Olivier Sename

Vicente Milanes

John Martinez

H Atoui

I Mahtout

D Gonzalez

Méthode de contrôle pour contrôler le déplacement d'un

Keywords: Procédé de supervision du fonctionnement d Lateral control, LPV control, Youla-Kucera parameterization, Switching control, Robust control, Autonomous vehicles

This PhD thesis is a result of a hard work, passion, and overnights to get it done on time. But I wasn't alone, God and the spirit of my family and friends were the mainstay for me to keep giving until the end.

I thank God always and forever for every little success that I have achieved in this thesis. With every drop from people, blessings were raining on me from every side.

Secondly, I would like to thank my parents, who supported me in every circumstance, and in every failure. If it wasn't for their steadfastness in my moments of failure, this work would not be as what you are reading now. I thank my sister Rawan for being there and I hope to read her PhD thesis one day. In mentioning my family, I will never forget to mention my sister and childhood friend Riham, I hope you are

Introduction

This thesis is developed within the framework of the PhD CIFRE (from french: Convention Industrielle de Formation par la Recherche) from November 2019 till October 2022. Research work has been accomplished between the SAFE team (which was previously Robust and Linear Systems team (SLR)) at the Gipsa-Lab laboratory at Grenoble-Alpes University for research in robust control, in collaboration with Renault research centre (from french: Technocentre).

Since the first foot-step in the world of automotive, huge number of scientists, engineers and researchers were attracted to this basic field of land transportation. During the 20th century, this field was subjected to many changes and developments especially in the autonomous domain. Intelligent and autonomous vehicles are at the heart of the societal concerns of future transportation. These cars, equipped with numerous sensors and actuators, will enhance road safety, streamline traffic, make transportation more accessible to people with disabilities, and participate in the development of new modes of transportation.

The way an automated vehicle behaves, takes a vital role in rating its acceptability. It is important as much as it is general since there exist infinite ways to travel a vehicle from point A to point B. Specifically, the performance evaluation of any vehicle is done by testing it in critical situations, i.e. how much it is able to deal with critical lane changes, collision avoidance, instability, sensor failure, autonomous starting mode, low speed precision, driver emergency, high slipping motions, high oscillations, aggressive maneuvers, etc. For each automated system in the vehicle, a controller is responsible to provide stability and performance for its motion.

In that context, the field of switching or multi-mode control architecture has been opened to solve variant tasks and achieve different driving performances. Such an architecture is resulted from a combination between several controllers to achieve an interpolated performance. The main problem arises when the transition between such control algorithms occurs, leading to undesirable instability responses. To cope with such difficult situations, this PhD is focused on complex dynamic environment that will lead to advanced control structures, enabling multi-controller transitions while taking into account sensors failures or traffic circumstances. The motivation of the thesis, objectives, manuscript organization, and its main contributions are presented below.

Motivation

Nowadays, systems are getting more and more complex leading to control algorithms able to consider online varying objectives for performance and safety. The field of autonomous systems, in particular autonomous vehicles, is indicative of such an evolution. Indeed, their driving capabilities have been recently improved for highly, and even fully, autonomous driving thanks to advanced control theory. A Chapter 1. Introduction fully autonomous car needs to perform several tasks including longitudinal control, lateral control, chassis control, etc.

It is known that the lateral dynamics of an autonomous vehicle varies significantly with respect to its longitudinal speed. Specifically, at low speeds, the lateral dynamics become harder to be controlled (due to approaching system singularity), whereas at high speeds, robustness and system stability decrease. On the other hand, even at nominal speeds, the lateral control aims to achieve various objectives such as lane tracking, lane changing, obstacle avoidance, etc. Consequently, various performances are required accordingly for different traffic situations that faces the vehicle. However, it is difficult to design a single controller covering the full speed range and achieving multiple objectives. This work focuses on developing advanced switching/interpolating control structures that can guarantee several control objectives over the full speed-range.

In the literature, various techniques have been proposed to obtain parametervarying switching/interpolating control structures. In this thesis, Linear Parameter-Varying (LPV) control approaches and Youla-Kucera (YK) parameterization are introduced to formulate different LPV-YK interpolating/switching control schemes, guaranteeing stable transition under arbitrary switching.

Objectives

The main objective of this PhD work is to design a general control configuration able to maintain different vehicle performances covering the low and high speed vehicle dynamics. In addition, it can handle different situations including lane-keeping, lane-changing, obstacle avoidance, etc.

Generally speaking, the focus is on obtaining interpolating/switching control algorithms running in a real experimental research platform validating the efficiency of the proposed solutions in a real traffic conditions. The main achievements of this study are summarized below:

• Review of the state-of-the-art on the field of switching and interpolation of LPV controllers, and its use in different control applications.

• Define and compare the design and implementation of the main LPV approaches (polytopic, gridding, and LFT), with experimental validation on autonomous vehicles. Advantages and disadvantages of each approach are presented also.

• Propose new LPV-YK switching/interpolating control structures to improve the closed-loop performance and the stability during the switching instants.

• Implement the proposed LPV-YK controllers on lateral control of autonomous vehicles, in simulation and experimental environments, showing the performance improvement.

• The improvement caused by the LPV-YK control approaches on the vehicle lateral performance is shown by either: 1) Adapting the closed-loop specification regarding the performed lateral task; or 2) Maintaining a constant closed-loop specification over the full speed range by switching between consecutive parameter subsets.

1.3. Manuscript Organization 3

Manuscript Organization

This PhD manuscript is organized in different eight chapters. A brief summary of each of the next chapters is represented below:

• Chapter 2, Toward Switching / Interpolating LPV Control: A Review: This chapter corresponds to a state-of-the-art review that was carried out on the switching/interpolation of Linear Parameter-Varying (LPV) control approaches. These concepts have been used to develop theoretical tools solving many control problems ranging from closed-loop stabilization, robust control, gain-scheduling control, switching control to interpolating control. Recent works are collected and classified providing the latest advances with main applications in different control fields. A final discussion is presented viewing the time evolution of the described switching LPV techniques and classifies them with respect to their applications.

• Chapter 3, Autonomous Vehicles Architecture: A historical background of Renault projects on automated vehicles is first presented in this chapter. Then, the architecture of the vehicle platform that has been used in our experiments is detailed. The vehicle modelling with the steering actuator dynamics are formulated. In addition, a brief literature review is shown on the lateral control which has been mainly tackled in this thesis.

• Chapter 4, Application of LPV approaches to Vehicle Lateral Control: This chapter presents the simulation and experimental implementation of the polytopic, grid-based and LFT LPV approaches to solve the lateral control problem on an electric Renault ZOE vehicle. A comparative analysis is shown among the three LPV approaches to the autonomous vehicle lateral control. The design of the three approaches from a theoretical point of view as well as their application on a real vehicle are presented, including a comparative analysis.

• Chapter 5, Advanced LPV-YK Control Structures: Theoretical Proofs: Four switching/interpolating control architectures based on LPV and Youla-Kucera (YK) parameterization are proposed in this chapter, aiming to achieve robust and adaptive closed-loop specifications (rising time, steady-state error, etc.

). An LPV-YK control architecture incorporates multiple LPV or LTI controllers which can switch/interpolate between them with ensuring stability. It can be used to interpolate between different control performances aiming to achieve different closed-loop specifications, or to switch over partitioned parameter subsets when dealing with systems having large parameter variations.

• Chapter 6, Application of the LPV-YK Structures on Vehicle Lateral Control:

This chapter uses the LPV-YK control structures, presented in the previous chapter, to improve the vehicle tracking performance over a large speed range. The performance of the vehicle lateral control is shown to be improved by either adapting the closed-loop specification regarding the performed lateral task, or by maintaining a constant closed-loop specification over the full speed range by switching between consecutive parameter subsets. The chapter first presents, for each approach, the implementation architecture with its designing steps, then, simulation and experimental results are presented to validate the efficiency of the proposed LPV-YK control structures.

• Chapter 7, About The Interpolation Logic of LPV-YK Control: An Example Using Reinforcement Learning: An RL-based LPV-YK interpolation scheme is proposed in this chapter, aiming to achieve an optimal performance with guaranteeing closed-loop quadratic stability. First, it is shown how the interpolating signals affect the closed-loop specifications by analyzing the closed-loop step responses. Then, the RL model is trained and simulated using our Renault's simulation environment. Simulation results are presented to validate the efficiency of the RL-based interpolation logic.

• Chapter 8, Discussion:

This chapter includes conclusions and most important remarks, with respect to the problems addressed in the current PhD manuscript. In addition, future research perspectives that could be derived from the results of this PhD work are mentioned.

Contributions

The main contributions in this thesis concern the LPV and YK control concepts with their implementation on the lateral control of autonomous vehicles, including:

Comparison of LPV approaches concerning modelling, control design, and experimental implementation on vehicle lateral control

1. Quasi-LPV models are structured using the three LPV approaches from the nonlinear bicycle model that vary with respect to the longitudinal velocity.

2. To limit the control input effort and to achieve the noise/disturbance rejection performance, LPV/H ∞ problems are solved using a set of LMIs.

3. The design and experimental validation of grid-based and LFT approaches is carried out for the first time on the lateral control of autonomous vehicles.

4. Input reference is adjusted as a function of the speed and lateral error, modifying the look-ahead parameter accordingly to deal with large lateral errors (lane changing) and the small ones (lane tracking).

5. Simulation and experimental results (on a Renault ZOE vehicle) are shown to compare the performance of the controllers concerning tracking, actuator limitations and noise/disturbance rejection.

Novel LPV-YK control structures design with stability proofs

Four LPV-YK control structures are proposed. The closed-loop stability of each approach is analyzed and proved.

1. The first two structures are designed to maintain several control objectives over the full speed-range. Such structures are considered to be more efficient than the LPV switching control systems since: 1) There is no limitation on the switching/interpolating signals;

2) The interpolated LPV controllers are predefined and designed separately without requiring any common condition or re-design; and 3) The generalized LPV-YK interpolation scheme is defined and proved to achieve closed-loop quadratic stability with smooth assumptions and LMI conditions. More details on both structures are shown below:

• Interpolation of polytopic-based LPV controllers: It is used to interpolate between already designed LPV controllers, knowing that all the interpolated LPV controllers should quadratically stabilize the LPV model for any parameter variations. • Interpolation of YK-based LPV controllers: This structure uses the LTI-YK parameterization to first design multiple quadratically stabilizing LPV controllers. Then interpolates between these YK-based LPV controllers providing a whole closed-loop quadratic stability.

2. The other two LPV-YK control structures are grid-based and partitioned polytopicbased LPV-YK. Their main objective is to switch smoothly between local YKbased LPV controllers, each is designed to be suitable over a certain parameter subregion. Their interest is to decrease the control design conservatism and limitations compared to the standard LPV control approaches, and improving the closed-loop performance. For more details:

• Switching of partitioned YK-based LPV controllers: It is designed aiming to solve the conservatism of the quadratic stability problem due to overbounding (in convex regions), large number of varying parameters, and their large range of variations. A YK-based LPV controller is designed in each parameter subset, and the switching appears at the intersection boundary between each two consecutive subsets. • Switching of grid-based LPV controllers: This structure is formulated to switch between multiple grid-based LPV controllers, each is designed to be useful over its subset. Each parameter subset is gridded and the local LPV controllers are required to be exponentially stabilizing.

Implementation of the proposed LPV-YK controllers on lateral control

1. The implementation steps of each LPV-YK control structure are detailed and some of them are simulated using our Renault simulator.

2. For the first time, the LPV-YK control structures are implemented and experimentally validated on a real Renault ZOE vehicle. Experimental tests on the vehicle lateral control enhance the stability of the closed-loop system, and show the different performances achieved by the vehicle.

Propose an RL-based LPV-YK interpolation scheme

An RL-based LPV-YK interpolation scheme is proposed aiming to achieve an optimal performance with guaranteeing closed-loop quadratic stability. Specifically:

1. Analyze how the interpolating signals affect the closed-loop specifications using step response analysis.

2. The RL model is trained and simulated using the Renault simulator for successive lane changes at different vehicle speeds.

Publications

The scientific contributions derived as a result of the developed research work are listed below:

Chapter 2

Toward Switching / Interpolating LPV Control: A Review

This chapter corresponds to a state-of-the-art review that has been carried out on the switching/interpolation of Linear Parameter-Varying (LPV) control approaches. These concepts have been used to develop theoretical tools solving many control problems ranging from closed-loop stabilization, robust control, gain-scheduling control, switching control to interpolating control. This chapter collects the recent works and classifies them providing the latest advances with main applications in different control fields. A final discussion presents the time evolution of the described LPV techniques and classifies them with respect to their applications. This chapter presents the literature review shown in:

• Toward Switching/Interpolating LPV Control: A Review, see [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF].

The rest of the chapter is organized as follows: Section 2.1 presents the fundamentals of the LPV modelling, control, and its applications. In Section 2.2, LPVswitched systems are represented including the switching methods: hysteresis and Average Dwell-Time (ADT). To solve the aggressive transient switching, the smooth LPV switching technique is explained in Section 2.3. Section 2.4 introduces the LPV control interpolation based on bumpless transfer. The fundamental of Youla-Kucera (YK) parameterization is depicted in Section 2.5 including some of its applications. In Section 2.6, the LPV-YK control interpolation is presented with its new applications in different domains. Finally, the concerned LPV switching/interpolating approaches as represented in a timeline with application classification in the last section.

LPV Systems

Since 1970s, robust control research field started [START_REF] Safonov | Origins of Robust Control: Early History and Future Speculations[END_REF]. From a practical point of view, linearization at different operating points was deeply investigated in the literature [START_REF] Dj Leith | On formulating nonlinear dynamics in LPV form[END_REF]. Linear systems are of several types such as Linear Time-Invariant (LTI), Linear Time-Varying (LTV), and Linear Parameter-Varying (LPV) systems. These systems can be formulated as LPV system which can be considered the most generalized form. LPV concept surged as a control technique that can use LTI synthesis tools to control a nonlinear model. The studies tended to improve robustness in addition to the optimal performance which can handle parameter variations in the plant model.

Consider the following LPV system: It can be shown that according to the following cases of ρ, different types of systems can be achieved:

G(ρ) :    ẋ(t) = A(ρ
• ρ=constant, the system is Linear Time-Invariant (LTI).

• ρ = ρ(t), where the variation of ρ with respect to time is known explicitly, the system is Linear Time-Varying (LTV).

• ρ = ρ(x(t)), the system is called quasi-Linear Parameter Varying (Quasi-LPV) system.

• ρ = ρ(t), which is an external parameter, the system is Linear Parameter-Varying (LPV).

Notice that if the model parameters can be measured online, a gain-scheduling between LTI controllers, designed at different operating points, can be used [START_REF] Shamma | Analysis and design of gain scheduled control systems[END_REF]. Otherwise, the unmeasurable parameters are considered as model uncertainties in the control design to reduce their effects on the closed-loop performance [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF].

For time-varying systems, it may not be possible to find a single Linear Time-Invariant (LTI) controller that can perform well for all operating conditions. Therefore, the Linear Parameter-Varying (LPV) control concept has been successfully developed to achieve a stable gain-scheduling [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF], self-scheduling [START_REF] Apkarian | Self-scheduled H ∞ control of linear parameter-varying systems: a design example[END_REF] or interpolation [START_REF] Wu | Induced L 2 norm control for LPV system with bounded parameter variation rates[END_REF] between LTI systems synthesized at different operating points. The importance of the LPV approach to control general nonlinear systems comes from the fact that the system can be written in the form of a quasi-LPV, where the parameter can vary as a function of states, inputs or outputs and not just considered as exogenous inputs [START_REF] Leith | Survey of gain-scheduling analysis and design[END_REF]. [START_REF] Mohammadpour | Control of linear parameter varying systems with applications[END_REF] and [START_REF]Robust control and linear parameter varying approaches: application to vehicle dynamics[END_REF] present successful application example of LPV/robust techniques to different domains. In the last decades, research studies have developed multivariable control systems using gain-scheduling techniques, see for instance the pioneering works [START_REF] Packard | Gain scheduling via linear fractional transformations[END_REF], [START_REF] Rugh | Research on Gain Scheduling[END_REF]. Gain-scheduling control is used when a plant changes significantly among different operating conditions; which is actually the case in many real applications, see [START_REF] Hoffmann | A Survey of Linear Parameter-Varying Control Applications Validated by Experiments or High-Fidelity Simulations[END_REF] and references therein.

LPV/H ∞ Control Design

LPV/H ∞ controllers have been widely used for different objectives: noise rejection, parameter uncertainties, significant parameter variations, vibration control, tracking control, etc. This section defines the design of an LPV/H ∞ controller based on Linear Matrix Inequality (LMI) satisfying the closed-loop stability. The robust H ∞ control approach is usually used to design an LPV control. The main objective of the H ∞ control is to minimize the L 2 induced gain from the external input w to the controlled output z. This is achieved by solving the following L 2 induced minimization problem:

∥z∥ 2 ≤ γ ∞ ∥w∥ 2 and γ ∞ > 0, to be minimized, represents how much the demanded performance is achieved. If γ ∞ < 1, the demanded performance is totally achieved by the controller.

Among the three approaches, each controller is designed by solving its corresponding LMI-based optimisation problem. For each approach, the complexity of LPV controller existence conditions is determined in terms of the size of LMIs and the number of decision variables. Consider a generalized plant P(ρ) as follows:

P(ρ) =    ẋ(t) = A p (ρ
-→   ẋ z y   =   A p (ρ) B 1 (ρ) B 2 (ρ) C 1 (ρ) D 11 (ρ) D 12 (ρ) C 2 (ρ) D 21 (ρ) D 22 (ρ)     x w u   (2.
3)

The LMI-based LPV/H ∞ control design in Appendix A shows infinite optimization problem due to the infinite possibilities of values of the scheduling parameters. Usually, the formulation of an LPV control problem requires to solve an infinite number of Linear Matrix Inequalities (LMIs) due to the parameter space. Methods have been proposed to reduce the problem to a finite set of LMIs: polytopic [START_REF] Apkarian | Self-scheduled H ∞ control of linear parameter-varying systems: a design example[END_REF], [START_REF] Georgo | System analysis, modelling and control with polytopic linear models[END_REF], grid-based [START_REF] Wu | Control of linear parameter varying systems[END_REF], and Linear Fractional Transformation (LFT) [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF] approaches that have been discussed in the literature. An informative review on the three approaches is drawn in [START_REF] Hoffmann | A Survey of Linear Parameter-Varying Control Applications Validated by Experiments or High-Fidelity Simulations[END_REF]. Currently, there exists a commercial LPV toolbox [START_REF] Hjartarson | LPV aeroservoelastic control using the LPVTools toolbox[END_REF] for use in MATLAB that focuses on model reduction and gridding-based LPV control. The synthesis of an LPV control can be formulated as an LMI optimization problem using a single Lyapunov function, either quadratic [START_REF] Packard | Gain scheduling via linear fractional transformations[END_REF] or parameter-dependent [START_REF] Wu | Induced L 2 Norm Control for LPV Systems with Bounded Parameter Variation Rates[END_REF]. Early synthesis methods have been limited to slow parameter variations [START_REF] Shamma | Gain scheduling: potential hazards and possible remedies[END_REF], however, methods have been proposed over the years allowing arbitrarily fast parameter variations [START_REF] Apkarian | Self-scheduled H ∞ control of linear parameter-varying systems: a design example[END_REF]- [START_REF] Wu | A generalized LPV system analysis and control synthesis framework[END_REF]. Moreover, it has been shown how incorporating bounds on the parameters' rate of variation can reduce control design conservatism, see for instance, [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF] and [START_REF] Wu | Gain-scheduling control of LFT systems using parameterdependent Lyapunov functions[END_REF]. A recent significant survey on LPV control approaches for suspension systems is shown in [START_REF] Sename | Review on LPV Approaches for Suspension Systems[END_REF].

Polytopic Approach

The polytopic approach is the most popular among the LPV control approaches [START_REF] Hoffmann | A Survey of Linear Parameter-Varying Control Applications Validated by Experiments or High-Fidelity Simulations[END_REF]. In many fields, engineers are interested in applying the gain-scheduling method based on optimized designs at different operating points. However, the application of this approach is limited to only a few scheduling parameters. This is due to the exponential growth of the vertices with the number of parameters. In many applications, it is unnecessary to consider the full hyperbox spanned by all combinations of extrema of the scheduling parameter values, as some portions are physically inadmissible. It may then be possible to find tighter convex sets that require fewer vertices.

Overbounding of the parameter set is considered as a main cause of conservatism. The operating region of the underlying LPV model is defined by a convex polytope containing the parameter trajectories. This convex parameter region may include vertices that are not attained by the real plant, resulting in conservatism. The reason is that the construction of the polytope is based on the assumption that all parameters vary independently, whereas they could be related to each other by inherent couplings. For example, the known bicycle model describing the lateral dynamics of an autonomous vehicle is parameterized by the scheduling parameters "v x " and "1/v x " [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF] (being v x the longitudinal speed). Such a situation might cause unstable models at the vertices of the polytope. In addition, the parameters could be physically correlated with each other, such that some combinations of extreme values of the parameters do not occur in real operation. For example, an LPV model describing the vertical flight dynamics of an aeroplane might be parameterized by the external scheduling signals (i.e., parameters) "speed" and "altitude." But usually, the maximum speed is not reached for minimum altitude and vice versa [START_REF] Kwiatkowski | PCA-Based Parameter Set Mappings for LPV Models With Fewer Parameters and Less Overbounding[END_REF].

Several solutions have been investigated in the literature to find a reduced convex parameter region. Several solutions has been investigated in the recent survey [START_REF] Li | Polytopic LPV approaches for intelligent automotive systems: State of the art and future challenges[END_REF]. [START_REF] Azuma | A New LMI Approach to Analysis of Linear Systems Depending on Scheduling Parameter in Polynomial Forms[END_REF] suggests to construct convex polyhedrons along given parameter trajectories, and solve the control design problem using affine parameter-dependent LMIs. Unfortunately, these methods often result in a huge number of vertices or nonconvex parameter sets and thus in increased computational burden. Scheduling Dimension Reduction (SDR) approach based on principal component analysis is proposed in [START_REF] Kwiatkowski | PCA-Based Parameter Set Mappings for LPV Models With Fewer Parameters and Less Overbounding[END_REF], [START_REF] Hoffmann | Synthesis of LPV Controllers With Low Implementation Complexity Based on a Reduced Parameter Set[END_REF], [START_REF] Hoffmann | Synthesis of LPV controllers with reduced implementation complexity[END_REF] which changes the coordinate basis. This method could be used to reduce the parameter set based on experimental data [START_REF] Kwiatkowski | PCA-Based Parameter Set Mappings for LPV Models With Fewer Parameters and Less Overbounding[END_REF], and yields the benefit of tailoring a control design to specific trajectories. Gridding the signal range can also be done [START_REF] Hoffmann | Synthesis of LPV Controllers With Low Implementation Complexity Based on a Reduced Parameter Set[END_REF], [START_REF] Hoffmann | Synthesis of LPV controllers with reduced implementation complexity[END_REF], to obtain a more comprehensive approximate representation without experimental data, which is especially useful if the plant is unstable. The multiconvexity approach has been proposed, extended to polynomial parameterdependent LMIs (PLMIs), and relaxed by incorporating slack variables in [START_REF] Apkarian | Parameterized LMIs in Control Theory[END_REF]. Other approaches are convex approximations and so-called difference convex representations [START_REF] Tuan | Relaxations of parameterized LMIs with control applications[END_REF]. In [START_REF] Koelewijn | Scheduling Dimension Reduction of LPV Models -A Deep Neural Network Approach[END_REF], a Deep Neural Network (DNN) approach is used to develop the SDR methods and achieve higher model accuracy under scheduling dimension reduction. In [START_REF] Zhao | Optimal switching surface design for statefeedback switching LPV control[END_REF] and [START_REF] Zhao | Switching LPV controller design under uncertain scheduling parameters[END_REF], authors develop an LPV switching controller and prove its stability when switching among the overlapped subsets of a polytopic parameter region.

Another reason, which makes the polytopic LPV synthesis conservative, is that it requires a constant Lyapunov function to ensure quadratic stability, which increases the problem conservatism. For more illustration, consider a closed-loop LPV system CL(ρ), where ρ belongs to a polytopic parameter region P. Then, the closed-loop quadratic stability is achieved according to the following theorem. Theorem 1. Define A cl (ρ) as the state matrix of the closed-loop system CL(ρ). The closedloop quadratic stability is achieved for every ρ ∈ P, and for any parameter variation ρ, if there exists a constant, positive definite Lyapunov matrix P satisfying condition (2.4).

PA cl (ρ) +

A T cl (ρ)P < 0 (2.4)
Later, a solution has been proposed in the literature to design LPV controllers based on parameter-varying Lyapunov function [START_REF] Wu | Induced L 2 Norm Control for LPV Systems with Bounded Parameter Variation Rates[END_REF].

Grid-based Approach

The grid-based approach is based on the definition of a mesh on the parameter region. A grid-based LPV system is formulated using a linear or nonlinear interpolation between the corresponding LTI systems at the gridded operating points. It is known that a higher density of gridded points is required for better interpolation performance. An advantage of this technique is that it is applicable for any LPV plants with general parameter-dependency, requiring neither polytopic nor LFT representations. Moreover, model elements like lookup tables can be introduced in the design, since gridding is not restricted to convex regions. Consequently, the Lyapunov variable can be parameterized by general parameter dependency [START_REF] Wu | Induced L 2 Norm Control for LPV Systems with Bounded Parameter Variation Rates[END_REF].

The implementation of the controller is computationally inexpensive, but may require large amounts of memory, to store the local controllers. The implementation scheme may consider an interpolation or a switching between local controllers. In general, this approach is limited to a few scheduling signals because of the exponential increase in grid points, hence memory requirements or interpolation complexity. Note, however, that if observer-based synthesis is performed, parameter-dependent state-feedback matrices of both observer and controller can be computed online each by a single matrix inversion without the need to store multiple controllers in memory [START_REF] Saupe | Applied LPV control exploiting the separation principle for the single axis positioning of an industrial manipulator[END_REF], [START_REF] Saupe | An Observer Based State Feedback LFT LPV Controller for an Industrial Manipulator[END_REF], [START_REF] Saupe | Linear Parameter Varying Control Design for Industrial Manipulators[END_REF].

Consider a closed-loop LPV system CL(ρ), where ρ belongs to a gridded parameter region P. Then, the closed-loop exponential stability is achieved according to the following theorem. Theorem 2. Define A cl (ρ) as the state matrix of the closed-loop system CL(ρ). The closedloop exponential stability is achieved for every ρ ∈ P, and for a bounded parameter variation ρ ∈ [ν, ν], if there exists a positive definite, parameter-dependent Lyapunov matrix P(ρ) satisfying condition (2.5).

Ṗ(ρ) + P(ρ)A cl (ρ) + A T cl (ρ)P(ρ) < 0 (2.5)

LFT Approach

An LPV-LFT system is formulated as a Linear Fractional Representation (LFR) containing the nominal LTI system and the model uncertainties block (see [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF] for more illustration). LFT control synthesis using D-scalings is presented in [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF] and [START_REF] Packard | Gain scheduling via linear fractional transformations[END_REF], which is then refined to D/G-scalings in [START_REF] Scorletti | Improved linear matrix inequality conditions for gain scheduling[END_REF], [START_REF] Scorletti | IMPROVED LMI CONDITIONS FOR GAIN SCHEDULING AND RELATED CONTROL PROBLEMS[END_REF], and [START_REF] Helmersson | Methods for robust gain scheduling[END_REF]. The underlying theory to LFT LPV controller synthesis essentially relies on the S-procedure [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF], and its variants and extensions, most notably the so-called full-block S-procedure (FBSP), [START_REF] Scherer | LPV control and full block multipliers[END_REF], [START_REF] Scherer | A full block S-procedure with applications[END_REF]. A general unifying approach to both LFT and grid-based LPV synthesis was presented in [START_REF] Wu | A generalized LPV system analysis and control synthesis framework[END_REF]. Historical surveys of the S-procedure can be found in [START_REF] Derinkuyu | On the S-procedure and Some Variants[END_REF] and [START_REF] Pólik | A Survey of the S-Lemma[END_REF]. The S-procedure was also used early on for robust/LPV analysis problems, already extending to less conservative results using parameter-dependent Lyapunov functions [START_REF] Feron | S-procedure for the analysis of control systems with parametric uncertainties via parameter-dependent Lyapunov functions[END_REF], [START_REF] Feron | Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions[END_REF]. It was much later, that LFT LPV synthesis was extended to parameter-dependent Lyapunov functions [START_REF] Wu | Gain-scheduling control of LFT systems using parameterdependent Lyapunov functions[END_REF], while the basic idea was first presented in [START_REF] Dettori | Robust stability analysis for parameter dependent systems using full block S-procedure[END_REF]. A general framework for the synthesis of uncertain dynamical systems described by standard linear fractional representations, and robust gainscheduling controllers by convex optimization techniques has been investigated in [START_REF] Veenman | A synthesis framework for robust gain-scheduling controllers[END_REF].

LPV Control Applications

The three LPV control approaches have been widely used and experimentally validated in different applications. Knowing that most of the real systems are accurately modeled as nonlinear, the LPV approaches have been chosen to simplify the control design problems providing both stability and multi-objective performance. Such an advantage has encouraged researchers and engineers to apply the LPV control approaches on real systems such as aircrafts, aero-engines, vehicles, and different electronic and mechatronic systems. Recently, the three LPV approaches have been compared and experimentally validated on a real automated vehicle in [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF].

Chapter 2. Toward Switching / Interpolating LPV Control: A Review

Polytopic Approach:

Since the polytopic controller has been considered to be the simplest to design and implement, plenty of works have investigated its application to several kinds of systems. Active Magnetic Bearing (AMB) system has been modelled and controlled using the polytopic approach in [START_REF] Lu | LPV control design and experimental implementation for a magnetic bearing system[END_REF], [START_REF] Lu | Linear parameter-varying techniques for control of a magnetic bearing system[END_REF], [START_REF] Simon Ghersin | LPV CON-TROL OF A MAGNETIC BEARING EXPERIMENT[END_REF], [START_REF] Jasper Witte | Robust and LPV control of an AMB system[END_REF], [START_REF] Ghersin | Classical, Robust and LPV Control of a Magnetic-bearing Experiment[END_REF], [START_REF] Narita | Gain scheduling controller synthesis for active magnetic bearing based on parameter dependent LMI with convexity condition[END_REF], [START_REF] Kuseyri | Adaptive vibration control of rotors with active magnetic bearings[END_REF]. The polytopic approach has been mainly chosen due to its high robustness in dealing with input disturbances and parameter uncertainties. An example of a more complex LPV model can be a three degrees-of-freedom (3-DOF) robotic manipulator that is introduced in [START_REF] Hoffmann | Benchmark problem -nonlinear control of a 3-DOF robotic manipulator[END_REF]. A polytopic controller that is scheduled affinely on 16 different scheduling parameters has been designed and simulated in MATLAB on an accurate nonlinear model. In [START_REF] Hoffmann | Benchmark problem -nonlinear control of a 3-DOF robotic manipulator[END_REF], [START_REF] Seyed | Gain-scheduled H infinity control of a robotic manipulator with nonlinear joint friction[END_REF], [START_REF] Patton | LPV fault estimation and FTC of a two-link manipulator[END_REF], [START_REF] Paraiso | Black-box versus grey-box LPV identification to control a mechanical system[END_REF], [START_REF] Hashemi | LPV modelling and control of a 2-DOF robotic manipulator using PCA-based parameter set mapping[END_REF], [START_REF] Hoffmann | Closed-loop stability and performance optimization in LPV control based on a reduced parameter set[END_REF], [START_REF] Hashemi | Lowcomplexity linear parameter-varying modeling and control of a robotic manipulator[END_REF], [START_REF] Jabali | Uncertain polytopic LPV modelling of robot manipulators and trajectory tracking[END_REF], [START_REF] Yu | Polytopic gain scheduled H ∞ control for robotic manipulators[END_REF], the design complexity of the robotic manipulators, due to its high number of scheduling parameters, has been tackled by employing different methods such as a principle component analysis-based reduction of parameter set.

Additionally, the polytopic controller has been widely used in different kinds of applications, including engine control [START_REF] Kwiatkowski | Linear parameter varying PID controller design for charge control of a spark-ignited engine[END_REF], [START_REF] White | Mixed H 2 /H ∞ Observer-Based LPV Control of a Hydraulic Engine Cam Phasing Actuator[END_REF], [START_REF] Castillo | Fresh Air Fraction Control in Engines Using Dynamic Boundary Stabilization of LPV Hyperbolic Systems[END_REF], miscellaneous mechatronic systems [START_REF] Chen | Compensating for spatially repetitive disturbance with linear parameter varying repetitive control[END_REF], [START_REF] Bergmann | Implementation of LPV H ∞ Loop-Shaping Control for a Variable Stiffness Actuator[END_REF], [START_REF] Boudaoud | Gain scheduled control strategies for a nonlinear electrostatic microgripper: Design and real time implementation[END_REF], [START_REF] Hoffmann | Active damping of container crane load swing by hoisting modulation -An LPV approach[END_REF], [START_REF] Vincent | Robust Gain Scheduled Control of a Hydrokinetic Turbine Part 2: Testing[END_REF], [START_REF] Sugiyama | SWITCHING FROM VELOCITY TO FORCE CONTROL FOR THE ELECTRO-HYDRAULIC SERVOSYSTEM BASED ON LPV SYSTEM MODELING[END_REF], [START_REF] Bolea | LPV VS MULTI-MODEL PI(D) GAIN-SCHEDULING APPLIED TO CANAL CONTROL[END_REF], [START_REF] Wijnheijmer | Modelling and LPV control of an electro-hydraulic servo system[END_REF], [START_REF] Forrai | Electromagnetic Actuator Control: A Linear Parameter-Varying (LPV) Approach[END_REF], [START_REF] Asadi | Fault Reconstruction of Islanded Nonlinear DC Microgrids: An LPV-based Sliding Mode Observer Approach[END_REF], vehicle motion control [START_REF] Hirano | Automated driving for an experimental vehicle using visual feedback control[END_REF], [START_REF] Inoue | Experimental results on the nonlinear H ∞ control via quasi-LPV representation and game theory for wheeled mobile robots[END_REF], [START_REF]Robust control and linear parameter varying approaches: application to vehicle dynamics[END_REF], [START_REF] Guo | Robust gain-scheduling automatic steering control of unmanned ground vehicles under velocity-varying motion[END_REF], [START_REF] Alcala | Gain-scheduling LPV Control For Autonomous Vehicles Including Friction Force Estimation and Compensation Mechanism[END_REF], academic test benches [START_REF] Benicio | Discrete-time L2 loopshaping control of a Maglev System using the LPV framework[END_REF], [START_REF] Rotondo | Quasi-LPV modeling, identification and control of a twin rotor MIMO system[END_REF], [START_REF] Kajiwara | LPV techniques for control of an inverted pendulum[END_REF], [START_REF] Ali | Discrete-Time LPV Controller Synthesis Using Dilated LMIs with Application to an Arm-Driven Inverted Pendulum[END_REF], [START_REF] Robert | An H ∞ LPV Design for Sampling Varying Controllers: Experimentation With a T-Inverted Pendulum[END_REF], noise cancelling headsets and vibration control test benches, [START_REF] Shu | Design of structured discrete-time LPV gain-scheduling controllers through state augmentation and partial state feedback[END_REF], [START_REF] Shu | Two-Parameter pLPV Modeling of Nonstationary Harmonically Related Multisine Disturbances for Reduced-Order Gain-Scheduling Control[END_REF], [START_REF] Ballesteros | Reduced-order two-parameter pLPV controller for the rejection of nonstationary harmonically related multisine disturbances[END_REF], [START_REF] Heins | GAIN-SCHEDULED STATE-FEEDBACK CONTROL FOR ACTIVE CANCELLATION OF MUL-TISINE DISTURBANCES WITH TIME-VARYING FREQUENCIES[END_REF], aerospace flight control [START_REF] Sato | Robust Gain-Scheduled flight controller using inexact scheduling parameters[END_REF], aero-engine systems [START_REF] Wang | Gain scheduling controller of the aero-engine based on LPV model[END_REF] temperautre control [START_REF] Pipino | Nonlinear temperature regulation of solar collectors with a fast adaptive polytopic LPV MPC formulation[END_REF], and computer sciences [START_REF] Sun | LPV Model and Its Application in Web Server Performance Control[END_REF].

Grid-based Approach:

The grid-based approach has been raised since the pioneering works [START_REF] Wu | Control of linear parameter varying systems[END_REF] and [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF] and the experimental results in flight control shown in [START_REF] Papageorgiou | Taking robust LPV control into flight on the VAAC Harrier[END_REF], with interesting results. This approach has been mainly used for systems that have shown high conservatism when using the polytopic approach, in particular, aircraft motion control [START_REF] Sato | Robust Gain-Scheduled flight controller using inexact scheduling parameters[END_REF] and [START_REF] Sato | Robust H/sub 2/ problem for LPV systems and its application to model-following controller design for aircraft motions[END_REF]. The grid-based approach has been used also in vehicle control for: 1) Automated lane guidance [START_REF] Hingwe | Linear parameter varying controller for automated lane guidance: experimental study on tractor-trailers[END_REF]; 2) Automated driving via visual feedback [START_REF] Hirano | Automated driving for an experimental vehicle using visual feedback control[END_REF]; 3) Longitudinal and lateral tracking controls [START_REF] Atoui | Parameter Varying Approach For A Combined (Kinematic + Dynamic) Model Of Autonomous Vehicles[END_REF]; 4) Brake control [START_REF] Van | The design of an H ∞ /LPV active braking control to improve vehicle roll stability[END_REF]; 5) Electric steering system [START_REF] Yamamoto | Design and experimentation of an LPV extended state feedback control on Electric Power Steering systems[END_REF]; and 6) Car engine control [START_REF] Kwiatkowski | LPV controller synthesis for charge control of a car engine -A hybrid evolutionary algebraic approach[END_REF].

Moreover, several applications are introduced in the literature using the gridding approach including AMB [START_REF] Morsi | Model Predictive Control Based on Linear Parameter-Varying Models of Active Magnetic Bearing Systems[END_REF], robotics [START_REF] Salimi | Gain-scheduling control of a cable-driven MRIcompatible robotic platform for intracardiac interventions[END_REF], [START_REF] Yu | Gain Scheduled LPV H ∞ Control Based on LMI Approach for a Robotic Manipulator[END_REF], engine control [START_REF] Zhang | Transient lean burn air-fuel ratio control using input shaping method combined with linear parameter-varying control[END_REF], [START_REF] Zhang | Linear Parameter-Varying Lean Burn Air-Fuel Ratio Control[END_REF], [START_REF] Jung | Calibratable linear parameter-varying control of a turbocharged diesel engine[END_REF], wind turbine [START_REF] Farzad A Shirazi | Wind turbine integrated structural and LPV control design for improved closed-loop performance[END_REF], and the control of a control moment gyroscope [START_REF] Hossam | LPV gain-scheduled control of a control moment gyroscope[END_REF], [START_REF] Theis | Self-Scheduled Control of a Gyroscope[END_REF], and aero-engine control [START_REF] Bei | Non-affine parameter dependent LPV model and LMI based adaptive control for turbofan engines[END_REF].

LFT Approach:

LFT LPV synthesis with full-block multipliers has been successfully applied in [START_REF] Dettori | Design and implementation of a gain scheduled controller for a compact disc player[END_REF], validating simulation results shown in [START_REF] Dettori | Gain scheduling control for a CD player mechanism using LPV techniques[END_REF]. Even though the LFT controllers have not been widely used compared to the polytopic and the gridding approaches, experimental results with industrial relevance were published between 2001 and 2014. These include CD players [START_REF] Dettori | LPV design for a CD player: an experimental evaluation of performance[END_REF], [START_REF] Dettori | LMI techniques for control with application to a compact disc player mechanism[END_REF], induction motors [START_REF] Trangbaek | Linear Parameter Varying Control of Induction Motors[END_REF], [START_REF] Bendtsen | Discrete-time LPV current control of an induction motor[END_REF], engine control [START_REF] Umut | Linear Parameter-Varying Modelling and Robust Control of Variable Cam Timing Engines[END_REF], [START_REF] Gauthier | SOME EXPERIMENTAL RESULTS OF AN H ∞ -LPV CONTROLLER APPLIED TO A DIESEL ENGINE COMMON RAIL IN-JECTION SYSTEM[END_REF], [START_REF] Gauthier | An LFT Approach to H Control Design for Diesel Engine Common Rail Injection System[END_REF], and a wafer stage [START_REF] Groot Wassink | LPV control for a wafer stage: beyond the theoretical solution[END_REF]. In addition, it is worth mentioning that the LFT approach has been experimentally validated on lateral control of autonomous vehicles [START_REF] Atoui | Design And Experimental Validation Of A Lateral LPV Control Of Autonomous Vehicles[END_REF].

Like the polytopic and gridding approaches, the LFT control synthesis has been tested on the AMB system [START_REF] Lauridsen | Design of robust AMB controllers for rotors subjected to varying and uncertain seal forces[END_REF], [START_REF] Sahinkaya | Cancelling Gyroscopic Effects in AMB Systems With Flexible Rotors via Modal Feedback[END_REF], flight control [START_REF] Poussot | Generation of a reduced-order LPV/LFT model from a set of large-scale MIMO LTI flexible aircraft models[END_REF], [START_REF] Biannic | Design and robustness analysis of fighter aircraft flight control laws[END_REF], and aero-engine [START_REF] Wang | Fault estimation for aero-engine LPV systems based on LFT[END_REF]. It has been widely used in the field of active noise cancellation headsets and vibration control [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF] that is also followed by [START_REF] Ballesteros | LPV gain-scheduled output feedback for active control of harmonic disturbances with time-varying frequencies[END_REF], [START_REF] Ballesteros | A frequency-tunable LPV controller for narrowband active noise and vibration control[END_REF], [START_REF] Ballesteros | Disturbance Rejection through LPV Gain-Scheduling Control with Application to Active Noise Cancellation[END_REF], [START_REF] Shu | Active vibration control for harmonic disturbances with time-varying frequencies through LPV gain scheduling[END_REF], [START_REF] Duarte | An LPV discrete-time controller for the rejection of harmonic time-varying disturbances in a lightweight flexible structure[END_REF]. In [START_REF] Lim | LPV Controller Design for Diesel Engine SCR Aftertreatment Systems Based on Quasi-LPV Models[END_REF], a diesel engine controller has been designed using the LFT approach.

LPV complexity

The application of LPV techniques on robotic manipulators has been always chosen to validate methods aiming to solve the high LPV complexities with large number of scheduling parameters [START_REF] Hoffmann | Synthesis of LPV Controllers With Low Implementation Complexity Based on a Reduced Parameter Set[END_REF], [START_REF] Hoffmann | Benchmark problem -nonlinear control of a 3-DOF robotic manipulator[END_REF], [START_REF] Hoffmann | Linear Parameter-Varying Control of Complex Mechanical Systems[END_REF], [START_REF] Hoffmann | Closed-loop stability and performance optimization in LPV control based on a reduced parameter set[END_REF]. Moreover, negative examples of applied LPV controllers that do not show an increase in performance are given in [START_REF] Bolea | LPV VS MULTI-MODEL PI(D) GAIN-SCHEDULING APPLIED TO CANAL CONTROL[END_REF], [START_REF] Wijnheijmer | Modelling and LPV control of an electro-hydraulic servo system[END_REF]. [START_REF] Bolea | LPV VS MULTI-MODEL PI(D) GAIN-SCHEDULING APPLIED TO CANAL CONTROL[END_REF] compares the LPV technique against a classical gain-scheduled PI control approach, which consists in designing LTI controllers for several operating points and then applying an interpolation strategy. In [START_REF] Wijnheijmer | Modelling and LPV control of an electro-hydraulic servo system[END_REF], a comparison is shown between an LPV control with a classical gain-scheduling approach and constant nonscheduled LTI controllers. The performance increase obtained by the LPV controller is considered insignificant. In both works, possible reasons are stated in the conservatism due to parameter-independent Lyapunov functions, which is not a restriction when classical gain-scheduling is applied [START_REF] Bolea | LPV VS MULTI-MODEL PI(D) GAIN-SCHEDULING APPLIED TO CANAL CONTROL[END_REF].

Nowadays, systems are getting more and more complex leading to control algorithms able to consider online varying objectives for performance and safety. For instance, the field of autonomous systems, in particular autonomous vehicles, is indicative of such an evolution. Indeed, their driving capabilities have been recently improved for highly, and even fully, autonomous driving thanks to advanced control theory. A fully autonomous car needs to perform several tasks including longitudinal control, lateral control, chassis control, etc. Moreover, the lateral dynamics of an autonomous vehicle varies significantly with respect to its longitudinal speed [START_REF] Pacejka | Tire and vehicle dynamics[END_REF], [START_REF] Rajamani | Vehicle dynamics and control[END_REF]. Specifically, at low speeds, the lateral dynamics becomes harder to be controlled (due to approaching system singularity), whereas at high speeds, robustness and system stability decrease [START_REF] Mariano | Nested PID steering control for lane keeping in autonomous vehicles[END_REF]. On the other hand, even at nominal speeds, the lateral control aims to achieve various objectives such as lane tracking, lane changing, obstacle avoidance, etc. Consequently, various performances are required accordingly for different traffic situations that faces the vehicle. However, it is difficult to design a single controller covering the full speed range and achieving multiple objectives. As shown below, some solutions have been investigated in the literature to design LPV switched/interpolated control systems.

Switching LPV Systems

Describing system dynamics over a large scheduling parameter region by a single LPV model would lead to significant modeling errors, and hence inevitably degrades the closed-loop system performance [START_REF] Balas | Robustness and performance trade-offs in control design for flexible structures[END_REF]. In addition, it could be very conservative to design a single LPV controller that aims to shape the closed-loop performance over a certain operating condition, or according to critical/fault situations.

Fig. 2.1 shows a parameter region P divided into multiple subregions, so that P = i∈ [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF][START_REF] Shamma | Analysis and design of gain scheduled control systems[END_REF] P i . The i th subregion is denoted by P i (ρ) (i ∈ Z N = {1, 2, . . . , N}).

The sub-systems for adjacent subregions are to be switched according to different laws, including scheduling-parameter-dependent laws, state-dependent laws or external switching signals. In 2004, Lu and Wu [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF] have proposed a switching LPV control design based on multiple parameter-dependent Lyapunov functions, each suitable for a specific parameter subregion, the switching allowing to achieve better performance. The switching stability has been studied for hysteresis switching and switching with average dwell-time strategies [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF].

Later, several research studies have been focused on relaxing the LPV switching stability constraints [START_REF] Deaecto | Dynamic output feedback H ∞ control of continuous-time switched affine systems[END_REF], [START_REF] Zhao | New Results on Stability of Slowly Switched Systems: A Multiple Discontinuous Lyapunov Function Approach[END_REF], [START_REF] Briat | Stability analysis of LPV systems with piecewise differentiable parameters[END_REF], [START_REF] Deaecto | Stability Analysis and Control Design of Discrete-Time Switched Affine Systems[END_REF], [START_REF] Blanchini | Stability results for linear parameter varying and switching systems[END_REF], [START_REF] Liberzon | Switching in systems and control[END_REF], [START_REF] Qu | LPV Modeling and Switched Control for EGR Valves with Dry Friction[END_REF], [150]. It is worth mentioning that, in the literature, it has been often proposed to design the multiple LPV controllers in the same form and order. Even though the local LPV controllers can have different order and form, however switching stability conditions will be difficult to derive. The assumption of same-order controllers has been always taken into consideration to primarily simplify the stability issue and controller synthesis.

The switching LPV controllers are usually designed offline, and their switching logic is determined in terms of scheduling parameter region division. The switching stability is often proved by non-increasing Lyapunov functions in the switching sequence. In [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF], hysteresis switching and average-dwell-time switching strategies have been introduced, and different control synthesis conditions have been proved for each strategy. In addition, the minimum switching strategy has been reported in [START_REF] Yuan | Robust switching output-feedback control of time-varying polytopic uncertain systems[END_REF]. These switching strategies impose constraints on Lyapunov matrices or switching signals on switching surfaces and achieve guaranteed switching stability. The switching strategies are illustrated more in the next section.

Switching strategies

Switching systems can be seen as a hybrid system of the continuous-time system and discrete-time switching signal. The neighboring controllers to be switched are always assumed to be in the same form and order to simplify the stability problem and controller synthesis complexity. It is noted that this dissertation also follows this assumption. In the literature, there are many studies about the switching strategies and switching stability conditions, in particular, for linear systems [START_REF] Hespanha | Stability of switched systems with average dwell-time[END_REF], [START_REF] Liberzon | Switching in systems and control[END_REF], [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF]. These switching strategies have been extended to switching LPV systems [START_REF] Lim | Analysis of hybrid linear parameter-varying systems[END_REF], [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF]. In this section, the state-of-art research about switching LPV system and control will be covered, then popular switching strategies and associated switching stability conditions of LPV systems will be given. Consider the closedloop autonomous LPV system without external inputs in (2.6) for analysis of switching stability,

CL i (ρ) : { ẋ(t) = A cl,i (ρ)x(t) (2.6)

Hysteresis Switching

Hysteresis switching requires to have an overlapped subregion between any two neighboring subregions. Referring to Figure 2.2, when ρ crosses the switching surface P 1,2 , the sub-system CL 1 (ρ) is switched to CL 2 (ρ), and when ρ crosses the switching surface P 2,1 , CL 2 (ρ) is switched back to CL 1 (ρ). Direct Lyapunov method is often used to prove the switching stability. Since the multiple controllers are designed on subregions, it is supposed that there is a family of positive definite Lyapunov matrices P i (ρ) dependent on the scheduling parameter ρ. Then parameterdependent Lyapunov functions are formulated as quadratic functions as

V i (x, ρ) = x T P i (ρ)x, ∀i (2.7)
where i represents the local active controller on the i th subregion P i , and its corresponding Lyapunov matrix P i (ρ) is used in formulating the Lyapunov function. The switching stability can be achieved by non-increasing Lyapunov functions during each switching event [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF]. The non-increasing condition is proved to be a sufficient condition but not a necessary condition. In literature, there are proven results to relax this conservative result, in which the multiple Lyapunov functions may increase its value during a time interval, only if the increment is bounded by certain kinds of continuous functions. Interested readers are recommended to the reference [START_REF] Hui Ye | Stability theory for hybrid dynamical systems[END_REF], [START_REF] Lin | Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results[END_REF]. Consider a switching event on a surface P i,j from i th subregion to j th subregion, if the hysteresis switching is used, the sufficient condition of globally exponentially stability of switching system is given by the following Theorem. Theorem 3. Consider the closed-loop LPV system in (2.6), if there exists a family of parameterdependent Lyapunov matrices satisfying condition (2.8), then the closed-loop exponential stability is achieved within local subsystems CL i (ρ) with (ρ, ρ) ∈ P i × V. Moreover, if condition (2.9) is satisfied on the switching surface, then global exponential stability is achieved on entire scheduling parameter region (ρ, ρ) ∈ P × V.

Ṗi (ρ) + P i (ρ)A cl,i (ρ) + A T cl,i (ρ)P i (ρ) < 0 (2.8) P i (ρ) ≥ P j (ρ), ρ ∈ P i,j , i, j ∈ N J , i ̸ = j (2.9)
Proof is illustrated in [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF].

Average-Dwell-Time (ADT) switching

The Average-Dwell-Time (ADT) switching strategy enforces the "slow-switching" property of switching signals so that the closed-loop system achieves global stability under the switching sequence. By ADT switching strategy, only a limited number of switches are allowed within a finite time interval [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF], [START_REF] Lim | Analysis and control of linear-parameter-varying systems[END_REF], [START_REF] Hespanha | Stability of switched systems with average dwell-time[END_REF]. [START_REF] Yan | Dwell time optimization in switching control of parameter varying time delay systems[END_REF], [START_REF] Zhang | L 2 -L ∞ Model Reduction for Switched LPV Systems With Average Dwell Time[END_REF], [START_REF] Yan | A switching control approach to stabilization of parameter varying time delay systems[END_REF], [START_REF] Huang | Average Dwell Time Based Smooth Switching Linear Parameter-Varying Proportional-Integral-Derivative Control for an F-16 Aircraft[END_REF] We assume that switching signal σ(t) renders N σ (T, t), the number of switching events within the time interval [t, T]. If there exist two positive numbers N 0 and τ a such that

N σ (T, t) ≤ N 0 + T -t τ a , ∀T ≥ t ≥ 0 (2.10)
where N 0 is the chatter bound to avoid chattering phenomenon, then a sufficient condition for ADT switching stability is given in the next Theorem. Theorem 4. Given positive scalar λ 0 and µ, if there exists a family of parameter-dependent Lyapunov matrices P i (ρ) satisfying condition (2.11) on each subregion (θ, θ) ∈ P i × V and condition (2.12) on switching surface, then the exponentially stability is achieved by switching signal with average dwell time τ a > lnµ λ 0 within the entire scheduling parameter region (θ, θ) ∈ P × V.

Ṗj (ρ) + λ 0 Ṗj (ρ) + P j (ρ)A cl,j (ρ) + A T cl,j (ρ)P j (ρ) < 0 (2.11) 1 µ P j (ρ) ≥ P i (ρ) ≥ µP j (ρ), ρ ∈ P i,j , i, j ∈ N J , i ̸ = j (2.
12)

The proof is illustrated in [START_REF] Liberzon | Switching in systems and control[END_REF], [START_REF] Hespanha | Stability of switched systems with average dwell-time[END_REF].

Applications

Since 2004, the switched LPV control has augmented the use of the LPV control approaches achieving better performance and lower conservatism. They have been simulated and experimentally validated in several applications, mainly in aircraft flight control [START_REF] Ligang | Switching control of morphing aircraft based on Qlearning[END_REF], [START_REF] Wang | Development of a switched polytopic LPV model for the aircraft performing high angle of attack maneuver[END_REF], [START_REF] Yang | Observer-based H ∞ resilient control for a class of switched LPV systems and its application[END_REF], [START_REF] He | Robust control of morphing aircraft based on switched polytopic system[END_REF], [START_REF] Lu | Reference tracking control of hypersonic vehicles using switched linear parameter-varying approach[END_REF], [START_REF] Li | Fault detection for switched linear parametervarying systems: An average dwell-time approach[END_REF], [START_REF] Lu | Control design for a hypersonic aircraft using a switched linear parameter-varying system approach[END_REF], [START_REF] Xu He | Control of switched LPV systems using common Lyapunov function method and an F-16 aircraft application[END_REF], [START_REF] Xu He | A blending method control of switched LPV systems with slow-varying parameters and its application to an F-16 aircraft model[END_REF], [START_REF] Lu | Switching-based fault-tolerant control for an F-16 aircraft with thrust vectoring[END_REF], [START_REF] Dong | Online Switching Control of LFT Parameter-Dependent Systems[END_REF], [START_REF] Lu | Switching LPV control of an F-16 aircraft via controller state reset[END_REF], and aero-engine control [START_REF] Guangdeng | Fault-tolerant Control of Switched LPV Systems: A Bumpless Transfer Approach[END_REF], [START_REF] Yang | Refined Anti-disturbance Control of Switched LPV Systems With Application to Aero-Engine[END_REF], [START_REF] Wang | Optimal control of aero-engine systems based on a switched LPV model[END_REF], [START_REF] Li | Dynamic Event-Triggered Finite-Time H ∞ Tracking Control of Switched LPV Aero-Engine Models[END_REF], [START_REF] Yang | H ∞ output tracking control for a class of switched LPV systems and its application to an aero-engine model[END_REF], [START_REF] Zhu | Event triggered control for a switched LPV system with applications to aircraft engines[END_REF], [START_REF] Zhu | H ∞ Tracking Control for Switched LPV Systems With an Application to Aero-Engines[END_REF], [START_REF] Zhu | H ∞ filtering for switched linear parameter-varying systems and its application to aero-engines[END_REF], [START_REF] Zhu | L 2 gain analysis of actuator saturation for switched LPV systems and applications to aero-engines[END_REF], [START_REF] Kongwei | State feedback control for switched LPV systems with actuator saturation and its application to aero-engines[END_REF], [START_REF] Postma | Air-Fuel Ratio Control of Spark Ignition Engines Using a Switching LPV Controller[END_REF], [START_REF] Tang | Switching LPV Control with Double-Layer LPV Model for Aero-Engines[END_REF], [START_REF] Jiang | Gain-scheduled control for morphing aircraft via switching polytopic linear parameter-varying systems[END_REF]. In addition, [START_REF] Li | Linear Parameter-Varying Model of an Electro-Hydraulic Variable Valve Actuator for Internal Combustion Engines[END_REF], [START_REF] Qu | LPV Modeling and Switched Control for EGR Valves with Dry Friction[END_REF], [START_REF] Qu | Engine EGR Valve Modeling and Switched LPV Control Considering Nonlinear Dry Friction[END_REF], [START_REF] Qu | Engine EGR Valve Modeling and Switched LPV Control Considering Nonlinear Dry Friction[END_REF], [START_REF] Qu | LPV Modeling and Switched Control for EGR Valves with Dry Friction[END_REF] apply the switched-LPV control on the Exhaust Gar Re-circulation (EGR) system. In mechatronics, the LPV-switched controller has been designed for different systems such as quadrotors [START_REF] Nie | Finite-time switched LPV control of quadrotors with guaranteed performance[END_REF], [START_REF] Nie | Switched linear parameter-varying tracking control for quadrotors with large attitude angles and time-varying inertia[END_REF], vehicle lateral dynamics [START_REF] Gagliardi | A Fault Detection Filter Design Method for Hybrid Switched Linear Parameter Varying Systems[END_REF], robotic manipulators [START_REF] Bagher | Uncertain polytopic LPV modelling of robot manipulators and trajectory tracking[END_REF], suspension control [START_REF] Zhang | Switched LPV Modeling and H 2 Control for Hybrid Suspension with Mass Variation[END_REF], [START_REF] Nguyen | A switched LPV observer for actuator fault estimation[END_REF], microsatellite [START_REF] Xu | Fault tolerant control with switched LPV method based on hysteresis strategy and an application to a microsatellite model[END_REF]. In addition to other applications as AMB [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF], [START_REF] He | Switching control for LPV polytopic systems using multiple Lyapunov functions[END_REF], [START_REF] Lu | Control design of switched LPV systems using multiple parameter-dependent Lyapunov functions[END_REF], wind turbine [START_REF] Xue | Hysteresis Switching LPV Control of a Wind Turbine System Covering Partial and Full Load Conditions[END_REF], [START_REF] Lescher | Switching LPV Controllers for a Variable Speed Pitch Regulated Wind Turbine[END_REF], hard disk drives [START_REF] Ehsan | Automated Multiple Robust Track-Following Control System Design in Hard Disk Drives[END_REF], flexible ball-screw drives [START_REF] Hanifzadegan | Switching Gain-Scheduled Control Design for Flexible Ball-Screw Drives[END_REF], mass-spring-damper [START_REF] Zhang | Stability and Stabilization of a Class of Multimode Linear Discrete-Time Systems With Polytopic Uncertainties[END_REF], [START_REF] Ehsan | A parameter set division and switching gain-scheduling controllers design method for time-varying plants[END_REF], glucose [START_REF] Colmegna | Automatic Glucose Control During Meals and Exercise in Type 1 Diabetes: Proof-of-Concept in Silico Tests Using a Switched LPV Approach[END_REF], [START_REF] Patricio | Reducing glucose variability due to meals and postprandial exercise in T1DM using switched LPV control: In silico studies[END_REF], [START_REF] Colmegna | Switched LPV Glucose Control in Type 1 Diabetes[END_REF], and academic simulation [START_REF] Deaecto | H ∞ state feedback switched control for discrete time-varying polytopic systems[END_REF].

Smooth LPV switching

It has been shown in [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF] that the switched LPV controllers may not provide a smooth transient response during switching, when aggressive performance is obtained at switching instants. Such a case may lead to mechanical damage, decrease material lifetime, or signal saturation which is out of real application objectives. Following this work, challenges have been raised to smooth the control response during switching, and to relax the limitations on the switching signals. Several research studies have been involved in solving the switching smoothness.

The leading cause of un-smooth switching control response is due to the sudden change of control variables during switching events. The ultimate reason is that un-smooth switching LPV control optimizes closed-loop system performance over each subregion, nevertheless switching smoothness between adjacent controllers is not considered. The system performance optimization over each subregion, but ignoring switching often leads to high-gain controllers with jumped controller gains. This can be easily validated by checking the control gain difference between two neighboring subregions over the switching surface.

The smooth switching LPV controller aims to minimize the gap between controller gains and achieves smooth switching, whereas the conventional LPV controller only considers the switching stability but not the switching smoothness. To this purpose, a new small subregion is considered between each two consecutive subregions which plays the role of smooth switching. Fig. 2.3 shows the developed partitioning expressed in the smooth LPV switching where:

P i := ρ ∈ R : ρ (i) < ρ < ρ (i) , i ∈ Z N P i,i+1 := ρ ∈ R : ρ (i) < ρ < ρ (i+1) , i ∈ Z N-1 (2.13)
Consequently, the smooth switching LPV controller can be described as:

K(ρ) = K i (ρ), if ρ ∈ P i , i ∈ Z N K i,i+1 (ρ), if ρ ∈ P i,i+1 , i ∈ Z N-1 (2.14)
Here, the controller K j is interpreted as a local LPV controller, to which the LPV controller K is switched when the parameter ρ lies in the sub-interval P i . On the other hand, the controller K i,i+1 plays a role in smoothly switching between two local LPV controllers K i and K i+1 while the parameter ρ is in a transitional sub-interval P i,i+1 .

Smooth LPV switching control design

In this section, an example of the requirements demanded to design a smooth LPV switching control is introduced, the conditions are illustrated more in [START_REF] Hanifzadegan | Smooth switching LPV controller design for LPV systems[END_REF].

For a given LPV plant in (2.1), with ρ ∈ P, ρ ∈ V (V := { ν ∈ R : ν ≤ ν ≤ ν }), and the subintervals in (2.13), the objective is to design an LPV controller with the switching rule in (2.14) that satisfies the following requirements for any trajectory ρ(.) with ρ(t) ∈ P and ρ(t) ∈ V for all t ≥ 0.

1. The closed-loop system containing the plant G(ρ) and the switching LPV controller K(ρ) (2.14) is exponentially stable.

Chapter 2. Toward Switching / Interpolating LPV Control: A Review 2. For zero initial condition, the L 2 -gain of the closed-loop system from the external input w to to the controlled output z is bounded by a specified value γ > 0.

3. The controllers switch continuously and smoothly between K i and K i,i+1 , as well as between K i,i+1 and K i+1 , i.e., for i ∈ Z N-1 , and for m = 0, 1, . . . , M for a specified integer M ≥ 0,

d m dt m K i (ρ(t)) = d m dt m K i,i+1 (ρ(t)), at ρ(t) = ρ (i) , d m dt m K i+1 (ρ(t)) = d m dt m K i,i+1 (ρ(t)), at ρ(t) = ρ (i+1) (2.15)
4. The rate of change of the controller matrix is bounded by a specified value η > 0 in the transitional subintervals, i.e.,

d dt K(ρ(t)) < η, ( 2.16)
for any ρ ∈ P i,i+1 , i ∈ Z N-1 , and any ρ ∈ V. Here, ∥.∥ denotes the l 2 -induced matrix norm, i.e., the largest singular value of a matrix.

An iterative descent algorithm has been introduced in [START_REF] Hanifzadegan | Smooth switching LPV controller design for LPV systems[END_REF] to solve a series of convex optimization problems by fixing a part of optimization variables. This method is used to find a feasible solution numerically for a non-convex optimization problem. The problem has been also augmented to two dimensional parameter regions.

In [START_REF] He | Smooth Switching LPV Control and Its Applications[END_REF], two smooth switching techniques, simultaneous design and sequential design, have been illustrated. These two approaches are discussed briefly as follows:

• Simultaneous design approach : All the switching controllers are designed at the same time to formulate a convex optimization problem [START_REF] He | Simultaneous Design of Smooth Switching State-Feedback LPV Control[END_REF]. Using the norm of deviation of controller parameters between two consecutive switching surfaces, a numerically tractable smoothness index is introduced into the cost function. The minimization of this smoothness index helps in reducing significantly the sharp changes in control states and outputs, but at the cost of degraded H 2 and H ∞ system performance. This means that there exists a trade-off relationship between switching smoothness and system performance. Thus, a tunable weighting coefficient is introduced in the cost function to balance the system performance and switching smoothness. This weighting coefficient is tuned until finding an optimal trade-off which provides smoothswitching LPV controller with acceptable system performance. In the simultaneous design approach, the control synthesis conditions depend on the adjacent subregions due to the switching stability condition. Consequently, a higher-dimensional optimization problem, with a higher amount of LMI constraints, decision variables, online computational load, and memory requirement are obtained when increasing the number of subregions [START_REF] He | Smooth-switching LPV control for vibration suppression of a flexible airplane wing[END_REF], [START_REF] Hoffmann | A Survey of Linear Parameter-Varying Control Applications Validated by Experiments or High-Fidelity Simulations[END_REF]. As a result, the simultaneous design would be practically infeasible for high-order systems with many divided subregions.

• Sequential design approach: This approach is proposed in [START_REF] He | Sequential Design of Switching <tex>H ∞ </tex> LPV State-Feedback Control[END_REF], [START_REF] He | A Sequential Design Approach for Switching H ∞ LPV Control[END_REF] to reduce the computational complexity found in the previous approach. Over each subregion, PLMIs for H ∞ are used to synthesize switching LPV controllers separately. While for each overlapped subregion, the Lyapunov matrix Moreover, other methods have been proposed in the literature for switching smoothness. In [START_REF] Zhao | Optimal switching surface design for statefeedback switching LPV control[END_REF] and [START_REF] Zhao | Switching LPV controller design under uncertain scheduling parameters[END_REF], the authors develop an LPV switching controller and prove its stability when switching among the overlapped subsets of a polytopic parameter region. More recently, [START_REF] He | Smooth Switching LPV Dynamic Output-feedback Control[END_REF] proposes a more developed smooth LPV switching control design. On the other hand, it is worth noticing that such algorithms have increased the complexity and the design constraints of the local controllers to achieve their objectives. These methods rely heavily on iterative computations to solve multi-objective convex or non-convex problems. In addition, the introduced smoothness index lacks physical meaning, and the smoothness constraints on controller matrices are selected through trial and error.

Applications

The additional smoothness to the transient response of the LPV switched controllers has encourages more researchers and engineers to implement it on simulated and real environments. The concerned applications include aircraft flight control [START_REF] He | Smooth-switching LPV control for vibration suppression of a flexible airplane wing[END_REF], [START_REF] He | Active Vibration Suppression of BWB Airplane Using Smooth Switching LPV Control[END_REF], [START_REF] Jiang | A systematic method of smooth switching LPV controllers design for a morphing aircraft[END_REF], [START_REF] Cai | Smooth control for a class of linear time-varying systems with input quantization and input saturation[END_REF], [START_REF] Cheng | Smooth switching linear parameter-varying fault detection filter design for morphing aircraft with asynchronous switching[END_REF], [START_REF] Huang | Average Dwell Time Based Smooth Switching Linear Parameter-Varying Proportional-Integral-Derivative Control for an F-16 Aircraft[END_REF], [START_REF] Yiru | Extended state observer based smooth switching control for tilt-rotor aircraft[END_REF], [START_REF] Song | Smooth Switching Output Tracking Control for LPV Systems[END_REF], [START_REF] Jiang | Smooth switching linear parameter-varying control for hypersonic vehicles via a parameter set automatic partition method[END_REF], [START_REF] Huang | Average Dwell Time Based Smooth Switching Linear Parameter-Varying Proportional-Integral-Derivative Control for an F-16 Aircraft[END_REF], aero-engine control [START_REF] Zhao | Optimal switching surface design for switching LPV control and its application to air-fuel ratio control of an automotive engine[END_REF], [START_REF] Liu | Smooth Switching Controller Design for Multiobjective Control Systems and Applications[END_REF], [START_REF] Zhao | A Control Design of Rotor Speed Regulation for an Aeroengine based on Smooth Switching Strategy[END_REF], wind turbine [START_REF] Chen | Smooth switching gain-scheduled control for large scale offshore wind turbine under full wind-speed conditions[END_REF], missile control [START_REF] Song | Smooth switching output tracking control for a class of linear time-varying systems: An LMI approach[END_REF], flexible ball-screw [START_REF] Hanifzadegan | Smooth switching LPV controller design for LPV systems[END_REF], and TORA (Translational Oscillator with Rotational Actuator) systems [START_REF] Chen | The smooth switching control for TORA system via LMIs[END_REF], [START_REF] Chen | The smooth switching gain-scheduled control with H ∞ performance to the application of TORA systems[END_REF].

In addition to the smooth switching approaches, an extensive research on bumpless transfer-based switching controllers has been investigated. The following section illustrates briefly the bumpless transfer method and its applications.

LPV Control Interpolation Based on Bumpless Transfer

A bumpless transfer aims to improve the transient switching behavior for a switched system. In order to reduce switching bumps, a bump constraint method has been contributed to realize bumpless transfer. The main objective is to study the switching bumps problem by modifying the original controllers, while the switching design is neglected. A description of most popular strategies for bumpless transfer can been found in [START_REF] Edwards | Anti-windup and bumplesstransfer schemes[END_REF], [START_REF] Mayuresh | A unified framework for the study of anti-windup designs[END_REF], [START_REF] William S Levine | The Control Handbook (three volume set)[END_REF] and [START_REF] Edwards | Anti-windup and bumplesstransfer schemes[END_REF]. One of the first schemes is proposed by Hanus for nonlinear plants [START_REF] Hanus | Conditioning technique, a general anti-windup and bumpless transfer method[END_REF]. The idea consists in pre-setting the off-line controller state for reducing the transient behavior at the switching time. Turner and Walker have generalized the results of Hanus for controllers which are not bi-proper [START_REF] Turner | Modified linear quadratic bumpless transfer[END_REF], [START_REF] Matthew | Linear quadratic bumpless transfer[END_REF]. For instance, a bumpless transfer of switching controllers is proposed in [START_REF] Dimon | Bumpless Transfer Between Observer-Based Gain Scheduled Controllers[END_REF] followed by some developments in [START_REF] Kinnaert | State resetting for bumpless switching in supervisory control[END_REF] and [START_REF] Safaei | On controller initialization in multivariable switching systems[END_REF].

Chapter 2. Toward Switching / Interpolating LPV Control: A Review A bumpless transfer method can be used also to overcome the problem of switching jumps in a scheduled robust model predictive control approach [START_REF] Khani | Smooth switching in a scheduled robust model predictive controller[END_REF]. A scheduled robust model predictive controller implements a set of local robust model predictive controllers based on an on-line switching strategy. This method could enlarge the domain of attraction efficiently but the transient response might be hampered by spikes appearing at the moment of switching between adjacent local controllers. The algorithms based on bumpless transfer could enhance the transient response by implementing some intermediate controllers augmented to the main control scheme to solve the problem without needing more computation.

Figure 2.4 represents a bumpless-transfer switching between the pre-designed off-line controllers controller 1 and controller 2 using two on-line controllers Q1 and Q2 . Let us define the signal

zi e (k) = α i (k) -e(k), (2.17)
which represents the difference between the i th off-line controller input α i (k) ∈ R m and the on-line controller input e(k) = r(k)y(k), where r(k) ∈ R m is the reference; and the signal zi

u (k) = u i (k) -u(k), (2.18)
that represents the difference between the i th off-line controller output u i (k) and the on-line controller output u(k), for any i ∈ I and for all k ∈ Z + . The idea in [START_REF] Matthew | Linear quadratic bumpless transfer[END_REF] is to minimize the following LQ criterion:

Ji = φi ( Ti f ) + 1 2 Ti f -1 ∑ k=t i [ zi T u (k) Wi u zi u (k) + zi T e (k) Wi e zi e (k)], (2.19)
where φi ( Ti f ) =

1 2 zi T u ( Ti f ) Xi zi u ( Ti f ),
Wi e = W iT e > 0 and Wi u = W iT u > 0 are the weighting matrices. t i is the switching time to the sybsystem corresponding to the mode i ∈ I, Ti f is the terminal time and Xi = Xi T is a terminal weighting matrix ∀i ∈ I. Since reference signals are not known a priori, an extension to an infinite horizon is required for practical implementation. This leads to a constant feedback matrix Qi that adds a condition on the i th off-line controller to achieve the desired transient behavior at the switching instant (Fig. 2.4).

In particular,

α i (k) = Qi   x c (k) u(k) e(k)   , (2.20) with Qi = (I m -Γ i B i T c Π i B i c ) -1 Γ i    (D i T c Wi u C i c + B i T c Π i A i c ) T -(D i T c Wi u + B i T c (I p -M i ) -1 U i ) T -( Wi e + B i T c (I p -M i ) -1 E i ) T   
T More details can be found in [START_REF] Malloci | Sur les systèmes à commutation à deux échelles de temps: une application au contrôle de guidage de bande dans un laminoir à chaud[END_REF]. LQ bumpless transfer has been one of most celebrated bumpless transfer methods on industrial MIMO applications [START_REF] Zheng | Steady-state bumpless transfer under controller uncertainty using the state/output feedback topology[END_REF], [START_REF] Zheng | Input/output structure of the infinite horizon LQ bumpless transfer and its implications for transfer operator synthesis[END_REF]. This success is due to different factors: the existence of several reliable numerical solvers for Riccati equations, the excellent convergence properties of LQ based feedback controllers, and the fact that no plant knowledge is needed. Nevertheless, the extension to an infinite horizon assumes that the tracking error e and the control signal u are constant. This approximation is effective only if these signals vary slowly enough, with reference to the system dynamics. Another drawback is concerned with the fact that this strategy guarantees the closed loop stability only around a specific operating point. In general, it is assumed that the closedloop stability of the whole process is maintained, if both on-line and off-line control loops are stable. This assumption is justified only if the operating point is subject to "slow" variations.

Applications

Several research studies have concerned the application of the LPV bumpless-transfer approach to AMB [START_REF] Li | Dynamic transfer of robust AMB controllers[END_REF], aircraft flight control [START_REF] Turner | Switched control of a vehicle/short take-off land aircraft: An application of linear quadratic bumpless transfer[END_REF], [START_REF] Dong | Online Switching Control Designs of LFT Systems[END_REF], [START_REF] Dong | Advanced design techniques in linear parameter varying control[END_REF], aero-engine control [START_REF] Liu | Design for aircraft engine multi-objective controllers with switching characteristics[END_REF], [START_REF] Yang | H ∞ bumpless transfer for switched LPV systems and its application[END_REF], [START_REF] Guangdeng | Fault-tolerant Control of Switched LPV Systems: A Bumpless Transfer Approach[END_REF], [START_REF] Yang | Bumpless Transfer H ∞ Anti-Disturbance Control of Switching Markovian LPV Systems Under the Hybrid Switching[END_REF], [START_REF] Zhao | Rate Bumpless Transfer Control for Switched Linear Systems With Stability and Its Application to Aero-Engine Control Design[END_REF], [START_REF] Shi | Bumpless Transfer Control for Switched Linear Systems and its Application to Aero-Engines[END_REF], [START_REF] Yang | H ∞ bumpless transfer reliable control of Markovian switching LPV systems subject to actuator failures[END_REF], [START_REF] Sun | Bumpless Transfer of Uncertain Switched System and Its Application to Turbofan Engines[END_REF], and wind turbine [START_REF] Raja | Parameterized Disturbance Observer Based Controller to Reduce Cyclic Loads of Wind Turbine[END_REF]. In addition, it has been implemented on inverted pendulum [START_REF] Yavari | Robust H2 outputfeedback bumpless transfer control of polytopic uncertain LPV systems[END_REF], water-tank level control [START_REF] Khani | Smooth switching in a scheduled robust model predictive controller[END_REF], academic examples [START_REF] Zong | Bumpless transfer fault detection for switched systems: a state-dependent switching approach[END_REF], [START_REF] Song | Bumpless H ∞ control for periodic piecewise linear systems[END_REF], [START_REF] Yame | Parameterization of Linear Controllers for Bumpless Switching in Multi-Controller Schemes[END_REF], and on electronic systems [START_REF] Zhao | Anti-Disturbance Bumpless Transfer Control for Switched Systems With its Application to Switched Circuit Model[END_REF], [START_REF] Steenis | An Approach to Bumpless Control for LPV Modeled Inverters in a Microgrid[END_REF].

YK Parameterization

Recently, a new smooth LPV switching technology is proposed based on Youla-Kucera (YK) parameterization [START_REF] Wu | Design of Smooth Switching LPV Controller Based on Coprime Factorization and H ∞ Performance Realization[END_REF]. Compared with the literature [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF], the novelty of this method is to introduce a parameter overlap division method to improve the smoothness of switching. The flexibility and freedom of the switching controller are increased, and the transient response disturbance during the switching is suppressed. The next section introduces a general review on YK parameterization.

The fundamentals of YK parameterization started in 1970s, developed by Youla [START_REF] Youla | Modern Wiener-Hopf design of optimal controllers Part I: The single-input-output case[END_REF], [START_REF] Youla | Modern Wiener-Hopf design of optimal controllers-Part II: The multivariable case[END_REF] and Kucera [START_REF] Kucera | Stability of Discrete Linear Feedback Systems[END_REF]. The interest behind YK concept is to parameterize a [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF]. Each one is parameterized by its corresponding YK dynamical parameter Q(s) [START_REF] Niemann | Dual Youla parameterisation[END_REF]. The stability is guaranteed depending on the parameter Q(s). Similarly, the dual YK parameterization provides all the linear plants stabilized by a given controller. The class of all the plants stabilized by a controller depends on the transfer function called dual YK dynamical parameter S(s), so G(S). This parameter could represent any plant variations. Hence, this useful way of parameterizing either plants, controllers or both is employed to solve many control issues.

According to the control objectives, three main configurations can be targeted:

• Controller parameterization: allows stable controller reconfiguration when some change occurs. It is also widely used in disturbance and noise rejection control. A number of successful applications can be found in the last two decades, being the most used approach in different control fields [START_REF] Tay | High performance control[END_REF].

• Plant parameterization: is employed to solve the problem of closed-loop identification. Some successful implementation can be found in Plug & Play control where the dynamics of new sensors or actuators are identified on real-time without system disconnection [START_REF] Niemann | Dual Youla parameterisation[END_REF].

• Simultaneous control and plant parameterization: provides a new control structure that changes according to new identified dynamics on the plant. This principle is mainly used in fault tolerant control and adaptive control [START_REF] Bendtsen | Multiple model adaptive control using dual youla-kucera factorisation[END_REF].

The current review mainly focuses on the controller parameterization part. Consider a SISO LTI plant G(s) connected to a controller K(s) in a stable feedback structure (see Fig. 2.5 for details). The closed-loop transfer function is obtained as:

CL(s) = G(s)K(s) 1 + G(s)K(s) (2.21)
Moreover, let's define a transfer function Q(s) from the reference r to the control input u, then:

Q(s) = K(s) 1 + G(s)K(s) (2.22)
Thus, if G(s) and Q(s) are known, the controller K(s) can be expressed as:

K(s) = Q(s) 1 -G(s)Q(s) (2.23)
From (2.22), it is shown that if K(s) stabilizes G(s), then Q(s) is stable and proper. Equivalently, if Q(s) is stable and proper, then K(s) given by (2.23) stabilizes G(s).

In addition, reformulating the closed-loop expression to be written in terms of Q(s), the closed-loop transfer function is affine with respect to Q(s) expressed as:

CL(s) = Q(s)G(s) (2.24)
From this point, the concept of a controller parameterized by a YK transfer function Q(s) is opened. Then, a set of stabilizing controllers can be parameterized in terms of all stable and proper functions Q(s) for any LTI plant G(s). The LTI-YK parameter Q(s) is designed using the doubly coprime factorisation [START_REF] Desoer | Feedback system design: The fractional representation approach to analysis and synthesis[END_REF], [START_REF] Vidyasagar | Control system synthesis: a factorization approach, part II[END_REF]. YK parameterisation uses the doubly coprime factorisation concepts to reduce the algebraic complexity of Q computation [START_REF] Desoer | Feedback system design: The fractional representation approach to analysis and synthesis[END_REF], [START_REF] Vidyasagar | Control system synthesis: a factorization approach, part II[END_REF]. Let an LTI controller K stabilizes the LTI plant G, then they can be factorized (from left and right) as a product of a stable transfer function matrix and a transfer function matrix with a stable inverse as shown below:

Doubly Coprime Factorisation

YK parameterisation uses the doubly coprime factorisation concepts to reduce the algebraic complexity of Q computation [START_REF] Desoer | Feedback system design: The fractional representation approach to analysis and synthesis[END_REF], [START_REF] Vidyasagar | Control system synthesis: a factorization approach, part II[END_REF]. Let an LTI controller K stabilizes the LTI plant G, then they can be factorized (from left and right) as a product of a stable transfer function matrix and a transfer function matrix with a stable inverse as shown below:

G = N M -1 = M-1 Ñ K = UV -1 = Ṽ-1 Ũ ∀j (2.25)
Lemma 1. If the coprime factors M, N, M, Ñ, U, V, Ũ, Ṽ ∈ RH ∞ (stable and proper), and they satisfy the following Bezout Identity:

Ṽ - Ũ -Ñ M M U N V = M U N V Ṽ - Ũ -Ñ M = I 0 0 I (2.26)
then, all the factorized LTI controller K stabilize G (proof in [START_REF] Tay | High performance control[END_REF]).

LTI-Q Parameterisation

Consider the LTI-YK parameter Q as a transfer function which characterizes the dynamic mapping of a controller K that stabilizes the plant G.

Lemma 2. Assume an LTI plant G = N M -1 stabilized by the LTI controller K = UV -1 , with M and N are coprime and ∈ RH ∞ , U and V are coprime and ∈ RH ∞ . Then the set of all stabilizing controllers K, for G, as a function of a stable filter YK parameter Q with appropriate dimensions is defined as:

K = K(Q) = (U + MQ)(V + NQ) -1 = ( Ṽ + Q Ñ) -1 ( Ũ + Q M) (2.27)
where Q ∈ RH ∞ by construction [START_REF] Tay | High performance control[END_REF].

Control reconfiguration is important to improve system performance, especially for multi-variable and multi-objective systems. Such complex systems may require some specifications which is difficult to handle using a single controller, such as desired bandwidth, time response, or robustness against modeling errors. an H ∞ multi-objective controller can hold several specifications in a single controller, however, it is known to be conservative when complexity increases. In addition to the previously mentioned switching methods: 1) Gain-scheduled approach [START_REF] Packard | Gain scheduling via linear fractional transformations[END_REF]; Adhoc technique [START_REF] Rugh | Research on Gain Scheduling[END_REF]; and 2) Bumpless transfer [START_REF] Zaccarian | The L 2 bumpless transfer problem for linear plants: Its definition and solution[END_REF], YK parameterization has been Chapter 2. Toward Switching / Interpolating LPV Control: A Review also used for switching objective, where a set of linear stabilizing controllers can be mapped onto a Q-based controller [START_REF] Hespanha | Switching between stabilizing controllers[END_REF].

On the other hand, YK parameterization for switching/interpolating controllers has shown several advantages: 1) It allows stable interpolation between unstable controllers [START_REF] Stoustrup | Starting up unstable multivariable controllers safely[END_REF]; 2) Interpolated controllers can be designed and tuned separately using different techniques (H ∞ , LQR, PID, ...) [START_REF] Tay | High performance control[END_REF]; 3) It facilitates adding new parts to an existing system online as Plug&Play control theory [START_REF] Stoustrup | Plug& play control: Control technology towards new challenges[END_REF]; and 4) The closedloop stability is guaranteed under arbitrary continuous/discontinuous interpolating signals between different stabilizing controllers without requiring a common Lyapunov function [START_REF] Hespanha | Switching between stabilizing controllers[END_REF]. In 80s, the first application of YK-based control parameterization has been performed [START_REF] Moore | On improving control-loop robustness of model-matching controllers[END_REF]. Specifically, two controllers K 1 and K 2 have been designed satisfying reference tracking and disturbance rejection respectively. Both controllers have been parameterized with respect their corresponding YK parameters Q 1 and Q 2 respectively.

Applications

A significant literature review on YK work including applications can be found in [START_REF] Mahtout | Advances in Youla-Kucera parametrization: A Review[END_REF]. The YK parameterization has been successfully used in several domains. A Q-based YK controller has been applied on the roll angle of an aircraft to compensate the well-known trade-off between performance and robustness [START_REF] Hespanha | Switching between stabilizing controllers[END_REF]. Another work has formulated a YK-based switching architecture to switch between a high performance controller when the measured angle is not noisy, to a robust controller when there are model uncertainties and external disturbances [START_REF] Hespanha | Towards the High Performance Control of Uncertain Processes via Supervision[END_REF]. In [START_REF] Cifdaloz | Control of distributed parameter systems subject to convex constraints: Applications to irrigation systems and Hypersonic Vehicles[END_REF], two classes, Irrigation systems and Hypersonic vehicles have been controlled based on YK parameterization. Both classes are formulated by hyperbolic partial differential equations. The YK controller switches between different LTI controllers to regulate time-domain specifications (i.e. peak value of control signal, overshoot). Thanks to YK parameterization, [START_REF] Trangbaek | Stable controller reconfiguration through terminal connections[END_REF] and [START_REF] Trangbaek | Stable controller reconfiguration through terminal connections -A practical example[END_REF] were able to introduce new components to the control control loop without affecting system stability. The controller reconfiguration is implemented on a livestock stable climate system, where a new temperature measurement come out during system operation and a new controller is added to the existent control loop.

In [START_REF] Vestergaard | Stepwise commissioning of a steam boiler with stability guarantees[END_REF], the YK concept has been proposed for a steam boiler to switch between two controllers with different objectives; the first aims to regulate the water level within min and max bounds, and the second control objective is to keep the feed water flow steady. In addition, noise/vibration control domains have investigated the YK controllers to attenuate noises and disturbances [START_REF] Landau | On the use of Youla-Kucera parametrization in adaptive active noise and vibration control-A review[END_REF]. Moreover, it has been also applied for different applications requiring high control precision and noise sensitivity such as: wafer scanning in semiconductors [START_REF] Xu Chen | Pseudo Youla-Kucera parameterization with control of the waterbed effect for local loop shaping[END_REF], data storage systems (reading/writing) [START_REF] Martinez | Improving playability of Blu-ray disc drives by using adaptive suppression of repetitive disturbances[END_REF], [START_REF] Chen | New Repetitive Control With Improved Steady-State Performance and Accelerated Transient[END_REF], [START_REF] Wu | Youla parameterized adaptive vibration suppression with adaptive notch filter for unknown multiple narrow band disturbances[END_REF], mechatronics [START_REF] Tomizuka | Dealing with periodic disturbances in controls of mechanical systems[END_REF], active suspension systems [START_REF] Doumiati | Road profile estimation using an adaptive Youla-Kučera parametric observer: Comparison to real profilers[END_REF], [START_REF] Doré Landau | Adaptive narrow band disturbance rejection applied to an active suspension-an internal model principle approach[END_REF] and biochemistry [START_REF] Valentinotti | Optimal Operation of Fed-Batch Fermentations via Adaptive Control of Overflow Metabolite[END_REF] where the regulation problem is to maximize the biomass productivity in the fed-batch fermentation of a specie of yeasts and the cell growth is an undesirable consequence and considered as an unstable disturbance. An application on steamboiler is represented in [START_REF] Vestergaard | Stepwise commissioning of a steam boiler with stability guarantees[END_REF].

Moreover, YK control schemes have been successfully used for Intelligent Transportation Systems (ITS) in the last years. A steering control of autonomous vehicles considering two LTI controllers designed separately (one for lane-changing and one for lane-tracking) [START_REF] Mahtout | Youla-Kucera Based Lateral Controller for Autonomous Vehicle[END_REF]. The YK control scheme of both controllers has shown interesting performance for small and large lateral errors. Moreover, [START_REF] Navas | Using Plug & Play control for stable ACC-CACC system transitions[END_REF], [START_REF] Navas | Mixing V2V-and non-V2V-equipped vehicles in car following[END_REF] have experimentally validated the YK control structure by switching between Cooperative Adaptive Cruise Control system (CACC) and ACC when the communication between vehicles fails.

LPV-YK Parameterization

Consider an LPV plant G(ρ) (2.1), with ρ is a scheduling parameter within a predefined range ρ ∈ [ρ, ρ], is formulated using either polytopic or grid-based approach. Let K(ρ) be an LPV controller designed to stabilize G(ρ) over the whole parameter region. Then, at each vertex or grid point ρ i , the local LTI controller K i is known to stabilize the local plant G i . Using YK parameterization, it is possible to parameterize each local LTI controller K i by a corresponding local LTI-YK parameter Q i , obtaining a parameterized LTI-YK local controller Ki . The interpolation or gain-scheduling between the obtained Ki (i ∈ [1, N]) achieves a gain-scheduling LPV-YK control scheme K(ρ) satisfying:

1. Stability of the closed-loop including LPV system under fast changes of the varying parameter ρ.

2. Recovering the local controllers K i in the operating points.

In [START_REF] Xie | Design of LPV control systems based on Youla parameterisation[END_REF], conceptions of doubly coprime factorisation and Youla parameterization of LTI systems have been extended to LPV systems with respect to quadratic stability using a state-space expression:

LPV Doubly Coprime Factorization

YK parameterization uses the doubly coprime factorization concepts to reduce the algebraic complexity of Q-computation. Assume an LPV controller K(ρ) stabilizes the LPV plant G(ρ), then they can be factorized (from left and right) as a product of a stable transfer function matrix and a transfer function matrix with a stable inverse as shown below:

G(ρ) = N(ρ)M -1 (ρ) = M-1 (ρ) Ñ(ρ) K(ρ) = U(ρ)V -1 (ρ) = Ṽ-1 (ρ) Ũ(ρ) (2.28)
Lemma 3. If the coprime factors M(ρ), N(ρ), M(ρ), Ñ(ρ), U(ρ), V(ρ), Ũ(ρ), Ṽ(ρ) are stable and proper, and they satisfy the following Bezout Identity:

Ṽ(ρ) -Ũ(ρ) -Ñ(ρ) M(ρ) M(ρ) U(ρ) N(ρ) V(ρ) = M(ρ) U(ρ) N(ρ) V(ρ) Ṽ(ρ) -Ũ(ρ) -Ñ(ρ) M(ρ) = I 0 0 I (2.29)
then, the factorized LPV controller K(ρ) stabilizes G(ρ) [START_REF] Xie | Design of LPV control systems based on Youla parameterisation[END_REF].

LPV-Q Parameterization

Consider the LPV-YK parameter Q(ρ) that maps its corresponding LPV controller K(ρ) to a parameterized Q-based LPV-YK controller K(ρ). Then, the following Lemma is proved to be satisfied [START_REF] Mahtout | Advances in Youla-Kucera parametrization: A Review[END_REF].

K(ρ) = (U(ρ) + M(ρ)Q(ρ))(V(ρ) + N(ρ)Q(ρ)) -1 = ( Ṽ(ρ) + Q(ρ) Ñ(ρ)) -1 ( Ũ(ρ) + Q(ρ) M(ρ)) (2.30)
where Q(ρ) is designed to be quadratically stable (see [START_REF] Xie | Design of LPV control systems based on Youla parameterisation[END_REF] for details).

In addition, [START_REF] Xie | Design of LPV control systems based on Youla parameterisation[END_REF] has designed an observer-based state-feedback LPV controller based on Youla parameterization. It has been proved that any quadratically stabilizing LPV controller can be parameterized based on YK concept, providing the closed-loop quadratic stability.

On the other hand, some works have proposed the YK concept to design a polytopic gain-scheduling controller with lower conservatism than the standard LPV design. [START_REF] Bianchi | Interpolation for gain-scheduled control with guarantees[END_REF] proposes a YK-based gain-scheduled controller by interpolating LTI controllers designed separately at the different vertices of a polytopic parameter region. The interpolation is performed as a function of the varying parameters of the LPV model. Closed-loop quadratic stability and performance are guaranteed at intermediate interpolation points of the convex domain. However, the proposed interpolation is preferably used when the system often remains at the designed operating points with rarely transitions from one to another. As a conclusion, the polytopic LPV controllers could perform better in fast parameter changes where a uniform performance is obtained in the designed convex region.

In [START_REF] Rasmussen | Stable Controller Interpolation and Controller Switching for LPV Systems[END_REF], a YK configuration is proposed to improve the performance of a polytopic LPV control. It introduces an LPV system which switches between a minimumphase and nonminimum-phase dynamics as a function of the parameter variations. An LPV controller based on the polytopic approach is designed as a nominal controller in the full parameter region. Then, two different LTI controllers (H ∞ and PID) are designed separately at certain operating conditions (one in minimum-phase region and another in nonminimum-phase region). These LTI controllers are then interpolated with the nominal LPV controller using an LTI-YK configuration. In [START_REF] Blanchini | Stable LPV Realization of Parametric Transfer Functions and Its Application to Gain-Scheduling Control Design[END_REF], a fixed pole-assignment application is introduced using an LPV YK-based method to preserve the closed-loop poles at the same location by interpolating between different controllers regardless the variations in the LPV system. Therefore, a unique performance is obtained along the variation of the parameters while keeping stability.

Recently, [START_REF] Atoui | Interpolation of multi-LPV control systems based on Youla-Kucera parameterization[END_REF] proposes a YK-based interpolation scheme between two LPV controllers, designed separately, to achieve a multi-objective control system. Each LPV controller is designed over a convex domain with a common Lyapunov function (following the approach in [START_REF] Packard | Gain scheduling via linear fractional transformations[END_REF]), to quadratically stabilize the plant model. However, this approach may be conservative since it requires all the local LPV controllers to be designed based on standard polytopic-based LPV approach.

Applications

LPV-YK control schemes have not been widely used in applications yet. However, some works have implemented them with remarkable results. In [START_REF] Xie | Design of LPV control systems based on Youla parameterisation[END_REF], an LPV model of mass-spring-damper system (with a varying stiffness) is controlled using quadratically stabilizing LPV-YK controller. Another application has been performaed on a MIMO LPV system of a quadruple tank [START_REF] Rasmussen | Stable Controller Interpolation and Controller Switching for LPV Systems[END_REF]. The objective is to control the water height in four tanks using two valves; the plant changes from minimum phase to a non minimum-phase with respect to the operating points (valves values). Two LTI-YK controllers are designed; one corresponding to a PID controller designed in the minimum phase operating point, and another H ∞ controller in the non-minimum phase operating point. These two controllers are parameterized on two significant operating points with respect to a polytopic LPV controller. The overall interpolation control scheme shows optimal performance in both operating conditions.

In [START_REF] Blanchini | Stable LPV Realization of Parametric Transfer Functions and Its Application to Gain-Scheduling Control Design[END_REF] the LPV Q-based controller is used in fixed pole assignment application, the Q-based controller switches among different controllers to place the closed-loop poles always at the same location independently of the varying parameter. Thus, the LPV closed-loop system achieves the same performance in the range of the varying parameter without loosing stability. A YK based feedback controller rejecting bounded disturbances is extended to cover LPV systems using LMI computation toolbox in [START_REF] Luca | Control of disturbed LPV systems in a LMI setting[END_REF].

To validate the smoothness of the LPV-YK control structure compared to the previous approaches, it has been implemented on the known AMB system in [START_REF] Wu | Design of Smooth Switching LPV Controller Based on Coprime Factorization and H ∞ Performance Realization[END_REF], [START_REF] Jasper Witte | Experimental results with stable and unstable LPV controllers for active magnetic bearing systems[END_REF], [START_REF] Balini | Performance Enhancement for AMB Systems Using Unstable H ∞ Controllers[END_REF], [START_REF] Daniel | A novel design approach for switched LPV controllers[END_REF], [START_REF] Balini | Synthesis and implementation of gain-scheduling and LPV controllers for an AMB system[END_REF], [START_REF] Wu | Switching Controller Parameterization Realization for an LPV Plant with H ∞ Performance[END_REF], [START_REF] Wu | Switching Linear Parameter-Varying Controller Design With H ∞ Performance Based on Youla Parameterization[END_REF], and aircraft flight control [START_REF] Wu | Switching fault-tolerant control of a flexible airbreathing hypersonic vehicle[END_REF], [START_REF] Schirrer | Convex Design for Lateral Control of a Blended Wing Body Aircraft[END_REF]. Different applications have been also concerned including wind turbine [START_REF] Wang | Adaptive Damping Control Scheme for Wind Grid-Connected Power Systems with Virtual Inertia Control[END_REF], hydro-active gas bearings [START_REF] Lukas | An application of gain-scheduled control using state-space interpolation to hydroactive gas bearings[END_REF], damping control [START_REF] Ma | Application of Dual Youla Parameterization Based Adaptive Wide-Area Damping Control for Power System Oscillations[END_REF], and several academic examples [START_REF] Trangbaek | Safe lpv controller switching[END_REF], [START_REF] Cai | Switching fault-tolerant control by Youla parametrization[END_REF], [START_REF] Luca | Control of disturbed LPV systems in a LMI setting[END_REF], [START_REF] Blanchini | On the LPV control design and its applications to some classes of dynamical systems[END_REF], [START_REF] Friedrich | A Simple Architecture for Arbitrary Interpolation of State Feedback[END_REF].

Discussion

This chapter has reviewed the evolution of the smooth LPV switching/interpolating control starting from the basic gain-scheduled LPV control, switched-LPV control, smooth LPV switching control, bumpless-transfer-based LPV switching, and the most recent LPV-YK interpolating control. From the beginning of 90s till the current year, different kinds of applications have been controlled based on them. The timeline of LPV switching/interpolating concepts is presented in Fig. 2.7, where the column represents the time evolution and the row shows the studied switching and interpolating LPV controls.

Time evolution of the theoretical contributions

In 90s, the LPV control approaches have been proposed referring to the known works: polytopic [START_REF] Apkarian | Self-scheduled H ∞ control of linear parameter-varying systems: a design example[END_REF], grid-based [START_REF] Wu | Induced L 2 Norm Control for LPV Systems with Bounded Parameter Variation Rates[END_REF], and LFT [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF]. The LMI approaches have been FIGURE 2.7: Timeline of known applications using LPV switching/interpolating approaches used to achieve either quadratic stability over convex parameter region (using constant Lyapunov function), or exponential stability over gridded parameter region (using parameter-dependent Lyapunov function). It is worth mentioning that the LPV approaches have shown lower closed-loop performance, than a local LTI one, and conservatism in some applications having large parameter regions, large number of parameters, or significant dynamic uncertainty. Since 2000, researchers have been involved in solving the conservatism problem using switching/interpolating concepts.

In [START_REF] Matthew | Linear quadratic bumpless transfer[END_REF], the first step to design a gain-scheduled LPV control based on bumplesstransfer has been raised. The advantage of bumpless-transfer is to reduce the difference of the control input just before and just after switching. [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF] has proposed a switched-LPV control design using multiple parameter-dependent Lyapunov functions. It shows good performance over each parameter sub-region, however, an unsmooth performance appears at the transient switching. Later, switching smoothness conditions have been added to the LMI optimization problem [START_REF] Chen | The smooth switching control for TORA system via LMIs[END_REF]. These three LPV switching approaches have increased the design complexity and stability studies. A parameterization approach has been also used to design a gain-scheduling LPV controller in [START_REF] Xie | Design of LPV control systems based on Youla parameterisation[END_REF]. It shows several advantages compared to the previous ones including lower design complexity, closed-loop stability can be proved easier, smooth switching, and most important that the local controllers can be designed separately with different state dimensions.

The importance of LPV control approaches, switched-LPV control, bumplesstransfer LPV interpolating, smooth-LPV switching, and LPV-YK control interpolation is shown regarding their implementation to several applications mainly Active Magnetic Bearing (AMB), aircraft motion, aero-engine, wind turbine, and others. Fig. 2.7 presents the applications with different colors and their time evolution.

LPV control

Several applications of the LPV control approaches are shown in the first column of the timeline in Fig. 2.7. Researchers have been interested in applying them to nonlinear models, by reformulating them to LPV models. For instance, the AMB system is nonlinear with respect to the rotor speed, which has been considered as a parameter varying. AMB systems are known to be highly nonlinear, multi-variable and inherently unstable systems. The nonlinearity is described in the relationship between the electro-magnetic force, current and length of the air gap. Moreover, The system (vibrational) dynamics depend on the rotational speed due to gyroscopic and electro-magnetic coupling. The AMB has been controlled using the three LPV approaches: polytopic [START_REF] Lu | Linear parameter-varying techniques for control of a magnetic bearing system[END_REF], [START_REF] Simon Ghersin | LPV CON-TROL OF A MAGNETIC BEARING EXPERIMENT[END_REF], grid-based [START_REF] Morsi | Model Predictive Control Based on Linear Parameter-Varying Models of Active Magnetic Bearing Systems[END_REF], and LFT [START_REF] Lauridsen | Design of robust AMB controllers for rotors subjected to varying and uncertain seal forces[END_REF]. Moreover, the aircraft speed and altitude have been chosen as varying parameters for aircraft motion control. In 2000, it has been first applied using the gridding approach [START_REF] Papageorgiou | Taking robust LPV control into flight on the VAAC Harrier[END_REF], then using the LFT approach in 2006 [START_REF] Dong | Online Switching Control Designs of LFT Systems[END_REF], and the polytopic in 2013 [START_REF] Sato | Robust Gain-Scheduled flight controller using inexact scheduling parameters[END_REF].

In 2012, the grid-based approach has been applied on wind turbine control [START_REF] Farzad A Shirazi | Wind turbine integrated structural and LPV control design for improved closed-loop performance[END_REF]. The nonlinearity in the wind turbine model is due to the aerodynamic interaction of the wind and the rotor blades. Therefore, the LPV model is formulated considering the rotor speed, absolute wind speed, and pitch angle variables as varying parameters. Since 2016, the three LPV approaches have been implemented on aero-engine control using the polytopic in 2016 [START_REF] Wang | Gain scheduling controller of the aero-engine based on LPV model[END_REF], the grid-based in 2019 [START_REF] Bei | Non-affine parameter dependent LPV model and LMI based adaptive control for turbofan engines[END_REF], and most recently in 2021 the LFT [START_REF] Wang | Fault estimation for aero-engine LPV systems based on LFT[END_REF]. The high pressure rotor speed is chosen as a varying parameter that changes over time. Furthermore, other domains have been interested in LPV control including electro-hydraulic servo system (2002) [START_REF] Sugiyama | SWITCHING FROM VELOCITY TO FORCE CONTROL FOR THE ELECTRO-HYDRAULIC SERVOSYSTEM BASED ON LPV SYSTEM MODELING[END_REF], induction motor (2003) [START_REF] Bendtsen | Discrete-time LPV current control of an induction motor[END_REF], canal control (2005) [START_REF] Bolea | LPV VS MULTI-MODEL PI(D) GAIN-SCHEDULING APPLIED TO CANAL CONTROL[END_REF], active-noise cancellation (2011) [START_REF] Heins | GAIN-SCHEDULED STATE-FEEDBACK CONTROL FOR ACTIVE CANCELLATION OF MUL-TISINE DISTURBANCES WITH TIME-VARYING FREQUENCIES[END_REF], mechanical systems (2014) [START_REF] Hoffmann | Linear Parameter-Varying Control of Complex Mechanical Systems[END_REF], and real experiment on autonomous driving (2020) [START_REF] Atoui | Design And Experimental Validation Of A Lateral LPV Control Of Autonomous Vehicles[END_REF].

Switching LPV control

The second column of the timeline concerns the switched-LPV control applications. As already mentioned, this approach has been contributed by Fen Wu in 2004 [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF] to decrease the conservatism of the LPV control design using multiple parameterdependent Lyapunov function. The application on the AMB system in [START_REF] Lu | Control design of switched LPV systems using multiple parameter-dependent Lyapunov functions[END_REF] and [START_REF] He | Switching control for LPV polytopic systems using multiple Lyapunov functions[END_REF] shows the performance improvement over the parameter subregions. Moreover, this approach has been widely used for aircraft flight motion between 2006 and 2020 in [START_REF] Lu | Switching LPV control of an F-16 aircraft via controller state reset[END_REF], [START_REF] Lu | Switching-based fault-tolerant control for an F-16 aircraft with thrust vectoring[END_REF], [START_REF] Xu He | Control of switched LPV systems using common Lyapunov function method and an F-16 aircraft application[END_REF], [START_REF] Li | Fault detection for switched linear parametervarying systems: An average dwell-time approach[END_REF], [START_REF] Wang | Development of a switched polytopic LPV model for the aircraft performing high angle of attack maneuver[END_REF], and [START_REF] Ligang | Switching control of morphing aircraft based on Qlearning[END_REF]. Similarly, it has been implemented on aero-engine control such as in [START_REF] Postma | Air-Fuel Ratio Control of Spark Ignition Engines Using a Switching LPV Controller[END_REF] (2012), [START_REF] Zhu | L 2 gain analysis of actuator saturation for switched LPV systems and applications to aero-engines[END_REF] (2016), [START_REF] Zhu | Event triggered control for a switched LPV system with applications to aircraft engines[END_REF] (2018), [START_REF] Wang | Optimal control of aero-engine systems based on a switched LPV model[END_REF] (2021). In addition to wind turbine control [START_REF] Lescher | Switching LPV Controllers for a Variable Speed Pitch Regulated Wind Turbine[END_REF], [START_REF] Xue | Hysteresis Switching LPV Control of a Wind Turbine System Covering Partial and Full Load Conditions[END_REF], and other domains like flexible ballscrew drives [START_REF] Hanifzadegan | Switching Gain-Scheduled Control Design for Flexible Ball-Screw Drives[END_REF], glucose control in type 1 diabetes [START_REF] Colmegna | Switched LPV Glucose Control in Type 1 Diabetes[END_REF], and engine Exhaust Gas Recirculation (EGR) [START_REF] Qu | Engine EGR Valve Modeling and Switched LPV Control Considering Nonlinear Dry Friction[END_REF].

Bumpless-Transfer LPV control Interpolation

Some applications on bumpless-transfer LPV interpolation are shown in the third column of the timeline. In 2002, an implementation on the AMB is shown [START_REF] Li | Dynamic transfer of robust AMB controllers[END_REF]. Later in 2006, two works have proposed the bumpless-transfer approach for aircraft flight control [START_REF] Dong | Online Switching Control Designs of LFT Systems[END_REF], [START_REF] Turner | Switched control of a vehicle/short take-off land aircraft: An application of linear quadratic bumpless transfer[END_REF]. Recently, bumpless-transfer LPV interpolation has been widely implemented on the aero-engine control starting from 2014 [START_REF] Liu | Design for aircraft engine multi-objective controllers with switching characteristics[END_REF], then 2019 [START_REF] Yang | H ∞ bumpless transfer for switched LPV systems and its application[END_REF], then 2020 [START_REF] Zhao | Rate Bumpless Transfer Control for Switched Linear Systems With Stability and Its Application to Aero-Engine Control Design[END_REF], and the newest in 2021 [START_REF] Shi | Bumpless Transfer Control for Switched Linear Systems and its Application to Aero-Engines[END_REF]. Others applications are mentioned including electronics [START_REF] Steenis | An Approach to Bumpless Control for LPV Modeled Inverters in a Microgrid[END_REF], inverted pendulum [START_REF] Yavari | Robust H2 outputfeedback bumpless transfer control of polytopic uncertain LPV systems[END_REF], and academic examples [START_REF] Yame | Parameterization of Linear Controllers for Bumpless Switching in Multi-Controller Schemes[END_REF], [START_REF] Zong | Bumpless transfer fault detection for switched systems: a state-dependent switching approach[END_REF].

Smooth LPV switching control

In the fourth column of the timeline, the applications on smooth LPV switching control are represented. This approach has been mostly applied on aircraft motion control since 2012 [START_REF] Song | Smooth Switching Output Tracking Control for LPV Systems[END_REF], and later in 2015 [START_REF] Jiang | Smooth switching linear parameter-varying control for hypersonic vehicles via a parameter set automatic partition method[END_REF], 2018 [START_REF] Cheng | Smooth switching linear parameter-varying fault detection filter design for morphing aircraft with asynchronous switching[END_REF], and the last one in 2019 [START_REF] He | Smooth-switching LPV control for vibration suppression of a flexible airplane wing[END_REF]. In 2016 and 2017, [START_REF] Liu | Smooth Switching Controller Design for Multiobjective Control Systems and Applications[END_REF] and [START_REF] Zhao | Optimal switching surface design for switching LPV control and its application to air-fuel ratio control of an automotive engine[END_REF] have proposed the smooth LPV switching control for aero-engine control. Moreover, it has been implemented on different applications such as wind turbine [START_REF] Chen | Smooth switching gain-scheduled control for large scale offshore wind turbine under full wind-speed conditions[END_REF] (2016), missile control [START_REF] Song | Smooth switching output tracking control for a class of linear time-varying systems: An LMI approach[END_REF] (2010), and TORA [START_REF] Chen | The smooth switching gain-scheduled control with H ∞ performance to the application of TORA systems[END_REF] (2014).

LPV-YK control Interpolation

The last column in the timeline shows the applications on the LPV-YK control interpolation. This approach has not been yet applied to aero-engine control. However, it has been widely used for AMB in [START_REF] Daniel | A novel design approach for switched LPV controllers[END_REF] (2010), [START_REF] Balini | Synthesis and implementation of gain-scheduling and LPV controllers for an AMB system[END_REF] (2012), [START_REF] Wu | Switching Controller Parameterization Realization for an LPV Plant with H ∞ Performance[END_REF] (2018), [START_REF] Wu | Switching Linear Parameter-Varying Controller Design With H ∞ Performance Based on Youla Parameterization[END_REF] (2020), and recently in [START_REF] Wu | Design of Smooth Switching LPV Controller Based on Coprime Factorization and H ∞ Performance Realization[END_REF] (2021). It has been also implemented on aircraft control [START_REF] Wu | Switching fault-tolerant control of a flexible airbreathing hypersonic vehicle[END_REF] (2012), [START_REF] Schirrer | Convex Design for Lateral Control of a Blended Wing Body Aircraft[END_REF] (2014), and wind turbine [START_REF] Wang | Adaptive Damping Control Scheme for Wind Grid-Connected Power Systems with Virtual Inertia Control[END_REF] (2022). In addition to different applications including hydroactive gas bearings [START_REF] Lukas | An application of gain-scheduled control using state-space interpolation to hydroactive gas bearings[END_REF], flexible-link robot [START_REF] Friedrich | A Simple Architecture for Arbitrary Interpolation of State Feedback[END_REF], and academic examples [START_REF] Trangbaek | Safe lpv controller switching[END_REF], [START_REF] Blanchini | On the LPV control design and its applications to some classes of dynamical systems[END_REF].

Table 2.1 lists the main advantages and disadvantages of each studied switching/interpolating approach. Autonomous Vehicles Architecture

Introduction

First, it is worth noting that the work in this thesis is not starting from scratch but it is a long work that has been carried out in the research department at Renault for developing an autonomous driving architecture. The current thesis has been done by the time of the running projects TORNADO and SAM Saclay. The prototype of TORNADO has been used for the experimental validation of the work presented here.

Even if we focus on the control part, it is worthy to review the full autonomous driving architecture to understand the input/output connections of our work by describing the connected modules.

The current chapter presents the architecture of the vehicle platform that has been used in our experiments. In addition a brief literature review is shown on the lateral control system which has been mainly tackled in this thesis. The field of autonomous vehicles has been expanding rapidly in the last decade, in order to meet road safety and environmental objectives. Most of the definitions and details that we present here are issued from [START_REF] Milanés | The Tornado Project: An Automated Driving Demonstration in Peri-Urban and Rural Areas[END_REF], the readers that would like to have more details can refer to it.

To validate the contributions raised by this thesis, a RENAULT ZOE robotized car has been used. The following Section 3.2 presents the Autonomous Vehicle (AV) architecture including its subsystems perception, map service, localization, world model, decision-making, navigation, supervisor, and control. In addition, a brief review on vehicle control is depicted in Section 3.3. Section 3.4 discusses the main parts in the AV architecture that are tackled and concerned in the later chapters.

Experimental Platform Description

This section presents the test vehicle concerning the scientific and technical development especially from perception, navigation and control point of view. The platform is a fully robotized Renault ZOE car (see Fig. 3.2) that has been prepared for Renault project available for the use in the Rambouillet Territory in France [START_REF] Milanés | The Tornado Project: An Automated Driving Demonstration in Peri-Urban and Rural Areas[END_REF].

Autonomous driving is computationally demanding and requires integrating different functions such as perception, map service and localization, world model, decision-making, navigation, and vehicle control into a unified system achieving intelligent behavior. The interaction and organization of these functional components

Perception

The automated car is equipped by five evenly distributed cameras and a 32-layer LiDAR located in the center of the roof rack, providing a 360°field of view as shown in Fig. 3.4, see [START_REF] Beltrán | Towards autonomous driving: a multi-modal 360°perception proposal[END_REF] for more details. Through a low-level fusion, the sensors information is fused to provide obstacle detection, classification, 3D box estimation, and tracking. Briefly speaking, the perception system uses a robust state-of-the-art framework and feeds the vision-based detections to a 3D box estimation method that makes use of LiDAR information to know their size and location. Then, the tracking algorithm exploits this spatial reasoning to add time consistency and enhance the reliability of the final detections, refer to Fig. 3.5.

Vision-based detection and classification

Computer vision approaches are used to take advantage of the feature-rich appearance information delivered by the cameras. Notice that thanks to the multi-camera setup, the system can identify agents in the entire range of interest, without blind spots. To improve the data fusion between images and LiDAR data, detections with a pixel-wise semantic mask is chosen, knowing that accurate extrinsic calibration is available.

At the end, a set of 2D bounding boxes with information about the category (e.g., car, pedestrian, bus etc.) is provided as an output. Each one contains a pixel-wise

3D box estimation

The LiDAR scanners is known to be highly accurate in measuring distances. So, it is used here to obtain geometrical information about the previously identified objects. A frustum PointNet approach [START_REF] Qi | Frustum PointNets for 3D Object Detection from RGB-D Data[END_REF] is used to recover the geometrical structure behind the raw LiDAR representation of the obstacle. This approach provides the size, location, and orientation of each obstacle using LiDAR information, and taking into consideration the missing parts due to occlusions or perspective.

This stage outcomes positioned cuboids representing the real geometry and distance from the ego-vehicle to every obstacle.

Tracking

The main objective of this stage is to add consistency to the 3D detections that have been expressed in the same frame. It is composed of three steps: 1) Movement estimation based on an Unscented Kalman Filter [START_REF] Wan | The unscented Kalman filter for nonlinear estimation[END_REF] for different pedestrians and vehicles movement models; 2) Ego-vehicle movement compensation, due to its displacement, is important to correct the misalignments on the subsequent detections. This is achieved using the GPS/INS system available in the vehicle; and 3) Data association aiming to correlate the detection in the current cycle with the set of already tracked agents, adding new instances whenever it is necessary.

Map Service

The map database is a High Definition Map that has been created specially for a certain experimentation. It represents the road network and includes topological, geometrical and semantic information. It is mainly composed of details about the interactions existing between roads at a lane level. The center and the boundaries of each lane is also found in the stored map with centimeter accuracy. In addition, information such as speed limit, marking types, and driving directions are also associated with each lane as attributes. Moreover, different information are included in the map service such as: 1) Intersection type, shapes or priority order; and 2) Traffic lights location and association to lanes. All these information are compiled as a geographical database within the vehicle. For the experimental tests in this thesis, the private test track in Satory is used, see FIGURE 3.6

Localization

The Localization component is responsible for centralizing the ego vehicle within the available map information. It provides position, heading, velocity and timing information, together with integrity information. To achieve its objective, it receives information from the perception, map service, and the GPS. The accuracy of localization is improved by performing map matching with landmarks detected. For instance, the localization system identifies detected lane markings with the corresponding lane markings found in the map database. After this identification, it corrects absolute pose estimation so that relative position of detected landmarks correspond with absolute position of lane markings in the map. Consequently, the system upgrades its covariance matrix. In order to prevent localization leaps, these corrections can be smoothed, preserving both requirements on absolute and relative accuracy. Finally, the output localization information includes position, heading, velocity and timing information, together with integrity information.

World Model

The main objective of the World Model component is to understand the situation of the ego-vehicle with its surrounding by combining the information from the perception and the map service. Since the perception system does not consider context information while providing the surrounding situation, it is important to associate it with the map service to better understand the current scene. The map service is used to:

• Find the global path to transport from the departure location A to the final destination B.

• Assist the navigation system module to generate a real time local trajectory while operating in autonomous mode.

• Provide context in which the vehicle navigates.

The World Model component is composed of four modules: Contextualization, Interaction, Intention, and Coherence module. For more illustration:

• Contextualization Module: The perceived objects from the perception system are matched with the map to identify their position with respect to the road and to enrich them with semantic information. For instance, it labels the perceived objects (received from the perception system) as pertinent or not pertinent. The objects are labelled as pertinent, such as pedestrians, if they can be located within the navigable space. On the other hand, the vehicles and other obstacles are considered as not pertinent when they are not located on the navigable space of the ego vehicle. In addition, it can deal with traffic lights detected using Vehicle to Infrastructure (V2I) communication since the perception system in our platform does not return traffic lights information.

• Interaction Module: It identifies the maneuvers that the ego vehicle is allowed to perform and determines the perceived objects with which it would interact. This process is performed with respect to the context in which the ego vehicle navigates and applicable traffic laws that are stored in a map database.

• Intention Module: It estimates the intention of the road users with respect to the other entities with which they may interact. For this purpose, it uses the state of the entities provided by the contextualization module and the data from the interaction module. While the interaction module identifies the possible entities that might interact with each other, the intention module estimates the behaviors that those entities are likely adopting.

• Coherence module: It computes whether any conflicting intentions are present.

Decision-Making

After a global route plan has been found, the autonomous vehicle must be able to navigate the selected route and interact with other traffic participants according to driving conventions and rules of the road. Given a sequence of road segments specifying the selected route, the Decision-Making system is responsible for selecting an appropriate driving behavior at any point of time based on the perceived behavior of other traffic participants, road conditions, and signals from infrastructure. To obtain a comfort and safe vehicle behaviour, the driving plan is modified to the expected interaction between the ego vehicle and the detected obstacles. The decision is taken according to several criteria: 1) the relative position of the obstacle: same lane as ego vehicle, opposite lane, partially on ego lane, side walk; 2) obstacle direction: moving along and backward or forward, moving perpendicular and to or away from the driving plan; and 3) obstacle type: pedestrian, vehicle. For example, when the vehicle is reaching the stop line before an intersection, the decision-making system will command the vehicle to come to a slow down, observe the behavior of other vehicles, bikes, and pedestrians at the intersection, and let the vehicle proceed once it is its turn to go. The maneuver associated with each perceived obstacle is created in an expert system manner. So, the computational complexity is kept low and a deterministic behaviour is obtained. Every maneuver adds constraints on the longitudinal (slow down, follow, stop) and lateral (avoid) dimensions which are taken into account in the local trajectory planning process.

Navigation

After the driving behavior is decided, which could be cruise-in-lane, change-lane, or turn-right, a path or trajectory that can be tracked by the low-level feedback controller is constructed. The main role of the Navigation system is to generate the local trajectory of the ego vehicle in real time. The resulting trajectory should be dynamically feasible for the ego vehicle, comfortable for the passengers, and avoid collisions with the existing obstacles. The trajectory generation data includes the geometric description of the desired path, lateral error tolerance and associated speed profile. This process can be described in three stages:

• Space definition: The high definition vector map provides a geometric description of each lane which is used to create lateral boundaries (i.e., a driving corridor) limits to the vehicle. According to the closest obstacle, a longitudinal boundary is set so that the decision system chooses a stop maneuver. Spatiotemporal footprint of each obstacle combined with an avoidance maneuver is computed based on estimated position and speed. The available driving space is therefore reduced according to the set of perceived obstacles.

• Path optimization: It aims to optimize the driving space in the short term horizon. The driving space is discretized into a set of vertices along regularly spaced. These vertices are then used to construct the path by connected splines as shown in Fig. 3.7. A second order geometric continuity [START_REF] Piazzi | Quintic G 2 -splines for the iterative steering of vision-based autonomous vehicles[END_REF] is applied to compute a convenient optimized path after splines combination.

• Speed profile: After constructing the geometric path, a speed profile is associated to the optimal path considering road geometry and vehicle dynamic constrains (i.e. maximum allowed speed or desired lateral acceleration in turns). Additional constraints such as pedestrian lines, approaching intersections, and obstacle avoidance are considered to modify the speed profile. The final trajectory including the path and speed profile is then sent to the control system as a set of way-points with information about vehicle position, heading, speed, acceleration and road curvature within a speed-based horizon.

Supervisor in System Management

The purpose of the Supervisor system is to monitor and control the components to ensure the safety and robustness of the operational components and to facilitate the integration, deployment, and for graceful degradation of software components. It selects the discrete behavior of the components, that is, to determine which continuous behavior each of the functional components must have at each moment of time.

The supervisor maintains different operational modes (e.g. autonomous/manual mode) and sends signals to the operational components to switch their states accordingly. Its main functions are as follows:

1. Safely deploys the software components and brings the system to the safe operational state.

2. Interacts with the driver to start or stop the autonomous mode.

3. Monitors the states of all the operational components.

4. Identifies the problems in the components and switches to alternative modes of operation to keep the system under safe state and to request the driver to take back the control. Facilitates the integration and maintenance of the overall system.

Vehicle Model

The main dynamics which are usually considered in the vehicle model are the longitudinal, lateral and vertical dynamics [START_REF] Pacejka | Tire and vehicle dynamics[END_REF], [START_REF] Rajamani | Vehicle dynamics and control[END_REF]. Thus, simplified models are taken at the beginning such as the bicycle model (two-wheels model), without considering the suspension system (see FIGURE 3.8). Notice that the vehicle model contains two types, the kinematic model and the dynamic model. The kinematic model assumes that the tire-slip of the vehicle with the ground is null. It is derived geometrically where no forces and tire-slips are considered, as if it is a point-object. This kind of models is mainly used at low speed systems such as mobile-robots as well as for the vehicles in automatic parking tasks. Moreover, kinematic model is very useful for motion planning (reference tracking or path-following) [START_REF] Paden | A survey of motion planning and control techniques for self-driving urban vehicles[END_REF]. Considering the tire-slips in the model is vital for high speed vehicles. Thus, the dynamic model is used to introduce the side-slip angle with the applied forces from the vehicle and on it. This model can better use the vehicle's capabilities for executing aggressive maneuvers that will be significant in planning motions with high accelerations and jerks. This section represents the kinematic and dynamic models of both longitudinal and lateral vehicle motions that have been briefly derived and verified in [START_REF] Pacejka | Tire and vehicle dynamics[END_REF] and [START_REF] Rajamani | Vehicle dynamics and control[END_REF].

Kinematic Model

As mentioned before, the kinematic model is derived geometrically as a point-object. Thus, observing the marked center of gravity in FIGURE 3.8, the longitudinal and the lateral vehicle motions in the global reference can be computed as:

   ẋ = v cos θ ẏ = v sin θ θ = w (3.1)
where • x and y are the cartesian coordinates of the vehicle's center of gravity in the global frame (x,y). (m)

• θ represents the rotation of the vehicle with respect to the xaxis and called the yaw angle. (rad)

• w is the yaw rate that represents the rotation rate of the vehicle. (rad/s)

Since the equations in (3.1) are derived in the global frame, the following rotation matrix is used to write them in terms of the vehicle's frame as follows:

  x e y e θ e   =   cos θ sin θ 0 -sin θ cos θ 0 0 0 1     x d -x y d -y θ d -θ   (3.2)
where • x d , y d and θ d are the coordinates of the reference to be tracked.

• x e , y e and θ e represent the longitudinal, lateral and rotational errors of the vehicle with respect to the demanded trajectory.

From (3.2), the kinematic-error model differential equations are derived to obtain:

   ẋe = wy e + v re f cos θ e -v x ẏe = -wx e + v re f sin θ e θe = w re f -w (3.3)
where v re f and w re f are the longitudinal and rotational velocity references in the vehicle's frame. Derivations can be found in [START_REF] Martí | LPV Control of an Autonomous Vehicle[END_REF].

Dynamic Model

The dynamic model is derived using physical concepts, the side-slip angle α will be considered from now on. Applying Newton's 2 nd law to the vehicle's longitudinal and lateral motions we get:

     vx = -F y f sin δ-µmg m + wv y + a vy = F y f cos δ+F yr m -wv x ẇ = F y f l f cos δ-F yr l r I (3.4)
where

• v x , v y and w are the body frame velocities, i.e. longitudinal, lateral and rotational velocities in the vehicle's frame.

• δ and a are the control inputs, steering angle and longitudinal acceleration respectively.

• F y f and F yr are the lateral forces applied on the front and rear tires respectively.

• I, m, l f and l r are the vehicle's inertia, mass and the distance from the center of gravity to the front and rear wheel axes respectively.

• µ and g are the friction coefficient and the gravity acceleration constant respectively. F y f and F yr can be modelled using Pacejka's tire model [START_REF] Pacejka | Tire and vehicle dynamics[END_REF] as follows:

F y f = c 3 sin(c 2 tan -1 (c 1 α f )), α f = δ -tan -1 ( v y v x - l f w v x ) F yr = c 3 sin(c 2 tan -1 (c 1 α r )), α r = -tan -1 ( v y v x -l r w v x ) (3.5)
where c 1 , c 2 and c 3 are constants.

It is worth mentioning that the above tire model shows a nonlinear-dependency of F y f and F yr with respect to α. Notice that the objective of this thesis, related to the internal project at RENAULT, concerns the passenger vehicles (or daily vehicles) which constrains the lateral acceleration, and consequently small slipping angle α. Thus, assuming small α, one can linearize it resulting the following relation:

F y f = C f (δ - v y v x - l f w v x ) F yr = C r (- v y v x + l r w v x ) (3.6)
where C f and C r represent the tires' stiffness of the front and rear wheels.

Steering Actuator Model

The dynamics of the steering actuator affects significantly the vehicle dynamics, leading to model uncertainties in the vehicle model. Thus, the steering actuator model is identified at the nominal longitudinal speed v x = 12 m/s as an LTI model.

A second order transfer function has been obtained in the following form:

G act = k s 2 + 2ζw n s + w 2 n e -T d s (3.7)
where k, ζ, w n , and T d are the static gain, the damping, the natural frequency and the time delay, respectively. The second order Padé approximation is chosen for the time delay. Consequently, the time delay term is then represented as:

e -T d s = 1 -T d s 2 + (T d s) 2 12 1 + T d s 2 + (T d s) 2 12
(3.8)

Control System

The first functionality of the Control system is to ensure stability when switching from manual to autonomous mode and vice-versa. During autonomous mode: 2. It observes continuously the health state of the surrounding systems and inner modules, and executes specific fail-safe policies or emergency actions triggered using vehicle safety devices (emergency stop button, override key, etc.).

3. It receives the vehicle localization and the planned trajectory to achieve accurate, safe, and comfort tracking.

The general control architecture is shown in FIGURE 3.9 including the longitudinal and lateral controllers.A dSPACE MicroAutoboxII (MABx) is used to embed the control algorithms using MATLAB/Simulink turning at 100 Hz. The MABx is connected to the vehicle via a CAN network. On-board sensors provide the system with information such as vehicle localization, speed, longitudinal acceleration, yaw rate, etc. The maximum capacity (100 Hz) is used for the communication with the steering, throttle, and brake actuators.

Longitudinal Control

A low-level, longitudinal controller is designed to control the throttle and brake actuators according to the speed reference received from the navigation system. Since the speed reference could include discontinuities which can affect the stability and performance of the low-level control. A high-level longitudinal controller is designed achieving high robustness and tracking capabilities. This is obtained using a Linear Parameter-Varying (LPV) control concept, considering the rate of the speed reference as a scheduling parameter.

Lateral Control

The lateral control is responsible to control the steering actuator aiming to track the required path maneuvers. To maintain human steering behavior, the look-ahead concept is used by measuring the lateral displacements from the reference trajectory at a distance in front of the vehicle. Such concept mainly uses the GPS, where a high integrity navigation system of an automated vehicle, based on the fusion between the Inertial Measurement Unit (IMU) and the GPS, is developed and implemented. This allows to predict the future lateral error and try to minimize it smoothly as the human driver does.

The used scheme for the lateral control, as shown in Fig. 3.10, consists of:

1. A feedforward term which concerns the reference trajectory by considering the road curvature and vehicle speed (see Fig. 3.11). The road curvature is estimated at a target point chosen in front by a look-ahead distance in order to predict the future lateral displacement.

2.

A feedback compensator which activates the steering actuator to minimize the current vehicle errors and correct its lateral and heading positions..

The lateral controller, applied to the automated ZOE, is designed using the LPV concept, having the vehicle longitudinal speed as the scheduling parameter. It can be designed using any control concept (pole-placement, PID, H ∞ , ...). The aim is to track the yaw rate reference w re f and respecting the actuator limitations.

The Look-Ahead System (LAS) block represented in Fig. 3.10 aims to generate a coherent yaw-rate reference w re f . The LAS uses the current vehicle situation measured by the sensors and the information from the navigation system. The main role of LAS is to improve the lane-tracking accuracy and driving comfort at the same time. w re f can be approximated as:

w re f = 2v x y L L 2 (3.9)
where y L is the predicted lateral error at the look-ahead distance L (see Fig. 3.11).

Related works on vehicle control

Vehicle automation was leveled from 0 to 5 according to the SAE J3016 standard. In this standard, levels were distributed as follows: Level 0, where the human driver has the full responsibility of all driving actions. Level 1 contains basic driving assistance like adaptive cruise control (ACC), anti-lock braking systems and electronic stability control [START_REF] Rajamani | Vehicle dynamics and control[END_REF]. Level 2 is more assisted than level 1, by considering advanced assistance such as hazard-minimizing lateral/longitudinal control, e.g., [START_REF] Gerdes | A unified approach to driver assistance systems based on artificial potential fields[END_REF], or emergency braking, see e.g., [START_REF] Brannstrom | Model-based threat assessment for avoiding arbitrary vehicle collisions[END_REF], [START_REF] Vahidi | Research advances in intelligent collision avoidance and adaptive cruise control[END_REF]. At level 3, the system is fully automatic in driving with environment monitoring under specific conditions. However the human driver is still able to take control when the driving actions are out of autonomous system's operational circumstance. Level 4 represents the vehicle with full automation as in the previous level but in addition to safe control of the vehicle in in case that the operator fails to control the intervention request. Finally, level 5 performs all the driving situations as fully autonomous, i.e. the vehicle moves without a driver. Autonomous driving aims to avoid accidents, reduce fuel consumption, improve traffic flow. It also provides passenger comfort in critical situations and make it possible to car travelling for everyone regardless of their abilities or conditions. Several studies in the literature have focused on the path tracking performance of the vehicle, their references differ by taking either online or offline motion plan. The topic of motion planning is taking a wide range of interest discussing different strategies to design a plan feasible to the vehicle dynamics (see [START_REF] Paden | A survey of motion planning and control techniques for self-driving urban vehicles[END_REF]).

Different vehicular models (linear or not) can be considered for control design objectives. For example, in [START_REF] Paden | A survey of motion planning and control techniques for self-driving urban vehicles[END_REF], the authors neglect the dynamics of the vehicle for path-planning design. In [START_REF] Navas | Stability analysis for controller switching in autonomous vehicles[END_REF], an LTI model is used including the dynamics of the vehicle, whereas in [START_REF] Alcala | Gain-scheduling LPV Control For Autonomous Vehicles Including Friction Force Estimation and Compensation Mechanism[END_REF] and [START_REF] Alcala | Autonomous Racing using Linear Parameter Varying -Model Predictive Control (LPV-MPC)[END_REF], the authors prefer to use a Linear Parameter Varying (LPV) bicycle model to track normal and racing car references, respectively. However, in the field of heavy vehicles, the vertical dynamics cannot be neglected, as observed in [START_REF] Van-Tan | H ∞ active anti-roll bar control to prevent rollover of heavy vehicles: a robustness analysis[END_REF]. In [START_REF] Atoui | Parameter Varying Approach For A Combined (Kinematic + Dynamic) Model Of Autonomous Vehicles[END_REF], a more complete model, considering kinematic and dynamic equations, rewritten in the LPV form, is proposed. In addition to increasing accuracy, the LPV approach allows the use of linear-like control design tools. In [START_REF] Wit | Autonomous ground vehicle path tracking[END_REF] and [START_REF] Alcala | Gain-scheduling LPV Control For Autonomous Vehicles Including Friction Force Estimation and Compensation Mechanism[END_REF], the authors design a cascaded control scheme for the kinematic and the dynamic controls. This strategy adds some limitations and constraints into the control design.

Vehicle control methodologies that include the vehicle, the path, and the driver are currently being developed at several research centers and automotive suppliers. The importance of the used control strategy lies in its simplicity and the achieved performance. The main objective in [START_REF] Atoui | Parameter Varying Approach For A Combined (Kinematic + Dynamic) Model Of Autonomous Vehicles[END_REF] is to track an offline trajectory safely by minimizing the external disturbances and noises (from the driver, sensors, etc.), and dealing with uncertainties due to some neglected (or unmodeled) dynamics or fixed parameters. Referring to these demanded performances, robust control is a suitable technique to achieve their goal where it is able to deal with model uncertainties and disturbance rejection, see [START_REF] Scherer | Theory of robust control[END_REF]. In [START_REF] Alcala | Gain-scheduling LPV Control For Autonomous Vehicles Including Friction Force Estimation and Compensation Mechanism[END_REF], the authors use the LMI-based LPV-LQR/H 2 cascaded controllers with an LPV UIO observer to estimate the road friction force and achieve less tracking error. The controller is computed offline using the polytopic approach. Moreover, in [START_REF] Alcala | Autonomous Racing using Linear Parameter Varying -Model Predictive Control (LPV-MPC)[END_REF], the authors apply the LPV-MPC approach, for a racing car, where the controller is computed online with reducing the computational cost.

Longitudinal control

Longitudinal control has been studied considering energy efficiency, fuel consumption and traveling time. For instance, Adaptive Cruise Control (ACC) strategies have been contributed for automated car-following. Recent results have proposed new spacing policies and new control architecture to decrease the time gap between vehicles, while ensuring the string stability [START_REF] Ploeg | Controller Synthesis for String Stability of Vehicle Platoons[END_REF]. Concerning the cruise control performances, the speed control algorithm [START_REF] Németh | Control design of traffic flow using look-ahead vehicles to increase energy efficiency[END_REF] has been augmented to enhance road stability and safety of the vehicle, see [START_REF] Mihály | Look-ahead control of road vehicles for safety and economy purposes[END_REF]. In addition, the look-ahead cruise control has been improved using a parameter-dependent model describing the traffic flow in terms of the velocity trajectory design. Hence, the design of a longitudinal control strategy becomes challenging.

ACC systems adjusts vehicle speed and provides a specified distance to the preceding vehicle by automatically controlling the throttle and brake actuators [START_REF] Xiao | A comprehensive review of the development of adaptive cruise control systems[END_REF]. ACC with the Stop&Go system offered by the Advanced Driver Assistance System (ADAS), makes driving easier and less stressful. Different control approaches have been proposed for ACC. The Model Predictive Control (MPC) is largely and commonly used, where a high level controller is proposed based on MPC approaches, taking into consideration the driver longitudinal ride comfort, driver permissible tracking range, and rear-end safety [START_REF] Li | Model Predictive Multi-Objective Vehicular Adaptive Cruise Control[END_REF]. In [START_REF] Flores | Fractional-order-based ACC/CACC algorithm for improving string stability[END_REF], a Fractional-Order Control (FOC) approach has been presented to design a feed-forward structure for ACC to enhance the car-following while ensuring the robustness/stability. A fuzzy longitudinal control is studied in [START_REF] Tsai | Fuzzy longitudinal controller design and experimentation for adaptive cruise control and stop&go[END_REF] to control the throttle and braking pedals, this control approach is using (IF...THEN) conditions based on experience and experimental results. [START_REF] Laib | String Stable H ∞ LPV Cooperative Adaptive Cruise Control with a Variable Time Headway[END_REF] introduces the Cooperative Adaptive Cruise Control (CACC), it is known as the extension of the ACC by enabling the wireless communication between vehicles (V2V) to control the time gap. An LPV control has been used to deal with the variations of the time headway while ensuring the convergence of the spacing errors towards zero and the attenuation of any disturbance propagating along the platoon.

Lateral control

Vehicle lateral control is a vital problem to achieve an on-road automated movement. It is important as much as it is critical where it concerns lane-keeping, lane-changing, obstacle avoidance, autonomous parking, etc. Most of the path-tracking methods in practice encounter challenges when high accuracy and robustness are required [START_REF] Snider | Automatic Steering Methods for Autonomous Automobile Path Tracking[END_REF]. The first role of the lateral control is to keep the vehicle within the boundaries of a lane, which is performed by adjusting the steering actuator to minimize the lateral error between the vehicle position and a target point in the generated reference. The question comes on how to choose the target point. If the target point is chosen to be the closest point (in the reference) to the vehicle, the actuator may not be able to minimize the corresponding lateral error in case of high speeds, high curvature changes, etc. This is due to delays in the used sensors and actuators. Then, a lookahead approach is raised to predict the future lateral error at a chosen look-ahead distance. Look-ahead systems use sensors as machine vision, radar, and LiDAR, to measure the lateral displacement in front of the vehicle.

Lateral control for overtaking or obstacle avoidance maneuvers has been also tackled in the literature. This concept took a part of the studies concerning the steering rate and the forward distance to be covered by ensuring safety with respect to the vehicles around. Fuzzy logic control [START_REF] Milanés | Intelligent automatic overtaking system using vision for vehicle detection[END_REF] and different optimization techniques, as in [START_REF] Lin | Active collision avoidance system for steering control of autonomous vehicles[END_REF], [START_REF] Yu | MPC-based Path Following Design for Automated Vehicles with Rear Wheel Steering[END_REF], and [START_REF] Chowdhri | Integrated nonlinear model predictive control for automated driving[END_REF], were used to maintain better scenarios for lane change and collision avoidance.

Look-ahead based lateral control

In 90s, the control concepts based on look-ahead systems have been raised. These concepts have been developed to mainly improve the efficiency and performance of the longitudinal and lateral vehicle controls. [START_REF] Yi | Vehicle trajectory prediction for adaptive cruise control[END_REF] proposes a new approach to improve the vehicle trajectory prediction for the Adaptive Cruise Control (ACC) system. It concludes that the higher look-ahead distance, the greater the prediction errors. Moreover, the look-ahead approach is also used to optimize traffic flow, minimize trip time [START_REF] Hellström | Look-ahead control for heavy trucks to minimize trip time and fuel consumption[END_REF], decrease fuel consumption and provide safety for heavy-duty vehicles [START_REF] Turri | Fuel-efficient heavy-duty vehicle platooning by look-ahead control[END_REF], [START_REF] Alam | Look-ahead cruise control for heavy duty vehicle platooning[END_REF], [START_REF] Turri | Cooperative Look-Ahead Control for Fuel-Efficient and Safe Heavy-Duty Vehicle Platooning[END_REF].

On the other hand, look-ahead systems are used for lateral objectives as shown in the pioneering works [START_REF] Ozguner | An analytical study of vehicle steering control[END_REF], [START_REF] Patwardhan | A general framework for automatic steering control: system analysis[END_REF]. In [START_REF] Kosecka | Vision-based lateral control of vehicles[END_REF] and [START_REF] Kosecka | A comparative study of vision-based lateral control strategies for autonomous highway driving[END_REF], the authors have recognized that the closed-loop stability is sensitive to the variation of speed. Specifically, when the vehicle speed increases, the closed-loop zeros and poles move toward the imaginary axis leading to poor damping of the poles. Increasing the look-ahead distance moves the zeros closer to the real axis, improving the damping of the closed-loop poles. As a result, the choice of a proper look-ahead distance is important for stability and performance of the closed-loop system. Practically, as much as the lookahead distance decreases, the vehicle will lose farer information which leads to periodic oscillations due to actuators and sensors delays. On the other hand, when the look-ahead distance increases, the vehicle may not be able to deal with near obstacles or maneuvers. The relation has been analyzed between the look-ahead distance and the longitudinal velocity, the road curvature and the processing delay of the vision. In [START_REF] Hingwe | A variable look-ahead controller for lateral guidance of four wheeled vehicles[END_REF], the authors introduce an equation which calculates the look-ahead distance as a function of vehicle specifications and longitudinal speed taking into account the distance between the vehicle and a bumper. [START_REF] Chen | Steering control of high speed vehicles: dynamic look ahead and yaw rate feedback[END_REF] proposes a dynamic look-ahead distance L(s) which varies with respect to speed. The aim was to obtain the lateral acceleration independent of the longitudinal speed which makes the design of the steering control much easier.

One of the most used schemes for the lateral control consists of:

1. A feedforward term which generates the desired yaw rate at a varying lookahead distance with respect to the vehicle's speed.

2. A feedback compensator that minimizes the current vehicle yaw rate error to reduce the lateral error.

In [START_REF] Tan | Design of a High-Performance Automatic Steering Controller for Bus Revenue Service Based on How Drivers Steer[END_REF], the steering rate is defined to be proportional to both heading and lateral errors, adapting look-ahead distance accordingly to vehicle speed. Thus, a Linear Time-Invariant (LTI) controller is designed for a fixed look-ahead distance and speed. Each one of those obtained LTI controllers has a different performance to be used in a certain situation. [START_REF] Mahtout | Youla-Kucera Based Lateral Controller for Autonomous Vehicle[END_REF] uses this approach to design a switching LTI controller based on Youla-Kucera (YK) parameterization. Two LTI controllers are designed separately having the same speed but different look-ahead distance parameter d; the first one with d = 30m for managing larger lateral errors smoothly, and the second with d = 15m to provide fast tracking capabilities. When the absolute lateral error increases from 0.6 to 3m, the controller switches its performance from the second to the first. A recent approach has been proposed in [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF], solving the same problem but by adapting the look-ahead distance within the generated yaw rate reference in terms of the absolute lateral error, and using a single robust LPV controller.

Recently, several studies aim to find the best tuning of the look-ahead distance L with respect to the vehicle speed v x . The studies in [START_REF] Skarpetis | Vehicle lateral control using a robust tracking controller based on vision look ahead system[END_REF], [START_REF] Park | Development of Steering Control System for Autonomous Vehicle Using Geometry-Based Path Tracking Algorithm[END_REF] and [START_REF] Roselli | H ∞ control with look-ahead for lane keeping in autonomous vehicles[END_REF] propose to tune L from numerical analysis. [START_REF] Tan | Design of a High-Performance Automatic Steering Controller for Bus Revenue Service Based on How Drivers Steer[END_REF] estimates L manually by analyzing the closed-loop poles with respect to speed, look-ahead distance, and lateral control feedback gains. In [START_REF] Hasegawa | Three look-ahead distance scheme for lateral control of vision-based vehicles[END_REF], the authors indicate three look-ahead distances (L 1 , L 2 , L 3 ) at a fixed speed. Then, a feedback lateral control uses vehicle lateral deviations at the indicated look-ahead points to improve the performance of the vehicle at different road curvatures. In [START_REF] Lee | Multirate active steering control for autonomous vehicle lateral maneuvering[END_REF], the look-ahead distance is formulated as a linear function of speed by choosing a suitable look-ahead time as a slope, with lower and upper bounds. The same methodology is also introduced in [START_REF] Kang | A comparative study of lane keeping system: Dynamic and kinematic models with look-ahead distance[END_REF] and [START_REF] Kang | Linear Parameter Varying Design for Lateral Control using Kinematics of Vehicle Motion[END_REF]. [START_REF] Kuwata | Motion Planning in Complex Environments Using Closed-loop Prediction[END_REF] and [START_REF] Kuwata | Real-Time Motion Planning With Applications to Autonomous Urban Driving[END_REF] propose a simplified adaptive method that tunes the look-ahead distance from the commanded speed instead of the measured one to improve a path prediction process. The tuning method parameters are obtained from a number of different experiences. Other works employ the fuzzy logic approach to tune the look-ahead distance [START_REF] Shan | CF-Pursuit: A Pursuit Method with a Clothoid Fitting and a Fuzzy Controller for Autonomous Vehicles[END_REF], [START_REF] Chen | A Robust Look-ahead Distance Tuning Strategy for the Geometric Path Tracking Controllers[END_REF]. They consider the road curvature and the current vehicle lateral error for the selection of the look-ahead distance.

As an overall, the look-ahead distance may be determined based on at least one of the following: vehicle speed, rotating speed, steering acceleration, steering angle, heading angle, lane curvature, current lateral error, predicted lateral error, distance to lane boundary, vehicle rotating performance, obstacle location. Tuning the lookahead distance based only on the vehicle speed is not sufficient, since it could face some critical situations as large lateral errors for example. Recently, [START_REF] Atoui | Real-Time Look-Ahead Distance Optimization for Smooth and Robust Steering Control of Autonomous Vehicles[END_REF] proposes a method to find, in real-time, an optimal look-ahead distance according to the vehicle speed and steering acceleration, and taking into consideration the vehicle heading error. This improves the vehicle performance when subjected to high noises at highways, and to large lateral errors that may occur at the autonomous mode initialization or sudden lane-changes.

Discussion

The current thesis focuses on the Control system, mainly the look-ahead based lateral control of autonomous vehicles. A general control architecture is formulated based on LPV and YK parameterization aiming to switch/interpolate between multiple controllers. Its objective is to maintain different lateral motion capabilities, at low and high speeds. The thesis contributions are validated using simulations and real-time implementations:

Chapter 4

Application of LPV Approaches to Vehicle Lateral Control

Introduction

This chapter presents, the real implementation of the polytopic, grid-based and LFT LPV approaches to solve the lateral control problem on an electric Renault ZOE vehicle. A comparative analysis is shown among the three LPV approaches to the autonomous vehicle lateral control. The design of the three approaches from a theoretical point of view as well as their application on a real vehicle are presented, including a comparative analysis. Specifically, comparison is carried out attending to the following criteria:

1. Designing procedure which is discussed in a theoretical point of view.

2. Controller performance when dealing with different situations (noise, disturbance, large lateral errors, low and high speeds, etc...). This is achieved by analysing the lateral errors (to ensure safety) and the control input efforts (to ensure comfortability and actuator limitation) from the simulation and real implementation results.

3. Implementation complexities when applying them on real applications and their effects on the computational costs.

The main contribution of this chapter is to develop, experimentally validate and compare three distinct LPV controllers which can deal, comfortably, with both lane change and lane tracking problems of a daily passenger vehicle. This can be summarized as follows:

• Quasi-LPV models are structured for the three approaches from the nonlinear bicycle model that vary with respect to the longitudinal velocity.

• To limit the control input effort and to achieve the noise/disturbance rejection performance, LPV/H ∞ problems are solved using a set of LMIs.

• The design and experimental validation of grid-based and Linear Fractional Transformation (LFT) approaches is carried out for the first time on the lateral control of autonomous vehicles.

• Input reference is adjusted as a function of the speed and lateral error, modifying the look-ahead parameter accordingly to deal with large lateral errors (lane changing) and the small ones (lane tracking).

• Simulation and experimental results (on a Renault ZOE vehicle) are shown to compare the performance of the controllers concerning tracking, actuator limitations and noise/disturbance rejection.

The simulation and experimental results shown in this chapter have been presented in:

• LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis, published in IEEE Transactions on Intelligent Transportation Systems (T-ITS), refer to [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF].

This chapter is organized as follows: In Section 4.2, a brief theoretical explanation on how LPV structures are formulated for the different approaches. The used control scheme of a full speed-range controller is represented in Section 4.3. Also, the optimization problems of the LPV/H ∞ control synthesis are discussed. Section 4.4 presents the implementation control scheme achieving multiple tasks (lane change and path tracking). Then, simulation results obtained by the designed controllers and the limitations of each approach are analysed. In section 4.5, experimental validation is demonstrated to ensure the reliability of the comparison done. Section 4.6 summarizes the comparison in general concerning all the chosen criteria, and some comments are introduced. Finally, section 4.7 collects some conclusions and remarks.

Model Formulation

The lateral vehicle dynamics is modelled using the well-known bicycle model [START_REF] Pacejka | Tire and vehicle dynamics[END_REF], [START_REF] Rajamani | Vehicle dynamics and control[END_REF]. Next subsections describe the plant model and the design of each LPV model.

Lateral Bicycle Model

In [START_REF] Pacejka | Tire and vehicle dynamics[END_REF] and [START_REF] Rajamani | Vehicle dynamics and control[END_REF], the nonlinear lateral bicycle model is derived as:

vy = F y f cos δ+F yr m -wv x ẇ = F y f l f cos δ-F yr l r I , ( 4.1)
where v x , v y and w are the longitudinal, lateral and rotational velocities in the vehicle's frame, respectively. δ is the control input, the steering front angle of the front tire. I, m, l f and l r are the vehicle's inertia, mass and the distance from the center of gravity to the front and rear wheel axes respectively. F y f and F yr are the lateral forces applied to the front and rear tires, respectively. Considering small side-slip angles and constraint lateral acceleration (≤ 5m/s 2 ). Then, the lateral forces can be approximated as follows:

F y f = C f (δ - v y v x + l f w v x ), F yr = C r (- v y v x + l r w v x ), (4.2)
where C f and C r are the stiffness of the front and rear wheel-tires respectively. Notice that the linear bicycle model with constraint lateral acceleration has been stated as a good approximation of the nonlinear model [START_REF] Gottmann | On the Influence of Rear Axle Steering and Modeling Depth on a Model Based Racing Line Generation for Autonomous Racing[END_REF]. Then this model could be sufficient for daily passenger vehicles.

LPV Model Structures

After choosing ρ(t) = v x ∈ R n ρ as a varying parameter, and assuming small steering front angles (sin(δ) ≈ δ), and small slip angles, the LPV state-space representation is written as:

G(ρ) ẋ(t) = A(ρ)x(t) + Bu(t) y(t) = Cx(t) + Du(t) (4.3)
where:

x(t) = v y w ∈ R n x , u(t) = δ, , y(t) = w, B = 1 m C f 1 I C f l f , A(ρ) =   - C r +C f mv x - l f C f -l r C r mv x -v x - l f C f -l r C r Iv x - l 2 f C f +l 2 r C r Iv x   . (4.4)
The parameter-dependency of the LPV model differs from one approach to another. Specifically, the control synthesis of the polytopic approach requires an affine parameter-dependency. However, the gridding approach does not require such assumption. Finally, the LFT model is defined to be a lower or upper Linear Fractional Representation (LFR) between a known LTI model and a varying-parameter block. The following discussion shows how each approach is formulated.

Polytopic Model

The polytopic model is defined in a convex hull bounded by the parameters extremums (see Fig. 4.1). It is formulated as a convex combination between the vertices of the polytope. Two conditions must be satisfied: 1) the input and output matrices should be independent of the varying parameters, this is usually solved by pre-filtering the input or output; and 2) the model must be affine with respect to the varying parameters. So, (4.4) is written as:

A pol (ρ) = - C r +C f m ρ 2 - C f l f -C r l r m ρ 2 -ρ 1 - C f l f -l r C r I ρ 2 - C f l 2 f +l 2 r C r I ρ 2 , ( 4.5)
where ρ 1 and ρ 2 are v x and 1 v x respectively. Figure 4.1 shows the polytope formed by the extremums of the two parameters. Notice that the parameter vector is represented as a convex combination between the 4 vertices, where ρ = The LPV model is then written as a convex combination of the LTI systems obtained at the vertices of the polytope:

2 np ∑ i=1 µ i ω i and 2 np ∑ i=1 µ i = 1, µ i ≥ 0 ∀i being ω 1 = (ρ 1 , ρ 2 ), µ 1 = |ρ 1 -ρ 1 | |ρ 1 -ρ 1 | × |ρ 2 -ρ 2 | |ρ 2 -ρ 2 | ω 2 = (ρ 1 , ρ 2 ), µ 2 = |ρ 1 -ρ 1 | |ρ 1 -ρ 1 | × |ρ 2 -ρ 2 | |ρ 2 -ρ 2 | ω 3 = (ρ 1 , ρ 2 ), µ 3 = |ρ 1 -ρ 1 | |ρ 1 -ρ 1 | × |ρ 2 -ρ 2 | |ρ 2 -ρ 2 | ω 4 = (ρ 1 , ρ 2 ), µ 4 = |ρ 1 -ρ 1 | |ρ 1 -ρ 1 | × |ρ 2 -ρ 2 | |ρ 2 -ρ 2 | (4.6)
A pol (ρ) B pol C pol D pol = 2 np ∑ i=1 µ i (ρ) A(ω i ) B C D (4.7)
where n p = 2 is the number of the varying parameters.

Grid-based Model

The grid-based LPV model is a model interpolated over a set of LTI models that are linearized at different operating points. This approach can be considered when the parameter dependency of the model is nonlinear (without model reformulation) [START_REF] Hoffmann | A Survey of Linear Parameter-Varying Control Applications Validated by Experiments or High-Fidelity Simulations[END_REF], [START_REF] Wu | Control of linear parameter varying systems[END_REF]. So, the only needed parameter here is ρ = v x . The varying parameter is gridded to a chosen number of n g grid points as shown in Fig. 4.2.

The gridding approach uses any kind of interpolation (linear or nonlinear) between the gridded models to compute the LPV model. Suppose that, at an instant, the longitudinal velocity ρ ∈ ρ k , ρ k+1 m/s, the linear interpolation of the statespace matrices:

A grid (ρ) B grid (ρ) C grid (ρ) D grid (ρ) = k+1 ∑ i=k α i (ρ) A i B i C i D i , ( 4.8)
where,

α k = ρ k+1 -ρ ρ k+1 -ρ k and α k+1 = ρ-ρ k ρ k+1 -ρ k (4.9)

LFT model

The LFT approach defines the model as the upper LFR between a known LTI model and a parameter block as shown in Fig. 4.3 [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF]. At each instant, the parameters in the block Θ are updated being the input of the LTI model M. In general, the upper LFT interconnection of the shown model is written as:

  ẋ z θ y   =   A B θ B C θ D θθ D θ1 C D 1θ D     x w θ u   w θ = Θz θ (4.10)
where Θ is the time-varying operator block introduced as:

Θ = blockdiag(θ 1 I r 1 , ..., θ k I r k ) (4.11)
being r i > 1 which presents the number of occurrences of the varying-parameter θ i . Notice that θ i can be normalized to be always ∈ [-1, 1], which makes M represents the nominal model when θ i = 0 ∀i.

Let us consider the system model (4.3), considering the parameter v x varying as:

v x = v x 0 + aθ (4.12)
where

v x 0 = v x min +v xmax 2
represents the nominal value of v x , and a =

v xmax -v x min 2
represents the rate of variation when θ varies in [-1,1]. Then, (4.3) can be rewritten under the LFT form (4.10) where:

Chapter 4. Application of LPV Approaches to Vehicle Lateral Control A =   - C r +C f mv x 0 - l f C f -l r C r mv x 0 -v x 0 - l f C f -l r C r Iv x 0 - l 2 f C f +l 2 r C r Iv x 0   B θ = -a v x 0 0 -a 2 v x 0 0 0 0 0 -a v x 0 C θ =       - C r +C f mv x 0 - l f C f -l r C r mv x 0 + v x 0 0 1 0 0 - l f C f -l r C r Iv x 0 - l 2 f C f +l 2 r C r Iv x 0       D θθ =      -a v x 0 0 -a 2 v x 0 0 0 0 0 0 0 1 0 0 0 0 0 -a v x 0      D θ1 = 0 4×1 , D 1θ = 0 1×4 , Θ = θ × I 4 (4.13)
And finally, the state-space matrices of G(ρ) are scheduled as:

A LFT (ρ) = A + B θ ∆ θ C θ B LFT (ρ) = B + B θ ∆ θ D θ1 C LFT (ρ) = C + D 1θ ∆ θ C θ D LFT (ρ) = D + D 1θ ∆ θ D θ1 ∆ θ = Θ(I -D θθ Θ) -1 (4.14)
It is worth noting that, in the LFT approach, the LPV model is converted as in Fig. 4.3 in a single LTI model feeded by a parameter dependent input. It differs from the polytopic and grid-based approach where a set of LTI models is handled. . Note that the input delay is described as a first order transfer function, so as a result, G act is a second order transfer function. Considering this structure, the controller K(ρ) (in Fig. 4.4) is designed for the three approaches using LPV/H ∞ concept. Notice that w re f represents the yaw rate reference. Control performance criteria in H ∞ control theory are given by frequency domain functions (check Appendix B). Two weighting functions W e and W u are used to achieve tracking and actuator limitations performances respectively. The objective is to achieve both performances with a trade-off between minimizing the lateral error and ensuring the driving comfort. Notice that the steering speed δ could be related to the driving comfort since it reflects how fast the front wheels are acting (e.g. at high values of δ), and how noisy they appear (when δ oscillates around zero at high frequencies). We know that when δ increases, the lateral acceleration increases, and consequently more aggressive actions are achieved. Moreover, a highly frequent oscillation of the steering speed around zero leads to noises in the front wheels, which encounters uncomfortable driving. As a result, W u is designed in a way to achieve low steering speeds with noise rejection.

LPV/H ∞ Control Design

Tracking specification (W e )

The weighting transfer function is chosen as:

W e (s) = s M s + w b s + w b ϵ (4.15)
where the parameters M s , w b and ϵ are tuned as follows:

• M s = 2, to ensure robustness at any frequency.

• w b ≥ 10, to choose the speed of rising time-response.

• ϵ ≤ 10 -4 , to represent the steady-state tracking error.

Specification on the control input limitations (W u )

A filter is used to minimize the effort of the steering actuator control δ c . This filter is designed as:

W u (s) = s + w bu M u ϵ u s + w b u (4.16)
The parameters M u , w b u and ϵ u are adopted as:

• M u represents the limitations on the maximum allowed effort of the actuators.

• w b u , is related to the actuator bandwidth.

• ϵ u ≤ 10 -2 , is concerned with the noise rejection from the control inputs at high frequencies.

Generalized Plant

Using the plant model and the weighting functions, Fig. 4.4 is converted to build a general control configuration as in Fig. 4.5a. The generalized plant P(ρ) is structured to consider the chosen weights in addition to the LPV model (G(ρ) × G act ). Thus the state vector of P(ρ) is x P = x x act x W e x W u T , and the controlled output For the polytopic and the gridding approaches, the state-space representation of P(ρ) (see Fig. 4.5a) has the form:

    ẋP e 1 e 2 r -y -n     =     A P (ρ) B 1 B 2 B 3 B 4 0 W e 0 -W e -W e G(ρ) 0 0 0 0 W u 0 1 0 -1 -G(ρ)           x P   r d n   u       . ( 4.17)
In the polytopic approach, there exist four generalized plants, each one is related to a corresponding vertex of the polytope. However, a set of n g P i 's are formulated along the gridded points for the gridding approach. Fig. 4.5b shows the generalized plant P(ρ) as an upper LFR between a parameterinvariant P and the parameter block Θ for designing an LPV/LFT controller. Using this approach, P is then structured as:

      ẋP z θ e 1 e 2 r -y -n       =       A P B Θ B 1 B 2 B 3 B 4 C θ D θθ D θ1 D θ2 D θ3 D θ4 0 0 W e 0 -W e -W e G(ρ) 0 0 0 0 0 W u 0 0 1 0 -1 -G(ρ)               x P w Θ   r d n   u         . ( 4

LPV/H ∞ Control Synthesis

For the rest of the paper, the longitudinal speed is considered to be bounded as:

v x ∈ 3, 30 m/s (4.19)
The main objective of the H ∞ control is to minimize the L 2 induced gain from the external input w to the controlled output z = e 1 e 2 T . This is achieved by solving

∥z∥ 2 ≤ γ ∞ ∥w∥ 2
and γ ∞ > 0, to be minimized, represents how much the demanded performance is achieved. If γ ∞ < 1, the demanded performance is totally achieved by the controller. Among the three approaches, each controller is designed by solving its corresponding LMI-based optimisation problem. A complete overview of the synthesis and implementation complexity of LPV approaches can be found in [START_REF] Hoffmann | Complexity of implementation and synthesis in linear parameter-varying control[END_REF]. For each approach, the complexity of LPV controller existence conditions is determined in terms of the size of LMIs and the number of decision variables. Note that the obtained controller K(ρ) is an LPV Dynamic Output Feedback Controller.

Polytopic Approach

Fig. 4.1 shows that the polytopic approach is considering the parameters v x and 1 v x as two independent parameters. However, they depend on each other as a function of the black curve (i.e. y = 1/x). Since the polytope is a very large set compared to the real parameter variation rule (and mainly due to w 4 ), this may lead to conservatism when solving the optimisation problem on the 4 vertices. Thus, a solution to this problem is drawn in [START_REF] Robert | A reduced polytopic LPV synthesis for a sampling varying controller: Experimentation with a T inverted pendulum[END_REF], where the number of vertices is reduced from 2 n p to n p + 1. So, the polytope in Fig. 4.1 can be reduced to C O {w 1 , w 2 , w 3 }, and then, the coefficients µ ′ i s will be:

µ 1 = 1 -(µ 2 + µ 3 ), µ 2 = |ρ 2 -ρ 2 | |ρ 2 -ρ 2 | , µ 3 = |ρ 1 -ρ 1 | |ρ 1 -ρ 1 | (4.20)
In the polytopic approach, the problem is solved in the framework of quadratic stability of the closed-loop system. This is obtained by solving a set of LMIs at the vertices of the polytope with a constant Lyapunov function [START_REF] Apkarian | Self-scheduled H ∞ control of linear parameter-varying systems: a design example[END_REF]. The existence conditions of this approach show the number of LMIs to grow by O(2 n p ). Also, the number of decision variables is computed to be n x (n x + 1) [START_REF] Hoffmann | Complexity of implementation and synthesis in linear parameter-varying control[END_REF]. As a result, a controller K i is obtained at each vertex and K(ρ) is computed as (in the case of non-reduced polytope and ∀ρ inside the polytope of Fig. 4.1):

A K (ρ) B K (ρ) C K (ρ) D K (ρ) = 2 np ∑ i=1 µ i (ρ) A K i B K i C K i D K i (4.21)

Grid-based Approach

Grid-based approach formulates the problem in the context of robust stability [START_REF] Briat | Linear Parameter-Varying and Time-Delay Systems[END_REF] by using a parameter-dependent Lyapunov function along the gridded axis [START_REF] Wu | Control of linear parameter varying systems[END_REF]. Notice that a basis function is chosen to write the Lyapunov function in terms of the varying parameters. An example on the lateral vehicle control problem, the Lyapunov function X(ρ) is chosen to be linearly dependent (order 1) on the varying

parameter ρ = v x : X(ρ) = X 0 + ρX 1 , (4.22)
where X 0 and X 1 are unknown constant matrices to be computed from the LMIs. From this point of view, one can see an advantage in the optimisation problem of this approach where it reduces the conservatism of the polytopic approach. Notice that the LMIs include the absolute maximum rate-change of the parameter. Studying the existence conditions of this control approach, the number of LMIs grows g ). To determine the number of decision variables, let us assume that the parameter-dependent Lyapunov matrix is parameterized as:

X(ρ) = X 0 + s ∑ i=q a i (ρ)X i , (4.23)
then the number of decision variables is 1 2 n x (n x + 1)(n X ρ + +1), where n X ρ = sq + 1. For more details, the LMIs are derived in Theorem 4.3.1 in [START_REF] Wu | Control of linear parameter varying systems[END_REF]. As a result, a set of controllers is obtained where each one corresponds to a frozen value in the gridded axis of the parameter. When ρ ∈ [ρ k , ρ k+1 ], the controller K(ρ) can be linearly interpolated as [START_REF] Hoffmann | A Survey of Linear Parameter-Varying Control Applications Validated by Experiments or High-Fidelity Simulations[END_REF]:

A K (ρ) B K (ρ) C K (ρ) D K (ρ) = k+1 ∑ i=k α i (ρ) A K i B K i C K i D K i , (4.24)
where,

α k = ρ k+1 -ρ ρ k+1 -ρ k and α k+1 = ρ-ρ k ρ k+1 -ρ k (4.25)

LFT Approach

The LPV/LFT controller synthesis mainly relies on the S-procedure [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF]. The solution of the LMI-based optimization problem is detailed in [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF], where continuous and discrete-time LMI problems are discussed. The LMI size is typically smaller than that in the polytopic approach since there is no anymore a set of LMIs to be solved for a set of parameter values (see Fig. 2 in [START_REF] Hoffmann | A Survey of Linear Parameter-Varying Control Applications Validated by Experiments or High-Fidelity Simulations[END_REF] for more details). However, a trade-off is found where the number of decision variables of LFT is much bigger than the polytopic approach when n ρ increases (n x (n x + 1) + 2n Θ (2n Θ + 1) where Θ ∈ R n Θ ×n Θ ). Control synthesis is formulated in two steps: 1) A quadruple (R, S, L 3 , J 3 ) is found -by solving 2 LMIs -to ensure the solvability of the control problem (check Theorem 5.1 in [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF]); and 2) The gain-scheduling controller is then computed using the quadruple and solving another LMI with one decision variable containing the controller matrices (Algorithm 6.1 in [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF]). Then, the designed controller is written in the form of a lower LFR F l (K, Θ), and obtained as follows:

K(ρ) = F l (K, Θ) = K 11 + K 12 Θ(I -K 22 Θ) -1 K 21 (4.26)
being K(ρ) defined as:

u zθ = K 11 K 12 K 21 K 22 r -y -n wθ (4.27)
where K 11 , K 12 , K 21 and K 22 are obtained from the LMI's. Thus, the state-space matrices of K(ρ) are computed as:

A K (ρ) = A K + B K θ ∆ θ C K θ B K (ρ) = B K + B K θ ∆ θ D K θ1 C K (ρ) = C K + D K 1θ ∆ θ C K θ D K (ρ) = D K + D K 1θ ∆ θ D K θ1 ∆ θ = Θ(I -D K θθ Θ) -1

Frequency Domain Analysis

This subsection analyses the designed controllers in frequency-domain in order to check if the requirements are satisfied by the controllers. This is achieved using several sensitivity functions compared to the weights designed previously. Additionally, an LTI controller designed at the fixed-nominal speed v x =14 m/s is added to the analysis to show the benefits of using parameter-varying approaches.

To observe the tracking performance, the error sensitivity function S is drawn as shown in Fig. 4.6. Sensitivity functions of the tracking error to the reference are drawn with the required performance 1 W e . Notice that each of the LPV approaches has several sensitivity functions where each one is referred to a corresponding value of speed (in gridding and LFT approaches) or a corresponding vertex (in polytopic approach). At low frequencies, all the controllers achieve the demanded steadystate tracking error. Also, all the controllers are respecting the requirements at high frequencies related to robust margin (max ∀w r-y r < 6 dB).

On the other hand, the analysis of the control input is carried out to evaluate the sensitivity to the noise. To do so, the control sensitivity function KS of each controller is computed. Fig. 4.7 shows the sensitivity functions of the control input δ with respect to the reference, and the requirements on the actuator limitations 1 W u . It is shown that all the controllers are respecting the demanded limitations with a small exceeding in the bandwidth of the LFT approach. One can appreciate how the gridding approach shows the best noise rejection at high frequencies compared to the others.

Simulation Results

The simulations are employed by a Renault simulator considering the electric Renault ZOE vehicle, developed in MATLAB/Simulink, and considering nonlinear vehicle and tire models. The simulations are performed in discrete-time domain with a sampling time T s = 10 ms. The proposed control strategies are implemented in simulation following the scheme in Fig. 4.8. The look-ahead system block is added to perform smoothly both lane-keeping and lane-changing (i.e. small and large lateral errors respectively). It modifies the yaw rate reference when having large lateral errors (initial error in autonomous starting mode, sudden lane-changes, etc.). Since the desired curvature is inversely proportional to the look-ahead distance [START_REF] Tan | Design of a High-Performance Automatic Steering Controller for Bus Revenue Service Based on How Drivers Steer[END_REF], a smooth or aggressive steering can be obtained by changing the look-ahead distance. Specifically, as much as the lateral error y e increases, the look-ahead distance d is increased linearly as a function of the lateral error as follows:

d(v x , ye) = sat( d nom (v x ) 2 × (1 + abs(y e )), [d nom (v x ), d max ]) (4.29)
where d nom (v x ) is the nominal look-ahead distance tuned at each speed v x for small lateral errors (y e ≤ 0.6m), d max is the maximum saturation of look-ahead distance chosen when y e ≥ 3m. d max is chosen depending on the facing situation where it should ensure that the vehicle is not losing forward information (we choose it here as d max = 40m). As discussed in subsection 4.3.4, the LPV approaches have different ways of implementation and interpolation. Each one differs with respect to its complexity and time consuming, which makes it worthy to see how they can be simplified and what will be the effects of such simplifications. Due to the limited number of real tests which can be done, this section discusses different ways of implementations for each approach separately and their impact on the system performance. The trajectory in Fig. 4.9 is used to test the lateral control performance. Vehicle speed input is recovered from a real test (see Fig. 4.10). Some critical situations were chosen to show the functionality of the look-ahead system and to compare the response of the different designed controllers, where:

• a large lateral error is presented at the initialization time,

• the reference positioning system (X,Y) is recovered from a real test with noises.

• the rate of the varying parameter v x is increased to be large enough at some times (t ∈ [70, 80]s),

• a high speed v x = 25 m/s is reached at time t = 80s,

• two successive aggressive maneuvers (with high lateral acceleration greater than 5 m/s 2 ) at high speed (around 23 m/s) are carried out between t = 105s and t = 115s.

LPV Approaches Limitations

This subsection describes the main limitations of each approach from the implementation point of view.

Polytopic Approach

As mentioned in paragraph 4.3.4, the polytope is reduced to decrease its problem conservatism. Thus, to analyse the benefit of this reduction, two polytopic LPV controllers (one with four vertices and the other with three vertices) are designed. Both of them are simulated separately and the results of the obtained lateral errors and steering input angles are represented in Fig. 4.11 and 4.12. Table 4.1, Fig. 4.11 and 4.12 show that both controllers achieve approximately the same lateral error and steering performance. This result is due to the fact that the bicycle model is not highly nonlinear in terms of the speed v x . Moreover, Fig. 4.11 and 4.12 show that the initial lateral error is minimized smoothly, thanks to the lookahead system which adapts the look-ahead distance as the lateral error increases (see Fig. 4.13). Notice that the look-ahead distance is no more adapted for t > 6s, i.e. d = d nom (v x ) ∀t > 6s except when lateral error exceeds 0.6m.

Finally, Table 4.1 shows the lateral error RMS for both polytopic controllers. One can appreciate the benefits of reducing the polytope, especially when having varying-parameters that depend on each other.

Gridding Approach

The gridding approach reduces the problem of conservatism by using a parametervarying Lyapunov function. Gridded controllers are often interpolated either linearly or nonlinearly to obtain the current-state controller. The benefit of this interpolation is that even if the parameter goes out of its extremum bounds that was considered in the control design (i.e. v x < 3 m/s), the controller still can be interpolated with negative coefficients. Designing a gridded-based LPV controller, the gridded axis of the varying parameter should be drawn. This axis can be gridded starting from two grid-points until infinity. Thus, it is worthy to study the effect of increasing the number of gridding points on the performance of the system. A set of tests is done at each chosen number of gridding points and the RMSs of the lateral errors are computed. The chosen number of gridding points are the even numbers from 2 to 40 grid-points (2,4,...,38,40). Fig. 4.14 shows the normalised RMS of the lateral errors from each test. Notice that all of them are normalized with respect to the test of n g = 2 to make it easier for comparison, being the first test RMS equals to 0.263. It is clearly shown that increasing the number of gridding points may not always improve the performance.

Thus, it can be concluded that, in the used model, the performance of the gridded controller is not significantly affected by the number of gridding points.

LFT Approach

The implementation of the LFT controller (4.28) is simpler where it consists of a state-space control system which varies with an input parameter ∆ θ , unlike the previous approaches that have a set of controllers to be interpolated between each other. However, an online inverse must be done at each sample time. This may lead to a limitation during simulation or specifically in real implementations. Thus, a method has been used in [START_REF] Gauthier | SOME EXPERIMENTAL RESULTS OF AN H ∞ -LPV CONTROLLER APPLIED TO A DIESEL ENGINE COMMON RAIL IN-JECTION SYSTEM[END_REF] to avoid online calculations. A look-up table of the inverted matrix is computed at each chosen grid-point along the parameter range, and then, an interpolation is done during simulation.

The controller with online inverse, the one with interpolation and the nominal controller (at v x = 17.5m/s) are tested (see Fig. 4.15 and 4.16 for details). Table 4.2 presents the RMS lateral error, where there is no significant difference between the tracking performances of the tested controllers. Fig. 4.15 shows that the performance of the interpolated controller (yellow line) differs a bit from the one with online inverse (red line) when the parameter variation rate increases (i.e. t ∈ [40, 60]s). On the other hand, it can be clearly observed that when the speed approaches the nominal speed (v x = 17.5m/s), the nominal LFT controller (blue line) acts more closely to the one with online inverse (red line) (t ∈ [135, 140]s). Notice that at high speeds (t ∈ [START_REF] Ballesteros | Reduced-order two-parameter pLPV controller for the rejection of nonstationary harmonically related multisine disturbances[END_REF][START_REF] Atoui | Parameter Varying Approach For A Combined (Kinematic + Dynamic) Model Of Autonomous Vehicles[END_REF]), the nominal LFT controller is showing a small difference in the lateral error, whereas the other two controllers are having exactly the same performance regarding also the steering action (see Fig. 4.16). As a result, the interpolated LFT controller showed a similar performance as the actual one which may help in reducing the computational cost in real implementations.

LPV Approaches Robustness Test

The most promising result of each LPV approach is retained and tested again with the LTI one (at v x = 17.5m/s) by modifying some vehicle parameters (vehicle mass and tire stiffness). The mass of the vehicle changes by the number of passengers. Moreover, the tire stiffness is hard to be estimated (see [START_REF] Berntorp | Tire-Stiffness and Vehicle-State Estimation Based on Noise-Adaptive Particle Filtering[END_REF] for instance), and it varies by time when the tire loses its quality. Then, the following test is important to study the robustness of the designed controllers in terms of such uncertainties. In this test, the mass of the vehicle is chosen arbitrarily to be increased by 400Kg and each front and rear tire stiffness is adjusted by 30%.

The obtained results are shown in Fig. 4.17 and 4.18. Again, one can appreciate the success of the look-ahead system which helps in minimizing the initial lateral error smoothly, i.e. low steering actuation (see Fig. 4.18). Fig. 4.17 and Table 4.3 show that the tested controllers achieve comparable minimization of the lateral error, with the best tracking performance achieved by the gridded controller. The interpolated LFT seems to have the lowest robustness where it maintains the highest RMS lateral error with more steering overshoots as observed in Fig. 4.18. The steering rate RMS of the reduced polytopic controller shows the highest value in Table 4.3, and it has the highest overshoot when performing the initial lateral error (Fig. 4.18). Finally, all the compared controllers are considered to be robust to the injected uncertainties thanks to the used H ∞ concept.

Real Implementation

The last four controllers, compared in simulation, are then tested on a robotized electric Renault ZOE shown in Fig. 4.19. This automated vehicle is adapted for lateral and longitudinal controls by computer-controlled steering and pedal actuators. Vehicle speed and the global coordinates are measurable using GPS and IMU, see more details in Chapter 3. The vehicle is employed using a dSPACE MicroAutoBox in discrete time sampled by T s = 10ms. As previously stated, actuator dynamics are modelled using an LTI model which influence uncertain dynamics between the real plant and the designed model. The test results of the designed controllers are discussed concerning their implementation, the acceptability of the resulted performance and the actuator limitations.

The tests are done in the closed track Satory shown in Fig. 4.20. This track contains bad road conditions and road-inclinations which allows to evaluate the controller robustness. The first part of the test describes the response of the controllers at high speeds. The second part concerns the precision of lateral control at optimal speeds chosen coherently depending on the road curvature. Fig. 4.21 shows the longitudinal speed evolution over time which is considered as an external parameter of the LPV mode. For the coming analysis, the limitations of the used steering actuator are as follows:

| δ| max = 0. lateral error, however Fig. 4.23 and 4.24 show that the control input effort is quiet different. One can observe that both the polytopic and the LTI controllers are sensitive to noises, especially at high speeds (when t ≤ 60 s). This can be justified by the conservatism problem of the polytopic approach, and the highly uncertain dynamics reached at high speeds worsen the LTI performance. In fact, this uncertainty is not caused only by the bicycle model, but by the actuator model. Since G act is used as an LTI system where actually it varies with speed and quiet changes at high speeds. Then, reaching speeds around 13 m/s and below, both controllers perform better with lower sensitivity to noises. On the other hand, the LFT and the gridding controllers show better performance in all situations with a little difference in the minimization of the lateral error. Regarding the steering front angle in Fig. 4.23, these two controllers have the same evolution with less noises even at high speeds. Fig. 4.24 shows the steering rate input for the tested controllers. Again the LFT and the gridding controllers are less noisy than the one obtained by the LTI and the polytopic controllers. Table 4.6 presents that the gridding approach shows the less control effort. Notice that these results support the frequency domain analysis in paragraph 4.3.5, especially the analysis of noise sensitivity at high frequencies.

Summary of Comparison

This section presents a general review on the comparison done between the three approaches under the main proposed criteria. It is divided into two main parts, each one discusses the main collected comparisons introduced above:

About The Three LPV Approaches

LPV Formulations

In section 4.2, the nonlinear model was reformulated using the LPV approaches. It has shown that the type of parameter-dependency of the model affects the shape and the complexity of the LPV state-space representation.

For the polytopic approach, the system should be written in a way to be affine with respect to the varying parameters, and the input and output matrices should be constant. This may increase the number of parameters which increases the conservatism of the optimisation problem. Regarding the LFT model in (4.13), a varying parameter leaded a 4-dimensional parameter block Θ by the cause of its number of occurrences. This explains a real problem facing the applications of LFT approach. Some research studies are done on how to reduce the LFT state-space model [START_REF] Wang | Model reduction of LFT systems[END_REF], especially by the researchers working on aircraft control. Finally, the gridding approach showed a simplicity in model formulation where a set of state-space matrices are obtained at a chosen number of operating points, regardless of the parameterdependency and its occurrences.

LPV/H ∞ Control Design

Section 4.3 introduced the H ∞ control design applied to the studied LPV approaches. It has stated the complexity of the existence conditions of each approach in terms of its LMI size and the number of decision variables. Regarding those LMIs, the polytopic approach has the simplest optimisation problem where the known concept of LTI/H ∞ is applied at the vertices of the polytope with a constant Lyapunov function. However, its LMI size grows exponentially when n p increases, O(2 n p ). This leads to conservatism which is decreased by reducing some vertices [START_REF] Robert | A reduced polytopic LPV synthesis for a sampling varying controller: Experimentation with a T inverted pendulum[END_REF], but still can not be totally erased. The LFT approach comes to decrease the size of LMIs in polytopic approach, but it requires bigger number of decision variables [START_REF] Hoffmann | A Survey of Linear Parameter-Varying Control Applications Validated by Experiments or High-Fidelity Simulations[END_REF]. The problem of conservatism can be solved using a parameter-dependent Lyapunov function which exists in the grid-based approach. The LMI size in this approach grows by O n ρ g , so it is much bigger than that in the polytopic and LFT approaches when having n g > 2.

Also, the complexity of the grid-based approach appears significantly when increasing the order of the parameter-dependent Lyapunov function (n X ρ ).

Implementation Part

This part plays a vital role in choosing the suitable controller of real applications, where there are many limitations in terms of computer power and costs.

Simulation Results

Section 4.4 has presented the performance of each control approach with its implementation limitation. The main concluding remarks obtained are:

• The reduction of the polytope decreases the conservatism of the optimisation problem, and thus obtaining better performance.

• Using the gridding approach, the system performance is independent from the number of chosen grid-points.

• The interpolated LFT controller could be used instead of its gain-scheduling (with inverse (4.28)) without losing its performance.

Real Implementation Procedure

Starting with the way of implementation of each approach, the polytopic controller is known to be interpolated as a convex combination of the designed LTI controllers at the polytopic vertices. Thus, those LTI controllers must be saved in a look-up table and then interpolated at each sampling time. It is similar with the gridding approach, but the set of gridded controllers will be linearly interpolated. However, the scheduling of the LFT approach is different, where the matrix ∆ θ contains a matrix which must be inverted online at each sample time, and this increases the computational time. Since the interpolated LFT shows a performance close to the gain-scheduling, it can be implemented using linear interpolation as the gridding approach.

Experimental Results

The figures of the experimental results show how the studied controllers deal with different situations. For example, the LTI and the polytopic controllers are more sensitive to noises at high speeds, i.e. when the system dynamics change significantly. However, the LFT and the gridding controllers are more robust to handle this kind of uncertainty when the approximated model used in the control design differs from the actual system.

As a result, although the grid-based approach needs to solve more LMIs for more decision variables compared to the polytopic one, it provides a general formulation for any parameter-dependency, and it is has the lowest conservatism among the three discussed approaches. On the other hand, the gain-scheduling structure of the LFT approach seems to be the most complex, and its LMIs contain huge number of decision variables compared to the other two. Thus, the grid-based approach could be proposed as a powerful and performant for LPV control applications. That is why it is widely used in aerospace applications [START_REF] Zhenxing | Robust LPV modeling and control of aircraft flying through wind disturbance[END_REF], [START_REF] Takarics | Flight control oriented bottom-up nonlinear modeling of aeroelastic vehicles[END_REF].

Conclusion

This chapter has proposed a theoretical and experimental comparison of the LPV approaches for the lateral control of autonomous vehicle. The gridded-based model has the simplest structure with less conservatism in optimisation among the others. The weighting parameters used in control design can physically translate the real actuator limitations to a filter added to the optimisation problem.

The practical limitations of each approach have been discussed by observing the simulation results that have been obtained from different chosen critical scenarios. Then the results obtained in real implementation have shown interesting results regarding the minimization of the lateral error, which encourages the application of LPV/robust approaches on autonomous vehicles. In fact, such approaches help to control parameter-variant systems and to handle with environmental disturbances (wind speed, bad road conditions/slopes, etc...).

As a consequence of the practical limitations, mainly at high speeds, the next chapters propose a switching scheme between multiple LPV controllers to improve the vehicle performance over the whole speed range. The switching strategy is based on Youla-Kucera (YK) Parameterization, where several LPV-YK control structures are designed and validated on the ZOE car.

Chapter 5

Advanced LPV-YK Control Structures: Theoretical Proofs

Introduction

Previous chapter compared the polytopic and the grid-based approach on the automated robotized platform. It has been concluded that a single Lyapunov function for models with large parameter variations (as ranging between low and fast vehicle speeds) leads to degraded performance at some parameter subsets. In addition, several kinds of lateral control performances may be needed such as lane-tracking, lane-changing, urgent maneuvers, parking, etc.

The current chapter proposes several switching/interpolating control architectures based on Youla-Kucera (YK) parameterization aiming to achieve robust and adaptive closed-loop specifications (rising time, steady-state error, ...). The LPV-YK control architecture incorporates multiple LPV or LTI controllers which can switch/ interpolate between them with ensuring stability. It can be used to interpolate between different control performances aiming to achieve different closed-loop specifications, or to switch over partitioned parameter subsets when dealing with large parameter variations. Indeed, as introduced in Chapter 2, the interest behind YK concept is to parameterize a set of linear stabilizing controllers K(Q) where each one is parameterized by its corresponding YK parameter Q [START_REF] Niemann | Dual Youla parameterisation[END_REF]. This kind of configuration is structured by mapping a set of linear stabilizing controllers onto a Q-based controller.

The main contributions in this chapter lie in the LPV world and are:

1. Design an interpolating LPV-YK control scheme to achieve different closedloop specifications.

2. Design a switching LPV-YK control scheme aiming to switch between partitioned parameter subsets.

The theoretical results in this chapter have been represented in:

• Interpolation of Multi-LPV Control Systems Based on Youla-Kucera Parameterization, published in Automatica, see [START_REF] Atoui | Interpolation of multi-LPV control systems based on Youla-Kucera parameterization[END_REF].

• Multi-Variable and Multi-Objective Gain Scheduled Control Based on Youla-Kucera Parameterization: Application to Autonomous Vehicles, under review in International Journal on Robust and Nonlinear Control.

• Advanced LPV-YK Control Design with Experimental Validation on Autonomous Vehicles, under review in Automatica.

Chapter 5. Advanced LPV-YK Control Structures: Theoretical Proofs

The following table summarizes briefly the LPV-YK structures presented in this chapter:

• The first two structures are proposed to achieve multiple closed-loop specifications, i.e. rising time, settling time, overshoot, etc. Both of them ensure closed-loop quadratic stability regardless the parameter variations and interpolating signals. The difference is just that the interpolated LPV controllers can be designed either using standard polytopic approach, or based on LTI-YK parameterization. Refer to Sections 5.5.1 and 5.5.2 for more illustration.

• The other structures are introduced aiming to provide robust performance over large parameter regions by designing a local LPV controller at each parameter subset. Quadratic or exponential closed-loop stability is possible depending on the parameter partitioning. Sections 5.6.3 and 5.6.4 illustrates the difference in more details. The structure is as follows: Section 5.2 states the motivations behind designing LPV-YK control structures. In Section 5.3, some fundamental lemmas are introduced to be used later . Section 5.4 defines a general LPV model with its parameter-varying region. Two LPV-YK control approaches for multi-objective control are presented in Section 5.5. Section 5.6 introduces two switching LPV-YK control structures over partitioned parameter regions. Finally, a discussion and summary is given in Section 5.7.

Motivation

YK parameterization has been successfully used for steering control of autonomous vehicles considering two LTI controllers designed separately (one for lane-changing and one for lane-tracking) [START_REF] Mahtout | Youla-Kucera Based Lateral Controller for Autonomous Vehicle[END_REF]. The YK control scheme of both controllers has shown interesting performance for small and large lateral errors. On the other hand, the YK controller is parameterized for a fixed-speed (LTI lateral dynamics). The objective in this chapter is to generalize the LPV-YK control structures that interpolates/switches between multiple LPV controllers obtaining a multi-variable and multi-objective switched/interpolated controller.

The LPV-YK control approaches are proposed in this thesis to improve the performance of the switched/interpolated LPV controllers, with lower conservatism and design complexities. Indeed, previous works which have shown successful and smooth LPV switching controllers as in [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF], [START_REF] Chen | The smooth switching control for TORA system via LMIs[END_REF], and [START_REF] Hanifzadegan | Smooth switching LPV controller design for LPV systems[END_REF], require the re-design of the local LPV controllers using proposed LMIs, depending on the switching signals (e.g. hysteresis switching, switching with average dwell-time, etc.). It is worth mentioning that the re-design of all local LPV controllers together may cause conservatism when increasing the number of controllers.

The theoretical developments presented here are of interest to:

1. Simplify the design of the LPV switching/interpolating control system by decreasing the complexity of the LMI conditions (no need to re-design the local LPV controllers), using parameterization instead.

2. Avoid any limitation on the switching signals without requiring the use of a constant Lyapunov function.

3. Smooth the control and state responses during the switching instants.

4. Ensure a Plug&Play structure, avoiding the re-design of the switching control scheme if one needs to add or remove any of the local LPV controllers.

Preliminaries

This section introduces some notations and assumptions regarding LPV systems and LTI-YK parameterization. In addition, useful concepts and several lemmas are reviewed.

State Transformation

The concept of state transformation is to transform the states of a system without affecting its input/output property. Consider a state transformation matrix T (of appropriate dimension) which transforms the state of system W (with state-space matrices A, B, C, D). Then, the transformed system W is computed as: (ii) there exist n p transformation matrices T i such that the LPV matrix

W : TAT -1 TB CT -1 D . ( 5
Ā(ρ) = n p ∑ i=1 α i (ρ) Āi = n p ∑ i=1 α i (ρ)T i A i T -1 i (5.2)
is quadratically stable ∀ρ ∈ J , where ρ =

n p ∑ i=1 α i (ρ)ρ i such that n p ∑ i=1 α i (ρ) = 1.
The proof is introduced in [START_REF] Bianchi | Interpolation for gain-scheduled control with guarantees[END_REF] as follows:

(i) ⇒ (ii). If A i is Hurwitz, then ∃X i > 0 such that X i A i + A T i X i < 0, i ∈ I n p .
According to [START_REF] Hespanha | Switching between stabilizing controllers[END_REF], it is always possible to find state transformations T i (e.g. T i = X 1/2 i ) such that, for a common P > 0, with Āi = T i A i T -1 i . Finding the coordinates α i (ρ), with ρ as a convex combination of the vertices of J , the LPV matrix (5.2) can be constructed. Based on α i ≥ 0, ∀i ∈ I n p , inequalities (5.3) and linearity,

P Āi + Āi P < 0, ∀i ∈ I n p (5.3)
P n p ∑ i=1 α i (ρ) Āi + n p ∑ i=1 α i (ρ) Āi T P < 0, (5.4)
and thus the quadratic stability of Ā(ρ) is proved.

(ii) ⇒ (i). Take ρ = ρ m , with ρ m one of the vertices of J ; then α m = 1, and α i = 0,∀i ̸ = m. Therefore, Ā(ρ) = Ām , and from (5.4) it can be concluded that Ām is Hurwitz, and thus A m .

YK Parameter Definition

Observing the control scheme in Fig. 5.1, assume a Single-Input-Single-Output (SISO) LTI plant G(s) connected to a controller K(s) in a stable feedback structure. Then, the closed-loop transfer function is obtained as:

CL(s) = G(s)K(s) 1 + G(s)K(s) (5.5)
Moreover, let's define a transfer function Q(s) from the reference r to the control input u, then:

Q(s) = K(s) 1 + G(s)K(s) (5.6)
Thus, if G(s) and Q(s) are known, the controller K(s) can be expressed as:

K(s) = Q(s) 1 -G(s)Q(s) (5.7)
From (5.6), it is shown that if K(s) stabilizes G(s), then Q(s) is stable and proper. Equivalently, if Q(s) is stable and proper, then K(s) given by (5.7) stabilizes G(s).

In addition, reformulating the closed-loop expression to be written in terms of Q(s), we obtain a transfer function affine with respect to Q(s) expressed as:

CL(s) = Q(s)G(s) (5.8)
From this point, the concept of a controller based on YK parameter Q(s) which is parameterized by a transfer function is opened. Then a set of stabilizing controllers can be parameterized in terms of all stable and proper functions Q(s) for any LTI plant G(s).

Doubly Coprime Factorisation

YK parameterisation uses the doubly coprime factorisation concepts to reduce the algebraic complexity of Q computation. Consider the LTI plant G stabilized by a set of LTI controllers K i , then they can be factorized (from left and right) as a product of a stable transfer function matrix and a transfer function matrix with a stable inverse as shown below:

G = N M -1 = M-1 Ñ K i = U i V -1 i = Ṽi -1
Ũi , ∀i (5.9) Lemma 6. If the coprime factors M, N, M, Ñ, U i , V i , Ũi , Ṽi ∈ RH ∞ ∀i, and if they satisfy the following Bezout Identity:

Ṽi -Ũi -Ñ M M U i N V i = M U i N V i Ṽi -Ũi -Ñ M = I 0 0 I (5.10)
then, all the factorized controllers K i stabilize G (proof in [START_REF] Tay | High performance control[END_REF]).

Q Parameterisation

Consider the YK parameter Q i as a transfer function which characterizes the dynamic switching between a base controller K 0 and K i ∀i.

Lemma 7. Assume an LTI plant G = N M -1 stabilized by the LTI controllers K i = U i V -1 i , ∀i, which have been already designed separately, with M and N are coprime, U i and V i are coprime, and M, N, U i and V i ∈ RH ∞ ∀i. Let us choose K 0 as a nominal controller factorised as K 0 = U 0 V -1 0 , then the set of parameterized LTI controllers Ki (∀i ̸ = 0) is defined as:

Ki = K(Q i ) = (U 0 + MQ i )(V 0 + NQ i ) -1 = ( Ṽ0 + Q i Ñ) -1 ( Ũ0 + Q i M)
where Q i ∈ RH ∞ by construction [START_REF] Tay | High performance control[END_REF].

Exponential Stability of A Triangular Matrix

The following Lemma shows that a triangular matrix is exponentially stable if its diagonal elements are exponentially stable with bounded off-diagonal elements. Notice that a matrix A(ρ) is said to be exponentially stable if there exists a positive definite matrix function X(ρ) such that π = Ẋ(ρ) + X(ρ)A(ρ) + A T (ρ)X(ρ) < 0 (5.11) Lemma 8. Assume that the matrices A 11 (ρ) and A 22 (ρ) are exponentially stable, every continuous and bounded block triangular matrix whose diagonal matrices consist of A 11 (ρ) and A 22 (ρ) is also exponentially stable.

To prove the following Lemma, consider an upper triangular matrix:

A(ρ) = A 11 (ρ) A 12 (ρ) 0 A 22 (ρ) (5.12)
According to the assumption, there exists bounded positive definite matrix functions X 1 (ρ) and X 2 (ρ), positive real numbers α 1 and α 2 , satisfying the following inequalities [START_REF] Ravi | H ∞ control of linear time-varying systems: A state-space approach[END_REF]:

Ẋ1 (ρ) + X 1 (ρ)A 11 (ρ) + A T 11 (ρ)X 1 (ρ) ≤ -α 1 I -α 2 I ≤ Ẋ2 (ρ) + X 2 (ρ)A 22 (ρ) + A T 22 (ρ)X 2 (ρ) < 0 (5.13)
Since the off-diagonal matrix A 12 (ρ) is assumed to be bounded, there exists a positive real number α 3 satisfying

X 1 (ρ)A 12 (ρ) ≤ α 3 I (5.14) Choose X(ρ) = X 1 (ρ) 0 0 λX 2 (ρ)
Then,

π = Ẋ1 (ρ) + X 1 (ρ)A 11 (ρ) + A T 11 (ρ)X 1 (ρ) X 1 (ρ)A 12 (ρ) A T 12 (ρ)X 1 (ρ) λ Ẋ2 (ρ) + X 2 (ρ)A 22 (ρ) + A T 22 (ρ)X 2 (ρ)
Now, let us find a positive real number λ > 0 which satisfies (5.11) such that π is negative definite. Using Schur complement, inequality (5.11) is equivalent to the following two inequalities:

π(2, 2) < 0 π = π(1, 1) -π(1, 2)π(2, 2) -1 π(2, 1) < 0 (5.15)
The first inequality holds for any λ > 0 (from (5.13)). Considering (5.13) and (5.14) in (5.15), then

π ≤ -α 1 I + λ -1 α 2 π(1, 2)π(2, 1) ≤ -α 1 I + λ -1 α 2 α 2 3 I (5.16)
Therefore, π < 0 for any λ > 0 satisfying α 2 α 2 3 /λ < α 1 . A similar proof could be deduced for the lower triangular matrices.

Notations.

For the rest of the thesis, general notations are considered as follows. I[a, b] denotes the integer set from a to b. R stands for the set of real numbers. R m×n is the set of real m × n matrices. The transpose of a real matrix M is denoted by M T . I and 0 denote an identity matrix and a zero matrix, respectively, of appropriate dimensions. diag(X 1 , X 2 , ..., X N ) denotes a matrix with matrices X 1 , X 2 , ..., and X N as diagonal blocks. It is worth noting that γ ∞ corresponds to the known γ-performance design, whereas γ is a switching/interpolating signal.

LPV Model Definition

In this section, a general LPV model is defined to be used in the following sections. Consider a Multi-Input-Multi-Output (MIMO) LPV system G(ρ) with m inputs and p outputs: where x(t) ∈ R n x , y(t) ∈ R p , u(t) ∈ R m , z(t) ∈ R n z are the state, output, input, controlled output vectors respectively. w(t) = r n d T ∈ R n w contains the exogenous inputs of the tracking reference r, measurable noise n and input disturbance d. ρ(t) := ρ ∈ R n p is a vector of n p measurable time-varying parameters.

G(ρ)    ẋ(t) = A(ρ(t))x(t) + B 1 (ρ(t))w(t) + B 2 (ρ(t))u(t) z(t) = C 1 (ρ(t))x(t) + D 11 (ρ(t))w(t) + D 12 (ρ(t))u(t) y(t) = C 2 (ρ(t))x(t) + D 21 (ρ(t))w(t) + D 22 (ρ(t))u(t) (5.

Polytopic-based LPV Model

To design quadratically stabilizing LPV controllers, the LPV model should be defined based on the polytopic approach, i.e. assuming affine parameter-dependency. Define a convex polytopic region P by the parameters extremums [ρ, ρ] as:

P := C O {w 1 , ..., w 2 np } (5.18)
where w i represent the vertices of P ∀i ∈ I[1, 2 n p ].

The polytopic approach for LPV control design requires two assumptions: 1) the system must be strictly proper (D 22 (ρ) = 0); and 2) the input and output matrices B 2 , C 2 , D 12 and D 21 must be parameter-independent [START_REF] Georgo | System analysis, modelling and control with polytopic linear models[END_REF]. For what follows, we then assume that the LPV system is given as:

G(ρ)    ẋ(t) = A(ρ)x(t) + B 1 (ρ)w(t) + B 2 u(t) z(t) = C 1 (ρ)x(t) + D 11 (ρ)w(t) + D 12 u(t) y(t) = C 2 x(t) + D 21 w(t) (5.19)
Note that the second assumption doesn't impose any serious constraints where, if needed, it can be fulfilled by filtering the input u and output y [START_REF] Apkarian | Self-scheduled H ∞ control of linear parameter-varying systems: a design example[END_REF]. Here, ρ belongs to P and is written as:

ρ = 2 np ∑ i=1 α i w i , (5.20)
where

2 np ∑ i=1 α i = 1, α i ≥ 0 ∀i
. Therefore, the system representation at any operating point ρ ∈ P is given as a convex combination of the state-space realizations of the LTI systems given at the vertices w i :

  A(ρ) B 1 (ρ) B 2 C 1 (ρ) D 11 (ρ) D 12 C 2 D 21 0   = 2 np ∑ i=1 α i (ρ)   A i B 1,i B 2 C 1,i D 11,i D 12 C 2 D 21 0   (5.21)

Multi-Objective Control Interpolation

This section aims to design an interpolation scheme between different multi-objective controllers based on LPV-YK parameterization. Nowadays, systems are getting more and more complex leading to control algorithms able to consider online varying objectives for performance and safety. For instance, the field of autonomous systems, in particular autonomous vehicles, is indicative of such an evolution. However, it is difficult to design a single controller covering a large parameter range and achieving multiple objectives. Fig. 5.2 shows an interpolation example between two control performances, where two controllers K (1) (ρ) and K (2) (ρ) are designed over the same convex parameter region P, satisfying closed-loop quadratic stability. It is assumed that each controller has been designed separately achieving different closed-loop specification such as rising time, actuator bandwidth, steady-state error, etc.

The following two sections introduce two LPV-YK control structures aiming to interpolate between different control objectives satisfying the closed-loop quadratic stability for any interpolating signal. To do so, the polytopic LPV model G(ρ) (5. [START_REF] Shamma | Gain scheduling: potential hazards and possible remedies[END_REF]) is used for both control architectures. The subscript i of a system/matrix/variable of an LPV system (e.g. G i , A i , w i ) denotes the local LTI system/matrix/variable at the i th vertex of a polytope P. The superscript (j) denotes the j th controller (e.g. K (j) ) in the set of the designed controllers. For example, A

k,i represents the LTI state matrix of the j th LPV controller at the i th vertex of P.

Interpolation between polytopic-based LPV controllers

Let K = {K (1) (ρ), K (2) (ρ), . . . , K (j) (ρ), . . . , K (ζ) (ρ)} be a finite set of quadratically stabilizing LPV controllers of G(ρ) (5.19) that are designed achieving different objectives and closed-loop specifications. Thus, ∀j ∈ I[0, ζ]:

K (j) (ρ) : A (j) k (ρ) B (j) k (ρ) C (j) k (ρ) D (j) k (ρ)
A (j) k (ρ) B (j) k (ρ) C (j) k (ρ) D (j) k (ρ) = 2 np ∑ i=1 α i (ρ) A (j) k,i B (j) k,i C (j) k,i D (j) k,i (5.23)
where

A (j) k (ρ) ∈ R n (j) k ×n (j) k , B (j) k (ρ) ∈ R n (j) k ×m k , C (j) k (ρ) ∈ R p k ×n (j) k and D (j) k (ρ) ∈ R p k ×m k .
Based on the statements on LPV concepts and YK parameterization, a quadratically stable interpolation procedure between ζ LPV controllers is formulated (as mentioned by the next theorem). The interpolation is performed using an interpolating signal vector γ = [γ 1 , . . . , γ j , . . . , γ ζ ].

Let's assume that (A.1.1). There exists an LPV output-feedback controller K (0) (ρ) which quadratically stabilizes G(ρ) over P (following the approach in [START_REF] Apkarian | Self-scheduled H ∞ control of linear parameter-varying systems: a design example[END_REF]), defined as:

K (0) (ρ) : A (0) k (ρ) B (0) k (ρ) C (0) k (ρ) D (0) k (ρ) (5.24) being, A (0) k (ρ) B (0) k (ρ) C (0) k (ρ) D (0) k (ρ) = 2 np ∑ i=1 α i (ρ) A (0) k,i B (0) k,i C (0) k,i D (0) k,i (5.25)
where A For illustration, Figure 5.3 represents an example of an interpolation between three polytopic-based LPV controllers K (0) (ρ), K (1) (ρ), and K (2) (ρ), along the convex parameter space P. The orange solid line represents the chosen nominal LPV controller K (0) (ρ). The blue/red solid lines are two LPV controllers K (1) (ρ)/K (2) (ρ) that are designed using the standard polytopic approach, achieving different required objectives. The overall interpolation is performed using the interpolating signal γ = [γ 1 , γ 2 ], and is represented by the LPV-YK controller K(ρ, γ).

(0) k (ρ) ∈ R n (0) k ×n (0) k , B (0)
k (ρ) ∈ R n (0) k ×m k , C (0)
k (ρ) ∈ R p k ×n (0) k and D (0) k (ρ) ∈ R p k ×m k .
Notice that for every ρ ∈ P and for every continuous/discontinuous interpolating signal γ, K(ρ, γ) quadratically stabilizes G(ρ). Some interpolation cases can be mentioned to show how K(ρ, γ) recovers a single gain-scheduled controller K (j) (ρ) by varying γ j :

• if γ j = 0 ∀j, K(ρ, γ) ≡ K (0) (ρ) • if γ j = 1 for j = c ∈ [1, ζ] and γ j = 0 ∀j ̸ = c, K(ρ, γ) ≡ K (c) (ρ)
• else, the performance of K(ρ, γ) is interpolated among K (j) (ρ) according to the chosen γ j .

The following theorem proves that K(ρ, γ) quadratically stabilizes G(ρ) for every ρ and for any continuous/discontinuous interpolating vector signal γ. Theorem 5. Consider an LPV plant G(ρ) (5.19), and given a set of LPV controllers K (5.22) that quadratically stabilizes G(ρ). Let us choose K (0) (ρ) as the nominal (central) controller for YK parameterization. Then, the parameterized LPV-YK controller K(ρ, γ) (5.28), represented in Fig. 5.4, stabilizes G(ρ) for any continuous/discontinuous interpolating signal γ = [γ 1 , γ 2 , . . . , γ ζ ], if there exist symmetric, positive definite matrices X g ∈ R n x ×n x ,

X k ∈ R n (0) k ×n (0) k such that: A(ρ)X g + X g A T (ρ) + B 2 W(ρ) + W T (ρ)B T 2 < 0 (5.26) A (0) k (ρ)X k + X k A (0)T k (ρ) + B (0) k (ρ)V(ρ) + V(ρ) T (ρ)B (0)T k (ρ) < 0 (5.27)
The state space matrices of the parameterized LPV-YK controller K(Q(ρ)) = K(ρ, γ) are written as:

Ãk (ρ, γ) =   A(ρ) + B 2 F g (ρ) -B 2 D q (ρ, γ)C 2 -B (0) k (ρ)C 2 -B q (ρ)C 2 -B 2 D q (ρ, γ)F (0) k (ρ) B 2 C q (ρ, γ) A (0) k (ρ) 0 -B q (ρ)F (0) k (ρ) A q (ρ)    Bk (ρ, γ) =   B 2 D q (ρ, γ) B (0) k (ρ) B q (ρ)   Ck (ρ, γ) = F g (ρ) -(D (0) k (ρ) + D q (ρ, γ))C 2 C (0) k (ρ) -D q (ρ, γ)F (0) k (ρ) C q (ρ, γ) Dk (ρ, γ) = D (0) k (ρ) + D q (ρ, γ) (5.28)
where A q (ρ) = diag(A

q , . . . , A

q , . . . ,

A (ζ) q ), B q (ρ) = B (1) q . . . B (j) q . . . B (ζ) q T , C q (ρ, γ) = γ 1 C (1) q . . . γ j C (j) q . . . γ ζ C (ζ) q , D q (ρ, γ) = ζ ∑ j=1 γ j D (j) q , (5.29) being F g (ρ) = W(ρ)X -1 g , F (0)
k (ρ) = V(ρ)X -1 k , and A (j) q , B (j) q , C (j)
q , and D

q the statespace matrices of Q (j) (ρ, γ) ∀j ∈ I[1, ζ] represented in (5.31). Notice that γ j is multiplied by the output of Q

(j) (ρ), i.e. Q (j) (ρ, γ) = γ j × Q (j) (ρ).
Proof 1. According to YK parameterisation concept, the parameterized controller can be formulated as a Linear Fractional Transformation (LFT) system [START_REF] Tay | High performance control[END_REF], i.e. K(j) (ρ, γ) = F l (J(ρ), Q (j) (ρ, γ)) ∀j (see Fig. 5.4), where J(ρ) and Q (j) (ρ, γ)) derived as shown in (5.30)- (5.31).

J(ρ) = 2 np ∑ i=1 α i (ρ)       A i + B 2 F g,i 0 0 B 2 -B (0) k,i C 2 A (0) k,i B (0) k,i 0 F g,i -D (0) k,i C 2 C (0) k,i D (0) k,i I -C 2 -F (0) k,i I 0      
(5.30) At each vertex w i , the closed-loop system CL i (γ) (5.33) is derived from the LFT interconnection between G i and Ki (γ) (see Fig. 5.5).

Q (j) (ρ, γ) = 2 np ∑ i=1 α i (ρ)       A i + B 2 D (j) k,i C 2 B 2 C (j) k,i B 2 [D (j) k,i - D (0) k,i ]F (0) k,i - B 2 C (0) k,i B (j) k,i C 2 A (j) k,i B (j) k,i F (0) k,i 0 0 A (0) k,i + B (0) k,i F (0) k,i B 2 [D (j) k,i - D (0) k,i ] B (j) k,i B (0) k,i γ[D (j) k,i C 2 - F g,i ] γC (j) k,i γ[(D (j) k,i - D (0) k,i )F (0) k,i - C (0) k,i ] γ[D (j) k,i - D (0) k,i ]       (5.31) Ācl,i (γ) = TA cl,i (γ)T -1 =       A i + B 2 F g,i B 2 C q,i -B 2 (F g,i -(D (0) k,i + D q,i (γ))C 2 ) B 2 (C (0) k,i - D q,i F (0) k,i ) 0 A q,i B q,i C 2 -B q,i F (0) k,i 0 0 A i + B 2 D (0) k,i C 2 B 2 C (0) k,i 0 0 B (0) k,i C 2 A (0) k,i      
(5.32)

Multi-Objective Control Interpolation

85

CL i (γ) =          A i + B 2 (D (0) k,i + D q,i (γ))C 2 B 2 (F g,i -(D (0) k,i + D q,i (γ))C 2 ) B 2 (C (0) k,i - D q,i (γ)F (0) k,i ) B 2 D q,i (γ)C 2 A i + B 2 (F g,i - D q,i (γ)C 2 ) -B 2 D q,i (γ)F (0) k,i B (0) k,i C 2 -B (0) k,i C 2 A (0) k,i B q,i C 2 -B q,i C 2 -B q,i F (0) k,i C 1,i + D 12 (D (0) k,i + D q,i (γ))C 2 D 12 (F g,i -(D (0) k,i + D q,i (γ))C 2 ) D 12 (C (0) k,i - D q,i (γ)F (0) k,i ) B 2 C q,i (γ) B 1,i + B 2 (D (0) k,i + D q,i (γ))D 21 B 2 C q,i (γ) B 2 D q,i (γ)D 21 0 B (0) k,i D 21 A q,i B q,i D 21 D 12 C q,i (γ) D 11,i + D 12 (D (0) k,i + D q,i (γ))D 21         (5.33) FIGURE 5.5: G -K LFT interconnection
Now, the closed-loop state matrix A cl (ρ, γ) =

2 np ∑ i=1 α i (ρ)A cl,i (γ) is quadratically stable
if there exist a symmetric, positive definite, parameter-invariant matrix X cl such that:

X cl A cl (ρ, γ) + A T cl (ρ, γ)X cl < 0 ∀γ (5.34)
To prove this, let T =

    I 0 0 0 0 0 0 I I -I 0 0 0 0 I 0     be a state transformation matrix which is applied to CL(ρ, γ), without changing its input-output nature, so T -1 =     I 0 0 0 I 0 -I 0 0 0 0 I 0 I 0 0     .
Due to the block-triangular form of Ācl,i (γ) (5.32) , (5.34) is satisfied if the following equations hold (check Lemma 2 in [START_REF] Xie | Design of LPV control systems based on Youla parameterisation[END_REF]):

2 np ∑ i=1 α i (ρ)(Y g (A i + B 2 F g,i ) + (A i + B 2 F g,i ) T Y g ) < 0 (5.35) 2 np ∑ i=1 α i (ρ)(Y q A q,i + A T q,i Y q ) < 0 (5.36) 2 np ∑ i=1 α i (ρ)(Y 0 A (0) i + A (0)T i Y 0 ) < 0 (5.37)
where Y g ∈ R n x ×n x , Y q ∈ R n q ×n q and Y 0 ∈ R (n x +n (0)

k )×(n x +n (0)
k ) are symmetric, positive definite, parameter-invariant matrices, and choosing X cl = T T diag(Y g , Y q , Y 0 ) T, and

A (0) i = A i + B 2 D (0) k,i C 2 B 2 C (0) k,i B (0) k,i C 2 A (0) k,i
(5.38)

Inequality (5.35) can be linearized by Y g = X -1 g which leads to (5.26) when choosing

W(ρ) = 2 np ∑ i=1 α i (ρ)F g,i X g . Due to the diagonal shape of A q (ρ), it is quadratically stable if each diagonal term A (j)
q (ρ) is quadratically stable. Regarding (5.31), the state matrix of each LPV-YK parameter A (j) q (ρ) is quadratically stable if there exist symmetric, positive definite, constant matrices P 1 ∈ R (n x +n (1) k )×(n x +n (1)

k ) and P 2 ∈ R n (0) k ×n (0) k such that: 2 np ∑ i=1 α i (ρ)(P 1 A (j) i + A (j)T i P 1 ) < 0 (5.39) 2 np ∑ i=1 α i (ρ)(P 2 (A (0) k,i + B (0) k,i F (0) k,i ) + (A (0) k,i + B (0) k,i F (0) k,i ) T P 2 ) < 0 (5.40) being, A (j) i = A i + B 2 D (j) k,i C 2 B 2 C (j) k,i B (j) k,i C 2 A (j) k,i
(5.41)

The condition in (5.39) is verified given that K (j) (ρ) ∈ K quadratically stabilizes G(ρ).

Moreover, the inequality (5.40) satisfies (5.27) by choosing

P 2 = X -1 k and V(ρ) = 2 np ∑ i=1 α i (ρ)F (0) k,i X k .
Thus, ∀j ∈ I[1, j] A (j) q (ρ), is quadratically stable, consequently A q (ρ), and (5.36) is verified. Finally, (5.37) is fulfilled given that K (0) (ρ) quadratically stabilizes G(ρ).

Notice that the closed-loop state matrix A cl (ρ, γ) is proved to be quadratically stable regardless the switching signal γ. It is worth mentioning that this is achieved without requiring an instantaneous design of all the local LPV controllers including the interpolating signals.

Interpolation between YK-based LPV controllers

Unlike the previous polytopic-based LPV-YK control, the current section proposes a more advanced one aiming to reduce the problem conservatism caused by the condition of requiring polytopic-based LPV controllers, i.e. relaxing assumption (A.1.2) and the LMI condition (5.27). Generally speaking, it has been found that it is not necessary to have Q (j) (ρ, γ) quadratically stable by design. It is only necessary to have each Q (j) i (γ) stable ∀w i . Then, a state transformation (as in Lemma 5) can be used to obtain a quadratically stabilizing LPV-YK parameter Q(ρ, γ). To do so, and to improve the closed-loop performance, the local LPV controllers are supposed now to be designed using YK-based gain-scheduling, i.e. the local interpolated LPV controllers are designed based on YK parameterization of their corresponding LTI controllers.

As a result, the current section expresses deeply the necessary LMI conditions to design a quadratically stabilizing LPV-YK interpolation scheme, whereas the LMIs in Theorem 5 are considered to be sufficient conditions.

Considering the polytopic-based LPV model G(ρ) (5. [START_REF] Shamma | Gain scheduling: potential hazards and possible remedies[END_REF]), assume that

(A.2

.1).

There exists an LPV output-feedback controller K (0) (ρ) which quadratically stabilizes G(ρ) over P (following the approach in [START_REF] Apkarian | Self-scheduled H ∞ control of linear parameter-varying systems: a design example[END_REF]), defined as:

K (0) (ρ) : A (0) k (ρ) B (0) k (ρ) C (0) k (ρ) D (0) k (ρ) (5.42) being, A (0) k (ρ) B (0) k (ρ) C (0) k (ρ) D (0) k (ρ) = 2 np ∑ i=1 α i (ρ) A (0) k,i B (0) k,i C (0) k,i D (0) k,i
(5.43)

= [γ 1 , γ 2 ], between K (0) (ρ) (for γ = [0, 0]), K (1) (ρ) (for γ = [1, 0]), and
K (2) (ρ) (for γ = [0, 1])
where A The objective of the proposed strategy is to:

(0) k (ρ) ∈ R n (0) k ×n (0) k , B (0)
k (ρ) ∈ R n (0) k ×m k , C (0)
k (ρ) ∈ R p k ×n (0) k and D (0) k (ρ) ∈ R p k ×m k . (A.2.2). At each vertex w i (i ∈ I[1, 2 n p ]) of
1. Design a set of gain-scheduled controllers K (j) (ρ), j ∈ I [1, ζ]. Each one is designed by interpolating its corresponding LTI controllers

K (j) i (i ∈ I[1, 2 n p ]) based on YK parameterization.

Create an overall interpolation scheme between the gain-scheduled controllers

K (j) (ρ) (j ∈ I[0, ζ])
, by an interpolating signal vector γ, which is referred to as K(ρ, γ), such that the resultant LPV-YK controller K(ρ, γ) quadratically stabilizes G(ρ) ∀ρ ∈ P and for every γ = [γ 1 , ..., γ j , ..., γ ζ ].

Figure 5.6 represents an example of an LPV-YK based interpolation between three gain scheduled-scheduled controllers K (0) (ρ), K (1) (ρ), and K (2) (ρ), along the convex parameter space P. The orange solid line represents the chosen nominal LPV controller K (0) (ρ), as defined by (A.2.1). The blue/red points represent the local LTI controllers K

(1)

i /K (2) i (i ∈ I[1, 2 n p ]) as defined in (A.2.
2). The blue/red dashed line is the gain-scheduled controller K (1) (ρ)/K (2) (ρ) that is designed by the YK-based interpolation of the LTI controllers

K (1) i /K (2)
i . The overall interpolation is performed using the interpolating signal γ = [γ 1 , γ 2 ], and is represented by the LPV-YK controller K(ρ, γ). It is worth mentioning that the difference between this figure and Fig. 5.3 is that the interpolated LPV controllers K (1) (ρ) and K (2) (ρ) are designed using the standard polytopic approach in Fig. 5.3, whereas here they are designed using YK-based gain-scheduling as proposed by [START_REF] Atoui | Interpolation of multi-LPV control systems based on Youla-Kucera parameterization[END_REF].

The LPV-YK controller K(ρ, γ) is designed based on a Linear Matrix Inequality (LMI) optimization problem, where it is defined as

K(ρ, γ) : Ãk (ρ, γ) Bk (ρ, γ) Ck (ρ, γ) Dk (ρ, γ) (5.44)
It is worth mentioning that for every ρ ∈ P and for every continuous/discontinuous interpolating signal γ, K(ρ, γ) quadratically stabilizes G(ρ).

The interpolation cases can be mentioned to show how K(ρ, γ) recovers a single gain-scheduled controller K (j) (ρ) by varying γ j :

• if γ j = 0 ∀j, K(ρ, γ) ≡ K (0) (ρ) = 2 np ∑ i=1 α i (ρ)K (0) i • if γ j = 1 for j = c ∈ [1, ζ] and γ j = 0 ∀j ̸ = c, K(ρ, γ) ≡ K (c) (ρ) = 2 np ∑ i=1 α i (ρ)K (c) i
• else, the performance of K(ρ, γ) is interpolated among K (j) (ρ) according to the chosen γ j .

The next theorem proves that the LPV-YK controller K(ρ, γ) (5.47) quadratically stabilizes G(ρ) ∀ρ, and for every continuous/discontinuous interpolating vector signal γ. Theorem 6. Consider an LPV plant G(ρ) (5.19), satisfying assumptions (A.2.1) and (A.2.2). Then, the following generalized LPV-YK controller K(ρ, γ) (5.47), as represented in Fig. 5.7, quadratically stabilizes G(ρ) for any ρ ∈ P and for any continuous/discontinuous interpolating signals γ j ∈ [0, 1] (j ≥ 1), if there exist symmetric, positive definite, constant matrices

X g ∈ R n x ×n x , X k,i ∈ R n (0) k ×n (0)
k , and matrices W i ∈ R m×n x and V i ∈ R m×n (0) k such that ∀i:

A i X g + X g A T i + B 2 W i + W T i B T 2 < 0 ∀w i (5.45) A (0) k,i X k,i + X k,i A (0)T k,i + B (0) k,i V i + V T i B (0)T k,i < 0 ∀w i (5.46)
The state-space matrices of K(ρ, γ) are

Ãk (ρ, γ) = 2 np ∑ i=1 α i (ρ)     A i + B 2 F g,i -B 2 Dq,i (γ)C 2 -B (0) k,i C 2 -Bq,i C 2 -B 2 Dq,i (γ)F (0) k,i B 2 Cq,i (γ) A (0) k,i 0 -Bq,i F (0) k,i Ãq,i     Bk (ρ, γ) = 2 np ∑ i=1 α i (ρ)    B 2 Dq,i (γ) B (0) k,i Bq,i    Ck (ρ, γ) = 2 np ∑ i=1 α i (ρ) F g,i -(D (0) k,i + Dq,i (γ))C 2 C (0) k,i -Dq,i (γ)F (0) k,i
Cq,i (γ)

Dk (ρ, γ) = 2 np ∑ i=1 α i (ρ)[D (0) k,i + Dq,i (γ)] (5.47) where ∀i ∈ I[1, 2 n p ],
q,i (Z (1) i ) -1 , . . . , Z (j) i A (j) q,i (Z (j) i ) -1 , . . . , Z (ζ) i A (ζ) q,i (Z (ζ) i ) -1 ), Bq,i = Z (1) i B (1) q,i . . . Z (j) i B (j) q,i . . . Z (ζ) i B (ζ) q,i T , Cq,i (γ) = γ 1 C (1) q,i (Z (1) i ) -1 . . . γ j C (j) q,i (Z (j) i ) -1 . . . γ ζ C (ζ) q,i (Z (ζ) i ) -1 , Dq,i (γ) = ζ ∑ j=1 γ j D (j)
q,i , (5.48)

A (j) q,i , B (j) q,i , C (j)
q,i and D (j) q,i are the state-space matrices of Q (j) (ρ) given in (5.54) at the polytopic vertices w i , F g,i = W i X -1 g , F

k,i = V i X -1 k,i , (0)

and Z (j)

i are state transformation matrices chosen to satisfy Lemma 5 such that Ā(j) q (ρ) =

2 np ∑ i=1 α i (ρ)Z (j) i A (j) q,i (Z (j) i ) -1 is quadratically stable ∀j ≥ 1.
Proof 2. According to YK parameterization concept, each parameterized controller can be formulated as a Linear Fractional Transformation (LFT) system [START_REF] Tay | High performance control[END_REF]. Then, the LPV-YK controller can be written as K(ρ, γ) = F l (J(ρ), Q(ρ, γ)) (see Fig. 5.7), where J(ρ) is represented in (5.49), and (5.54) satisfying Lemma 5, to be shown later in the proof.

Q(ρ, γ) = ζ ∑ i=1 γ j (t) Q(j) (ρ) being Q(j) (ρ) a transformed system of Q (j) (ρ)
J(ρ) = 2 np ∑ i=1 α i (ρ)       A i + B 2 F g,i 0 0 B 2 -B (0) k,i C 2 A (0) k,i B (0) k,i 0 F g,i -D (0) k,i C 2 C (0) k,i D (0) k,i I -C 2 -F (0) k,i I 0       (5.49)
The following proof is achieved by two steps:

1. Prove that the LPV-YK parameter Q(ρ, γ) is quadratically stable ∀ρ ∈ P, ∀γ.

2. Prove the closed-loop quadratic stability ∀ρ ∈ P, ∀γ.

Multi-Objective Control Interpolation

91

Step 1: Q(ρ, γ) can be proved to be quadratically stable ∀ρ ∈ P and ∀γ as follows: Assuming (A.2.2), and since inequality (5.46) is satisfied, there exist symmetric, positive definite matrices S

(j) i ∈ R (n x +n (j) k )×(n x +n (j) k ) and R i ∈ R n (0) k ×n (0) k such that ∀i, ∀j ≥ 1: S (j) i A (j) i + A (j)T i S (j) i < 0 (5.50) (A (0) k,i + B (0) k,i F (0) k,i )R i + R i (A (0) k,i + B (0) k,i F (0) k,i ) T < 0 (5.51) being, A (j) i = A i + B 2 D (j) k,i C 2 B 2 C (j) k,i B (j) k,i C 2 A (j) k,i
(5.52)

with R i = X k,i and F (0)

k,i = V i X -1 k,i . Thus, A (j)
q,i is Hurwitz ∀i, j and there exist symmetric, positive definite matrices X (j)

q,i = diag(S (j) i , R i ) ∈ R n (j) q,i ×n (j)
q,i such that:

X (j) q,i A (j) q,i + (A (j) q,i ) T X (j) q,i < 0 ∀i, ∀j (5.53)
Following Lemma 5, for any j ≥ 1, there exist transformation matrices Z (j)

i such that the transformed system Q(j) (ρ) is quadratically stable ∀ρ ∈ P (j) , choose Z

(j) i = (X (j) q,i ) 1/2 . As a result, Q(ρ, γ) = ζ ∑ j=1 γ j Q(j) (ρ) is quadratically stable ∀ρ ∈ P and ∀γ ∈ [0, 1]. Q (j) (ρ) = 2 np ∑ i=1 α i (ρ)   A (j) q,i B (j) q,i C (j) q,i D (j) q,i   = 2 np ∑ i=1 α i (ρ)       A i + B 2 D (j) k,i C 2 B 2 C (j) k,i B 2 [D (j) k,i - D (0) k,i ]F (0) k,i - B 2 C (0) k,i B (j) k,i C 2 A (j) k,i B (j) k,i F (0) k,i 0 0 A (0) k,i + B (0) k,i F (0) k,i B 2 [D (j) k,i - D (0) k,i ] B (j) k,i B (0) k,i [D (j) k,i C 2 - F g,i ] C (j) k,i [(D (j) k,i - D (0) k,i )F (0) k,i - C (0) k,i ] [D (j) k,i - D (0) k,i ]       (5.54) Ācl (ρ, γ) = 2 np ∑ i=1 α i (ρ)TA cl,i (γ)T -1 = 2 np ∑ i=1 α i (ρ)       A i + B 2 F g,i B 2 Cq,i (γ) -B 2 (F g,i -(D (0) k,i + Dq,i (γ))C 2 ) B 2 (C (0) k,i -Dq,i (γ)F (0) k,i ) 0 Ãq,i Bq,i C 2 -Bq,i F (0) k,i 0 0 A i + B 2 D (0) k,i C 2 B 2 C (0) k,i 0 0 B (0) k,i C 2 A (0) k,i      
(5.55)

5.5. Multi-Objective Control Interpolation 93 CL(ρ, γ) = 2 np ∑ i=1 α i (ρ)          A i + B 2 (D (0) k,i + Dq,i (γ))C 2 B 2 (F g,i -(D (0) k,i + Dq,i (γ))C 2 ) B 2 (C (0) k,i -Dq,i (γ)F (0) k,i ) B 2 Dq,i (γ)C 2 A i + B 2 (F g,i -Dq,i (γ)C 2 ) -B 2 Dq,i (γ)F (0) k,i B (0) k,i C 2 -B (0) k,i C 2 A (0) k,i Bq,i C 2 -Bq,i C 2 -Bq,i F (0) k,i C 1,i + D 12 (D (0) k,i + Dq,i (γ))C 2 D 12 (F g,i -(D (0) k,i + Dq,i (γ))C 2 ) D 12 (C (0) k,i -Dq,i (γ)F (0) k,i ) B 2 Cq,i (γ)
B 1,i + B 2 (D (0) k,i + Dq,i (γ))D 21 B 2 Cq,i (γ)
D 11,i + D 12 (D (0) k,i + Dq,i (γ))D 21         (5.56) (γ)
Step 2: The closed-loop system CL(ρ, γ) (5.56) is derived from the LFT interconnection between G(ρ) and K(ρ, γ) (see Fig. 5.

5). Its state matrix

A cl (ρ, γ) = 2 np ∑ i=1 α i (ρ)A cl,i (γ)
is quadratically stable if there exist a symmetric, positive definite, constant matrix X cl such that:

X cl A cl (ρ, γ) + A T cl (ρ, γ)X cl < 0 ∀γ (5.57) Now , let T =     I 0 0 0 0 0 0 I I -I 0 0 0 0 I 0   
 be a state transformation matrix which is applied to CL(ρ, γ)

without changing its input-output nature, with T -1 =     I 0 0 0 I 0 -I 0 0 0 0 I 0 I 0 0     .
Due to the block-triangular form of Ācl (ρ, γ) (5.55) , (5.57) is satisfied if the following equations hold (check Lemma 2 in [START_REF] Xie | Design of LPV control systems based on Youla parameterisation[END_REF]):

2 np ∑ i=1 α i (ρ)(Y g (A i + B 2 F g,i ) + (A i + B 2 F g,i ) T Y g ) < 0 (5.58) 2 np ∑ i=1 α i (ρ)(Y q Ãq,i + ÃT q,i Y q ) < 0 (5.59) 2 np ∑ i=1 α i (ρ)(Y 0 A (0) i + A (0)T i Y 0 ) < 0 (5.60)
where Y g ∈ R n x ×n x , Y q ∈ R n (1) q ×n

(1) q and Y 0 ∈ R (n x +n (0)

k )×(n x +n (0)
k ) are symmetric, positive definite, parameter-invariant matrices, choosing X cl = T T diag(Y g , Y q , Y 0 ) T, with

A (0) i = A i + B 2 D (0) k,i C 2 B 2 C (0) k,i B (0) k,i C 2 A (0) k,i (5.61)
Therefore,

• Inequality (5.58) is equivalent to (5.45) by choosing Y g = X -1 g and W i = F g,i X g .

• Since Q(ρ, γ) is proved to be quadratically stable, inequality (5.36) is satisfied.

• (5.60) is fulfilled given that K (0) (ρ) quadratically stabilizes G(ρ).
It is worth mentioning that the problem complexity refers mainly to find a nominal LPV controller K (0) (ρ) which must quadratically stabilize G(ρ), i.e. assumption (A.2.1). The rest is carried out using classical LTI control approaches. This shows an interest since a quadratically stabilizing gain-scheduled controller K (j) (ρ) can be designed based on an interpolation of LTI controllers, with lower conservatism compared to the standard polytopic design. In addition, the interpolation of several gain-scheduled controllers, with any finite continuous/discontinuous interpolating signals γ j , provides a general multi-variable and multi-objective controller K(ρ, γ) based on LPV-YK concept.

The next sections introduce LPV-YK control structures aiming to achieve stable smooth switching between parameter partitions.

Switching between Parameter Subsets

This section proposes two different LPV-YK structures aiming to switch between different parameter subsets. This is usually needed when dealing with large number of varying parameters or large variations. Thus, the aim is not to change the closedloop performance, but to maintain a closed-loop robust performance over the wide parameter variations. The following two subsections discuss the limitations of both polytopic and grid-based approaches, and some solutions that have been proposed in the literature.

Polytopic Approach

As discussed in Chapter 2, the polytopic approach is known to be the most conservative one [START_REF] Hoffmann | A Survey of Linear Parameter-Varying Control Applications Validated by Experiments or High-Fidelity Simulations[END_REF]. One of the main causes is that the polytopic LPV synthesis requires a constant Lyapunov function to ensure quadratic stability, which increases the problem conservatism.

Moreover, the overbounding of the parameter set is considered as a main cause of conservatism. The operating region of the underlying LPV model is defined by a convex polytope containing the parameter trajectories. This convex parameter region may include vertices that are not attained by the real plant, resulting in conservatism. The reason is that the construction of the polytope is based on the assumption that all parameters vary independently (in the affine parameter dependency formulation), whereas they could be related to each other by inherent couplings. For example, the known bicycle model describing the lateral dynamics of an autonomous vehicle is parameterized by the scheduling parameters "v x " and "1/v x " [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF] (being v x the longitudinal speed). Such a situation might cause unstable models at the polytopic vertices. In addition, the parameters could be physically correlated with each other, such that some combinations of extreme values of the parameters do not occur in real operation. For example, an LPV model describing the vertical flight dynamics of an aeroplane might be parameterized by the external scheduling signals (i.e., parameters) "speed" and "altitude." But usually, the maximum speed is not reached for minimum altitude and vice versa [START_REF] Kwiatkowski | PCA-Based Parameter Set Mappings for LPV Models With Fewer Parameters and Less Overbounding[END_REF].

Several solutions have been investigated in the literature to find a reduced convex parameter region, as presented in the recent survey [START_REF] Li | Polytopic LPV approaches for intelligent automotive systems: State of the art and future challenges[END_REF]. [START_REF] Kapsalis | LPV/Gain-Scheduled Lateral Control Architectures for Autonomous Vehicles[END_REF] has recently proposed a methodology for polytopic vertices reduction. [START_REF] Azuma | A New LMI Approach to Analysis of Linear Systems Depending on Scheduling Parameter in Polynomial Forms[END_REF] suggests to construct convex polyhedrons along given parameter trajectories, and solve the control design problem using affine parameter-dependent LMIs. Unfortunately, these methods often result in a huge number of vertices or nonconvex parameter sets and thus in increased computational burden. Scheduling Dimension Reduction (SDR) approach is proposed in [START_REF] Kwiatkowski | PCA-Based Parameter Set Mappings for LPV Models With Fewer Parameters and Less Overbounding[END_REF] which reduces the parameter set based on experimental data, and yields the benefit of tailoring a control design to specific trajectories. In [START_REF] Koelewijn | Scheduling Dimension Reduction of LPV Models -A Deep Neural Network Approach[END_REF], a Deep Neural Network (DNN) approach is used to develop the SDR methods and achieve higher model accuracy under scheduling dimension reduction. In [START_REF] Zhao | Optimal switching surface design for statefeedback switching LPV control[END_REF] and [START_REF] Zhao | Switching LPV controller design under uncertain scheduling parameters[END_REF], authors develop an LPV switching controller and prove its stability when switching among the overlapped subsets of a polytopic parameter region.

Concerning YK methods, in [START_REF] Rasmussen | Stable Controller Interpolation and Controller Switching for LPV Systems[END_REF], a YK configuration is proposed to improve the performance of a polytopic LPV control. On the other hand, [START_REF] Bianchi | Interpolation for gain-scheduled control with guarantees[END_REF] proposes a YK-based gain-scheduled controller by interpolating LTI controllers designed separately at the different vertices of a polytopic parameter region. The interpolation is performed as a function of the varying parameters of the LPV model. Closed-loop quadratic stability and performance are guaranteed at intermediate interpolation points of the convex domain. In [START_REF] Blanchini | Stable LPV Realization of Parametric Transfer Functions and Its Application to Gain-Scheduling Control Design[END_REF], a fixed pole-assignment application is introduced using an LPV YK-based method to preserve the closed-loop poles at the same location by interpolating between different controllers.

In addition, in our previous work [START_REF] Atoui | Interpolation of multi-LPV control systems based on Youla-Kucera parameterization[END_REF], an LPV-YK control scheme has been proposed to interpolate between two LPV controllers, each one designed over a full polytopic region, to achieve multiple control performances. On the other hand, all the previous YK-based solutions are still conservative for systems having overbounding polytopic parameter region.

Grid-based Approach

The grid-based approach uses a parameter-dependent Lyapunov function to solve a set of parameter-dependent LMIs (pLMIs). Since a single parameter-dependent Lyapunov function could not be efficient for complex designs and large parameter regions, a first solution has been proposed in [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF]. The objective is to design multiple LPV controllers based on multiple parameter-varying Lyapunov functions, each suitable for a specific parameter subregion, and switch between them to achieve better performance. The switching stability has been studied for hysteresis switching and switching with average dwell-time strategies. This methodology enhances the use of the switched LPV techniques in several applications, see for instance [START_REF] Lu | Switching LPV control of an F-16 aircraft via controller state reset[END_REF], [START_REF] Lescher | Switching LPV Controllers for a Variable Speed Pitch Regulated Wind Turbine[END_REF], [START_REF] Postma | Air-Fuel Ratio Control of Spark Ignition Engines Using a Switching LPV Controller[END_REF] , [START_REF] Colmegna | Switched LPV Glucose Control in Type 1 Diabetes[END_REF].

On the other hand, it has been stated in [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF] that the switched LPV controllers may not provide a smooth transient response during switching, where aggressive performance is obtained at switching instants. Such a case may lead to mechanical damage, decrease material lifetime, or signal saturation which is out of real application objectives. Following this work, several research studies have been involved in solving the switching smoothness. For instance, a bumpless transfer of switching controllers is proposed in [START_REF] Dimon | Bumpless Transfer Between Observer-Based Gain Scheduled Controllers[END_REF] followed by some developments in [START_REF] Kinnaert | State resetting for bumpless switching in supervisory control[END_REF], [START_REF] Chen | The smooth switching control for TORA system via LMIs[END_REF], and [START_REF] Safaei | On controller initialization in multivariable switching systems[END_REF].

Finally, a smooth switching LPV controller has been proposed in [START_REF] Hanifzadegan | Smooth switching LPV controller design for LPV systems[END_REF]. It is designed in considering adjustable interpolation functions and a higher order differential control signal. An iterative descent algorithm is applied to optimize three decision variables (the parameter-dependent Lyapunov functions, the local controllers, and the interpolation functions). It also augments the problem to two dimensional parameter regions. This concept is developed in the recent works [START_REF] He | Simultaneous Design of Smooth Switching State-Feedback LPV Control[END_REF] and [START_REF] He | Smooth Switching LPV Dynamic Output-feedback Control[END_REF], however, it increases the complexity and the design constraints of the local LPV controllers to achieve their objectives.

Notations.

The following notations are used in the next two subsections 5.6.3 and 5.6.4. The subscript i of a system/matrix/variable of an LPV plant (e.g. G(ρ)) denotes the LPV system/matrix/variable of the plant when the varying-parameter ρ ∈ P i , where P i is a subset of the full parameter region P. In addition, the second subscript j, when used, corresponds to the j th vertex of a polytope. For example, A k,ij represents the state matrix of the i th LPV local controller in the polytope P i at the j th vertex of P i .

Partitioned Gain-scheduled Control

This section proposes a partitioned gain-scheduled controller which solves the overbounding and conservatism problems of the standard polytopic controller. It presents an LPV-YK control scheme which:

1. Formulates a YK-based gain-scheduling between LTI controllers that have been already designed separately at the polytopic vertices of each parameter subset.

2. Switches between the formulated gain-scheduling controllers over the partitioned polytopic subregions.

The closed-loop system is proved to guarantee the quadratic stability for any continuous/discontinuous switching signals in terms of a set of Linear Matrix Inequalities (LMIs).

Define P 0 as a convex polytope that contains all the parameter trajectories ρ ∈ R n p . Let P i , i ∈ Z N = {1, . . . , N}, be the convex polytopic subsets, that could intersect by a boundary or surface, defined in P 0 along the parameter trajectories, i.e. ρ ∈ P = i≥1 P i ⊂ P 0 . Each P i (i ≥ 0) is defined as:

P i := C O {w i1 , ..., w i2 np } (5.62)
where w ij represent the polytopic vertices of P i ∀j ∈ I[1, 2 n p ]. ρ is then scheduled as:

∀ρ ∈ P i , ρ = 2 np ∑ j=1 α ij (ρ)w ij , i ∈ {1, . . . , N} (5.63)
and ∀i , ρ 2 ], with ρ 2 = 1/ρ 1 . The convex region P 0 is represented in solid orange, and the convex subregions P i (i ≥ 1) are bounded within dashed green polygons.

2 np ∑ j=1 α ij (ρ) = 1, α ij (ρ) ≥ 0 ∀i, j, being α ij (ρ) the
The LPV representation of G(ρ) is defined over P 0 as a convex combination of the state-space realizations of the local LTI systems G 0j at the vertices w 0j :

  A(ρ) B 1 (ρ) B 2 C 1 (ρ) D 11 (ρ) D 12 C 2 D 21 0   = 2 np ∑ j=1 α 0 j(ρ)   A 0j B 1,0j B 2 C 1,0j D 11,0j D 12 C 2 D 21 0   (5.64)
Notice that G(ρ) could be defined equivalently in terms of the LTI plants G ij (at w ij ) as, for ρ ∈ P i :

  A(ρ) B 1 (ρ) B 2 C 1 (ρ) D 11 (ρ) D 12 C 2 D 21 0   = 2 np ∑ j=1 α ij (ρ)   A ij B 1,ij B 2 C 1,ij D 11,ij D 12 C 2 D 21 0   (5.65)
Now, assume that

(A.3

.1).

There exists an LPV output-feedback controller K 0 (ρ) which quadratically stabilizes G(ρ) (using the standard polytopic approach in [START_REF] Apkarian | Self-scheduled H ∞ control of linear parameter-varying systems: a design example[END_REF]) over the full parameter region P 0 , defined as: Regarding the example in Fig. 5.8, the local LTI controllers at the intersecting boundaries w i3 (at K i3 ) and w (i+1)2 (at K (i+1)2 ), ∀i ∈ I[1, N -1], are designed similarly. Consequently, the switching between two successive subsets (P i and P i+1 ) undergoes using the same LTI controller (K i3 ≡ K (i+1)2 ). In such a case, the state and control input energies, just before and just after switching, are not affected.

K 0 (ρ) : A k,0 (ρ) B k,0 (ρ) C k,0 (ρ) D k,0 (ρ) (5.66) being, A k,0 (ρ) B k,0 (ρ) C k,0 (ρ) D k,0 (ρ) = 2 np ∑ j=1 α 0j (ρ) A k,0j B k,0j C k,0j D k,0j (5.67)
The objective of the work in this section is to:

1. Design multiple YK-based gain-scheduled controllers Ki (ρ), i ∈ I[1, N]. Each one is designed by interpolating its corresponding LTI controllers K ij (j ∈ I[1, 2 n p ]) based on YK concept (refer to Fig. 5.8).

2. Create an overall switched LPV-YK controller Kσ (ρ) to switch between Ki (ρ), such that Kσ (ρ) quadratically stabilizes G(ρ) ∀ρ ∈ P and for every continuous/discontinuous switching signal σ(t). The switched closed-loop system is represented as

CL σ (ρ) : Ãcl,σ (ρ) Bcl,σ (ρ) Ccl,σ (ρ) Dcl,σ (ρ) (5.68)
To illustrate the proposed approach, Fig. 5.8 represents an example of a convex parameter set partitioning. The dashed black curve represents the actual operating conditions of the parameters (ρ 2 = 1/ρ 1 ). The orange solid polygon represents the nominal LPV controller K 0 (ρ) designed using standard polytopic approach over the convex region P 0 . The blue points represent the local LTI controllers K ij (j ∈ I[1, 4], i ∈ I[1, N]) designed separately to stabilize G ij at w ij using any LTI control approaches. The green dashed polygons are the subsets chosen along the parameter trajectory. Over each subset P i , a gain-scheduled controller Ki (ρ) is designed based on the YK interpolation of the local LTI controllers K ij . The overall switching scheme between the Ki (ρ) is represented by the LPV-YK controller Kσ (ρ).

The LPV-YK controllers Ki (ρ) are designed based on a Linear Matrix Inequality (LMI) optimization problem, where they are defined as

Ki (ρ) : Ãk,i (ρ) Bk,i (ρ) Ck,i (ρ) Dk,i (ρ) (5.69)
The next theorem proves the quadratic stability of the closed-loop system CL σ (ρ) (5.68) for every ρ ∈ P. Then, the switched LPV-YK controller Kσ (ρ) (5.69)-(5.72), represented in Fig. 5.9, quadratically stabilizes G(ρ) for any ρ ∈ P and for any continuous/discontinuous switching signals

γ i ∈ [0, 1] (∀i ≥ 1), if there exist symmetric, positive definite, constant matrices X g ∈ R n x ×n x , X k,ij ∈ R n k,ij ×n k,ij , and matrices W j ∈ R m×n x and V ij ∈ R m k ×n k,ij such that: A 0j X g + X g A T 0j + B 2 W j + W T j B T 2 < 0 ∀w 0j (5.70) A k,0 (w ij )X k,ij + X k,ij A T k,0 (w ij ) + B k,0 (w ij )V ij + V T ij B T k,0 (w ij ) < 0 ∀w ij (5.71)
with the state-space matrices of Ki (ρ) are

Ãk,i (ρ) = 2 np ∑ j=1 α ij (ρ)   A ij + B 2 F g,ij -B 2 Dq,ij C 2 -B k,0 (w i,j )C 2 -Bq,ij C 2 -B 2 Dq,ij F k,ij B 2 Cq,ij A k,0 (w i,j ) 0 -Bq,ij F k,ij Āq,ij   Bk,i (ρ) = 2 np ∑ j=1 α ij (ρ) B 2 Dq,ij B k,0 (w i,j ) Bq,ij Ck,i (ρ) = 2 np ∑ j=1 α ij (ρ) F g,ij -(D k,ij + D q,ij )C 2 C k,0 (w i,j ) -D q,ij F k,ij Cq,ij Dk,i (ρ) = 2 np ∑ j=1 α ij (ρ)[D k,0 (w i,j ) + Dq,ij ] (5.72) ∀i ∈ I[1, N],
where

Āq,ij = Z ij A q,ij (Z ij ) -1 , Bq,ij = Z ij B q,ij
, and Cq,ij = C q,ij (Z ij ) -1 ∀i, j, and Z ij are state transformation matrices chosen, to satisfy Lemma 5, such that Āq,i (ρ) =

2 np ∑ j=1 α j (ρ)Z ij A q,ij (Z ij ) -1 is quadratically stable ∀i ≥ 1. A q,ij
, B q,ij , C q,ij and D q,ij are the state-space matrices of Q i (ρ) (5.74) at the polytopic vertices w ij . In addition, F g,j =

W j X -1 g , F k,ij = V ij (X k,ij ) -1 , and F g,ij = F g (w ij ).
Proof 3. As mentioned in the previous proofs, each parameterized controller can be formulated as a Linear Fractional Transformation (LFT) system [START_REF] Tay | High performance control[END_REF]. Each LPV-YK controller can be written as Ki (ρ) = F l (J(ρ), Qi (ρ)) ∀ρ ∈ P i (see Fig. 5.9), where J(ρ) is represented in (5.73), and Qi (ρ) is the transformed system of Q i (ρ) (5.74).

J(ρ) = 2 np ∑ j=1 α i j(ρ)     A ij + B 2 F g,ij 0 0 B 2 -B k,0 (w ij )C 2 A k,0 (w ij ) B k,0 (w ij ) 0 F g,ij -D k,0 (w ij )C 2 C k,0 (w ij ) D k,0 (w ij ) I -C 2 -F k,ij I 0     (5.73)
5.6. Switching between Parameter Subsets

101 Q i (ρ) = 2 np ∑ j=1 α ij (ρ)     A ij + B 2 D k,ij C 2 B 2 C k,ij B 2 [D k,ij - D k,0 (w ij )]F k,ij - B 2 C k,0 (w ij ) B k,ij C 2 A k,ij B k,ij F k,ij 0 0 A k,0 (w i ij) + B k,0 (w ij )F k,ij B 2 [D k,ij - D k,0 (w ij )] B k,ij B k,0 (w ij ) D k,ij C 2 - F g,ij C k,ij (D k,ij - D k,0 (w ij ))F k,ij - C k,0 (w ij ) D k,ij - D k,0 (w ij )     (5.74) Ācl,i (ρ) = 2 np ∑ j=1 α ij (ρ)TA cl,ij (γ)T -1 = 2 np ∑ j=1 α ij (ρ)     A ij + B 2 F g,ij B 2 Cq,ij -B 2 (F g,ij -(D k,0 (w ij ) + Dq,ij )C 2 ) B 2 (C k,0j (w ij ) -Dq,ij F k,ij ) 0 Āq,ij Bq,ij C 2 -Bq,ij F k,ij 0 0 A ij + B 2 D k,0 (w ij )C 2 B 2 C k,0 (w ij ) 0 0 B k,0 (w ij )C 2 A k,0 (w ij )     (5.75)
The following proof is achieved by two steps: 1) Prove that the LPV-YK parameter Qi (ρ) is quadratically stable ∀ρ ∈ {P i } i∈Z N ; and 2) Prove the closed-loop quadratic stability ∀ρ ∈ {P i } i∈Z N , ∀σ.

Step 1: [START_REF] Boudaoud | Gain scheduled control strategies for a nonlinear electrostatic microgripper: Design and real time implementation[END_REF]) is satisfied, it can be shown that the triangular elements of A q,ij ∀i, j are Hurwitz, and consequently, A q,ij is Hurwitz ∀i, j. According to Lemma 5, for any i ≥ 1, there exist transformation matrices Z ij such that the transformed system Qi (ρ) is quadratically stable ∀ρ ∈ P i , choose Z ij = (X q,ij ) 1/2 . As a result, Qi (ρ) is quadratically stable ∀ρ ∈ P i .

Knowing that: 1) K ij stabilizes G ij ∀i ≥ 1, ∀j ∈ [1, 2 n p ]; and 2) (5.
Step 2:

The switched closed-loop system CL σ (ρ) is derived from the LFT interconnection between G(ρ) and Kσ (ρ) (see Fig. 5.9).

The switched closed-loop state matrix A cl,σ (ρ) =

2 np ∑ j=1 α σj (ρ)A cl,σj is quadratically stable if:
1. There exist symmetric, positive definite, constant matrices X cl,i such that ∀i ≥ 1

X cl,i Ãcl,i (ρ) + ÃT cl,i (ρ)X cl,i < 0 ∀ρ ∈ P i (5.76) 2. ∀t k ∈ [0, T], V(x cl (t k )) ≤ V(x cl (t - k ))
Taking Due to its block-triangular form, (5.76) is then satisfied if the following equations hold (check Lemma 2 in [START_REF] Xie | Design of LPV control systems based on Youla parameterisation[END_REF]):

2 np ∑ j=1 α ij (ρ)(Y g,i (A ij + B 2 F g,ij ) + (A ij + B 2 F g,ij ) T Y g,i ) < 0 (5.77) 2 np ∑ j=1 α ij (ρ)(Y q,i Āq,ij + ĀT q,ij Y q,i ) < 0 (5.78) 2 np ∑ j=1 α ij (ρ)(Y i A ij + A T ij Y i ) < 0 (5.79)
where Y g,i ∈ R n x ×n x , Y q,i ∈ R n q ×n q and Y i ∈ R (n x +n k,0 )×(n x +n k,0 ) are symmetric, positive definite, constant matrices, choosing X cl,i = T T diag(Y g,i , Y q,i , Y i ) T, with

A ij = A ij + B 2 D k,0 (w ij )C 2 B 2 C k,0 (w ij ) B k,0 (w ij )C 2 A k,0 (w ij ) (5.80)
• Inequality (5.77) is equivalent to (5.70) by choosing Y g,i = X -1 g and W j = F g,ij X g for every ρ ∈ P i .

• Since Qi (ρ)) is proved to be quadratically stable ∀i, inequality (5.78) is satisfied.

• (5.79) is fulfilled given that K 0 (ρ) quadratically stabilizes G(ρ) over P 0 , and consequently over any P i ⊂ P 0 .

Therefore condition (5.76) is satisfied. Assume a sequence of finite switching time over the interval [0, T] is t 0 , t 1 , . . . , t n with t 0 = 0, knowing that the closed-loop Lyapunov function is defined as V(x cl ) = x T cl X cl,σ x cl . From the YK basic concept, the closed-loop is written as F l (G, J, Q). Consider a switching between any two adjacent subsets P i and P i+1 ∀i ≥ 1 at time t k , then, the closed-loop dynamics switches from

F l (G, J, Q i,3 ) to F l (G, J, Q i+1,2 ), or vise-versa. According to (A.3.2), K i,3 ≡ K i+1,2 ∀i ≥ 1, consequently Q i,3 ≡ Q i+1,2 . Then, for any switching time t k , F l (G, J, Q i,3 ) = F l (G, J, Q i+1,2 ), and thus V(x cl (t k )) = V(x cl (t - k )
). As a result, the switched closed-loop system CL σ (ρ) is quadratically stable ∀ρ ∈ P, and for any switching signal σ.

Grid-based LPV-YK Control

In this section, a switching between consecutive parameter partitions is attained with a closed-loop exponential stability. Recall that the grid-based LPV approach has shown interesting results for our experimental results shown in Chapter 5. However, to achieve robust closed-loop performance with large parameter variations, multiple grid-based LPV controllers can be designed each suitable for its corresponding parameter subset. Specifically, the nominal LPV controller K 0 (ρ) and the others K i (ρ) (i ∈ {1, . . . , N}) exponentially stabilize the LPV plant. This is performed without asking an affine parameter dependency assumption of the LPV model. Consider the general LPV system G(ρ) (5.17) with m inputs and p outputs. All the state-space data are continuous functions of the parameter vector ρ. Assume that ρ is in a compact set P ⊂ R s with its parameter variation rate bounded by ν k ≤ ρk ≤ ν k for k = 1, 2, ..., s. Moreover, let us assume the following:

• (A(ρ), B 2 (ρ), C 2 (ρ)) triple is parameter-dependent stabilizable and detectable ∀ρ ∈ P.

• B T 2 (ρ) D T 12 (ρ) and C 2 (ρ) D 21 (ρ) have full row ranks ∀ρ ∈ P.

• D 22 (ρ) = 0.
Suppose that the parameter set P is covered by a finite number of closed subsets {P i } i∈Z N , where the index set Z N = {1, 2, . . . , N}, and P = P i . At the boundaries between each adjacent subsets, there exist at least a single intersecting boundary or an intersecting surface. Now, assume that (A.4.1). There exists an LPV output-feedback controller K 0 (ρ) which exponentially stabilizes G(ρ) at the full parameter region P 0 := P. (following the grid-based approach in [START_REF] Wu | Induced L 2 Norm Control for LPV Systems with Bounded Parameter Variation Rates[END_REF]), (A.4.2). Over each parameter subset {P i } i∈Z N , there exists an LPV controller K i (ρ) predesigned separately and exponentially stabilize G(ρ) over {P i } i∈Z N . Each K i (ρ) is designed, based on grid-based approach, to achieve a suitable performance in its corresponding parameter region {P i } i∈Z N The defined LPV controllers K i (ρ) (i ≥ 0) are described over P i as

K i (ρ) : A k,i (ρ, ρ) B k,i (ρ) C k,i (ρ) D k,i (ρ) , i ∈ {0, Z N } (5.81)
where

A k,i (ρ, ρ) ∈ R n k,i ×n k,i , B k,i (ρ) ∈ R n k,i ×m k , C k,i (ρ) ∈ R p k ×n k,i and D k,i (ρ) ∈ R p k ×m k .
The closed-loop system performance is achieved in each parameter subregion and under given switching/interpolating logics via YK parameterization concept. The switching occurs when the parameter trajectory hits one of the subsets boundaries. A switching signal could be any continuous/discontinuous switching signal γ. The LPV closed-loop system of each pre-defined controller K i (ρ) over P i can be described by

CL i (ρ) : A cl,i (ρ, ρ) B cl,i (ρ) C cl,i (ρ) D cl,i (ρ) , i ∈ {0, Z N } (5.82)
where x T cl,i = x T x T k,i ∈ R n x +n k,i (i ≥ 0). For any i ∈ {0, Z N }, the closed-loop LPV system (5.82) is required to satisfy the bounded real lemma over P i with a performance level γ ∞,i , i.e. ∥z∥ 2 < γ ∞,i ∥w∥ 2 and there exists a symmetric, positive definite matrix functions X cl,i (ρ), each of them is smooth over the corresponding parameter subset P, such that

      A T cl,i (ρ)X cl,i (ρ) + X cl,i (ρ)A cl,i (ρ) + s ∑ k=1 ±{ν k , ν k } ∂X cl,i ∂ρ k X cl,i (ρ)B cl,i (ρ) C T cl,i (ρ) B T cl,i (ρ)X cl,i (ρ) -γ ∞,i I n w D T cl,i (ρ) C cl,i (ρ) D cl,i (ρ) -γ ∞,i I n z       < 0 (5.83)
The objective of grid-based LPV-YK control structure is to obtain exponential stability of the closed-loop system based on YK parameterisation. Moreover, a smooth interpolation scheme K(ρ, γ) is formulated between multiple pre-designed LPV controllers K i (ρ) (i ∈ Z N ), using the switching signal γ = [γ 1 , ..., γ i , ..., γ N ] , where each K i (ρ) is designed to be suitably used for a certain parameter subregion P i . This could be achieved by two steps:

1. Parameterize each LPV controller K i (ρ) with respect to the nominal LPV controller K 0 (ρ), by an LPV-YK parameter Q i (ρ).

2. At each boundary of two adjacent subsets P i and P i+1 , the interpolating signals γ i and γ i+1 are adjusted in a way to switch smoothly from K i (ρ) to K i+1 (ρ)

or vise-versa. As a result, the overall parameterized LPV-YK controller K(ρ, γ) stabilizes G(ρ) ∀ρ ∈ P and for every continuous/discontinuous interpolating signals γ i , i ∈ Z N .

In order to illustrate the approach, Figure 5.10 shows the partitioned parameter region P with intersecting boundaries. The orange solid line represents the chosen nominal LPV controller K 0 (ρ), as defined by (A.4.1). The blue solid lines represent the local LPV controllers K i (ρ) (i ∈ Z N ) as defined in (A.4.2). The overall switching controller is performed using the interpolating signal γ = [γ 1 , ..., γ N ] (γ i ∈ [0, 1] ∀i), and is represented by the LPV-YK controller K(ρ, γ). These LPV and LPV-YK controllers are designed in the next section.

The overall interpolation scheme which is referred to K(ρ, γ) (see Fig. 5.11) is designed based on a Parametric Linear Matrix Inequality (PLMI) optimization problem, where its state-space matrices are represented as

K(ρ, γ) : Ãk (ρ, γ) Bk (ρ, γ) Ck (ρ, γ) Dk (ρ, γ) (5.84)
being γ(ρ) the vector of the parameter-dependent switching signals γ i (ρ) (i ∈ Z N ) that are chosen here as follows:

• if γ i (ρ) = 0 ∀i, K(ρ, γ) ≡ K 0 (ρ). • For any ρ ∈ P m , γ m (ρ) = 1 and γ i (ρ) = 0 ∀i ̸ = m, which implies that K(ρ, γ) is equivalent to F l (J(ρ), Q m (ρ)) that recovers K m (ρ) (refer to Fig. 5.11).
The following theorem aims to prove that K(ρ, γ) exponentially stabilizes G(ρ) for every ρ ∈ P and for every continuous/discontinuous interpolating signals γ i .

Theorem 8. Consider an LPV plant G(ρ) (5.17), and that the assumptions (A.4.1) and (A.4.2) are satisfied. Let K 0 (ρ) be the nominal LPV controller designed over the full parameter region P 0 . Then, the parameterized LPV-YK controller K(ρ, γ) (5.84)-(5.87), as represented in Fig. 5.11, exponentially stabilizes G(ρ), with an achieved performance ∥z∥ 2 < γ ∞ ∥w∥ 2 , where γ ∞ = max{γ ∞,i } i∈Z N , for any continuous/discontinuous bounded switching signal γ i ∈ [0, 1], if there exist symmetric positive definite matrix functions X g (ρ) ∈ R n x ×n x , X k,0 (ρ) ∈ R n k,0 ×n k,0 , and matrices V(ρ) and W(ρ) such that for any ρ ∈ P:

A(ρ)X g (ρ) + X g (ρ)A T (ρ) + s ∑ j=1 ±{ν j , ν j } ∂X g ∂ρ j + B 2 (ρ)V(ρ) + V T (ρ)B T 2 (ρ) < 0 (5.85) A k,0 (ρ)X k,0 (ρ) + X k,0 (ρ)A T k,0 (ρ) + s ∑ j=1 ±{ν j , ν j } ∂X k,0 ∂ρ j + B k,0 (ρ)W(ρ) + W T (ρ)B T k,0 (
ρ) < 0 (5.86) And ∀ρ ∈ P, the state-space matrices of K(ρ, γ) in (5.84) are

Ãk (ρ, γ) =   A(ρ) + B 2 (ρ)F g (ρ) -B 2 (ρ)D q (ρ, γ)C 2 (ρ) -B 2 (ρ)D q (ρ, γ)F k,0 (ρ) -B k,0 (ρ)C 2 (ρ) A k,0 (ρ) -B q (ρ)C 2 (ρ) -B q (ρ)F k,0 (ρ) B 2 (ρ)C q (ρ, γ) 0 A q (ρ)   , FIGURE 5.11: Generalized YK configuration Bk (ρ, γ) = B 2 (ρ)D q (ρ, γ) B k,0 (ρ) B q (ρ) T , Ck (ρ, γ) = F g (ρ) -(D k,0 (ρ) + D q (ρ, γ))C 2 (ρ) C k,0 (ρ) -D q (ρ, γ)F k,0 (ρ) C q (ρ, γ) , Dk (ρ, γ) = D k,0 (ρ) + D q (ρ, γ).
(5.87) where A q (ρ) = diag(A q,1 , . . . , A q,i , . . . , A q,N ), B q (ρ) = B q,1 . . . B q,i . . . B q,N T ,

C q (ρ, γ) = γ 1 C q,1 . . . γ i C q,i . . . γ N C q,N , D q (ρ, γ) = N ∑ i=1 γ i D q,i , (5.88)
being A q,i , B q,i , C q,i , and D q,i the state-space matrices of Q i (ρ) represented in (5.91), F g (ρ) = V(ρ)X -1 g (ρ), and F k,0 (ρ) = W k,0 (ρ)X -1 k,0 (ρ). Proof 4. According to YK concept, the parameterized controller can be formulated as a Linear Fractional Transformation (LFT) system [START_REF] Tay | High performance control[END_REF], i.e. K(ρ, γ) = F l (J(ρ), Q(ρ, γ)) (see Fig. 5.11), where J(ρ) is defined as:

J(ρ) =     A(ρ) + B 2 (ρ)F g (ρ) 0 0 B 2 (ρ) -B k,0 (ρ)C 2 (ρ) A k,0 (ρ, ρ) B k,0 (ρ) 0 F g (ρ) -D k,0 (ρ)C 2 (ρ) C k,0 (ρ) D k,0 (ρ) I -C 2 (ρ) -F k,0 (ρ) I 0     (5.89) and Q(ρ, γ) = N ∑ i=1 γ i (ρ)Q i (ρ) (see Q i (ρ) in (5.91)). Specifically, a defined LPV controller K m (ρ) (m ∈ [1, N]) can be formulated as F l (J(ρ), Q m (ρ))), i.e. K m (ρ) ≡ F l (J(ρ), Q(ρ, γ)) for γ m = 1 and γ i = 0 ∀i ̸ = m.
The following proof is achieved by two steps: 1) Prove that the LPV-YK parameter Q(ρ, γ) is exponentially stable ∀ρ ∈ P and ∀γ; and 2) Prove the closed-loop exponential stability ∀ρ ∈ P, ∀γ.

Step 1:

Q i (ρ) can be proved to be exponentially stable ∀ρ ∈ {P i } i≥1 as follows: Following (A.4.2) and (5.86), there exist symmetric positive definite matrix functions P cl,i , and P k,0 such that ∀ρ ∈ {P

i } i≥1 A T cl,i (ρ)P cl,i (ρ) + P cl,i (ρ)A cl,i (ρ) + s ∑ k=1 ρk ∂P cl,i (ρ) ∂ρ k < 0 (5.90a) A F k (ρ)P k,0 (ρ) + P k,0 (ρ)A T F k (ρ) + s ∑ k=1 ρk ∂P k,0 (ρ) ∂ρ k < 0 (5.90b) where A F k (ρ) = A k,0 (ρ) + B k,0 (ρ)F k,0 (ρ).
(5.90a) can be deduced from (5.83) by choosing P cl,i (ρ) = X cl,i (ρ), and (5.90b) can be satisfied from (5.86) by choosing P k,0 (ρ) = X k,0 (ρ) and W k,0 (ρ) = F k,0 (ρ)X k,0 (ρ). Following Lemma 8, A q,i (ρ), consequently Q i (ρ), is exponentially stable over

P i ∀i ≥ 1. As a result, Q(ρ, γ) = N ∑ i=1 γ i Q i (ρ) is exponentially stable over P i = P for every bounded signals γ ∈ [0, 1].
Step 2:

The closed-loop system CL(ρ, γ) (5.92) is derived from the LFT interconnection between G(ρ) (5.17) and K(ρ, γ) (5.87) (see Fig. 5.11).

Chapter 5. Advanced LPV-YK Control Structures: Theoretical Proofs

Q i (ρ) =      A(ρ) + B 2 (ρ)D k,i (ρ)C 2 (ρ) B 2 (ρ)C k,i (ρ) B 2 (ρ)[D k,i (ρ) - D k,0 (ρ)]F k - B 2 (ρ)C k,0 (ρ) B k,i (ρ)C 2 (ρ) A k,i (ρ, ρ) B k,i (ρ)
F k,0 0 0 A k,0 (ρ) + B k,0 (ρ)F k,0 (ρ) B 2 (ρ)[D k,i (ρ) - D k,0 (ρ)] B k,i (ρ) B k,0 (ρ) D (1)
k,i C 2 - F g,i C k,i (ρ) [D k,i (ρ) - D k,0 (ρ)]F k,0 (ρ) - C k,0 (ρ) D k,i (ρ) - D k,0 (ρ)      (5.91) CL(ρ, γ) =       A(ρ) + B 2 (ρ)(D k,0 (ρ) + D q (ρ, γ))C 2 (ρ) B 2 (ρ)(F g (ρ) -(D k,0 (ρ) + D q (ρ, γ))C 2 (ρ)) B 2 (ρ)(C k,0 (ρ) - D q (ρ, γ)F k (ρ)) B 2 (ρ)D q (ρ, γ)C 2 (ρ) A(ρ) + B 2 (ρ)(F g (ρ) - D q (ρ, γ)C 2 (ρ)) -B 2 (ρ)D q (ρ, γ)F k (ρ) B k,0 (ρ)C 2 (ρ) -B k,0 (ρ)C 2 (ρ)
A k,0 (ρ, ρ) B q (ρ)C 2 (ρ) -B q (ρ)C 2 (ρ) -B q (ρ)F k (ρ) C 1 (ρ) + D 12 (ρ)(D k,0 (ρ) + D q (ρ, γ))C 2 (ρ) D 12 (ρ)(F g (ρ) -(D k,0 (ρ) + D q (ρ, γ))C 2 (ρ)) D 12 (ρ)(C k,0 (ρ) - D q (ρ, γ)F k (ρ)) B 2 (ρ)C q (ρ, γ) B 1 (ρ) + B 2 (ρ)(D k,0 (ρ) + D q (ρ, γ))D 21 (ρ) B 2 (ρ)C q (ρ, γ) B 2 (ρ)D q (ρ, γ)D 21 (ρ) 0 B k,0 (ρ)D 21 (ρ) A q (ρ, ρ) B q (ρ)D 21 (ρ) D 12 (ρ)C q (ρ, γ) D 11 (ρ) + D 12 (ρ)(D k,0 (ρ) + D q (ρ, γ))D 21 (ρ)       (5.92) Ācl (ρ, γ) = TA cl (ρ, γ)T -1 =     A(ρ) + B 2 (ρ)F g (ρ) B 2 (ρ)C q (ρ, γ) -B 2 (ρ)(F g (ρ) -(D k,0 (ρ) + D q (ρ, γ))C 2 (ρ)) B 2 (ρ)(C k,0 (ρ) - D q (ρ, γ)F k (ρ)) 0 A q (ρ, ρ) B q (ρ)C 2 (ρ) -B q (ρ)F k (ρ) 0 0 A(ρ) + B 2 (ρ)D k,0 (ρ)C 2 (ρ) B 2 (ρ)C k,0 (ρ) 0 0 B k,0 (ρ)C 2 (ρ)
A k,0 (ρ, ρ)     (5.93)
The closed-loop state matrix A cl (ρ, γ) is exponentially stable if ∀ρ ∈ P, ρ ∈ [ν, ν], there exists a symmetric, positive definite matrix function X cl (ρ) such that ∀γ:

A T cl (ρ, γ)X cl (ρ) + X cl (ρ)A cl (ρ, γ) + s ∑ k=1 ρk ∂X cl ∂ρ k < 0 (5.94) Let T =     I 0 0 0 0 0 0 I I -I 0 0 0 0 I 0   
 be a state transformation matrix which is applied to CL(ρ, γ)

without changing its input-output nature, with T -1 =     I 0 0 0 I 0 -I 0 0 0 0 I 0 I 0 0     .
Due to the block-triangular form of Ācl (ρ, γ) (5.93), (5.94) is satisfied if the following equations hold ∀ρ ∈ P (check Lemma 8):

Y g (ρ)(A(ρ) + B 2 (ρ)F g (ρ)) T + (A(ρ) + B 2 (ρ)F g (ρ))Y g (ρ) + s ∑ k=1 ρk ∂Y g ∂ρ k < 0 (5.95a) Y q (ρ)A T q (ρ) + A q (ρ)Y q (ρ) + s ∑ k=1 ρk ∂Y q ∂ρ k < 0 (5.95b) A T cl,0 (ρ)Y cl,0 (ρ) + Y cl,0 (ρ)A cl,0 (ρ) + s ∑ k=1 ρk ∂Y cl,0 ∂ρ k < 0 (5.95c)
where Y g (ρ) ∈ R n x ×n x , Y q (ρ) ∈ R n q ×n q and Y cl,0 (ρ) ∈ R (n x +n k,0 )×(n x +n k,0 ) are symmetric, positive definite matrix functions, choosing X cl (ρ) = T T diag(Y g , Y q , Y cl,0 ) T.

Therefore,

• Inequality (5.95a) can be deduced from (5.85) by choosing Y g (ρ) = X g (ρ) and considering that F g (ρ) = V(ρ)X -1 g (ρ) in (5.85).

• (5.95b) is satisfied since Q(ρ, γ) has been proved in Step 1 to be exponentially stable over P.

• (5.95c) is fulfilled given that K (0) (ρ) exponentially stabilizes G(ρ) according to (A.4.1). Now, assume a sequence of finite switching time over the interval [0, T] is t 0 , t 1 , . . . , t N with t 0 = 0, knowing that the closed-loop Lyapunov function is V(x cl ) = x T cl X cl (ρ)x cl . From the YK basic concept [START_REF] Tay | High performance control[END_REF], a parameterized controller K = F l (J, Q) recovers the performance of its actual controller K. Thus, the closed-loop performance of F l (G, K) is equivalent to the performance of F l (G, K). In the current work, it can be deduced that the closed-loop performance of CL(ρ, γ) (5.92) is equivalent to that of CL i (ρ) (5.82) within each parameter subset P i . Notice that according to (5.83), the following inequality describes the performance of CL

i (ρ) ∀ρ ∈ {P i } i∈Z N d dt (x T cl,i X cl,i (ρ)x cl,i ) + 1 γ ∞,i z T z -γ ∞,i w T w < 0 (5.96)
On the other hand, it is worth mentioning that X cl (ρ) is independent of the switching signal γ, so, for any switching time t k , V(x cl (t k )) = V(x cl (t - k )), and thus

V(x cl (t k )) ≤ V(x cl (t - k )) (5.97)
Chapter 5. Advanced LPV-YK Control Structures: Theoretical Proofs Given the initial condition x cl (0) = 0, from (5.96)-(5.97), it can be shown that the inequality

V(x cl ) + 1 γ ∞ z T z -γ ∞ w T w < 0, γ ∞ = max{γ ∞,i } i∈Z N (5.98)
holds within each parameter subset. Integrate on both sides, we get

V(x cl (T)) -V(x cl (0)) + 1 γ ∞ ∥z∥ 2 2 -γ ∞ ∥w∥ 2 2 < 0 Since V(x cl (T)) ≥ 0 and V(x cl (0)) = 0, ∥z∥ 2 < γ ∞ ∥w∥ 2 is achieved.
Therefore, the closed-loop is exponentially stable with an achieved performance ∥z∥ 2 < γ ∞ ∥w∥ 2 , where γ ∞ = max{γ ∞,i } i∈Z N .

Discussion

This chapter has presented four LPV-YK control structures:

1. Interpolation of polytopic-based LPV controllers: It is used to interpolate between already designed LPV controllers, knowing that all the interpolated LPV controllers should quadratically stabilize the LPV model for any parameter variations.

Interpolation of YK-based LPV controllers:

This structure uses the LTI-YK parameterization to first design multiple quadratically stabilizing LPV controllers. Then interpolates between these YK-based LPV controllers providing a whole closed-loop quadratic stability.

Switching of partitioned YK-based LPV controllers:

It is designed aiming to solve the conservatism of the quadratic stability problem due to overbounding (in convex regions), large number of varying parameters, and their large range of variations. A YK-based LPV controller is designed in each parameter subset, and the switching appears at the intersection boundary between each two consecutive subsets.

Switching of grid-based LPV controllers:

This structure is formulated to switch between multiple grid-based LPV controllers, each is designed to be useful over its subset. Each parameter subset is gridded and the local LPV controllers are required to be exponentially stabilizing.

Notice that one can combine multiple parameter subsets with multiple closedloop specifications in one structure. Such a structure can be the most generalized control scheme including parameter-variations, robustness, and different control objectives. For instance, one can partition P into N parameter subsets, and design ζ YK-based LPV controllers in each subset P i . As a result, the LPV-YK controller K(ρ, γ) switches and interpolates between (

N × ζ) LPV controllers K(j) i (ρ), i ∈ I[1, N], j ∈ I[1, ζ].

Chapter 6

Application of the LPV-YK Structures on Vehicle Lateral Control 6.1 Introduction

The main thesis motivation behind considering an interpolation scheme between multiple LPV controllers is the application to autonomous vehicles. Indeed, several studies have involved the LPV control approaches to solve the lateral tracking problem over the full speed-range (speed as the varying parameter) [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF]. However, it is not sufficient to achieve different tracking performances (e.g. smooth and aggressive). On the other hand, [START_REF] Mahtout | Youla-Kucera Based Lateral Controller for Autonomous Vehicle[END_REF] has proposed an LTI-YK control structure for both lane-tracking and lane-changing control performances, but works only for a constant speed.

Indeed, the driving capabilities have been recently improved for highly, and even fully, autonomous driving thanks to advanced control theory. A fully autonomous car needs to perform several tasks including longitudinal control, lateral control, chassis control, etc. Moreover, the lateral dynamics of an autonomous vehicle varies significantly with respect to its longitudinal speed [START_REF] Pacejka | Tire and vehicle dynamics[END_REF], [START_REF] Rajamani | Vehicle dynamics and control[END_REF]. Specifically, at low speeds, the lateral dynamics becomes harder to be controlled (due to approaching system singularity), whereas at high speeds, robustness and system stability decrease [START_REF] Mariano | Nested PID steering control for lane keeping in autonomous vehicles[END_REF]. Even at nominal speeds, the lateral control aims to achieve various objectives such as lane tracking, lane changing, obstacle avoidance, etc. Consequently, various performances are required accordingly for different driving situations that are facing the vehicle.

The following chapter uses the LPV-YK control structures, presented in the previous chapter, to improve the vehicle tracking performance over a large speed range. First, the implementation architecture with its designing steps are shown, then, simulation and experimental results are presented to validate the efficiency of the proposed LPV-YK control structures.

The simulation and experimental results shown in this chapter have been presented in:

• Multi-Variable and Multi-Objective Gain Scheduled Control Based on Youla-Kucera Parameterization: Application to Autonomous Vehicles, under review in International Journal on Robust and Nonlinear Control.

• Advanced LPV-YK Control Design with Experimental Validation on Autonomous Vehicles, under review in Automatica. This chapter is organized as follows: Section 6.2 states the formulated LPV models for the lateral bicycle model. In Section 6.3, the generalized LPV model including the weighting functions is introduced. Section 6.4 defines LPV/H ∞ control design problem to be used later. The general LPV-YK control scheme is defined in Section 6.5. Section 6.6 presents the implementation of the Interpolation of polytopic-based LPV controllers. The interpolation of YK-based LPV controllers is depicted in Section 6.7. The partitioned LPV-YK controller is presented in Section 6.8. Section 6.9 shows the implementation of the grid-based LPV-YK controller. Finally, a summary and conclusion is given in Section 6.10.

LPV Model Formulation

The following section describes the LPV model that is used for each of the LPV-YK control structures, considering both the general and affine parameter-dependency.

Lateral Bicycle Model

Recall that the lateral bicycle model has been used to control the lateral motion of our automated car. Notice that the chosen bicycle model is one type of Pacejkas' models only [START_REF] Pacejka | Tire and vehicle dynamics[END_REF]. As previously mentioned in Chapter 3, the bicycle model is represented as shown in Fig. 6.1, with vy =

F y f cos δ+F yr m -wv x ẇ = F y f l f cos δ-F yr l r I (6.1)
F y f and F yr can be modelled using Pacejka's tire model [START_REF] Pacejka | Tire and vehicle dynamics[END_REF] as follows:

F y f = c 3 sin(c 2 tan -1 (c 1 α f )) α f = δ -tan -1 ( v y v x - l f w v x ) F yr = c 3 sin(c 2 tan -1 (c 1 α r )) α r = -tan -1 ( v y v x -l r w v x ) (6.2)
where c 1 , c 2 and c 3 are constants.

Generalized LPV Model

113

Assuming small α (refer to Section 3.2.8), the tire forces are linearized resulting the following:

F y f = C f (δ - v y v x - l f w v x ) F yr = C r (- v y v x + l r w v x ) (6.3)
where C f and C r represent the tires' stiffness of the front and rear wheels.

LPV Bicycle Model Structures

Referring to Chapter 4, taking ρ(t) = v x ∈ R n ρ as a varying parameter, and assuming small steering front angles (sin(δ) ≈ δ), and small slip angles, the LPV state-space representation can be written as:

Σ(ρ) ẋ(t) = A Σ (ρ)x(t) + B Σ u(t) y(t) = C Σ x(t) + D Σ u(t) (6.4)
where:

x(t) = v y w ∈ R n x , u(t) = δ, , y(t) = w, B Σ = 1 m C f 1 I C f l f , A Σ (ρ) =   - C r +C f mρ - l f C f -l r C r mρ -ρ - l f C f -l r C r Iρ - l 2 f C f +l 2 r C r Iρ   . (6.5)

Grid-based Model

The grid-based LPV model is a model interpolated over a set of LTI models that are linearized at different operating points. This approach considers the general parameter dependency of the model (without model reformulation). Thus, the model in (6.4)-(6.5) can be used for the grid-based LPV-YK control structure. Refer to Section 4.2.2 for more details.

Polytopic Model

Since the polytopic model must be affine with respect to the varying parameters. So, the state matrix in (6.5) is reformulated as:

A Σ (ρ) = - C r +C f m ρ 2 - C f l f -C r l r m ρ 2 -ρ 1 - C f l f -l r C r I ρ 2 - C f l 2 f +l 2 r C r I ρ 2 , ( 6.6)
where

ρ = [ρ 1 , ρ 2 ] = [v x , 1 v x ].
Refer to Section 4.2.2 for more details.

Generalized LPV Model

This section shows the components of the generalized LPV model G(ρ), that is used later for control design.

Actuator model

The actuator model is included to achieve a more realistic model. Thus, the bicycle model is extended by considering an identified steering actuator dynamics Σ act . Recall from Chapter 3 that the steering actuator model is identified at the nominal longitudinal speed v x = 12 m/s as an LTI model. A second order transfer function has been obtained in the following form:

Σ act = k s 2 + 2ζw n s + w 2 n e -T d s (6.7)
where k, ζ, w n , and T d are the static gain, the damping, the natural frequency and the time delay, respectively. The time delay term-e -T d s is represented as:

e -T d s = 1 -T d s 2 + (T d s) 2 12 1 + T d s 2 + (T d s) 2 12
(6.8)

Weighting functions

In this thesis, the LPV and LTI controllers are designed using the robust H ∞ concept. For control design purpose, two weighting transfer functions W e (s) and W u (s) are designed to present the closed-loop tracking performance and the actuator limitations respectively. Then, the state-space representation of G(ρ) is obtained from the generalized plant shown in Fig. 6.2. Having a second order Σ(ρ), Σ act of order 4, W e (s) and W u (s) of first order each, then, the obtained generalized LPV model G(ρ) is of order 8.

LPV/H ∞ Control Design

In this chapter, all the LPV lateral controllers K(ρ) are designed using the LPV/H ∞ control design approaches, either polytopic or grid-based. The objective of the lateral control is to track a given yaw-rate reference r = w re f with rejecting the measurement noises n. Fig 6 .3 presents the control design scheme where w r = [w re f n] T is a vector of the yaw-rate reference w re f and measurement noises n, δ c is the steering control input, and w is the measured yaw-rate of the vehicle. z = [z 1 z 2 ] T is the controlled output vector containing the required yaw-rate tracking performance z 1 = W e × (w re fw) and steering actuator limitations

z 2 = W u × δ c .
The objective is to minimize the L 2 induced gain from the external input w r to the controlled output z = z 1 z 2 T . This is achieved by solving the following L 2 induced minimization problem: and γ ∞ > 0, to be minimized. Note that the obtained controller K(ρ) is an LPV Dynamic Output Feedback Controller.

∥z∥ 2 ≤ γ ∞ ∥w r ∥ 2

General LPV-YK Control Scheme

In this section, a general implementation scheme of an LPV-YK controller K(ρ, γ) is defined as shown in Fig. 6.4. Note that the LFT representation of the generalized LPV-YK controller in Fig. 5.11 can be equivalently reformulated as in Fig. 6.4. This scheme is considered to be general among all the LPV-YK control structures that are discussed in the previous chapter. Among these structures, the definitions of M(ρ), N(ρ), U 0 (ρ), and V -1 0 (ρ) are similar. The only difference between a structure and another is the LPV-YK parameter Q(ρ, γ). The implementation scheme of the LPV-YK control is formulated using the doubly coprime factorization [START_REF] Vidyasagar | Control system synthesis: a factorization approach, part II[END_REF].

LPV Coprime Factorization

LPV-YK parameterisation uses the LPV coprime factorisation concept to reduce the algebraic complexity of Q computation. Consider the LPV plant G(ρ) stabilized by a set of LPV controllers K i (ρ), then they can be factorized (from left and right) as a product of a stable transfer function matrix and a transfer function matrix with a stable inverse as shown below:

G(ρ) = N(ρ)M -1 (ρ) = M-1 (ρ) Ñ(ρ) K i (ρ) = U i (ρ)V -1 i (ρ) = Ṽ-1 i (ρ) Ũi (ρ), ∀i (6.9)
where the LPV coprime factors are computed such that M(ρ), N(ρ), M(ρ), Ñ(ρ), U i (ρ), V i (ρ), Ũi (ρ), Ṽi (ρ) are stable and proper, and satisfying the following Bezout Identity:

Ṽi (ρ) -Ũi (ρ) -Ñ(ρ) M(ρ) M(ρ) U i (ρ) N(ρ) V i (ρ) = M(ρ) U i (ρ) N(ρ) V i (ρ) Ṽi (ρ) -Ũi (ρ) -Ñ(ρ) M(ρ) = I 0 0 I (6.10)
The LPV coprime factors are computed at each vertex using the state-space representations written in (6.11)-(6.12).

M(ρ) U i (ρ) N(ρ) V i (ρ) :     A(ρ) + B 2 (ρ)F g (ρ) 0 B 2 (ρ) 0 0 A k,i (ρ) + B k,i (ρ)F k,i (ρ) 0 B k,i (ρ) F g (ρ) C k,i (ρ) + D k,i (ρ)F k,i (ρ) I D k,i (ρ) C 2 (ρ) F k,i (ρ) 0 I     (6.11) Ṽi (ρ) -Ũi (ρ) -Ñ(ρ) M(ρ) :     A(ρ) + B 2 (ρ)D k,i (ρ)C 2 (ρ) B 2 (ρ)C k,i (ρ) -B 2 (ρ) B 2 (ρ)D k,i (ρ) B k,i (ρ)C 2 (ρ) A k,i (ρ) 0 B k,i (ρ) F g,i (ρ) -D k,i (ρ)C 2 (ρ) -C k,i (ρ) I -D k,i (ρ) C 2 (ρ) -F k,i (ρ) 0 I    
(6.12)

LPV-YK Control Structure

Choosing K 0 (ρ) = U 0 (ρ)V -1 0 (ρ) = Ṽ-1 0 (ρ) Ũ0 (ρ) (or K (0) (ρ) as notated in some structures) as the nominal LPV controller, the LPV-YK controller K(ρ, γ) in Fig. 6.4 is then expressed as:

K(ρ, γ) = U 0 (ρ) + M(ρ) Q(ρ, γ) V 0 (ρ) + N(ρ) Q(ρ, γ) -1 = Ṽ0 (ρ) + Q(ρ, γ) Ñ(ρ) -1 Ũ0 (ρ) + Q(ρ, γ) M(ρ) (6.13)
The next sections define the LPV coprime factors M(ρ), N(ρ), U 0 (ρ), V 0 (ρ), and the LPV-YK parameter Q(ρ, γ), for each of the discussed LPV-YK control structures.

Interpolation of polytopic-based LPV controllers

This section presents the steps done to design an LPV-YK controller aiming to interpolate between polytopic-based LPV controllers, as has been shown in Fig. 5.3. Simulation results are also presented after the design steps. The objective of this method is to:

1. Design three LPV controllers K (j) (ρ), j ∈ I[0, 2]. Each one is designed based on the standard polytopic approach.

2. Create an overall interpolation scheme between the gain-scheduled controllers K (j) (ρ) (j ∈ I[0, 2]), by an interpolating signal vector γ, which is referred to as K(ρ, γ), such that the resultant LPV-YK controller K(ρ, γ) quadratically stabilizes G(ρ) ∀ρ ∈ P and for every γ = [γ 1 , γ 2 ].

Lateral Control Design

Here, the longitudinal speed is bounded in [START_REF] Dj Leith | On formulating nonlinear dynamics in LPV form[END_REF][START_REF] Tuan | Relaxations of parameterized LMIs with control applications[END_REF]m/s. Indeed it has been observed that considering speeds lower than 3m/s, the performance of lateral controllers deteriorates significantly (due to the conservatism of the polytopic approach). G(ρ) is written as a convex combination of the vertices of the triangular polytope

P = C O {(ρ 1 , ρ 2 ), (ρ 1 , ρ 2 ), (ρ 1 , ρ 2 )} = C O {(3, 1 30
), (3,

), (30,

)} (6.14) with the state-space representation as described in (6.4)- (6.6). Notice that the fourth vertex has been removed to decrease the problem conservatism. Referring to Fig. 5.3, assumptions (A.1.1)-(A.1.2), and Theorem 5 in Section 5.5.1, a highly robust LPV controller K (0) (ρ), and two gain-scheduled controllers K (1) (ρ) and K (2) (ρ) are designed to perform different required performances. The three controllers are designed using the standard polytopic-based LPV/H ∞ concept following a method similar to [START_REF] Apkarian | Self-scheduled H ∞ control of linear parameter-varying systems: a design example[END_REF], having the weighting functions as follows:

• K (0) (ρ): A slow transient response and noise rejection performances are required using the weighting functions W

e and W

u :

W (0) e (s) = s+2 2s+0.002 , W (0)
u (s) = s+5 0.01s+5 (6.15) The following closed-loop step responses are obtained: • K (1) (ρ): needs to perform smooth lateral transitions which is important to provide comfort riding. This is achieved using the following weighting functions W

e and W

u : W

e (s) = s+2 2s+0.002 , W

u (s) = s+10 0.01s+10 (6.16) The following closed-loop step responses are obtained:

• K (2) (ρ): to perform fast lateral transitions to handle the vehicle when facing aggressive maneuvers and lateral oscillations. The chosen weighting functions W

e and W

u are:

W (2) e (s) = s+20 2s+0.02 , W (2)
u (s) = 10s+25 0.1s+500 (6.17)

The following closed-loop step responses are obtained:

Design the LPV-YK Control Structure

The following steps are done to design the LPV-YK control:

1. According to the method explained in Section 5.5.1 and Theorem 5, the LPV polytopic-based state-feedback controllers F g (ρ) and F

k (ρ) are designed using the LMIs (5.26)- (5.27). After finding the decision variables W(ρ) and X g from (5.26), and V(ρ) and X k from (5.27), F g (ρ) and F (0) k (ρ) are computed as

F g (ρ) = W(ρ)X -1 g , F (0) k (ρ) = V(ρ)X -1 k (6.18)
2. The state-space representations of M(ρ), N(ρ), U (0) (ρ) and V (0) (ρ) are computed as illustrated in Section 6.5.1. Q(ρ, γ) is formulated as:

Q(ρ, γ) = 2 ∑ j=1 γ j Q (j) (ρ) (6.19)
where Q (j) (ρ) are obtained from (5.31), ∀j ∈ {1, 2}.

3. Here, the interpolating signal γ(t) is a vector of dimension two [γ 1 (t), γ 2 (t)],

where each γ j multiplies its corresponding Q (j) (ρ). In this example, and based on an experimental experience, γ 2 (t) is chosen to vary according to the demanded control tasks. Linear relations are proposed between different variables as follows:

• if θ e ≤ 0.1, γ 2 (t) = sat(-y L + 1.4 + 0.1 δ, [0, 1]) • if θ e > 0.1, γ 2 (t) = sat(-0.7y e + 1.4, [0, 1]) • γ 1 (t) = 1 -γ 2 (t)
where θ e and y e represent the heading and lateral errors between the vehicle and the current point on the reference path, respectively. δ is the steering speed, and y L is the lateral error at a look-ahead distance L [START_REF] Kosecka | Vision-based lateral control of vehicles[END_REF].

More illustration about the choice of the interpolating signals γ j is shown in the next chapter.

Simulation Results

The parameterized LPV-YK controller K(ρ, γ) is simulated on a nonlinear full car model designed for the Renault ZOE vehicle. The simulation is done using a part of a real trajectory map (Satory) (see Fig. 3.6) where its coordinates are obtained from a pre-recorded map using the positioning system mounted on the robotized ZOE car. The simulation in performed with MATLAB/Simulink with fixed sampling-time= 0.01s using the scheme shown in Fig. 6.8. The real vehicle speed from this recording is multiplied by 1.7 gain and used as a speed profile reference, to test the controllers within critical high speeds.

A scenario is chosen to cover several lateral tasks and critical situations as follows:

1. The vehicle is required to start its Autonomous Driving (AD) with a large lateral error (y e > 2 m). 3. When t ∈ [START_REF] Helmersson | Methods for robust gain scheduling[END_REF][START_REF] Lu | LPV control design and experimental implementation for a magnetic bearing system[END_REF]s, the vehicle is subjected to some sensor noises (due to real measurements) on a straight stretch.

4. An obstacle is detected when t ∈ [START_REF] Lu | LPV control design and experimental implementation for a magnetic bearing system[END_REF][START_REF] Paraiso | Black-box versus grey-box LPV identification to control a mechanical system[END_REF]s, the navigation modifies the trajectory suddenly as two successive lateral steps (each one of 3 m), aiming to avoid a collision.

The scenario is tested with different controllers:

• The nominal highly robust controller: K(ρ, [0, 0]) ≡ K (0) (ρ),

• The smooth tracker: K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0]) ≡ K (1) (ρ),

• The aggressive tracker: K(ρ, [0, 1]) ≡ K (2) (ρ),

• The proposed LPV-YK controller: K(ρ, γ) with γ = [γ 1 , γ 2 ] varies in real-time Fig. 6.9 shows the speed profile for all the tests. Fig. 6.10 shows the (X-Y) coordinates of the reference trajectory and the vehicle positioning response of the different tested controllers. Fig. 6.11a depicts the lateral error from the reference trajectory to the vehicle Center of Gravity (CoG), and the steering control input is shown in Fig. 6.11b. Fig. 6.11c represents the steering speed which reflects the driving comfort. Fig. 6.12 shows the evolution of the interpolating signal γ for K(ρ, γ).

The results are studied along time evolution as follows:

• At t = 0s, it is shown in Fig. 6.11a that only some of the tested controllers are activated to start with high lateral error (y e > 2.5m), since both controllers K (0) (ρ) and K(ρ, [0, 1]) can't deal with large lateral errors (observed from pretesting). On the other hand, K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0]) and K(ρ, γ) perform better with the initial high lateral error, where K(ρ, γ) shows a lower overshoot (see Fig. 6.11a when t ∈ [2, 5]s) with a smoother steering action (see Fig. 6.11b when t ∈ [2, 4]s). • When t ∈ [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF][START_REF] Helmersson | Methods for robust gain scheduling[END_REF]s, during the four successive turns, the four tested controllers have almost similar tracking performance (check Fig. 6.11a), where there exists low steering noises using the fast controller K(ρ, [0, 1]) as shown in Fig. 6.11c.

• When t ∈ [START_REF] Scherer | LPV control and full block multipliers[END_REF][START_REF] Dettori | Robust stability analysis for parameter dependent systems using full block S-procedure[END_REF]s, on the straight highway, Fig. 6.11a shows that the slow controllers (K (0) (ρ), K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0])) have lateral oscillations due to actuator limitations (for smoothness). On the other hand, the fast controller K(ρ, [0, 1]) and K(ρ, γ) could handle the vehicle. This is achieved due to the use of the steering speed in the proposed equations of γ 2 , γ changes to [0.3, 0.7].

• Finally, when t ∈ [52, 62]s, Fig. 6.10 shows that only K(ρ, [0, 1]) could not perform a fast double lane-change to overcome an obstacle (represented as "x"). Although the nominal and the smooth controllers K (0) (ρ) and K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0]) could succeed in performing it, however, the vehicle performs high lateral oscillations. On the other hand, K(ρ, γ) shows better performance without any lateral overshoots (see Fig. 6.11a) and with a very smooth and optimized steering action (see Fig. 6.11b and 6.11c when t ∈ [52, 62]s).

Thanks to the variations of γ which reflects the needed lateral task to the proposed parameterized controller K(ρ, γ), obtaining a combination between the designed performances.

The next section introduces the improvement obtained by interpolating between YK-based LPV controllers instead of polytopic-based.

Interpolation of YK-based LPV controllers

In this section, a more advanced interpolation scheme is introduced, where the interpolation LPV controllers are designed based on YK parameterization as has been discussed in Section 5.5.2. The objective of this method is to:

1. Design two of gain-scheduled controllers K (j) (ρ), j ∈ I [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF][START_REF] Safonov | Origins of Robust Control: Early History and Future Speculations[END_REF]. Each one is designed by interpolating its corresponding LTI controllers K (j) i (i ∈ I[1, 2 n p ]) based on YK parameterization. The nominal controller K (0) (ρ) is design based on the standard polytopic LPV approach. 6.7. Interpolation of YK-based LPV controllers 123 2. Create an overall interpolation scheme between the gain-scheduled controllers K (j) (ρ) (j ∈ I[0, 2]), by an interpolating signal vector γ, which is referred to as K(ρ, γ), such that the resultant LPV-YK controller K(ρ, γ) quadratically stabilizes G(ρ) ∀ρ ∈ P and for every γ = [γ 1 , γ 2 ].

Lateral Control Design

The first improvement that can be mentioned about using YK-based LPV controllers instead of polytopic-based, is that the control design conservatism can be decreased. Consequently, in this section, the longitudinal speed can be considered in [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF][START_REF] Tuan | Relaxations of parameterized LMIs with control applications[END_REF]m/s (while it has been in [START_REF] Dj Leith | On formulating nonlinear dynamics in LPV form[END_REF][START_REF] Tuan | Relaxations of parameterized LMIs with control applications[END_REF]m/s in Section 6.7). G(ρ) is written as a convex combination of the vertices of the polytope

P = C O {(ρ 1 , ρ 2 ), (ρ 1 , ρ 2 ), (ρ 1 , ρ 2 )} = C O {(1, 1 30
), (1,

), (30,

)} (6.20)

with the state-space representation as is described in (6.4)-(6.6). Referring to Fig. 5.6, assumptions (A.2.1)-(A.2.2) and Theorem 6 in Section 5.5.2, a highly robust LPV controller K (0) (ρ), and two gain-scheduled controllers K (1) (ρ) and K (2) (ρ) are designed to perform different required performances as follows:

• The polytopic-based LPV controller K (0) (ρ) is designed using the standard polytopic approach. A slow transient response and noise rejection performances are required for the nominal controller K (0) (ρ) using weighting functions W

e and W

u as follows:

W (0) e (s) = s+2 2s+0.002 , W (0)
u (s) = s+5 0.01s+5 (6.21)
The closed-loop step responses are obtained as shown in Fig. 6.13. Amplitude FIGURE 6.13: Closed-loop step response with K (0) (ρ)

• LTI controllers K (1)
i are designed separately using LTI/H ∞ concept at each vertex w i to perform smooth lateral transitions which is important to provide comfort riding. This is achieved using the following weighting functions W

e (s) = s+2 2s+0.002 , W

u (s) = s+10 0.01s+10 (6.22) The closed-loop step responses are obtained as shown in Fig. 6.14. (1)

1 , K (1)
2 , and K

3 at the polytopic vertices

• LTI controllers K (2)
i are designed separately using LTI/H ∞ concept at each vertex w i to perform fast lateral transitions to handle the vehicle when facing aggressive maneuvers and lateral oscillations. The chosen weighting functions are W

(2) e and W

(2) u ∀i as:

W (2) e (s) = s+20 2s+0.02 , W (2)
u (s) = 10s+25 0.1s+500 (6.23)
The closed-loop step responses are obtained as shown in Fig. 6.15. (2)

1 , K (2)
2 , and K

(2)

at the polytopic vertices

It is worth noting the difference in closed-loop step responses between the ones in Fig. 6.6 and Fig. 6.14, and Fig. 6.7 and Fig. 6.15. The difference of using polytopicbased control design and LTI control design is clearly observed at each polytopic vertex.

Design the LPV-YK Control Structure

The following steps are done to design the LPV-YK control shown in Fig. 6 {1, 2, 3}) and X g from (5.45), and V i and X k,i (i ∈ {1, 2, 3}) from (5.46). Then ,

F g (ρ) and F (0)
k (ρ) are computed as

F g (ρ) = 3 ∑ i=1 W i X -1 g , F (0)
k (ρ) = 3 ∑ i=1 V i X -1 k,i (6.24)
2. The state-space representations of M(ρ), N(ρ), U (0) (ρ) and V (0) (ρ) are computed as illustrated in Section 6.5.1. Q(ρ, γ) is formulated as:

Q(ρ, γ) = 2 ∑ j=1 γ j Q (j) (ρ) (6.25)
where Q (j) (ρ) are obtained from (5.54), ∀j ∈ {1, 2}.

3. Here, the interpolating signal γ(t) = [γ 1 (t), γ 2 (t)] is chosen as in the previous section as follows:

• if θ e ≤ 0.1, γ 2 (t) = sat(-y L + 1.4 + 0.1 δ, [0, 1]) • if θ e > 0.1, γ 2 (t) = sat(-0.7y e + 1.4, [0, 1]) • γ 1 (t) = 1 -γ 2 (t)
The following two sub-sections introduce simulation and experimental results of the designed LPV-YK control structure.

Simulation Results

The parameterized LPV-YK controller K(ρ, γ) is simulated on a nonlinear full car model designed for a Renault ZOE vehicle. The performed simulation is similar to the one in the previous section, using the scheme shown in Fig. 6.8. The scenario, the test track, the longitudinal speed profile are exactly the same used in the previous section.

The scenario is tested with different controllers:

• The nominal highly robust controller: K(ρ, [0, 0]) ≡ K (0) (ρ),

• The smooth tracker: K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0]) ≡ K (1) (ρ),

• The aggressive tracker: K(ρ, [0, 1]) ≡ K (2) (ρ), • The proposed LPV-YK controller: K(ρ, γ) with γ = [γ 1 , γ 2 ] varies in real-time Fig. 6.16 shows the speed profile for all the tests. Fig. 6.17a depicts the lateral error from the reference trajectory to the vehicle Center of Gravity (CoG), and the steering control input is shown in Fig. 6.17b. Fig. 6.17c represents the steering speed which reflects the driving comfort. Fig. 6.18 shows the evolution of the interpolating signal γ for K(ρ, γ), and Fig. 6. [START_REF] Shamma | Gain scheduling: potential hazards and possible remedies[END_REF] shows the (X-Y) coordinates of the reference trajectory and the vehicle positioning response of the different tested controllers.

• At t = 0s, it is shown in Fig. 6.17a that only some of the tested controllers are activated to start at high lateral error (y e > 2.5m), since both controllers K (0) (ρ) and K(ρ, [0, 1]) can't deal with large lateral errors (observed from pre-testing).

On the other hand, K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0]) and K(ρ, γ) perform better with the initial high lateral error, where K(ρ, γ) shows a lower overshoot (see Fig. 6.17a when t ∈ [2, 5]s) with a smoother steering action (see Fig. 6.17b when t ∈ [2, 4]s).

• When t ∈ [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF][START_REF] Helmersson | Methods for robust gain scheduling[END_REF]s, during the four successive turns, the four tested controllers have almost similar tracking performance (check Fig. 6.17a), where there exists low steering noises using the fast controller K(ρ, [0, 1]) as shown in Fig. 6.17c.

• When t ∈ [START_REF] Scherer | LPV control and full block multipliers[END_REF][START_REF] Dettori | Robust stability analysis for parameter dependent systems using full block S-procedure[END_REF]s, on the straight highway, Fig. 6.17a shows that the slow controllers (K (0) (ρ), K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0])) have lateral oscillations due to actuator limitations (for smoothness). On the other hand, the fast controller K(ρ, [0, 1]) could handle the situation, but demanding more steering effort with noises (see Fig. 6.17c when t ∈ [42, 48]s). However, due to the use of the steering speed in the proposed equations of γ 2 , γ changes to [0.3, 0.7], and K(ρ, γ) achieves a perfect trade-off between decreasing the lateral oscillations (in Fig. 6.17a) and relaxing the steering action (in Fig. 6.17b and 6.17c).

• Finally, when t ∈ [52, 62]s, it is clear in Fig. 6.19 that both K (0) (ρ) and K(ρ, [0, 1]) could not perform a fast double lane-change to overcome an obstacle (represented as "x"). Although the smooth controller K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0]) could succeed in performing it, however, the vehicle performs high lateral oscillations. On the other hand, K(ρ, γ) shows better performance without any lateral overshoots (see Fig. 6.17a) and with a very smooth and optimized steering action (see Fig. 6.17b and 6.17c when t ∈ [52, 62]s).

Thanks to the variations of γ which reflects the needed lateral task to the proposed parameterized controller K(ρ, γ), obtaining a combination between the designed performances. Table 6.1 summarizes the conclusions behind the simulation results that are considered similar between the YK-based and polytopic-based interpolation, with much performance improvement using the YK-based interpolation.

Experimental Results

The LPV-YK controller K(ρ, γ) is tested on the robotized electric Renault ZOE vehicle. Previously in Chapter 4, we have tested a polytopic-based LPV controller K 1 (ρ) (see Section 4.5) which has been designed with the weights W (1) e and W [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF] u (i.e. the same weights used to design K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0])), and using the standard polytopic optimisation problem [START_REF] Packard | Gain scheduling via linear fractional transformations[END_REF]. Notice that the interpolating signal vector γ is switched manually during the test. This experimental test mainly aims to: The longitudinal speed is assumed to vary within the range v x [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF][START_REF] Tuan | Relaxations of parameterized LMIs with control applications[END_REF] m/s. Since the system dynamics changes significantly in this speed range, it could be conservative to design a single LPV controller over the full parameter region. Thus, the parameter region is divided into 5 subsets as:

v x ∈ [5, 10] ∪ [10, 15] ∪ [15, 20] ∪ [20, 25] ∪ [25, 30] (6.26)
The objective of this method is to: 2. Create an overall switched LPV-YK controller Kσ (ρ) to switch between Ki (ρ), such that Kσ (ρ) quadratically stabilizes G(ρ) ∀ρ ∈ P and for every continuous/discontinuous switching signal σ(t).

Lateral Control Design

G(ρ) is written here as a convex combination of the vertices of a triangular polytope P 0 = C O {(ρ 1 , ρ 2 ), (ρ 1 , ρ 2 ), (ρ 1 , ρ 2 )} as shown in Fig. 6.23 with the state-space representation as is described in (6.4)- (6.6). This section aims to design the partitioned polytopic-based LPV-YK control. Five triangular convex subsets (P 1 . . . P 5 ) are chosen along the parameter trajectory as shown in Fig. 6.23. The nominal LPV controller K 0 (ρ) is designed, based on (A.3.1), using the weighting functions W e,0 and W u,0 defined as: W e,0 (s) = s+2 2s+0.002 , W u,0 (s) = s+5 0.01s+5 (6.27)

The following closed-loop step responses are obtained as shown in Fig. 6.24.

In addition, at the polytopic vertices w ij ∀i ∈ I[1, 5], ∀j ∈ I[1, 3], LTI controllers K ij are designed, based on (A.3.2), separately using the same weighting functions W e and W u chosen to present the tracking performance and the actuator limitations respectively. as: The closed-loop step responses are shown in Fig. 6.25.

Design the LPV-YK Control Structure

Referring to Fig. 6.23, assumptions (A.3.1)-(A.3.2) and Theorem 7 in Section 5.6.3, the design of the partitioned polytopic-based LPV-YK (shown in Fig. 6.4) is done according to the following steps:

1. According to the LMIs in Theorem 7, the LPV polytopic-based state-feedback controller F g (ρ), and the LTI state-feedback controllers F k,i (ρ), ∀i ∈ I[1, 5], are designed. After finding the decision variables W 0j (j ∈ {1, 2, 3}) and X g from (5.70), and V ij and X k,ij (i ∈ I[1, 5],j ∈ {1, 2, 3}) from (5.71). Then, F g (ρ) and

F (0) k (ρ) are computed as F g (ρ) = 3 ∑ j=1 W 0j X -1 g , F (0)
k,i (ρ) = 3 ∑ j=1 V ij (X k,ij ) -1 , ∀i ∈ I[1, 5] (6.29)
2. The state-space representations of M(ρ), N(ρ), U 0 (ρ) and V 0 (ρ) are computed as illustrated in Section 6.5.1. Q(ρ, γ) is defined here as Q σ (ρ), recall that σ represents the switching signals which changes at the intersecting boundaries of the parameter subsets. Q i (ρ) are obtained from (5.74), ∀j ∈ {1, 5}. It is worth mentioning that the switching signal is called σ instead of γ, since the signal does not appear in the closed-loop state matrix, whereas the closed-loop state matrix switches from one to another.

3. Here, the switching signal σ depends only on the varying parameter ρ, where switching occurs at the intersecting boundaries of the parameter subsets.

The current structure has been directly experimentally validated on the ZOE vehicle as shown in the next shown.

Grid-based LPV-YK controller

This section implements the grid-based LPV-YK controller, proposed in Section 5.6.4, to the lateral dynamics of an autonomous vehicle, aiming to achieve robust performance. Designing steps and experimental results are depicted in the following subsections.

As in the previous section, the parameter region is divided into 5 subsets as: Each parameter subset is gridded with six grid points spaced equally. The objective of grid-based LPV-YK control structure is to obtain exponential stability of the closed-loop system based on YK parameterisation. Moreover, a smooth interpolation scheme K(ρ, γ) is formulated between multiple pre-designed LPV controllers K i (ρ) (i ∈ I[1, 5]), using the switching signal γ = [γ 1 , , ..., γ 5 ] , where each K i (ρ) is designed to be suitably used for a certain parameter subregion P i . This could be achieved by two steps:

1. Parameterize each LPV controller K i (ρ) (i ∈ I[1, 5
]) with respect to the nominal LPV controller K 0 (ρ), by an LPV-YK parameter Q i (ρ).

2. At each boundary of two adjacent subsets P i and P i+1 , the interpolating signals γ i and γ i+1 are adjusted in a way to switch from K i (ρ) to K i+1 (ρ) or viseversa. As a result, the overall parameterized LPV-YK controller K(ρ, γ) stabilizes G(ρ) ∀ρ ∈ P and for every continuous/discontinuous interpolating signals γ i , i ∈ I[1, 5].

Lateral Control Design

In this section, G(ρ) is not required to have affine parameter-dependency, so it is written as shown in (6.4)-(6.5), over the parameter region presented in Fig. 6.30.

• The nominal LPV controller K 0 (ρ) is designed using W e,0 and W u,0 in (6.27) over the full parameter region P satisfying (A.4.1).

The following closed-loop step responses are obtained: 1. Following the conditions of Theorem 8, the LPV state-feedback gains F g (ρ) and F k,0 (ρ) are designed using the LMIs (5.85)- (5.86), where the multiple parameterdependent Lyapunov functions at each subset are specified as affine and smooth functions of scheduling parameters. That is,

X g (ρ) = X 0 g + X 1 g ρ, X k,0 (ρ) = X 0 k,0 + X 1 k,0 ρ,
where matrices X j g and X j k,0 , j = 0, 1 are the optimization variables to be determined.

2. The state-space representations of M(ρ), N(ρ), U 0 (ρ) and V 0 (ρ) are computed as illustrated in Section 6.5.1. Q(ρ, γ) is formulated as:

Q(ρ, γ) = 5 ∑ i=1 γ i Q i (ρ) (6.31)
where Q i (ρ) are obtained from (5.91), ∀i ∈ I[1, 5].

3. γ i (ρ) is switched between {0,1} when ρ(t) touches the switching boundaries. Q i (ρ) are obtained from (5.74), ∀j ∈ {1, 5}. Then, the LPV-YK controller K(ρ, γ) is ready to be implemented.

The current structure has been directly experimentally validated on the ZOE vehicle as shown in the next shown.

Experimental Results

To validate the proposed grid-based LPV-YK control, the vehicle performance is analyzed and compared to an LPV-switched controller designed using the Theorem presented in [START_REF] Lu | Switching LPV control designs using multiple parameterdependent Lyapunov functions[END_REF]. Similarly to the previous section, a restarting mode of the navigation systems appears at the end of the highway, and the results analysis doesn't concern the boarded part marked "NAV". The scenario in this experiment is exactly the same to the previous experiment with the reference trajectory and the longitudinal speed are as shown in Fig. 6.26 and 6.27 respectively. Fig. 6.33a depicts the lateral error from the reference trajectory to the vehicle Center of Gravity (CoG). Fig. 6.33b represents the yaw rate of the vehicle, and the steering control input is shown in Fig. 6.33c.

Both controllers achieve minimized lateral error as show in Fig. 6.33a. However, Fig. 6.33b and 6.33c emphasize a clear difference in performances during switching from one subset to another. The zoomed parts in Fig. 6.33c show that the switching effect is negligible using the proposed grid-based LPV-YK controller, which is not the case for the switched LPV controller.

Moreover, it is worth mentioning that the control input response at high speeds is not noisy. However, it has been shown previously in Section 4.5 that designing a single LPV grid-based controller over the whole parameter region may cause noisy performance at high speed.

Conclusion

This chapter has implemented the LPV-YK control structures proposed in the previous chapter. First, it has been used to improve the system performance, while dealing with various objectives and situations. The application to the autonomous vehicle lateral control is carried out. The simulation shows interesting results regarding the efficiency of the proposed method of providing high performance and ensuring safety at critical situations. In addition, experimental results are shown to validate its real performance by testing the approach on a real Renault ZOE vehicle.

Then, the partitioned and grid-based LPV-YK control structures have been implemented to maintain unique closed-loop specifications over the whole parameter region. Both structures have shown remarkable experimental results when compared to the standard LPV control appraoches.

Finally, it is worth mentioning that that the LPV-YK control structure facilitates adding any new controller, by introducing its corresponding LPV-YK parameter to the YK configuration (as shown in Fig. 5.9), without recalling all the control design procedure. This enhances real applications, such as in industries, where the systems are subjected to frequent instrumentation changes such as autonomous vehicles.

An interest appears to study the optimal choice of γ, and how to find always the best combination of a larger set of controllers, which will improve more the presented work. This study will be investigated in the next chapter.

Chapter 7

About The Interpolation Logic of LPV-YK Control: An Example Using Reinforcement Learning

Introduction

In this chapter, the structure of interpolation between YK-based LPV controllers is concerned aiming to find an optimal interpolation according to the required objectives. In Section 6.7, the interpolating signal γ has been chosen according to our previous experience in autonomous vehicle lane-tracking and lane-changing. The interpolation has shown good results, however, it could not be optimal. Since the whole closed-loop system can be complex including the LPV-YK control and the nonlinear vehicle model, Artificial Intelligence (AI) tools are proposed to learn how to find the optimal solution of γ.

Reinforcement Learning (RL) algorithms are highly recommended to find optimal actions according to a certain reward function. The reward function can be designed according to engineering experience, and the optimization objective is to maximize the reward function regarding the current vehicle and surrounding environment situations. Thanks to the stability conditions of the LPV-YK control, the interpolating signal γ can be any ad-hoc physically based, continuous/discontinuous, external or internal signal. Consequently, using the RL method does not affect the stability of the closed-loop system.

The current chapter proposes an RL-based LPV-YK interpolation scheme aiming to achieve an optimal performance with guaranteeing closed-loop quadratic stability. First, it is shown how the interpolating signals affect the closed-loop specifications using time-domain analysis. Then, the RL model is trained and simulated using the same simulation environment as in Section 6.7.

The results in this chapter has been represented in:

• Intelligent Control Switching for Autonomous Vehicles based on Reinforcement Learning, published in 2022 33rd IEEE Intelligent Vehicles (IV) Symposium Conference, refer to [START_REF] Atoui | Intelligent Control Switching for Autonomous Vehicles based on Reinforcement Learning[END_REF].

The chapter is organized as follows: Section 2 discusses the related works to our contribution. Section 3 defines some preliminaries on the fundamentals of Reinforcement Learning. A proof on control performance recovery of the LPV-YK control structure is shown in Section 4. A time-domain analysis on the affect of the interpolating signal on the closed-loop performance is depicted in Section 5.

Related Works

In the last decades, autonomous driving has involved different main domains such as automatic control theory and Artificial Intelligence (AI). For instance, [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF] has presented the design and experimental validation of three Linear Parameter-Varying (LPV) control approaches for autonomous vehicles. On the other hand, a review has investigated Deep Reinforcement Learning (DRL) algorithms and provides automated driving tasks where DRL methods have been employed [START_REF] Ravi | Deep Reinforcement Learning for Autonomous Driving: A Survey[END_REF]. Each of these domains has proposed different kinds of approaches to solve the longitudinal and lateral control problems of autonomous driving.

On the other hand, recent studies have proposed AI learning tools due to its powerful performance, such as Reinforcement Learning (RL) [START_REF] Ravi | Deep Reinforcement Learning for Autonomous Driving: A Survey[END_REF], [START_REF] Li | Reinforcement Learning and Deep Learning based Lateral Control for Autonomous Driving[END_REF]. [START_REF] Wasala | Trajectory based lateral control: A Reinforcement Learning case study[END_REF] presents a deep RL based approach for lateral motion control. Deep RL has proved to be able to solve many control tasks in different fields, however there is no stability guarantees for its behavior in real-world scenarios. Several studies have been investigated to provide stability guarantees either by adding dynamic constraints [START_REF] Han | Reinforcement learning control of constrained dynamic systems with uniformly ultimate boundedness stability guarantee[END_REF] or by combining deep RL with a theoretical optimization control [START_REF] Berkenkamp | Safe Model-based Reinforcement Learning with Stability Guarantees[END_REF], [START_REF] Rosenstein | Reinforcement learning with supervision by a stable controller[END_REF]. Moreover, recent works have proposed RL algorithms for decision-making and performance optimization for vehicles lane change [START_REF] Zhao | Imitation of Real Lane-Change Decisions Using Reinforcement Learning[END_REF], [START_REF] Ye | Automated Lane Change Strategy using Proximal Policy Optimizationbased Deep Reinforcement Learning[END_REF], [START_REF] Wang | A Reinforcement Learning Based Approach for Automated Lane Change Maneuvers[END_REF].

As mentioned before, automated steering control should be able to address any circumstances such as activating the vehicle away from its reference trajectory with a significant error, or performing step lane changes. For ensuring stability and performance in such situations, a single controller could not be able to satisfy all critical situations. This chapter aims to propose an LPV-YK control interpolation based on RL to handle situations as large lateral deviations. Such deviations can be caused either by a first system activation, avoidance maneuver or even a localization discontinuity.

Preliminaries

This section presents some fundamentals of Reinforcement Learning to be used later in the chapter, readers can refer to [START_REF] Richard | Reinforcement learning: An introduction[END_REF] for more details.

Reinforcement Learning Elements

In order to explain the elements of the RL algorithms some definitions for the interaction to achieve a goal will be explained. The learner and decision-maker is called the agent, for e.g. the ego vehicle. The agent interacts with what is called the environment which includes everything outside the agent, in this case is the racetrack, the obstacles and the surrounding vehicles. The agent selects actions and the environment changes accordingly, which make them interact continually. This interaction is received by the agent with some representation of the environment described as State which includes information about its situation, it could be information about coordinates or speed of other vehicles, features of the road, among others. Refer to Fig. 7.1 to visualize the connection between these components.

The main sub-elements of RL algorithms are: • Policy: Is a mapping from perceived states of the environment to actions to be taken in those actions. It is sufficient to determine the behavior, policies may be stochastic.

• Reward: Defines the goal that is send as a number by the environment. The objective of the agent is to maximize the cumulative reward received over the long run. This value depends on the agents current action and the current state of the agent's environment at any time. The only way the agent can influence the reward signal is through its actions, which can have a direct effect on the total reward, or an indirect effect through changing the environment's state. The policy may be changed to select the action that will be followed by a higher reward on that situation in the future. Generally, reward signals may be stochastic functions of the state of the environment and the actions taken.

• Value function: Specifies what is good in the long run defined as episode. The value of a state can be described as the total amount of reward an agent can expect to accumulate over the future, starting from that state. Whereas rewards determine the immediate, intrinsic desirability of environmental states, values indicate the long-term desirability of states after taking into account the states that are likely to follow, and the rewards available in those states.

• Model: Is a representation of the behavior of the environment. When an action is made given a state the model might predict the resultant next state and reward due to this action. The model is used for planning and to consider possible future situations before they actually happen. There are two types of methods, model-free that are trial-and-error learners and model-based that integrate model and planning.

Reinforcement Learning Theory

The interaction between the agent and the environment occur at a sequence of discrete time steps t in which it receives some representation of the environment's state S t ∈ S in the S set of possible states, and it selects an action A t ∈ A(S t ) where A(S t ) is the set of actions available in state S t . One time step later, in part as consequence of its action, the agent receives a numerical reward R t+1 ∈ R ⊂ R and finds itself in a new state S t+1 . The Fig. 7.1 represents the agent-environment interaction. At each time step, the agent implements a mapping from states to probabilities of selecting each possible action. This mapping is called the agent's policy and is denoted π t , where π t (a | s) is the probability that A t = a if S t = s. Reinforcement learning methods specify how the agent changes its policy as a result of its experience. The agent's goal, roughly speaking, is to maximize the total amount of reward it receives over the long run. Chapter 7. About The Interpolation Logic of LPV-YK Control: An Example Using Reinforcement Learning

An agent can increase the long-term reward by exploiting knowledge learned about the discounted sum of expected future rewards of different state-action pairs. The learning agent has to exploit what it already knows in order to obtain rewards, but it also has to explore the unknown in order to make better action selections in the future. One of the main challenges in reinforcement learning is managing the trade-off between exploration and exploitation [START_REF] Ravi | Deep Reinforcement Learning for Autonomous Driving: A Survey[END_REF].

For some stochastic control problems when the models for sequential decision making outcomes are uncertain, Markov Decision Processes (MDP) are used. The MDP model consists of decision epochs, states S, actions A, rewards R, and transition probabilities T; a tuple < S, A, T, R >. Choosing an action a in a state s generates a reward R(s,a) and determines the state at the next decision epoch s' through a transition probability function T(s,a,s'). Policies are instructions of which action to choose under any occurrence at every future decision. The agent look for policies which are optimal. The mathematical representation of the policy which is a mapping from the state space to a probability over the set of actions, and π t (a | s) represents the probability of choosing action a at state s. The goal is to find the optimal policy π * at time k, defined as:

π * = arg max π E π { H-1 ∑ k=0 γ k R(s k , a k ) | s 0 = s} := arg max π V π (s) (7.1)
Where γ is the discount factor that controls how an agent consider future rewards. When γ is low the agent will maximize short term rewards, on the contrary with high values of γ the agent will try to maximize rewards over a longer time frame. Eq. (7.1) represents the highest expected sum of discounted rewards in a time horizon H in the MDP. From the models directly, RL agents may learn value function estimates, policies and/or environment. Finding a policy π that maximizes the expected discounted sum of rewards over trajectories in the state space is what solving a RL task means.

Actor Critic Approach

Actor-critic methods are hybrid methods that combine value-based and policy-based algorithms. One actor is the one that selects the actions and this is the policystructure. After an action is made by the 'actor', the estimated value function evaluates the action and this is known as the 'critic'. The value-based methods are modelfree Temporal Difference (TD), are methods that can learn directly from raw experience without a model of the environment's dynamics and learn estimates of the utility of individual state-action pairs represented in Eq. (7.2). This scalar signal is the sole output of the critic and drives all learning in both actor and critic, as shown in Fig. 7.2

Q π (s, a) = E π { H-1 ∑ k=0 γ k R(s k , a k ) | s 0 = s, a 0 = a} (7.2)
Q-learning will learn (near) optimal state-action values provided a big number of samples are obtained for each pair. Agents implementing Q-learning update their Q values according to the update rule of Eq. ( 7.3): The policy-based methods aim to estimate the optimal policy directly, and the value is a secondary. Typically, a policy π θ is parameterized as a neural network. Policy gradient methods use gradient descent to estimate the parameters of the policy that maximize the expected reward. The result can be a stochastic policy where actions are selected by sampling, or a deterministic policy. When selecting actions, exploration is performed by adding noise to the actor policy. To stabilize learning a replay buffer is used to minimize data correlation. A separate actor-critic specific target network is also used. Normal Q-learning is adapted with a restricted number of discrete actions the optimal Q-value and optimal action as Q * and a * .

Q(s, a) ← -Q(s, a) + α r + γ max α ′ ∈A Q(s ′ , a ′ ) -Q(s, a) (7.3)
Q * (s, a) = max π Q π (s, a) , a * = arg max aQ * (s, a) (7.4)
By correcting the Q-values towards the optimal values using the chosen action, the policy is updated towards the optimal action proposition. Thus two separate networks work at estimating Q * and π * .

Soft-Max Policy Approximation

In policy gradient methods, the policy π(a|s, θ) can be formulated in any way, as long as it is differentiable with respect to its parameters θ; i.e. as long as ∇π(a|s, θ) (the column vector of partial derivatives of π(a|s, θ) with respect to the components of θ) exists and is finite for all s ∈ S, a ∈ A(s), and θ ∈ R d ′ . In practice, parameterized numerical preferences h(s, a, θ) ∈ R, for each state-action pair, is chosen if the action space is discrete and not too large. Usually, an exponential soft-max distribution is used so that the actions having the highest preferences, in each state, are given the highest probabilities to be selected, the policy is then formulated as follows: Chapter 7. About The Interpolation Logic of LPV-YK Control: An Example Using Reinforcement Learning

• The highly robust polytopic-based LPV controller K (0) (ρ) is designed using the standard polytopic approach, achieving a slow transient response and noise rejection performances.

• The smooth YK-based LPV controller K (1) (ρ) designed to perform smooth lateral transitions to provide comfort riding.

• The fast YK-based LPV controller K (2) (ρ) achieving fast lateral transitions to handle the vehicle when facing aggressive maneuvers and lateral oscillations.

Figs. 7.3a, 7.3b, 7.3c show the closed-loop step responses of each LPV controller at the three polytopic vertices. In the following, the objective is to analyze the change of the closed-loop step response, at each vertex separately, when γ 1 changes from 0 to 1, while always ensuring γ 2 = 1γ 1 . It is shown in these figures that the closed-loop specifications (i.e. rising time, overshoot, settling time, etc.) stay always between the two interpolated controllers K

(1) i and K

(2) i wherever γ j 's (j = {1, 2}) vary between 0 and 1 with γ 1 + γ 2 = 1. To clarify this point, Figs. 7.4a and 7.4b show the monotonic decreasing of both rising and settling times when changing from the slow controller K

i , with respect to γ = [γ 1 , 1γ 1 ]. Notice that the colored points (from blue to red) in these figures correspond to the same legends in the Fig. 7.3a. Thus, multiple closed-loop specifications can be achieved from the interpolation of two LPV controllers, each corresponds to a certain value of γ, which can be used later to choose the required specifications during online implementation.

RL-based LPV-YK Interpolation Architecture

Reinforcement learning can teach how an agent act by interacting with its environment in order to maximize the expected cumulative rewards for a certain task. There are two general categories of RL algorithms, namely value-based method and policy-based method. While value-based methods can approximate the value function using neural networks in an off-policy way, the primary advantage of policybased methods, such as the actor-critic [START_REF] Grondman | A Survey of Actor-Critic Reinforcement Learning: Standard and Natural Policy Gradients[END_REF], is that they can directly optimize the quantity of advantage, while remaining stable during function approximations. Our study thus focuses on policy-based RL methods. A detailed explanation of the actorcritic methods can be found in chapter 13 in [START_REF] Richard | Reinforcement learning: An introduction[END_REF].

Problem Description

In Section 6.7, it has been shown that using the LPV-YK interpolation is important to maintain an optimal performance for different lateral maneuvers, such as large lateral deviations. In our formulation, we focus on situations where explicit lane change intentions are already given by the trajectory planner as a 3 meters lateral step. Our task is to decide when and how to interpolate between the two designed controllers based on states of the ego vehicle with respect to the environment and the reference trajectory. Once a decision is made by the RL model, the LPV-YK control scheme achieves a new closed-loop performance and regulates the lateral movements. The applicable switching signal policy to be learned should incorporate the following three functionalities:

• Avoiding collisions with the lanes boundaries

System Architecture

The overall system architecture for enabling automated switching is shown in Fig. 7.5. The system is composed of two main components:

1. Learner Model: uses the actor-critic algorithm to train the switching logic of the LPV-YK control scheme (agent), to learn a high-level policy while interacting with the environment.

2. Simulation Environment: includes the road network and different task scenarios, is generated using a RENAULT simulator on MATLAB, for a real automated ZOE car, that has been already used in the previous chapters, and it interacts with the training agent.

The vehicle and controllers information can be accessed, taking into account the vehicle and control dynamics generated in the simulation model. To enable safe, smooth, and efficient driving behaviors on highways, the learner model first receives the vehicle current state with respect to the reference trajectory from the simulator, which are passed through the policy network. Next, the learner model determines the high-level switching policy, which then sends the switching action back to the simulator to model the controllers' and vehicle's movement in the next time step, and compute the corresponding reward. Notice that the controller output is the steering angle δ.

Simulation Environment

The simulation environment in presented in Fig. 7.6. For a chosen value of the interpolating signal γ, the dynamics of the LPV-YK controller K(ρ, γ) is adapted The vehicle model contains the nonlinear dynamics of the automated ZOE vehicle. This model provides all the needed observations and the state-space variables to be used in the learner model. The simulation environment is designed on MAT-LAB/Simulink with a fixed step-size T s = 0.01s. The frequency of the interpolating signal γ is considered here to be similar to the closed-loop frequency. However, it can be taken slower for smoother interpolation.

Learner Model

The learner model is chosen here to be an RL model using the policy-based actorcritic method. The pseudo-code for the actor-critic algorithm can be expressed as follows [START_REF] Richard | Reinforcement learning: An introduction[END_REF]: Algorithm 1 Actor-Critic method to estimate the policy π θ ≈ π * Input: a differentiable policy parameterization π(a|s, θ) Input: a differentiable state-value function parameterization v(s, w) Algorithm parameters: λ w ∈ [0, 1], λ θ ∈ [0, 1], α w > 0, α θ > 0, α R > 0 Initialize R ∈ R (e.g., to 0) Initialize state-value weights w ∈ R d and policy parameters θ ∈ R d ′ (e.g., to 0) Initialize S ∈ S (e.g., to s 0 ) z w ← 0 (d-component eligibility trace vector) z θ ← 0 (d ′ -component eligibility trace vector) for Loop forever (for each time step) do A ∼ π(.|S, θ) Take action A, observe S ′ (next state), R (reward) ∆ TD ← R -R + v(S ′ , w)v(S, w) R ← R + α R∆ TD z w ← λ w z w + ∇ v(S, w) z θ ← λ θ z θ + ∇ ln π(S, w) w ← w + α w ∆ TD z w θ ← θ + α θ ∆ TD z θ S ← S ′ end for Two Deep Neural Network (DNN) models are to be trained expressing the policy π and the state-value function v. Their weighting parameters θ and w, the average reward R are initialized by zeros. The initial state s 0 can be chosen depending on the initial state of the vehicle with respect to its surroundings. At each time-step:

1. The action is estimated from the trained policy (actor). This action is entered to the Simulation Environment resulting the reward R and the next state S ′ .

2. The Temporal-Difference error ∆ TD is then computed, and the average reward R is updated.

3. The weighting parameters of both actor and critic DNNs are updated using the gradient-descent method.

4. The current state is updated to the next state, and continue to the next timestep.

State Space

Before getting into the definition of the state space, we introduce an observation space to help modeling the surrounding environment. The observation of a vehicle consists of six variables {v x , y L , θ e , y e , δ, y L × θ e }, where v x , y L , θ e , y e , δ are the vehicle speed, lateral error at the look-ahead distance L, vehicle heading error, vehicle lateral error, and the steering speed, respectively. y L × θ e is used to identify if the vehicle is going toward or away from the trajectory depending on its sign. In real traffic flow, a driver can neither observe nor process all the above observations precisely and continuously. A human can possibly observe whether he is in close or far distance, approaching or moving away. Therefore, it is suggested here to discretize the observations into grids (4 grids for each). As a result, we end up with a state space S including 4 6 = 4096 states.

Action Space

In this study, the action space A represents the switching signal γ which is discretized as {0, 0.2, 0.4, 0.6, 0.8, 1}. So, there are six actions achieving different closedloop performances, and thus different vehicle behavior. Notice that we have already presented the possible closed-loop performances resulting from each γ action in the previous sections.

Reward Function

The reward function is designed to incorporate key objectives of this study, which is to develop an automated switching strategy to provide safety, efficiency, and comfort. Specifically, 1. Efficiency : evaluation of the relative distance toward the reference trajectory.

2. Comfort : evaluation of steering speed, which relates proportionally the yaw acceleration.

3. Safety : evaluations of collision risks and high lateral overshoots.

The sub-reward functions can be expressed as:

Actor-Critic Model

A one-step softmax actor-critic episodic algorithm is used, see chapter 13 in [START_REF] Richard | Reinforcement learning: An introduction[END_REF] for more details. The state-action feature vector is formulated using a tile coding with 1 tilings and four tiles for each state. Having 4 6 states and 6 actions, results in 6 × 1 × 4 6 = 24576 features. Notice that only one tiling has been chosen due to computation cost, and since it has achieved our requirements as presented in the next section. The actor and critic weights, Θ and W respectively, are initialized by zeros. The discount factor is 1. The actor and critic step-sizes are α Θ = 0.0625 and α W = 1 respectively.

Simulation Experiment

The simulation is done using the nonlinear full car model designed for a Renault ZOE vehicle. The scenario is chosen as the one used in [START_REF] Atoui | Real-Time Look-Ahead Distance Optimization for Smooth and Robust Steering Control of Autonomous Vehicles[END_REF]. This scenario covers several lateral steps at different speeds. Fig. 7.8 depicts the planned and the controlled trajectories, and the longitudinal speed profile is shown in Fig. 7.9. The first step is to train the actor-critic model. After training, tests are performed to compare the resultant performance using the RL-based switching with another switching logic. The averaged RMS of the TD-error is computed and represented in Fig. 7.7. It is shown how the TD-error is optimized and decreases along the number of episodes. In order to evaluate the trained policy of γ RL , it is compared to the switching logic already used Section 6.7, which has been designed according to our experience.

Testing Results

The parameterized LPV-YK controller K(ρ, γ) is structured as shown in Section 6.7 and simulated on the Renault simulator with the trained switching policy. The simulation is done on a designed trajectory containing consecutive lateral deviation steps over different longitudinal speeds.

The chosen scenario is tested with different controllers: K(ρ, [1; 0]) = K (1) (ρ) (smooth tracker), K(ρ, [0; 1]) = K (2) (ρ) (aggressive tracker), the LPV-YK controller with experienced switching logic K(ρ, γ f ), and the proposed RL-based LPV-YK switching control K(ρ, γ RL ). First, it is important noticing that neither K (1) (ρ) nor K (2) (ρ) could achieve an optimal performance regarding the lateral error optimization and steering comfort. Specifically, Fig. 7.10a shows the high lateral oscillations caused by K (1) (ρ), whereas K (2) (ρ) leads to steering speed noises, as shown in Figs. [START_REF] Apkarian | Self-scheduled H ∞ control of linear parameter-varying systems: a design example[END_REF].10b and 7.10c, which affects the riding comfort.

On the other hand, the interpolation between both controllers using the LPV-YK control scheme achieves better performance on both tracking and steering speed noise attenuation, i.e. reaching more optimized performance. This is shown regarding the performance of K(ρ, γ f ) and K(ρ, γ RL ) compared to K (1) (ρ) and K (2) (ρ). According to Fig. 7.11, it is presented how both switching logic γ f and γ RL have similar evolution with respect to time. This means that the trained RL-based switching model is close to what have been previously proposed based on our experience.

Regarding the zoomed parts in Figs. 7.10a, 7.10b, and 7.10c, it is shown that the overall vehicle performance is even better using the RL-based LPV-YK switching control. The steering angle and the steering speed are clearly shown to be smoother and optimized. Concerning the lateral error, Fig. 7.10a presents how K(ρ, γ RL ) performs the smoothest lateral deviations with lowest overshoot. The RMS are calculated for both K(ρ, γ f ) and K(ρ, γ RL ) obtaining the following: RMS f y e = 0.47, RMS RL y e = 0.452 (7.15)

Conclusion

This chapter has proposed an new combination between reinforcement learning and automatic control tools, to ensure both stability and high performance for autonomous vehicles. The LPV-YK switching control scheme is chosen since it ensures the closed-loop stability for any continuous/discontinuous interpolating signal. This signal can be used to incorporate any ad-hoc physically-based interpolation. Chapter 7. About The Interpolation Logic of LPV-YK Control: An Example Using Reinforcement Learning

A rising topic, such as reinforcement learning, is used to supervise the switching logic to provide an optimized performance. Instead of estimating the switching logic depending on our experience and knowledge, a softmax actor-critic RL model has been used to observe the vehicle states with respect to the reference trajectory, and choose the best switching scenario.

The LPV-YK switching control with the highly performant RL model has provided interesting results in simulation. It has achieved safety, efficiency, and comfort compared to other simulated controllers. It is important mentioning that the LPV-YK control structure can be generalized to multiple controllers, and facilitates adding any new controller without recalling all the control design procedure. This can encourage to design a family of multiple controllers, and train an RL-based supervisor to interpolate between them depending on the vehicle-surrounding situation.

As future work, an interest appears to experimentally validate the proposed system architecture on the automated ZOE car. Moreover, the proposed architecture could aid also to strengthen the link between the decision-making and the control systems which leads to higher vehicle performance.

Chapter 8 Discussion

In this chapter, a summary and some concluding remarks are presented concerning the proposed LPV-YK controllers with their benefits on the lateral control of autonomous vehicles. In addition, several future perspectives are reported to aid the continuation of the thesis work.

Thesis Summary

The deployment of autonomous driving aims to avoid accidents, reduce fuel consumption, improve traffic flow. It also provides safety and passenger comfort in critical situations and make it possible to car travelling for everyone regardless of their abilities or conditions. However, control of autonomous vehicles is not trivial since it is required to handle various situations that may lead to use the maximum capabilities of the vehicle. This thesis have developed multi-variable and multiobjective control schemes aiming to achieve various control performances covering the whole vehicle speed range.

A general summary of each of the included chapters is shown below:

• Chapter 1 has introduced the main objectives and novelties of this thesis. In addition, the thesis organization is mentioned. Resulted publications such as journals, conferences, and patents are presented at the end of the chapter.

• Chapter 2 has reviewed the use of different switching and interpolating LPV control approaches in control fields. The most applied applications have been illustrated and classified over time evolution. The discussed approaches can be described as follows: 1) Standard LPV control: polytopic, grid-based, LFT: designed using one single (constant or parameter-dependent) Lyapunov function; 2) Switched-LPV control: designed using multiple parameter-dependent Lyapunov functions, and leads to aggressive controller states and output jumps at the switching instants; 3) Smooth LPV switching: designed sequentially or instantaneously, and uses iterative methods to smooth the switching transitions; 4) LPV interpolation based-on bumpless-transfer: tries to minimize the controller states or output differences at the switching transitions; and 5) LPV-YK control interpolation: designed by parameterizing the local controllers with respect to a single nominal controller. A timeline is provided at the end classifying their main applications.

• Chapter 3 has presented the autonomous vehicle architecture including the perception, motion planning, and control. A specific literature review is done on lateral control of autonomous vehicles based on the look-ahead approach.

In addition, the simulation and experimental environments, that have been used in the work of this thesis, have been introduced.

• Chapter 4 has introduced a theoretical and experimental comparison of the LPV approaches for the lateral control of autonomous vehicles. It has been shown that the grid-based model has the simplest structure with less conservatism in optimization among the others. The weighting parameters used in the LPV/H ∞ control design can physically translate the real actuator limitations to a filter added to the optimization problem. The practical limitations of each approach have been discussed by observing the simulation results which were obtained from different chosen critical scenarios. Then, the real experiments showed interesting results regarding the minimization of the lateral error, which encourages the application of LPV/robust approaches on autonomous vehicles. A comparison table with pros and cons of each approach has been represented at the end. In fact, such approaches help to control parameter-variant systems and to handle with environmental disturbances (wind speed, bad road conditions/slopes, etc...).

• Chapter 5 has proposed four LPV-YK switching/interpolating methods to obtain smooth interpolation between different closed-loop specifications, and switching between consecutive parameter subregions. The switching signal can be parameter-dependent or any ad-hoc continuous/discontinuous signal.

The main advantages behind these approaches, compared to the previous LPVswitched controllers in the literature are: 1) The pre-designed LPV controllers are parameterized with respect to a nominal LPV controller, instead of requiring the redesign in new constrained LMIs; 2) The design conditions proposed for LPV-YK switching control are more smooth and can be satisfied assuming just the LPV stabilizability of the system; 3) The interpolated/switched controllers can be designed using different control approaches, i.e. PID, H ∞ , etc; and finally 4) The proposed LPV-YK control schemes don't pose any conditions or limitations on the switching signals.

• Chapter 6 has shown the implementation steps of the LPV-YK control schemes proposed in the previous chapter. These approaches improve the system performance, while dealing with various objectives and situations. An application to the autonomous vehicle lateral control has been carried out. The simulations have shown interesting results regarding the efficiency of the proposed methods of providing high performance and ensuring safety at critical situations. In addition, experimental results have been presented to validate their real performance by testing the approaches on a real Renault ZOE vehicle. Finally, notice that the LPV-YK control structures facilitate adding any new controller, by introducing its corresponding LPV-YK parameter to the YK configuration, without recalling all the control design procedure. This enhances real applications, such as in industries, where the systems are subjected to frequent instrumentation changes such as autonomous vehicles.

• Chapter 7 has proposed a new combination between Reinforcement Learning (RL) and automatic control tools, to ensure both stability and high performance for autonomous vehicles. The LPV-YK switching control scheme is chosen since it ensures the closed-loop stability for any continuous/discontinuous interpolating signal. This signal can be used to incorporate any ad-hoc physicallybased interpolation. A rising topic, such as reinforcement learning, is used to supervise the switching logic to provide an optimized performance. Instead of estimating the switching logic depending on our experience and knowledge, a softmax actor-critic RL model has been used to observe the vehicle states with respect to the reference trajectory, and choose the best switching scenario. The LPV-YK switching control with the highly performant RL model has provided interesting results in simulation. It has achieved safety, efficiency, and comfort compared to other simulated controllers. It is important mentioning that the LPV-YK control structure can be generalized to multiple controllers, and facilitates adding any new controller without recalling all the control design procedure. This encourages to design a family of multiple controllers, and train an RL-based supervisor to interpolate between them depending on the vehicle-surrounding situation.

Conclusions and Future Perspectives

To continue developing the work of this thesis, different future works are highly encouraged to be investigated regarding the obtained conclusions:

• The LPV-YK parameterization has shown significant improvement in the field of LPV switching control, where it simply ensures closed-loop stability without requiring an instantaneous design of the local LPV controllers. In addition, it achieves negligible switching effect, on the systems states and control inputs, at the switching instants. On the other hand, it is worth mentioning that it is important to analyze the output response of the LPV-YK parameter Q(ρ, γ) which affects the performance of the LPV-YK controller K(ρ, γ). A future work can be done to study the choice of the dynamics of Q(ρ, γ), i.e. how to choose the best design of the LPV state-feedback gains F g (ρ) and F k,0 (ρ), and how they affect the closed-loop performance.

• The LPV approaches, specifically the polytopic one, have achieved lower performance at high vehicle speeds which is resulted from dynamic model uncertainties. The actuator model has been identified as a speed-independent model, whereas in reality, it changes its dynamics with respect to the vehicle speed. Thus, it is suggested to identify the steering actuator model as an LPV model in the future works to obtain a more realistic LPV vehicle model. Then, implement again the LPV approaches and analyze the performance improvement at high vehicle speeds.

• The proposed LPV-YK control structures have shown interesting simulation and experimental results. In comparison with the LPV controllers, they have achieved better driving comfort at high vehicles speeds. Moreover, smoother switching transitions are resulted using the LPV-YK controllers. A next step can be done by testing them in more complex environments, i.e. friction drop, gust wind, higher lateral accelerations.

• Concerning the multi-objective LPV-YK control structures, their importance is shown by implementing them in different driving situations such as lane changing, lane tracking, collision avoidance. This has been achieved by regulating the interpolating signals accordingly with the vehicle-surrounding situations. For more development, an interest appears to study the optimal choice of the interpolating signals γ, and how to find always the best combination of a larger set of controllers, which will improve more the presented work.

• The LPV-YK switching control with the highly performant RL model has provided interesting results in simulation. It has achieved safety, efficiency, and comfort compared to other simulated controllers. Such a combined architecture can be used in the future to strengthen the link between the decisionmaking and the control systems which leads to higher vehicle performance.

Résumé

De nos jours, les systèmes deviennent de plus en plus complexes, ce qui nécessite des algorithmes de contrôle capables de prendre en compte en ligne des objectifs variables de performance et de sécurité.

  approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Renault's History on Autonomous Vehicles . . . . . . . . . . . . . . . . 3.2 Renault ZOE automated vehicle . . . . . . . . . . . . . . . . . . . . . . . 3.3 Functional components of the automated ZOE . . . . . . . . . . . . . . 3.4 Sensor setup in the automated ZOE Vehicle [318] . . . . . . . . . . . . . 3.5 Pipeline for 3D object detection. Objects are detected in the images from the cameras. Later, LiDAR data is included to infer the geometrical properties of the objects. Finally, all the detections are merged, and a tracking stage adds temporal consistency across consecutive frames [318] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Satory private test track . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Sampling of driving space with vertices (blue dots) arranged in layers. An example of a circle spline that connects second and third layer is shown in solid black. Continuity with first and fourth layer are shown in dotted black lines. [318] . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Vehicle Bicycle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Control system architecture . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Lateral control closed-loop scheme . . . . . . . . . . . . . . . . . . . . . 3.11 Lateral control based on look-ahead concept . . . . . . . . . . . . . . . 3.12 Adaptive Cruise Control (ACC) scheme . . . . . . . . . . . . . . . . . . 4.1 The polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The gridded matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Linear fractional representation of a parameter-varying system . . . . 4.4 Control design scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Sensitivity Functions S = w re f -w w re f . . . . . . . . . . . . . . . . . . . . . . . 4.7 Sensitivity Functions KS = δ w re f . . . . . . . . . . . . . . . . . . . . . . . 4.8 Control implementation scheme . . . . . . . . . . . . . . . . . . . . . . 4.9 Reference Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Speed profile measured from a real test (m/s) . . . . . . . . . . . . . . 4.11 Lateral error using the polytopic approaches (m) . . . . . . . . . . . . . 4.12 Steering front angle δ using the polytopic approaches (rad) . . . . . . . 4.13 Look-ahead distance d (m) . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1

FIGURE 2 . 1 :

 21 FIGURE 2.1: Multiple parameter subregions

FIGURE 2 . 2 :

 22 FIGURE 2.2: Hysteresis switching

FIGURE 2 . 3 :

 23 FIGURE 2.3: Smooth LPV switching

FIGURE 2 . 4 :

 24 FIGURE 2.4: Bumpless Transfer with I = {1, 2} [242]

FIGURE 2 . 5 :

 25 FIGURE 2.5: Negative feedback loop

2

 2

  .

FIGURE 2 . 6 :Lemma 4 .

 264 FIGURE 2.6: Parameterized LPV-YK control scheme

Fig. 3 .

 3 1 shows the projects timeline achieved by Renault since 2013 to the present time. The main general objective of these projects was to offer a taxi pool for people transportation to their work, home or a Mall. Before the beginning of this thesis work, Renault has already developed several autonomous driving prototypes such as: PAMU (in 2013), KAIROS (in 2014), TRAJAM (in 2015), SYMBIOZ (in 2017), and ROUEN Autonomous Lab (in 2018).

FIGURE 3 . 1 :

 31 FIGURE 3.1: Renault's History on Autonomous Vehicles

FIGURE 3 . 6 :

 36 FIGURE 3.6: Satory private test track

FIGURE 3 . 7 :

 37 FIGURE 3.7: Sampling of driving space with vertices (blue dots) arranged in layers. An example of a circle spline that connects second and third layer is shown in solid black. Continuity with first and fourth layer are shown in dotted black lines. [318]

FIGURE 3 . 8 :

 38 FIGURE 3.8: Vehicle Bicycle Model

FIGURE 3 . 9 :

 39 FIGURE 3.9: Control system architecture

FIGURE 3 . 10 :

 310 FIGURE 3.10: Lateral control closed-loop scheme

FIGURE 3 .

 3 FIGURE 3.12: Adaptive Cruise Control (ACC) scheme

FIGURE 4 . 1 :

 41 FIGURE 4.1: The polytope

FIGURE 4 . 2 :

 42 FIGURE 4.2: The gridded matrices

FIGURE 4 . 3 :

 43 FIGURE 4.3: Linear fractional representation of a parameter-varying system

Fig 4 . 4

 44 Fig 4.4 presents the control design scheme where an LTI actuator model G act (first order transfer function + input delay) is added to the bicycle model dynamics (δ c is the controller output). Note that the input delay is described as a first order transfer function, so as a result, G act is a second order transfer function. Considering this structure, the controller K(ρ) (in Fig.4.4) is designed for the three approaches using LPV/H ∞ concept. Notice that w re f represents the yaw rate reference.Control performance criteria in H ∞ control theory are given by frequency domain functions (check Appendix B). Two weighting functions W e and W u are used to achieve tracking and actuator limitations performances respectively. The objective is to achieve both performances with a trade-off between minimizing the lateral error and ensuring the driving comfort. Notice that the steering speed δ could be related to the driving comfort since it reflects how fast the front wheels are acting (e.g. at high values of δ), and how noisy they appear (when δ oscillates around zero at high frequencies). We know that when δ increases, the lateral acceleration increases, and

FIGURE 4 . 4 :

 44 FIGURE 4.4: Control design scheme

  (A) General control configuration (B) LFT general control configuration z = e 1 e 2 T represents the objective function to be optimized when designing the controller. w = r d n T is the exogenous input, where r, d and n are the desired reference, input disturbance and the measurable noises respectively.

  .18)

4. 3 .

 3 LPV/H ∞ Control Design 57 the following L 2 induced minimization problem:

Chapter 4 .

 4 Application of LPV Approaches to Vehicle Lateral Control with O(n n ρ

FIGURE 4 . 6 :

 46 FIGURE 4.6: Sensitivity Functions S = w re f -w w re f

FIGURE 4 . 7 :

 47 FIGURE 4.7: Sensitivity Functions KS = δ w re f

FIGURE 4 FIGURE 4 . 10 :

 4410 FIGURE 4.8: Control implementation scheme

1 FIGURE 4 . 11 :FIGURE 4 . 12 : 30 FIGURE 4 . 13 :

 141141230413 FIGURE 4.11: Lateral error using the polytopic approaches (m)

FIGURE 4 .

 4 FIGURE 4.14: Normalized RMS of the lateral error with respect to different number of gridding points n g

FIGURE 4 . 15 : 1 FIGURE 4 . 16 :

 4151416 FIGURE 4.15: Lateral error using the LFT approaches (m)

3 FIGURE 4 . 17 :FIGURE 4 . 18 :

 3417418 FIGURE 4.17: Lateral error for robustness comparison (m)

FIGURE 4 . 20 :FIGURE 4 . 21 :FIGURE 4 . 22 :

 420421422 FIGURE 4.20: Satory trajectory plan and vehicle position

FIGURE 5 . 1 :

 51 FIGURE 5.1: Negative feedback loop

17

 17

  )

FIGURE 5 . 2 :

 52 FIGURE 5.2: Interpolation of multiple control performances

(A. 1 . 2 )

 12 . A group of ζ LPV controllers K (j) (ρ) (5.22)-(5.23) (j ∈ I[1, ζ]) have been designed separately to quadratically stabilize G(ρ) achieving different objectives and closed-loop specifications for all operating conditions.

FIGURE 5 . 4 :

 54 FIGURE 5.4: Polytopic-based LPV-YK general configuration

FIGURE 5 . 6 :

 56 FIGURE 5.6: Interpolation between three LPV controllers; two YK-based gain-scheduled controllers K (1) (ρ) and K(2) (ρ), and a polytopic-based LPV controller K (0) (ρ) along the convex parameter space P. The LPV-YK controller K(ρ, γ) interpolates, using γ = [γ 1 , γ 2 ], between K (0) (ρ) (for γ = [0, 0]), K(1) (ρ) (for γ = [1, 0]), and K (2) (ρ) (for γ = [0, 1])

  the polytope P, a group of ζ local LTI controllers K (j) i (j ∈ I[1, ζ]) have been designed separately to stabilize G i achieving different objectives and performances for all operating conditions.

FIGURE 5 . 7 :

 57 FIGURE 5.7: LPV-YK general configuration

1 ,

 1 scheduling coefficients in the convex region P i . Fig. 5.8 shows an example of a convex parameter region defined by two varying parameters ρ 1 ∈ [ρ ρ 1 ], and ρ 2 ∈ [ρ 2

FIGURE 5 . 8 :

 58 FIGURE 5.8: Partitioned polytopic regions

Theorem 7 .

 7 Consider an LPV plant G(ρ) (5.19), satisfying assumptions (A.3.1) and (A.3.2).

FIGURE 5 . 9 :

 59 FIGURE 5.9: Partitioned polytopic-based generalized LPV-YK configuration

  the same state transformation matrix T = state matrix Ācl,i (ρ) is represented in (5.75).

FIGURE 5 .

 5 FIGURE 5.10: LPV-YK gridded controller

FIGURE 6 . 1 :

 61 FIGURE 6.1: Vehicle Bicycle Model

FIGURE 6 . 2 :

 62 FIGURE 6.2: Generalized Plant G(ρ)

FIGURE 6 . 3 :

 63 FIGURE 6.3: LPV/H ∞ Control Design Scheme

FIGURE 6 . 5 :

 65 FIGURE 6.5: Closed-loop step response with K (0) (ρ)

FIGURE 6 . 6 :FIGURE 6 . 7 :

 6667 FIGURE 6.6: Closed-loop step response with K (1) (ρ)

FIGURE 6 30 FIGURE 6 . 9 :

 63069 FIGURE 6.8: Simulation scheme

Chapter 6 .FIGURE 6 . 10 :

 6610 FIGURE 6.10: Simulation: Planned and controlled trajectories

6. 6 . 122 Chapter 6 . 1 FIGURE 6 . 12 :

 612261612 FIGURE 6.11: Simulation results

FIGURE 6 . 14 :

 614 FIGURE 6.14: Closed-loop step response with K

FIGURE 6 . 15 :

 615 FIGURE 6.15: Closed-loop step response with K

  .

4 : 1 . 6 . 7 . 30 FIGURE 6 . 16 :

 416730616 FIGURE 6.16: Simulation: Parameter-varying longitudinal speed v x (m/s)

126 Chapter 6 .FIGURE 6 . 19 :

 1266619 FIGURE 6.17: Simulation results

FIGURE 6 . 20 : 60 FIGURE 6 . 21 :FIGURE 6 .

 620606216 FIGURE 6.20: Experimental planned and controlled trajectories

1 .

 1 Design multiple YK-based gain-scheduled controllers Ki (ρ), i ∈ I[1, 5]. Each one is designed by interpolating its corresponding LTI controllers K ij (j ∈ I[1, 2 n p ]) based on YK concept (refer to Fig. 6.23).

FIGURE 6 . 24 :

 624 FIGURE 6.24: K 0 (ρ) ∈ P 0

FIGURE 6 . 26 :FIGURE 6 . 27 : 4 FIGURE 6 .

 62662746 FIGURE 6.26: Experimental planned and controlled trajectories

ρ ∈ [ 5 ,

 5 10] ∪[START_REF] Mohammadpour | Control of linear parameter varying systems with applications[END_REF][START_REF] Georgo | System analysis, modelling and control with polytopic linear models[END_REF] ∪[START_REF] Georgo | System analysis, modelling and control with polytopic linear models[END_REF][START_REF] Wu | A generalized LPV system analysis and control synthesis framework[END_REF] ∪[START_REF] Wu | A generalized LPV system analysis and control synthesis framework[END_REF][START_REF] Li | Polytopic LPV approaches for intelligent automotive systems: State of the art and future challenges[END_REF] ∪ [25, 30] (6.30)

FIGURE 6 . 31 : 32 0 2 (E) K 5 (ρ) ∈ P 5 FIGURE 6 . 32 : 1 , 3 ] 6 . 9 . 2

 6313225563213692 FIGURE 6.31: K 0 (ρ) ∈ P 0

142 Chapter 6 .

 1426 FIGURE 6.33: Grid-based LPV-YK control

  Section 6 represents the RL-based LPV-YK interpolation architecture. The training and testing Chapter 7. About The Interpolation Logic of LPV-YK Control: An Example Using Reinforcement Learning results of the RL model are shown in Section 7. Finally, some concluding remarks are given in Section 8.

FIGURE 7 . 1 :

 71 FIGURE 7.1: The agent-environment interaction in RL

FIGURE 7 . 2 :

 72 FIGURE 7.2: The actor-critic architecture

7. 6 . 3 FIGURE 7 . 3 :FIGURE 7 . 4 :

 637374 FIGURE 7.3: Closed-loop step responses at the three polytopic vertices with different γ values

FIGURE 7 . 6 :

 76 FIGURE 7.6: Simulation Environment

100 ( 7 . 13 ) 2 FIGURE 7 . 7 :

 100713277 FIGURE 7.7: Averaged RMS of the TD-error

Fig. 7 . 25 FIGURE 7 . 9 : 2 (FIGURE 7 . 10 1 FIGURE 7 . 11 :

 7257927101711 FIGURE 7.8: Planned and controlled trajectories

  Chapter 2. Toward Switching / Interpolating LPV Control: A Review where x(t) ∈ R n x , y(t) ∈ R p , u(t) ∈ R m , z(t) ∈ R n z are the state, output, input, controlled output vectors respectively. w(t) contains the exogenous inputs, and ρ ∈ R n p is a vector of n p measurable varying parameters that belongs to P. P is the parameter space which can be represented by the below set: P := {ρ(.) := [ρ 1 . . . ρ n p ] T ∈ R n p and ρ i ∈ [ρ

)x(t) + B 1 (ρ)w(t) + B 2 (ρ)u(t) z(t) = C 1 (ρ)x(t) + D 11 (ρ)w(t) + D 12 (ρ)u(t) y(t) = C 2 (ρ)x(t) + D 21 (ρ)w(t) + D 22 (ρ)u(t) (2.1) i , ρ i ]∀i = 1, ...., n p }

  2.4. LPV Control Interpolation Based on Bumpless Transfer21 is formulated by convexly combining Parameter-Dependent Lyapunov Matrix (PDLM) on adjacent subregions. In addition, this method guarantees that the overlapped subregion has intermediate performance between its neighboring subregions. An individual LPV controller is designed for each subregion in sequential order, instead of synthesizing all controllers simultaneously. By iteratively solving the reduced-dimensional optimization problem for each subregion, switching controllers with guaranteed H ∞ performance on all subregions and overlapped subregions can be obtained.

TABLE 2 .

 2 1: Overview of discussed approaches

	Switching/Interpolating approach	Advantages	Disadvantages
	LPV control	Simple and fast design conditions	High conservatism for large number of parameters and large ranges
	Switching LPV control	Decreased conservatism and providing better performance at each parameter subset	Aggressive transient response at switching instants
	Bumpless-Transfer LPV	Continuity of the control input is	Local controllers are blended to guarantee
	control Interpolation	ensured at switching instants	closed-loop and switching stability
	Smooth LPV switching	Smoothness can be guaranteed from	Complex design conditions including the design
	control	the design by a matrix norm	of the switching signals for switching stability
	LPV-YK control Interpolation	Controllers are designed separately, new parts can be added to an existing system online, no constraints on the switching signals	Transient response is directly affected by the designed YK parameter

TABLE 4 .

 4

	1: RMS of the lateral error using the polytopic approaches
		4-Vertices Polytope Reduced Polytope
	RMS	0.98	1.01

TABLE 4 .

 4 2: RMS of the lateral error using the LFT approaches

		Nominal LFT LFT with online inverse Interpolated LFT
	RMS	0.1849	0.1683	0.17

TABLE 4 .

 4

		3: RMS of the lateral error for robustness comparison
	LTI	Reduced Polytopic	Gridding (n g = 16)	Interpolated LFT
	RMS 0.257	0.266	0.251	0.33

2750 2800 2850 2900 2950 3000 3050 Y (m) Trajectory Plan Vehicle Position

	300	400	500	600	700	800	900	1000 1100
					X (m)

2, |δ| max = 0.4, (4.30) Fig. 4.22 and Table 4.5 show that all the controllers succeed to minimize the 4.5. Real Implementation 67 FIGURE 4.19: Renault ZOE automated vehicle

TABLE 4 .

 4 4: RMS of the steering front rate for robustness comparison

		LTI	Reduced Polytopic	Gridding (n g = 16)	Interpolated LFT
	RMS 0.017		0.018		0.017	0.017
		0.15
	Steering Front Angle (rad)	-0.05 0 0.05 0.1					Polytopic LTI Gridding LFT
		-0.1
		0	20	40	60	80	100
					Time (s)
	FIGURE 4.23: Experimental steering front angle of the LTI and LPV
				controllers (rad)

TABLE 4 .

 4

	5: RMS of the lateral error for experimental comparison
		Polytopic	LTI	Gridding	LFT
	RMS	0.1473	0.1105	0.1025	0.1096

TABLE 4 .

 4

	6: RMS of the steering front rate for experimental compari-
			son
		Polytopic	LTI	Gridding	LFT
	RMS	0.0263	0.0149	0.0107	0.0129

TABLE 4 .

 4 7: Overview of discussed approaches

	Controller	Parameter dependency	Conservatism with large np	LMI size/ memory growth	Number of decision variables	Simulation results	Experimental results
	Polytopic	Affine	High	O(2 2 )	6	Polytopic reduction leads better performance	High sensitivity to noises
	LFT	Rational	Medium	O(2)	78	Matrix inverse can be replaced by interpolation using a look-up table	Low sensitivity to noises
	Grid-based	General	Low	O(ng)	9	Performance and ng are not proportional	Low sensitivity to noises

TABLE 5 .

 5

		1: Overview of discussed approaches
	Objective	Section	LPV-YK Structure	Closed-loop stability
	Multiple closed-loop specifications	Section 5.5.1 Section 5.5.2	Interpolation of polytopic-based LPV controllers Interpolation of polytopic-based LPV controllers	Quadratic Quadratic
	Maintain robust performance	Section 5.6.3 Section 5.6.4	Switching of partitioned YK-based LPV controllers Switching of grid-based LPV controllers	Quadratic Exponential

  Le domaine des systèmes autonomes, en particulier les véhicules autonomes, est révélateur d'une telle évolution. En effet, leurs capacités de conduite ont été récemment améliorées pour une conduite hautement, voire totalement, autonome grâce à une théorie de contrôle avancée. Une voiture entièrement autonome doit effectuer plusieurs tâches dont le contrôle longitudinal, le contrôle latéral, le contrôle du châssis, etc. La présente thèse propose différentes structures de contrôle multi-objectifs visant à adapter les performances du véhicule en couvrant toute la plage de vitesse. Cette thèse présente de nouvelles architectures de contrôle pour commuter/interpoler entre plusieurs controleurs linéaires à paramètres variants (LPV), en considérant la vitesse du véhicule comme un paramètre variable. Le schéma de commutation ou d'interpolation est réalisé sur la base du paramétrisation de Youla-Kucera (YK). L'architecture de contrôle LPV-YK peut incorporer plusieurs contrôleurs LPV ou LTI qui peuvent commuter/interpoler entre eux en garantissant la stabilité. Elle peut être utilisée pour interpoler entre différentes performances de contrôle visant à atteindre différentes spécifications en boucle fermée dans différentes situations de conduite (c'est-à-dire changement de voie, évitement d'obstacles), ou pour basculer entre des sous-ensembles de paramètres cloisonnés lorsqu'on a affaire à de grandes variations de paramètres (c'est-à-dire à des vitesses de véhicule faibles et élevées). La stabilité en boucle fermée est garantie pour tout signal d'interpolation continu/discontinu en termes d'un ensemble d'inégalités linéaires matricielles (LMI). Les structures de contrôle proposées peuvent aider les systèmes multi-variables et multi-objectifs à atteindre des performances élevées dans différentes conditions de fonctionnement et différentes situations critiques, quel que soit le taux d'interpolation. Des exemples numériques sont simulés pour montrer l'importance des méthodes proposées pour atteindre différents objectifs pour le contrôle latéral des véhicules autonomes. En outre, les différentes structures de contrôle ont été testées sur un véhicule Renault ZOE réel pour valider leurs performances réelles et les comparer aux contrôleurs LPV standard.

	Mots Clés
	Contrôle latéral; Contrôle LPV; Paramétrisation Youla-Kucera; Contrôle de switch;
	Contrôle robuste; Véhicules autonomes.

Acknowledgements

Inaccurate tracking performance and conservative K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0]) ≡ K (1) (ρ) [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0] Smooth tracker Good tracking performance with smooth steering Oscillatory and cannot perform well at high lateral accelerations K(ρ, [0, 1]) ≡ K (2) (ρ) [0,1] Aggressive tracker

Fast tracking performance and could achieve high lateral accelerations

Too sensitive to noises K(ρ, γ) variant as in Fig. 6.18

Multiple objectives by varying the interpolating vector γ.

All the mentioned advantages and even more by choosing the optimal combination of controllers by γ No bad performance is observed 1. Compare both controllers K 1 (ρ) and K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0]).

2. Observe the controller response when γ switches aggressively as a step function.

The test is done in a part of Satory test-track shown in Fig. 6.20. Fig. 6.21 shows the variation of the measured longitudinal speed in Kph which is considered to be coherent with respect to the road curvature. The vehicle starts on a straight highway using the smooth controller K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0]). Then, it switches to the faster controller K(ρ, [0, 1]) when approaching two successive maneuvers (at t = 20s) aiming to achieve the lowest lateral error. After exiting the successive maneuvers, it switches again to the smooth controller K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0]) (at t = 50s) and enters a second maneuver to compare its performance to the previously tested polytopic LPV controller K 1 (ρ).

Regarding Figs. [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF].22a and 6.22b, it is clearly shown that the vehicle performance is not affected during the switching times (at t = 20s and t = 50s) with negligible transient response.

• When t ∈ [START_REF] Wu | A generalized LPV system analysis and control synthesis framework[END_REF][START_REF] Lu | LPV control design and experimental implementation for a magnetic bearing system[END_REF]s, K(ρ, [0, 1]) achieves smaller lateral error compared to K 1 (ρ) (see Fig. 6.22a), but with low steering noises as shown in Fig. 6.22b. Notice that these steering noises has been observed also in simulation section for K(ρ, [0, 1]).

• When t ∈ [65, 85]s, K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0]) shows lower lateral error during maneuvering compared to K 1 (ρ). This appears since the LPV-YK controller K(ρ, [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF]0]) is less conservative than the polytopic LPV controller K 1 (ρ).

The next two sections implements the LPV-YK control structure to switch between multiple parameter subsets aiming to maintain a unique closed-loop performance over the whole parameter region.

Partitioned LPV-YK controller

This section implements the partitioned LPV-YK controller, proposed in Section 5.6.3, to the lateral dynamics of an autonomous vehicle, since it contains a non-affine scheduling parameter which causes an overbounding convex parameter region when implementing the polytopic approach. Designing steps and experimental results are depicted in the following subsections.

Experimental Results

To validate the improvement caused by the proposed partitioned polytopic-based LPV-YK control, the vehicle performance is analyzed and compared to a gain-scheduled LPV-YK controller. The gain-scheduled LPV-YK controller is referred to a YK-based polytopic LPV controller that is designed over the whole parameter region P 0 (for instance, as the design on K (1) (ρ) in the previous section). Notice that, in all the tests, a restarting mode of the navigation systems appears at the end of the highway. Consequently, the results analysis doesn't concern the boarded part marked "NAV".

The experiments shown here have been carried out on the Renault ZOE vehicle. The test results of the designed controllers are discussed concerning their implementation and the analysis of the obtained performance. The tests are done in a private test track in Satory as shown in Fig. 6.26. The first part of the test describes the response of the controllers at a straight highway with high speeds. The second part concerns the precision of lateral control at optimal speeds chosen coherently depending on the road curvature.

The longitudinal speed is considered as an external parameter of the LPV mode that is shown in Fig. 6.27. Fig. 6.28 presents the operating parameter subset that switches according to the speed evolution. Fig. 6.29a depicts the lateral error from the reference trajectory to the vehicle Center of Gravity (CoG). Fig. 6.29b represents the yaw rate of the vehicle, and the steering control input is shown in Fig. 6.29c.

It is shown in Fig. 6.29a that both controllers, the proposed partitioned and gainscheduled LPV-YK, could successfully achieve lateral error minimization. On the other hand, the gain-scheduled LPV-YK controller leads to higher steering oscillations, compared to the partitioned LPV-YK controller, as shown in Figs. [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF].29b and 6.29c.

When t ∈ [28.3, 28.5]s, it is observed in Fig. 6.28 switches hysterically back and forth between the subsets P 3 and P 4 . However, the hysterical switching of the partitioned LPV-YK controller between both subsets didn't affect the performance of the steering input and consequently the yaw rate (see the zoomed part in Figs. [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF].29c and 6.29b). This is expected since, at that instants, the operating controllers at that intersecting boundary are dynamically equivalent, i.e. F l (G, J 33 , Q 33 ) ≡ F l (G, J 42 , Q 42 ). Chapter 7. About The Interpolation Logic of LPV-YK Control: An Example Using Reinforcement Learning π(a|s, θ) = e h(s,a,θ) ∑ b e h(s,b,θ) , (

where e = 2.71828 is the base of the natural logarithm. This policy parameterization is known as sof-max in action preferences. Notice that the action preferences h(s, a, θ) can be parameterized by a DNN or can be any linear function in features. An example which is usually used:

where x(s, a) is a state-action pair feature.

An advantage of the soft-max policy approximation is that the approximate policy can approach a deterministic policy.

Different Kinds of Interpolating Signals

In the autonomous vehicle field, an interpolation scheme with any ad-hoc interpolating signal can be beneficial to link between the decision-making and control systems, i.e. as a control supervisor. It can be expressed as a list of characters, floats or binaries which describes the decision taken. For example:

• As a character: it indicates the general task which should be taken by the vehicle, i.e. smooth/aggressive lane tracking, slow/fast lane change, far/ emergency collision avoidance, parking, autonomous-mode initialization, decrease lateral acceleration noises, sudden brake, etc.

• As a float: it represents all the ego vehicle information, i.e. position, lateral/longitudinal speed, lateral/longitudinal acceleration, vehicle parameters' change (mass, tire stiffness estimation, etc.), and others. Moreover, it gives the surrounding information such as: fixed/moving objects using cameras and Li-DAR (vehicles, pedestrians, etc.), road conditions (holes, friction, slope, wet/ humid, traffic lights/ marks, lane lines, etc.), environment condition (weather, wind speed, etc.)

• As a binary: it displays driving conditions due to a certain situation, i.e. sport/ classic drive, emergency calls (heart attack), etc.

In this chapter, the interpolating signal is considered to be float ranging in [0, 1].

Considering the LPV-YK control structure in Section 6.7, the next section proves the control recovery of each already designed LTI controller.

Local LTI Control Performance Recovery

Regarding Fig. 5.7, the interpolation cases can be mentioned to show how K(ρ, γ) recovers a single gain-scheduled controller K (j) (ρ) by varying γ j : • else, the performance of K(ρ, γ) is interpolated among K (j) (ρ) according to the chosen γ j .

In this section, the performance recovery of each pre-designed local LTI controller K (j) i (i ∈ [1, 2 n p ], j ≥ 1), from the LPV-YK controller K(ρ, γ), is verified. Let us choose an LTI controller K

Since the input-output performance of Q and Q is similar (according to the state transformation concept), Q(0

l . According to [START_REF] Tay | High performance control[END_REF], (7.7) can be written as

l , after some derivations:

Substitute and shape it, then

l ) (7.10) Knowing that K

l , and applying the Bezout identities (1), we get:

Therefore, it is shown that the performance achieved by the LPV-YK controller K(ρ, γ) at any vertex w l is equivalent to the performance of its corresponding LTI controller

Variation of Closed-loop Performance With Respect to The Interpolating Signal

The objective of this section is to show how the closed-loop performance changes with respect to the interpolating signal γ. The analysis performed here can help later understand the change in the LPV-YK control performance while changing γ to certain values. The LPV-YK control structure designed in Section 6.7 is used here.

Recall that the objective of this structure is to interpolate between three LPV controllers K (j) (ρ), j ∈ {0,

This controller is obtained after solving the following LMIs in (X(ρ), Y(ρ), A(ρ), B(ρ), C(ρ), D(ρ))

Where:

Then, we obtain the dynamical controller K(ρ) as follows (for

where M(ρ) and N(ρ) are defined such that M(ρ

Appendix B

H ∞ Control Theory

The H ∞ robust control is one of the most controllers used nowadays. The word robust means that, if a system is influenced by input external disturbances, model uncertainties, or sensors'/actuators' failure, the H ∞ controller will ensure the stability of the system to remain in the safe zone. The H ∞ control design is a mathematical optimization problem where it can be applied to multivariable systems. • P is represented as a state space called the generalized plant which contains the LTI or LPV model system in addition to the weights that are added to the controlled outputs to achieve the required performances.

• K is the H ∞ , static state-feedback or dynamic output-feedback, controller to be designed.

• w is the external input vector containing tracking references, input or output disturbances, or noises on the measured output.

• u is the controlled input to the LTI (LPV) model.

• y is the measured output from the LTI (LPV) model which will result an outputfeedback dynamic controller K (a state-feedback controller is obtained if y = x p where x p is the state vector of the generalized plant P).

• e is the controlled output which describes the objective of the controller.

Appendix B. H ∞ Control Theory

The main objective of the mathematical optimization is to minimize the H ∞ norm of the energy-to-energy gain, which is the closed loop function T ew (s) in (B.1), from the external input w(t) to the controlled output e(t) (L 2 to L 2 norm). The H ∞ norm is defined as the maximum singular value of a function along the whole frequencies (In SISO systems, it is the highest magnitude reached by the transfer function). It is computed numerically using several algorithms where LMI method is the most popular one.

Where γ ∞ , to be minimized, represents how the demanded performance is achieved.

Abstract

Nowadays, systems are getting more and more complex leading to control algorithms able to consider online varying objectives for performance and safety. The field of autonomous systems, in particular autonomous vehicles, is indicative of such an evolution. Indeed, their driving capabilities have been recently improved for highly, and even fully, autonomous driving thanks to advanced control theory.

A fully autonomous car needs to perform several tasks including longitudinal control, lateral control, chassis control, etc. The current thesis proposes different multiobjective control structures aiming to adapt the vehicle performance with covering the full speed range. This thesis presents novel control architectures to switch/interpolate between multiple Linear Parameter-Varying (LPV) controllers, by considering the vehicle speed as the varying parameter. The switching or interpolation scheme is achieved based on Youla-Kucera (YK) parameterization. The LPV-YK control architecture can incorporate multiple LPV or LTI controllers which can switch/ interpolate between them with ensuring stability. It can be used to interpolate between different control performances aiming to achieve different closed-loop specifications in different driving situations (i.e. lane change, obstacle avoidance), or to switch over partitioned parameter subsets when dealing with large parameter variations (i.e. at low and high vehicle speeds). The closed-loop stability is guaranteed for any continuous/discontinuous interpolating signals in terms of a set of Linear Matrix Inequalities (LMIs). The proposed control structures can help multi-variable and multiobjective systems to achieve high performances at different operating conditions and different critical situations regardless of the interpolation rate. Numerical examples are simulated to show the importance of the proposed methods to achieve different objectives for lateral control of autonomous vehicles. In addition, the different control structures have been tested on a real Renault ZOE vehicle to validate their real performance, and compare them with the standard LPV controllers.