𝑛3 � 2 2	⎠ ⎟ ⎟ ⎟ ⎞	,	(5.41)
	where,
			⎩ � � � � � � � 𝜁𝜁 3 𝜔𝜔 𝑛𝑛3 ′ = � 𝐺𝐺 2 𝐺𝐺 0 2�𝐺𝐺 0 𝐺𝐺 2 ′ = 𝐺𝐺 1 � ⎨ � � 𝜔𝜔 𝑛𝑛3 = 𝜔𝜔 𝑛𝑛2 ⎧ � � � 𝜁𝜁 3 = 𝜁𝜁 2 � � � 𝐺𝐺 0 𝜔𝜔 3 = � 𝑞𝑞	,	(5.42)
	with,
	3 (𝑠𝑠) = ⎩ 𝐺𝐺 2 = 𝑊𝑊 𝑦𝑦𝑓𝑓 /𝑀𝑀 𝑌𝑌 𝐺𝐺 (𝑠𝑠) 𝑈𝑈 𝛽𝛽 (𝑠𝑠) = 1 𝑠𝑠 � � � � ⎨ � � 𝑉𝑉 𝑥𝑥0 𝑏𝑏 𝑦𝑦 𝐿𝐿 𝑓𝑓 𝑊𝑊 𝑦𝑦𝑓𝑓 -𝑃𝑃𝑃𝑃𝐸𝐸 𝑊𝑊 𝑦𝑦𝑓𝑓 �𝑉𝑉 𝑥𝑥0 𝐺𝐺 2 (𝑠𝑠) 𝑠𝑠 + 𝑉𝑉 𝑦𝑦 (𝑠𝑠) 𝑈𝑈 𝛽𝛽 (𝑠𝑠) ⎧ 𝐺𝐺 0 = 𝑀𝑀 𝑃𝑃 𝑧𝑧 � � 𝐺𝐺 1 = 𝑏𝑏 𝑧𝑧 𝑉𝑉 𝑥𝑥0 𝑊𝑊 𝑦𝑦𝑓𝑓 -𝑃𝑃𝑃𝑃𝐸𝐸 𝐿𝐿 𝑓𝑓 𝑊𝑊 𝑦𝑦𝑓𝑓 𝑀𝑀 𝑃𝑃 𝑧𝑧 𝑉𝑉 𝑥𝑥0 .	� .	(5.39) (5.43)

Acknowledgement