Mr Benoît

Mr Valter

J S Leite

Mr Rezak Ayad

Laurence Oudin-Arnoult

A wise man once told me that the most important part of a work is this one, that I'm humbling writing now, with the hope of not forgetting someone who contributed to the development of this manuscript.

First, I would like to thank Ms. Zsófia Lendek and Mr. Dalil Ichalal, for accepting

is a must, living in France, making friends it is essential, especially when you are far from your family. So, I would like to thank Jéssica, Jerônymo, Mariana e Anderson for your friendship and for sharing your lives with us, you were extremely important in our adaptation to this new life. Getting through my manuscript required more than academic support, so, I would like to thank my COACH's Team mates, especially Mr. Nadhir Messai, Alan, and Khalil, thank you for all the fruitful discussions and for your patience with my systematic manner. I would like to also thank the colleagues from CReSTIC who I had the pleasure of exchanging some words, especially Alexandre, Gwendoline, Noémie, Paul, Rafiq, Soizic, Stéphane and Tom, who I spent most of my lunch times, talking about the French culture, agriculture, countryside expressions and giving some free Babyfoot lessons. Thank you for making my day life on the lab lighter and for sharing with us some especial moments such as your marriage Alex and Gwen. I would like to extend my thanks to the old friends who made themselves close even being thousands of kilometers far away.

Most importantly, none of this could have happened without my family support. I must express my very profound gratitude to my wife Viviane who boarded the idea of living this adventure with me. Thank you darling for being present through all the difficult moments and for sharing especial memories with me. I wish to extend the thank to her parents and her sister who gave us daily doses of strength. I would like to thank my parents Simone and Edison, and my brother Juliano for providing me with unfailing support and continuous encouragement throughout my years of study. Thank you for being always present, you are my stronghold and my biggest source of motivation.

Finally, I want to record my gratitude to one and all, who directly or indirectly helped on this work.

ii The n-dimensional real space R n×m

The set of real matrices of dimensions n × m M T The transpose of a matrix M M (ℓ)

The ℓth row of a matrix M e 0 Represents a matrix of zeros, defined as e 0 = 0 n×pn ∈ R n×pn , ∀p ∈ Z + S ν (dashed black) and unstable initial conditions obtained with previous works (× [START_REF] Lv | Fuzzy PID controller design for uncertain networked control systems based on T-S fuzzy model with random delays[END_REF] 3.4 Comparison between the trajectories of the inverted-pendulum on a cart, under sampled-data controllers with gains obtained from different methods for the relaxation of the asynchronous double sums of the design conditions. . . . . . . . 3.5 Simulation of the states trajectories of the inverted pendulum on a cart with the respectively sampled input control signal for a fixed sampling period η k = η = 50ms.

3.7

Closed-loop state responses and control signal of the nonlinear model (3.52) under the sampled-data controller design from the approximated T-S fuzzy model taken from [START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF]

General Introduction

In our everyday life as well as in industry, the place that takes autonomous systems is continuously growing. Such devices are intended to autonomously perform simple daily or even complex tasks (e.g. in manufacturing, services, entertainment...), making accessible some applications, in terms of reduced costs or even viability, that were not previously possible. For instance, in the past few decades, due to the globalization of industrial production, the manufacturing world witnessed growing research toward more efficient processes with increasing requirements in quality associated with the need for cost cuts to maintain the competitiveness, testifying to the increasing employment of robots and specialized machines in production plants [START_REF] Brecher | Automation technology as a key component of the Industry 4.0 production development path[END_REF]. In this context, Industry 4.0 emerged as a step further in integrating automated systems. Using programmable machines, process data analysis, and network interconnected systems makes viable to manufacture complex products that previously demanded numerous unskilled workers with only few skilled ones. These brought back competitiveness in manufacturing technological products in developed countries, where the salaries wages are higher than in underdeveloped countries, previously chosen because of their weak labors' protection, which consequently implied in a reduced outlay [START_REF] Naghavi | Asymmetric Labor Markets, Southern Wages and the Location of Firms[END_REF]. This paradigm change reduced the overall costs and the transportation losses between the production site and the consumers market, which partially addresses the demand for environment-friendly production. Also, in our daily life, technological devices become more and more essential or unavoidable since they are now spreading into many common areas. For instance, we are facing an invasion of Internet of Things (IoT) systems, such like, in home automation, where autonomous vacuum cleaners or many other connected devices are intending to simplify our lives. Another example is the ongoing revolution of transportation with autonomous electric cars, or the use of Unmanned Aerial Vehicles (UAV) for making deliveries or even for entertainment, and so on. This contextualization is intended to give an overview of the field where this thesis research is inserted. In this regard, it is important to highlight that most of the above mentioned technological devices involve continuous-time dynamics, usually mathematically modeled through a set of differential equations. Moreover, to perform the task they are designed for, they usually need to be controlled. Hence, with the advance of actual technologies, their controllers are often implemented on digital devices that process sampled data during scan cycles to compute the adequate control signals, see e.g. [START_REF] Ogata | Engenharia de Controle Moderno[END_REF][START_REF] Åström | Computer-Controlled Systems: Theory and Design[END_REF]. Additionally, the dynamics of these plants are often nonlinear and their sensors and actuators can be connected to the controllers through communication channels. In this context, there are two main points to be addressed. The control of nonlinear systems and the hybrid characteristics of the closed-loop dynamics of such systems where continuous and discrete-time signals are involved [START_REF] Liu | Networked Control Under Communication Constraints[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF].

Among the nonlinear control theory researchers, Takagi and Sugeno have provided, in the mid 80's, an interesting way to represent nonlinear systems [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF]. Initially based on the fuzzy IF-THEN rules formalism, Takagi-Sugeno (T-S) models provide polytopic approximations of nonlinear systems as weighted sums of linear subsystems. Then, with the introduction of the sector nonlinearity decomposition [START_REF] Kawamoto | An Approach to Stability Analysis of Second Order Fuzzy Systems[END_REF][START_REF] Morère | Mise en oeuvre de lois de commande pour les modèles flous de type Takagi-Sugeno[END_REF]Tanaka and Wang, 2001), when analytic nonlinear models are available, T-S models can exactly match nonlinear ones on a compact set of their state space. Such a feature filled the gap with other well-known polytopic approaches that were developed at the same time in the control community, namely the quasi-LPV models, introduced in [START_REF] Shamma | Analysis and design of gain scheduled control systems[END_REF][START_REF] Shamma | Gain-scheduled missile autopilot design using linear parameter varying transformations[END_REF], which are now well-known as equivalent to T-S models.

A vast literature is available for various T-S model-based control problems, for instance dealing with continuous-time controller design, see e.g. [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Cherifi | Uncertain TS model-based robust controller design with D-stability constraints-A simulation study of quadrotor attitude stabilization[END_REF][START_REF] Cherifi | Global non-quadratic D-stabilization of Takagi-Sugeno systems with piecewise continuous membership functions[END_REF], discrete-time ones, see e.g. (Xie et al., 2017a;Lopes et al., 2020b), T-S systems with time-delays, see e.g. [START_REF] Peng | Improved delay-dependent robust stabilization conditions of uncertain T-S fuzzy systems with time-varying delay[END_REF][START_REF] Li | New results on delay-dependent robust stability criteria of uncertain fuzzy systems with state and input delays[END_REF]Bourahala et al., 2017[START_REF] Bourahala | Relaxed Non-Quadratic Stability Conditions for Takagi-Sugeno Systems with Time-Varying Delays: A Wirtinger's Inequalities approach[END_REF], or also sampled-data control, see e.g. [START_REF] Yoneyama | Robust H ∞ control of uncertain fuzzy systems under time-varying sampling[END_REF][START_REF] Zhang | H ∞ control design for network-based T-S fuzzy systems with asynchronous constraints on membership functions[END_REF][START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF].

Indeed, thanks to their convex polytopic structures, stability conditions and controller design conditions for T-S systems are usually studied via Lyapunov approaches and solved in the Linear Matrix Inequality (LMI) framework [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF][START_REF] Skelton | A unified algebraic approach to linear control design[END_REF]. Nevertheless, these LMI-based results provide only sufficient conditions and so suffer from conservatism, which reduction is an important and common challenge in the quasi-LPV/T-S community, see e.g. [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF][START_REF] Guerra | Fuzzy control turns 50: 10 years later[END_REF][START_REF] Lam | A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis[END_REF]Nguyen et al., 2019a) and references therein. Indeed, successive improvements are found in the literature to relax the LMI conditions, from the use of basic quadratic Lyapunov functions [START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF] to non-quadratic ones [START_REF] Jadbabaie | A reduction in conservatism in stability and L 2 gain analysis of Takagi-Sugeno fuzzy systems via linear matrix inequalities[END_REF][START_REF] Blanco | Non quadratic stability of nonlinear systems in the Takagi-Sugeno form[END_REF][START_REF] Tanaka | A Multiple Lyapunov Function Approach to Stabilization of Fuzzy Control Systems[END_REF][START_REF] Morère | Mise en oeuvre de lois de commande pour les modèles flous de type Takagi-Sugeno[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF], by relaxing the parameterized double fuzzy summation structure of LMI conditions [START_REF] Kim | New approaches to relaxed quadratic stability condition of fuzzy control systems[END_REF][START_REF] Tuan | Parameterized Linear Matrix Inequality Techniques in Fuzzy Control System Design[END_REF][START_REF] Xiaodong | New approaches to H ∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI[END_REF], or even considering T-S descriptors [START_REF] Taniguchi | Model Construction, Rule reduction and robust compensation for generalized form of Takagi-Sugeno fuzzy systems[END_REF][START_REF] Tanaka | A Multiple Lyapunov Function Approach to Stabilization of Fuzzy Control Systems[END_REF][START_REF] Marx | Stability and L 2 -Norm Bound Conditions for Takagi-Sugeno Descriptor Systems[END_REF][START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF], or introducing descriptor redundancy [START_REF] Tanaka | A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions[END_REF][START_REF] Guelton | Robust dynamic output feedback fuzzy Lyapunov stabilization of Takagi-Sugeno systems -A descriptor redundancy approach[END_REF][START_REF] Bouarar | Robust non-quadratic static output feedback controller design for Takagi-Sugeno systems using descriptor redundancy[END_REF]. Similarly to these previous related works, this thesis aims at providing relaxed LMI-based controller design procedures for stabilizing T-S models, especially when continuous-time nonlinear systems are controlled by digital devices.

As mentioned above, the control algorithms are nowadays often implemented on digital devices, (i.e. evolving in a discrete-time basis), rather than continuous-time analog ones. However, many physical systems evolve in continuous-time while the action of digital controllers are updated only at specific sampling time instants, e.g. mechatronic systems, embedded systems, networked control systems, and so on. In this context, Figure 1 illustrates an usual control topology of a basic sampled-data system driven by a discrete-time state-feedback controller, which can be implemented into a platform based on an Arithmetic Logic Unit (ALU), such as an industrial Programmable Logic Controller (PLC), a computer, or a microcontroller board, which can be embedded, installed on a control panel with dedicated communication wires, or connected via a data network. Then, based on the sampled measurements x(t k ) of the system's state x(t), provided by an Analog to Digital Converter (ADC), the input control signal u(t k ) is fed-back to the continuous-time nonlinear plant through a Digital to Analog Converter (DAC) and a Zero-Order Hold (ZOH). From the literature, it is possible to gather the main approaches employed to meet such requirements in which practical continuous-time applications are driven by digital controllers into three groups according to the characterization of the sampling interval, the strategy adopted for the project of digital controllers and the associated stability guarantees [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF]):

• Emulation: It consists on the discretization of continuous-time controllers designed ignoring samplings. This is often a common choice in practical applications. However, from the theoretical point-of-view and for a rigorous research perspective, it is worth to highlight that, estimating the maximal allowable sampling period would be very conservative or even analytically tricky [START_REF] Di Ferdinando | Sampled-data emulation of dynamic output feedback controllers for nonlinear time-delay systems[END_REF].

• Direct Design: From a discretized realization of the plant model, a discrete-time controller is designed, disregarding the continuous-time behavior between consecutive samples (Lopes et al., 2020b;[START_REF] Ogata | Discrete-Time Control Systems[END_REF]. It is to be noticed that discretization procedures lead to approximations. Indeed, standard discretization procedure usually consider the well-known Euler transforms [START_REF] Ogata | Discrete-Time Control Systems[END_REF] and recent works in this topic aim at improving the discretization procedure to improve discrete-time models' accuracy, e.g. by considering Taylor series expansions [START_REF] Braga | Improved Discretization Method for Uncertain Linear Systems: A Descriptor System Based Approach[END_REF], Tustin bilinear transforms [START_REF] Ogata | Discrete-Time Control Systems[END_REF][START_REF] Åström | Computer-Controlled Systems: Theory and Design[END_REF] or using the Cayley-Hamilton Theorem [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF]. How-ever, it is worth to point-out that such discrete-time approach may fail to capture the inter-sampling behavior of the continuous-time plant and so stabilize it, especially when large or aperiodic sampling intervals are required [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF].

• Sampled-data Design: To circumvent the aforementioned drawback of the Direct Design approach, the goal is to take into account the inter-sampling behavior of the continuoustime system for the design of the discrete-time controller with some closed-loop stability guarantees. Among the sampled-data analysis methodologies [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF], the most celebrated approaches are the input-delay one [START_REF] Fridman | New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems[END_REF] and the study of the problem in the Hybrid framework [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF][START_REF] Hetel | Stabilization of linear impulsive systems through a nearly-periodic reset[END_REF][START_REF] Goebel | Hybrid dynamical systems[END_REF].

Recognizing the attractiveness of both the Direct Design and Sampled-Data design techniques, a brief survey about these approaches is presented in the sequel. This work investigates these techniques to address the sampled-data controller's design problem for nonlinear systems described by T-S models. In this context, the Direct Design has the appeal of having a consolidated framework that allows us to focus the investigation on the input constraints issues and set-point tracking problem. In contrast, the input-delay approach for Sampled-Data design appears as an interesting and recent alternative to improve stability guarantees under large and/or aperiodic sampling intervals.

Because of their easy implementation on industrial controllers like PLCs, the direct discretetime design for sampled-data systems is the most commonly used technique in industry. In the T-S model-based framework, it is the subject of many studies, see e.g. [START_REF] Guerra | LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF][START_REF] Gonzalez | An improved robust stabilization method for discretetime fuzzy systems with time-varying delays[END_REF][START_REF] Ichalal | A method to avoid the unmeasurable premise variables in observer design for discrete time TS systems[END_REF][START_REF] Lendek | Local stabilization of discrete-time TS descriptor systems[END_REF]. Also, most industrial applications involve practical constraints such like input saturation, set-point tracking, and so on.

Classical linear Proportional-Integral-Derivative (PID) controllers have been widely recognized both in academic and industrial environments, because of their easy implementation aligned with their nice performance properties. Indeed, the proportional action allows to tune the transient speed, the integral action provides the closed-loop system with robustness against model or static errors, and the derivative part helps to improve settling time and stability of the closedloop system. A number of tuning rules for conventional PID controllers can be found in the literature, see e.g. [START_REF] O'dwyer | Handbook of PI and PID controller tunning rules[END_REF]. The simplicity, robustness and the success of this classical linear controller has influenced other control research fields, including fuzzy control systems.

For instance, in [START_REF] Qiao | PID type fuzzy controller and parameters adaptive method[END_REF], a PID-type fuzzy controller has been introduced, that has attracted lots of attention in scientific community. Moreover, several fuzzy PI control applications can be found in different fields such as, among others, in power electronics [START_REF] Nouri | DSP-based implementation of self-tunning fuzzy controller for three-level boost converter[END_REF][START_REF] Tamilarasi | Fuzzy PI Control of Symmetrical and Asymmetrical Multilevel Current Source Inverter[END_REF], wind energy generation [START_REF] Ounnas | Optimal Reference Model Based Fuzzy Tracking Control for Wind Energy Conversion System[END_REF], level control [START_REF] Gao | Setpoints Compensation for Nonlinear Industrial Processes with Disturbances Based on Fuzzy Logic Control[END_REF][START_REF] Kmetóvá | Neuro-fuzzy control of exothermic chemical reactor[END_REF], power train [START_REF] Dragos | Alternative Control Solutions for Vehicles with Continuously Variable Transmission. A Case Study[END_REF], robotics [START_REF] Fateh | Robust Voltage Control of Electrical Manipulators in Task-Space[END_REF], electrical motors [START_REF] Precup | Generic two-degree-of-freedom linear and fuzzy controllers for integral process[END_REF][START_REF] Chen | Speed Control of Vane-Type Air Motor Servo System Using Proportional-Integral-Derivative-Based Fuzzy Neural Network[END_REF], magnetic levitation [START_REF] Sun | Hopf bifurcation analysis of maglev vehicle-guideway interaction vibration system and stability control based on fuzzy adaptive theory[END_REF][START_REF] Yu | T-S Fuzzy Control for Magnetic Levitation Systems Using Quantum Particles Swarm Optimization[END_REF][START_REF] Yu | T-S Fuzzy Control of Magnetic Levitation Systems Using QEA[END_REF] and spacecraft applications [START_REF] Sari | Adaptive fuzzy PID control strategy for spacecraft attitude control[END_REF]. Despite an abundant literature in the T-S model-based framework, we observe that a reduced number of works are concerned with the discrete-time counterpart of fuzzy PI controllers implementation [START_REF] Precup | Generic two-degree-of-freedom linear and fuzzy controllers for integral process[END_REF][START_REF] Preitl | Sensitivity study of a class of fuzzy control systems[END_REF][START_REF] Preitl | Sensitivity Analysis of Low Cost fuzzy controlled servo systems[END_REF]. Note that the discrete-time counterpart of PI controllers can be convenient to be implemented in cheap digital processors, for instance in embedded or industrial dedicated applications. In this context, discrete-time PI-type controllers, can be implemented in a Parallel Distributed Compensator (PDC) control scheme or in a non-PDC structure for conservatism reduction [START_REF] Gonzalez | An improved robust stabilization method for discretetime fuzzy systems with time-varying delays[END_REF]. Usually, the PDC structure requires less computational effort allowing to use less expensive processors [START_REF] Laurain | Avoiding Matrix Inversion in Takagi-Sugeno-Based Advanced Controllers and Observers[END_REF]. Moreover, the above mentioned works do not handle actuators' saturation, which can lead to small regions of attraction and poor performances in real world applications [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF][START_REF] Kong | Disturbance-observer-based fuzzy model predictive control for nonlinear processes with disturbances and input constraints[END_REF]. In this case, a local stability analysis is required [START_REF] Nguyen | An augmented system approach for LMIbased control design of constrained Takagi-Sugeno fuzzy systems[END_REF][START_REF] Klug | Fuzzy dynamic output feedback control through nonlinear Takagi-Sugeno models[END_REF][START_REF] Du | Fuzzy Control for Nonlinear Uncertain Electrohydraulic Active Suspensions With Input Constraint[END_REF], which motivated recent works on T-S model-based control to handle input or state constraints [START_REF] Nguyen | An augmented system approach for LMIbased control design of constrained Takagi-Sugeno fuzzy systems[END_REF][START_REF] Fan | Mechatronics and Robotics Engineering for Advanced and Intelligent Manufacturing, volume 3, chapter DOB Tracking Control for Systems with Input Saturation and Exogenous Disturbances via T-S Disturbance Modelling[END_REF]Li et al., 2016a;[START_REF] Kong | Disturbance-observer-based fuzzy model predictive control for nonlinear processes with disturbances and input constraints[END_REF], domain of validity fuzzy models [START_REF] Lendek | Local stabilization of discrete-time TS descriptor systems[END_REF][START_REF] Klug | Fuzzy dynamic output feedback control through nonlinear Takagi-Sugeno models[END_REF] or even sensors' saturation [START_REF] Zhang | Distributed Filtering for Discretetime T-S Fuzzy Systems with Incomplete Measurements[END_REF].

A consequence of actuators' saturation is the region of attraction in which one must carefully analyze the effects of exogenous inputs in the closed-loop system. Indeed, exogenous signals, such as set-point changes, may drive the trajectories of the system outside of the region of attraction. In this case, both the performance and the stability can be impaired. This effect has been investigated in [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF] where non-PDC PI-like controller has been proposed, but with quite small achieved region of attraction. Moreover, in [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF], a slow enough time-variation of the state vector of the system is required. However, these bad effects on stability and performance may be mitigated by using an anti-windup action [START_REF] Zaccarian | Modern Anti-windup Synthesis: control augmentation for actuator saturation[END_REF][START_REF] Mehdi | A novel anti-windup framework for cascade control systems: An application to underactuated mechanical systems[END_REF]. It is worth to say that to the best of the author's knowledge, no works in the previous literature handle these issues for the design of discrete-time fuzzy PI-like controllers. This point will be the subject of the first contribution of this thesis, developed in Chapter 2.

Nevertheless, as mentioned above, one of the main drawbacks of the Direct Design approach remains on the fact that discretization procedures bring approximations, leading to loss of information regarding the continuous-time systems' dynamics. Indeed, it is worth to highlight that the so designed discrete-time controller does not strictly guarantee the convergence of the continuous system but rather for the discrete-time model, which is particularly true even when the sampling period cannot be small enough to represent the continuous-time plant correctly. Moreover, note that the digital controller part must synchronize the sampling instants, receives the sampled measures from the ADC, computes the control action and sends it to the DAC (see Fig-

ure 1). These tasks are often assumed to be done on a constant sampling interval τ s basis, which has allowed the development of the discrete-time model-based approach for sampled-data systems. However, this assumption is not always accurate since the intervals between two successive sampling instants may vary due to practical constraints. Indeed, even in a point-to-point digital control topology, clock inaccuracy and system architecture characteristics, such like real-time scheduling, can induce jitters, imperfect synchronization and/or computation delays (Witten-mark et al., 1995;[START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. When dealing with Networked Controlled Systems (NCS), in which sampled-data systems are controlled through communication networks [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]Fridman, 2014a), aperiodic sampling intervals are almost inevitable. This is why, several approaches consider triggering the sampling intervals to reduce the network usage [START_REF] De Souza | Co-design of an eventtriggered dynamic output feedback controller for discrete-time LPV systems with constraints[END_REF]. To circumvent these problems, the Sampled-Data Design approach is investigated in Chapter 3 and Chapter 4. Emerging as a promising research topic in control theory, it consists on the investigation of the overall closed-loop stability of continuous-time plants driven by sampled-data controllers, see, e.g. [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF]. In this approach, the control signals are kept constant during the sampling period, and evolve according to discrete samples of the continuous plant, taking into consideration the inter-sampling behavior of the system. An elegant and powerful way to do so is to rewrite the closed-loop dynamics as a continuous-time system with input-time varying delay, also known as a time-delay approach for stabilization of sampled-data systems [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF]. Furthermore, it is widely known from the literature that most real applications exhibit nonlinear dynamics, and if many efforts have been made

by prior authors to stabilize linear dynamical systems from sampled-data measurements, this is not the case for nonlinear systems. Thus, this problem emerges as an exciting subject for our research, which was first explored in the T-S model-based framework in [START_REF] Nishikawa | Design of output feedback controllers for sampled-data fuzzy systems[END_REF]. to be known and introduced in the design conditions [START_REF] Zhang | H ∞ control design for network-based T-S fuzzy systems with asynchronous constraints on membership functions[END_REF][START_REF] Lam | Stabilization of Nonlinear Systems Using Sampled-Data Output-Feedback Fuzzy Controller Based on Polynomial-Fuzzy-Model-Based Control Approach[END_REF][START_REF] Jia | Event-triggered fuzzy H ∞ control for a class of nonlinear networked control systems using the deviation bounds of asynchronous normalized membership functions[END_REF][START_REF] Zhang | Network-based output tracking control for T-S fuzzy systems using an event-triggered communication scheme[END_REF]; enlargement scheme has been introduced in the stabilization criteria in [START_REF] Zhu | An Improved Input Delay Approach to Stabilization of Fuzzy Systems Under Variable Sampling[END_REF][START_REF] Zhu | H ∞ stabilization criterion with less complexity for nonuniform sampling fuzzy systems[END_REF]; the ratio bounds (assumed to be known) of asynchronous membership functions have been considered in [START_REF] Marouf | T-S fuzzy controller design for stabilization of nonlinear networked control systems[END_REF][START_REF] Pan | Event-triggered fuzzy control for nonlinear networked control systems[END_REF][START_REF] Peng | Observer-Based Non-PDC Control for Networked T-S Fuzzy Systems With an Event-Triggered Communication[END_REF] or the upper bounds of the time-derivatives of the membership functions [START_REF] Wang | Fuzzy-Model-Based Sampled-Data Control of Chaotic Systems: A Fuzzy Time-Dependent Lyapunov-Krasovskii Functional Approach[END_REF][START_REF] Wang | An Improved Result on Exponential Stabilization of Sampled-Data Fuzzy Systems[END_REF][START_REF] Kim | Sampled-data control of fuzzy systems based on the intelligent digital redesign method via an improved fuzzy Lyapunov functional approach[END_REF]; [START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF] have implemented a structured vertex separator to reduce the number of LMIs constraints. From these studies, it is clearly shown how important is to take into account the membership functions information into the closed-loop sampled-data stability conditions to reduce their conservatism. However, in the sequel of this thesis, we will show that this brings local results, which have not been considered yet, and so require further meticulous investigation of the closed-loop domain of attraction. Another significant way to reduce the conservatism consists on the convenient choice of a Lyapunov-Krasowskii Functional (LKF), then introducing free weighting matrices, and employing some bounding techniques based on integral terms and modified inequalities [START_REF] Han | Sampled-data Robust H ∞ Control for T-S Fuzzy Time-delay Systems with State Quantization[END_REF]. A looped LKF was first proposed for the sampled-data stabilization of linear systems in [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF], then extended to the T-S model-based framework [START_REF] Lee | New Methods of Fuzzy Sampled-Data Control for Stabilization of Chaotic Systems[END_REF][START_REF] Zeng | Sampled-data stabilization of chaotic systems based on a T-S fuzzy model[END_REF][START_REF] Hua | Stabilization of T-S Fuzzy System With Time Delay Under Sampled-Data Control Using a New Looped-Functional[END_REF]. Indeed, looped LKF appears to be relevant for sampled-data systems as it allows to cope at once with the discrete-time and continuous-time nature of the involved quantities. Following • Lopes, A. N. D., Arcese, L., [START_REF] Lopes | Sampled-data Controller Design with Application to the Quanser AERO 2-DOF Helicopter[END_REF]. Sampled-data Controller Design with Application to the Quanser Aero 2-DOF Helicopter, 2020 IEEE International Conference on Automation, Quality and Testing, Robotics (AQTR), pp. 1-6.

• Lopes, A. N. D., Guelton, K., Arcese, L., Leite, V. J. S. and Bourahala, F. (2020a).

Finsler-based Sampled-data Controller Design for Takagi-Sugeno systems, 21th IFAC World Congress, pp. 1-6.

• Bourahala, F., Guelton, K. and Lopes, A. N. D. (2019)

Traduction en Français de l'Introduction Générale:

Dans notre vie quotidienne comme dans l'industrie, la place que prennent les systèmes autonomes ne cesse de croître. Ces dispositifs sont destinés à effectuer de manière autonome des tâches aussi simples que complexes (e.g. dans la fabrication, les services, le divertissement...), rendant accessibles certaines applications, en termes de réduction des coûts ou même de faisabilité, qui n'étaient pas envisageables auparavant. À titre d'illustration, au cours des dernières décennies, en raison de la mondialisation de la production industrielle, le monde manufacturier a été témoin de nombreuses avancées vers des procédés plus efficaces avec des exigences croissantes en matière de qualité, associées à la nécessité de réduire les coûts pour maintenir la compétitivité, avec l'utilisation croissante de robots et/ou de machines spécialisées dans les usines de production [START_REF] Brecher | Automation technology as a key component of the Industry 4.0 production development path[END_REF]. Dans ce contexte, l'Industrie 4.0 est apparue comme une étape supplémentaire dans l'intégration des systèmes automatisés. L'utilisation de machines programmables, d'analyses de données de processus et de systèmes interconnectés en réseau permet la fabrication de produits complexes qui exigeaient auparavant de nombreux travailleurs non qualifiés pour seulement quelques autres qualifiés. L'Industrie 4.0 a pour objectif de ramener de la compétitivité pour la fabrication de produits technologiques. Notamment, dans les pays développés, l'automatisation permet de réduire les coût de fabrication alors que les salaires sont plus élevés que dans d'autres pays moins développés, parfois plébiscités en raison des bas salaires, de la faible protection sociale des travailleurs, et de l'allègement des contraintes environnementales [START_REF] Naghavi | Asymmetric Labor Markets, Southern Wages and the Location of Firms[END_REF]. Ce changement de paradigme doit permettre de réduire les coûts globaux, de réduire les pertes liées au transport entre le site de production et les consommateurs, répondant en partie à la demande de production respectueuse de l'environnement. Par ailleurs, dans notre vie quotidienne, les dispositifs technologiques deviennent de plus en plus essentiels ou inévitables. Par exemple, l'Internet des objets (IoT, appareils connectés) envahit notre quotidien par la domotique (aspirateurs autonomes, ampoules connectées...), dans l'intention de simplifier nos vies. Un autre exemple est la révolution en cours dans les transports, avec l'arrivée probable de véhicules électriques autonomes, ou l'utilisation de véhicules aériens sans pilote (UAV) afin d'effectuer des livraisons, voire pour du divertissement, etc.

Cette contextualisation vise à donner un aperçu du large domaine dans lequel cette thèse s'inscrit. À cet égard, il est important de souligner que la plupart des dispositifs technologiques mentionnés ci-dessus possèdent une dynamique d'évolution en temps continu, habituellement modélisée mathématiquement par un ensemble d'équations différentielles. Aussi, afin d'accomplir les tâches pour lesquelles ils sont conçus, ils doivent habituellement être contrôlés. Par conséquent, avec le développement des nouvelles technologies, de tels contrôleurs sont souvent mis en oeuvre à l'aide de dispositifs numériques qui traitent les données de manière discrète, sur la base de l'échantillonnage de signaux mesurés, pour fournir la commande adéquate, e.g. [START_REF] Ogata | Engenharia de Controle Moderno[END_REF][START_REF] Åström | Computer-Controlled Systems: Theory and Design[END_REF]. En outre, la dynamique de ces systèmes est souvent non linéaire et leurs capteurs ou actionneurs peuvent être connectés à la partie commande par des canaux de communication numérique. Dans ce contexte, deux points principaux sont à aborder : Le contrôle des systèmes non linéaires, et les caractéristiques hybrides de la dynamique en boucle fermée de ces systèmes où des signaux continus et discrets sont impliqués [START_REF] Liu | Networked Control Under Communication Constraints[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF].

Parmi les nombreuse approches de la théorie du contrôle non linéaire, Tomohiro Takagi et Michio Sugeno ont fourni, au milieu des années 80, une approche intéressante pour représenter les systèmes non linéaires [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF]. Initialement basés sur le formalisme flou de règles SI-ALORS, les modèles Takagi-Sugeno (T-S) fournissent des approximations sous forme polytopique des systèmes non linéaires, i.e. sous forme de sommes pondérées de modèles linéaires locaux. Puis, avec l'introduction de la décomposition en secteurs non linéaires [START_REF] Kawamoto | An Approach to Stability Analysis of Second Order Fuzzy Systems[END_REF][START_REF] Morère | Mise en oeuvre de lois de commande pour les modèles flous de type Takagi-Sugeno[END_REF]Tanaka and Wang, 2001), lorsque des modèles non linéaires analytiques sont disponibles, les modèles T-S peuvent représenter exactement des modèles non linéaires sur un compact de leurs espaces d'état. Une telle caractéristique a comblé le fossé qui les séparait d'autres approches polytopiques bien connues et qui étaient développées en parallèle au sein de la communauté automaticienne, à savoir les modèles quasi-LPV, introduits dans [START_REF] Shamma | Analysis and design of gain scheduled control systems[END_REF][START_REF] Shamma | Gain-scheduled missile autopilot design using linear parameter varying transformations[END_REF], qui sont aujourd'hui communément connus comme équivalents aux modèles T-S.

Une vaste littérature est disponible pour divers problèmes de contrôle basés sur le modèle T-S; par exemple la conception de contrôleurs à temps continu, e.g. [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Cherifi | Uncertain TS model-based robust controller design with D-stability constraints-A simulation study of quadrotor attitude stabilization[END_REF][START_REF] Cherifi | Global non-quadratic D-stabilization of Takagi-Sugeno systems with piecewise continuous membership functions[END_REF]; les systèmes à temps discrets, e.g. (Xie et al., 2017a;Lopes et al., 2020b); les systèmes T-S avec retards, e.g. [START_REF] Peng | Improved delay-dependent robust stabilization conditions of uncertain T-S fuzzy systems with time-varying delay[END_REF][START_REF] Li | New results on delay-dependent robust stability criteria of uncertain fuzzy systems with state and input delays[END_REF]Bourahala et al., 2017[START_REF] Bourahala | Relaxed Non-Quadratic Stability Conditions for Takagi-Sugeno Systems with Time-Varying Delays: A Wirtinger's Inequalities approach[END_REF]; ou la synthèse de contrôleur à base de signaux échantillonnées, e.g. [START_REF] Yoneyama | Robust H ∞ control of uncertain fuzzy systems under time-varying sampling[END_REF][START_REF] Zhang | H ∞ control design for network-based T-S fuzzy systems with asynchronous constraints on membership functions[END_REF][START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF]. En effet, grâce à leurs structures polytopiques convexes, les conditions de stabilité et les conditions de synthèse pour les systèmes T-S sont généralement étudiées via la seconde méthode de Lyapunov et résolues sur la base d'inégalités linéaires matricielles (LMIs) [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF][START_REF] Skelton | A unified algebraic approach to linear control design[END_REF]. Néanmoins, ces résultats fondés sur des contraintes LMIs ne fournissent que des conditions suffisantes et souffrent donc de conservatisme, dont la réduction est un défi important et commun au sein de la communauté traitant les approches quasi-LPV/T-S, e.g. [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF][START_REF] Guerra | Fuzzy control turns 50: 10 years later[END_REF][START_REF] Lam | A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis[END_REF]Nguyen et al., 2019a). De ce fait, des améliorations successives ont été proposées afin de relâcher les conditions LMIs; depuis l'utilisation de fonctions quadratiques de Lyapunov [START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF] jusqu'aux fonctions non quadratiques [START_REF] Jadbabaie | A reduction in conservatism in stability and L 2 gain analysis of Takagi-Sugeno fuzzy systems via linear matrix inequalities[END_REF][START_REF] Blanco | Non quadratic stability of nonlinear systems in the Takagi-Sugeno form[END_REF][START_REF] Tanaka | A Multiple Lyapunov Function Approach to Stabilization of Fuzzy Control Systems[END_REF][START_REF] Morère | Mise en oeuvre de lois de commande pour les modèles flous de type Takagi-Sugeno[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]; en relâchant la structure en double somme paramétrée des conditions LMIs [START_REF] Kim | New approaches to relaxed quadratic stability condition of fuzzy control systems[END_REF][START_REF] Tuan | Parameterized Linear Matrix Inequality Techniques in Fuzzy Control System Design[END_REF][START_REF] Xiaodong | New approaches to H ∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI[END_REF]; en considérant la classe des modèles descripteurs T-S [START_REF] Taniguchi | Model Construction, Rule reduction and robust compensation for generalized form of Takagi-Sugeno fuzzy systems[END_REF][START_REF] Tanaka | A Multiple Lyapunov Function Approach to Stabilization of Fuzzy Control Systems[END_REF][START_REF] Marx | Stability and L 2 -Norm Bound Conditions for Takagi-Sugeno Descriptor Systems[END_REF][START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF]; ou encore en tirant parti de la redondance des descripteurs [START_REF] Tanaka | A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions[END_REF][START_REF] Guelton | Robust dynamic output feedback fuzzy Lyapunov stabilization of Takagi-Sugeno systems -A descriptor redundancy approach[END_REF][START_REF] Bouarar | Robust non-quadratic static output feedback controller design for Takagi-Sugeno systems using descriptor redundancy[END_REF]. À l'instar de ces travaux antérieurs, cette thèse vise à fournir des procédures de conception relâchées sous la forme de LMIs pour la stabilisation des modèles T-S, en particulier lorsque les systèmes non linéaires en temps continu sont contrôlés par des dispositifs numériques.

Comme mentionné ci-dessus, les algorithmes de contrôle sont aujourd'hui souvent mis en oeuvre sur des dispositifs numériques (i.e. évoluant dans une base de temps discret), plutôt que sur des dispositifs analogiques. Cependant, de nombreux systèmes physiques évoluent en temps continu, tandis que l'action des contrôleurs numériques n'est mise à jour qu'à des instants précis et échantillonnés. C'est le cas, par exemple, des systèmes mécatroniques, des systèmes embarqués, des systèmes contrôlés en réseaux, etc. Dans ce contexte, la Figure 1 illustre une topologie de contrôle usuelle d'un système continu, piloté par un contrôleur par retour d'état échantillonné. Celui-ci peut être implémenté dans un Automate Programmable Industriel (API), un ordinateur ou encore une carte microcontrôleur, et éventuellement déporté via un réseau de communication. De plus, sur la base des mesures échantillonnées x(t k ) de l'état du système x(t), fournies par un convertisseur analogique-numérique (ADC), le signal de contrôle d'entrée u(t k ) est transmis au système continu non linéaire à commander via un convertisseur numériqueanalogique (DAC) et maintenu par un bloqueur d'ordre zéro (ZOH).

À partir de la littérature, on constate que les principales approches employées pour répondre à ces exigences, dans lesquelles les applications évoluant en temps continu sont pilotées par des contrôleurs numériques, peuvent être classifiées en trois approches décrites ci-dessous, selon la façon dont est caractérisé l'intervalle d'échantillonnage, la stratégie adoptée pour la synthèse du contrôleur numérique, ou encore les garanties de stabilité associées [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF]) :

• Émulation : Cette approche consiste en la discrétisation de contrôleurs continus, synthétisés sur la base d'un modèle continu ignorant les contraintes d'échantillonnage. Il s'agit ici d'un choix courant dans les applications pratiques. Toutefois, du point de vue théorique et dans une perspective de recherche rigoureuse, il convient de souligner que l'estimation de la période maximale d'échantillonnage admissible serait très conservative, voire difficile à obtenir (Di Ferdinando and Pepe, 2019).

• Synthèse Directe : A partir d'une réalisation discrète du modèle du système à contrôler, une loi de commande en temps discret est synthétisée, sans tenir compte du comportement continu du système entre deux instants d'échantillonnage consécutifs (Lopes et al., 2020b;[START_REF] Ogata | Discrete-Time Control Systems[END_REF]. Notons que les procédures de discrétisation conduisent à des approximations. En effet, la procédure de discrétisation standard considère généralement la célèbre transformation d'Euler [START_REF] Ogata | Discrete-Time Control Systems[END_REF]. Aussi, des travaux récents sur ce sujet considèrent d'autres procédures de discrétisation pour améliorer la précision des modèles à temps discret, e.g. en considérant les décompositions en série de Taylor [START_REF] Braga | Improved Discretization Method for Uncertain Linear Systems: A Descriptor System Based Approach[END_REF], la transformation bilinéaire de Tustin [START_REF] Ogata | Discrete-Time Control Systems[END_REF][START_REF] Åström | Computer-Controlled Systems: Theory and Design[END_REF], ou en utilisant le Théorème de Caley-Hamilton [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF]. Cependant, il convient de souligner qu'une telle approche à temps discret peut ne pas capturer fidèlement le comportement du système continu entre les instants d'échantillonnage successifs, et donc le contrôleur ainsi synthétisé peut échouer à stabiliser le système continu, en particulier lorsque de grands intervalles d'échantillonnage ou des intervalles apériodiques sont nécessaires [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF].

• Synthèse continue-échantillonnée : Pour contourner l'inconvénient susmentionné de l'approche de Synthèse Directe, l'objectif est de prendre en compte le comportement inter-échantillonnage du système à temps continu pour la conception du contrôleur à temps discret avec certaines garanties de stabilité en boucle fermée. Parmi les méthodologies d'analyse des données échantillonnées [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF], les approches les plus répandues considère la réécriture de la boucle fermée sous la forme d'un système à retard sur l'entrée [START_REF] Fridman | New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems[END_REF] ou l'étude du problème selon le formalisme des systèmes hybrides [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF][START_REF] Hetel | Stabilization of linear impulsive systems through a nearly-periodic reset[END_REF][START_REF] Goebel | Hybrid dynamical systems[END_REF].

Reconnaissant l'attrait des techniques de Synthèse Directe et de Synthèse continueéchantillonnée, un bref état de l'art sur ces approches est présenté dans la suite. Ce travail étudie ces techniques pour résoudre le problème de synthèse du contrôleur basé sur des données échantillonnées pour les systèmes non linéaires décrits par les modèles T-S. Dans ce contexte, la Synthèse Directe dispose d'un cadre consolidé qui nous permet de nous concentrer sur les problèmes de contraintes sur l'entrée et le problème de suivi des points de consigne. En revanche, l'approche de Synthèse continue-échantillonnée apparaît comme une alternative intéressante et récente à l'élaboration de conditions garantissant la stabilité en boucle fermée pour des intervalles d'échantillonnage importants et/ou apériodiques.

En raison de leur mise en oeuvre facile sur des contrôleurs industriels tels que les APIs, la Synthèse Directe à temps discret est la technique la plus couramment utilisée dans l'industrie. Dans le cadre des modèle T-S, cette approche fait l'objet de nombreuses études, e.g. [START_REF] Guerra | LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF][START_REF] Gonzalez | An improved robust stabilization method for discretetime fuzzy systems with time-varying delays[END_REF][START_REF] Ichalal | A method to avoid the unmeasurable premise variables in observer design for discrete time TS systems[END_REF][START_REF] Lendek | Local stabilization of discrete-time TS descriptor systems[END_REF]. En outre, la plupart des applications industrielles possèdent des contraintes pratiques telles que la saturation sur les entrées, le suivi des points de consigne, etc. Les contrôleurs Proportionnel-Intégrale-Dérivée (PID) linéaires classiques sont largement employés en raison de leur facilité de mise en oeuvre. En effet, l'action proportionnelle permet de régler le temps de réponse, l'action intégrale fournit au système en boucle fermée une robustesse contre les erreurs statiques, et la partie dérivée contribue à réduire les oscillations du système en boucle fermée. Un certain nombre de règles de réglage pour les contrôleurs PID conventionnels peuvent être trouvées dans la littérature, e.g. [START_REF] O'dwyer | Handbook of PI and PID controller tunning rules[END_REF].

La simplicité, la robustesse et le succès de ces contrôleurs linéaires classiques ont influencé d'autres domaines de recherche en automatique, y compris les systèmes de contrôle flou. Par exemple, dans [START_REF] Qiao | PID type fuzzy controller and parameters adaptive method[END_REF], un contrôleur flou de type PID a été introduit, qui a attiré beaucoup d'attention dans la communauté scientifique. De plus, plusieurs applications de contrôleurs flous PI existent dans différents domaines tels que, entre autres, l'électronique de puissance [START_REF] Nouri | DSP-based implementation of self-tunning fuzzy controller for three-level boost converter[END_REF][START_REF] Tamilarasi | Fuzzy PI Control of Symmetrical and Asymmetrical Multilevel Current Source Inverter[END_REF], la production d'énergie éolienne [START_REF] Ounnas | Optimal Reference Model Based Fuzzy Tracking Control for Wind Energy Conversion System[END_REF], le contrôle de niveau [START_REF] Gao | Setpoints Compensation for Nonlinear Industrial Processes with Disturbances Based on Fuzzy Logic Control[END_REF][START_REF] Kmetóvá | Neuro-fuzzy control of exothermic chemical reactor[END_REF], la robotique [START_REF] Fateh | Robust Voltage Control of Electrical Manipulators in Task-Space[END_REF], les moteurs électriques [START_REF] Precup | Generic two-degree-of-freedom linear and fuzzy controllers for integral process[END_REF][START_REF] Chen | Speed Control of Vane-Type Air Motor Servo System Using Proportional-Integral-Derivative-Based Fuzzy Neural Network[END_REF], la lévitation magnétique [START_REF] Sun | Hopf bifurcation analysis of maglev vehicle-guideway interaction vibration system and stability control based on fuzzy adaptive theory[END_REF][START_REF] Yu | T-S Fuzzy Control for Magnetic Levitation Systems Using Quantum Particles Swarm Optimization[END_REF][START_REF] Yu | T-S Fuzzy Control of Magnetic Levitation Systems Using QEA[END_REF] et des applications spatiales [START_REF] Sari | Adaptive fuzzy PID control strategy for spacecraft attitude control[END_REF]. Malgré une littérature abondante dans le cadre des modèles T-S, nous observons qu'un nombre réduit d'études sont concernées par la contrepartie discrète des contrôleurs PI flous et de leur implémentation [START_REF] Precup | Generic two-degree-of-freedom linear and fuzzy controllers for integral process[END_REF][START_REF] Preitl | Sensitivity study of a class of fuzzy control systems[END_REF][START_REF] Preitl | Sensitivity Analysis of Low Cost fuzzy controlled servo systems[END_REF]. Notons que, la contrepartie de temps discret du contrôleur PI peut être intéressante pour être mis en oeuvre sur des processeurs numériques bon marché, ou pour des applications intégrées ou industrielles dédiées. Dans ce contexte, des contrôleurs flous PI à temps discret peuvent être implémentés dans un schéma de Compensation Parallèle Distribuée (PDC) ou dans une structure non-PDC pour la réduction du conservatisme [START_REF] Gonzalez | An improved robust stabilization method for discretetime fuzzy systems with time-varying delays[END_REF]. Habituellement, la structure PDC nécessite moins d'effort de calcul permettant d'utiliser des processeurs moins coûteux [START_REF] Laurain | Avoiding Matrix Inversion in Takagi-Sugeno-Based Advanced Controllers and Observers[END_REF]. En outre, les travaux mentionnés ci-dessus ne prennent pas en compte la saturation des actionneurs, ce qui peut conduire à de petites régions d'attraction et de mauvaises performances dans les applications réelles [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF][START_REF] Kong | Disturbance-observer-based fuzzy model predictive control for nonlinear processes with disturbances and input constraints[END_REF]. Dans ce cas, une analyse minutieuse de la stabilité locale est requise [START_REF] Nguyen | An augmented system approach for LMIbased control design of constrained Takagi-Sugeno fuzzy systems[END_REF][START_REF] Klug | Fuzzy dynamic output feedback control through nonlinear Takagi-Sugeno models[END_REF][START_REF] Du | Fuzzy Control for Nonlinear Uncertain Electrohydraulic Active Suspensions With Input Constraint[END_REF], qui a motivé des travaux récents sur le contrôle des modèles T-S soumis à des contraintes d'entrée ou d'état [START_REF] Nguyen | An augmented system approach for LMIbased control design of constrained Takagi-Sugeno fuzzy systems[END_REF][START_REF] Fan | Mechatronics and Robotics Engineering for Advanced and Intelligent Manufacturing, volume 3, chapter DOB Tracking Control for Systems with Input Saturation and Exogenous Disturbances via T-S Disturbance Modelling[END_REF]Li et al., 2016a;[START_REF] Kong | Disturbance-observer-based fuzzy model predictive control for nonlinear processes with disturbances and input constraints[END_REF][START_REF] Lendek | Local stabilization of discrete-time TS descriptor systems[END_REF][START_REF] Klug | Fuzzy dynamic output feedback control through nonlinear Takagi-Sugeno models[END_REF][START_REF] Zhang | Distributed Filtering for Discretetime T-S Fuzzy Systems with Incomplete Measurements[END_REF]. Une conséquence de la saturation des actionneurs est la région d'attraction dans laquelle il faut analyser soigneusement les effets des entrées exogènes dans le système en boucle fermée. En effet, les signaux exogènes, tels que les changements de points de consigne, peuvent entraîner les trajectoires du système en dehors de la région d'attraction. Dans ce cas, la performance et la stabilité peuvent être altérées. Cet effet a été étudié dans [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF] où un contrôleur de type PI non-PDC a été proposé, mais conduisant à une région d'attraction relativement restreinte. En outre, dans [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF], l'hypothèse d'une variation suffisamment lente du vecteur d'état du système est rendue nécessaire. Cependant, ces effets néfastes sur la stabilité et les performances peuvent être atténués en utilisant une action anti-windup [START_REF] Zaccarian | Modern Anti-windup Synthesis: control augmentation for actuator saturation[END_REF][START_REF] Mehdi | A novel anti-windup framework for cascade control systems: An application to underactuated mechanical systems[END_REF]. Il est intéressant de dire que, à la connaissance de l'auteur, aucune étude antérieure ne considère ces questions pour la synthèse de contrôleurs PI flous à temps discret. Ce point fera l'objet de la première contribution de cette thèse, développée dans le Chapitre 2.

Néanmoins, comme cela a été mentionné ci-dessus, l'un des principaux inconvénients de l'approche de Syntèse Directe repose sur le fait que les procédures de discrétisation apportent des approximations, conduisant à la perte d'informations concernant la dynamique des systèmes en temps continu. En effet, il convient de souligner que, dans ce cas, le contrôleur à temps discret ainsi conçu ne garantit pas strictement la convergence du système continu, mais seulement celle du système discret, ce qui est particulièrement vrai lorsque la période d'échantillonnage ne peut pas être assez petite pour représenter correctement la dynamique en temps continu.

De plus, notons que la partie commande numérique (voir Figure 1) doit synchroniser les instants d'échantillonnage, recevoir les mesures échantillonnées de l'ADC, calculer l'action de commande et l'envoyer au DAC. Ces tâches sont souvent réalisées sous l'hypothèse d'un intervalle d'échantillonnage τ s constant, ce qui a permis le développement de l'approche de Syntèse Directe. Toutefois, cette hypothèse n'est pas toujours exacte puisque les intervalles successifs entre deux instants d'échantillonnage peuvent varier en raison de contraintes pratiques. En effet, même dans une topologie de contrôle numérique pointà-point, l'imprécision de l'horloge et les caractéristiques de l'architecture du système, comme la programmation en temps réel, peuvent induire une synchronisation imparfaite et/ou des retards de calcul [START_REF] Wittenmark | Timing problems in real-time control systems[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. De plus, lorsqu'il s'agit de systèmes contrôlés en réseau (NCS), les intervalles d'échantillonnage apériodiques sont presque inévitables [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]Fridman, 2014a). Par ailleurs, plusieurs approches envisagent le déclenchement d'intervalles d'échantillonnage variables pour réduire l'utilisation du réseau [START_REF] De Souza | Co-design of an eventtriggered dynamic output feedback controller for discrete-time LPV systems with constraints[END_REF]. Pour contourner ces problèmes, l'approche de Synthèse continue-échantillonnée est étudiée dans le Chapitre 3 et le Chapitre 4. Émergeant comme un sujet de recherche prometteur dans la théorie du contrôle, cette approche consiste à étudier la stabilité globale en boucle fermée des systèmes en temps continu pilotés par des contrôleurs basés sur des données échantillonnées [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF]. Dans cette approche, les signaux de contrôle sont maintenus constants pendant la période d'échantillonnage et évoluent selon des échantillons discrets, en tenant compte du comportement inter-échantillons du système continu. Une façon élégante et puissante de le faire est de réécrire la dynamique en boucle fermée en tant que système à temps continu avec des retards variables sur l'entrée [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF]. En outre, puisque la plupart des applications réelles présentent une dynamique non linéaire, si de nombreux travaux existent pour la stabilisation des systèmes dynamiques linéaires à partir de données échantillonnées, ce n'est pas le cas pour les systèmes non linéaires. Ainsi, ce problème apparaît comme un sujet passionnant, qui a initialement été exploré dans le cadre des modèles T-S dans [START_REF] Nishikawa | Design of output feedback controllers for sampled-data fuzzy systems[END_REF]. En outre, dans le contexte spécifique des contrôleurs basés sur des données échantillonnées, dans le cadre des modèles T-S, cela conduit à l'hybridation temporelle de la dynamique en boucle fermée, i.e. à l'apparition de fonctions d'appartenance asynchrones. Cet aspect a retenu l'attention de nombreux auteurs qui ont proposés des réductions du conservatisme. Par exemple, dans [START_REF] Yoneyama | Robust H ∞ control of uncertain fuzzy systems under time-varying sampling[END_REF], une fonctionnelle de Lyapunov-Krasovskii (LKF) associée à des techniques de relaxation basées sur la formule de Leibniz-Newton et l'introduction de matrices de décision libres, ont été considérées. Ensuite, considérant que les fonctions d'appartenance retardées impliquées dans le contrôleur flou ne correspondent pas à celles existant dans la représentation T-S du système non linéaire continu, les limites supérieures des erreurs asynchrones sont supposées connues et introduites dans les conditions de synthèse [START_REF] Zhang | H ∞ control design for network-based T-S fuzzy systems with asynchronous constraints on membership functions[END_REF][START_REF] Lam | Stabilization of Nonlinear Systems Using Sampled-Data Output-Feedback Fuzzy Controller Based on Polynomial-Fuzzy-Model-Based Control Approach[END_REF][START_REF] Jia | Event-triggered fuzzy H ∞ control for a class of nonlinear networked control systems using the deviation bounds of asynchronous normalized membership functions[END_REF][START_REF] Zhang | Network-based output tracking control for T-S fuzzy systems using an event-triggered communication scheme[END_REF][START_REF] Zhu | An Improved Input Delay Approach to Stabilization of Fuzzy Systems Under Variable Sampling[END_REF][START_REF] Zhu | H ∞ stabilization criterion with less complexity for nonuniform sampling fuzzy systems[END_REF]; les limites du ratio (supposées connues) des fonctions d'appartenance asynchrones ont été considérées dans [START_REF] Marouf | T-S fuzzy controller design for stabilization of nonlinear networked control systems[END_REF][START_REF] Pan | Event-triggered fuzzy control for nonlinear networked control systems[END_REF][START_REF] Peng | Observer-Based Non-PDC Control for Networked T-S Fuzzy Systems With an Event-Triggered Communication[END_REF] ou encore les limites supérieures des dérivées temporelles des fonctions d'appartenance dans [START_REF] Wang | Fuzzy-Model-Based Sampled-Data Control of Chaotic Systems: A Fuzzy Time-Dependent Lyapunov-Krasovskii Functional Approach[END_REF][START_REF] Wang | An Improved Result on Exponential Stabilization of Sampled-Data Fuzzy Systems[END_REF][START_REF] Kim | Sampled-data control of fuzzy systems based on the intelligent digital redesign method via an improved fuzzy Lyapunov functional approach[END_REF]; un séparateur de sommets structurés pour réduire le nombre de contraintes LMI a été proposé dans [START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF]. À partir de ces études, il est clairement démontré à quel point il est important de tenir compte des fonctions d'appartenance dans les conditions de stabilité des données échantillonnées en boucle fermée pour réduire leur conservatisme. Cependant, dans la suite de cette thèse, nous montrerons que cela conduit à des résultats locaux, qui n'ont pas encore été considérés, et donc nécessitent une étude plus méticuleuse du domaine d'attraction en boucle fermée. Une autre façon significative de réduire le conservatisme consiste à choisir une LKF plus adéquate, puis à introduire des matrices de décision libres, et à employer quelques techniques de majoration des termes sous la forme d'intégrale [START_REF] Han | Sampled-data Robust H ∞ Control for T-S Fuzzy Time-delay Systems with State Quantization[END_REF]. Une LKF "bouclée" a été proposée pour la stabilisation des données échantillonnées des systèmes linéaires dans [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF], puis étendu au cadre des modèle T-S [START_REF] Lee | New Methods of Fuzzy Sampled-Data Control for Stabilization of Chaotic Systems[END_REF][START_REF] Zeng | Sampled-data stabilization of chaotic systems based on a T-S fuzzy model[END_REF][START_REF] Hua | Stabilization of T-S Fuzzy System With Time Delay Under Sampled-Data Control Using a New Looped-Functional[END_REF]. En effet, l'utilisation d'une LKF "bouclée" s'avère pertinente pour les systèmes de commande à base de données échantillonnées car il permet de faire face à la fois à la nature discrète et continue des quantités concernées. Sur la base de ces idées, dans le Chapitre 3, à partir du choix d'une LKF "bouclée" adéquate et de l'application du Lemme de Finsler, de nouvelles conditions de synthèse LMIs sont obtenues, ainsi que la proposition d'un schéma de relaxation dédié pour les structures à double somme asynchrones. Ces résultats sont ensuite étendus dans le Chapitre 4. À partir d'une fonctionnelle de Lyapunov-Krasovskii non quadratique (NQLKF) "bouclée", des conditions de synthèse relâchées pour une classe de systèmes descripteurs non linéaires réguliers soumis à la saturation des actionneurs sont proposées. De plus, tout au long de ce dernier chapitre, il est souligné que la méthodologie de commande à base de données échantillonnées n'est valide que localement dans le cadre des modèles T-S. En effet, il existe trois sources principales de contraintes locales : le domaine de validité du modèle T-S, l'hypothèse effectuée sur les dérivées temporelles des fonctions d'appartenance, et les contraintes sur l'entrée dues à la saturation des actionneurs. Dans chacun de ces cas, la caractérisation d'une estimation du domaine d'attraction en boucle fermée, où la stabilité en boucle fermée peut être garantie, doit être effectuée avec un examen minutieux. Notons que, à la connaissance de l'auteur, ce point d'intérêt a été ignoré par la plupart des auteurs dans les études précédentes, ce qui constitue une contribution importante de ce travail de thèse.

En résumé, le reste de ce manuscrit est organisé comme suit. Dans le Chapitre 1, les préliminaires sur la commande des modèle T-S sont fournis pour mieux appréhender la contribution de cette thèse de doctorat. Le Chapitre 2 présente la première contribution de ce travail de thèse, qui consiste en la proposition de nouvelles conditions LMI pour la synthèse de lois de commande PI en suivi de consigne, avec action anti-windup, pour les modèles T-S en temps discret soumis à des saturations sur l'entrée. Ensuite, l'approche de Synthèse continu-échantillonnée, pour la stabilisation des modèles T-S continus par des contrôleurs discrets est considérée dans Chapitre 3, où des réductions du conservatisme sont proposées à partir de la sélection d'une LKF "bouclée" adéquate et de la proposition d'un nouveau schéma de relaxation pour les structures à double somme asynchrones. Ces résultats sont étendus dans le Chapitre 4 à la classe de systèmes T-S descripteurs soumis à des saturations des actionneurs, ainsi qu'à une étude minutieuse du domaine d'attraction en boucle fermée qui en résulte. Enfin, dans la conclusion générale, les perspectives de ces travaux sont discutées.

Chapter 1

Preliminaries on T-S model-based approaches and sampled-data control

Résumé en Français : Préliminaires sur les approches à base de modèles T-S et la commande échantillonnée.

Ce chapitre donne une vue d'ensemble des méthodes et des techniques employées dans cette thèse. Il présente également quelques investigations préliminaires dans le but de mieux cerner le problème étudié et les verrous scientifiques associés pour établir clairement les objectifs de ce travail de thèse. Ainsi, le principal objectif de cette thèse est de proposer de nouvelles conditions relâchées sous la forme d'inégalités matricielles linéaires (LMI) pour la synthèse de contrôleurs basés sur des données échantillonnées, dans le but de garantir la stabilité asymptotique des systèmes non linéaires à temps continu, commandés par des contrôleurs numériques.

Parmi les approches dévolues au contrôle des systèmes non linéaires, nous nous focaliserons sur celles relevant des modèles de type quasi-LPV/T-S [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF][START_REF] Shamma | Gain-scheduled missile autopilot design using linear parameter varying transformations[END_REF]. En effet, ceux-ci permettent de représenter fidèlement les systèmes non linéaires par des modèles polytopiques convexes sur un compact de leurs espaces d'états. Ainsi dans ce chapitre, après avoir présenté la classe des modèles de type quasi-LPV/T-S à temps continu et les différentes méthodes pour les obtenir, nous présentons comment définir les modèles à temps discret qui leur sont associés avec une période d'échantillonnage constante. Ensuite, à partir de la seconde méthode de Lyapunov, les conditions élémentaires de stabilité et de synthèses de contrôleurs sont présentées sous forme de LMIs, dans le cas continu et le cas discret (Tanaka et al., 2001). Puisque de telles conditions ne sont que suffisantes, des méthodes classiquement employées pour réduire le conservatisme sont présentées [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF].

Sur la base de ces travaux pionniers pour les systèmes quasi-LPV/T-S, les limitations de la synthèse de contrôleurs pour les modèles à temps discret sont discutées ainsi que certaines contraintes pratiques telles que la saturation des actionneurs. En effet, une approche basée sur les modèles à temps discret ne donne des résultats satisfaisant que pour des périodes d'échantillonnage relativement petites au regard de la dynamique du système considéré. Aussi, une alternative intéressante est présentée. Celle-ci consiste à représenter la dynamique en boucle fermée par un système à retard variable sur l'entrée [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF], lorsqu'un système continu est contrôlé sur la base de données échantillonnées. Notons que cette approche convient également lorsque de larges intervalles apériodiques d'échantillonnage sont nécessaires [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. Sur la base de l'ensemble des considérations mises en lumière dans ce chapitre, en soulignant certaines limites et défis dans le cadre de la stabilisation des systèmes T-S à temps continu à partir de contrôleurs numériques, ce chapitre se termine par une conclusion où les principales contributions de cette thèse, présentées dans les chapitres suivants sont introduites. À savoir : la synthèse de contrôleurs Proportionnel-Intégral (PI) locaux pour le suivi de consignes continues par morceaux des modèles T-S à temps discret dans le Chapitre 2; la réduction de conservatisme pour la synthèse de contrôleur échantillonné pour les systèmes T-S à temps continu par l'approche à retard variable sur l'entrée dans le Chapitre 3; L'extension de cette approche pour le cas des systèmes T-S descripteurs soumis à des saturations sur l'entrée, conjointement à l'estimation du domaine d'attraction en boucle fermée, dans le Chapitre 4.

Introduction

This chapter gives an overview of the methods and techniques employed in this thesis. It also presents some preliminary investigations, done to fill gaps in the comprehension of the studied problem, which helped to define the methodology and establishing the goals. At the end of this chapter, it should be clear that our main target in this thesis is to propose relaxed Linear Matrix Inequalities (LMI) based conditions for the design of sampled-data controllers, in order to guarantee the asymptotic stability of continuous-time nonlinear systems driven by digital devices.

Among the nonlinear control theory, we focus on quasi-LPV/T-S approaches because of their faculty to accurately represent nonlinear systems as convex polytopic systems. Hence, in this chapter, after presenting the class of continuous-time quasi-LPV/T-S models and the different ways to obtain them, we present how to get their discrete-time realization from constant sampling periods. Then, based on the second Lyapunov method, the basic stability and controller design conditions are presented in terms of LMIs. Also, because such conditions are only sufficient, the most common ways to reduce their conservatism are presented. From these basic approaches, the limitations of discrete-time model-based controller design will be discussed, together with some practical constraints like actuators' saturation. Finally, an elegant alternative to discrete-time model based approaches, suitable when large aperiodic sampling intervals are required, namely the input time-varying delay approach for sampled-data systems, is presented, with highlights on the fact that it is only locally suitable in the T-S model based framework.

Based on the proposed materials and consideration, pointing-out some limitations and challenges involved in the context of stabilizing continuous-time T-S systems from sampled-data controllers, this chapter ends with a conclusion where the main contributions of this thesis, brought in the next chapters, are introduced.

Presentation of Quasi-LPV / Takagi-Sugeno models

In this thesis, we consider the class of affine-in-control nonlinear dynamical systems represented by their nonlinear state space models given by:

ẋ(t) = f x (x(t), u(t)) y(t) = g y (x(t), u(t)) (1.1)
where x(t) ∈ R n , u(t) ∈ R m and y(t) ∈ R q are respectively the system's state, input and output time-varying vectors; f x : R n+m → R n and g y : R n+m → R q are nonlinear functions of the state and input vectors'.

Note that (1.1) describes the nonlinear input-output relationships of a nonlinear system where the first equation, called the state space equation, consists in a finite number of coupled first-order ordinary differential equations, while the second equation, called the output equation, represents the direct static transfer from the state and input to the outputs. Moreover, in the sequel of this thesis, rather than output feedback control approach, which is the subject of many specific studies from the literature (see e.g. [START_REF] Zerar | Linear fractional transformation based H ∞ output stabilization for Takagi-Sugeno fuzzy models[END_REF][START_REF] Guelton | Robust dynamic output feedback fuzzy Lyapunov stabilization of Takagi-Sugeno systems -A descriptor redundancy approach[END_REF][START_REF] Chadli | LMI Solution for Robust Static Output Feedback Control of Discrete Takagi-Sugeno Fuzzy Models[END_REF][START_REF] Bouarar | Robust non-quadratic static output feedback controller design for Takagi-Sugeno systems using descriptor redundancy[END_REF][START_REF] Estrada-Manzo | An alternative LMI static output feedback control design for discrete-time nonlinear systems represented by Takagi-Sugeno models[END_REF] and reference therein), we will stay in the context of state feedback control, with the following assumption, where the characterization of the output equation is not required.

Assumption 1.1. In the sequel of this thesis, we assume that the nonlinear system (1.1) is affine-in-control without direct transfer from the input to the output, such that (1.1) can be reduced to:

ẋ(t) = f (x(t)) + B(x(t))u(t) y(t) = g(x(t)) (1.2)
where f : R n → R n , B : R n → R n×m and g : R n → R q are nonlinear functions depending only on the state entries.

Among the nonlinear control approaches, Takagi-Sugeno (T-S) fuzzy models [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF] have been widely considered in the past few decades because of their ability to represent a large class of nonlinear systems as convex polytopic systems. A vast literature is available for various T-S model-based control problems, for instance dealing with continuoustime controller design (see e.g. [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Cherifi | Non-PDC controller design for Takagi-Sugeno models via line-integral Lyapunov functions[END_REF][START_REF] Cherifi | Uncertain TS model-based robust controller design with D-stability constraints-A simulation study of quadrotor attitude stabilization[END_REF]), discrete-time ones (see e.g. [START_REF] Guerra | Discrete Tagaki Sugeno models for control: Where are we?[END_REF][START_REF] Xie | Control synthesis of discrete-time TS fuzzy systems: reducing the conservatism whilst alleviating the computational burden[END_REF]), or also the control of T-S systems with time-delays (see e.g. [START_REF] Peng | Improved delay-dependent robust stabilization conditions of uncertain T-S fuzzy systems with time-varying delay[END_REF][START_REF] Li | New results on delay-dependent robust stability criteria of uncertain fuzzy systems with state and input delays[END_REF]Bourahala et al., 2017)). Historically, T-S fuzzy models [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF] propose to represent affine-in-control nonlinear models (1.2) by the inference of r fuzzy rules with, as conclusion part, local analytic linear state space models. Thereby, the i-th rules (i ∈ I r ) of a T-S fuzzy model can be described in the following form:

Rule i: IF z 1 (t) is M i1 and • • • and z p (t) is M ip , THEN ẋ(t) = A i x(t) + B i u(t) y(t) = C i x(t) (1.3)
where, for i = 1, . . . , r and j = 1, . . . , p, z j (t) are premise variables depending only (for control purpose) on the states and/or on the output of the system and M ij are fuzzy sets; the matrices

A i ∈ R n×n , B i ∈ R n×m and C i ∈ R
w i (z(t)) = p j=1 M ij (z j (t)) ≥ 0, i ∈ I r , (1.4)
where

z(t) = z 1 (t) z 2 (t) • • • z p (t)
T is the vector of premises.

Hence, we can define the normalized membership function as:

α i (z(t)) = w i (z(t)) r j=1 w j (z(t)) (1.5)
which holds, ∀t, the convex sum properties α i (z(t)) ≥ 0 and

r i=1 α i (z(t)) = 1.
Then, considering such fuzzy inferences, yields the T-S model in its compact defuzzified form given by:

           ẋ(t) = r i=1 α i (z(t)) (A i x(t) + B i u(t)) y(t) = r i=1 α i (z(t))C i x(t)
(1. [START_REF] Shamma | Analysis and design of gain scheduled control systems[END_REF], also belong to the class of polytopic systems. Their main difference with the above described T-S models relies on the fact that LPV models are weighted by time-varying parameters that does not necessary or analytically depend on the states or input variables of the systems. Another interesting class of polytopic systems, namely quasi-LPV systems, has been introduced in [START_REF] Shamma | Gain-scheduled missile autopilot design using linear parameter varying transformations[END_REF] to provide an exact representation of an affine-in-control nonlinear system on a compact set of its state space. It is worth to point-out that the scheduling parameters of quasi-LPV systems depend on the state (or eventually the input) variables, similarly to T-S models. Indeed, as it will be shown in the next section, with the development of the sector nonlinearity approach for obtaining T-S models that exactly match nonlinear ones [START_REF] Kawamoto | An Approach to Stability Analysis of Second Order Fuzzy Systems[END_REF]Tanaka and Wang, 2001;[START_REF] Morère | Mise en oeuvre de lois de commande pour les modèles flous de type Takagi-Sugeno[END_REF], the gap has been filled so that they are nowadays known as strictly equivalent to quasi-LPV models.

Construction of a T-S fuzzy Model

The construction of the T-S fuzzy model is an essential step for the analysis and synthesis of a Fuzzy Model-Based (FMB) control system. In a practical fashion, the fuzzy T-S models can be obtained through an identification or linearization procedure or from a convex polytopic transformation usually based on the sector nonlinearity. In the sequence, these three approaches are summarized.

Identification: Initially proposed in the pioneer works of Takagi and Sugeno [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF], basically consists on the identification from input/output data of the structure and parameters of the system around a set of predefined operation points. Then, these local models are blended together with chosen membership functions (gaussian, triangular, trapezoidal, etc.), which parameters can be further adjusted, as proposed in [START_REF] Gasso | Fuzzy rule based optimisation: a pruning and merging approach[END_REF].

If these approaches are very appealing when an analytic nonlinear model of the plant is unavailable, let us recall that such obtained T-S models are only approximately representing the nonlinear systems, so those results (stability analysis, controller and observer design...) should be considered with caution.

Linearization: This method is based on the linearization of a nonlinear model around a finite number of operating points [START_REF] Ma | Analysis and design of fuzzy controller and fuzzy observer[END_REF]Tanaka and Wang, 2001). Then a T-S representation is obtained by the interconnection of the local linear polytopes by some well chosen membership functions (gaussian, triangular, trapezoidal, etc.). We must highlight that this approach also results in an approximation of the nonlinear system and usually the number of models depends on the complexity of the system, the specified modeling precision and the choice of the membership functions.

Sector Nonlinearity Approach:

The idea of using the sector nonlinearity for the construction of the fuzzy T-S representation is based on the idea that, for any continuous scalar nonlinear functions f : R → R, with f (0) = 0, there exists

D x ⊆ R such that, ∀x ∈ D x we have f (x) ∈ a 1 a 2 x
where a 1 and a 2 are finite scalars. Then, when D x = R, the nonlinear sector is said to be global, otherwise, when D x ⊂ R (e.g. 1.1 (Tanaka and Wang, 2001).

D x = {x ∈ [-d, d]}, with d > 0) it is local, as illustrated in Figure
For a large majority of systems it is reasonable to have a well defined operation region and consequently to have bounded state variables. So, it is wise to use this information for the fuzzy T-S model construction, which will be exactly matching the nonlinear model in a compact subset of its state space, to generalize the above definition of nonlinear sectors for x(t) ∈ R n , given by:

D x = {x(t) ∈ R n | Lx(t) ≤ d} (1.7)
where κ is the number of bounds of the state variables, which constitutes the entries of the vector d ∈ R κ and L ∈ R κ×n is a state component selection matrix. [START_REF] Morère | Mise en oeuvre de lois de commande pour les modèles flous de type Takagi-Sugeno[END_REF].

Lemma 1.1. Let f s : D x → [ fs , fs ] a bounded function. There always exists a pair of scalar functions w 1 : D x → [0, 1] and w 2 : D x → [0, 1] and two scalars a and b such that:

f s (x) = w 1 (x)a + w 2 (x)b,
where w 1 (x) + w 2 (x) = 1.

For instance, let us assume that the nonlinear affine-in-control systems (1.2) can be rewritten as:

ẋ(t) = A(x(t))x(t) + B(x(t))u(t) y(t) = C(x(t))x(t) (1.8)
where the matrices A(x(t)) ∈ R n×n , B(x(t)) ∈ R n×m and C(x(t)) ∈ R q×n contain p nonlinear entries of the state variables, denoted f j : D x → [ fj , fj ] (j ∈ I p ). Then, ∀x(t) ∈ D x , with a, b ≥ 0 we can write:

f j (x(t)) = f j (x(t)) - fj fj - fj w 1 j (x(t)) fj + fj -f j (x(t)) fj - fj w 2 j (x(t))
fj , (1.9)

with w 1 j (x(t)) + w 2 j (x(t)) = 1, w 1 j (x(t)) ≥ 0 and w 2 j (x(t)) ≥ 0. So, we may readily get a T-S model in the compact form (1.6) that exactly matches (1.8) on D x with the following r = 2 p normalized membership functions α i (x(t))(i ∈ I r ) and matrices defining the vertices: introduced. This last point will be illustrated through simulation examples along the thesis.

                                   α 1 (x(t)) = w 1 1 (x(t))w 1 2 (x(t)) . . . w 1 p-1 (x(t))w 1 p (x(t)), M 1 = M ( f1 , f2 , . . . , fp-1 , fp ), α 2 (x(t)) = w 1 1 (x(t))w 1 2 (x(t)) . . . w 1 p-1 (x(t))w 2 p (x(t)), M 2 = M ( f1 , f2 , . . . , fp-1 , fp ), α 3 (x(t)) = w 1 1 (x(t))w 1 2 (x(t)) . . . w 2 p-1 (x(t))w 1 p (x(t)), M 3 = M ( f1 , f2 , . . . , fp-1 , fp ), α 4 (x(t)) = w 1 1 (x(t))w 1 2 (x(t)) . . . w 2 p-1 (x(t))w 2 p (x(t)), M 4 = M ( f1 , f2 , . . . , fp-1 , fp ), . . . α r-2 (x(t)) = w 1 1 (x(t))w 2 2 (x(t)) . . . w 1 p-1 (x(t))w 2 p (x(t)), M r-2 = M ( f1 , f2 , . . . , fr-1 , fp ), α r-1 (x(t)) = w 2 1 (x(t))w 1 2 (x(t)) . . . w 2 p-1 (x(t))w 1 p (x(t)), M r-1 = M ( f1 , f2 , . . . , fp-1 , fp ), α p (x(t)) = w 2 1 (x(t))w 2 2 (x(t)) . . . w 2 p-1 (x(t))w 2 p (x(t)), M r = M ( f1 , f2 , . . . , f-1 , fp ), ( 1
In the sequence, two examples give an overview on the procedure to obtain a fuzzy T-S representation for a nonlinear system from the sector nonlinearity approach.

Example 1.1. Let us consider the following academic example of a nonlinear system:

ẋ1 (t) = 2x 1 (t) + x 1 (t)x 2 (t) + (2 -cos x 2 (t))u(t) ẋ2 (t) = -2x 1 (t) cos x 2 (t) + x 2 (t) (1.11) Defining x(t) = x 1 (t) x 2 (t)
T , it can be rewritten as:

ẋ(t) = A(x(t))x(t) + B(x(t))u(t)
(1.12) with:

A(x(t)) = 2 x 1 (t) -2 cos x 2 (t) 1 and B(x(t)) = 2 -cos x 2 (t) 0
which includes two nonlinear functions f 1 (x(t)) = x 1 (t) and f 2 (x(t)) = cos x 2 (t), that are dependent on the state variables. From (1.9), to obtain a T-S representation, ∀(x 1 (t), x 2 (t)) ∈

[-b, a] × R = D x , we can write:

f 1 (x(t)) = x 1 (t) + b a + b w 1 1 (z(t)) a + a -x 1 (t) a + b w 2 1 (z(t)) (-b) and f 2 (x(t)) = cos x 2 (t) + 1 2 w 1 2 (z(t)) 1 + 1 -cos x 2 (t) 2 w 2 2 (z(t)) (-1)
So, from (1.10), we readily get the following normalized membership functions and vertices:

                                 α 1 (x(t)) = w 1 1 (x(t))w 1 2 (x(t)), A 1 = 2 a -2 1 , B 1 = 1 0 , α 2 (x(t)) = w 1 1 (x(t))w 2 2 (x(t)), A 2 = 2 a 2 1 , B 2 = 3 0 , α 3 (x(t)) = w 2 1 (x(t))w 1 2 (x(t)), A 3 = 2 -b -2 1 , B 3 = B 1 , α 4 (x(t)) = w 2 1 (x(t))w 2 2 (x(t)), A 4 = 2 -b 2 1 , B 4 = B 2 ,
which define the following T-S models with r = 4:

ẋ(t) = 4 i=1 α i (z(t))(A i x(t) + B i u(t)) (1.13)
where, ∀i ∈ I r , α i (z(t)) ≥ 0 and

4 i=1 α i (z(t)) = 1.
Example 1.2 (1 Degree of Freedom (DOF) Inverted Pendulum). The first step when projecting model based controllers is to obtain a good representation of the original system. Let us consider the problem of stabilizing the inverted pendulum depicted in Figure 1.3. The dynamical behavior of this system can be described by the following differential equation:

mgL sin θ(t) -kL θ(t) + u(t) = mL 2 θ(t)
where θ(t) denotes the angular position regarding the vertical axis, u(t) is the control input torque, g = 9.8 m/s 2 is the gravitational acceleration, m = 0.5 kg refers to the pendulum mass, L = 0.8 m is associated with the length of the Pendulum and last, k = 0.2 SI is the coefficient of friction at the pivot point. Defining the state vector x T (t) = θ(t) θ(t) , a state-space representation of (1.3) can be given as:

ẋ(t) = 0 1 g L sin x 1 (t) x 1 (t) -k mL x(t) + 0 1 mL 2 u(t) (1.14)
Taking benefit from the fact z(x 1 (t)) = (sin x 1 (t))/x 1 (t) ∈ [ρ, 1] is always bounded (e.g., for

x 1 (t) ∈ R, ρ ≈ -0.217234), we get:

z(x(t)) = z(x(t)) -ρ 1 -ρ α 1 (x(t)) 1 + 1 -z(x(t)) 1 -ρ α 2 (x(t)) ρ (1.15)
which readily gives the following globally matching T-S model with r = 2:

ẋ(t) = 2 i=1 α i (x(t)) (A i x(t) + B i u(t)) (1.

16)

with:

A 1 = 0 1 g L -k mL , A 2 = 0 1 g L ρ -k mL and B 1 = B 2 = 0 1 mL 2 .

Discrete-time T-S fuzzy models

Until now, we presented T-S models in the continuous-time framework. However, some dynamical systems may also evolve in the discrete-time. This is the case of digital systems, where the dynamics is characterized by discrete sequences of k ∈ N. Also, recall that most of continuous-time systems are nowadays driven by digital devices. In this case, design procedures are often based on a discretized representation of their continuous-time dynamics. This being said, a discrete-time version of T-S models is given by the following recurrence equation:

           x k+1 = r i=1 α i (z k ) (A i x k + B i u k ) y k = r i=1 α i (z k )C i x k
(1.17)

where x k ∈ R n , u k ∈ R m and y k ∈ R q denote respectively the state, the input and the output vectors taken at the discrete instant k; ∀i ∈ I r , A i ∈ R n×n , B i ∈ R n×m and C i ∈ R q×n are real constant matrices describing the local discrete-time dynamics of the system.

In this thesis, we are mainly focused on the control of continuous-time systems by digital controllers. Therefore, let us now discuss the way to get a discrete-time T-S model (1.17) from the continuous-time ones (1.6). One of the most considered approach consists in the forward Euler discretization of the continuous-time system [START_REF] Chen | Linear System Theory and Design[END_REF]Tanaka and Wang, 2001). Hence, let:

ẋ(t) = lim τs→0 x(t + τ s ) -x(t) τ s , (1.18)
Then, when τ s → 0, from the continuous-time T-S model (1.6), we can write:

           x(t + τ s ) = r i=1 α i (z(t)) ((I + τ s A i )x(t) + τ s B i u(t)) y(t) = r i=1 α i (z(t))C i x(t) (1.19)
Therefore, assuming a sampling period τ s , at sampling instant t k = kτ s (k ∈ N), the discrete-time T-S model (1.17) approximates the continuous-time one (1.6) with:

A i = (I + τ s A i ), B i = τ s B i and C i = C i . (1.20)
As a matter of fact, it is often assumed that the error introduced by discretization is negligible when a sufficiently small enough sampling period is selected regarding the highest-frequency component of a system time response. In this context, the Shannon sampling Theorem brings some rules for choosing an appropriated sampling period.

Theorem 1.1 (Shannon sampling Theorem). [START_REF] Åström | Computer-Controlled Systems: Theory and Design[END_REF] For a uniformly sampled digital signal processing system, an analog signal can be perfectly recovered as long as the sampling rate is at least twice as large as the highest-frequency component of the analog signal to be sampled.

However, identifying the highest-frequency component of a system time-response can be complex, and usually, a more empirical rule is adopted. From this approach, the constant sampling period τ s is often taken in order to produce a ratio of 4 to 10 times between the rise time τ r and the fixed sampling interval τ s , i.e., to have the sampling rate 4 to 10 times faster than the system dominant time constant τ c in the case of first-order systems [START_REF] Åström | Computer-Controlled Systems: Theory and Design[END_REF].

Moreover, it is important to highlight that the choice of the sampling period τ s has a great influence on the discrete-time realization, and, sometimes due to the control project limitations such as cost reduction or energy consumption, it is necessary to find a suitable trade-off between catching all of the dynamics of the plant or alleviating the hardware requirements for the control device. Some other approaches exist to obtain a discrete-time approximation of a continuoustime system with better accuracy than the forward Euler approach, e.g. using Tustin bilinear transforms [START_REF] Ogata | Discrete-Time Control Systems[END_REF][START_REF] Åström | Computer-Controlled Systems: Theory and Design[END_REF], using Taylor series expansions or the Cayley-Hamilton theorem [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF], or more recently using a descriptor systembased approach in the T-S framework [START_REF] Braga | Improved Discretization Method for Uncertain Linear Systems: A Descriptor System Based Approach[END_REF]. Even if appealing, these will be left-out from this thesis since it is not our purpose to compare every discretization approaches.

Remark 1.3. The choice of the sampling period τ s may affect the accuracy of the discretetime representation of a continuous-time system. Therefore, when considering discrete-time realization for the stability analysis, control or observer design, conclusions regarding to the continuous-time system must be taken with care, especially when τ s cannot be chosen small enough. Indeed, in this case, it is well-known that discrete-time models fail to capture the intersampling behavior of dynamical continuous-time systems [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. This absence of guarantee for large sampling period can be seen as a drawback of such approaches, especially when the controller is implemented on a low-cost device or when considering networked control systems to drive plants, and especially when they involve fast dynamics.

The usual way to obtain a T-S discrete-time model from a continuous one being presented, the following example is proposed to illustrate the effect of the choice of the sampling period τ s on the accuracy of the discrete-time realization. (1.17) with r = 2 and the following matrices:

A 1 = 1 τ s g L τ s 1 -k mL τ s , A 2 = 1 τ s g L τ s ρ 1 -k mL τ s and B 1 = B 2 = 0 τs mL 2 (1.21)
If obtaining the matrices (1.21) via Euler discretization for the discrete-time T-S fuzzy model

(1.16) happens to be an easy task, the choice of the constant sampling instant is not so evident since we are dealing with an unstable nonlinear systems. Indeed, from the best of author's knowledge, there is no generic approach in the literature to find the minimal sampling interval required to precisely describe this class of systems. Thus, two approaches are explored to give some thoughts on how to choose a reasonable value for the fixed sampling period τ s .

First, the open-loop time response of the continuous system (1.14) is compared with the one for the discrete-time T-S fuzzy model (1.17 Hold (ZOH), and an error of about 10%. Then, our choice is confined to 1ms or 5ms, if the first option delivery an accuracy of at about 1%, it also requires 5 times faster samplings than the second, while the error from it raise to less than 3%. So, thinking in terms of compromise, it seems that τ s = 5ms is a reasonable choice for the constant sampling period.

In other hand, let us highlight that, because the introduction of a controller makes the closedloop dynamics having different characteristics (constant of time, highest frequency components...)

than the open-loop, the above chosen sampling period may be irrelevant for closed-loop systems.

To illustrate this fact, let us consider here a continuous-time linear state feedback controller u(t) = Kx(t), with K = 4.24 0.16 , to stabilize this 1 DOF inverted pendulum around its unstable equilibrium point (θ = 0) with closed-loop eigenvalues λ 1,2 = -0.5 ± i (3)/2. Then, the continuous-time dynamics is discredited using the forward Euler approach with different values of τ s picked as 20ms, 50ms, 100ms, 200ms and 300ms. For all these cases, the closed-loop time-responses are depicted in Figure 1.5 with there respective normalized discretization errors.

From these results, we now notice that choosing τ s = 20ms brings a normalized error which is less than 3%, which is almost the same than in the open-loop test but with a four times larger sampling period. This illustrate the difficulty to choose the most suitable sampling period in this discrete-time controller design framework.

Basic Stability Analysis and Stabilization of T-S models

In this section, the goal is to present the basic stability conditions for T-S models, then to present their extension to controller design. Because T-S models belong to the class of nonlinear systems, we will first recall some basics of the Lyapunov theory, which is commonly considered in this framework.

Second Lyapunov method for the stability analysis of dynamical systems

One of the most important results in the analysis of dynamical system was the theory proposed by Aleksandr Lyapunov, at the end of the 19th century in its original thesis "The general problem of the stability of motion", translated from Russian in [START_REF] Lyapunov | The general problem of the stability of motion[END_REF]. The main idea relies in the fact that if a dynamical system loses energy over time and the energy is never restored, then the system must grind to a stop and reach some final resting state.

Let us consider an autonomous nonlinear system given in the continuous-time framework by:

ẋ(t) = f (x(t)) (1.22)
or in the discrete-time framework by:

x k+1 = f (x k ) (1.23)
where the origin O is, without loss of generality by a change of origin, assumed to be an equilibrium point.

The concept of stability is strictly related to the Lyapunov stability theory. This theory establishes that systems whose trajectories asymptotically converge to an equilibrium point progressively lose energy in a monotonic way. Lyapunov generalizes the notion of energy using a V : R n → R + function called "Lyapunov candidate function" which depends on the system's states. This function is usually a norm or a distance. The main theorems, adapted from [START_REF] Khalil | Nonlinear Systems[END_REF] for the continuous and discrete-time framework, are given in the sequence.

V (0) = 0 and V (x) > 0, ∀x = O (1.24)
Then, the continuous-time system (1.22) (or the discrete-time system (1.23)) is asymptotically stable (at least locally at the origin O) and V is said to be a Lyapunov function if, ∀x(t) = O: (1.25) or:

V (x) < 0, in the continuous-time framework,
V (x k+1 ) -V (x k ) < 0, in the discrete-time framework, (1.26)
Moreover, if V is radially unbounded, then the continuous-time system (1.22) (or the discretetime system (1.23)) is globally asymptotically stable if:

||x|| → ∞ ⇒ V (x) → ∞ (1.27)
Next, another definition of the Lyapunov stability conditions is presented which uses comparison functions, known as class K functions and defined as follows.

Definition 1.1 (Class K comparison functions [START_REF] Khalil | Nonlinear Systems[END_REF]). A continuous function β :

[0, a) → [0, ∞) is said to belong to class K if it is strictly increasing and β(0) = 0. It is said to belong to class K ∞ if a = ∞ and β(r) → ∞ as r → ∞.
β 1 (||x||) ≤ V (x) ≤ β 2 (||x||), (1.28) and: V (x) ≤ -β 3 (||x||), in the continuous-time framework, (1.29)
or:

V (x k+1 ) -V (x k ) ≤ -β 3 (||x||), in the discrete-time framework, (1.30)
then the system is asymptotically stable (at least locally at the origin O) and V is a Lyapunov function.

Moreover, if Ω = R n and β 1 , β 2 and β 3 are K ∞ functions, V is radially unbounded and the system is globally asymptoticallly stable.

From Theorem 1.2, a continuously differentiable function V (x) satisfying (1.24) and (1.25) (or (1.26)) is called a Lyapunov function and, for some c > 0, V (x) = c is called an equipotential level set of Lyapunov. Therefore, the condition V ≤ 0 implies that when a trajectory crosses an equipotential of Lyapunov, the system's trajectories necessary move towards a lower equipotential and so, the set

L(c) = {x ∈ R n |V (x)
≤ c} is said to be contractive and the trajectories can never come out again. That property is often used when only local stability can be guaranteed, i.e. when (1.27) cannot be satisfied, to characterize an estimate of the stability domain of attraction.

The main definition of the Lyaponov stability conditions which will be considered in this thesis being presented, the next subsection provides the pioneer results on the stability analysis of T-S models.

Stability Analysis and Stabilization of T-S Models

In the context of polytopic models, Linear Matrix Inequalities (LMIs) are a fundamental tool for the stability analysis and synthesis of controllers for nonlinear systems and can be easily implemented through convex optimization techniques (e.g., see [START_REF] Skelton | A unified algebraic approach to linear control design[END_REF][START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF][START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF]). Since the introduction of semidefinite programming (SDP), several problems from control theory have been formulated as a convex optimization problem and numerically solved. They seldom include a linear objective function subject to a constraint written as an affine combination of symmetric matrices [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF].

Together, the Lyapunov functions were extensively employed for the stability study of polytopic systems [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]Tanaka and Wang, 2001;[START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF]. The main advantage of this combination is the possibility of expressing the stability criteriums in terms of LMIs. This last has been efficiently numerically solved using interior-point methods, and from this approach, an optimal solution is guaranteed. Today, several software toolboxes are available to write LMIs problems like the Control Robust Toolbox for MATLAB ® [START_REF] Gahinet | LMI Control Toolbox for Use with MATLAB[END_REF] and YALMIP [START_REF] Löfberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF] and solve them via interior-point algorithms, such as SeDuMi [START_REF] Sturm | Using sedumi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF] and Mosek (ApS, 2019).

In the next subsections, we will provide the basic LMI-based quadratic stability conditions for autonomous T-S models, then the pioneer LMI-based results for T-S model-based quadratic design of dedicated controllers.

Quadratic stability analysis:

Let us focus on the stability analysis of the following autonomous T-S models:

ẋ(t) = r i=1 α i (z(t))A i x(t) (1.31)
or in their discrete-time version:

x t+k = r i=1 α i (z k )A i x k (1.32)
Then, to apply Theorem 1.2, consider the following Quadratic Lyapunov Function (QLF) candidate: • In continuous-time framework:

V (x) = x T P x, with x ≡ x(t) or x ≡ x k , (1.33) which satisfies (1.24) if P ∈ R n×n is
V (x(t)) = ẋT (t)P x(t) + x T (t)P ẋ(t) = x T (t) r i=1 α i (z(t)) A T i P + P A i x(t) < 0 (1.34)
• In discrete-time framework:

V (x k-1 ) -V (x k ) = x T k+1 P x k+1 -x T k P x k = x T k r i=1 α i (z k ) A T i P A i + P x k < 0 (1.35)
The T-S models (1.31) and (1.32) being convex polytopic models, we always have α i (x) ≥ 0 (∀i ∈ I r ), yielding to the first quadratic stability conditions given as a convex optimization procedure to be solved as LMIs and summarized by the following theorem.

Theorem 1.4. [START_REF] Tanaka | Stability analysis and design of fuzzy control systems[END_REF] The autonomous continuous-time T-S model (1.31)

(or the discrete-time T-S model (1.32)
) is asymptotically stable (at least locally) if there exists 0 < P = P T ∈ R n×n such that the following LMI's are satisfied for all i ∈ I r :

A T i P + P A i < 0, in the continuous-time framework,
(1.36)

A T i P A i -P < 0, in the discrete-time framework.
(1.37)

The basic quadratic stability conditions having been established, the following subsection presents their extension to controller design.

Quadratic design of stabilizing PDC controllers:

T-S models belonging to the class of convex polytopic systems, assuming that the state are available from measurement, and assuming that the premise variables depend only on the state, a convenient way to drive them is to employ a gain-scheduled polytopic controller based on the same interconnection structure (membership functions), namely the Parallel-Distributed-Compensation (PDC) state feedback controllers given by, for continuous-time T-S models (1.6):

u(t) = r j=1 α j (z(t))K j x(t) (1.38)
and for discrete-time T-S models (1.17):

u k = r j=1 α j (z k )K j x k (1.39)
Substituting the PDC control laws (1.38) and (1.39) into (1.6) and (1.17), respectively expresses the continuous-time and discrete-time closed-loop dynamics as:

ẋ(t) = r i=1 r j=1 α i (z(t))α j (z(t)) (A i + B i K j ) x(t) (1.40)
and:

x k+1 = r i=1 r j=1 α i (z k )α j (z k ) (A i + B i K j ) x k (1.41)
Therefore, assuming a candidate quadratic Lyapunov function (1.33) and applying Theorem 1.2, we can readily get that the closed-loop dynamics (1.40) and (1.41) are respectively asymptotically stable (at least locally) if:

r i=1 r j=1 α i (z(t))α j (z(t)) (A i + B i K j ) T P + P (A i + B i K j ) < 0 (1.42)
and:

r i=1 r j=1 α i (z k )α j (z k ) (A i + B i K j ) T P (A i + B i K j ) -P < 0 (1.43)
From the Schur complement, (1.43) is equivalent to:

r i=1 r j=1 α i (z k )α j (z k ) -P ⋆ A i + B i K j -P -1 < 0 (1.44)
Then, let M j = K j P -1 , by congruence of (1.42) with X = P -1 and (1.43) with diag{X, I}, and from the fact that:

r i=1 r j=1 α i (.)α j (.)Γ ij = r i=1 α 2 i (.)Γ ij + 1 2 r i=1 r j=1 α i (.)α j (.)(Γ ij + Γ ji ),
we get the conditions expressed in the following theorem.

Theorem 1.5. [START_REF] Tanaka | Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs[END_REF] The (or (1.39)) if there exists 0 < X = X T ∈ R n×n such that the following LMI's are satisfied for all (i, j) ∈ I 2 r :

Γ ii < 0 (1.45) Γ ij + Γ ji < 0, ∀i < j (1.46)
with:

Γ ij = A i X + XA T i + B i M j + M T j B T i , in the continuous-time framework, (1.47)
and:

Γ ij = -X ⋆ A i X + B i M j -X
, in the discrete-time framework.

(1.48)

In that case, the PDC controller gains and Lyapunov matrices can be recovered by the change of variables K j = M j X -1 and P = X -1 .

Remark

(c) = {x ∈ R n | V (x) ≤ c} ⊆ D x .
The conditions of Theorem 1.4 and Theorem 1.5 do not require any information about the membership functions, which holds the convex sum properties. This implies that these conditions are only sufficient ones and so suffer from conservatism [START_REF] Sala | Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem[END_REF]. In other words, not finding a solution for a convex optimization problem from these conditions does not mean that the considered T-S model is unstable or cannot be stabilized. Different ways to reduce the conservatism of LMI-based conditions will be discussed in the next subsection.

Before going to the next subsection, let us highlight some limitations from the use of discretetime T-S model based approach to control a continuous-time system. Indeed, because the discretization of a continuous-time system involves approximations, it is well-known that such designed discrete-time controller ensures the stability of the discrete-time model, but not necessarily of the original continuous-time system it is dedicated to, especially when large sampling period are considered. This happens because the inter-sampling behavior of a continuous-time system is lost in the above detailed discrete-time framework [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. To circumvent such drawback, appealing approaches appeared during the last few decades, especially the input-delay approach for sampled-data systems [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF], which will be further presented and the focus of our contributions in Chapters 3 and 4.

Some usual ways to relax LMI-based conditions for T-S models:

As mentioned above, the basic conditions expressed for the stability analysis and controller design suffer from conservatism. Among the sources of conservatism investigated, we recall in the following the currently investigated ones in the T-S model-based framework, which will be considered to get the new results proposed in this thesis.

First, note that the closed-loop dynamics (1.40) and (1.41) involve a convex double sum structure and so, also the Lyapunov conditions (1.42) and (1.44). Hence, checking the negativity of all their vertices make the results very pessimistic. A first relaxation of such double sum constraint is presented in Theorem 1.5. However, further relaxation improvements have been proposed in the literature, see e.g. [START_REF] Kim | New approaches to relaxed quadratic stability condition of fuzzy control systems[END_REF][START_REF] Tuan | Parameterized Linear Matrix Inequality Techniques in Fuzzy Control System Design[END_REF]Xiaodong and Qingling, 2003). In this thesis, we will consider Tuan's relaxation lemma given below, since it is commonly considered as a good compromise between complexity and conservatism reduction [START_REF] Sala | On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems[END_REF][START_REF] Lam | A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis[END_REF].

Lemma 1.2. [START_REF] Tuan | Parameterized Linear Matrix Inequality Techniques in Fuzzy Control System Design[END_REF]: For (i, j) ∈ I 2 r , let Γ ij be matrices of appropriate dimensions. The inequality

r i=1 r j=1 α i (.)α j (.)Γ ij < 0 is satisfied if the following conditions hold: ∀i ∈ I r : Γ ii < 0 (1.49) ∀ (i, j) ∈ I 2 r , i = j : 2 r -1 Γ ii + Γ ij + Γ ji < 0 (1.50)
Another widely investigated source of conservatism is the choice of a candidate Lyapunov function. Indeed, with the QLF (1.33), solving the conditions of Theorem 1.4 or 1.5 means that we need to find a common decision variable P , solution to a set of several LMI constraints, which is indeed restrictive. To relax such conditions, one of the most considered approach is to employ a Non-Quadratic Lyapunov Function (NQLF) candidate introduced by [START_REF] Jadbabaie | A reduction in conservatism in stability and L 2 gain analysis of Takagi-Sugeno fuzzy systems via linear matrix inequalities[END_REF], for instance for controller design in the continuous-time case:

V (x(t)) = x T (t) r i=1 α i (z(t))P i -1 x(t) (1.51)
with the non-PDC control law:

u(t) = r i=1 α i (z(t))K i   r j=1 α j (z(t))P j   -1 x(t) (1.52)
The application of Theorem 1.2 leads to the non-quadratic closed-loop stability condition given by:

r i=1 r j=1 α i (z(t))α j (z(t)) A i P j + P j A T i + B i K j + K T j B T i - r k=1 αk (z(t))P k < 0 (1.53)
Note that (1.53) involves the derivatives of the membership functions (this vanishes in the discrete-time framework as shown in [START_REF] Guerra | LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form[END_REF]). To cope with this, assuming that | αk (z(t)| ≤ φ k , we can bound (1.53), as firstly proposed in [START_REF] Jadbabaie | A reduction in conservatism in stability and L 2 gain analysis of Takagi-Sugeno fuzzy systems via linear matrix inequalities[END_REF][START_REF] Blanco | Non quadratic stability of nonlinear systems in the Takagi-Sugeno form[END_REF], by:

r i=1 r j=1 α i (z(t))α j (z(t)) A i P j + P j A T i + B i K j + K T j B T i + r k=1 φ k P k < 0 (1.54)
Then, refinements have been proposed to further reduce the conservatism of (1.54), see e.g. [START_REF] Tanaka | A Multiple Lyapunov Function Approach to Stabilization of Fuzzy Control Systems[END_REF][START_REF] Tanaka | A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions[END_REF][START_REF] Morère | Mise en oeuvre de lois de commande pour les modèles flous de type Takagi-Sugeno[END_REF][START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF]. Among them, we provide below the conditions proposed in [START_REF] Morère | Mise en oeuvre de lois de commande pour les modèles flous de type Takagi-Sugeno[END_REF] since they also constitute a good compromise between complexity and conservatism reduction.

Lemma 1.3. (Mozelli et al., 2009) Assume that | ḣk | ≤ φ k , ∀k ∈ I r .
The T-S fuzzy system

(1. 6) is stable if the following LMIs are satisfied:

P i = P T i > 0, i ∈ I r , P i + X ≥ 0, i ∈ I r-1 }, Pφ + 1 2 A T i P j + P j A i + A T j P i + P i A j < 0, i ≤ j,
(1.55)

where (i, j) ∈ I 2 r , Pφ = r-1 k=1 φ k (P k + X)
, and φ k are scalars.

Furthermore, it is important to highlight that, assuming the bounds of the time-derivatives make the results valid only locally in the regard of the compact subset:

D φ = r k=1 {x(t) ∈ R n : | αk (z)| φ k } (1.56)
which, combined with the domain of validity of the T-S model D x , allows to provide an estimate of the domain of attraction by searching the outermost level set c > 0 such that

L(c) = {x ∈ R n |V (x) ≤ c} ⊆ D x ∩ D φ .
Remark 1.5. In the above non-quadratic context, we need to estimate the bounds of the timederivatives which are parameters to solve the LMI conditions. This task is in general difficult or even impossible if global stability results are required. Otherwise, in [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF], a systematic approach has been proposed to avoid setting these parameters while providing an estimate of the closed-loop domain of attraction. Some other approaches consider the use of Line-Integral Lyapunov functions [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF], but restricted to second order systems [START_REF] Guelton | Comments on Fuzzy Control Systems Design via Fuzzy Lyapunov Functions[END_REF][START_REF] Guelton | Some Refinements on Stability Analysis and Stabilization of Second Order T-S Models Using Line-Integral Lyapunov Functions[END_REF], or consider Lyapunov functions involving the mean values of the membership functions [START_REF] Marquéz | Asymptotically necessary and sufficient conditions for Takagi-Sugeno models using generalized non-quadratic parameterdependent controller design[END_REF][START_REF] Cherifi | Global non-quadratic D-stabilization of Takagi-Sugeno systems with piecewise continuous membership functions[END_REF], appealing when membership functions are piecewise continuous, which is not the purpose of this thesis. This being said, in some special cases, the bounds of the time-derivatives of the membership functions can be analytically computed (see e.g. Chapter 3,Subsection 3.5.2) or, when considering local results and input limitations, an estimate of the bounds can be numerically computed as it will be shown in Chapter 4.

Finally, another way of interest for this thesis to reduce the conservatism is the introduction of slack decision variables into the stability conditions. This can be done, for instance, by considering descriptor redundancy approaches [START_REF] Tanaka | A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions[END_REF][START_REF] Guelton | Robust dynamic output feedback fuzzy Lyapunov stabilization of Takagi-Sugeno systems -A descriptor redundancy approach[END_REF][START_REF] Bouarar | Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi-Sugeno descriptors[END_REF][START_REF] Bouarar | Robust non-quadratic static output feedback controller design for Takagi-Sugeno systems using descriptor redundancy[END_REF], by applying Peaucelle's transforms [START_REF] Peaucelle | A new robust D-stability condition for real convex polytopic uncertainty[END_REF][START_REF] Cherifi | Global non-quadratic D-stabilization of Takagi-Sugeno systems with piecewise continuous membership functions[END_REF] or the Finsler's Lemma [START_REF] Skelton | A unified algebraic approach to linear control design[END_REF][START_REF] Oliveira | Robust state feedback LMI methods for continuous-time linear systems: Discussions, extensions and numerical comparisons[END_REF][START_REF] Jaadari | New controllers and new designs for continuous-time Takagi-Sugeno models[END_REF]Bourahala and Guelton, 2017), or S-procedure [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF][START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] presented below.

Lemma 1.4. Finsler's Lemma [START_REF] Skelton | A unified algebraic approach to linear control design[END_REF].

Let ξ ∈ R n , G ∈ R m×n and Q = Q T ∈ R n×n such that rank(G) < n.
The following statements are equivalent:

ξ T Qξ < 0, ∀ξ ∈ {ξ ∈ R n : ξ = 0, Gξ = 0} (1.57) ∃R ∈ R n×m : Q + RG + G T R T < 0 (1.58)
Lemma 1.5. S-procedure [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Let T 0 , . . . , T p ∈ R n×n be symmetric matrices. If there exists λ 1 ≥ 0, . . . , λ p ≥ 0, such that

T 0 - p i=1 λ i T i > 0 (1.59) then x T T 0 x > 0, ∀x = 0 such that x T T i x ≥ 0, i = 1, . .

. , p

When p = 1, the converse holds if there exists x 0 such that x T 0 T 1 x 0 > 0.

Preliminaries on specific control problems investigated in the sequel of the thesis

The basics on T-S models and their stability analysis and controller design being now recalled, let us now introduce the preliminaries related to the more specific control problems investigated in the sequel of this thesis. First, because most physical systems are subject to actuators' limitation, the preliminary concepts which will be used in this thesis to deal with such issues will be presented [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. Then, because it has been shown that some issues may occur when applying discrete-time model-based controller design for continuous-time systems, we present some preliminaries on input-delay approaches for sampled-data controller design [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF].

Preliminaries on controller design with actuators' limitations

Among the practical characteristics sometimes miscarried in controller design procedures, the saturation of actuators is one of the most critical aspects [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF][START_REF] Hu | Composite quadratic Lyapunov functions for constrained control systems[END_REF]. When not correctly considered, actuators saturation can lead to limit-cycles, the emergence of new equilibrium points, degeneration in performance, or the induction of unstable behavior even on stable systems. Indeed, since it is present in nearly all systems and processes, actuator saturation remains an essential investigation subject. In this context, let us consider the following nonlinear affine-in-control system (1.2), where the input signal is constrained such that:

ẋ(t) = A(x(t))x(t) + B(x(t))satu(t) (1.60)
where A(x(t)) ∈ R n×n and B(x(t)) ∈ R n×m are matrices that may nonlinearly depend only on the state entries, and sat(u (ℓ) (t)) is a generic saturation function, whereū(ℓ) and ū(ℓ) are the minimum and maximal value allowed for the ℓ-th control signal, which define the compact set of input values denoted by:

D u = {u(t) ∈ R m : -ū ≤ sat(u(t)) ≤ ū} (1.61)
where the inequalities are to be understood component-wise.

To illustrate the saturation function, let us consider the state-feedback control law (1.38), rewritten as u(t) = K(x(t))x(t) to alleviate the mathematical expressions, constrained by the saturation function:

sat(K(x(t))x(t)) (ℓ) =        ū(ℓ) if (K(x(t))x(t)) (ℓ) > ū(ℓ) (K(x(t))x(t)) (ℓ) if -ū(ℓ) ≤ (K(x(t))x(t)) (ℓ) ≥ ū(ℓ) -ū(ℓ) if (K(x(t))x(t)) (ℓ) < -ū(ℓ) (1.62)
According to the saturation function, when actuators reach their saturation level, a nonlinearity is introduced into the closed-loop dynamics, leading to degradation of the closed-loop performance (which may be even unstable), even when the open-loop system is stable. To deal with such issue, three main approaches are often considered. The first one considers that the saturation is written in terms of polytopic models [START_REF] Hu | Composite quadratic Lyapunov functions for constrained control systems[END_REF], in the second, saturation regions are defined [START_REF] Molchanov | Criteria of asymptotic stability of differential and difference inclusions encountered in control theory[END_REF] and in the third, the actuator saturation is treated as a sector nonlinearity [START_REF] Khalil | Nonlinear Systems[END_REF]. In the sequel of this work, the developments are limited to the third approach since it has been found less conservative than polytopic approaches, especially when the characterization of the closed-loop domain of attraction is needed [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF].

Generally speaking, the stabilization of nonlinear systems under saturated actuators can be considered as a Lur'e problem [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], with the dead-zone sector-nonlinearity defined as follows.

Definition 1.2. The dead-zone is a saturated signal defined as

ψ(u(t)) = sat(u(t)) -u(t), with ψ(u(t)) ∈ R q such that, ∀ℓ = 1, . . . , q: ψ(u (ℓ) (t)) =        ū(ℓ) -u (ℓ) (t) if u (ℓ) (t) > ū(ℓ) 0 if -ū(ℓ) ≤ u(t) ≤ ū(ℓ) -u (ℓ) (t) -ū(ℓ) if u (ℓ) (t) < -ū(ℓ)
,

The closed-loop dynamics can be expressed as:

ẋ(t) = (A(x(t)) + B(x(t))K(x(t)))x(t) + B(x(t))ψ(u(t)), (1.63)
Hence, the design of the controller's gain K(x(t)) will only ensure the local closed-loop stability of (1.63) and the goal is to provide relaxed margins regarding the input saturation, as illustrated in Figure 1.6, given by:

-ūλ (ℓ) = -ū(ℓ) 1 -λ (ℓ) ≤ u (ℓ) (t) ≤ ū(ℓ) 1 -λ (ℓ) = ūλ (ℓ)
where λ (ℓ) ∈ [0, 1). Therefore, the design of a controller which locally stabilizes the closed-loop dynamics (1.63) can be done with the second Lyapunov method (see e.g. Theorem 1.2), while considering the following Lemma to cope with the dead-zone nonlinearity ψ(u(t)) (see Chapter 2 and 4 for more developments in the context of this thesis).

u (ℓ) (t) sat(u (ℓ) (t)) -ū(ℓ) -ū(ℓ) ū(ℓ) ū(ℓ) (a) u (ℓ) (t) ψ(u (ℓ) (t)) λ (ℓ) u (ℓ) -ūλ (ℓ) -ū(ℓ) ū(ℓ) ūλ (ℓ) (b)
Lemma 1.6. Generalized Sector Condition, Lemma 1.6 in [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. Consider the vector ū ∈ R m and signals u ∈ R m and ω ∈ R m belonging to the set S given by:

S(ū) = {u ∈ R m , ω ∈ R m : |u (ℓ) -ω (ℓ) | ≤ ū(ℓ) , ℓ ∈ I m } (1.
(c) = {x ∈ R n |V (x) ≤ c} ⊆ D x ∩ D φ ∩ D u .
This motivated several recent works on T-S model-based control to handle such constraints. For instance, [START_REF] Lendek | Local stabilization of discrete-time TS descriptor systems[END_REF][START_REF] Klug | Fuzzy dynamic output feedback control through nonlinear Takagi-Sugeno models[END_REF] focus on handling the domain of validity of T-S models. Input or state constraints have been considered in [START_REF] Nguyen | An augmented system approach for LMIbased control design of constrained Takagi-Sugeno fuzzy systems[END_REF][START_REF] Fan | Mechatronics and Robotics Engineering for Advanced and Intelligent Manufacturing, volume 3, chapter DOB Tracking Control for Systems with Input Saturation and Exogenous Disturbances via T-S Disturbance Modelling[END_REF]Li et al., 2016a;[START_REF] Kong | Disturbance-observer-based fuzzy model predictive control for nonlinear processes with disturbances and input constraints[END_REF]. Our works in the 2 nd and 4 th chapters will follow the same guidelines.

Preliminaries on the input delay approach for sampled-data control

To circumvent the above mentioned drawbacks and limitations of T-S model-based discretetime controllers applied to continuous-time systems, an appealing approach considers on rewriting the overall continuous-time closed-loop dynamics as a dynamical system with input-time varying delays [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. This subsection aims at presenting the preliminaries about this approach, projected in the T-S model framework, to better apprehend our contributions in Chapter 3 and 4.

ẋ(t) = r i=1 r j=1 α i (z(t))α j (z(t k )) (A i x(t) + B i K j x(t k )) (1.65)
It is important to highlight that, with the sequence of hold sampling instants:

0 = t 0 < t 1 < . . . < t k < t k+1 < . . . , lim k→∞ t k = ∞, (1.66)
the sampling interval can be either constant t k+1 -t k ≡ τ s or variable, with k-dependent

η k = t k+1 -t k .
Hence, the following assumption is done.

Assumption 1.2. We assume that the aperiodic sampling intervals η k are bounded such that they admit a maximal allowable value η, i.e.:

η k = t k+1 -t k ≤ η, ∀k ∈ N.
(1.67)

Remark 1.6. In many real applications, the assumption (1.2) is accurate since the intervals between two successive sampling instants may be varying due to practical constraints. Even in a point-to-point digital control topology, clock inaccuracy and system architecture characteristics such as real-time scheduling can induce jitters, imperfect synchronization, and computation delays [START_REF] Wittenmark | Timing problems in real-time control systems[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. Over the past decades, there has been an expansion in the study of Networked Controlled Systems (NCS), in which sampled-data systems are controlled through communication networks [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]Fridman, 2014a). In these circumstances, aperiodic sampling intervals are almost inevitable due to constraints induced by the network, e.g. event-triggering [START_REF] De Souza | Co-design of an eventtriggered dynamic output feedback controller for discrete-time LPV systems with constraints[END_REF][START_REF] Rouamel | Mixed Actual and Memory Data-based Event-Triggered H ∞ Control Design for Networked Control Systems[END_REF], network-induced time-varying delays and packet dropouts [START_REF] Nafir | Improved robust H ∞ stability analysis and stabilisation of uncertain and disturbed networked control systems with network-induced delay and packets dropout[END_REF], and so on.

The main idea of the input-delay approach consist in rewriting sampled signals maintained by a ZOH as continuous-time signal involving a time-varying delay. To do so, let us consider the sawtooth functions, illustrated in Figure (1.7), defined by: Note that, from Assumption 1.2, the sawtooth function (1.68) has the following properties:

τ (t) = t -t k , ∀t ∈ [t k , t k+1 ), ∀k ∈ N (1.68)
τ (t) ∈ [0, η k ) ⊆ [0, η), ∀t ∈ [t k , t k+1 ), ∀k ∈ N (1.69)
and:

τ (t) = 1, ∀t ∈ [t k , t k+1 ), ∀k ∈ N (1.70)
Therefore, from (1.68), any sampled signals s(t k ) maintained by a ZOH can be rewritten as s(t-τ (t)), which allows us to rewrite the closed-loop dynamics (1.65) as the following continuoustime system with time-varying delays:

ẋ(t) = r i=1 r j=1 α i (z(t))α j (z(t -τ (t))) (A i x(t) + B i K j x(t -τ (t))) (1.71)
Rewriting the closed-loop dynamics is such a fashion make usual tools dedicated to the stability analysis and controller design for systems involving time-varying delays suitable for the design of sampled-data controllers. Two main approaches are often considered to deal with time-varying delays from the use of Lyapunov-Razumikhin Functionals (LRF) or Lyapunov-Krasovskii Functionals (LFK). In the linear system control framework, LKF were first proposed to cope with slow varying time-delays, i.e. τ (t) < 1 [START_REF] Niculescu | Robust exponential stability of uncertain systems with time-varying delays[END_REF]. Unfortunately, such approach were not suitable for fast time-varying delays, i.e. when τ (t) = 1, so the stability issue was first investigated via LRF (see e.g. [START_REF] Hale | Introduction to Functional Differential Equations[END_REF]), because no restrictions on the derivative of the delay in this case, but it usually leads to conservative results. More recently, robust stability conditions for systems involving fast input time-varying delay (i.e. τ (t) = 1 almost everywhere, as illustrated in Figure 1.7) were provided via LKF in (Fridman andShaked, 2002, 2003;[START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF]. From these pioneer works, the time-delay approach became popular to cope with sampled-data systems and NCSs (Fridman, 2014a), especially bringing several conservatism improvements in the linear system framework, see e.g. [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF]Fridman, 2014a,b;[START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF][START_REF] Bourahala | Improved robust H ∞ stability analysis and stabilization of uncertain systems with stochastic input time-varying delays[END_REF][START_REF] Nafir | Improved robust H ∞ stability analysis and stabilisation of uncertain and disturbed networked control systems with network-induced delay and packets dropout[END_REF][START_REF] Rouamel | Mixed Actual and Memory Data-based Event-Triggered H ∞ Control Design for Networked Control Systems[END_REF].

Among the several improvements made with the choice of a convenient LKF in sampled-data controller design, looped LKF inspired by [START_REF] Seuret | Stability analysis for sampled-data systems with a time-varying period[END_REF][START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF] retains our attention because it allows to take into account the characteristic of the sawtooth function τ (t), especially when τ (t) = 1 (∀t = t k ), to provide derivative-dependent stability conditions. Such a looped LKF is of the form [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]:

V (x(t)) = x T (t)P x(t) + (η k -τ (t)) t t-τ (t) ẋT (s)R ẋ(s)ds (1.72)
which presents the advantage of being left and right continuous at each sampling instants t k , i.e.:

V (x(t - k )) = V (x(t + k )) = V (x(t k )) = x T (t k )P x(t k ) (1.73)
Therefore, as illustrated on Figure 1.8, assuming

P = P T > 0, (1.72), ∀t ∈ [t k , t k+1 ), if V (x(t)) < 0, then, ∀t ∈ R + , V (x(t)
) is monotonously decreasing and the considered system with input time-varying delay is asymptotically stable. In the context of T-S model-based sampled-data control, several attempts to reduce the conservatism have been done from different choices of an LKF, see e.g. [START_REF] Yoneyama | Robust H ∞ control of uncertain fuzzy systems under time-varying sampling[END_REF][START_REF] Zhu | Stabilization for sampled-data neural network-based control systems[END_REF][START_REF] Zhang | H ∞ control design for network-based T-S fuzzy systems with asynchronous constraints on membership functions[END_REF][START_REF] Zhu | An Improved Input Delay Approach to Stabilization of Fuzzy Systems Under Variable Sampling[END_REF][START_REF] Gunasekaran | Robust Sampled-data Fuzzy Control for Nonlinear Systems and Its Applications: Free-Weight Matrix Method[END_REF][START_REF] Zhu | H ∞ stabilization criterion with less complexity for nonuniform sampling fuzzy systems[END_REF][START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF]. Among them, only few authors have considered looped LKF [START_REF] Zhu | An Improved Input Delay Approach to Stabilization of Fuzzy Systems Under Variable Sampling[END_REF][START_REF] Zhu | H ∞ stabilization criterion with less complexity for nonuniform sampling fuzzy systems[END_REF][START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF], but with significant conservatism reduction. To further relax the conditions, refinements on the choice of a convenient looped LKF are proposed in Chapter 3 and 4 where LMI-based conditions will be proposed with the help of the following lemmas.

Lemma 1.7. [START_REF] Fridman | Tutorial on Lyapunov-based methods for time-delay systems[END_REF] For any matrix P = P T > 0 with appropriate dimensions, τ (t) ∈ [0, η k ) and τ (t) = 1, the following inequality holds:

t t-τ (t) x T (s)P x(s)ds ≥ η -1 k t t-τ (t) x T (s)dsP t t-τ (t)
x(s)ds

(1.74)

Lemma 1.8. [START_REF] Zhang | Novel delay-derivative-dependent stability criteria using new bounding techniques[END_REF] For any constant matrix R ∈ R n×n , R = R T > 0, a scalar function τ (t) with 0 < τ (t) ≤ η and a vector function ẋ : [-η, 0] → R n such that the integration concerned is well defined, let

t t-τ (t) ẋ(s)ds = Eψ(t) (1.75)
where E ∈ R n×k and ψ(t) ∈ R k . Then the following inequality holds for any matrix

M ∈ R n×k - t t-τ (t) ẋT (s)R ẋ(s)ds ≤ ψ T (t)Υ 1 ψ(t) (1.76)
where

Υ 1 = -E T M -M T E + τ (t)M T R -1 M .
Lemma 1.9. [START_REF] Kim | Further improvement of jensen inequality and application to stability of time-delayed systems[END_REF]. Let us consider a quadratic function

f (η(t)) = a 2 η(t) 2 + a 1 η(t) + a 0 ,
where a 2 , a 1 and a 0 ∈ R. If:

(i) f (0) < 0, (ii) f (η) < 0 (iii) f (0) -η2 a 2 < 0, (1.77) then f (η(t)) < 0, ∀η(t) ∈ [0, η].
Moreover, note that the sampled-data closed-loop dynamics (1.71) involves a mismatching double sum interconnection structure and so, applying the second Lyapunov method will lead to closed-loop stability conditions given in terms of parameterized LMIs involving summation structure such like:

r i=1 r j=1 α i (x(t))α j (x(t -τ (t)))Γ ij < 0 (1.78)
Figure 1.9 illustrates the mismatch between the continuous-time membership function α i (x(t))

of plant and the sampled ones, maintained by a ZOH such that

α i (x(t k )) = α i (x(t -τ (t))).
Figure 1.9: Mismatch between controller's and T-S model's membership function.

Hence, to provide relaxed parameter independent LMI-based conditions from (1.78), Lemma 1.2 or other usual double sum relaxation schemes cannot be employed directly. To circumvent this drawback, most of the authors assume that, ∀ρ ∈ I r , there exists σ ρ such that |α ρ (x(t -

τ (t))) -α ρ (x(t))| ≤ σ ρ ,
in order to provide mismatching double sum relaxation scheme, such like the one given by the following recent lemma.

Lemma 1.10. [START_REF] Koo | An improved digital redesign for sampled-data fuzzy control systems: Fuzzy Lyapunov function approach[END_REF]: For any real matrix

Γ ij = Γ T ij for (i, j) ∈ I 2 r and the continuous-time functions a i (t), b i (t) for i ∈ I r with (a i (t), b i (t)) ∈ [0, 1] 2 , r i=1 a i (t) = r i=1 b i (t) = 1 and considering |a i (t) -b i (t)| ≤ σ i , the inequality: r i=1 r j=1 a i (t)b j (t)Γ ij < 0,
holds, if there exists some matrices Y i and Z i , such that

Γ ii + Z i < 0, ∀i ∈ I r , Γ ij + Γ ji + Z i + Z j < 0, ∀(i, j) ∈ I 2 r , i < j, Γ ij + Y i > 0, ∀(i, j) ∈ I 2 r , r j=1 σ j (F ij + Y i ) -Z i < 0, ∀i ∈ I r .
(1.79)

Note that Lemma (1.10) consists in an extension to the case of mismatched membership functions of the first proposed double sum relaxation scheme in [START_REF] Tanaka | Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs[END_REF], which were found more conservative than the further proposed double sum relaxation scheme in Lemma (1.2) [START_REF] Tuan | Parameterized Linear Matrix Inequality Techniques in Fuzzy Control System Design[END_REF]. Hence, we believe that there is still space for conservatism improvements.

Moreover, let us highlight an important fact. Assuming that there exists σ ρ such that |α ρ (x(tτ (t))) -α ρ (x(t))| ≤ σ ρ to get LMI-based conditions for sampled-data controller means that such assumption need to be at least post-verified from extensive numerical closed-loop simulation, which is not an easy task to do. Hence, because many authors assume these parameters without post-verification, they may loose asymptotic stability guarantee if it occurs that |α ρ (x(t-τ (t)))α ρ (x(t))| is not bounded by σ ρ ∀t. Therefore, we believe that more investigations are required to cope with that issue. As a matter of fact, it is easy to show that these parameters are related to the bounds of the time-derivatives of the membership functions. Indeed, we can write, ∀ρ ∈ I r :

α ρ (x(t)) -α ρ (x(t -τ (t))) = t t-τ (t) αρ (x(s))ds (1.80)
Therefore, assuming that, ∀ρ ∈ I r , there exists φ ρ such that | αρ (x(t))| ≤ φ ρ , yields:

-φ ρ η ≤ -φ ρ τ (t) ≤ t t-τ (t) αρ (x(s))ds ≤ φ ρ τ (t) ≤ φ ρ η (1.81)
Moreover, since, ∀t, α ρ (x(t)) ∈ [0, 1], we have:

-1 ≤ α ρ (x(t)) -α ρ (x(t -τ (t))) ≤ 1 (1.82)
Consequently, from (1.80)-(1.82), we always have

|α ρ (x(t -τ (t))) -α ρ (x(t))| ≤ σ ρ with σ ρ = min{1, φ ρ η}.
Based on such considerations, in Chapter 3, an improved double sum relaxation Lemma, extending Tuan's lemma 1.2 to the case of mismatching sampled-data membership functions will be provided and compared with Lemma 1.10. Let us point-out that if a solution can be found with σ ρ = 1, then the assumption of a known φ ρ is no more required (unless non-quadratic Lyapunov Functions are employed). However, σ ρ = 1 being the more conservative case, we will often have to provide φ ρ , e.g. as explained in Remark 1.5, such that ηφ ρ ≤ 1, leading to local results constrained by D φ given in (1.56). Nevertheless, it is worth to say that, to the best of the authors' knowledge, there exists no previous results dealing with the estimation of the closed-loop domain of attraction in the context of sampled-data controller design for T-S models.

This being understood as a major issue, this point will be the subject of one of the contributions proposed in Chapter 4, from the convenient choice of a looped non-quadratic LKF.

Conclusion

In this chapter, some preliminaries on T-S model-based control have been presented. Ce chapitre présente la synthèse de contrôleurs pour les systèmes à temps discret, dont les bases ont été exposées dans le chapitre précédent. Dans ce cas, lorsque de tels contrôleurs discrets sont appliqués. à un système continu, les signaux de commande sont maintenus par un bloqueur d'ordre zéro entre deux instants d'échantillonage. Nécessitant une période d'échantillonnage fixe, cette approche a fait l'objet de nombreuses études depuis les années 1950, conduisant à une théorie bien établie pour les systèmes à temps invariant (LTI). Cette approche est donc encore largement répandue pour contrôler des systèmes industriels puisqu'elle permet une mise en oeuvre rapide sur des automates programmables, avec des performances acceptables et une bonne précision lorsque les pé-riodes d'échantillonnage constantes sont suffisamment petites par rapport à la dynamique du système à contrôler.

Plus précisément, dans ce chapitre, nous nous intéressons au suivi de trajectoire à partir de points de consigne pour les modèles de type T-S à temps discret sujet à la saturation des actionneurs. Cette dernière contrainte menant à l'obtention d'un résultat local, nous proposons une analyse de l'estimation de la région d'attraction en boucle fermée, en prenant en compte les entrées exogènes induites par le changement des points de consigne, menant à des changements d'origine de la dynamique en boucle fermée. En effet, en négligeant une telle analyse, les évolutions des points de consigne peuvent amener le système à suivre des trajectoires en dehors de la région d'attraction, de sorte que tant les performances que la stabilité de la boucle fermée peuvent être compromises. Cet aspect a été étudié dans [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF] où un contrôleur non-PDC de type PI a été proposé, mais possédant une région d'attraction modeste. De plus, dans [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF], une condition requise est que la variation temporelle du vecteur d'état du système soit lente. Cependant, comme le montrent [START_REF] Zaccarian | Modern Anti-windup Synthesis: control augmentation for actuator saturation[END_REF][START_REF] Mehdi | A novel anti-windup framework for cascade control systems: An application to underactuated mechanical systems[END_REF], de tels inconvénients peuvent être limités en considérant une action anti-windup. Néanmoins, à la connaissance de l'auteur, aucun travail antérieur ne semble avoir abordé ce point pour la synthèse de contrôleurs de type PI à temps discret dans le cadre des modèles de type T-S.

Par conséquent, la principale contribution de ce chapitre est de proposer une méthode d'optimisation convexe pour la synthèse de contrôleurs flous de type PI à temps discret avec une action anti-windup pour une stabilisation locale et un suivi de points de con-signe pour les systèmes non linéaires représentés par des modèles T-S. En particulier, la synthèse du contrôleur flou de type PI consiste en une boucle interne avec retour d'état associée à une boucle avec retour de sortie. Ce schéma de contrôle permet de garantir à terme une erreur nulle entre la sortie et les points de consigne. De plus, la saturation du signal de commande est abordée à partir des conditions données dans [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] et une optimisation de la région d'attraction est proposée. Un ensemble de résultats de simulation et d'expérimentation est présenté et discuté. Les résultats expérimentaux ont été obtenus sur un banc d'essai (système de réservoirs connectés), inspiré de [START_REF] Johansson | The Quadruple-Tank Process: A Multivariable Laboratory Process with an Adjustable Zero[END_REF], et disponible au CEFET-MG (Divinópolis, Brésil). Les résultats obtenus montrent l'intérêt de l'approche présentée dans ce chapitre par rapport aux travaux antérieurs présentés dans [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF][START_REF] Wang | H ∞ fuzzy PID control for discrete time-delayed T-S fuzzy systems[END_REF].

Introduction

This chapter investigates the design of discrete-time controllers for continuous-time systems based on the direct discrete-time model-based approach, whose basics have been presented in Chapter 1, Sections 1.2.2 and 1.3.2. Following this method, discrete-time fuzzy T-S models are employed to represent nonlinear systems for the design of discrete-time controllers, which are then implemented using a ZOH. Despite imposing a fixed sampling period, this procedure has been studied since the 1950s, leading to a mature discrete-time control theory for Linear Time-Invariant (LTI) systems. This is why it is still widely employed to control many industrial or practical applications, allowing easy implementation on digital devices like PLCs, with reasonable performances and good accuracy when small enough fixed sampling periods can be set with regard to the system's dynamics to be controlled (see Remark 1.3 and Example 1.3).

More specifically, in this chapter, we are concerned with the set-point tracking of discretetime T-S models subject to actuators' saturation. The latter constraints bringing locality, therefore we propose a careful analysis of the estimation of the closed-loop region of attraction by taking into account the effects of exogenous inputs induced by the change of the set-points in the closed-loop dynamics. Indeed, neglecting such analysis, set-point changes may drive the trajectories of the system outside of the region of attraction, so that both the performances and the closed-loop stability can be impaired. This effect has been investigated in [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF] where non-PDC PI-like controller has been proposed, but with quite small region of attraction. Moreover, in [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF] it is required a slow enough time-variation of the state vector of the system. However, as shown in [START_REF] Zaccarian | Modern Anti-windup Synthesis: control augmentation for actuator saturation[END_REF][START_REF] Mehdi | A novel anti-windup framework for cascade control systems: An application to underactuated mechanical systems[END_REF], such drawbacks can be mitigated by considering an anti-windup action.

In fuzzy model-based control, PID controllers have been studied with various goals including:

(a) to compare the performance [START_REF] Ounnas | Optimal Reference Model Based Fuzzy Tracking Control for Wind Energy Conversion System[END_REF][START_REF] Kmetóvá | Neuro-fuzzy control of exothermic chemical reactor[END_REF][START_REF] Blažič | Globally stable model reference adaptive control based on fuzzy description of the plant[END_REF] with fuzzy PID controllers; (b) to control systems modeled through T-S approach [START_REF] Estrada-Manzo | An alternative LMI static output feedback control design for discrete-time nonlinear systems represented by Takagi-Sugeno models[END_REF][START_REF] Gao | Setpoints Compensation for Nonlinear Industrial Processes with Disturbances Based on Fuzzy Logic Control[END_REF][START_REF] Yu | T-S Fuzzy Control for Magnetic Levitation Systems Using Quantum Particles Swarm Optimization[END_REF][START_REF] Yu | T-S Fuzzy Control of Magnetic Levitation Systems Using QEA[END_REF][START_REF] Wang | H ∞ fuzzy PID control for discrete time-delayed T-S fuzzy systems[END_REF][START_REF] Yi | Constrained PI Tracking control for the output pdfs based on T-S fuzzy model[END_REF]; (c) for output feedback with T-S fuzzy observers [START_REF] Lin | Implementation of fuzzy controller for measuring instantaneous arterial blood pressure via tissue control method[END_REF](d) to blend local gains through T-S rules [START_REF] Yi | Statistic Tracking Control for Non-Gaussian Systems Using T-S Fuzzy Model[END_REF] to improve the performance of a PI structure with both state and output feedback; (e) to develop self-tuning techniques by i) 2.2. Problem statement blending conventional PI controllers adjusted for local operational points or for different control objectives, such as disturbance rejection and reference tracking [START_REF] Sun | Hopf bifurcation analysis of maglev vehicle-guideway interaction vibration system and stability control based on fuzzy adaptive theory[END_REF][START_REF] Nouri | DSP-based implementation of self-tunning fuzzy controller for three-level boost converter[END_REF][START_REF] Mishra | Stiction Combating Intelligent Controller Tuning: A Comparative Study[END_REF][START_REF] Fattah | Performance and Comparison Analysis of Speed Control of Induction Motors using Improved Hybrid PID-Fuzzy Controller[END_REF][START_REF] Dragos | Alternative Control Solutions for Vehicles with Continuously Variable Transmission. A Case Study[END_REF][START_REF] Fateh | Robust Voltage Control of Electrical Manipulators in Task-Space[END_REF], ii) particle swarm approach [START_REF] Bouallegue | PID type fuzzy logic controller tuning based on particle swarm optimization[END_REF] iii) Cuckoo search parameter fuzzy PID optimization [START_REF] Hamzaa | Cuckoo search algorithm based design of interval Type-2 Fuzzy PID Controller for Furuta pendulum system[END_REF], etc. Nevertheless, to the best of the author's knowledge, no works in the previous literature handle the use of anti-windup action to improve the estimation of the region of attraction for the design of discrete-time PI-like set-point tracking controllers under actuators' saturation in the Takagi-Sugeno framework.

Therefore, the main contribution of this chapter is to propose a convex optimization method for the design of discrete-time fuzzy PI-like controllers with a fuzzy anti-windup gain, for the local stabilization and set-point tracking of nonlinear systems represented by discrete-time T-S models. In particular, the proposed fuzzy PI control scheme consists of an internal state-feedback loop combined with an output feedback one. The proposed topology is dedicated to ensure null output tracking errors for piecewise constant set-point references. Additionally, the saturation of the control signal is handled by the generalized sector condition [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], and a maximization procedure is proposed to enlarge the provided estimates of the closed-loop domain of attraction.

A set of simulation and experimental results is presented and discussed, obtained from the application of the proposed methodology to an industrial benchmark, namely the interactive tank system available at the CEFET-MG (Divinópolis, Brazil), inspired by [START_REF] Johansson | The Quadruple-Tank Process: A Multivariable Laboratory Process with an Adjustable Zero[END_REF], illustrating the improvements achieved with the approach presented in this chapter with regard to previous works [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF][START_REF] Wang | H ∞ fuzzy PID control for discrete time-delayed T-S fuzzy systems[END_REF].

Problem statement

In this chapter, we consider the class of nonlinear dynamic systems with input saturations represented by:

ẋ(t) = f (x(t)) + g(x(t))satu(t) y(t) = Cx(t) (2.1)
where u(t) ∈ R m is the control vector, sat : R m → R m is the centered vector valued function, defined as:

satu (ℓ) (t) = sgn(u (ℓ) (t)) min{ū (ℓ) , | u (ℓ) (t) |}, ∀ℓ ∈ I m
with symmetric bounds ū(ℓ) that restricts the ℓ th control input, the state vector x(t) ∈ R n and the output signals vector y(t) ∈ R q . The functions f : R n → R n and g : R m → R n belong to the subset D x ⊆ R n . We assume that the output matrix C ∈ R q×n is known and system (2.1) is controlled through a discrete-time control law with a sample-time of τ s seconds.

In this chapter, we will consider the discrete-time control of (2.1). Hence, from Euler discretization with an appropriate (fixed) sampling period τ s and applying the sector nonlinearity approach (Tanaka and Wang, 2001), the following discrete-time T-S model (with r rules) can be obtained to represent (2.1) on D x in the discrete-time framework:

     x k+1 = r i=1 α i (z k )(A i x k + B i satu k ) y k = Cx k (2.2)
where z k is the vector of premise variables, which is assumed to depend only on the state variables, for i ∈ I r , A i ∈ R n×n and B i ∈ R n×m are known real matrices, and α i ≥ 0 are the membership functions satisfying the convex sum property

r i=1 α i = 1
In the sequel, for all the matrices with fuzzy summation structures, we will adopt the following notations: The goal of this chapter is to provide an LMI-based methodology for the design of a discretetime controller such that the output y k tracks piecewise constant reference signals y r k with null error. To this end, we use the topology shown in Figure 2.1, where a (discrete-time) integral action over the set-points tracking error is included.

M α = r i=1 α i (z k )M i , M αα = r i=1 r j=1 α i (z k )α j (z k )M ij (2.
From the proposed topology, a PDC fuzzy PI-like control law can be readily derived as:

u k = K I α v k -K P α x k (2.4)
where K P α ∈ R m×n and K I α ∈ R m×q are fuzzy control gains to be designed, which keep the fuzzy sum structure (2.3), and v k corresponds to an additional state induced by the integral action in the whole closed-loop system (see the blue dashed box in Figure 2.1);

+ + + + + + - - - ū -ū e k satu k z -1 I q z -1 I q K P α K w (α)
Anti-windup action Integral action Proportional action

K I α y r k v k v k-1 y k x k u k u Ik u P k System (2.1)
Figure 2.1: Control topology with discretized integral and anti-windup actions.

Moreover, the windup effect can appear in the integral action due to the saturating actuators.

To mitigate such an impact, we proposed a non-PDC anti-windup action (see the green dashed box in Figure 2.1) delivered by the signal a w,k , which is computed as:

a w,k = K w (α)(satu k -u k ) (2.5) with K w (α) = E c α S -1
α and where E c α ∈ R m×q and S α ∈ R q×q are the anti-windup gains, which keep the fuzzy sum structure (2.

3).

In what follows, we assume that the output of the system can be driven in certain region such that, for some piecewise constant bounded reference signals y r k belonging to:

W = {y r k ∈ R q : y r k ≤ β}, β ∈ R + , (2.6)
the output error e k = y r k -y k → 0 when k → +∞, while the state belongs to a bounded region of attraction (y r k is attractive) during transients, i.e. x k ∈ D a ⊆ D x , centered on the origin O of the considered plant.

The goal of this chapter is to develop a convex optimization based methodology to design the fuzzy gains K I α , K P α , and K w (α) such that the overall closed-loop set-points tracking control plant represented in Figure 2.1, i.e. the system (2.1) under the control signal satu k , with u k given in (2.4) and y r k ∈ W, is locally stable. Indeed, due to the saturating actuators, the convergence of the set-point tracking error e k , and so the stationary point achieved by the state, will be impacted by the initial state of the system. Therefore, the region of attraction centered on the origin O, denoted by D a , can be defined by the set of initial conditions such that the tracking error trajectories goes to zero, i.e.:

D a = {x(0) ∈ R n ; e k = y r k -Cx k → 0 as k → ∞}.
(2.7)

As it is well-known in the literature, the complete characterization of the region of attraction D a is a challenging task because it may be non-convex and even open [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF].

Usually an estimate of this region, D * a ⊆ D a , is computed instead. Therefore, in this chapter, we are concerned with both the following problems: P 1 : (Closed-loop stabilization) For y r k = 0, the objective is to compute the fuzzy gains K I α , K P α , and K w (α), and to estimate the larger set of initial conditions D * a ⊆ D a such that, ∀k ≥ 0, x k ∈ D * a and the closed-loop system is stable, i.e. k → ∞ ⇒ x k → O.

P 2 : (Closed-loop set-points tracking) ∀y r k ∈ W, the objective is to compute the fuzzy gains K I α , K P α , and K w (α) that maximize β, and to estimate the larger related set of initial conditions D * a ⊆ D a such that, ∀k ≥ 0, x k ∈ D * a and k → ∞ ⇒ e k → 0, when y r k is constant over a sufficiently large number of samples.

Preliminary results

In this preliminary results section, the goal is to introduce the overall closed-loop dynamics including the integral and anti-windup actions, then to propose a way to deal with the saturation and so, to provide an useful preliminary lemma for the estimation of the domain of attraction.

One way to implement the integral action in discrete-time systems is to follow the counterpart path of what is done in continuous-time domain [START_REF] Kailath | Linear Systems[END_REF]. This is shown in the block diagram presented in Figure 2.1, where the integral action can be written as

v k = e k + v k-1 + a w,k-1
, where v k is the signal preceding the fuzzy integral gain K I α . Thus, in our approach we propose a fuzzy PI-like controller with an internal state proportional control loop, combined with an integral tracking error one and an anti-windup action. Hence, by considering (2.2) and

(2.5), we can express v k+1 as:

v k+1 = y r k -CA α x k -CB α satu k + v k + K w (α)(satu k -u k ) (2.8) Let ξ k = x T k v T k
T be an extended state vector, we can combine (2.2)-(2.4) and (2.8) as:

ξ k+1 = A α ξ k + B α satu k + B e α (satu k -u k ) + B w y r k (2.9)
where:

A α = A α 0 -CA α I , B α = B α -CB α , B w = 0 I T , B e α = 0 K wT α T .
Then, the fuzzy PI control law can be rewritten as:

u k = -K α ξ k (2.10) with K α = K P α -K I α .
Moreover, by using the decentralized dead-zone function ψ(u k ) = satu k -u k to deal with the saturation of the actuators, we can rewrite (2.9) as:

ξ k+1 = (A α -B α K α )ξ k + (B α + B e α )ψ(u k ) + B w y r k (2.11)
Furthermore, to deal with the saturation problem and to cope with the above defined decentralized dead-zone function, an auxiliary set S is considered as:

S = {ξ k ∈ R n+q : ∀ℓ ∈ I r , | -(K α(ℓ) + G α(ℓ) )ξ k | ≤ ū(ℓ) } (2.12)
where G α is an appropriate fuzzy matrix (i.e. having the same structure as (2.3)).

Thus, S is a set where the modulus of the control signal component -K α(ℓ) ξ k can be greater than ū(ℓ) , but not the auxiliary signal

-(K α (ℓ) + G α (ℓ) )ξ k .
This means that the proposed design allows the saturation of the actuator, yielding a more energetic control signal than those obtained by saturation avoidance approach, see e.g. [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. In this context, the following lemma is provided to deal with the nonlinearity ψ(u k ).

Lemma 2.1. : If ξ ∈ S, then the relation

ψ(u k ) T T α [ψ(u k ) -G α ξ k ] ≤ 0 (2.13)
is verified for all diagonal matrices 0 < T α ∈ R m×m .

Proof. : Straightforward from adapting Lemma 1.6 in [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] by considering the convex sum structure of the matrices T α and G α .

In the sequel, the local stability of the overall closed-loop dynamics (2.11) will be investigated by means of the following fuzzy Lyapunov candidate function:

V (ξ k ) = ξ T k W -1 α ξ k (2.14)
where W α is a fuzzy Lyapunov matrix (i.e. holding the fuzzy structure (2.3)).

If we can find the matrices W i = W T i > 0, ∀i ∈ I r , such that (2.14) is a Lyapunov function, then it can be used to define the Lyapunov level set L V (µ), with 0 < µ < ∞, such that:

L V (µ) = ξ k ∈ R (n+q) : V (ξ k ) ≤ µ -1 .
(2.15)

The next lemma will be used to compute an estimate D * a of the region of attraction, composed of the intersection of all the level sets defined by a fuzzy Lyapunov function. The details are presented in Section 4.

Lemma 2.2. [START_REF] Jungers | Gain-scheduled output control design for a class of discrete-time nonlinear systems with saturating actuators[END_REF]: Suppose that (2.14) is a Lyapunov function. Then an estimate D * a of the region of attraction D a is given by the level set L V (µ) such that:

D * a = L V (µ) = ξ k ∈D ξ E(W -1 α , µ) = i∈Ir E(W -1 i , µ), (2.16
)

with E(W -1 i , µ) = {ξ k ∈ R n+1 : ξ T k W -1 i ξ k ≤ µ -1 }. Thus, it means that D *
a can be computed by using only the vertices W -1 i , i ∈ I r .

Main Results

Based on the materials presented in the previous sections, the goal is now to proposed LMIbased conditions for the design of the PI-like controller and anti-windup gains such that P 1 and P 2 are satisfied. The main result is summarized by the following theorem.

Theorem 2.1. Consider the overall closed-loop dynamics (2.11) with the known upper bounds of the decentralized saturation vector ū ∈ R m + , and that there exist positive scalar parameters η, τ 1 and τ 2 , the matrices 0 (n+q) , and E c i ∈ R m×q , for i = 1, . . . , r, such that the following LMI-based conditions are verified for p = 1, . . . , r, j = p, . . . , r, and

< W T i = W i ∈ R (n+q)×(n+q) , the diagonal matrices 0 < S i ∈ R m×m , the matrices U ∈ R (n+q)×(n+q) , Y i ∈ R m×(n+q) , Z i ∈ R m×
ℓ = 1, . . . , m:       -W i Γ (p,j) 12 Γ (p,j) 13 B w ⋆ Γ (p,j) 22 (Z T p + Z T j )/2 0 ⋆ ⋆ -S p -S j 0 ⋆ ⋆ ⋆ -τ 2 I q       < 0, (2.17) W p -U -U T Y T p(ℓ) + Z T p(ℓ) ⋆ -ηū 2 p(ℓ)
≤ 0, (2.18)

-(1 -τ 1 )δ + ητ 2 < 0, (2.19) with Γ (p,j) 12 = [(A p + A j )U + (B p Y j + B j Y p )]/2, (2.20) Γ (p,j) 22 = τ 1 [(W p + W j )/2 -U -U T ], ( 2
B e i = 0 E cT i T .
(2.23)

Then the fuzzy PI-like control law:

u k = -K P α -K I α x k v k
and the anti-windup control action:

a w,k = K w (α)(satu k -u k )
with the fuzzy gain matrices computed as:

K P α -K I α = - r i=1 α i (x k )Y i U -1 (2.24)
and: 2. for any y r k verifying y r k ≤ δ -1/2 = β, the trajectories of the closed-loop system, for all initial conditions x(0) taken in D * a do not leave such a set.

K w (α) = r i=1 α i (x k )E c i r i=1 α i (x k )S i -1 , ( 2
Proof. Let the functions defined by κ i : [0, ∞) → [0, ∞), i ∈ {0, 1, 2}, be K ∞ and strictly increasing functions with κ i (0) = 0 and lim a→∞ κ i (a) = ∞. The inequality (2.17) ensures that W i > 0 and:

0 <κ 1 ( ξ k )≤V (ξ k , α k ) = ξ T k P (α k )ξ k < κ 2 ( ξ k ) (2.26) with P (α k ) = N i=1 α ki W -1 i .
From the S-procedure, we can write:

∆V (ξ k ) = V (ξ k+1 ) -V (ξ k ) < -κ 0 ( ξ k ) (2.27)
for ξ k belonging to the trajectory of (2.11).

Then, similarly to the approach proposed in [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], ∀ξ k such that ξ T k P (α k )ξ k ≥ η -1 , for any y r k ∈ W and for all u k ∈ S, the closed-loop tracking problem P 2 of (2.11) is ensured if:

∆V (ξ k ) + (1 -τ 1 )(ξ T k W -1 α ξ k -η -1 ) + τ 2 (δ -1 -y r k T Ry r k ) -2ψ(u k ) T T α [ψ(u k ) -G α ξ k ] ≤ 0 (2.28) with 0 < τ 1 < 1, τ 2 > 0, κ 1 ( ξ k ) = ǫ ξ k 2 with small enough ǫ > 0, and κ 2 ( ξ k ) = s i=1 λ max (P i ) ξ k 2 .
Now, for the closed-loop stabilization problem P 1 , i.e. y r k = 0, the inequality (2.28) reduces to:

∆V (ξ k ) + (1 -τ 1 )(ξ T k W -1 α ξ k -η -1 ) ≤ (1 -τ 1 )(κ 2 ξ k -η -1 ) = -κ 0 ( ξ k ) ≤ 0.
(2.29)

As it is demonstrated later, the inequality (2.18) implies that the set E(P, η) is inside the set S and thus, the generalized sector condition in Lemma 2.1 is ensured, i.e.:

ψ(u k ) T T α [ψ(u k ) -G α ξ k ] ≤ 0.
(2.30) Thus, ∀ξ k ∈ int(E(P, η)) and ∀y r k ∈ W, the inequality (2.28) ensures:

∆V (ξ k ) ≤ -κ 0 ( ξ k ) (2.31)
Let us assume that, for k = k 1 , ξ k 1 ∈ ∂E(P, η) and y r k ∈ W, then ∆V (ξ k1 ) < -κ 0 ξ k . This implies that ξ k1+1 ∈ int(E(P, η)) and allows us to conclude that the verification of (2.28) ensures that the trajectories starting in E(P, η) do not leave such a set for all y r k ∈ W, and thus, E(P, η) is a robustly positively invariant set for (2.11).

Therefore, to verify (2.28), we perform the following sequence of operations:

i) multiply (2.17) by α i , α p , and α j , then sum it up for i = 1, . . . , r, p = 1, . . . , r and j = p, . . . , r;

ii) replace Y α , Z α , and S α by -K α U , G α U and T -1 α , respectively; iii) using the fact -U T W -1 α U ≤ W α -U -U T and, with W + standing for W α k+1 , get:

Θ =      -W + θ 12 B α T -1 α + B eα B ω α ⋆ θ 22 U T α G T α 0 ⋆ ⋆ -2T -1 α 0 ⋆ ⋆ ⋆ -τ 2 I q     
, with:

θ 12 = A α U -B α K α U, and θ 22 = -τ 1 (U T W -1 α U );
iv) thanks to the regularity of U we use the similarity transformation Θ = T T ΘT , with:

T = diag{I n , U -1 , T T α , I q }; v) replace W α by P -1 α , (A α -B α K α ) by A clα
, and E cα T α = K w (α); vi) apply the Schur's complement, then pre-and post-multiply respectively by:

ς T k = ξ T k ψ(u k ) T y r k T
and ς k , then recover (2.11) to get:

ξ T k+1 P (α k+1 )ξ k+1 -τ 1 ξ T k P (α k )ξ k -τ 2 y r k T y r k -2ψ(u k ) T T α (ψ(u k ) -G α ξ k ) < 0 vii) then, from the fact that ∆V (ξ k ) = ξ T k+1 P (α k+1 )ξ k+1 -ξ T k P (α k )ξ k , we get: ∆V (ξ k ) + (1 -τ 1 )ξ T k P (α k )ξ k -τ 2 y r k T y r k -2ψ(u k )T α (ψ(u k ) -G α ξ k ) ≤ 0.
(2.32)

In this last inequality, the term

ψ(u k )T α (ψ(u k ) -G α ξ k ) is ensured to be negative if u k belongs
to S, which is ensured by (2.18). To verify this, follow the sequence:

viii) multiply (2.18) by α k(m) and sum it up for m = 1, . . . , r;

ix) replace W (α k ) -U -U T by -U T W (α k ) -1 U , also Y α by -K α U , and Z α by G α U ;
x) pre-and post-multiply the resulting condition by diag{U -T , 1} and its transpose;

xi) apply the Schur complement then pre-and post multiply the result by ξ T k and ξ k to get: To conclude this proof, we observe that the region of attraction can be computed through the intersection of the ellipsoidal sets as D * a = i∈Ir E(W -1 i , 1) by using Lemma 2.2.

-ξ T k P (α k )ξ k + ξ T k (-K α(ℓ) -G α(ℓ) ) T (ηū 2 (ℓ) ) -1 × (-K α(ℓ) -G α(ℓ) )ξ k ≤ 0, ( 2
Therefore, Theorem 2.1 provides a solution to both problems P 1 and P 2 . It is worth to say that we can choose η = 1 without loss of generality. Moreover, the parameter τ 1 relates the exponential stability: from (2.32) with the Lemma 2.1 and y r k = 0, we can conclude that

∆V (ξ k ) ≤ -(1 -τ 1 )V (ξ k ) ⇒ V (ξ k+1 ) ≤ τ 1 V (ξ k ).
Thus, the smaller τ 1 , the faster the trajectory convergence. Consequently, a search on τ 1 ∈ (0, 1] is useful to improve the convergence rate of the closed-loop. Moreover, the design condition proposed in [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF]) can be recovered from Theorem 2.1 by imposing K w (α) = 0. In this case, we get the following Corollary.

Corollary 2.1. [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF] Consider the overall closed-loop dynamics (2.11) with K w (α) = 0, the known upper bounds of the decentralized saturation vector ū ∈ R m + , and that there exist positive scalar parameters η, τ 1 , the matrices 0 (n+q) , and Z i ∈ R m× (n+q) , i = 1, . . . , r, such that (2.18)-(2.19) and the following LMI-based conditions are verified for p = 1, . . . , r, j = p, . . . , r, and i = 1, . . . , q:

< W T i = W i ∈ R (n+q)×(n+q) , the diagonal matrices 0 < S i ∈ R m×m , the matrices U ∈ R (n+q)×(n+q) , Y i ∈ R m×
      -W i Γ (p,j) 12 Γ13 Γ14 ⋆ Γ (p,j) 22 0.5(Z T p + Z T j ) 0 ⋆ ⋆ -S p -S j 0 ⋆ ⋆ ⋆ -τ 2 I q       < 0 (2.34) with Γ13 = 0.5(B p S j + B p S j ), Γ14 = 0.5(B ω p + B ω j ), Γ (p,j) 12
given by (2.20), and Γ (p,j) 22

by (2.21).

Then the fuzzy PI-like control law (2.10) with gains computed by:

K i = -Y i U -1 (2.35)
ensures:

1. for y r k = 0, the local asymptotic stability for the respective fuzzy closed-loop system, for all initial conditions belonging to (2.36)

D * a = i∈Ir E(W -1 i ,
S ν :                  min ν, W i , S i , Y i , Z i , U i ∈ I r ν s.t
Another interesting objective can be to maximize the amplitude bound of the tracking reference signal y r k , i.e. to maximize the allowable amplitude changes in the reference signal, such that the trajectories starting in D * a do not leave it. In this case, the following convex optimization procedure can apply with Theorem 2.1:

S δ :        min δ, W i , S i , Y i , Z i , U, E ci δ s.t.
(2.17), (2.18), and (2.19).

(2.37) Remark 2.4. The optimization procedures S ν and S δ can be used without the anti-windup design. In this case, it is enough to replace (2.17) by (2.34), and thus, employing Corollary 2.1 in these procedures.

Illustrative Examples

In what follows, before going to the experimental validation of the proposed approach on an industrial oriented process, a numerical example is presented to illustrate the relevance of the proposed design procedure, considering the bounds of the actuators saturation, and to compare the obtained results with recent relevant works from the literature [START_REF] Lv | Fuzzy PID controller design for uncertain networked control systems based on T-S fuzzy model with random delays[END_REF] and [START_REF] Wang | H ∞ fuzzy PID control for discrete time-delayed T-S fuzzy systems[END_REF].

Numerical example

Assume the T-S fuzzy model given by (2.2)-(2.3) and investigated in [START_REF] Lv | Fuzzy PID controller design for uncertain networked control systems based on T-S fuzzy model with random delays[END_REF] with the matrices:

A 1 = 0 1 0.1 -2 , A 2 = 0 1 0 -0.5 , A 3 = 0.6 -1 1 0 , B 1 = 0 1.1 , B 2 = 0.1 0.4 , B 3 = 0.2 0.3 , C 1 = -0.1 0.3 T , C 2 = 0.4 -0.1 T , C 3 = 0.3 0.5 T ,
and the normalized membership functions given by:

α 1 (x 1,k ) = 1, α 2 (x 1,k ) = α 3 (x 1,k ) = 0, for ||x 1,k || > 5, α 1 (x 1,k ) = 0, α 2 (x 1,k ) = (5 -||x 1,k ||)/5, α 3 (x 1,k ) = 1 -α 2 , otherwise.
(2.38)

Although [START_REF] Lv | Fuzzy PID controller design for uncertain networked control systems based on T-S fuzzy model with random delays[END_REF] considers time-delay, this part is left-out from this example as it is not the concern of our study.

Additionally, let us assume that the control signal is bounded by ±10. Our objective is to design a controller solving the Problem P 1 with K w (α) = 0, η = 1, τ 1 = 0.9999, and y r k = 0, thus, maximizing the estimate of the region of attraction. Therefore, we use the convex optimization procedure S ν in (2.36) with the conditions proposed in Corollary 2.1 with C m , m = 1, . . . , 3. The achieved estimate of the region of attraction is shown in Figure 2.2, where the points belonging to the intersection of the ellipses (black dashed line) correspond to the optimized border of the estimated region of attraction. However, when the design procedure proposed in [START_REF] Lv | Fuzzy PID controller design for uncertain networked control systems based on T-S fuzzy model with random delays[END_REF] is considered, the convergence of the trajectories cannot be ensured.

Indeed, in this case, many initial conditions lead to unstable trajectories. Some of them are marked in blue × in Figure 2.2. A similar comparison is proposed with a recent fuzzy PID controller proposed in [START_REF] Wang | H ∞ fuzzy PID control for discrete time-delayed T-S fuzzy systems[END_REF]. Using the conditions proposed in (Wang et al., 2019, Th. 2) the convergence of the trajectories cannot be ensured for several initial conditions. Some examples of initial conditions yielding a divergent trajectory are marked with a red circle, •, in Figure 2.2. This clearly shows the relevance of our approach in terms of conservatism reduction and its potential, especially in real industrial applications, will be illustrated in the following section with the nonlinear control level of a two interactive tanks system.

To conclude this numerical example, let us consider the optimization procedure (2.37) to maximize the bound variation of y r k by using both Theorem 2.1 and Corollary 2.1. By employing η = 1 and τ 1 = 0.21, the maximum achievable bounds were β = 167.97 with Corollary 2.1, and β = 182.30 with Theorem 2.1. Thus, thanks to the anti-windup control action, the maximal variation of the reference signal has augmented by 7.86%. This clearly demonstrates the advantage of considering such a control strategy. Moreover, let us point-out that the approaches proposed in [START_REF] Lv | Fuzzy PID controller design for uncertain networked control systems based on T-S fuzzy model with random delays[END_REF] and [START_REF] Wang | H ∞ fuzzy PID control for discrete time-delayed T-S fuzzy systems[END_REF]) cannot be used, because they cannot handle amplitude bounded exogenous signals.

Experimental Validation

This section is devoted to illustrate and validate the proposed PI-like fuzzy controller design conditions on a real experimental setup, consisting of the level control problem of coupled tanks, with standard industrial equipment. First, we will present the experimental setup, then its physical modeling and finally the experimental results followed by a discussion.

Experimental setup:

The considered test-bench consists of an interactive tank system inspired by [START_REF] Johansson | The Quadruple-Tank Process: A Multivariable Laboratory Process with an Adjustable Zero[END_REF], shown in In what follows, a specific configuration of two tanks, as illustrated in Figure 2.4, has been used. In the following experiments, the manual valves FV-01 (connection between the tanks TQ-01 and TQ-02) and FV-02 (outflow) are kept with constant opening. Let u * (t) be the power delivered to the pump BA-01 given as a percentage of its maximal capability, the tank TQ-01 receives the controlled flow q i,1 (satu * (t)) and is connected through FV-01 to the tank TQ-02 that has an outlet flow q o (h 2 (t)). The continuous states of the system are h 1 (t) and h 2 (t), respectively the levels of TQ-01 and TQ-02. The cross section area of TQ-02 is constant (a 2 = 3.019 × 10 -5 cm 2 ) while TQ-01 has a nonlinear variable cross section area due to solid body placed inside it as indicated in Figure 2.4. In this case, the area is given by:

a 1 (h 1 (t)) = 8.1r 2 5 - 300r cos(0.025π(h 1 (t) -8) -0.4)) 2.75 √ 2π × exp -((h 1 (t) -8)10 -2 -0.4) 2 0.605 (2.39)
where r = 31cm is the radius of the tank.

BA-01

q i,1 (u 1 (t)) R 12 (∆h)
FV-01 FV-02 TQ-01 TQ-02

q o (h 2 (t)) h 1 h 2 Figure 2.4: Coupled Tank System Schematic.
Hence, because of the nonlinearity (2.39), it is convenient to employ the T-S modeling of the whole system, and so the application of the proposed approach for PI-like fuzzy controller design.

T-S fuzzy modeling of the considered two tanks system:

The dynamical model of the considered coupled tank system is obtained from the mass balance equations, which are resulting in:

ḣ1 (t) = K b satu * (t) a 1 (h 1 (t)) - h 1 (t) -h 2 (t) a 1 (h 1 (t))R 12 (h 1 (t), h 2 (t)) (2.40) ḣ2 (t) = h 1 (t) -h 2 (t) R 12 (h 1 (t), h 2 (t))a 2 - 1 a 2 q o (h 2 (t)) (2.41)
where K b = 16.998 is a gain for the pump BA-01 and R 12 (h 1 , h 2 ) = (0.412(h 1 (t) -h 2 (t)) + 11.488)10 -3 is the flow resistance between the two tanks, the outlet flow is modeled as q o (h 2 (t)) = (12.241h 2 (t) + 868.674) cm 3 /s.

All the above-given values of parameters and scalars functions were identified from experimental data.

Let T be the sampling time period used in the digital control of the plant,

h k = h 1,k h 2,k
T and u * k be respectively the discrete-time state vector and input signal. By using Euler discretization methods, a discrete-time model representing (2.40) is given by:

h k+1 = 1 -z 1,k z 3,k z 1,k z 3,k z 1,k a 2 1 - z 1,k +z 2,k a 2 h k + T K b z 3,k 0 satu * k (2.42)
with:

z 1,k = T R 12 (h 1,k , h 2,k ) , (2.43) z 2,k = T q 0 (h 2,k ) h 2,k , (2.44) z 3,k = (a 1 (h 1,k )) -1 , (2.45)
and a 1 (h 1 ) computed by (2.39).

In what follows, to obtain a T-S fuzzy model of this discretized dynamic equation we choose the following operating range (in centimeters) for the state variables verifying 20 ≤ h 1 ≤ 74 and 12 ≤ h 2 ≤ 50. Taking into account that both tanks have a maximal level of 75cm, the level of tank TQ-01 can vary from 26.7% up to 98.7%, and the level for tank TQ-02 from 16% up to 66.7% of the maximal allowed level (75cm). These choices have been made to avoid tank overflows as well as to avoid turbulent flows at low levels due to the resistance of valve FV-01.

Therefore, we assume z 1,k , z 2,k , and z 3,k as the premise variables such that: z3 ], z 3 = 0.0004, z3 = 0.0048.

       z 1,k ∈ [z 1 , z1 ], z 1 = 124.6572, z1 = 216.3098; z 2,k ∈ [z 2 , z2 ], z 2 = 118.4579, z2 = 338.5220; z 3,k ∈ [z 3 ,
(2.46)

By applying the sector nonlinearity approach (Tanaka and Wang, 2001), for j = 1, . . . , 3, we can rewrite (2.46) as:

z j,k = w j 1,k z j + w j 2,k zj , (2.47)
where w 1 j,k ≥ 0, w 2 j,k ≥ 0, w 1 j,k + w 2 j,k = 1 and:

w j 1,k = zj -z j,k zj -z j , and w j 2,k = z j,k -z j zj -z j .
(2.48)

This allows to derive an exact T-S model (2.2) representing the dynamic equation of h k+1

having N = 8 vertices with the matrices:

A 1 = 1 -z 1 z 3 z 1 z 3 z 1 a 2 1 - z 1 +z 2 a 2 , A 2 = 1 -z 1 z3 z 1 z3 z 1 a 2 1 - z 1 +z 2 a 2 A 3 = 1 -z 1 z 3 z 1 z 3 z 1 a 2 1 - z 1 +z 2 a 2 , A 4 = 1 -z 1 z3 z 1 z3 z 1 a 2 1 - z 1 +z 2 a 2 A 5 = 1 -z1 z 3 z1 z 3 z1 a 2 1 -z1 +z 2 a 2 , A 6 = 1 -z1 z3 z1 z3 z1 a 2 1 -z1 +z 2 a 2 A 7 = 1 -z1 z 3 z1 z 3 z1 a 2 1 -z1 +z 2 a 2 , A 8 = 1 -z1 z3 z1 z3 z1 a 2 1 -z1 +z 2 a 2 B 1 = B 3 = B 5 = B 7 = T K b z 3 0 T , B 2 = B 4 = B 6 = B 8 = T K b z3 0 T ,
with the following membership function vector

α k = w 1 1,k w 2 1,k w 3 1,k w 1 1,k w 2 1,k w 3 2,k w 1 1,k w 2 2,k w 3 1,k w 1 1,k w 2 2,k w 3 2,k . . . w 1 2,k w 2 1,k w 3 1,k w 1 2,k w 2 1,k w 3 2,k w 1 2,k w 2 2,k w 3 1,k w 1 2,k w 2 2,k w 3 2,k T where, ∀i = 1, . . . , 8, α k(i) ≥ 0, 8 i=1 α k(i) = 1
, and the functions w j i,k computed as in (2.48) with z j,k given by (2.43)-(2.45).

Even that both levels h 1 and h 2 are measured from pressure sensors, only h 2 is controlled.

Therefore, to design the PI-like PDC controller, the output matrix C = 0 1 will be considered to implement the design conditions expressed in Theorem 2.1 or Corollary 2.1. Moreover, there is no operational interest to drive the considered system to the points (h 1 , h 2 ) = (0, 0) since it represents the empty tanks situation. Hence, to drive our experiment and to test the proposed anti-windup scheme, we assume that the input signal is saturated such that u * k ∈ [u * , ū * ], where u * = 30% and ū * = 70% of the capability of the pump. This case illustrates that the actuator saturation bounds may arise not only by physical limitations, but also for operational security reasons. Thus, to apply our control strategy, a change of origin is required and given by:

x k = x 1,k = h 1,k -h O 1 x 2,k = h 2,k -h O 2
, and

u k = u * k -u O (2.

Controller design and closed-loop experiments:

In this subsection, the proposed design methodology of the fuzzy controller is considered to solve both problems P 1 and P 2 . Due to the change of origin (2.49), an offset of the reference signal to be tracked also occurs, i.e.:

y r k = y * r k -h O 2 (2.50)
where y * r k denotes the absolute reference signal before the change of origin. Let us first consider the stabilization problem P 1 , i.e. y r k = 0, and evaluate the proposed fuzzy PI-like controller design conditions with and without the anti-windup action. This can be achieved by solving the optimization procedure S ν given in (2.36) with the design conditions expressed in Theorem 2.1 and Corollary 2.1, accordingly to Remark 2.4. We show in Figure 2.6 the obtained estimates R e of the regions of attraction from Theorem 2.1 and Corollary 2.1. Each plotted points correspond to initial conditions ξ 0 = x 1,0 x 2,0 v 0 T taken in R e with an equally spaced grid by 0.5. Let us highlight that the number of points in R e with the anti-windup action (Theorem 2.1) raised 2.55 × 10 7 , whereas without the anti-windup action (Corollary 2.1) it is only 6.97 × 10 5 points, i.e. about 2.7% of the region computed with antiwindup action. Thus, this illustrates the significant improvement when the anti-windup action is considered. Furthermore, the presented results have been obtained under the assumption of a fuzzy Lyapunov function (2.14). Obviously, if a quadratic approach is considered as quote in Now, let us consider the set-point tracking problem P 2 , where the reference signal y r k to be tracked is piecewise constant over a sufficient number of samples. By solving the proposed conditions with and without the anti-windup action, respectively Theorem 2.1 and Corollary 2.1, the goal is now to maximize the allowable amplitude range β of the set-point y r k , which means minimizing δ with the optimization procedure S δ given in(2.37) and Remark 2.4, around its considered origin (see (2.50)). The related regions of attraction, shown in Figure 2.7, have been computed similarly as in the previous case. Once again, it clearly appears that the estimated region of attraction obtained from Theorem 2.1, i.e. including the integral and the anti-windup action, is much bigger than the one obtained from Corollary 2.1, i.e. without the anti-windup action. Indeed, with the same equally spaced grid by 0.5, the region of attraction contains 1.54 × 10 7 points with Theorem 2.1 and 2.36 × 10 5 points with Corollary 2.1, i.e. about 1.54% of the region computed with anti-windup. This illustrates, one more time, the benefit when the anti-windup scheme is considered. Moreover, as expected, the regions achieved by solving Theorem 2.1 for Problem P 2 are about 60.3% smaller than the ones obtained for Problem P 1 . Indeed, let us recall that the set-point tracking problem is more constrained than the simple stabilization one. A particular set-point tracking trajectory is also plotted in Figure 2.7 (see black dots), this will be discussed latter. For the considered coupled tanks systems with the whole set-points tracking control scheme depicted in Figure 2.1, the following PI-like and anti-windup gains have been obtained with the parameters η = 1 and τ 1 = 0.94 from Theorem 2.1 and the optimization procedure S δ given in (2.37), which provides a maximal allowable bound of the reference signal |y r k | ≤ β = 6.64cm around the origin defined in (2.49): For this simulation, the resulting trajectory ξ k is also shown in Figure 2.7 (see the above mentioned particular trajectory depicted by black dots). It clearly appears that it stays inside the estimated region of attraction D * a for Theorem 2.1 (blue dots), and so remains stable. Moreover, it also clearly shows that this trajectory overpasses the border of the estimated region of attraction for Corollary 2.1, i.e. without anti-windup action. One more time, these observations illustrate the significant improvement of our proposal with the fuzzy anti-windup action.

K P 1 = -0.4363 3.3327 , K I1 = 0.1076, S 1 = 0.8234, E c1 = 1.
To conclude this experimental results subsection, let us go back to Corollary 2.1 associated with the optimization procedure S δ given in(2.37), which provides the following gains when the anti-windup scheme is not considered: K P 1 = 0.4778 6.3142 , K I1 = 0.4911, K P 2 = 3.7216 9.1654 , K I2 = 0.9489, K P 3 = 0.8313 6.4683 , K I3 = 0.5139, K P 4 = 3.9880 11.1556 , K I4 = 1.0414, K P 5 = 0.1946 4.9830 , K I5 = 0.3571, K P 6 = 4.2113 9.9432 , K I6 = 1.0356, K P 7 = 1.3820 6.1739 , K I7 = 0.5303, K P 8 = 4.1947 10.1137 , K I8 = 1.0239.

In this case, the obtained maximal allowable bound of the reference signal is only |y r k | ≤ β = 1.41cm around the origin, i.e. a reduced range in the set-points tracking of the second tank. Whereas, Figure 2.9 presents experimental results with the same set-points reference as for the previous experiment. Obviously, the latter overpasses the allowed reference bounds for the present case, but it allows us to highlight that, when these bounds are not respected, we may have bad transient response due to higher windup effects and, even if the set-points are reached after a longer settling times, there is no guarantee about the overall closed-loop stability. Therefore, the experimental tests show the occurrence of actuators saturation, which indicates that the controller delivers the maximum allowed energy to control the system, leading to better regulation and transient performances, while ensuring local stability.

Conclusions

In this chapter, we proposed a discrete-time fuzzy PI like controller with fuzzy anti-windup action to stabilize nonlinear discrete-time systems. The saturating actuators are considered in the design method and an estimate of the region of attraction is provided. A fuzzy Lyapunov function is used to certify the discrete-time closed-loop stability and to provide an estimate of the region of attraction. Moreover, the proposed design method can take into account bounded exogenous signals, providing for instance bounds for the set-point variation and ensuring that the closed-loop trajectories do not leave the region of attraction during set-point changes. Our approach has been shown less conservative than recent relevant result from the literature through a numerical example, then applied experimentally to a nonlinear level control system where the measures and control signals are performed with an industrial programmable logic controller (PLC).

The experimental results show the efficiency of the proposal and clearly indicates how the fuzzy anti-windup action can improve the time behavior of the closed-loop system. Additionally, the experimental tests show the occurrence of actuators saturation, which indicates that the controller delivers the maximum allowed energy to control the system, leading to a better regulation and transient performances, while ensuring local stability. The proposed PI-like control methodology with anti-windup action is given as general design conditions, which is not restricted to the test-bench of the coupled tank system described in this chapter. Indeed, it is clear that our approach can found a rich field of applications in the industrial processes where saturation may occur, such like furnaces with oil or gas fuel, chemical and petroleum processes, etc.

However, let us recall that some limitations arise when considering discrete-time control approaches for continuous-time plants. Indeed, if the results presented in this chapter are attractive for the design of point-to-point controllers, they require a constant sampling period, which is challenging to be guaranteed, especially when considering networked controllers, event-triggering schemes or sometimes when using low-cost embedded digital devices. Moreover, this technique frequently implies the choice of a hidebound small sampling period to ensure that the discretetime model will catch all the plant dynamics. The aforementioned points certainly reduce the appeal of employing the direct discrete-time approach, especially when continuous-time systems involves fast dynamics driven by low cost digital devices. Moreover, these approaches cannot be employed in the context of Networked Controlled Systems (NCS) or when event-triggering control techniques are considered, since these often involves aperiodic sampled control signals.

In order to cope with such aperiodicity in sampled-data control of continuous-time plant, an appealing approach is to rewrite the closed-loop dynamics as a continuous-time system with time-varying input delays, see e.g. [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]). This will be the focus of our investigations in the next chapters, with the design of sampled-data controllers for continuous-time nonlinear systems represented by T-S fuzzy models.

Chapter 3

Aperiodic Sampled-data Controllers Design for Continous-time T-S Models

Résumé en Français : Synthèse de contrôleurs avec échantillonage apériodique pour les modèles T-S décrits en temps continu.

Dans le chapitre précédent, une approche permettant la synthèse de contrôleurs pour les modèles T-S à temps discret a été proposée dans le but de la commande de systèmes non linéaires, possiblement décrits en temps continu. Si cette approche est intéressante en raison de sa simplicité conceptuelle pour une mise en oeuvre facilitée pour de nombreuses applications industrielles, il est important de rappeler que cette approche n'est valable que lorsqu'une période d'échantillonnage fixe, et suffisamment petite par rapport à la dynamique du système, est définie. Cependant, il est fréquent qu'une telle contrainte ne puisse être vérifiée. Par exemple, même dans le cadre d'un contrôleur numérique basé sur un protocole point à point, une période d'échantillonnage apériodique peut survenir en raison de l'incertitude sur la fréquence de l'horloge ou, plus généralement, l'architecture d'un

système numérique temps réel, peut entraîner des retards de calcul ou une synchronisation imparfaite [START_REF] Wittenmark | Timing problems in real-time control systems[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. De plus, avec le développement des applications utilisant l'Internet des Objets (IoT), les données échantillonnées transitent sur des réseaux de communication où des intervalles d'échantillonnage apériodiques sont généralement inévitables (Fridman, 2014a;[START_REF] De Souza | Co-design of an eventtriggered dynamic output feedback controller for discrete-time LPV systems with constraints[END_REF].

De plus, notons que lorsque de grandes périodes d'échantillonnage sont considérées, l'approche basée sur un modèle à temps discret ne garantit plus la stabilité du système complet en boucle fermée [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. Brièvement présentée dans le Chapitre 1, une approche exploitant un retard sur l'entrée pour les contrôleurs basés sur des données échantillonnées est apparue comme un sujet de recherche prometteur en théorie du contrôle. Cette approche consiste en une réécriture de la dynamique en boucle fermée sous la forme d'un système à temps continu avec un retard variable sur l'entrée [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF] [START_REF] Yoneyama | Robust H ∞ control of uncertain fuzzy systems under time-varying sampling[END_REF]. Ensuite, étant donné que les fonctions d'appartenance possédant un retard impliquées dans le contrôleur ne correspondent pas précisément à celles impliquées dans le système à temps continu à contrôler, les bornes supérieures des erreurs asynchrones des fonctions d'appartenance ont été introduites dans l'obtention des conditions à vérifier [START_REF] Zhang | H ∞ control design for network-based T-S fuzzy systems with asynchronous constraints on membership functions[END_REF][START_REF] Koo | An improved digital redesign for sampled-data fuzzy control systems: Fuzzy Lyapunov function approach[END_REF]. Dans [START_REF] Zhu | An Improved Input Delay Approach to Stabilization of Fuzzy Systems Under Variable Sampling[END_REF], une amélioration de cette approche a été introduite pour les critères de stabilisation. De plus, les plages de variation des fonctions d'appartenance avec des intervalles d'échantillonnage variables ont été considérées dans [START_REF] Zhu | H ∞ stabilization criterion with less complexity for nonuniform sampling fuzzy systems[END_REF]. Plus récemment, un séparateur de sommets structuré a été utilisé afin de réduire le nombre de contraintes LMIs [START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF]. Cependant, comme expliqué dans la Section 1.4.2, il est important de noter que, dans le cadre des modèles de type T-S, la dynamique du système en boucle fermée implique une structure asynchrone possédant une double somme empêchant l'application des schémas de relaxation usuels (e.g. Lemme 1.

. Si de nombreux efforts ont été réalisés pour la stabilisation des systèmes dynamiques linéaires à partir de données échantillonnées, la plupart des applications réelles possède des dynamiques non linéaires, qui motive l'extension de ces approches dans le cadre non linéaire, notamment pour les modèles T-S. Pour la synthèse de contrôleurs basée sur des données échantillonnées, une façon classique d'évaluer le conservatisme est de chercher le plus grand intervalle d'échantillonnage admissible [0, η] qui garantit la stabilisation du système en boucle fermée. Dans le cadre de la synthèse de contrôleurs basée sur des données échantillonnées pour les modèles de type T-S, des réductions successives du conservatisme ont été obtenues. Par exemple, le choix d'une fonctionnelle de Lyapunov-Krasovskii avec l'utilisation de techniques de relaxation basées sur la formule de Leibniz-Newton et de l'introduction de matrices de décision libres a été considéré dans

2). Pour surmonter cet obstacle, la plupart des auteurs des études mentionnées précédemment ont borné les incertitudes associées, mais sans prêter beaucoup d'attention à leur caractérisation, et en particulier le lien avec les dérivées temporelles des fonctions d'appartenance. Il est donc raisonnable de penser que des améliorations significatives de réduction du conservatisme sont envisageables.

Par conséquent, dans ce chapitre, nous suivons le même objectif que les études mentionnées précédemment, afin de réduire le conservatisme, en proposant de nouvelles conditions à base de LMIs pour la synthèse de contrôleurs échantillonnés pour les modèles T-S décrits en temps continu. Pour cela, inspirée des travaux de [START_REF] Briat | A looped-functional approach for robust stability analysis of linear impulsive systems[END_REF], une fonctionnelle de Lyapunov-Krasovskii « bouclée » est définie et les conditions de stabilité en boucle fermée sont obtenues en appliquant le lemme de Finsler (Lemme 1.4) et les inégalités étendues de Jensen (Lemme 1.8).

Par ailleurs, dans ce chapitre, un schéma de relaxation générique, étendant le lemme de Tuan (Lemme 1. [START_REF] Yoneyama | Robust H ∞ control of uncertain fuzzy systems under time-varying sampling[END_REF][START_REF] Zhu | Stabilization for sampled-data neural network-based control systems[END_REF][START_REF] Zhang | H ∞ control design for network-based T-S fuzzy systems with asynchronous constraints on membership functions[END_REF][START_REF] Zhu | An Improved Input Delay Approach to Stabilization of Fuzzy Systems Under Variable Sampling[END_REF][START_REF] Gunasekaran | Robust Sampled-data Fuzzy Control for Nonlinear Systems and Its Applications: Free-Weight Matrix Method[END_REF][START_REF] Zhu | H ∞ stabilization criterion with less complexity for nonuniform sampling fuzzy systems[END_REF][START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF], à partir d'un modèle simplifié d'un pendule inversé sur un chariot [START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF]. Enfin, une validation expérimentale de l'approche de synthèse d'un correcteur basé sur des données échantillonnées est réalisée sur la plateforme Quanser ® AERO [START_REF] Quanser | Quanser aero user manual[END_REF], illustrant l'efficacité de cette approche sur un système réel.

2), est proposé pour tenir compte de la structure à double somme asynchrone apparaissant dans la dynamique en boucle fermée du modèle de type T-S continu commandé par un contrôleur PDC échantillonné. L'avantage de l'approche proposée dans ce chapitre par rapport à une approche basée directement sur un modèle T-S à temps discret est illustré au travers de l'exemple d'un pendule inversé en simulation (Exemple 1.2). Puis une comparaison, montrant des améliorations significatives pour la réduction du conservatisme, est effectuée par rapport aux résultats en lien de la littérature

Introduction

In the previous chapter, a discrete-time T-S model-based controller design approach has been considered for the control of continuous-time nonlinear systems. If this approach is appealing for its conceptual simplicity with regard to many industrial applications, it is important to recall that it is only suitable when a small enough fixed sampling period can be set in regard to the plant dynamics. However, it often occurs that such requirements cannot be fulfilled. For instance, 3.1. Introduction even in a point-to-point digital control topology, aperiodic sampling may arise because of clock inaccuracy and system architecture characteristics such as real-time scheduling, which can induce jitters, imperfect synchronization or computation delays [START_REF] Wittenmark | Timing problems in real-time control systems[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. Furthermore, with the growing of Internet of Things (IoT) applications, sampled-data systems are controlled through communication networks where aperiodic sampling intervals are almost inevitable (Fridman, 2014a;[START_REF] De Souza | Co-design of an eventtriggered dynamic output feedback controller for discrete-time LPV systems with constraints[END_REF]. Nonetheless, it is also well-known that when large sampling periods have to be considered, the direct discrete-time model-based approach may fail to guarantee the closed-loop inter-sampling stability [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF].

As introduced in Chapter 1, Section 1.4.2, the input delay approach for sampled-data control emerged as a promising research topic in control theory. It consists of rewriting the closed-loop dynamics as a continuous-time system with input time-varying delay [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. If many efforts have been done for the stabilization of linear dynamical systems from sampled-data measurements, most of real applications exhibit nonlinear dynamics, which motivate our interest in extending these approaches to the T-S model-based framework.

When dealing with sampled-data control, a convenient way to check the conservatism of the design conditions is to search for the maximal allowable sampling interval [0, η], which ensures the closed-loop dynamics stabilization. In the context of the sampled-data control of T-S models, successive conservatism improvements have been obtained. For instance, a Lyapunov-Krasovskii functional (LKF) and relaxation techniques based on the Leibniz-Newton formula and free-weighting matrix has been considered in [START_REF] Yoneyama | Robust H ∞ control of uncertain fuzzy systems under time-varying sampling[END_REF]. Then, since the delayed membership functions involved in the controller part do not match the ones involved in the continuous-time plant to be controlled, the upper bounds of the asynchronous errors of the membership functions have been introduced in the design conditions [START_REF] Zhang | H ∞ control design for network-based T-S fuzzy systems with asynchronous constraints on membership functions[END_REF][START_REF] Koo | An improved digital redesign for sampled-data fuzzy control systems: Fuzzy Lyapunov function approach[END_REF]. In [START_REF] Zhu | An Improved Input Delay Approach to Stabilization of Fuzzy Systems Under Variable Sampling[END_REF], an enlargement scheme has been introduced in the stabilization criteria. Furthermore, the variation ranges of membership functions within variable sampling intervals have been considered in [START_REF] Zhu | H ∞ stabilization criterion with less complexity for nonuniform sampling fuzzy systems[END_REF]. More recently, a structured vertex separator has been used to reduce the number of LMIs constraints [START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF].

However, as explained in Section 1.4.2, it is important to highlight that, in the T-S modelbased framework, the closed-loop dynamics involve an asynchronous double summation structure, which prevents from applying classical relaxation schemes (e.g. Lemma 1.2). To cope with such issue, most of the authors of the above-mentioned studies assumed bounds of these mismatches, without paying much attention to their characterization, especially their relationship to the time-derivatives of the membership function. In this regard, we believe that there is still space for significant conservatism improvements. Therefore, in this chapter, we follow the same goal as the above-mentioned studies to reduce the conservatism, i.e. the main objective is to propose new relaxed LMI-based conditions for the design of stabilizing aperiodic sampled-data controllers for continuous-time T-S systems.

To achieve this goal, a convenient augmented looped LKF candidate is selected, together with the application of bounding lemmas, such as extended Jensen's inequalities (Lemma 1.8 and the Finsler's Lemma (Lemma 1.4). Moreover, in this chapter, a generic relaxation scheme, extending Tuan's Lemma (Lemma 1.2), is proposed to handle the mismatched double summation structure of parameter dependent closed-loop stability conditions, which occurs in the considered T-S model-based sampled-data control context. The benefit of the proposed sampled-data approach vs conventional discrete-time T-S model-based design is illustrated through the 1-DOF inverted pendulum benchmark, previously introduced as Example 1.2. Then a comparison is proposed showing significant conservatism improvements regarding several previous related results from the literature [START_REF] Yoneyama | Robust H ∞ control of uncertain fuzzy systems under time-varying sampling[END_REF][START_REF] Zhu | Stabilization for sampled-data neural network-based control systems[END_REF][START_REF] Zhang | H ∞ control design for network-based T-S fuzzy systems with asynchronous constraints on membership functions[END_REF][START_REF] Zhu | An Improved Input Delay Approach to Stabilization of Fuzzy Systems Under Variable Sampling[END_REF][START_REF] Gunasekaran | Robust Sampled-data Fuzzy Control for Nonlinear Systems and Its Applications: Free-Weight Matrix Method[END_REF][START_REF] Zhu | H ∞ stabilization criterion with less complexity for nonuniform sampling fuzzy systems[END_REF][START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF], from the benchmark of a simplified model (approximated 2 rules T-S fuzzy model drawn from [START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF]) of an inverted pendulum on a cart. Finally, an experimental validation of the proposed sampled-data design methodology is performed on the Quanser ® AERO platform [START_REF] Quanser | Quanser aero user manual[END_REF], illustrating its effectiveness on a practical system.

Considered problem statement for T-S model-based sampleddata control design

Let us consider a continuous-time T-S system given by:

ẋ(t) = r i=1 α i (z(t)) (A i x(t) + B i u(t)) (3.1)
where z(t) = z 1 (t) ... z p (t) ∈ R p is a known vector of premise variables which only depends (for control purpose) on the entries of the state vector x(t) ∈ R n , u(t) ∈ R m is the control input vector, A i ∈ R n×n , B i ∈ R n×m are known constant matrices describing the dynamics of each polytope and α i (z(t)) ≥ 0 are the membership functions satisfying the convex properties

r i=1 α i (z(t)) = 1.
In this chapter, we consider the stabilization of T-S systems (3.1) from the following sampleddata PDC control law:

u(t) = r i=1 α i (z(t k ))K i X -1 x(t k ) (3.2)
where K i ∈ R m×n and X -1 ∈ R n×n , for i ∈ I r , are the controller gain matrices to be designed.

With such sampled-data controllers, a Zero Order Hold (ZOH) is employed, ∀t ∈ [t k , t k+1 ), to maintain x(t k ) from the aperiodic sampling instants t k ≥ 0 such that:

t k+1 -t k = η k ≤ η (3.3)
where the inner sampling intervals bounds η k > 0 can be non-uniform over samples with a maximal allowable sampling period η to be estimated.

For actual t ∈ [t k , t k+1 ), let:

τ (t) = t -t k ∈ [0, η k ) (3.4)
where τ (t) = 1.

The control law (3.2) can be rewritten as:

u(t) = r i=1 α i (z(t -τ (t)))K i X -1 x(t -τ (t)) (3.5)
Substituting (3.5) in (3.1) gives the following closed-loop dynamics expressed as a T-S system, involving input time-varying delays and mismatching membership functions:

ẋ(t) = r i=1 r j=1 α i (z(t))α j (z(t -τ (t)))(A i x(t) + B i K j X -1 x(t -τ (t))) (3.6)
In the sequel, to lighten mathematical expressions, we will use the following notations for fuzzy summations of matrices:

M α = r i=1 α i (z(t))M i , M ᾱ = r i=1 α i (z(t -τ (t)))M i
and:

M α ᾱ = r i=1 r j=1 α i (z(t))α j (z(t -τ (t)))M ij
Hence, (3.6) yields:

ẋ(t) = A α x(t) + B α K ᾱX -1 x(t -τ (t)) (3.7)
Note that the sampled-data closed-loop dynamics involves an asynchronous (mismatching) double fuzzy summation structure ( r i=1 r j=1 α i (z(t))α j (z(t -τ (t))) . . . ), which makes the design of sampled-data controllers (3.5) harder (more conservative) than in the standard continuous or discrete-time cases. This chapter aims at providing less pessimistic conditions to address this issue according to the following problem statement (divided in two points P 1 and P 2 ).

Problem statement:

P 1 : Provide relaxed parameterized LMI-based conditions for the design of the gain matrices K i and X such that the sampled-data closed-loop dynamics (3.7) is asymptotically stable.

P 2 : Propose a generic relaxation scheme to solve inequalities parameterized as Λ α ᾱ < 0, which constitutes a double fuzzy summation structure with asynchronous (mismatching) membership functions.

The next section is devoted to address P 1 , while improvements for P 2 will be proposed in section 3.4.

LMI-based sampled-data controller design

In the sequence, the goal is to provide parameterized LMI-based conditions satisfying P 1 (of the above problem statement). The closed-loop dynamics (3.7) resulting in a T-S system with input time-varying delay, we propose to derive these conditions from the following LKF:

V (t) = V 1 (t) + V 2 (t) + V 3 (t) + V 4 (t) (3.8)
where:

V 1 (t) = x(t) T Lx(t) (3.9) V 2 (t) = (η k τ (t) -τ 2 (t))ζ T (t)M ᾱζ(t) (3.10) V 3 (t) = (η k -τ (t))ρ T (t)N ᾱρ(t) (3.11) V 4 (t) = (η k -τ (t)) t t-τ (t)
χ T (s)P α ᾱχ(s)ds (3.12) with χ(t) = col {x(t), ẋ(t)} and:

ζ(t) = col x(t), x(t -τ (t)), t t-τ (t)
x(s)ds,

t t-τ (t) ẋ(s)ds , ρ(t) = col t t-τ (t)
x(s)ds,

t t-τ (t)
ẋ(s)ds .

We motivate the choice of this LKF candidate by the fact that it belongs to the class of looped-functionals, which is a feature that have been shown appealing in the context of sampleddata control for linear systems [START_REF] Briat | Commande et Observation Robustes des Systèmes LPV Retardées[END_REF]Seuret, 2012, 2015;[START_REF] Seuret | Stability analysis of uncertain sampled-data systems with incremental delay using looped-functionals[END_REF]. Indeed, it is worth to mention that, assuming L = L T > 0, the whole LKF (3.8) is continuous and positive at each sample time

t k since we have V 1 (t k ) > 0 and V ℓ (t - k ) = V ℓ (t k ) = 0, for ℓ = 2, .
.., 4. Furthermore, the only requirement is that the LKF (3.8) must be monotonously decreasing during the inner sampling intervals, i.e. ∀t ∈ [t k , t k+1 ). This implies that, ∀t ∈ [0, +∞), it is positive and monotonously decreasing, and so the closed-loop dynamics (3.7) is guaranteed to be stable. In addition, the extended vectors and structures of the decision matrices in (3.10)- (3.12) have been chosen to conveniently avoid sparsity, according to the development of the relaxed LMI-based conditions summarized in the following Theorem, with the help of an extended version of Jensen's inequality [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF] and the Finsler lemma [START_REF] Skelton | A unified algebraic approach to linear control design[END_REF] (see Lemma 1.8 and Lemma 1.4 given in Chapter 1). Theorem 3.1. : Let (i, j) ∈ I 2 r and assume that there exists the scalars φ i > 0 such that ∀t, | αi (t)| ≤ φ i . For aperiodic sampling periods η k ≤ η (η to be maximized), the T-S fuzzy model (3.1) is stabilized by the sampled-data PDC controller (3.2) if there exists the matrices

0 < L = LT ∈ R n×n , Mj = M T j ∈ R 4n×4n , Nj = N T j ∈ R 2n×2n , P11ij = P T 11ij ∈ R n×n , P22j = P T 22j ∈ R n×n , P12j = P T 12j ∈ R n×n , X ∈ R n×n , K j ∈ R m×n , Ȳij ∈ R 4n×n
and the scalars ε 1 , ε 2 and ε 3 , such that the following parameterized LMI-based condition is satisfied:

Λαᾱ =        Λ11 α ᾱ * * * * 0 Λ22 α ᾱ * * * 0 η Ȳαᾱ -η P22ᾱ * * 0 0 0 Λ44 α ᾱ * 0 0 0 0 -P11αᾱ        < 0 (3.13) with: Λ11 α ᾱ = Φ0 Σα ᾱ + I ε Ḡαᾱ + ḠT α ᾱI T ε , Λ44 α ᾱ = Λ11 α ᾱ -τ 2 Φ2 2 ᾱ Λ22 α ᾱ = η2 Φ 2 2 ᾱ + η( Φ1 Σ ᾱ -Pαᾱ ) + Φ0 Σα ᾱ + I ε Ḡαᾱ + ḠT α ᾱI T ε , I ε = I ε 1 I ε 2 I ε 3 I T , Ḡαᾱ = A i X B i K ᾱ 0 -X , Φ1 Σ ᾱ = H ηE T 1 Mᾱ E 2 -E T 1 Mᾱ E 1 -H E T 4 Nᾱ E 5 , Φ0 Σα ᾱ =ηE T 1 Mᾱ E 1 + H ηE T 4 Nᾱ E 5 -H E T Ȳαᾱ +      η P11αᾱ -P12ᾱ 0 0 L + η P12ᾱ 0 P12ᾱ 0 0 0 0 -η -1 k P11αᾱ 0 * 0 0 η P22αᾱ      , Mᾱ =      M11ᾱ M12ᾱ M13ᾱ M14ᾱ * M22ᾱ M23ᾱ M24ᾱ * * M33ᾱ M34ᾱ * * * M44ᾱ      , Pαᾱ =      P11αᾱ 0 0 P12ᾱ 0 0 0 0 0 0 0 0 * 0 0 P22ᾱ      , E 1 =      e 1 e 2 e 3 e 1 -e 2      , E 2 =      e 4 e 0 e 1 e 4      , E 4 = e 3 e 1 -
V (t) = V1 (t)) + V2 (t)) + V3 (t)) + V4 (t)) < 0 (3.14)
To provide less conservative stability conditions, let us consider the following extended state vector:

ξ(t) = col x(t), x(t -τ (t)), t t-τ (t)
x(s)ds, ẋ(t) (3.15)

The time-derivative of V 1 (t) is straightforwardly obtained as:

V1 (t) = 2x T (t)L ẋ(t) = ξ T (t)Φ 0 1 ξ(t), Φ 0 1 =      0 0 0 L 0 0 0 0 0 0 0 0 L 0 0 0      (3.16)
Then, for V 2 (t), the time-derivative is computed as:

V2 (t) = (η k -2τ (t))ζ T (t)M ᾱζ(t) + 2(η k τ (t) -τ 2 (t))ζ T (t)M ᾱ ζ(t) (3.17) Assuming ζ(t) = E 1 ξ(t) and ζ(t) = E 2 ξ(t)
, with E 1 and E 2 given above, (3.17) can be rewritten as:

V2 (t) = τ 2 (t)ξ T (t)Φ 2 2 ᾱξ(t) + τ (t)ξ T (t)Φ 1 2 ᾱξ(t) + ξ T (t)Φ 0 2 ᾱξ(t) (3.18) with Φ 2 2 ᾱ = -H E T 1 M ᾱE 2 , Φ 1 2 ᾱ = H η k E T 1 M ᾱE 2 -E T 1 M ᾱE 1 and Φ 0 2 ᾱ = η k E T 1 M ᾱE 1 .
In a similar way, the time-derivative of V 3 (t) is given by: (3.19) and since ρ(t) = E 4 ξ(t) and ρ(t) = E 5 ξ(t) we can write:

V3 (t) = -ρ T (t)N ᾱρ(t) + 2(η k -τ (t))ρ T (t)N ᾱ ρ(t),
V3 (t) = τ (t)ξ T (t)Φ 1 3 ᾱξ(t) + ξ T (t)Φ 0 with Φ 0 3 ᾱ = H η k E T 4 N ᾱE 5 -E T 4 N ᾱE 4 and Φ 1 3 ᾱ = -H E T 4 N ᾱE 5
. Taking the time-derivative of V 4 (t), we get:

V4 (t) = (η k -τ (t))χ T (t)P α ᾱχ(t) - t t-τ (t)
χ T (s)P α ᾱχ(s)ds (3.21) Assuming P α ᾱ = P 11α ᾱ P 12 ᾱ * P 22 ᾱ leads to:

V4 (t) =(η k -τ (t))χ T (t)P α ᾱχ(t) - t t-τ (t)
x T (s)P 11α ᾱx(s)ds

- t t-τ (t)
ẋT (s)P 22 ᾱ ẋ(s)ds -

2 t t-τ (t)
x T (s)P 12 ᾱ ẋ(s)ds (3.22) That is to say:

V4 (t) =(η k -τ (t))χ T (t)P α ᾱχ(t) - t t-τ (t)
x T (s)P 11α ᾱx(s)ds

- t t-τ (t)
ẋT (s)P 22 ᾱ ẋ(s)ds -x T (t)P 12 ᾱx(t) + x T (t -τ (t))P 12 ᾱx(t -τ (t))

(3.23)

Assuming P 11α ᾱ > 0 and applying Lemma 1.7 on the first integral term, we have:

V4 (t) ≤(η k -τ (t))χ T (t)P α ᾱχ(t) -η -1 k t t-τ (t) x T (s)dsP 11α ᾱ t t-τ (t)
x(s)ds

- t t-τ (t)
ẋT (s)P 22 ᾱ ẋ(s)ds -x T (t)P 12 ᾱx(t) + x T (t -τ (t))P 12 ᾱx(t -τ (t))

(3.24)

Note that: Hence, assuming P 22 ᾱ > 0, which is constant ∀t ∈ [t k , t k+1 ), and applying Lemma 1.8 on (3.24),

for any matrix Y α ᾱ we have:

V4 (t) ≤ (η k -τ (t))χ T (t)P α ᾱχ(t)+ξ T (t) -E T Y α ᾱ -Y T α ᾱE +τ (t)Y T α ᾱP -1 22 ᾱY α ᾱ ξ(t) -η -1 k t t-τ (t) x T (s)dsP 11α ᾱ t t-τ (t)
x(s)ds-x T (t)P 12 ᾱx(t)+x T (t -τ (t))P 12 ᾱx(t -τ (t)), (3.26) or, equivalently:

V4 (t) ≤ τ (t)ξ T (t)Φ 1 4α ᾱξ(t) + ξ T (t)Φ 0 4α ᾱξ(t) (3.27)
with:

Φ 1 4α ᾱ = Y T α ᾱP -1 22 ᾱY α ᾱ -Pαᾱ , Pαᾱ =      P 11α ᾱ 0 0 P 12 ᾱ 0 0 0 0 0 0 0 0 * 0 0 P 22 ᾱ     
and:

Φ 0 4α ᾱ = -H E T Y α ᾱ +      η k P 11α ᾱ -P 12 ᾱ 0 0 η k P 12 ᾱ 0 P 12 ᾱ 0 0 0 0 -η -1 k P 11α ᾱ 0 * 0 0 η k P 22 ᾱ      , requiring P 12 ᾱ = P T 12 ᾱ.
So, from (3.16), (3.18), (3.20) and (3.27), the inequality (3.14) is satisfied if:

P(τ (t)) =τ 2 (t)ξ T (t)Φ 2 2 ᾱξ(t) + τ (t)ξ T (t)(Φ 1 2 ᾱ + Φ 1 3 ᾱ + Φ 1 4α ᾱ)ξ(t) + ξ T (t)(Φ 0 1 + Φ 0 2 ᾱ + Φ 0 3 ᾱ + Φ 0 4α ᾱ)ξ(t) < 0 (3.28)
Let us recall that acoording Lemma 1.9, (3.28) can be checked if:

P(0) < 0, (3.29) P(η) < 0, (3.30) P(0) -τ 2 (t)ξ T (t)Φ 2 2 ᾱξ(t) < 0, (3.31)
Then, the inequalities (3.29) and (3.30) are verified if, ∀ξ T (t) = 0 :

ξ T (t) Φ 0 1 + Φ 0 2 ᾱ + Φ 0 3 ᾱ + Φ 0 4α ᾱ ξ(t) < 0 (3.32) and ξ T (t) η2 Φ 2 2 ᾱ + η(Φ 1 2 ᾱ + Φ 1 3 ᾱ + Φ 1 4α ᾱ) + Φ 0 1 + Φ 0 2 ᾱ + Φ 0 3 ᾱ + Φ 0 4α ᾱ ξ(t) < 0 (3.33)
Now, before dealing with (3.32) and (3.33), we will introduce the closed-loop dynamics into the stability conditions. To do so, note that (3.28) can be rewritten as:

(3.7) is equivalent to G α ᾱξ(t) = 0, with G α ᾱ = A α B α K ᾱX -1 0 -I . Moreover,
ξ T (t) τ 2 (t)Φ 2 2α ᾱ + τ (t)(Φ 1 Σ ᾱ + Φ 1 4α ᾱ) + Φ 0 Σα ᾱ ξ(t) < 0 (3.34) with Φ 1 Σ ᾱ = Φ 1 2 ᾱ + Φ 1 3 ᾱ and Φ 0 Σα ᾱ = 3 q=1 Φ 0 q ᾱ + Φ 0 4α ᾱ.
So, we can apply the Finsler's Lemma (Lemma 1.4) and the inequality (3.34) is satisfied if there exists R ∈ R 4n×n such that:

τ 2 (t)Φ 2 2α ᾱ + τ (t)(Φ 1 Σ ᾱ + Φ 1 4α ᾱ) + Φ 0 Σα ᾱ + RG α ᾱ + G T α ᾱR T < 0 (3.35)
Hence, (3.32) and (3.33) are satisfied if the following inequalities hold:

Φ 0 Σα ᾱ + RG α ᾱ + G T α ᾱR T < 0 (3.36) η2 Φ 2 2α ᾱ + η(Φ 1 Σ ᾱ + Φ 1 4α ᾱ) + Φ 0 Σα ᾱ + RG α ᾱ + G T α ᾱR T < 0 (3.37) Let X regular and R = X -1 ε 1 X -1 ε 2 X -1 ε 3 X -1 T .
To deal with (3.36), pre-and post-multiplying it respectively by diag{ X X X X } T and its transpose, we obtain:

Λ11 α ᾱ = Φ0 Σα ᾱ + I ε Ḡαᾱ + ḠT α ᾱI T ε < 0 (3.38)
Then, to deal with (3.37), apply first the Schur complement on Φ 1 4α ᾱ, then pre-and postmultiplying it respectively by diag{ X X X X X } T and its transpose, we obtain:

Λ22 α ᾱ * η Ȳαᾱ -η P22ᾱ < 0 (3.39) with: Λ22 α ᾱ = η2 Φ2 2α ᾱ + η( Φ1 Σ ᾱ -Pαᾱ ) + Φ0 Σα ᾱ + I ε Ḡαᾱ + ḠT α ᾱI T ε ,
and where, in (3.38) and (3.39),

I ε = I ε 1 I ε 2 I ε 3 I T , Ḡαᾱ = A α X B α K ᾱ 0 -X .
Then, to deal with (3.31), pre-and post-multiplying it respectively by diag{ X X X X } T and its transpose, we obtain: Finally, concatenating (3.38), (3.39), (3.40), P 11α ᾱ > 0 and P 22 ᾱ > 0 into the same parameterized LMI (Λ α ᾱ < 0), we obtain the conditions expressed in Theorem 3.1.

Λ44 α ᾱ = Λ11 α ᾱ -τ 2 Φ2 2 ᾱ < 0 (3.
Before proceeding with the analysis of the results from implementing sampled-data controllers to drive T-S models, it is important to highlight that the inequality (3.13) involves mismatching parameters (membership functions) associated with the sampled-data PDC control law (3.2) and with the continuous-time T-S model (3.1). Indeed, it is known that the input signal is computed only at the discrete-time instants while the system continuously evolves, and this phenomenon produces a mismatch between their membership functions during intersamplings. Obviously, solving the conditions of Theorem 3.1 for all vertices i and j would be extremely conservative and, due to the membership functions mismatches, we cannot apply directly double fuzzy sums relaxation schemes such like Tuan's Lemma (see Lemma 1.2). To circumvent the mismatch problem, many authors assume |α i -ᾱi | ≤ ρ i (see e.g. [START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF]). Moreover, Lemma 1.10 has been proposed as a dedicated relaxation scheme in [START_REF] Koo | An improved digital redesign for sampled-data fuzzy control systems: Fuzzy Lyapunov function approach[END_REF]. However, the latter doesn't take into consideration some particular characteristics of the membership functions mismatches, especially the fact that their upper bounds ρ i can be expressed with regards to the upper bounds of the membership functions' derivatives (| αi | < φ i , ∀i ∈ I r ,) and the maximal allowed sampling period (η). Such considerations may help to reduce the conservatism, as proposed in the next section, where an extension of Tuan's Lemma is proposed in the T-S model-based sampled-data controller design framework.

Remark 3.1. The conditions expressed in Theorem 3.1 are not strictly LMI because of the parameters ε 1 , ε 2 and ε 3 . However, as stated in many previous works applying the Finsler's Lemma, see e.g. [START_REF] Oliveira | Robust state feedback LMI methods for continuous-time linear systems: Discussions, extensions and numerical comparisons[END_REF]Bourahala et al., 2017;[START_REF] Cherifi | Uncertain TS model-based robust controller design with D-stability constraints-A simulation study of quadrotor attitude stabilization[END_REF][START_REF] Cherifi | Global non-quadratic D-stabilization of Takagi-Sugeno systems with piecewise continuous membership functions[END_REF], these parameters are usually tuned offline by grid search.

Remark 3.2. The proposed sampled-data controller design methodology for T-S systems includes linear systems as a special case by removing parameter dependency for all decision variables

in the LMI-based conditions expressed in Theorem 3.1. This particular case will be further investigated in this Chapter, providing an experimental validation of the proposed sampled-data controller design procedure on the Quanser ® AERO 2-DOF Helicopter test-bed.

Relaxation scheme for T-S model-based sampled-data controller design

Let us recall that standard double fuzzy sums relaxation scheme, such Lemma 1.2 [START_REF] Tuan | Parameterized Linear Matrix Inequality Techniques in Fuzzy Control System Design[END_REF], cannot be directly employed in the context of sampled-data control since the closed-loop dynamics (3.7) involves a double fuzzy sum structure with mismatched membership functions (α ᾱ). To circumvent this drawback, one may apply Lemma 1.10 ( [START_REF] Koo | An improved digital redesign for sampled-data fuzzy control systems: Fuzzy Lyapunov function approach[END_REF].

However, as previously mentioned, the latter does not take into consideration some particular characteristics of the membership functions mismatches, especially the fact that their upper bounds σ i can be expressed with regard to the upper bounds of the membership functions' derivatives (| αi | < φ i , ∀i ∈ I r ,) and the maximal allowed sampling period (η). Therefore, with the aim of providing less conservative conditions, we propose the following Theorem as an extension of Tuan's Lemma in the context T-S model-based sampled-data controller design. In this context, the following Lemma will be useful to address the above stated problem.

Lemma 3.1. [START_REF] Xie | Output feedback H ∞ control of systems with parameter uncertainty[END_REF]: Let X and Y be matrices of appropriate dimensions. For any matrix T > 0, the following inequality is true:

X T Y + Y T X ≤ X T T X + Y T T -1 Y (3.41)
Theorem 3.2. (Lopes et al., 2020a): For (i, j) ∈ I 2 r , let Λ ij be matrices of appropriate dimensions and assume, ∀t, | αi (t)| ≤ φ i . The inequality Λ α ᾱ < 0 is satisfied if there exists diagonal matrices T ij > 0 such that conditions (3.42) and (3.43) hold with: (3.43) with:

Γ ii < 0, ∀i ∈ I r , (3.42) 2 r -1 Γ ii + Γ ij + Γ ji < 0, ∀ (i, j) ∈ I 2 r , i = j,
Γ ij =       Λ ij + r-1 16 T ij ( * ) . . . ( * ) σ 1 Λij1 -T ij 0 0 . . . 0 . . . 0 σ r-1 Λijr-1 0 0 -T ij      
where, ∀ρ ∈ I r-1 , Λijρ = Λ iρ + Λ jρ -Λ ir -Λ jr and σ ρ = min{1, φ ρ η}.

Proof 3.2. Using the short hand notation for memberships functions

α i = α i (z(t)) and ᾱi = α i (z(t -τ (t)))
we have:

Λ α ᾱ = r i=1 r j=1 α i ᾱj Λ ij = r i=1 r j=1 α i α j   Λ ij + r ρ=1 (ᾱ ρ -α ρ )Λ iρ   = r i=1 r j=1 α i α j   Λ ij + r ρ=1 ᾱρ -α ρ 2 (Λ iρ + Λ jρ )   (3.44) Since r ρ=1 (ᾱ ρ -α ρ ) = 0 ⇔ (ᾱ r -α r ) = - r-1 ρ=1 (ᾱ ρ -α ρ ), ∀(i, j) we can write: r ρ=1 ᾱρ -α ρ 2 (Λ iρ + Λ jρ ) = r-1 ρ=1 ᾱρ -α ρ 2 Λijρ (3.45)
with Λijρ = Λ iρ + Λ jρ -Λ ir -Λ jr . Note that ∀ρ ∈ I r we have:

-1 ≤ α ρ -ᾱρ ≤ 1 (3.46)
Moreover, by assuming ∀t, | αρ (t)| ≤ φ ρ and since τ (t) ∈ [0, η k ) with η k ≤ η, we also have:

-φ ρ η ≤ α ρ -ᾱρ = t t-τ (t) αρ (s)ds ≤ φ ρ η (3.47)
Thus, from (3.46) and (3.47), we can assert that:

-1 ≤ ᾱρ -α ρ σ ρ ≤ 1 with σ ρ = min{1, φ ρ η} (3.48)
Let us now rewrite (3.45) as: 3.49) where:

r-1 ρ=1 ᾱρ -α ρ 2 Λijρ = H e     1 4 I . . . I r-1 times I ∆ α ᾱ∇ ij     (
∆ α ᾱ =     ᾱ1 -α 1 σ 1 0 0 0 . . . 0 0 0 ᾱr-1 -α r-1 σ r-1     and ∇ ij =     σ 1 Λij1 . . . σ r-1 Λijr-1    
From Lemma 3.1, for any matrices T ij > 0, it yields: Thus,considering (3.44),(3.45) and (3.50) and applying Lemma 1.2, then applying the Schur complement, we obtain the conditions expressed in Theorem 3.2.

r-1 ρ=1 ᾱρ -α ρ 2 Λijρ ≤ r -1 16 T ij + ∇ T ij ∆ α ᾱT -1 ij ∆ α ᾱ∇ ij (3.50) Now, let T ij > 0 be diagonal matrices, since ∆ α ᾱ is also diagonal and ∆ α ᾱ∆ α ᾱ ≤ I, ∆ α ᾱT -1 ij ∆ α ᾱ = ∆ α ᾱ∆ α ᾱT -1 ij ≤ T -1 ij .
From now, two main points are to be verified: The effectiveness of the sampled-data control design conditions proposed in Theorem 3.1 and the conservatism reduction induced by the asynchronous double fuzzy sums relaxation scheme proposed in Theorem 3.2. This will be done in the next section.

Illustrative Examples

To illustrate the effectiveness of the proposed sampled-data controller design methodology for continuous-time T-S fuzzy models, three examples are considered. The first one, consisting on a 1-DOF inverted pendulum, is considered to highlight the benefit of sampled-data controller design over conventional discrete-time controller approach when the plant to be driven evolves in continuous-time. The second example is based on an approximated two rules fuzzy model of an inverted pendulum on a cart, drawn from the literature, e.g., [START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF]. It is considered for comparison purpose since it has been largely considered in previous sampled-data related studies, especially to evaluate their successive conservatism improvements. Finally, because the proposed design conditions also holds for linear systems, an experimental validation of the sampled-data controller design procedure, with comparison to conventional continuous-time linear control approaches, is proposed on the Quanser ® AERO device in its 2-DOF helicopter configuration.

T-S model-based sampled-data controller design vs conventional discretetime approach

For this first example, we consider the dynamic equations of a 1-DOF inverted pendulum with ρ = -0.217234. To do so, we apply Theorem 3.1 together with Theorem 3.2, which require to estimate the bounds of the time-derivative of the membership functions, φ 1 and φ 2 .

To determine these parameters, let us assume that x 2 (t) ∈ [-π, π], from the definition of the membership functions (1.15), we have:

∀i ∈ I 2 , | αi (t)| ≤ φ i = sup {x 1 ∈R,x 2 ∈[-π,π]} x 1 cos(x 1 ) -sin(x 1 ) (1 -ρ)x 2 1 x 2 = 1.1258 Rad.s -1
Then, solving the conditions of Theorem 3.1 with Theorem 3.2 using YALMIP and SeDuMi [START_REF] Löfberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF] showing that the implementation of the latter for the 1-DOF inverted pendulum is 2.09 times more efficient in terms of control input energy consumption than the discrete-time controller. Finally, let us recall that the proposed SD controller design methodology summarized by Theorem 3.1 guarantees the closed-loop stability of the continuous-time system, while the conventional DT approach cannot guarantee the inter-sampling stability of continuous-time plants, especially for large sampling periods [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. Indeed, in Figure 3.2, we can observe that the DT Lyapunov function applied to the continuous-time model is not monotonously decreasing during two successive sampling instants. This concludes this first example, showing the effectiveness of this chapter's proposal with regards to conventional discrete-time control approaches. x T (t)Lx(t) computed for the CT system and DT model.

Conservatism comparison: Inverted pendulum on a cart benchmark

To illustrate the effectiveness of the proposed sampled-data control approach for T-S models, let us consider the benchmark of an inverted pendulum mounted on a cart depicted in Figure 3.3.

For brevity, the time t will be omitted in the mathematical expression of this section. The nonlinear dynamics of this well-known mechatronic system is given by [START_REF] Cannon | Dynamics of physical systems[END_REF][START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF]:

   ẋ1 (t) = x 2 (t), ẋ2 (t) = g sin x 1 (t)-amlx 2 2 (t) sin(2x 1 (t))/2-a cos x 1 (t)u(t) 4l/3-aml cos 2 x 1 (t)
, (3.52) where x 1 (t) and x 2 (t) are respectively the angle position (rad) and the velocity (rad/s) of the pendulum from the erect position, g = 9.8 m/s 2 is the acceleration of the gravity, m = 2 kg is Note that when x 1 (t) = ±π/2 the system is uncontrollable.

In this subsection, an approximated T-S fuzzy model (3.1) with two fuzzy rules (r = 2) of (3.52), drawn from [START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF], is considered for comparison purpose since it has been used in several related previous sampled-data controller design studies in the the literature (see the references given in Table 3.2). This approximated T-S fuzzy model with two rules, valid for |x 1 (t)| < π/2 and |x 2 (t)| ≤ π, is specified by the matrices: (3.53) and the triangular membership functions:

A 1 = 0 1 g 4l/3-aml 0 , A 2 = 0 1 2g π(4l/3-amlβ 2 ) 0 , B 1 = 0 -a 4l/3-aml , B 2 = 0 -aβ 4l/3-amlβ 2 , β = cos(88 • ),
α 1 (x 1 (t)) = 1 -2 π x 1 (t), if 0 ≤ x 1 (t) < π 2 1 + 2 π x 1 (t), if -π 2 < x 1 (t) < 0 and α 2 (x 1 (t)) = 1 -α 1 (x 1 (t)) (3.54)
In this context, note that, from the triangular membership function we have

| α1 | = | α2 | = 2 π | ẋ1 | and, since | ẋ1 | = |x 2 | ≤ π, we always have | αi | ≤ 2 = φ i (i ∈ I 2 ).
Let us remember that, because of the asynchronous double fuzzy summation structure of the parameterized LMIs proposed in Theorem 3.1, conventional relaxation schemes such like Tuan's Lemma [START_REF] Tuan | Parameterized Linear Matrix Inequality Techniques in Fuzzy Control System Design[END_REF]) cannot be directly applied. In this regard, we have proposed a specific relaxation scheme given by Theorem 3.2. Note also that, to the best of our knowledge, except Lemma 1.10 proposed in [START_REF] Koo | An improved digital redesign for sampled-data fuzzy control systems: Fuzzy Lyapunov function approach[END_REF], there was no generic relaxation scheme to cope with this issue in previous related works. This makes Lemma 1.10 proposed in [START_REF] Koo | An improved digital redesign for sampled-data fuzzy control systems: Fuzzy Lyapunov function approach[END_REF] suitable to be compared with Theorem 3.2, in order to solve the parameterized LMIs proposed in Theorem 3.1. The result is shown in Table 3.1, where the biggest value of the maximal allowable sampling interval was found with Theorem 3.1 (η = 50ms), outperforming the results obtained with Lemma 1.10 (η = 35ms) by 45.57% (the parameters ε 1 , ε 2 and ε 3 were tuned by a grid search for both results). This confirms the superiority of Theorem 3.2 regarding to Lemma 1.10 ( [START_REF] Koo | An improved digital redesign for sampled-data fuzzy control systems: Fuzzy Lyapunov function approach[END_REF] in terms of conservatism reduction. In order to illustrate these first results, with the initial condition x( 0 In addition, Table 3.2 lists the maximal allowable upper bound η obtained from several previous related results from the literature. Once again, we observe that the present approach outperform these previous results by at least 19.05%, which highlights the significant improvement raised by Theorem 3.1, together with Theorem 3.2, in terms of conservatism. [START_REF] Yoneyama | Robust H ∞ control of uncertain fuzzy systems under time-varying sampling[END_REF] 9 [START_REF] Zhu | Stabilization for sampled-data neural network-based control systems[END_REF] 13 [START_REF] Zhang | H ∞ control design for network-based T-S fuzzy systems with asynchronous constraints on membership functions[END_REF] 16 [START_REF] Zhu | An Improved Input Delay Approach to Stabilization of Fuzzy Systems Under Variable Sampling[END_REF] 19 [START_REF] Gunasekaran | Robust Sampled-data Fuzzy Control for Nonlinear Systems and Its Applications: Free-Weight Matrix Method[END_REF] 22 [START_REF] Zhu | H ∞ stabilization criterion with less complexity for nonuniform sampling fuzzy systems[END_REF] 24 [START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF] 42 Theorem 3.1 50

Now, let us consider the maximal allowed sampling interval with η = 50ms, for which only closed-loop domain of attraction would be necessary. This issue will be the focus of the next chapter. Furthermore, it is worth pointing out that the sampled-data controller obtained in this subsection only applies for the approximated two rules fuzzy model provided by [START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF] from which it is designed. As a matter of fact, nothing guarantees that it stabilizes the original nonlinear system (3.52). To assert this fact, applying this two rules sampled-data T-S fuzzy controller to the original nonlinear model (3.52) of the inverted pendulum, with the same initial conditions x = π/3 0 T , provides an unstable closed-loop behavior, as shown in Figure 3.7. This highlights the limitation of using such approximated standard T-S fuzzy models, which should be circumvented by considering an exact T-S modeling approach. This will be investigated in the next Chapter, for instance by using sector nonlinearity approach and, when relevant (e.g. for mechanical plants), by extending the proposed results to the class of T-S descriptors.

Linear case study with experimental validation

In this section, we are focused on the real-time implementation of a sampled-data controller employing the gains obtained from the design conditions provided in Theorem 3.1 to drive an unmanned aerial vehicle (UAV) benchmark, which presents some interesting characteristics such as its versatility, maneuverability, and ease of use. Namely, we choose for these experiments the Quanser ® AERO benchmark configured as a dual-rotor helicopter [START_REF] Quanser | Quanser aero user manual[END_REF]. For this real system, some works can be found in the literature dealing with reinforcement learning strategies [START_REF] Fandel | Development of Reinforcement Learning Algorithm for 2-DOF Helicopter Model[END_REF], model reference adaptive controllers (Arabi and Yucelen, 2019b,a) et al., 1996).

Reyhanoglu, 2019). Nevertheless, all the above mentioned works consider that the controller evolves in a continuous-time framework and, to the best of the authors knowledge, no previous studies can be found dealing with the design of a sampled-data controller for this system.

In this context, this section presents a real-time experimental validation of the sampled-data controller design procedure proposed in Theorem 3.1. The effectiveness of this proposal is illustrated in simulation, then validated experimentally, and compared to conventional continuoustime PID and Linear-Quadratic Regulators (LQR), which design procedure can be found in the Quanser ® AERO laboratory guide [START_REF] Quanser | Quanser aero user manual[END_REF]Lopes et al., 2020b).

2-DOF helicopter dynamical model:

The Quanser ® AERO, in its essence, is a dual-motor experiment, a reconfigurable platform created for serving advanced control research, simplifying the experimentation of various aerospace systems, from 1-DOF and 2-DOF helicopter to half-quadrotor. Although, it can also be used for teaching control concepts at the undergraduate level. In Figure 3.8 the free-body diagram of the considered benchmark is presented in its 2-DOF Helicopter configuration is presented:

This system is configured as a conventional dual-rotor helicopter with two identical highefficiency rotors that produce the thrust forces F p (t) and F y (t) acting at points with distances r p and r y from the z-axis along the x-axis, respectively. Hence, one propeller generates a torque around the y-axis leading to a pitch (θ(t)) motion, while the other one deals with a yaw (ψ(t)) motion (around the z-axis).

The dynamical model of this benchmark was taken from [START_REF] Quanser | Quanser aero user manual[END_REF], and was developed as a simple linear model that takes into account the coupling between the pitch and yaw axis. The equations of motion of the system are given by:

J p θ(t) + D p θ(t) + K sp θ = τ p (t) J u ψ(t) + D y ψ(t) = τ y (t) (3.55)
where the torques acting on the pitch (τ p (t)) and yaw (τ y (t)) axes are assumed proportional to the inputs voltages V p (t) and V y (t) of the DC-motors such that: .56) with the parameters listed in Table 3.3.

τ p (t) = K pp V p (t) + K py V y (t) and τ y (t) = K yp V p (t) + K yy V y (t), ( 3
A =       0 0 1 0 0 0 0 1 - Ksp Jp 0 - Dp Jp 0 0 0 0 - Dy Jy       and B =       0 0 0 0 Kpp Jp Kpy Jp Kyp Jy Kyy Jy       .
Recall that linear systems such like (3.57) constitute a special case of T-S fuzzy models (3.1) with r = 1. Hence, Theorem 3.1 can be directly applied (removing the parameter dependency of the decision variables in α and ᾱ) for the design of the following sampled-data controller:

u(t) = F x(t k ) = F x(t -τ (t)) (3.58)
where F = KX -1 ∈ R m×n is the controller gain matrix to be designed and τ (t) defined in (3.4).

Before going to the simulation and experimental results, the goal of this example is also to provide some comparisons with conventional linear continuous-time control design procedures proposed in the Quanser ® AERO laboratory guide [START_REF] Quanser | Quanser aero user manual[END_REF], namely a Proportional-Derivative (PD) controller and an LQR-based design state space linear controller, given as follows.

First, decoupled continuous-time PD control laws are proposed for each pitch and yaw axis as [START_REF] Quanser | Quanser aero user manual[END_REF]:

u θ (t) = -K Pp θ(t) -K Dp θ(t) u ψ (t) = -K Py ψ(t) -K Dy ψ(t) (3.59)
with the gains : Then, because the above mentioned PD controllers does not cope with coupling effects, the following linear continuous-time state feedback controller is also proposed [START_REF] Quanser | Quanser aero user manual[END_REF]:

K Pp =
u(t) = -K LQR x(t), (3.60)
The design of the gain K LQR is based on the LQR approach where the minimization of the following cost function is considered:

J(u(t)) = ∞ 0 x(t) T Qx(t) + u(t) T Ru(t) dt (3.61)
with:

Q =     
200 0 0 0 0 75 0 0 0 0 0 0 0 0 0 0 Let us highlight that these controllers are designed in continuous-time and are implemented in the Quanser ® lab assuming a very small fixed sampling period (usually η k = 2ms), which is satisfactory for a pedagogical tool to teach basics on this topics. Nevertheless, since the embedded electronics of the Quanser ® AERO provides only digital computations, from the theoretical point of view, this is not accurate since the inter-sampling stability is not guaranteed in these two cases, especially for large sampling periods.

For the following developments, the direct discrete-time controller design approach is not investigated. This decision relies on the employment of sampling periods greater than those found through the search of a reasonable rate based on the Nyquist and Shanon Criterion for the Quanser ® AERO. In a preliminary study, a time constant of τ c = 0.7643s was obtained for the linear system, and according to the literature, a proper choice for the sampling period value should be τ s ∈ [0.0764, 0.19]. Consequently, the results obtained with a larger sampling period could bring hidden issues like the growth of the Lyapunov function, as discussed in Chapter 1.

Simulation results with a large sampling period:

In this section, we provide some simulation results considering a large sampling period of η = 4.5s. As mentioned in Remark 3.2, a particular case of Theorem 3.1 can be considered for the design of the sampled-data controller (3.58). The result, obtained via YALMIP [START_REF] Löfberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF] and SEDUMI [START_REF] Sturm | Using sedumi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF] solver in MATLAB ® with the scalar parameters ǫ 1 = 3, ǫ 2 = 1 and ǫ 3 = 300, is given by the following sampled-data control gain matrix: F = 0.0432 1.1617 -0.1687 -0.1789 0.0530 -0.6085 -0.2070 0.0937

In this context, Figure 3.9 shows the closed-loop continuous-time responses of the pitch and yaw axis under the design sampled-data control law (3.58) with the initial conditions x(0) = 10 45 0 0 T . We can verify that the closed-loop sampled-data control systems is successfully stabilized in simulation.

Now, let us consider that the continuous-time controllers (3.59) and (3.60) are implemented on a digital device with the same huge sampling period of 4.5s. As mentioned above, since this controller does not cope with inter-sampling behavior, the closed-loop systems should be unstable as shown in the simulations depicted in Figure 3.10.

These simulations confirm the significance of considering the proposed sampled-data control methodology for large sampling periods. The next subsection provides an experimental validation.

Experimental validation of the proposed sampled-data controller design:

In the previous subsection, simulation results of the proposed sampled-data control strategy have been proposed with a large sampling period of 4.5s. Nevertheless, in practice, some unmodeled phenomena (frictions, motors dead zone which make then static under a tension smaller than 0.4V,...) does not allow a direct implementation with the same values. Hence, to provide a fair comparison with a realistic sampling period, we chose for experimental validation η k = 150ms. In this case, Theorem 3.1 has a solution with ǫ 1 = 0.25, ǫ 2 = 10 and ǫ 3 = 0.37, which provide the following sampled-data controller (3.58) gain matrix: -28.4589 34.6979 -27.5479 21.9293 -34.9268 -18.1751 -33.8088 -11.4868 .

F =
The experimental results are shown in Figure 3.11 where the time response and the input signals (Motors' voltage) are depicted for the PD, the LQR and the sampled-data controller.

We can notice that, while the PD and LQR control plants that are unstable with a sampling period η k = 150ms, the sampled-data controller successfully stabilizes the Quanser ® AERO. Finally, to take benefit of the fact that our conditions hold for aperiodic sampling periods, we proposed the following simple strategy to trigger it. Note that V 1 (t k ) ≤ V 1 (0). The purpose is to update the sampling period η k at each sampling instant t k so that, when

V 1 (t k ) is close to V 1 (0),
then the updated sampling period η k is small (2ms), and when it is close to 0, then η k is large (0.15s). This strategy can be implemented with an event-triggering mechanism to determine the next sampling instant η k+1 for which a new measurement of the states is required to compute the control law (3.58). In this context, to implement the sampled-data control approach based on events, the following simple rule can be proposed:

η k+1 = 0.002 -0.15 V 1 (0) V 1 (t k ) + 0.15 (3.62)
The results from implementing the event-triggering mechanism with the triggering rule defined on (3.62) are depicted in Figure 3.12, in which the simulation results are compared with the experimental ones under the proposed aperiodic sampled-data control scheme.

As expected, the closed-loop system is properly stabilized. However, we can also see some differences during the transients from the simulation and the experimentation. Indeed, in addition to experimental artifacts and unmodeled effects, these can also be due to the actuators saturation, which often occur in practice (here during the first few sampling intervals where the inputs saturate at the motors' voltage limit of 24V ) and which are not taken into account in the present controller design methodology. This will be another subject considered in the next chapter.

Conclusion

In this chapter, new LMI-based conditions have been proposed to design sampled-data PDC controllers for continuous-time T-S fuzzy models. Conservatism improvement regarding previous works has been achieved from choosing an appropriated LKF, then by employing useful bounding lemmas together with a second order quadratic polynomial constraints. In addition, generic conditions have been proposed to relax double fuzzy sums with asynchronous MFs involved in T-S model-based sampled-data control plants.

The effectiveness of the proposed sampled-data design conditions and their superiority regarding conventional discrete-time controller design approaches, like those considered in Chapter 2, were illustrated with the 1-DOF inverted pendulum example. Also, the conservatism reduction brought by the design conditions has been compared with previous results and illustrated through the well-known benchmark of an inverted pendulum on a cart. However, the simulation results pointed out that the designed controller guarantees the stability of the approximated T-S model, but fails to stabilize the original nonlinear model. To cope with the previous issue, a new design condition is proposed in the next chapter, extending the results for descriptor systems suitable for describing an exact representation of the nonlinear system through fuzzy T-S modeling for mechanical systems with the sector nonlinearity approach.

Then, the design and the implementation of the proposed sampled-data controller for the Quanser ® AERO 2-DOF helicopter system has been proposed. The results have been validated in simulation as well as experimentally, and the designed sampled-data controller has shown its superiority regarding previous designed continuous-time controllers, especially for large sampling periods. However, these experimental results also highlight the importance of taking into account practical constraints such like input saturation, which will also constitutes one of the improvements proposed in the next chapter.

Finally, let us also point-out that one of the main limitations of the sampled-data controller design proposed in this chapter are generally only local for T-S systems, excepted in the special case where D x = R n and σ i = 1 in Theorem 3.2, which is conservative. Hence, further investigations of the sampled-data closed-loop domain of attraction appears to be necessary and will be one of the main focus of the next chapter.

Chapter 4

Local Aperiodic Sampled-data Control of T-S Descriptors with Input Constraints

Résumé en Français : Contrôle local à base de données apériodiquement échantillonnées des modèles T-S descripteurs avec saturation sur l'entrée.

Ce chapitre vise à fournir quelques extensions permettant de surmonter les obstacles mentionnés dans le chapitre précédent. En effet, à partir des conditions données dans les Théorèmes 3.1 et 3.2 pour la synthèse de correcteurs échantillonnées pour les modèles T-S, on peut conclure que les résultats ne sont valables que localement (voir la fin du Chapitre 1). De plus, à partir de l'exemple proposé dans la Section 3.5.2, il a été montré que lorsque les limites du conservatisme sont atteintes, l'utilisation d'un modèle T-S approximé pour effectuer la synthèse du correcteur peut échouer à stabiliser le système non linéaires initialement considéré. Enfin, à partir des résultats expérimentaux présentés dans la Section 3.5.3, il est important de rappeler que la saturation des actionneurs est à prendre en compte.

Sur la base de ces considérations, dans ce chapitre, la méthodologie présentée précédemment est étendue à la classe des modèles descripteurs de type T-S [START_REF] Taniguchi | Fuzzy descriptor systems: stability analysis and design via LMIs[END_REF][START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF], qui ont montré tout leur intérêt pour représenter avec précision les systèmes mécaniques [START_REF] Guelton | Estimation des caractéristiques du mouvement humain en station debout[END_REF][START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF][START_REF] Bouarar | Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi-Sugeno descriptors[END_REF][START_REF] Arceo | Nonlinear convex control of the Furuta pendulum based on its descriptor model[END_REF][START_REF] Quintana | Real-time parallel distributed compensation of an inverted pendulum via exact Takagi-Sugeno models[END_REF][START_REF] Dang | Robust stabilizing controller design for Takagi-Sugeno fuzzy descriptor systems under state constraints and actuator saturation[END_REF][START_REF] Nguyen | Fuzzy Descriptor Tracking Control with Guaranteed L ∞ Error-Bound for Robot Manipulators[END_REF], et par conséquent les systèmes mécatroniques commandés de fait par des contrôleurs basés sur des données échantillonnées. De plus, puisque les modèles descripteurs T-S incluent les modèles T-S standards à titre de cas particuliers, ils sont souvent utilisés pour représenter plus précisément une plus grande classe de systèmes, tout en réduisant le conservatisme [START_REF] Estrada-Manzo | Improvements on nonquadratic stabilization of continuous-time Takagi-Sugeno descriptor models[END_REF][START_REF] Aps | The MOSEK optimization toolbox for MATLAB manual[END_REF]. Par ailleurs, rappelons que les saturations sur l'entrée ont été considérées dans le Chapitre 2, dans le cadre des modèles T-S décrits en temps discret.

Mis à part les travaux de [START_REF] Lamrabet | Sampled data Control for Takagi-Sugeno Fuzzy Systems with Actuator Saturation[END_REF], qui visent à proposer des conditions quadratiques relâchées mais sans aucune considération du domaine d'attraction en boucle fermée, il n'existe pas d'autres travaux pré-existants, traitant du problème de la stabilisation locale des modèles T-S continus avec saturations sur l'entrée à base données échantillonnées. Par la suite, inspirées des travaux présentés dans [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF][START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF], de nouvelles conditions prenant en compte la saturation des actionneurs sont proposées, garantissant ainsi la stabilité locale du système en boucle fermée. Afin de tenir compte de la nature locale de ces conditions, due aux contraintes induites par la saturation et/ou au domaine de validité restreint du modèle T-S, nous proposons une analyse minutieuse du domaine d'attraction du système en boucle fermée, qui peut-être réalisée grâce au choix effectué d'une fonctionnelle non quadratique de Lyapunov-Krasovskii « bouclée » (NQLKF) appropriée. Dans cette optique, deux méthodologies sont proposées. Premièrement, une procédure hors ligne est présentée pour la stabilisation des modèles descripteurs de type T-S commandés par un contrôleur basé sur des données échantillonnées, en exploitant la NQLKF pour l'estimation du domaine d'attraction. Ensuite, pour des cas particuliers de modèles T-S standards, une procédure d'optimisation et d'obtention systématique est proposée afin d'élargir le domaine d'attraction en boucle fermée.

Enfin, notons que, pour des intervalles d'échantillonnage proches de la limite maximum autorisée, on obtient une estimation du domaine d'attraction significativement réduite. Afin de pallier ce problème, une nouvelle méthodologie d'ajustement des paramètres de la loi de commande est proposée. Celle-ci exploite les principes du déclenchement par événements (event-triggering), selon les équipotentielles des fonctions de Lyapunov obtenues pour plusieurs intervalles d'échantillonnage. Cette méthode, illustrée au travers de l'exemple de simulation du pendule inversé sur un chariot, permet d'élargir plus encore l'estimation obtenue du domaine d'attraction en boucle fermée.

Introduction

This chapter aims at providing some extensions to cope with the limitations pointed-out in the previous chapter. Indeed, from the proposed sampled-data controller design approach for T-S models given in Theorems 3.1 and 3.2, it can be concluded that these results only hold locally (see also Chapter 1, end of Section 1.4.2). Also, from the example provided in Section 3.5.2, it has been shown that, when reaching the limits of conservatism, the proposed approach may fail to stabilize the original nonlinear system, if its T-S model representation is not sufficiently accurate (e.g. when using fuzzy approximations). Finally, from the experimental results provided in Section 3.5.3, the importance of handling the actuators' saturation have been highlighted.

Based on these considerations, in this chapter, the proposed methodology is extended to the class of regular Takagi-Sugeno descriptors [START_REF] Taniguchi | Fuzzy descriptor systems: stability analysis and design via LMIs[END_REF][START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF], which have been shown particularly efficient to accurately represent mechanical plants [START_REF] Guelton | Estimation des caractéristiques du mouvement humain en station debout[END_REF][START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF][START_REF] Bouarar | Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi-Sugeno descriptors[END_REF][START_REF] Arceo | Nonlinear convex control of the Furuta pendulum based on its descriptor model[END_REF][START_REF] Quintana | Real-time parallel distributed compensation of an inverted pendulum via exact Takagi-Sugeno models[END_REF][START_REF] Dang | Robust stabilizing controller design for Takagi-Sugeno fuzzy descriptor systems under state constraints and actuator saturation[END_REF][START_REF] Nguyen | Fuzzy Descriptor Tracking Control with Guaranteed L ∞ Error-Bound for Robot Manipulators[END_REF], and so mechatronic systems where sampled-data control is inherent.

Moreover, because T-S descriptors include as a special case standard T-S models, these are often used to more accurately represent a larger class of systems while reducing the conservatism of their design conditions [START_REF] Estrada-Manzo | Improvements on nonquadratic stabilization of continuous-time Takagi-Sugeno descriptor models[END_REF][START_REF] Aps | The MOSEK optimization toolbox for MATLAB manual[END_REF].

Recall that input saturation has been considered in Chapter 2, but in the discrete-time modelbased control framework. Despite the work in [START_REF] Lamrabet | Sampled data Control for Takagi-Sugeno Fuzzy Systems with Actuator Saturation[END_REF], which intended to provide relaxed quadratic conditions but missing the characterization of the domain of attraction, no previous works were found dealing with the sampled-data stabilization for T-S models subject to actuators' saturation. In the following, similarly to [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF][START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF], a generalized sector condition is employed to cope with the actuators' saturation problem, while keeping the local closed-loop sampled-data stability guaranteed.

Therefore, acknowledging the local nature of the proposed sampled-data synthesis, either associated with constraints induced by the saturation, or also due to the restrictions brought by the region of validity of the T-S model, we propose, along this chapter, a careful analysis of the estimation of the sampled-data closed-loop domain of attraction, from the convenient choice of a looped non-quadratic LKF (NQLKF). In that purpose, two methodologies are proposed. First, an offline procedure is presented for T-S descriptors-based sampled-data stabilization, taking full advantage of the NQLKF to enlarge the ensured limits of operation of the controlled system.

Then, in the special case of standard T-S models, an optimization procedure is provided to give a systematic enlargement of the estimation of the closed-loop domain of attraction.

Finally, because, for a given maximal allowed upper bound of the sampling intervals, the obtained estimation of the domain of attraction can be quite small, a new gain scheduled eventtriggering mechanism, based on the characterization of several Lyapunov level sets, is proposed to further enlarge the resulting closed-loop sampled-data domain of attraction.

Considered class of systems and problem statement

In this chapter, by extension of the class of nonlinear systems considered in the previous chapters, we will consider nonlinear descriptors with input (actuators) saturation given by:

E x(t) ẋ(t) = A x(t) x(t) + B x(t) sat(u(t)) (4.1)
where:

• x(t) ∈ R n is the state vector, E(x(t)) ∈ R n×n , A(x(t)) ∈ R n×n and B(x(t)) ∈ R n×m are matrices with nonlinear entries (only depending on the state variables for control purpose), bounded on a compact subset D x of the state space defined by:

D x = x(t) ∈ R n | L (j) x(t) ≤ z (j) ⊆ R n (4.2)
with L ∈ R κ×n , z ∈ R κ , and j ∈ I κ ,

• u(t) ∈ R m is the control input vector, which may saturate according to actuators limitations, i.e., sat(u(t)) is a decentralized vector valued function with components defined by:

sat(u (ℓ) (t)) = sign(u (ℓ) (t)) min(|u (ℓ) (t)|, ū(ℓ) ), ∀ℓ ∈ I m , ( 4.3)
which also allows us to define the set:

D u = {u(t) ∈ R m | -ū ≤ sat(u(t)) ≤ ū} (4.4)
where ū is a vector of R m with positive components, i.e. ū(ℓ) ≥ 0, ∀ℓ ∈ I m .

From the well-known sector nonlinearity approach (Tanaka and Wang, 2001), nonlinear descriptors (4.1) with input saturation can be exactly rewritten, ∀x(t) ∈ D x , as T-S descriptors given by:

r i=1 α i x(t) E i ẋ(t) = r i=1 α i x(t) (A i x(t) + B i sat(u(t))) (4.5)
where, for i

∈ I r , E i ∈ R n×n , A i ∈ R n×n , B i ∈ R n×m are constant matrices, α i (x(t)) ≥ 0 are convex membership functions satisfying r i=1 α i (x(t)) = 1.
Remark 4.1. T-S descriptors (4.5) include as special case the class of standard T-S models (3.1) (without input saturation) by considering E i = I, ∀i ∈ I m . Moreover, note that descriptor systems are often used in the literature to model singular systems, i.e. when E(x(t)) is not invertible (see e.g. [START_REF] Marx | Stability and L 2 -Norm Bound Conditions for Takagi-Sugeno Descriptor Systems[END_REF][START_REF] Li | Admissibility analysis for Takagi-Sugeno fuzzy singular systems with time delay[END_REF][START_REF] Chang | Derivative-Based Fuzzy Control Synthesis for Singular Takagi-Sugeno Fuzzy Systems with Perturbations[END_REF]). However, in the sequel of this chapter, we will only focus on the case where E(x(t)) is assumed to be regular (invertible) since it has been shown useful to reduce the conservatism in T-S model-based design [START_REF] Taniguchi | Fuzzy descriptor systems: stability analysis and design via LMIs[END_REF][START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF][START_REF] Tanaka | A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions[END_REF]Bouarar et al., 2007aBouarar et al., ,b, 2008[START_REF] Bouarar | Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi-Sugeno descriptors[END_REF][START_REF] Estrada-Manzo | Improvements on nonquadratic stabilization of continuous-time Takagi-Sugeno descriptor models[END_REF]. Furthermore, it is well-known that (regular) descriptors are particularly adequate to represent the dynamics of mechanical plants, by avoiding ill-conditioned matrix inversion, while reducing the number of vertices of resulting T-S models (see e.g [START_REF] Guelton | Estimation des caractéristiques du mouvement humain en station debout[END_REF]Schulte andGuelton, 2006, 2009;[START_REF] Guelton | An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form[END_REF][START_REF] Bouarar | Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi-Sugeno descriptors[END_REF][START_REF] Seddiki | Concept and Takagi-Sugeno descriptor tracking controller design of a closed muscular chain lower-limb rehabilitation device[END_REF][START_REF] Vermeiren | Motion control of planar parallel robot using the fuzzy descriptor system approach[END_REF][START_REF] Arceo | Nonlinear convex control of the Furuta pendulum based on its descriptor model[END_REF][START_REF] Quintana | Real-time parallel distributed compensation of an inverted pendulum via exact Takagi-Sugeno models[END_REF][START_REF] Dang | Robust stabilizing controller design for Takagi-Sugeno fuzzy descriptor systems under state constraints and actuator saturation[END_REF][START_REF] Nguyen | Nonlinear Tracking Control with Reduced Complexity of Serial Robots: A Robust Fuzzy Descriptor Approach[END_REF][START_REF] Nguyen | Fuzzy Descriptor Tracking Control with Guaranteed L ∞ Error-Bound for Robot Manipulators[END_REF]). Indeed, let us consider a class of d-link mechanical system, which dynamics can be described by:

M q(t) q(t) + C q(t), q(t) q(t) + G q(t) = Γ(t) (4.6)
where q(t) ∈ R d is the vector of generalized coordinates, M q(t) ∈ R d×d is the inertia matrix, C q(t), q(t) ∈ R d×d is the Coriolis/centripetal matrix, G q(t) is the vector of gravitation and Γ(t) ∈ R d is the vector of generalized forces. The vector of gravitation being smooth, we can get Ḡ q(t) ∈ R d×d such that G q(t) = Ḡ q(t) q(t). Thus, denoting x(t) = q T (t) qT (t)

T ∈ R n , n = 2d
, the dynamical model (4.6) can be rewritten in the affine-in-control continuous-time nonlinear state space descriptor form (4.1) with:

E(x(t)) = I 0 0 M(x(t)) , A(x(t)) = 0 I -Ḡ(x(t)) -C(x(t)) and B(x(t)) = 0 B(x(t))
.

Finally, note that, since the inertia matrix M (x(t)) of a mechanical plant is always positive definite, the descriptor matrix E(x(t)) is always invertible. Now, due to the discrete nature of the controllers, let t k ≥ 0 be the sampling instants such that t k+1 -t k ≤ η k ≤ η, where the inner sampling interval η k > 0 can be non-uniform (aperiodic) with a maximal allowable sampling interval η. A Zero-Order Hold (ZOH) is employed to maintain

the control signal ∀t ∈ [t k , t k+1 ). Hence, for actual t ∈ [t k , t k+1 ), let τ (t) = t -t k ∈ [0, η k )
, where τ (t) = 1, and consider the following sampled-data PDC control law for the stabilization of T-S descriptors (4.5):

u(t) = r i=1 α i (x(t -τ (t)))K i X -1 x(t -τ (t)) (4.7)
where K i ∈ R m×n and X -1 ∈ R n×n are the sampled-data controller gain matrices to be designed.

From (4.5) and (4.7), and using the following notations:

M α = r i=1 α i (z(t))M i , M ᾱ = r i=1 α i (z(t -τ (t)))M i
and:

M α ᾱ = r i=1 r j=1 α i (z(t))α j (z(t -τ (t)))M ij
to lighten mathematical expressions, the closed-loop sampled-data dynamics can be expressed by the following T-S descriptor model with saturated inputs and time-varying input delays:

E α ẋ(t) = A α x(t) + B α sat(K ᾱX -1 x(t -τ (t))) (4.8)
In this work, to cope with the saturation, let us consider the dead-zone nonlinearity defined by (see Definition 1.2 for more details):

ψ(u(t)) = sat(u(t)) -u(t) (4.9)
From this definition, the closed-loop sampled-data dynamics with input saturation (4.8) can be rewritten as the following T-S descriptor with input time-varying delays and with an additional dead-zone nonlinearity:

E α ẋ(t) = A α x(t) + B α K ᾱX -1 x(t -τ (t)) + B α ψ(K ᾱX -1 x(t -τ (t))) (4.10)
From now, our goal is to design the gain matrices K ᾱ and X that guarantee the closedloop dynamics (4.8) to be stable. This will be done in the next section by extending the results provided in Chapter 3, considering the application of Theorem 3.2. Recall that the latter assume that the time-derivatives of the membership functions are bounded such that the results are valid inside the following set:

D φ = r k=1 {x(t) ∈ R n : | αk (z)| φ k } (4.11)
Hence, considering the validity domain D x of the T-S descriptor defined by (4.2), the input constraints resulting to the set D u defined in (4.4), and the restrictions implied by the set D φ defined in (4.11), the sampled-data closed-loop dynamics (4.10) can only be locally guaranteed, which makes necessary the estimation of the domain of attraction.

Remark 4.2. In previous literature dealing with continuous-time T-S model-based non-quadratic

control, estimating the bounds φ k (∀k ∈ I r ) of the time-derivatives of the membership functions

h k (x(t)
) is, in general, commonly considered as a hard or even impossible task, since this has to be made priory to the design of the closed-loop dynamics, see e.g. [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Guelton | Some Refinements on Stability Analysis and Stabilization of Second Order T-S Models Using Line-Integral Lyapunov Functions[END_REF][START_REF] Cherifi | Global non-quadratic D-stabilization of Takagi-Sugeno systems with piecewise continuous membership functions[END_REF]. However, in the local context of our study, assuming E(x(t))

regular ∀x(t) ∈ D x , when considering known input constraints, i.e. when u(t) belongs to a known closed compact subset D u ⊂ R m , and x(t) belongs to a known closed compact subset D x ⊂ R n , such bounds can be easily estimated from numerical computation by:

φ k = sup {x∈Dx,u∈Du} |∇ T k (x)E -1 (x) A(x)x + B(x)u ≥ |∇ T k (x(t))E -1 x(t) A x(t) x(t) + B x(t) sat(u(t)) = ∇ T k (x(t)) ẋ(t)| = αk (x(t)) (4.12)
This being said, the main objectives of this chapter are summarized in the following problem statement.

Problem statement:

P 1 : Provide relaxed LMI-based conditions for the design of the gain matrices K i and X such that the sampled-data closed-loop dynamics (4.10) with input saturation is locally asymptotically stable for the largest as possible admissible upper bound η of the non-uniform sampling intervals η k .

P 2 : Provide a methodology to estimate the domain of attraction D * a of the closed-loop system (4.10). P 3 : Based on the satisfaction of P 1 and P 2 , provide a gain scheduled event-triggering mechanism to further enlarge the closed-loop domain of attraction.

Main results

In the sequence, the goal is to provide parameterized LMI-based conditions and methodologies to satisfy P 1 , P 2 and P 3 of the above given problem statement. First, the main result is proposed for the class of descriptor systems (4.1), then the special case of standard T-S systems (3.1) is considered to provide a systematic approach for the estimation of the closed-loop domain of attraction. Finally, a gain-scheduled event-triggering mechanism will be proposed to further enlarge the closed-loop domain of attraction.

LMI-based local non-quadratic sampled-data controller design for T-S descriptors

Consider the class of T-S descriptor (4.1) and the closed-loop dynamics presented in (4.10).

To provide parameterized LMI-based conditions satisfying P 1 and P 2 , we proposed the following Non-Quadratic Lyapunov-Krasovskii Function (NQLKF) candidate:

V (t) = 4 ℓ=1 V ℓ (t) (4.13)
where:

V 1 (t) = x T (t)P α x(t), (4.14) V 2 (t) = (η k -τ (t))ρ T (t)Q α ᾱρ(t), (4.15) V 3 (t) = (η k -τ (t)) t t-τ (t)
ρT (s)R α ᾱ ρ(s)ds, (4.16) with:

V 4 (t) = (η k τ (t) -τ 2 (t))ζ T (t)S ᾱζ(t), (4.17
ζ(t) = col x(t), x(t -τ (t)), t t-τ (t)
x(s)ds,

t t-τ (t)
ẋ(s)ds and

ρ(t) = col t t-τ (t)
x(s)ds,

t t-τ (t)
ẋ(s)ds .

The following theorem summarizes the proposed LMI-based conditions to address the abovedefined problem statements P 1 and P 2 for the sampled-data stabilization of T-S descriptors (4.5) with input saturation.

Theorem 4.1. Let (i, j) ∈ I 2 r and let assume that ∀t, | αi (t)| ≤ φ i . For given symmetric bounds ū ∈ R m of the input vector and for non-uniform sampling intervals η k ≤ η (η to be maximized), the T-S descriptor (4.5) is locally asymptotically stabilized by the sampled-data PDC controller (4.7) with saturation (see (4.3)), if there exists a diagonal positive matrix D j ∈ R m×m , the

matrices 0 < Pi = P T i ∈ R n×n , Sj = ST j ∈ R 4n×4n , Qij = QT ij ∈ R 2n×2n , R11ij = RT 11ij ∈ R n×n , R22j = RT 22j ∈ R n×n , R12j ∈ R n×n , X ∈ R n×n , K j ∈ R m×n , F j ∈ R m×n , Ȳij ∈ R n×4n , M 1 ij = M 1T ij ∈ R n×n , M 2 ij = M 2T ij ∈ R 2n×2n
, and the scalars ε 1 , ε 2 , and ε 3 , such that the following parameterized inequalities are satisfied:

       Λ11 α ᾱ η Ȳ T α ᾱ       < 0, (4.18) -Pρ + M 1 α ᾱ 0 ⋆ Qρᾱ + M 2 α ᾱ < 0, ∀ρ ∈ I r , (4.19) Pα ⋆ (K ᾱ -F ᾱ) (ℓ) ū2 (ℓ) ≥ 0, ∀ℓ ∈ I m , (4.20) with: Λ11 α ᾱ = -η 2 H E T 4 Sᾱ E 5 + η Φ1αᾱ + Φ0αᾱ + r ρ=1 φ ρ E T 0 ( Pρ + M 1 α ᾱ) + Ψᾱ , Λ33 α ᾱ = Φ0αᾱ + r ρ=1 φ ρ (E T 0 ( Pρ + M 1 α ᾱ)) + Ψᾱ , Λ55 α ᾱ = Λ33 α ᾱ + η2 H E T 4 Sᾱ E 5 , Φ0αᾱ = H P T α Āαᾱ + ηE T 1 Qαᾱ E 2 -E T 3 Ȳαᾱ -E T 1 Qαᾱ E 1 + ηE T 2 Rαᾱ E 2 + Wαᾱ + ηE T 4 Sᾱ E 4 , Φ1αᾱ = -H E T 1 Qαᾱ E 2 -ηE T 4 Sᾱ E 5 + E T 4 Sᾱ E 4 -E T 2 Rαᾱ E 2 , Wαᾱ = -e T 1 R12ᾱ e 1 + e T 2 R12ᾱ e 2 -η -1 k e T 3 R11αᾱ e 3 , Pα = Pα 0 0 0 0 I ε 1 I ε 2 I ε 3 I I , Āαᾱ = 0 0 0 I 0 A α X B α K ᾱ 0 -E α X B α D α , Ψᾱ = -E T 6 0 F T ᾱ D ᾱ ⋆ 2D ᾱ E 6 , E 0 = e 1 e 0 , E 1 = e 3 e 1 -e 2 , E 2 = e 1 e 4 , E 3 = e 1 -e 2 , E 4 =      e 1 e 2 e 3 e 1 -e 2      , E 5 =      e 4 e 0 e 1 e 4      , E 6 =
e 2 e 5 .

Furthermore, an estimate of the domain of attraction D * a of the designed closed-loop sampled-data descriptor is obtained by maximizing c such that:

D * a = {x(0) ∈ R n | ∃c = max c, L(c) ⊆ D u ∩ D x ∩ D φ } (4.21) with L(c) = {x ∈ R n | x T P ᾱx ≤ c}.
Furthermore, an estimate of the domain of attraction taken Proof. ..., 4. In this context, to ensure the asymptotic stability of the closed-loop dynamics (4.10), because the LKF is continuous ∀t ∈ [0, +∞), we only need to provide that it is monotonously decreasing ∀t ∈ [t k , t k+1 ), i.e.:

at t = t 0 is obtained by optimizing V 1 (x) = c ≤ 1, i.
Let P α = P T α > 0. At the sampling instant t k we have V (t k ) > 0 because V 1 (t k ) > 0 and V ℓ (t - k ) = V ℓ (t k ) = 0, for ℓ = 2,
V (t) = 4 ℓ=1 Vℓ (t) < 0 (4.22)
Nevertheless, at this point, the constraints introduced due to the actuators' saturation must be consider on the design. So, from Lemma 2.1, the sector conditions is introduced with (4.22), providing that x(t-τ (t)) ∈ S((K ᾱ -F ᾱ)X -1 , ū), for any diagonal matrix D ᾱ > 0, such that:

V(t)-2ψ(K ᾱX -1 x(t-τ (t))) T D -1 ᾱ (ψ(K ᾱX -1 x(t-τ (t)) +F ᾱX -1 x(t-τ (t))) < 0 (4.23)
In the remaining of this proof, let us consider the extended vector:

ξ(t) = col x(t), x(t -τ (t)), t t-τ (t) x(s)ds, ẋ(t), ψ(K ᾱX -1 x(t -τ (t))) .
Thus, the inequality (4.23) can be rewritten as:

V (t) + ξ T (t)Ψ ᾱξ(t) < 0 (4.24)
with:

Ψ ᾱ = -E T 6 0 X -T F T ᾱ D -1 ᾱ ⋆ 2D -1 ᾱ E 6 (4.25)
Then, considering the arbitrary scalars ε 1 , ε 2 , and ε 3 , and a regular matrix X, let:

Pα = P α 0 0 0 0 X -1 ε 1 X -1 ε 2 X -1 ε 3 X -1 D -1 α ,
which, combined with E 0 defined in Theorem 4.1, allow us to rewrite (4.14) as:

V 1 (t) = ξ T (t)E T 0 Pα ξ(t) (4.26)
where the symmetric condition E T 0 Pα = P T α E 0 is satisfied. Hence, the time-derivative of V 1 (t) can be written as:

V1 (t) = 2ξ T (t) P T α E 0 ξ(t) + ξ T (t)E T 0 Ṗα ξ(t) (4.27)
Note that the closed-loop dynamics (4.10) can be rewritten as:

E 0 ξ(t) = Ãαᾱ ξ(t), with Ãαᾱ = 0 0 0 I 0 A α B α K ᾱX -1 0 -E α B α . (4.28)
Then, equation (4.27) can be rewritten as:

V1 (t) = ξ T (t) H( P T α Ãαᾱ ) + E T 0 Ṗα ξ(t) (4.29)
Now, taking the time-derivative of V 2 (t) leads to:

V2 (t) = -ρ T (t)Q α ᾱρ(t) + (η k -τ (t))(2ρ T (t)Q α ᾱ ρ(t) + ρ T (t) Qαᾱ ρ(t)) (4.30)
With ρ(t) = E 1 ξ(t) and ρ(t) = col {x(t), ẋ(t)} = E 2 ξ(t), equation (4.30) can be rewritten as:

V2 (t) = -τ (t)ξ T (t) H E T 1 Q α ᾱE 2 + E T 1 Qαᾱ E 1 ξ(t) + ξ T (t) H η k E T 1 Q α ᾱE 2 -E T 1 Q α ᾱE 1 + η k E T 1 Qαᾱ E 1 ξ(t) (4.31)
Next, the time-derivative of V 3 (t) is given by:

V3 (t) = (η k -τ (t)) ρT (t)R α ᾱ ρ(t) - t t-τ (t)
ρT (s)R α ᾱ ρ(s)ds (4.32)

Let:

R α ᾱ = R 11α ᾱ R 12 ᾱ ⋆ R 22α ᾱ ,
and assume:

R 11α ᾱ = R T 11α ᾱ > 0, R 22α ᾱ = R T 22α ᾱ > 0 (4.33)
Equation (4.32) is equivalent to:

V3 (t) =(η k -τ (t)) ρT (t)R α ᾱ ρ(t) - t t-τ (t) x T (s)R 11α ᾱx(s)ds - t t-τ (t) ẋT (s)R 22α ᾱ ẋ(s)ds -2 t t-τ (t) x T (s)R 12 ᾱ ẋ(s)ds (4.34) Note that t t-τ (t)
ẋT (s)ds = E 3 ξ(t) and, ∀t ∈ [t k , t k+1 ), R 12 ᾱ is constant. Applying Lemma 1.7

and Lemma 1.8 we get:

V3 (t) ≤(η k -τ (t)) ρT (t)R α ᾱ ρ(t) -η -1 k t t-τ (t) x T (s)R 11α ᾱ t t-τ (t)
x(s)ds

-x T (t)R 12 ᾱx(t) + x T (t -τ (t))R 12 ᾱx(t -τ (t)) + ξ T (t) τ (t)Y T α ᾱR -1 22α ᾱY α ᾱ -E T 3 Y α ᾱ -Y T α ᾱE 3 ξ(t) = τ (t)ξ T (t) Y T α ᾱR -1 22α ᾱY α ᾱ -E T 2 R α ᾱE 2 ξ(t) + ξ T (t) η k E T 2 R α ᾱE 2 -H E T 3 Y α ᾱ + W α ᾱ ξ(t) (4.35)
with W α ᾱ = -e T 1 R 12 ᾱe 1 + e T 2 R 12 ᾱe 2 -η -1 k e T 3 R 11α ᾱe 3 . Now, taking the time-derivative of V 4 (t), we obtain:

V4 (t) = (η k -2τ (t))ζ T (t)S ᾱζ(t) + 2(η k τ (t) -τ 2 (t))ζ T (t)
P(τ (t)) = -τ 2 (t)ξ T (t)H E T 4 S ᾱE 5 ξ(t) + τ (t)ξ T (t) Φ 1α ᾱ + Y T α ᾱR -1 22α ᾱY α ᾱ -E T 1 Qαᾱ E 1 ξ(t) + ξ T (t) Φ 0α ᾱ + E T 0 Ṗα + η k E T 1 Qαᾱ E 1 ξ(t) < 0 (4.38)
with:

Φ 1α ᾱ = H η k E T 4 S ᾱE 5 -E T 1 Q α ᾱE 2 -E T 4 S ᾱE 4 -E T 2 R α ᾱE 2 , Φ 0α ᾱ =H P T α Ãαᾱ + η k E T 1 Q α ᾱE 2 -E T 3 Y α ᾱ -E T 1 Q α ᾱE 1 + η k E T 2 R α ᾱE 2 + W α ᾱ + η k E T 4 S ᾱE 4 -E T 9 Ψ ᾱE 9 .
Let us recall that acoording Lemma 1.9, (4.38) can be checked if:

P(0) < 0, ( 4
Φ 0α ᾱ + E T 0 Ṗα < 0, (4.42) -η 2 H E T 4 S ᾱE 5 + η Φ 1α ᾱ + Y T α ᾱR -1 22α ᾱY α ᾱ -E T 1 Qαᾱ E 1 + Φ 0α ᾱ + E T 0 Ṗα + ηE T 1 Qαᾱ E 1 < 0 (4.43)
and

Φ 0α ᾱ + E T 0 Ṗα + η2 H E T 4 S ᾱE 5 < 0, (4.44)
Recall that r ρ=1 αρ (x(t)) = 0, and for any slack matrix M q α ᾱ (q = 1, 2), (4.45) we get:

r ρ=1 αρ (x(t))M q α ᾱ = 0. With | αρ (x(t))| ≤ φ ρ , if: P ρ + M 1 α ᾱ > 0 and Q ρ ᾱ + M 2 α ᾱ > 0, ∀ρ ∈ I r ,
E T 0 Ṗα ≤ r ρ=1 φ ρ E T 0 (P ρ + M 1 α ᾱ) (4.46)
and:

η k E T 1 Qαᾱ E 1 ≤ η k r ρ=1 φ ρ E T 1 (Q ρ ᾱ + M 2 α ᾱ)E 1 . (4.47)
Thus, from (4.46) and (4.47), applying the Schur complement, the inequalities (4.42) and (4.43)

are respectively satisfied if:

Φ 0α ᾱ + r ρ=1 φ ρ (E T 0 (P ρ + M 1 α ᾱ)) < 0, (4.48) Λ 11 α ᾱ η k Y T α ᾱ ⋆ -η k R 22α ᾱ < 0, (4.49) with Λ 11 α ᾱ = -η 2 H E T 4 S ᾱE 5 + ηΦ 1α ᾱ + Φ 0α ᾱ + r ρ=1 φ ρ E T 0 (P ρ + M 1 α ᾱ).
Taking the congruence of (4.33) and (4.45) by X p (X) (with appropriate p), where X p (X) denotes a diagonal block matrix filled p times in diagonal with X ∈ R n×n . Then pre-and postmultiplying (4.44) and (4.48) respectively by diag{X, X, X, X, D ᾱ} T and its transpose followed by pre-and post-multiplying (4.49) respectively diag{X, X, X, X, D

of variables Z = X T ZX, Z = {P α , Q α ᾱ, R 11α ᾱ, R 12 ᾱ, R 22α ᾱ . . . }.
In the sequel of the proof, let us focus on the restriction brought by the saturated actuators.

Recall that the NQLKF (4.22) is continuous and monotonously decreasing if the previous inequalities hold. Therefore, for every x(t k ) reaching D a the systems will be stable and, without loss of generality, the NQLKF taken at t k allows to define the following level sets:

L(1) = {x(t -τ (t)) ∈ R n | x(t -τ (t)) T P α x(t -τ (t)) ≤ 1} (4.50)
Moreover, from the application of Lemma 1.6, we must provide that x(t -τ (t)) is such that (K ᾱ -F ᾱ)X -1 x(t -τ (t)) belongs to S(ū) (see eq. (1.64)). This holds if both the following conditions hold:

If x(t -τ (t)) T P α x(t -τ (t)) ≤ 1, then x(t) ∈ L(1) (4.51)
and, with

Z ᾱ = ((K ᾱ -F ᾱ)X -1 ) T (ℓ) ((K ᾱ -F ᾱ)X -1 ) (ℓ) /ū 2 (ℓ) : If x(t -τ (t)) T Z ᾱ(ℓ) x(t -τ (t)) ≤ 1, then (K ᾱ -F ᾱ)X -1 x(t -τ (t)) ∈ S(•) (4.52) Then L(1) ⊂ S(K ᾱX -1 -F ᾱ, ū) if: 1 ≥ x(t -τ (t)) T P α x(t -τ (t)) ≥ x(t -τ (t)) T Z ᾱ(ℓ) x(t -τ (t)) (4.53)
which is granted from the Schur complement if: To conclude this subsection, let us point out that the design conditions given in Theorem 4.1 do not guarantee the global asymptotic stability for the closed-loop system driven by the designed sampled-data controller (4.7). Indeed, from the above-presented design conditions, three main

P α ⋆ ((K ᾱ -F ᾱ)X -1 ) (ℓ) ū2 (
D * a = {x(0) ∈ R n | ∃c = max c, L(c) ⊆ D u ∩ D x ∩ D φ }
Hence, the computation of D * a can be done through a manual adjusting procedure after solving the LMI conditions of Theorem 4.1. It is important to highlight that it requires the graphical representation of D x and D φ , which makes tricky and time-consuming the optimization of all involved parameters by trial and errors. In order to reduce the complexity brought by such a manual procedure, in the next subsection, we propose a systematic LMI-based methodology to get the estimate D * a of the region of attraction from an unitary invariant level set approach, but for standard T-S fuzzy models instead of descriptor ones.

Systematic estimation of the domain of attraction for standard T-S models-based sampled-data controller design

In this section, let us consider the class of standard T-S models (3.1) instead of T-S descriptors (4.1). Our goal is to provide complementary conditions to obtain a systematic estimation D * a of the sampled-data closed-loop region of attraction, i.e. to prevent from using the manual optimization procedure described above. To do so, let us recall that such a region is constrained by the regions D u , D x and D φ , respectively defined in (4.4), (4.2) and (4.11).

Before going to the main result of this subsection, let us recall that the gradient of α k (x(t)) for x(t) is given by:

∇ k (x(t)) = ∂α k (x(t)
< Pi = P T i ∈ R n×n , Sj = ST j ∈ R 4n×4n , Qij = QT ij ∈ R 2n×2n , R11ij = RT 11ij ∈ R n×n , R22ij = RT 22ij ∈ R n×n , R12ij ∈ R n×n , X ∈ R n×n , K j ∈ R m×n , F j ∈ R m×n , Ȳij ∈ R n×4n , Ūij = Ū T ij ∈ R 3n×3n , M 1 ij = M 1T ij ∈ R n×n , M 2 ij = M 2T ij ∈ R 2n×2n
, and the scalars ε 1 , ε 2 , and ε 3 , such that the following optimization problem is satisfied:

       max Pα>0, Qα> Rαᾱ,Sᾱ, Kᾱ, Ȳαᾱ, M 1 α ᾱ, M 2 α ᾱZα>0
Trace(X) subject to the PLMIs (4.18), (4.19), (4.20), (4.58) and (4.59), (4.57)

Pα X T L T (j) L (j) X b 2 (j) ≥ 0, ∀j ∈ I κ , (4.58) Pα ⋆ ∇s G α ᾱ φ 2 ≥ 0, ∀s ∈ I r .
(4.59)

In this case, the estimation of the domain of attraction D * a is systematically obtained as:

D * a = L(1) = {x(0)∈R n |x T P α x ≤ 1} ⊆ D u ∩ D x ∩ D φ (4.60)
which edge is readily given by the unit equipotential x T P α x = 1.

Proof. Starting from the conditions of Theorem 4.1 for the special case of standard T-S models (4.5) (i.e. assuming E α = I), we want to provide an optimization procedure to enlarge D * a = L(1) expressed in (4.60). Note that L(1) ⊆ D u is already granted by (4.20). Then, without loss of generality, assuming that the initial instant is also the first sampling instant (i.e. t = t 0 = 0), L(1) is defined by:

x T (0)X -T Pα X -1 x(0) ≤ 1 (4.61)
From the definition (4.2), the edge of D x is given by:

x T (0)L T (j) L (j) x(0) b 2 (j) ≤ 1, ∀j ∈ I κ (4.62)
Hence, from (4.61) and (4.62), L(1) ⊆ D x is granted by:

X -T Pα X -1 - L T (j) L (j) b 2 (j) ≥ 0, ∀j ∈ I κ (4.63)
which gives, after congruence by X T and X:

Pα - X T L T (j) L (j) X b 2 (j) ≥ 0, ∀j ∈ I κ (4.64)
Then, applying the Shur complement, we get (4.58).

Now, let us focus on D φ constrained by:

| αk (x(t)))| = |∇ T k (x(t)) ẋ(t)| ≤ φ k , ∀k ∈ I r (4.65)
which edge, for the initial instant t = t 0 = 0, is characterized by:

x T (0)X -T G T α ᾱ∇ T k (x(0))∇ k (x(0))G α ᾱX -1 x(0) φ 2 k ≤ 1 (4.66) with G α ᾱ = (A α X + B α K ᾱ).
From (4.61) and (4.66), the following inequality guarantees L(1) ⊆ D φ :

X -T Pα X -1 - X -T G T α ᾱ∇ T k (x(0))∇ k (x(0))G α ᾱX -1 φ 2 k ≥ 0 (4.67)
After congruence by X T and X, it yields:

Pα - G T α ᾱ∇ T k (x(0))∇ k (x(0))G α ᾱ φ 2 k ≥ 0 (4.68)
Applying the Shur complement, we get:

Pα ⋆ ∇ k (x(0))G α ᾱ φ 2 ≥ 0 (4.69)
Note that the condition (4.69) depends on the initial conditions. Nevertheless, knowing that

x(0) ∈ D x , and considering (4.56), (4.69) is satisfied if:

2 n s=1 ℏ s (z(t)) Pα ⋆ ∇s G α ᾱ φ 2 ≥ 0 (4.70)
which is verified if the conditions (4.59) are satisfied.

Note that the completion of (4.20), (4.58) and (4.59) together ensures (4.60). Furthermore, to enlarge the area of L(1) and consequently the estimation of the region of attraction D * a , minimizing the trace of P α leads to maximize the trace of X under the constraint (4.61), since Pα = X -T P α X -1 . This defines the optimization problem (4.57).

Remark 4.5. Unfortunately, because of the dependency of the closed-loop dynamics for computing the estimate of the region of attraction from D φ (see. (4.70)), Theorem 4.2 cannot be directly employed for descriptors systems. At this moment, the literature to cope with descriptors is not extensive. However, in [START_REF] Lendek | Local stabilization of discrete-time TS descriptor systems[END_REF], the authors brought some interesting ideas in the discrete-time framework, which will be investigated for extension in our further works.

Remark 4.6. Note that to obtain a systematic characterization of L(1) ⊆ D φ , we employed a convexification procedure of the gradient (4.56). This may lead to conservatism since it increases the number of LMI vertices to be solved simultaneously. An interesting special case occurs when the ∇ s (s ∈ I 2 n ) are symmetric. Indeed, in this special case, ∀s ∈ I 2 n , conditions (4.59) are equivalent by Schur complement, alleviating the pessimism introduced by such convexification. In all the other cases, such convexification procedure should be employed with discretion, i.e. when the T-S fuzzy model involves few rules or when the membership function involves few premises variables. Otherwise, the manual procedure given in Theorem 4.1 should be preferred.

Gain scheduled event-triggering mechanism to enlarge the closed-loop domain of attraction

From the application of Theorem 4.1 or Theorem 4.2, let us point-out that D * a , obtained for a given η, is a contractive Lyapunov level set for the designed closed-loop dynamics. Taking benefit of such a feature, it is possible to propose a simple gain scheduled event-triggering control mechanism to maximize the guaranteed domain of attraction in sampled-data control design.

To do so, let η k ∈ {η 1 , η2 , . . . , ηq } be an ordered sequence (0 < η1 < η2 < • • • < ηq ) of chosen values for triggering the sampling intervals η k ≤ η. Then, for each of these values, solve (offline) the conditions of Theorem 4.1 or Theorem 4.2 and store their solutions, ∀ν ∈ I q and ∀i ∈ I r , as P ν i , K ν i and X ν . Therefore, for each P ν i , we can get L ν (c ν ), with cν ≤ 1 (manually optimized when employing Theorem 4.1) or cν = 1 (with Theorem 4.2), and we define:

D * a = q ν=1
L q (c q ) (4.71)

For initial conditions starting in D * a and then at each sampling instant t k , let:

q(t k ) = sup ν∈Iq {x T (t k )P ν α x(t k ) ≤ c ≤ 1} (4.72)
We define the following activation function, ∀ν ∈ I q :

µ ν (x(t k )) = 1, if ν = q(t k ) 0, otherwise (4.73)
Then, we propose the gain scheduled event-triggering sampled-data controller given by:

u(t) = q ν=1 µ ν (x(t k ))K ν ᾱX -1 ν x(t k ) (4.74)
where the actual sampling interval η k = t k+1 -t k = ηq(t k ) is updated at each sampling instants

t k .
Therefore, with the considered continuous-time T-S descriptor (4.5), closing the loop with the control law (4.74) ensures the asymptotic stabilization for all initial conditions taken in D * a given in (4.71).

The benefit of using such a simple gain scheduled event-triggering mechanism will be illustrated through simulation examples in the next section.

Illustrative Examples

In this section, to evaluate the effectiveness of the LMI-based sampled-data controller design conditions proposed in Theorems 4.1 and 4.2, we will consider benchmark models of the inverted pendulum on a cart described in Chapter 3 (see section 3.5.2), which nonlinear dynamics are given by (3.52). First, for comparison purpose and to highlight the improvements raised by the conditions proposed in this chapter, we will consider the approximated two rules T-S fuzzy model investigated in Section 3.5.2. Then, because it has been shown that this approximated models is not suitable to guarantee the stabilization of the full nonlinear model of the inverted pendulum (3.52) (especially when large sampling intervals are considered), a matching T-S descriptor model is then studied. For both these cases, the estimations of the domain of attraction will be provided with discussion for their enlargement.

Benchmark of the approximated fuzzy model of the inverted pendulum

To illustrate the effectiveness of the proposed sampled-data control approach with saturated actuators, let us first consider as benchmark the same approximated two rules T-S fuzzy model of an inverted pendulum on a cart, drawn from [START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF] and presented in Section [START_REF] Löfberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF] and MOSEK (ApS, 2019), with ε 1 = 5.6, ε 2 = 1.2, ε 3 = 0.33 and ū = 150N provides the following sampled-data PDC controller gains and Lyapunov matrices with the obtained maximal sampling interval η = 50ms: [START_REF] Yoneyama | Robust H ∞ control of uncertain fuzzy systems under time-varying sampling[END_REF] 9 [START_REF] Zhu | Stabilization for sampled-data neural network-based control systems[END_REF] 13 [START_REF] Zhang | H ∞ control design for network-based T-S fuzzy systems with asynchronous constraints on membership functions[END_REF] 16 [START_REF] Zhu | An Improved Input Delay Approach to Stabilization of Fuzzy Systems Under Variable Sampling[END_REF] 19 [START_REF] Gunasekaran | Robust Sampled-data Fuzzy Control for Nonlinear Systems and Its Applications: Free-Weight Matrix Method[END_REF] 22 [START_REF] Zhu | H ∞ stabilization criterion with less complexity for nonuniform sampling fuzzy systems[END_REF] 24 [START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF] 42 However, recall that the conditions of Theorem 4.1 and Theorem 4.2 only guarantee the local stabilization. Therefore, the domain of attraction of the closed-loop sampled-data system must be characterized. In this regard, it is important to highlight that, with exception to our preliminary works [START_REF] Lopes | Sampled-data Controller Design for Mechatronic Systems Described by Takagi-Sugeno Descriptors[END_REF] and (Lopes et al., 2021a), no other related works on sampled-data controller design provide such analysis. are obtained from solving the conditions of Theorem 4.2: This shows that the choice of ū influences a lot the enlargement of the domain of attraction.

K 1 = 8.
K 1 = 0.
Reciprocally, such analysis may be useful for designers to scale the actuators. To illustrate this feature, Figure 4.3 shows the domains of attraction obtained for several values of ū (i.e., ū ∈ [150, 200, 300, 400, 500, 750, 1000, 1250, 1500, 2000]), with the largest admissible value of η = 50ms. This figure shows that, for ū ≥ 1500N , there is no major improvement as, for η = 50ms, the successive estimations D * a ⊆ D u keep closely the same shape and reach the edge of D x ∩ D φ . It is to be noticed that the above described tests have been obtained for the maximal value η = 50ms, which constitutes the limit of conservatism achieved, i.e. the most constraining case for solving the LMI-based conditions of Theorem 4.1 and Theorem 4.2. This critical case explains why we get P 1 close to P 2 , making the shape of D * a constrained to be almost an ellipsoid. In the opposite, it is also possible to set a smaller value of η < 50ms to relax the LMI-based conditions of Theorem 4.1 and Theorem 4.2. To illustrate this possibility, Figure 4.4 shows the estimation D * a of the domain of attraction obtained for η = 1ms and for several values of ū (i.e., ū ∈ [150,200,500,1000,1500,2000,5000,6000,7500,10000]). It is observed that, for small values of η and large values of ū, the shape of D * a is now non-quadratic. At this step, the above tests clearly show the influence of the input saturation on the estimation D * a of the domain of attraction. Let us now focus on the influence of assuming maximal allowable sampling intervals. Figure 4.5 shows the estimates D * a obtained with ū = 10000N for several values of η (i.e., η ∈ [1, 5, 10, 15, 20, 25, 35, 45, 40, 50]). This also clearly shows that the choice of η influences the shape of D * a .

- Now, from this last test, let us recall that each D * a , obtained for each η, is contractive for its respective closed-loop dynamics since it represents the Lyapunov level set L(1). Taking benefit of such a feature, we consider the gain scheduled event-triggering mechanism proposed in Subsection 4.3.3 to further extend the guaranteed domain of attraction in sampled-data control design. To do so, solving Theorem 4.2 for each η k ∈ [1,5,10,15,20,25,35,45,50], then storing the results as P ν i , K ν i and X ν (ν ∈ I 12 and ∀i ∈ I 2 ). We can apply the gain scheduled event-triggering controller conditions taken at the border of D * a . We clearly see that, the closed-loop system is stable and, from the first few samples, the selected η k is small, then increases until the maximal allowed value η = 50ms (obtained from Theorem 4.2). These shows that this control strategy is high demanding in terms of computation only for the first few samples (during transients), while it quickly becomes undemanding with large sampling intervals. However, we can also notice an important chattering when the maximal allowed sampling period η k = 50ms is selected. This mainly due to the fact that this value is critical in terms of conservatism of the proposed LMIbased conditions, making the designed closed-loop closer to the limit of stability. To mitigate this effect, an easy way is to avoid going to such critical values. Figure 4.7 shows the trajectories of the closed-loop system, the input signals and the evolution of the scheduled sampling intervals, with transient improvements from the proposed gain scheduled event-triggering strategy with , 5, 10, 15, 20, 25, 35]. All of these tests and simulation illustrate the potential of the proposed sampled-data con-troller design for T-S fuzzy systems with significant improvements regarding to the previous results presented in Chapter 3 and from the literature. However, as mentioned in Chapter 3

η k ∈ [1
(see the end of Subsection 3.5.2), the sampled-data controller obtained in this subsection only apply for the approximated model (4.75) from which it is designed. As a matter of fact, there is no guarantee that it stabilizes the original nonlinear system. (3.52). Even if it is useful for comparison purpose, this highlights the limitation of using such approximated standard T-S fuzzy models, which should be circumvent by considering an exact T-S model in the descriptor form, as it will be shown in the next subsection.

Remark 4.7. From the conditions proposed in [START_REF] Lamrabet | Sampled data Control for Takagi-Sugeno Fuzzy Systems with Actuator Saturation[END_REF], which is the unique previous study found aiming at proposing sampled-data controller design for T-S systems under saturating actuators, we surprisingly obtained a maximal η = 164ms. Unfortunately, when performing the simulation with the gains obtained from [START_REF] Lamrabet | Sampled data Control for Takagi-Sugeno Fuzzy Systems with Actuator Saturation[END_REF], we observed an unstable closed-loop behavior with a strongly saturating input. Moreover, in this paper, no characterization of the closed-loop domain of attraction is provided and, because the authors did not select a looped LKF, this task is pretty much harder or even impossible. Hence, failing to reproduce the claimed results in [START_REF] Lamrabet | Sampled data Control for Takagi-Sugeno Fuzzy Systems with Actuator Saturation[END_REF], we assume irrelevant a fair comparison with this work.

Matching T-S descriptor model-based sampled-data controller design

In this subsection, we will consider the full nonlinear model of the inverted pendulum on a cart drawn from [START_REF] Cannon | Dynamics of physical systems[END_REF][START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design issues[END_REF], given by the dynamic equation (3.52).

Let us consider the state vector x(t) = x 1 (t) x 2 (t) T , where x 1 (t) is the angular position of the inverted pendulum with regard to the erect position, x 2 (t) = ẋ1 (t) is the angular velocity (rad/s). The nonlinear dynamics (3.52) can be rewritten as a nonlinear descriptor (4.1) with:

E(x(t)) = 1 0 0 4l 3 -aml cos 2 x 1 (t) , A(x(t)) = 0 1 g sin x 1 (t) x 1 (t) -aml 2 x 2 (t) sin(2x 1 (t)) , B(x(t)) = 0 -a cos x 1 (t) .
(4.76)

where g = 9.8 m/s 2 is the acceleration of the gravity, m = 2 kg is the mass of the pendulum, M = 8 kg is the mass of the cart, l = 0.5 m is the half length of the pendulum, u(t) is the input actuator force (N ) applied to the cart and a = 1/(m + M ).

Recall that the inverted pendulum is not controllable for x 1 = π 2 . Hence, to fairly cope with the previous example, in order to derive a T-S descriptor model (4.5) matching (4.76) we chose to restrict the validity domain D x to |x 1 | ≤ θ 0 = 22π/45 rad and |x 2 | ≤ π rad/s. Moreover, note that the nonlinear matrices (4.76) involve four nonlinear entries, bounded on D x , given by:

f 1 (x 1 (t)) = cos 2 x 1 (t) ∈ [ f1 , f1 ], f1 = cos 2 θ 0 , f1 = 1, f 2 (x(t)) = x 2 (t) sin(2x 1 (t)) ∈ [ f2 , f2 ], f2 = -π, f2 = π, f 3 (x 1 (t)) = sin x 1 (t) x 1 (t) ∈ [ f3 , f3 ], f3 = sin θ 0 θ 0 , f3 = 1, f 4 (x 1 (t)) = cos x 1 (t) ∈ [ f4 , f4 ], f4 = cos θ 0 , f4 = 1.
(4.77)

spectively for η = 1ms and η = 13ms. We can notice that these are pretty small, but of course, recall that ū = 150N , which is a really small value of the input saturation regarding to the overall mass of the considered benchmark (9kg, including the inverted pendulum and the cart). Hence, we may expect better results enlarging ū as proposed in the next case. (not optimized here just to compare the results with the previous case) and the following time response and the overall behavior of the successive Lyapunov-Krasovskii functions (evaluated at each sampling instant t = t k and selected according to the gain scheduled event-triggering mechanism) of the designed sampled-data closed-loop system with the initial conditions x T (0) = -0.9610 3.0550 , taken at the edge of D * a . Recall that each level sets L q (1) (q ∈ I 8 ) composing D * a are contractives. This is why the behavior of the successive Lyapunov-Krasovskii functions, selected from the event-triggering laws (4.72) and (4.74), shows jumps when the state trajectories cross these successive level sets, while otherwise monotonously decreasing. All these simulations confirm the effectiveness of the proposed sampled-data controller design procedure and event-triggering mechanism for T-S descriptors. descriptors has not been previously addressed in the literature, excepted in our preliminary result [START_REF] Lopes | Sampled-data Controller Design for Mechatronic Systems Described by Takagi-Sugeno Descriptors[END_REF], this simulation study confirms the benefits of the proposed T-S descriptor model-based sampled-data controller design methodology (with or without input saturation).

Conclusion

In this chapter, from a convenient NQLKF candidate, relaxed sampled-data controller design

General Conclusion and Perspectives

Along this thesis, we are concerned with providing relaxed LMI-based design conditions to Then, some limitations of the standard discrete-time approach have been pointedout and the input-delay approach for sampled-data controllers has been introduced as an elegant alternative to cope with such issues. Once again, the main limitations of this approach in the T-S model framework, such like the locality of the results or the overall conservatism of the design conditions, have been highlighted. Summarizing, the critical analysis presented in this chapter allowed the maturation of the problem statement considered in this thesis, which can be classified into two balanced research lines presented as follows.

First, acknowledging that discrete-time model-based approaches are mainly considered in industrial applications because of their versatility and easy implementation in industrial controllers, some improvements for discrete-time T-S model-based controller design were proposed in Chapter 2, with the goal of providing relaxed sampled-data set-points tracking controller design conditions for discrete-time nonlinear models under saturating actuators. In this chapter, a convex optimization methodology to design discrete-time PI-like controllers is presented. The main contribution of Chapter 2 extends our preliminary results in [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF], by including an anti-windup control action to dismiss the slow state variation requirement and, as a consequence, achieving less conservative estimates for both the region of attraction and the allowable amplitude changes in the set-point. In this context, the proposed approach can access an amplitude bound for exogenous signals such that the trajectories of the closed-loop system remain in the region of attraction, ensuring local asymptotic stability through a non-quadratic Lyapunov function. Also, the proposed fuzzy PI-like control law holds a PDC structure while the considered anti-windup fuzzy gain has a non-PDC one, which can be efficiently implemented in embedded industrial applications [START_REF] Sousa | Affordable control platform with MPC application[END_REF]. The proposed approach has been validated in simulation as well as experimentally on the level control of a two coupled nonlin-ear tanks system available at the CEFET-MG, with significant improvements regarding [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF].

If the discrete-time T-S model-based controller design approach was considered because of its appealing conceptual simplicity, especially when the implementation of the controller for industrial applications is required, it was also highlighted that this approach is only suitable when a small enough fixed sampling period can be set regarding the plant dynamics. However, recall that assuming fixed sampling periods is sometimes practically unrealistic. Indeed, even in a point-to-point digital control topology, aperiodic sampling may arise because of clock inaccuracy and/or system architecture characteristics such as real-time scheduling, which can induce jitters, imperfect synchronization or computation delays [START_REF] Wittenmark | Timing problems in real-time control systems[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. Also, when dealing with Networked Controlled Systems (NCS), in which sampled-data systems are controlled through communication networks [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]Fridman, 2014a), aperiodic sampling intervals are almost inevitable. Nonetheless, one may also argue that, when large sampling periods have to be considered, the direct discrete-time model-based approach may fail to guarantee the closed-loop inter-sampling stability [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. To circumvent these drawbacks, the alternative of the input-delay approach for sampled-data control has been investigated in Chapter 3 and Chapter 4 in the continuous-time T-S model-based framework. From this approach, the nonlinear sampled-data closed-loop dynamics is rewritten as a continuous-time T-S system with time-varying input delay, as proposed in the linear case, see e.g. [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF][START_REF] Seuret | Stability analysis for sampled-data systems with a time-varying period[END_REF].

Associated with the fact that most of real applications exhibits nonlinear dynamics, in the aforementioned chapters an extension of this approach for T-S model based framework is presented, and the conservatism reduction is verified by searching the maximal allowable sampling interval [0, η], for which the closed-loop dynamics stability is guaranteed.

In Chapter 3, our first contribution to the synthesis of sampled-data controllers for continuoustime T-S models, based on the input-delay approach, has been presented. Following the previous related works, in this chapter, relaxed LMI-based conditions for the design of stabilizing aperiodic sampled-data controllers for continuous-time T-S systems has been proposed. To achieve this goal, a convenient augmented looped LKF candidate has been selected, together with the application of bounding lemmas, such as extended Jensen's inequalities, the Finsler's Lemma and some convex quadratic polynomial constraints. Additionally, a generic relaxation scheme, extending the well-known Tuan's Lemma, has been proposed to handle the asynchronous dou- showing a significant enlargement of the maximal allowed sampling interval regarding to several previous related results [START_REF] Yoneyama | Robust H ∞ control of uncertain fuzzy systems under time-varying sampling[END_REF][START_REF] Zhu | Stabilization for sampled-data neural network-based control systems[END_REF][START_REF] Zhang | H ∞ control design for network-based T-S fuzzy systems with asynchronous constraints on membership functions[END_REF][START_REF] Zhu | An Improved Input Delay Approach to Stabilization of Fuzzy Systems Under Variable Sampling[END_REF][START_REF] Gunasekaran | Robust Sampled-data Fuzzy Control for Nonlinear Systems and Its Applications: Free-Weight Matrix Method[END_REF][START_REF] Zhu | H ∞ stabilization criterion with less complexity for nonuniform sampling fuzzy systems[END_REF][START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF], using the benchmark of a simplified T-S fuzzy model of an inverted pendulum on a cart. Also, an experimental validation of the proposed sampled-data design methodology is performed on the Quanser ® AERO platform [START_REF] Quanser | Quanser aero user manual[END_REF], illustrating its effectiveness on a practical system. If the results of this chapter have been shown effective to reduce the conservatism of the design conditions by enlarging the maximal admissible sampling interval, it is also pointed-out that such approach suffers from locality, which require further investigations of the closed-loop domain of attraction, as proposed in the next chapter.

In Summarizing our contribution in the field of sampled-data control for T-S models, let us point-out that, to the best of the author's knowledge, no previous results were found addressing the following points:

• the T-S model-based sampled-data controller design with actuators' saturation; excepted the work of [START_REF] Lamrabet | Sampled data Control for Takagi-Sugeno Fuzzy Systems with Actuator Saturation[END_REF], which did not provide any estimations of the closedloop domain of attraction,

• the resulting estimation of the sampled-data closed-loop domain of attraction with enlargement procedures, obtained thanks to the consideration of a convenient looped LKF; except very recent papers [START_REF] Coutinho | Local Sampled-Data Gain-Scheduling Control of quasi-LPV Systems[END_REF][START_REF] Ma | Exponential stabilization of sampled-data fuzzy systems via a parameterized fuzzy Lyapunov-Krasovskii functional approach[END_REF], published in the late stage of this thesis, who provide some interesting results that would merit further investigations and comparison with the present proposal,

• the extension of the T-S model-based sampled-data controller design methodology to the class of nonlinear descriptors, while these have been shown interesting to model mechatronic systems.

Also, along the manuscript, the proposed simulations and experimental results have established the merit of the proposed T-S model-based sampled-data approaches, in both the discrete-time and continuous-time framework. However, there are still some limitations of the proposed approaches, which can be discussed to provide some perspectives of this work.

Regarding the Direct Design approach for discrete-time systems, even if an abundant literature already exists, there are still points of interests that catch our attention for further improvements. First of all, like many previous studies in the discrete-time framework, we consider the discrete-time realization of a continuous-time T-S model based on the forward Euler discretization approach [START_REF] Chen | Linear System Theory and Design[END_REF]. However, this discretization leading to approximations, there is space for improvements. In this context, other discretization approaches exist, like Tustin bilinear transforms [START_REF] Ogata | Discrete-Time Control Systems[END_REF][START_REF] Åström | Computer-Controlled Systems: Theory and Design[END_REF], using Taylor series expansions or the Cayley-Hamilton Theorem [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF], or more recently using a descriptor system-based approach in the T-S framework [START_REF] Braga | Improved Discretization Method for Uncertain Linear Systems: A Descriptor System Based Approach[END_REF]. Hence, a direct perspective would be to evaluate the improvement raised by these approaches to our discrete-time PI-like tracking controller design conditions, proposed in Chapter 2, especially with the expectation of handling nonlinear systems involving faster continuous-time dynamics, or with the aim to further enlarge the resulting estimation of the closed-loop domain of attraction. Then, another perspective that have not been investigated during this PhD work, is event-triggering in the discrete-time framework, see e.g. [START_REF] De Souza | Co-design of an eventtriggered dynamic output feedback controller for discrete-time LPV systems with constraints[END_REF]. Such approach could be interesting for some industrial applications where fast sampling periods with regard to the plant dynamics can be set to mitigate computational or network workloads. Furthermore, in Chapter 2, we have considered the tracking PI-like controller design problem while investigating the changes in the resulting closed-loop domain of attractions with regard to the changes of set-points (i.e.

piecewise constant desired trajectories). A very interesting perspective would be to propose an extension to the more general case of dynamic trajectory tracking, but in this case the investigation of the closed-loop domain of attraction would be much more complex since its estimate should be dynamically evolving with regard to the desired trajectories at each sampling instant.

Of course, many further extensions of the proposed strategies in the discrete-time framework can be done, for instance to cope with output feedback control, NCS with network-induced delays, packet dropouts, cyber-attacks, and so on. relaxed LMI-based conditions often implies an increase of their computational cost. This is particularly true when dealing with sampled-data control since one important way to relax the conditions comes from the strategy employed to cope with their mismatched double-fuzzy sum structure. To this end, we have proposed an extension of the well-known Tuan's Lemma [START_REF] Tuan | Parameterized Linear Matrix Inequality Techniques in Fuzzy Control System Design[END_REF] but with an increase of the size of the LMIs. This can be regarded as a drawback but, this also allows to reduce the number of LMIs, i.e. balancing the computational cost.

Additionally, to further reduce the conservatism, we proposed looped Lyapunov-Krasovskii functionals for sampled-data controller design, as well as usual bounding lemmas (Extended Jensen's inequality, Finsler lemma...), which introduce several slack decision variables. In the expectation to balance the computational complexity, we may consider, as a first perspective of this work, reducing the number of decision variables (especially free weighting matrices) while considering some more recent and tighter bounding techniques dedicated to time-delayed systems like the use of Wirtinger's or Bessel-Legendre inequalities, see e.g. [START_REF] Seuret | Wirtinger-based integral inequality: Application to timedelay systems[END_REF][START_REF] Ariba | Stability analysis of time-delay systems via Bessel inequality: A quadratic separation approach[END_REF][START_REF] Seuret | Stability of Linear Systems With Time-Varying Delays Using Bessel-Legendre Inequalities[END_REF][START_REF] Bourahala | Relaxed Non-Quadratic Stability Conditions for Takagi-Sugeno Systems with Time-Varying Delays: A Wirtinger's Inequalities approach[END_REF][START_REF] Nafir | Improved robust H ∞ stability analysis and stabilisation of uncertain and disturbed networked control systems with network-induced delay and packets dropout[END_REF]. Of course, the high computational costs brought by complex LMI conditions can be regarded as a drawback for this kind of studies. Nevertheless, let us highlight that such computations are done offline.

Moreover, let us assume that this concern can be alleviated with the continuous growing of computational capabilities of our daily use computers (i.e. personal computer or, when available, high performance computers). [START_REF] Xie | Output feedback H ∞ control of systems with parameter uncertainty[END_REF], which can be further relaxed thanks to some of its extensions [START_REF] Briat | Commande et Observation Robustes des Systèmes LPV Retardées[END_REF]. Also, our approach implies an overall extension of the size of the LMI conditions to be checked, and so possibly affects the computational costs or conservatism for large-scale systems. In this context, some mitigation would be welcome, for instance by considering the introduction of more information about the membership functions, like a recently proposed approach to cope with the bounds of their timederivatives [START_REF] Gunasekaran | Robust Sampled-data Fuzzy Control for Nonlinear Systems and Its Applications: Free-Weight Matrix Method[END_REF]. Also, from the proposed looped LKF, the closed-loop

Lyapunov stability conditions leads to a quadratic inequality (second order polynomial in τ (t)), whose negativeness is handled by some convexity conditions [START_REF] Zhu | Sampled-Data Fuzzy Stabilization of Nonlinear Systems Under Nonuniform Sampling[END_REF], which are somewhat conservative. In a recent work (de Oliveira and Souza, 2020), some necessary and sufficient conditions have been proposed to cope with such quadratic constraints. However, we intended to apply them in our context but, with our sampled-data design approach, we faced sparsity in the obtained LMI-based constraints, and so an unsolvable convex optimization problem. Once again, following this way with more scrutiny can be interesting to further reduce the conservatism. Another point which can merit further refinement is the gain scheduled event-triggering mechanism proposed to enlarge the estimation of the guaranteed closed-loop domain of attraction. Indeed, noticing that the Lyapunov level set is found inside a compact set of the state space defining the validity domain of the T-S model-based design, further improvements can be obtained by optimizing this domain of validity for each scheduling conditions. This point will be the subject of our next investigations. Finally, in the prospect of future practical applications, the proposed sampled-data controller design methodology can be extended to cope with external disturbances, for instance by considering a H ∞ criterion (see e.g. [START_REF] Kim | Decentralized sampled-data fuzzy controller design for a VTOL UAV[END_REF][START_REF] Bourahala | Improved robust H ∞ stability analysis and stabilization of uncertain systems with stochastic input time-varying delays[END_REF], by considering static output-feedback controllers [START_REF] Bouarar | Robust non-quadratic static output feedback controller design for Takagi-Sugeno systems using descriptor redundancy[END_REF], extending recently proposed event-triggering approaches [START_REF] Rouamel | Mixed Actual and Memory Data-based Event-Triggered H ∞ Control Design for Networked Control Systems[END_REF]

Traduction en Français de la Conclusion Generale et des Perspectives:

Au sein de ce manuscrit de thèse de doctorat, des conditions de synthèse à base de LMIs relâchées ont été proposées pour la stabilisation d'une classe de systèmes non linéaires décrits en temps continu, représentés par des modèles Takagi-Sugeno, et contrôlés par des dispositifs numériques. Dans le Chapitre 1, les préliminaires sur la synthèse de lois de commande pour les modèles T-S ont été présentés, ainsi que les différentes façons d'obtenir une représentation T-S. En outre, conformément à l'objectif principal de cette thèse, des notions de base sur la discrétisation des modèles T-S en temps continu ont été présentées, ainsi que les bases sur la stabilité et les conditions LMIs de synthèse des contrôleurs, dans les cadres continu ou discret. Ensuite, certaines limites de l'approche standard à temps discret ont été soulignées et l'approche dite "à retard sur l'entrée" pour la synthèse de contrôleurs échantillonnés pour les systèmes continus a été introduite comme une alternative élégante pour faire face à de tels problèmes. Encore une fois, les principales limites de cette approche dans le cadre des modèles T-S, comme l'obtention de résultats locaux ou le conservatisme des conditions de synthèse, ont été soulignées. En résumé, l'analyse critique présentée dans ce chapitre a permis la maturation de l'énoncé des problèmes considérés dans cette thèse, qui peut être classée en deux lignes de recherche présentées comme suit.

Tout d'abord, reconnaissant que les approches basées sur des modèles à temps discret sont principalement considérées dans les applications industrielles en raison de leur polyvalence et de leur facilité de mise en oeuvre, certaines améliorations pour les mo dèles T-S discret ont été proposées dans le Chapitre 2. L'objectif était de fournir une méthodologie d'optimisation convexe pour concevoir des contrôleurs discrets de type PI. La principale contribution du Chapitre 2 est d'etendre nos résultats préliminaires [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF], en incluant une action anti-windup pour s'affranchir de l'hypothèse d'une variation lente de l'état et, en conséquence, fournir des estimations moins conservatives de la région d'attraction, ainsi que des changements d'amplitude admissibles des point de consigne. Dans ce contexte, l'approche proposée permet d'estimer l'amplitude maximale de signaux exogènes afin que les trajectoires du système en boucle fermée restent dans la région d'attraction, assurant une stabilité asymptotique locale grâce à une fonction de Lyapunov non quadratique. En outre, la loi de commande floue de type PI proposée adopte une structure PDC alors que le gain de l'action anti-windup adopte une structure non-PDC, qui peuvent être efficacement mis en oeuvre dans les applications industrielles embarquées [START_REF] Sousa | Affordable control platform with MPC application[END_REF]. L'approche proposée a été validée en simulation et expérimentalement à partir du contrôle de niveau d'un système de deux réservoirs non linéaires couplés disponible au CEFET-MG, avec des améliorations significatives vis-à-vis de résultats préliminaires présentés dans [START_REF] Lopes | On the integral action of discrete-time fuzzy T-S control under saturated actuator[END_REF].

Si l'approche de synthèse du contrôleur basé sur un modèle T-S à temps discret a été envisagée en raison de sa simplicité conceptuelle attrayante, en particulier lorsque la mise en oeuvre du contrôleur pour des applications industrielles est nécessaire, il a également été souligné que cette approche n'est appropriée que lorsqu'une période d'échantillonnage suffisamment petite et fixe peut être déterminée au regard de la dynamique continue du système à commander. Toutefois, il faut rappeler que l'hypothèse de périodes d'échantillonnage fixes est parfois irréaliste dans la pratique. En effet, même lorsqu'une topologie de contrôle numérique point-à-point est considérée, l'échantillonnage apériodique peut survenir en raison de l'inexactitude de l'horloge et/ou des caractéristiques de l'architecture du système telles que l'ordonnancement en temps réel, ce qui peut induire une synchronisation imparfaite ou des retards de calcul [START_REF] Wittenmark | Timing problems in real-time control systems[END_REF][START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. En outre, lorsqu'il s'agit de systèmes contrôlés au travers d'un réseau de communication, la considération d'intervalles d'échantillonnage apériodique sont souvent inévitables [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]Fridman, 2014a). Néanmoins, on peut également faire valoir que, lorsque de grandes périodes d'échantillonnage doivent être prises en compte, l'approche directe fondée sur un modèle à temps discret peut ne pas garantir la stabilité en boucle fermée du système continu entre les instants d'échantillonnage [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. Pour contourner ces inconvénients, l'approche alternative de la Synthèse continueéchantillonnée a été étudiée dans les Chapitres 3 et 4 pour la stabilisation des modèles T-S continus. À partir de cette approche, la dynamique non linéaire en boucle fermée est réécrite comme un système T-S à temps continu avec un retard variable sur l'entrée, tel que proposé pour les systèmes linéaires, e.g. [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF][START_REF] Seuret | Stability analysis for sampled-data systems with a time-varying period[END_REF].

Associé au fait que la plupart des applications réelles présentent une dynamique non linéaire, dans les chapitres susmentionnés, une extension de cette approche pour le cadre des modèle T-S est présentée, et la réduction du conservatisme est vérifiée en recherchant l'intervalle d'échantillonnage maximal autorisé [0, η], pour lequel la stabilité dynamique en boucle fermée est garantie.

Dans le Chapitre 3, notre première contribution à la synthèse des contrôleurs échantillonnés pour les modèles T-S continus, basée sur l'approche de la Synthèse continueéchantillonnée, a été présentée. Dans la continuité des travaux existants, des conditions à base de LMIs relâchées ont été proposées pour la synthèse de contrôleurs à échantillonnage apériodique stabilisant les systèmes T-S décrits en temps continu. Pour atteindre cet objectif, une LKF "bouclée" augmentée a été sélectionnées, associée à l'application de lemmes de majoration, tels que des inégalités étendues de Jensen, le Lemme de Finsler et des contraintes polynomiales quadratiques convexes. En outre, un schéma de relaxation générique a été proposé. Celui-ci permet l'extension du célèbre Lemme de Tuan, pour tenir compte de la structure en double somme des conditions de stabilité en boucle fermée, dépendantes de paramètres asynchrones. Similairement aux précédentes études pour la commande échantillonnée des modèles T-S continus, la réduction du conservatisme est étudiée dans ce chapitre en cherchant l'intervalle d'échantillonnage maximal autorisé [0, η], pour lequel les conditions de synthèse demeurent faisables. L'avantage de la méthode de Synthèse continue-échantillonnée vis-à-vis de la méthode de Synthèse Directe à base de modèles T-S discrets a été illustré par la simulation d'un modèle de pendule inversé à un degré de liberté. Ensuite, une comparaison a été fournie, montrant un élargissement significatif de l'intervalle d'échantillonnage maximal autorisé par rapport à plusieurs résultats connexes précédents [START_REF] Yoneyama | Robust H ∞ control of uncertain fuzzy systems under time-varying sampling[END_REF][START_REF] Zhu | Stabilization for sampled-data neural network-based control systems[END_REF][START_REF] Zhang | H ∞ control design for network-based T-S fuzzy systems with asynchronous constraints on membership functions[END_REF][START_REF] Zhu | An Improved Input Delay Approach to Stabilization of Fuzzy Systems Under Variable Sampling[END_REF][START_REF] Gunasekaran | Robust Sampled-data Fuzzy Control for Nonlinear Systems and Its Applications: Free-Weight Matrix Method[END_REF][START_REF] Zhu | H ∞ stabilization criterion with less complexity for nonuniform sampling fuzzy systems[END_REF][START_REF] Cheng | H ∞ stabilization for sampling fuzzy systems with asynchronous constraints on membership functions[END_REF], en utilisant l'exemple d'un modèle T-S approximé d'un pendule inversé sur un chariot. Enfin, une validation expérimentale de la méthodologie de Synthèse continue-échantillonnée proposée est effectuée sur la plate-forme AERO de Quanser ® (Quanser, 2016), illustrant son efficacité sur un système réel. Si les résultats de ce chapitre se sont révélés efficaces pour réduire le conservatisme des conditions de synthèse de correcteur en élargissant l'intervalle d'échantillonnage maximal admissible, il est également souligné que cette approche souffre de n'être valide que localement, ce qui justifie des investigations plus approfondies du domaine d'attraction en boucle fermée, comme proposé dans le chapitre suivant.

Dans le Chapitre 4, des conditions relâchées de Synthèse continue-échantillonnée pour la classe de systèmes T-S descripteurs réguliers, soumis à la saturation des actionneurs, ont été proposées. En effet, en dépit des modèles T-S standard, cette classe de systèmes descripteurs est souvent considérée comme représentant de façon plus précise une plus grande classe de systèmes non linéaires, en particulier pour les systèmes mécatroniques, où le contrôle à base de données échantillonnées est une caractéristique inhérente. Le défi de ce chapitre consiste donc à fournir des conditions LMIs relâchées pour la Synthèse continue-échantillonnée des modèles T-S descripteurs soumis à saturation sur l'entrée, en étendant les conditions proposées dans le Chapitre 3, ainsi qu'en proposant une analyse minutieuse du domaine d'attraction en boucle fermée. Tout d'abord, pour la classe de descripteurs T-S soumis à la saturation sur l'entrée, une procédure hors ligne a été proposée pour estimer le domaine d'attraction en boucle fermée qui résulte de la Synthèse continue-échantillonnée. Ensuite, pour le cas particulier des modèles T-S standards, une procédure d'optimisation systématique à base de contraintes LMI a été proposée pour réaliser une telle estimation. Cependant, il a été souligné que cette estimation systématique peut apporter du conservatisme (voir Remarque 4.6), et il faut donc l'utiliser avec précaution, pour des classes particulières de modèles T-S comportant un faible nombre de sommets ou des fonctions d'appartenance triangulaires. En outre, parce que, pour une limite supérieure donnée des intervalles d'échantillonnage, l'estimation obtenue du domaine d'attraction peut être relativement restreinte, une nouvelle méthodologie d'ajustement des paramètres de la loi de commande est proposée. Celle-ci exploite les principes du déclenchement par événements (event-triggering), selon les équipotentielles des fonctions de Lyapunov obtenues pour plusieurs intervalles d'échantillonnage, permettant ainsi d'étendre plus encore l'estimation obtenue du domaine d'attraction en boucle fermée. Tout au long de ce chapitre, les résultats ont été validés en simulation au travers de l'exemple d'un pendule inversé, d'abord, avec le même modèle T-S approximé que celui pris en compte dans le Chapitre 3, afin de mettre en évidence l'amélioration du conservatisme et d'illustrer l'approche systématique proposée dans le Chapitre 4. Ensuite, comme il a été montré que les contrôleurs conçus à partir d'un modèle T-S approché peuvent ne pas stabiliser le modèle non linéaire complet, une modélisation exacte sous la forme d'un modèle T-S descripteur a été proposée. Le contrôleur à temps discret ainsi conçu a permis de stabiliser avec succès le modèle non linéaire continu précis du pendule inversé, mais avec une estimation du domaine d'attraction en boucle fermée relativement petite pour l'intervalle obtenu d'échantillonnage maximal autorisé. Par conséquent, en élargissant davantage l'estimation du domaine d'attraction, l'avantage du mécanisme proposé pour l'ajustement des paramètres de la loi de commande est illustré.

En résumant notre contribution dans le domaine de Synthèse continue-échantillonnée pour les modèles T-S, soulignons que, à la connaissance de l'auteur, très peu de résultats antérieurs portant sur les points suivants existent :

• la synthèse de contrôleurs basés sur des données échantillonnés apériodiques pour les modèles T-S avec saturation des actionneurs ; excepté le travail de [START_REF] Lamrabet | Sampled data Control for Takagi-Sugeno Fuzzy Systems with Actuator Saturation[END_REF], où aucune considération n'est portée sur l'estimation du domaine d'attraction en boucle fermée,

• l'estimation du domaine d'attraction en boucle fermée et ses procédures d'optimisation, obtenue grâce à la prise en compte d'une LKF "bouclée" adéquate ; à l'exception des très récents articles publiés par [START_REF] Coutinho | Local Sampled-Data Gain-Scheduling Control of quasi-LPV Systems[END_REF][START_REF] Ma | Exponential stabilization of sampled-data fuzzy systems via a parameterized fuzzy Lyapunov-Krasovskii functional approach[END_REF] à la fin du déroulement de cette thèse, et qui fournissent des résultats intéressants qui mériteraient d'être approfondis et comparés avec notre proposition,

• l'extension de la méthodologie de Synthèse continue-échantillonnée pour la classe des modèles T-S descripteurs, qui s'avère intéressante pour traiter de la commande des systèmes mécatroniques.

Tout au long de ce manuscrit, les simulations proposées et les résultats expérimentaux ont permis d'illustrer le mérite des approches proposées de commande échantillonnée pour les modèle T-S, tant dans le cadre du temps discret que dans le cadre du temps continu. Cependant, les approches proposées présentent encore certaines limitations, qui peuvent être discutées pour fournir des perspectives de ce travail.

En ce qui concerne l'approche de Synthèse Directe pour les systèmes décrits en temps discret, même si une abondante littérature existe déjà, il y a encore des points d'intérêt qui retiennent notre attention pour de futures améliorations. Tout d'abord, comme de nombreuses études précédentes dans le contexte du temps discret, nous considérons la discrétisation d'un modèle T-S en temps continu via la célèbre transformation d'Euler [START_REF] Chen | Linear System Theory and Design[END_REF]. Cependant, la discrétisation conduisant à des approximations, des améliorations sont toujours possibles. Dans ce contexte, il existe d'autres approches de discrétisation, comme la transformation bilinéaire de Tustin [START_REF] Ogata | Discrete-Time Control Systems[END_REF][START_REF] Åström | Computer-Controlled Systems: Theory and Design[END_REF], en utilisant des décompositions en série de Taylor ou le Théorème de Caley-Hamilton [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF]. Par conséquent, une perspective directe serait d'évaluer l'amélioration potentielle de ces approches avec les conditions de Synthèse Directe proposé dans le Chapitre 2, en particulier dans l'optique de leur application à des systèmes non linéaires impliquant des dynamiques continues plus rapides, ou dans le but d'élargir davantage l'estimation résultante du domaine d'attraction en boucle fermée. Ensuite, une autre perspective qui n'a pas été étudiée au cours de ce travail de doctorat est le déclenchement d'événements dans le cadre du temps discret, e.g. [START_REF] De Souza | Co-design of an eventtriggered dynamic output feedback controller for discrete-time LPV systems with constraints[END_REF]. Une telle approche pourrait être intéressante pour certaines applications industrielles où des périodes d'échantillonnage rapides peuvent être définies vis-à-vis de la dynamique du système à contrôler et, le cas échéant, d'atténuer la charge dans le cas de la commande en réseau. En outre, dans le Chapitre 2, on a considéré la Synthèse Directe de contrôleurs discrets de type PI tout en étudiant les effet des changements de points de consigne (i.e. trajectoires désirées constantes par morceaux) sur l'estimation du domaine d'attractions en boucle fermée. Une perspective très intéressante serait de proposer une extension au cas plus général de suivi de trajectoire dynamique, mais dans ce cas l'étude du domaine d'attraction en boucle fermée serait beaucoup plus complexe puisque son estimation devrait évoluer dynamiquement par rapport aux trajectoires souhaitées, et ceci à chaque instant d'échantillonnage. Bien entendu, de nombreuses autres extensions des stratégies proposées dans le cadre de la synthèse en temps discret peuvent être réalisées, par exemple pour le retour de sortie, pour les NCS avec des retards induits par le réseau, des pertes de paquets, des cyber-attaques, etc. Á présent, portons notre attention sur l'approche de Synthèse continue-échantillonnée proposée pour la stabilisation des modèles quasi-LPV/T-S en temps continu. Si les résultats présentés dans le Chapitre 3 et le Chapitre 4 ont démontré l'attrait de la stratégie proposée, il faut également admettre sa principale limitation qui consiste en une complexité et un coût de calcul élevé des conditions à base de LMIs. En effet, comme dans de nombreuses tentatives pour réduire le conservatisme des conditions de synthèse pour les modèle T-S, fournir des conditions LMIs relâchées entraîne généralement une augmentation du coût de calcul. Cela est particulièrement vrai dans le cas de la Synthèse continue-échantillonnée, car une façon importante de réduire le conservatisme vient de la stratégie employée pour faire face à la structure de double somme asynchrone. À cette fin, nous avons proposé une extension du célèbre Lemme de Tuan [START_REF] Tuan | Parameterized Linear Matrix Inequality Techniques in Fuzzy Control System Design[END_REF], mais qui résulte en une augmentation de la taille des LMIs. Cela peut être considéré comme un inconvénient, mais cela permet également de réduire le nombre de contraintes LMIs. De plus, pour réduire encore le conservatisme, nous avons proposé des fonctions Lyapunov-Krasovskii "bouclées" pour la synthèse de contrôleurs échantillonnés, associées à l'utilisation de lemmes de majoration usuels (inégalité étendue de Jensen, Lemme de Finsler...), qui introduisent plusieurs variables de décision libres. Dans l'espoir de réduire la complexité des calculs, nous pouvons envisager, comme première perspective de ce travail, de réduire le nombre de variables de décision (en particulier les matrices de décision libres) tout en tenant compte de certaines techniques plus récentes proposées pour l'analyse des systèmes avec retards, comme par exemple l'utilisation des inégalités de Wirtinger ou de Bessel-Legendre [START_REF] Seuret | Wirtinger-based integral inequality: Application to timedelay systems[END_REF][START_REF] Ariba | Stability analysis of time-delay systems via Bessel inequality: A quadratic separation approach[END_REF][START_REF] Seuret | Stability of Linear Systems With Time-Varying Delays Using Bessel-Legendre Inequalities[END_REF][START_REF] Bourahala | Relaxed Non-Quadratic Stability Conditions for Takagi-Sugeno Systems with Time-Varying Delays: A Wirtinger's Inequalities approach[END_REF][START_REF] Nafir | Improved robust H ∞ stability analysis and stabilisation of uncertain and disturbed networked control systems with network-induced delay and packets dropout[END_REF]. Bien entendu, les coûts de calcul élevés induits par des conditions LMIs complexes peuvent être considérés comme un inconvénient pour ce type d'études. Néanmoins, soulignons que de tels calculs sont effectués hors ligne. De plus, nous pouvons envisager que cette préoccupation peut être atténuée par la croissance continue des capacités de calcul de nos ordinateurs personnels ou, lorsqu'ils sont disponibles, de supercalculateurs tel que le centre de calcul ROMEO de l'Université de Reims Champagne-Ardenne. Pour illustrer ce point, notons qu'avec le modèle T-S approximé du pendule inversé considéré dans la section 4.4.1, il a fallu 12, 97 secondes pour résoudre les conditions du Théorème 4.2 (nous avons utilisé pour ce test un ordinateur portable HP datant de fin 2016 avec un processeur Intel Core I7 de 2, 6 GHz et 16 Go de mémoire RAM). Aussi, pour résoudre les conditions du Théorème 4.1 avec le modèle complet sous la forme T-S descripteur du pendule inversé (donné dans la section 4.4.2), la charge de calcul était beaucoup plus importante, i.e. une moyenne de 45 minutes par test, avec le même ordinateur.

De nombreux autres points mériteraient une certaine attention pour rêlacher davantage les conditions LMIs proposées pour la synthèse du contrôleur échantillonné pour les mo dèles T-S. Tout d'abord, afin de tenir compte de la structure en double somme asynchrone des conditions LMIs dépendantes des paramètres, nous avons proposé une extension du Lemme de Tuan, résumée dans le Théorème 3.2. Celle-ci est basée sur l'application du Lemme de Peterson [START_REF] Xie | Output feedback H ∞ control of systems with parameter uncertainty[END_REF], qui peut être encore relâchée, par exemple en consi dérant ses extensions proposées dans [START_REF] Briat | Commande et Observation Robustes des Systèmes LPV Retardées[END_REF]. En outre, notre approche implique une augmentation de la taille des conditions LMIs à vérifier, et affecterait donc éventuellement le coût de calcul ou le conservatisme pour les systèmes de grande taille. Dans ce contexte, certaines mesures de relaxation seraient les bienvenues, par exemple en envisageant l'introduction de plus d'informations sur les fonctions d'appartenance, telle que l'approche récemment proposée pour faire face aux limites de leurs dérivés temporelles [START_REF] Gunasekaran | Robust Sampled-data Fuzzy Control for Nonlinear Systems and Its Applications: Free-Weight Matrix Method[END_REF]. De plus, à partir des LKF "bouclée" proposées, les conditions de stabilité en boucle fermée conduisent à une inégalité quadratique (polynôme de second ordre en τ (t)), dont la négativité est traitée par des conditions de convexité [START_REF] Zhu | Sampled-Data Fuzzy Stabilization of Nonlinear Systems Under Nonuniform Sampling[END_REF], qui peuvent s'avérer quelque peu conservatives. Dans une étude récente (de Oliveira and Souza, 2020), des conditions nécessaires et suffisantes ont été proposées pour faire face à de telles contraintes quadratiques. Cependant, nous avions l'intention de les appliquer dans notre cadre théorique, mais nous nous sommes heurtés à l'obtention de contraintes LMIs "creuses", et qui ont donc conduit à un problème d'optimisation convexe insoluble. Encore une fois, poursuivre nos efforts pour résoudre ce problème pourrait s'avérer très intéressant afin de réduire davantage le conservatisme. Un autre point qui mérite d'être affiné est le mécanisme de déclenchement d'événements programmé proposé pour augmenter l'estimation du domaine d'attraction en boucle fermée. En effet, remarquant que l'ensemble des équipotentielles de Lyapunov se trouve à l'intérieur d'un ensemble compact de l'espace d'état définissant le domaine de validité du modèle T-S, d'autres améliorations peuvent être obtenues en optimisant ce domaine de validité pour chaque condition de synthèse. Ce point fera l'objet de nos prochains travaux. Enfin, dans la perspective de futures applications pratiques, la méthodologie proposée pour la synthèse de contrôleurs échantillonnés peut être étendue pour faire face aux perturbations externes, par exemple en considérant un critère H ∞ (e.g. [START_REF] Kim | Decentralized sampled-data fuzzy controller design for a VTOL UAV[END_REF][START_REF] Bourahala | Improved robust H ∞ stability analysis and stabilization of uncertain systems with stochastic input time-varying delays[END_REF], en considérant le retour de sortie statique [START_REF] Bouarar | Robust non-quadratic static output feedback controller design for Takagi-Sugeno systems using descriptor redundancy[END_REF], en étendant les approches de déclenchement d'événements que nous avons récemment proposées [START_REF] Rouamel | Mixed Actual and Memory Data-based Event-Triggered H ∞ Control Design for Networked Control Systems[END_REF] au cas des modèle T-S ou, de manière similaire à ce qui a été proposé dans le Chapitre 2, en étendant les conditions de Synthèse continue-échantillonnée pour des contrôleurs échantillonnées de type PI. Cependant, pour ce dernier point, une première investigation de l'auteur a soulevé certains verrous qui doivent encore être débloquées, en particulier pour traiter l'hybridation du temps nécessaire pour implémenter les actions intégrales et/ou anti-windup sur des dispositifs de commande numérique.

Ceci étant dit, à partir de ces perspectives sur la Synthèse continue-échantillonnée de contrôleur échantillonné pour la stabilisation des systèmes non linéaires décrits en temps continu et représentés par des modèles T-S, la porte est maintenant grande ouverte pour un large éventail d'autres développements.

Contributions à la Synthèse de Contrôleurs Échantillonnés pour les Modèles Takagi-Sugeno

Ce travail étudie la stabilisation des systèmes non linéaires à temps continu pilotés par des dispositifs numériques, dans le but de relâcher les conditions de synthèse et d'améliorer l'estimation du domaine d'attraction en boucle fermée pour des intervalles d'échantillonnage aussi grands que possible. Tout d'abord, la commande des systèmes non linéaires soumis à la saturation des actionneurs, représentés par des modèles Takagi-Sugeno (T-S) à temps discret avec des contraintes sur l'entrée, est étudiée. L'accent est mis sur la conception de contrôleurs de type PI en temps discret avec une compensation anti-windup pour suivre des points de consigne constants par morceaux. Parallèlement, une procédure d'optimisation est proposée pour l'élargissement du domaine d'attraction garanti en boucle fermée. Ensuite, pour contourner certains inconvénients des approches basées sur un modèle à temps discret, tels que les approximations du modèle et l'exigence d'un intervalle d'échantillonnage fixe, l'approche consistant à réécrire la dynamique en boucle fermée sous la forme d'un système continu à retard variable sur l'entrée est considérée. Par conséquent, à partir d'une fonction de Lyapunov-Krasovskii (LKF) bouclée et l'application de lemmes de majoration, de nouvelles conditions LMI relâchées sont proposées pour la synthèse de contrôleurs échantillonnées stabilisant les modèles T-S à temps continu. Cependant, dans ce contexte, des fonctions d'appartenance asynchrones apparaissent dans la formulation de la boucle fermée et donc dans les conditions de synthèse, qui ne peuvent être traitées avec les schémas de relaxation usuels. Pour faire face à ce problème, une extension du Lemme de Tuan est proposée. Par la suite, l'approche de contrôle échantillonnées proposée est étendue à la classe plus générale des descripteurs T-S décrits en temps continu et soumis à la saturation des entrées. En soulignant que, dans le cadre des modèles T-S à temps continu, les contrôleurs échantillonnées conçus ne garantissent que localement la stabilité en boucle fermée, une analyse minutieuse du domaine d'attraction en boucle fermée est proposée. En outre, après avoir remarqué que, pour de larges intervalles d'échantillonnage admissibles, l'estimation résultante du domaine d'attraction en boucle fermée peut être assez petite, un nouveau mécanisme d'ajustement de la loi de commande par déclenchement d'événement est proposé pour élargir davantage cette estimation. Tout au long du manuscrit, des simulations et des résultats expérimentaux établissent le mérite des approches proposées pour la commande échantillonnée des modèles T-S.

Modèles Takagi-Sugeno, Commande échantillonnée, Synthèse LMI, Saturation des actionneurs, Stabilisation locale.

Contributions to the Design of Sampled-Data Controllers for Takagi-Sugeno Models

This work investigates the stabilization of continuous-time nonlinear systems driven by digital devices, with the goal of relaxing the design conditions and enhancing the estimation of the closed-loop domain of attraction for largest as possible sampling intervals. First, the control of nonlinear systems subject to actuators saturation, represented by discrete-time Takagi-Sugeno (T-S) models with input constraints, is investigated. The focus relied on the design of discrete-time PI-like controllers with an anti-windup compensation to track piece-wise constant set-points. Meanwhile, an optimization procedure is proposed for the enlargement of the guaranteed closed-loop domain of attraction. Then, to circumvent some drawbacks of the discrete-time model-based approaches, such as model approximations and the requirement of a fixed sampling interval, the input-delay approach for sampled-data systems is investigated. Therefore, from a convenient looped Lyapunov-Krasovskii Functional (LKF) and the application of some bounding lemmas, new relaxed LMI-based conditions are proposed for the design of sampled-data controllers to stabilize continuous-time T-S models. However, in this context, asynchronous membership functions arise from the closed-loop formulation and so from the design conditions, which cannot be handle with usual relaxation schemes. Hence, to cope with such issue, an extension of the well known Tuan's Lemma is proposed in this context of sampled-data control for continuous-time T-S models. Furthermore, the proposed sampled-data control approach is extended to the more general class of T-S descriptors subject to input saturation. Highlighting that, in the continuous-time T-S model-based framework, the designed sampled-data controllers only locally guarantee the closed-loop stability, a careful analysis of the closed-loop domain of attraction is proposed. Also, after noticing that, for large admissible sampling intervals, the resulting estimation of the closed-loop domain of attraction may be quite small, a new gain-scheduled event-triggering mechanism is proposed to further enlarge such estimation. Along the manuscript, simulations and experimental results establish the merit of the proposed approaches for the sampled-data control of T-S models.

  Represents the matrix 0 n×(j-1)n I n×n 0 n×(p-j)n ∈ R n×pn , ∀j ∈ I p diag{M 1 , . . . , M n } Denotes a block diagonal matrix.sign(x)The sign function of x ∈ R rank(M )The rank of a matrix M H(M ) Stands for M + M T col{a 1 , . . . , a n } Denotes a column vector with scalar entries ⋆ Stands for symmetric blocks in the expression of a matrix iii French abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Considered problem statement for T-S model-based sampled-data control design 3.3 LMI-based sampled-data controller design . . . . . . . . . . . . . . . . . . . . . . 3.4 Relaxation scheme for T-S model-based sampled-data controller design . . . . . . 3.5 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 T-S model-based sampled-data controller design vs conventional discretetime approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Conservatism comparison: Inverted pendulum on a cart benchmark . . . 3.5.3 Linear case study with experimental validation . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Local Aperiodic Sampled-data Control of T-S Descriptors with Input Constraints French abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Considered class of systems and problem statement . . . . . . . . . . . . . . . . . 4.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3.1 LMI-based local non-quadratic sampled-data controller design for T-S descriptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Systematic estimation of the domain of attraction for standard T-S modelsbased sampled-data controller design . . . . . . . . . . . . . . . . . . . . . 4.3.3 Gain scheduled event-triggering mechanism to enlarge the closed-loop domain of attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Benchmark of the approximated fuzzy model of the inverted pendulum . 4.4.2 Matching T-S descriptor model-based sampled-data controller design . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Conclusion and Perspectives French Translation of the General Conclusion and Perspectives . . . . . . . . . . diagram of a continuous-time plant controlled by a discretetime state-feedback controller processed on a digital device. . . . . . . . . . . . . 1.1 Global sector nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Local sector nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 1-DOF Inverted Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Open-loop response for discrete-time fuzzy T-S models taken from different sampling periods τ s compared with the continuous-time version. . . . . . . . . . . . . 1.5 Comparison between the time response of the closed-loop system regarding its discrete-time fuzzy T-S model considering different sampling periods τ s together with the normalized error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Saturation and dead-zone nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Representation of the time-varying delay (τ (t) = t -t k ) with a sawtooth function (o marks the left-limits of right-continuous piecewise functions). . . . . . . . . . . 1.8 Illustration of a decreasing looped LKF. . . . . . . . . . . . . . . . . . . . . . . . 1.9 Mismatch between controller's and T-S model's membership function. . . . . . . 2.1 Control topology with discretized integral and anti-windup actions. . . . . . . . . 2.2 Region of attraction achieved with Corollary 2.1 and the optimization procedure

  , • (Wang et al., 2019)). . . . . . . . . . . . . . . . . . . . . . 2.3 Interactive Tank System used in the experiments available at the CEFET-MG, Divinópolis, Brazil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Coupled Tank System Schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Open-loop validation of the discrete-time T-S model in simulation from data measurements on the real system. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Estimate of the regions of attraction D * a for the stabilization problem P 1 obtained from Theorem 2.1 (•) and Corollary 2.1 (•). . . . . . . . . . . . . . . . . . . . . . vii 2.7 Estimate of the regions of attraction D * a for the set-point tracking problem P 2 obtained from Theorem 2.1 (•) and Corollary 2.1 (•); particular set-points tracking trajectory (•). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Time responses obtained with the PDCaw controller. . . . . . . . . . . . . . . . . 2.9 Time responses obtained with the PDC controller. . . . . . . . . . . . . . . . . . 3.1 Time evolution of the continuous-time model (1.2), driven by the fuzzy discretetime (DT) PDC controller proposed in Chapter 2 and with the Sampled-Data (SD) PDC controller projected from Theorem 3.1, with η = τ s = 228ms. . . . . . 3.2 Discrete-time state-feedback control input signal u(t k ) and Lyapunov function x T (t)Lx(t) computed for the CT system and DT model. . . . . . . . . . . . . . . 3.3 Inverted Pendulum mounted on a cart. . . . . . . . . . . . . . . . . . . . . . . . .

3. 6

 6 Simulation of the states trajectories of the inverted pendulum on a cart with the respectively sampled input control signal computed for variable sampling intervals η k ∈ [0, η]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Simple free-body diagram of a 2-DOF helicopter system. . . . . . . . . . . . . . . 3.9 Time response of Quanser ® AERO model under the Sampled Data Controller. . . 3.10 Time response of Quanser ® AERO model, subjected to the LQR Controller (3.60) (-) and the PD Controllers (3.59) (• • • ) under a sampling period of 4.5s. . . . . . 3.11 Comparison of the time responses of the Quanser ® AERO under a sampling period of 150ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Time responses of the Quanser ® AERO under triggered sampling period. . . . . . 4.1 Simulation of the state trajectories of the inverted pendulum on a cart with the designed sampled controller, together with the respective saturated input control signal, updated at fixed sampling instants η k = η = 50ms. . . . . . . . . . . . . . 4.2 Estimates of the region of attraction for two sampled-data controllers designed from Theorem 4.2 to stabilize the T-S model (4.75), assuming different actuators' saturation limits (ū), with its resulting closed-loop trajectories for several initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 4.3 Estimates of the Region of Attraction for the maximal allowed sampling interval found from Theorem 4.1, (η = 50ms) assuming various actuators' saturation limits, (i. e. considering different values for ū). . . . . . . . . . . . . . . . . . . . 4.4 Estimates of the Region of Attraction for a sampling interval η = 1ms found from Theorem 4.2, assuming various actuators' saturation limits, (i. e. considering different values for ū). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Estimates of the Region of Attraction for several maximum allowable sampling interval η found from Theorem 4.2, assuming ū = 10000N . . . . . . . . . . . . . . 4.6 Estimates of the Region of Attraction D * a (highlighted in gray), control signal and selected sampling period η k for the gain scheduled event-triggering sampled-data controller proposed in (4.74) and designed from Theorem 4.2 for the 2 rules T-S model with a maximal allowed sampling interval η = 50ms. . . . . . . . . . . . . 4.7 Estimates of the Region of Attraction D * a (highlighted in gray), control signal and selected sampling period η k for the gain scheduled event-triggering sampled-data controller proposed in (4.74) and designed from Theorem 4.2 for the 2 rules T-S model with a maximal allowed sampling interval η = 35ms. . . . . . . . . . . . . 4.8 Plots of w 1 3 (x), w 1 4 (x), w 2 4 (x) and w 2 4 (x) for x 1 ∈ D x illustrating that w 1 3 (x) closely matches w 1 4 (x) (and so w 2 3 (x) and w 2 4 (x)). . . . . . . . . . . . . . . . . . . . . . . 4.9 (a) Estimated domain of Attraction D * A for the full closed-loop nonlinear model (3.52) under the designed sampled-data controller designed from the T-S descriptor (4.1) (with (4.76)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 (a) Estimated domain of Attraction D * A for the full closed-loop nonlinear model (3.52) under the sampled-data controller designed from the T-S descriptor (4.1) (with (4.76)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Estimates of the Region of Attraction D * a (highlighted in gray) for the gain scheduled event-triggering sampled-data controller proposed in (4.74) and designed from Theorem 4.2 for the descriptor (4.1) system with (4.76) and the maximal allowed sampling interval η = 8ms. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Closed-loop state responses and overall behaviour of the successively selected Lyapunov functions of the nonlinear model (3.52) under the sampled-data controller design from the T-S descriptor (4.1) (with (4.76)). . . . . . . . . . . . . . . . . . ix

D

  is cr e te -T im e F u zz y C o n tr o lle r Con tinuo us-T ime Non linea r Plan t Digital to Analog Converter with a Zero-Order Hold Analog to Digital Converter

Figure 1 :

 1 Figure 1: Illustration and block diagram of a continuous-time plant controlled by a discrete-time state-feedback controller processed on a digital device.

Further, in the

  specific context of T-S sampled-data controllers, due to the time hybridization of the closed-loop dynamics, asynchronous membership functions arise. This aspect kept the attention of many authors, who have proposed continuous relaxation improvements for the design of sampled-data controllers for T-S systems. For instance, in (Yoneyama, 2010), Lyapunov-Krasovskii Functionals (LKF) and relaxation techniques based on the Leibniz-Newton formula and free-weighting matrix were considered. Moreover, note that the delayed membership functions involved in the fuzzy controller are mismatching the ones existing in the T-S representation of the continuous-time plant. Hence, to handle this issue, several attempts have been proposed: the upper bounds of the asynchronous errors of the membership functions have been assumed

these ideas, in Chapter 3 , 3 ,

 33 from the choice of a convenient looped LKF and the application of the Finsler's Lemma, new LMI conditions are obtained for the design of sampled-data PDC controllers, together with the proposition of a dedicated relaxation scheme for asynchronous double summation structures. Then, these results are extended in Chapter 4. From a looped Non-Quadratic Lyapunov-Krasovskii Functional (NQLKF), relaxed controller design conditions for a class of regular nonlinear descriptors subject to actuators' saturation are proposed. Also, throughout this last chapter, it is highlighted that sampled-data control methodology is only valid locally in the T-S model based-framework. Indeed, there are three main sources of locality: the domain of validity of the T-S model, the assumption of the bounds for the time-derivatives of the membership functions to cope with the asynchronous membership functions (and also when considering a NQLKF), and the input constraints due to the actuators' saturation. In any of these cases, the characterization of an estimate of the closed-loop domain of attraction, where the closed-loop stability is guaranteed, is to be done with scrutiny. Particularly, to the best of the author's knowledge, this point of interest has been disregarded by previous authors in the context of stability analysis and controller design for sampled-data nonlinear systems expressed by T-S models, and is an important contribution brought by this work. Summarizing, the remaining of this manuscript is organized as follows. In Chapter 1, the preliminaries on T-S model-based control are provided to better apprehend the contribution of this PhD thesis. Chapter 2 presents the first contribution, which consists on the anti-windup T-S model-based PI-like set-point tracking control for nonlinear systems with saturated actuators, with enlargement procedure of the guaranteed closed-loop domain of attraction. Then, the input delay approach for the sampled-data stabilization of T-S models is proposed in Chapter where conservatism reduction is proposed from the selection of a convenient looped LKF and the proposition of a new relaxation scheme for asynchronous double fuzzy summation structures. These results are extended in Chapter 4 to the class of T-S descriptors subject to actuators' saturation, together with a meticulous investigation of the resulting sampled-data closed-loop domain of attraction. Finally, this manuscript ends with a general conclusion, where perspective of these works are discussed.Author's Publications Related to this WorkJournal papers:• Lopes, A. N. D., Leite, V. J. S., Silva, L. F. P. and Guelton, K. (2020). Anti-windup TS Fuzzy PI-like Control for Discrete-time Nonlinear Systems with Saturated Actuators, International Journal of Fuzzy Systems 22: 46-61. • Lopes, A. N. D., Guelton, K., Arcese, L. and Leite, V. J. S. (2021). Local Sampled-data Controller Design for T-S Fuzzy Systems with Saturated Actuators, IEEE Control Systems Letters 5(4): 1169-1174. Papers published in conferences proceedings: • Lopes, A. N. D., Guelton, K., Arcese, L. and Leite, V. J. (2021a). Sampled-data Controller Design for Mechatronic Systems Described by Takagi-Sugeno Descriptors, 2021 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM), pp. 653-660. • Lopes, A. N. D., Arcese, L. and Guelton, K. (2021). Synthèse non-quadratique de contrôleurs échantillonnés pour les modèles flous de type T-S décrits en temps continu, Rencontres Francophones sur la Logique Floue et ses Applications (LFA2021), Paris, France, pp. 117-124. • Rouamel, M., Bourahala, F., Lopes, A. N. D., Nafir, N. and Guelton, K. (2021). Mixed Actual and Memory Data-based Event-Triggered H ∞ Control Design for Networked Control Systems, 4th IFAC Conference on Embedded Systems, Computational Intelligence and Telematics in Control, pp. 1-6.

Figure 1 . 2 :

 12 Figure 1.1: Global sector nonlinearity.

Figure

  Figure 1.3: 1-DOF Inverted Pendulum.

Example 1. 3 .

 3 Consider the continuous-time T-S model of the 1-DOF inverted pendulum presented in (1.16). Assuming a sampling period τ s allows us to obtain a discrete-time T-S model

  ) with (1.21) and different values of τ s picked as 1ms, 5ms, 20ms and 50ms. The open-loop responses are depicted in Figure 1.4 to illustrate the compromise between accuracy and sampling effort. For that simulation, the system initial condition x(0) = π/6 0 T . To compare these first results, Figure 1.4 also plots the normalized error between the state variable θ(t) of the continuous-time response and its discrete counterpart θ(t k ) for each discrete-time model. It shows that the discrete-time fuzzy model with a sampling period of τ s = 50ms does not represent the inverted-pendulum dynamics around the equilibrium point θ = π, meanwhile, the choice of τ s = 20ms produced a closely-related behavior. However, seeing the zoom box, we can notice a kind of induced delay, probably related to the Zero-Order

Figure 1 Figure 1

 11 Figure 1.4: Open-loop response for discrete-time fuzzy T-S models taken from different sampling periods τ s compared with the continuous-time version.

Theorem 1. 2 .

 2 Let x = O be an equilibrium point for (1.22) in the continuous-time framework (or (1.23) in the discrete-time framework). Let V : R n → R be a continuously differentiable function. It is said to be a candidate Lyapunov function if:

Theorem 1. 3 .

 3 Let x = O be an equilibrium point of (1.22) in the continuous-time framework (or (1.23) in the discrete-time framework) and Ω a neighborhood of the origin O. Let V : R n → R be a candidate Lyapunov function. If there exists three class K functions β 1 , β 2 and β 3 such that, ∀x ∈ Ω:

  a symmetric and positive definite Lyapunov candidate matrix. The continuous-time T-S model (1.31) or the discrete-time T-S model (1.32)) are asymptotically stable (at least locally) if, ∀x = O:

  continuous-time T-S model (1.6) (or the discrete-time T-S model (1.17)) is asymptotically stabilized (at least locally) by the PDC control law (1.38)

  1.4. It is worth to highlight that, when T-S models are obtained from global sectors from nonlinear systems, we get D x = R n and the satisfaction of the conditions in Theorem 1.4 or Theorem 1.5 guarantee global asymptotical stability since the quadratic Lyapunov function (1.33) satisfies (1.27). However, when T-S models are obtained from local sectors from nonlinear systems, it implies D x ⊂ R n , so the result turns local and an estimate of the domain of attraction can be characterized by searching the outermost level set c > 0 such that L

Figure 1 . 6 :

 16 Figure 1.6: Functions: (a) saturation, and (b) Dead-zone non-linearity in a local sector.

  64) Then the dead-zone function satisfies ψ(u) T D(ψ(u) + ω) ≤ 0 for any m × m diagonal matrix D > 0. Moreover, when dealing with local results, it is fair to provide an estimate of the closedloop domain of attraction based on the knowledge of the bounds of the input signals, i.e. the domain of admissible input values D u , defined in (1.61). In the context of T-S fuzzy models, this has to be taken into consideration together with the other constraints brought by modeling considerations, i.e. the domain of validity D x of the considered T-S model defined in (1.7), or brought by the choice of the Lyapunov function, e.g. the subset D φ defined in (1.56). Combining these constraints allows to provide an estimate of the domain of attraction, e.g. by searching the outermost level set c > 0 such that L

Figure 1 . 7 :

 17 Figure 1.7: Representation of the time-varying delay (τ (t) = t -t k ) with a sawtooth function (o marks the left-limits of right-continuous piecewise functions).

Figure 1

 1 Figure 1.8: Illustration of a decreasing looped LKF.

  First, from a continuous-time nonlinear systems, the usual ways to get a T-S representation have been surveyed. Also, because the main goal of this thesis is to investigate the control of continuoustime systems driven by digital devices, basics on the discretization of continuous-time T-S models have been presented, together with the basic stability and LMI-based controller design conditions in both the continuous and discrete-time frameworks. The limitations of standard discretetime controllers, designed from discrete-time T-S model, have been pointed-out since it fails to accurately cope with the inter-sampling behavior of continuous-time systems, especially when large sampling periods are employed, or when aperiodic sampling occurs. Then, it has beenshown that an elegant way to cope with such issues is to reconsider the closed-loop sampleddata dynamics as a continuous-time system with input time-varying delays but, once again, with some limitation in the T-S model framework, such as the locality of the results or the overall conservatism of the design conditions. In addition, because most real applications involve input limitations, some usual ways to cope with actuators saturation have been presented. All of these concerns motivate the contributions presented in the sequel of this thesis, which can be classified into two balanced research lines presented as follows.First, acknowledging that discrete-time model-based approach are mainly considered in industrial applications because of their easy implementation on industrial controller like Programmable Logic Computers (PLCs), some improvements of actual discrete-time T-S modelbased approach are proposed in Chapter 2, especially to cope with practical goals and constraints like set-point tracking under saturating actuators. Under these constraints, the designed controller only locally guarantees the stabilization of the considered discrete-time T-S model, so the investigation and maximization of the closed-loop domain of attraction will be proposed. These results will be experimentally validated on an industrial benchmark, namely an interactive tank system, having slow dynamics, making relevant the use of discrete-time approaches.Then, as an alternative to discrete-time model-based control design, the input time-varying 1.5. Conclusion delay approach for continuous-time sampled-data systems is investigated in the remaining of the thesis. In Chapter 3, based on the choice of a convenient looped quadratic LKF, LMIbased relaxed conditions are proposed for the design of sampled-data PDC controllers, together with the proposition of a new relaxation scheme for double summation structures involving mismatching membership functions. Then, in Chapter 4, some extensions are proposed to the class of T-S descriptors, known as efficient to cope with mechanical systems, involving actuators saturation, in the looped non-quadratic LKF framework, together with the proposition of strategies to estimate and maximize the closed-loop domain of attraction. These theoretical contributions are validated and compared to previous related results, in simulation with the nonlinear benchmark of an inverted pendulum on a cart. Chapter 2 Anti-windup PI-like Controllers Design for Discrete-time T-S Models Résumé en Français : Synthèse de contrôleurs PI avec action Antiwindup pour les modèles T-S à temps discret.

  y r k = 0 the local asymptotic stability of the respective fuzzy closed-loop for all initial conditions belonging to D * a

Figure 2

 2 Figure2.2: Region of attraction achieved with Corollary 2.1 and the optimization procedure S ν (dashed black) and unstable initial conditions obtained with previous works (×[START_REF] Lv | Fuzzy PID controller design for uncertain networked control systems based on T-S fuzzy model with random delays[END_REF], •[START_REF] Wang | H ∞ fuzzy PID control for discrete time-delayed T-S fuzzy systems[END_REF]).

Figure 2 . 3 .

 23 This plant is composed of four 200 liters tanks, TQ-01 to TQ-04, actuated by two water pumps of 1cv, BA-01 and BA-02, fed by two 400 liters reservoirs and controlled by two independent three-phase inverters. Each levels of the tanks are measured by pressure sensors, whose signals are collected by a Siemens PLC which ensures the interlock security and also sends the control signals to the pumfrag (actuators). The experiments were performed by running the control algorithms on a notebook running Linux (Ubuntu 16.04 distribution), connected to the PLC through an open-source Python-based interface[START_REF] Sousa | Affordable control platform with MPC application[END_REF].

Figure 2 . 3 :

 23 Figure 2.3: Interactive Tank System used in the experiments available at the CEFET-MG, Divinópolis, Brazil.

  49) where h O 1 = 0.537m and h O 2 = 0.271m correspond to the new origin O, i.e. the stationary point achieved for a constant input u O = 70.8722 set at 50% of the variation range of the pump BA-01, leading to u k ∈ [-ū, ū] with ū = 20%. To conclude this modeling subsection, an experimental validation of the obtained discretetime T-S fuzzy model (2.2) representing the equation h k+1 at the change of origin (2.49) is proposed from data measurements on the real system. To do so, a sample time period of T = 4s has been chosen such that the maximum of the absolute error between the open-loop response of the real coupled tank system and the one of the obtained discrete-time T-S fuzzy model is less than 1% of the level ranges, see Figure 2.5. This sampled time period is used in the next subsection to conduct the closed-loop control experiments.

Figure 2

 2 Figure 2.5: Open-loop validation of the discrete-time T-S model in simulation from data measurements on the real system.

Remark 2. 3 ,

 3 we may expect more conservative results. Indeed, in the quadratic framework with anti-windup action, we get a smaller estimate of the region of attraction, which is about 31.5% of the corresponding region with anti-windup action and the proposed fuzzy Lyapunov function (the plot of the quadratic region is left-out since it doesn't bring new information).

Figure 2 . 6 :

 26 Figure 2.6: Estimate of the regions of attraction D * a for the stabilization problem P 1 obtained from Theorem 2.1 (•) and Corollary 2.1 (•).

Figure 2 . 7 :

 27 Figure 2.7: Estimate of the regions of attraction D * a for the set-point tracking problem P 2 obtained from Theorem 2.1 (•) and Corollary 2.1 (•); particular set-points tracking trajectory (•).

  2279,K P 2 = 1.3274 3.4812 , K I2 = 0.1941, S 2 = 1.1240, E c2 = 0.6694, K P 3 = -0.3887 2.7935 , K I3 = 0.0934, S 3 = 0.5286, E c3 = 1.2103, K P 4 = 0.7631 3.7546 , K I4 = 0.1698, S 4 = 2.2098, E c4 = 1.5900, K P 5 = 0.1552 3.0274 , K I5 = 0.1074, S 5 = 1.0855, E c5 = 0.8385, K P 6 = 1.6208 3.2999 , K I6 = 0.2017, S 6 = 1.0203, E c6 = 0.7700, K P 7 = 1.8054 3.3107 , K I7 = 0.1997, S 7 = 0.8278, E c7 = 0.7859,K P 8 = 1.3545 2.6136 , K I8 = 0.1772, S 8 = 1.9622, E c8 = 1.4185.With these gains, a comparison of the time-response obtained in simulation as well as the experimental data obtained from the real system is shown in Figure2.8 with a set-points reference signal reaching the obtained bounds |y r k | ≤ β = 6.64cm. From this figure, we can see the presence of the fuzzy anti-windup action (Figure 2.8(a)) whenever the actuator saturation occurs (Figure 2.8(c)), modifying the closed-loop behavior to preserve stability.

Figure 2 . 8 :

 28 Figure 2.8: Time responses obtained with the PDCaw controller.

Figure 2

 2 Figure 2.9: Time responses obtained with the PDC controller.

  40) and all decision matrices inside Φ2 2α ᾱ, Φ1 Σ ᾱ, Φ1 4α ᾱ and Φ0 Σα ᾱ belong to the bijective change of variables: D = X T DX, D = {L, M 11 ᾱ, . . . , M 44 ᾱ, N 11 ᾱ, . . . , N 22 ᾱ, P 11α ᾱ, P 12 ᾱ, P 22α ᾱ}.

  and its fuzzy T-S representations (with r = 2) presented in Examples 1.2 in the continuoustime framework, and 1.3 in the discrete-time framework. Let us first consider the design of a PDC sampled-data controller (3.5) in order to stabilize the continuous-time model of the 1-DOF inverted pendulum, exactly represented by the 2 rules T-S model (1.16) for all x 1 (t)

3 .

 3 on the MATLAB ® environment, with ε 1 = 0.35, ε 2 = 3.3, ε 3 = 0.23, we found a maximal allowed sampling interval η = τ s = 228ms and the following gain matrices for the sampled-data PDC controller (3result, consider now the discrete-time T-S model of the 1-DOF inverted pendulum given in Example 1.3. Applying Theorem 1.5 with the same sampling period τ s = 228 ms, we obtain the following gain matrices of a PDC discrete-time controller (1.39): K 1 = -10.0379 -2.6384 and K 2 = -8.6133 -2.6384 and a discrete-time Lyapunov function V (x k ) = x T k Px k with: To provide a fair comparison, both designed Sampled-Data (SD) controller and Discrete-Time (DT) one are applied to the continuous-time model of the inverted-pendulum. The simulation was performed with an ODE45 solver to implement the continuous-time evolution of the 1-DOF inverted pendulum represented by (1.2), and the controller was executed with a zero-order hold over the interval t ∈ [0, 4], which is given by: This results in ||u(t)|| 2 = 9.17 for the DT controller and ||u(t)|| 2 = 4.38 for the SD controller,

  Discrete-time state-feedback control input signal u(t k ) and Lyapunov function

Figure 3 . 3 :

 33 Figure 3.3: Inverted Pendulum mounted on a cart.

Figure 3 . 4 :

 34 Figure 3.4: Comparison between the trajectories of the inverted-pendulum on a cart, under sampled-data controllers with gains obtained from different methods for the relaxation of the asynchronous double sums of the design conditions.

Theorem 3 .

 3 1 (with Theorem 3.2) provides a solution with guaranteed sampled-data closed-loop stability. Figure3.5 shows the closed-loop state trajectories of the approximated T-S fuzzy model inverted pendulum from the initial condition x(0) = π/4 0 T with a fixed sampling periodη k = η = 50ms.In addition, because our proposal also guarantees the closed-loop stability for aperiodic sampling intervals, Figure3.6 shows the same simulation but with random sampling intervals such that η k ∈ [0, 50ms]. These simulations illustrate the effectiveness of the proposed sampled-data controller design methodology for T-S fuzzy models.

Figure 3 . 5 :Figure 3 . 6 :

 3536 Figure 3.5: Simulation of the states trajectories of the inverted pendulum on a cart with the respectively sampled input control signal for a fixed sampling period η k = η = 50ms.

Figure 3 .

 3 Figure 3.8: Simple free-body diagram of a 2-DOF helicopter system.

  107.7148, K Dp = 52.4365, K Py = 54.1163, K Dy = 19.5924, designed from standard decoupled second-order transfer functions models and some performances index based on the natural frequency w n , damping ratio ζ, peak time t p and overshoot specification P O .

Figure 3 . 9 :Figure 3 .

 393 Figure 3.9: Time response of Quanser ® AERO model under the Sampled Data Controller.

Figure 3 .

 3 Figure 3.11: Comparison of the time responses of the Quanser ® AERO under a sampling period of 150ms.

Figure 3 .

 3 Figure 3.12: Time responses of the Quanser ® AERO under triggered sampling period.

  )

  e. by finding the largest equipotential included in D u ∩ D x ∩ D φ with D u , D x and D φ , respectively defined by (4.4), (4.2) and (4.11).

  points lead to a local solution: the region of linearity (D u ) of the control inputs inside their bounds due to the actuators' saturation, the domain of validity of the fuzzy T-S model (D x ), and the constraints due to the bounds φ of the time-derivative of the membership functions (D φ ) introduced by the NQLKF (4.13) and Theorem 3.2. Therefore, the characterization of the region of attraction (D a ) must be provided to guarantee a safe operation of the controlled system. Nevertheless, finding the ultimate edge of the full region of attraction (D a ) is a hard or even impossible task. Hence, in Theorem 4.1 we propose a procedure to provide an estimate D * a of such a region. It consists on finding the largest Lyapunov level set L(c) = {x ∈ R n |x T P ᾱx ≤ c} with c < 1 such that:

  ℏ s (z(t)) = 1, and ∇s ∈ R n constant vectors of vertices. The following theorem summarizes the proposed sampled-data controller design conditions for the stabilization of standard T-S models (3.1), including an optimization procedure for the systematic computation of D * a . Theorem 4.2. Let (i, j) ∈ I 2 r and let assume that ∀t, | αi (t)| ≤ φ i . With ∇s (s ∈ I 2 n ) defined in (4.56), for given symmetric bounds ū ∈ R m of the input vector and for non-uniform sampling intervals η k ≤ η (η to be maximized), the standard T-S model (3.1) (i.e. assuming E α = I) is locally asymptotically stabilized by the sampled-data PDC controller (4.7) with saturation (see (4.3)), if there exists a diagonal positive matrix D j ∈ R m×m , the matrices 0

  To verify the effectiveness of the above designed sampled-data controller law (4.7), a simulation is performed in MATLAB ® with the two rules T-S model with actuators saturation.For this simulation, with the initial condition x 0 = 0.5 0 T , the state trajectories and control signals are depicted in Figure4.1. We can observe that the sampled-data controller successfully stabilizes the T-S model although, during the first few sampling intervals, it saturates.

Figure 4

 4 Figure 4.2(b) provides an estimate of the region of attraction D * a for the previously designed sampled-data controller. This one is found really small regarding to D x and D φ , because of the small value of ū = 150N (regarding to the model parameters) considered for the input constraints. Hence, to enlarge the domain of attraction, one may enlarge ū.Figure 4.2(a) shows

  Figure 4.2(a) shows the results obtained with ū = 1500N . In this case, the following Lyapunov and gain matrices

Figure 4

 4 Figure 4.1: Simulation of the state trajectories of the inverted pendulum on a cart with the designed sampled controller, together with the respective saturated input control signal, updated at fixed sampling instants η k = η = 50ms.

Figure 4 Figure 4 . 3 :

 443 Figure 4.2: Estimates of the region of attraction for two sampled-data controllers designed from Theorem 4.2 to stabilize the T-S model (4.75), assuming different actuators' saturation limits (ū), with its resulting closed-loop trajectories for several initial conditions.

  (4.74) to the approximated T-S fuzzy model (4.75) with guaranteed closed-loop stabilization for initial conditions taken in D * a given in (4.71). Figure 4.6 depicts the trajectories of the closed-loop system, the input signals and the evolution of the scheduled sampling intervals with the proposed gain scheduled event-triggering strategy, with two initial

Figure 4

 4 Figure 4.4: Estimates of the Region of Attraction for a sampling interval η = 1ms found from Theorem 4.2, assuming various actuators' saturation limits, (i. e. considering different values for ū).

Figure 4

 4 Figure 4.5: Estimates of the Region of Attraction for several maximum allowable sampling interval η found from Theorem 4.2, assuming ū = 10000N .

Figure 4 . 6 :

 46 Figure 4.6: Estimates of the Region of Attraction D * a (highlighted in gray), control signal and selected sampling period η k for the gain scheduled event-triggering sampled-data controller proposed in (4.74) and designed from Theorem 4.2 for the 2 rules T-S model with a maximal allowed sampling interval η = 50ms.

Figure 4 . 7 :

 47 Figure 4.7: Estimates of the Region of Attraction D * a (highlighted in gray), control signal and selected sampling period η k for the gain scheduled event-triggering sampled-data controller proposed in (4.74) and designed from Theorem 4.2 for the 2 rules T-S model with a maximal allowed sampling interval η = 35ms.

Figure 4

 4 Figure 4.9: (a) Estimated domain of Attraction D * A for the full closed-loop nonlinear model (3.52) under the designed sampled-data controller designed from the T-S descriptor (4.1) (with (4.76)).

Figure 4 .

 4 Figure 4.11: Estimates of the Region of Attraction D *a (highlighted in gray) for the gain scheduled event-triggering sampled-data controller proposed in (4.74) and designed from Theorem 4.2 for the descriptor (4.1) system with (4.76) and the maximal allowed sampling interval η = 8ms.

  conditions for a class of regular nonlinear descriptors subject to actuators' saturation have been proposed. It is noticed that such class of systems is interesting to accurately represent mechatronic systems, where sampled-data control is an inherent characteristic. Based on the generalized sector condition, such nonlinear closed-loop sampled data dynamics can be rewritten as regular T-S descriptors with time-varying delays and constrained inputs. The challenge being to provide relaxed LMI-based conditions for this class of systems, extending the conditions proposed in Chapter 3, useful bounding lemmas have been employed together with a second order polynomial constraints. Also, it has been highlighted that such a sampled-data controller design methodology is only valid locally in the T-S model based-framework. Therefore, a careful analysis of the closed-loop domain of attraction has been provided along this chapter. First, for the class of T-S descriptors subject to input saturation, an offline procedure has been proposed to estimate the resulting closed-loop domain of attraction. Then, for the particular case of standard T-S models, a systematic LMI-based optimization procedure has been proposed to realize such an estimate. However, it has been pointed-out that such systematic estimation may bring conservatism (see Remark 4.6), so it must be used with care, only for some particular appropriate cases (small number of T-S vertices, triangular membership functions...). Furthermore, because, for a given maximal allowed upper bound of the sampling intervals, the obtained estimation of the domain of attraction can be quite small, a new gain scheduled event-triggering mechanism, based on the characterization of several Lyapunov invariant level sets, has been proposed to further enlarge the resulting closed-loop sampled-data domain of attraction.The effectiveness of these theoretical results has been illustrated through the example of an inverted pendulum on a cart. First, to show the conservatism improvements with regard to the results obtained in Chapter 3 and previous related studies from the literature, the local sampled-data controller design for an approximated 2 rules fuzzy model of this benchmark has been proposed, together with extensive estimations of the closed-loop domain of attraction, for several cases (different upper-bounds of the sampling intervals and/or input saturation).The obtained results shown a significant conservative reduction in regard to the previous ones.However, recalling that the sampled-data controller designed from the approximated standard T-S model may fail to stabilize the full nonlinear model of the inverted pendulum, a tighter closely matching T-S descriptor model has been proposed, which succeeded to guarantee the closed-loop stability, establishing the merit of the proposed methodology. However, for the obtained maximal allowed upper bound of the sampling intervals, the resulting estimation of the closed-loop domain of attraction has been found to be quite small, an illustration of the proposed gain scheduled event-triggering mechanism has been provided, showing its efficiency to enlarge the guaranteed sampled-data closed-loop domain of attraction.

  stabilize a class of continuous-time nonlinear systems, represented by Takagi-Sugeno models, and driven by digital devices. In this context, Chapter 1 has presented the preliminaries on T-S model-based controller design, detailing important steps, such as the usual ways to get a T-S representation. Also, aligned to the main goal of this thesis, basic notions on the discretization of continuous-time T-S models have been presented, together with the basics on stability and LMI-based controller design conditions, in both the continuous and discrete-time frameworks.

  ble summation structure for parameter dependent closed-loop stability conditions, which occurs in the considered continuous-time T-S model-based sampled-data control context. As usual in related sampled-data control studies, the conservatism reduction is investigated in this chapter by searching the maximal allowable sampling interval [0, η], for which the LMI-based conditions remain feasible. The benefit of the proposed sampled-data controller design methodology with regard to conventional discrete-time T-S model-based design has been illustrated through the simulation of a 1-DOF inverted pendulum benchmark. Then a comparison has been provided,

  Chapter 4, from a convenient NQLKF candidate, further relaxed sampled-data controller design conditions for a class of regular T-S descriptors subject to actuators' saturation have been proposed. Indeed, despite standard T-S models, such a class of systems is often considered to more accurately represent a larger class of nonlinear systems, especially for mechatronic systems, where constrained sampled-data control is an inherent characteristic. Based on the generalized sector condition, the nonlinear closed-loop sampled data dynamics can be rewritten as regular T-S descriptors with time-varying delays and constrained inputs. The challenge was to provide relaxed LMI-based conditions for this class of systems, extending the conditions proposed in Chapter 3, together with a careful analysis of the closed-loop domain of attraction. First, for the class of T-S descriptors subject to input saturation, an offline procedure has been proposed to estimate the resulting closed-loop domain of attraction. Then, for the particular case of standard T-S models, a systematic LMI-based optimization procedure has been proposed to realize such an estimate. However, it has been pointed-out that such systematic estimation may bring conservatism (see Remark 4.6), so it must be used with care, only for some particular appropriate cases (small number of T-S vertices, triangular membership functions...). Furthermore, because, for a given maximal allowed upper bound of the sampling intervals, the obtained estimation of the domain of attraction can be quite small, a new gain scheduled event-triggering mechanism, based on the characterization of several Lyapunov invariant level sets, has been proposed to further enlarge the resulting closed-loop sampled-data domain of attraction. Along this chapter, the results have been validated in simulation through the benchmark of an inverted pendulum, first, with the same approximated T-S fuzzy model considered in Chapter 3, to fairly highlight the conservatism improvement and to illustrate the proposed systematic approach to estimate the sampled-data closed-loop domain of attraction. Then, because it has been shown that controllers designed from such an approximated T-S fuzzy model may fail to stabilize the full nonlinear model, a closely matching T-S descriptor model has been proposed. From it, the so designed sampled-data controller has been shown to successfully stabilize the full inverted pendulum nonlinear model, but in a quite small guaranteed estimation of the closed-loop domain of attraction for the obtained maximal allowable sampling interval. Hence, to further enlarge the resulting sampled-data closed-loop of attraction, the benefit of the proposed gain scheduled event triggering mechanism is illustrated.

  Now, let us focus more on the input-delay approach considered for the stabilization of continuous-time quasi-LPV/T-S models. If the results presented in Chapter 3 and Chapter 4 have shown the appeal of the proposed strategy, one must fairly infer its main limitation, which is the extensive computational complexity of the proposed LMI-based conditions. Indeed, like many attempts to reduce the conservatism in the T-S model-based framework, providing

  to the T-S model-based framework, or, similarly to what has been proposed in Chapter 2, by extend-ing the input-delay approach to cope with PI-like sampled-data controllers. However, for this last point, from the author's preliminary investigation, some issues are still to be unlocked, especially to deal with the necessary time hybridization of the integral and/or anti-windup actions. This being said, from these prospects on T-S model-based sampled-data controller design for continuous-time nonlinear systems, the door is now open for a wide range of further deve lopments.

Takagi-

  Sugeno models, Sampled-data control, LMI-based design, Saturated Actuators, Local Stabilization.

  q×n are known constant matrices describing the dynamics of each local linear models.

	For the antecedent part of each i-th fuzzy rule, a weighting function w i (t) is assigned. It
	characterizes the instantaneous contribution of each sub-model of the T-S model, and this func-
	tion depends on the grade of membership of the premise variables z p regarding a fuzzy subset
	M ij such as:

6) Remark 1.1. Note that

	T-S models (1.6) belong to the class of polytopic systems (i.e., a collec-
	tion of linear systems blended together by time-varying functions). In the control literature, other
	class of polytopic systems are often considered to deal with nonlinear or quasi-linear control. For
	instance, Linear Parameter Varying (LPV) systems, introduced in

  1); 2. for any y r k verifying ||y r k || ≤ δ -1/2 = β, the trajectories of the mentioned closed-loop system, for all initial conditions x(0) taken in D * a , do not leave such a set. Conditions of Corollary 2.1 also provide a solution for problems P 1 and P 2 , but The conditions proposed in Theorem 2.1 and Corollary 2.1 encompass the quadratic stability based approach, i.e., we can impose W i = W , for i ∈ I r in conditions (2.17), (2.18), (2.21), and (2.34). In this case, it is expected that the estimate of the region of attraction is more restricted than in the case with fuzzy Lyapunov functions. The solutions provided by Theorem 2.1 and Corollary 2.1 can be used to optimize objectives associated with problems P 1 and P 2 . A particular interest consists in maximizing the estimate of the region of attraction, which can be achieved by the following convex optimization procedure for Theorem 2.1:

	Remark 2.2. such a solution is usually more conservative than that achieved by Theorem 2.1 because no anti-
	windup action is considered. In this sense, whenever Corollary 2.1 has a solution, the feasibility
	of conditions (2.17)-(2.19) are ensured, but not the contrary.
	Remark 2.3.

Table 3 .

 3 1: Comparison of maximal η obtained from Theorem 3.1 with the selected asynchronous double sums relaxation scheme.

	Method	ε 1	ε 2	ε 3	K 1 K 2	X	η
	Lemma 1.10 (Koo et al., 2017) Theorem 3.2	8.25 5 0.28 5.5 3 0.31	32.8209 -43.5959 46.7316 -25.2741 0.2054 0.0214 1.2536 11.5638	0.6989 -2.1227 -2.0760 6.4966 0.0062 -0.0176 -0.0172 0.0702	35ms 50ms

Table 3 .

 3 2: Comparison of maximal η obtained with related previous studies.

	Method	η (ms)

Table 3 .

 3 3: Quanser ® AERO 2-DOF Helicopter Parameters.

	Parameter

Then, from

(3.55

) and

(3.56)

, the following linear state-space model of the 2-DOF Helicopter can be obtained, considering the state vector x T (t) = θ(t) ψ(t) θ(t) ψ(t) and the input

vector u T (t) = V p (t) V y (t) : ẋ(t) = Ax(t) + Bu(t)

(3.57)

with:

  ᾱ, X} T and its transpose,

	we obtain the conditions (4.18) (which embed (4.33), (4.44), (4.48) and (4.49)) and (4.19) (which
	embeds (4.45)) where the decision matrices inside Φ1αᾱ and Φ0αᾱ belong to the bijective change

  That is to say, pre-and post-multiplying (4.54) by diag{X, 1} T and its transpose, respectively, we get the conditions (4.20). Finally, note that the conditions of Theorem 4.1 only guarantee the local asymptotic stability of the designed closed-loop descriptor because the T-S descriptor (4.5) matches (4.1), ∀x(t) ∈ D x , and the conditions (4.18), and (4.19) are valid assuming that ∀t, | αi (t)| ≤ φ i . Hence, the domain of attraction D a of the designed closed-loop system must verify D a ⊆ D x ∩ D φ (recall that D a ⊆ D u is granted by (4.20)). Since V ℓ (0) = 0, for ℓ = 2, ..., 4, an estimate D * a of the domain of attraction D a is given by (4.21). Note that the LKF (3.8) considered in Chapter 3 is a special case of the NQLKF (4.13) (proposed to prove Theorem 4.1) with P α = L, Q α ᾱ = N ᾱ, R α ᾱ = P ᾱ and S ᾱ = Mα .

	Remark 4.3. The conditions (4.18) and (4.19) of Theorem 4.1 are parameterized inequalities,
	which can be solved via LMI tools from the application of Theorem 3.2. Furthermore, the obtained
	conditions are not strictly LMI because of the parameters η, ε 1 , ε 2 and ε 3 . However, similarly
	to what has been proposed in Chapter 3 (see Remark 3.1), these parameters can be usually tuned
	offline by grid search.
	Remark 4.4. Moreover, standard T-S models (3.1) being a special case of T-S descriptors (4.1) with E α = I,
	Theorem 1 in (Lopes et al., 2021) or Theorem 3.1 (when input saturations are not considered)
	constitute special cases of Theorem 4.1 (assuming M 1 ij = -P , and M 2 ij = -Qj to alleviate condi-tions (4.19)). Consequently, applying Theorem 4.1 instead of these special cases, for descriptors
	or standard T-S models, with or without input saturation, will provide the less conservative
	results.
	ℓ)	≥ 0	(4.54)

  Assuming that, ∀k ∈ I r , α k (x) are continuous with regard to x ∈ D x , then we have, ∀ℓ ∈ I n ,

		∂x 1	)	, . . . ,	∂α k (x(t)) ∂x n	T	(4.55)
	∂α ℓ (x(t)) ∂xn	∈ [ δℓ , δℓ ]. Therefore, from a sector nonlinearity decomposition, we can write:
		∇ k (x(t)) =	s=1 2 n	ℏ s (z(t)) ∇s	(4.56)
	with, ∀s ∈ I 2 n , ℏ s (z(t)) ≥ 0, 2 n s=1

  3.5.2. For the sake of our study let us assume the actuator saturation as u(t) ≤ ū. This model, valid in D x defined by |x 1 (t)| < π/2 and |x 2 (t)| ≤ π, is given by: |x 2 (t)| ≤ π defining D φ in (4.11), we always have | ḣi (x 1 (t))| ≤ 2 = φ i . Therefore, solving the conditions of Theorem 4.1, using YALMIP

	2
	ẋ(t) =	h i (x(t)) (A i x(t) + B i sat(u(t))) ,	(4.75)
	i=1
	with the matrices A 1 , A 2 , B 1 and B 2 given in (3.53), and the membership functions α 1 and α 2
	given in (3.54).
	Recall that, | ḣ1 (x 1 (t))| = | ḣ2 (x 1 (t))| = 2 π | ẋ1 (t)| and, since | ẋ1 (t)| =

  asynchronous double summation structure of the parameter-dependent LMI-based design conditions, we have proposed an extension of the Tuan's Lemma, summarized in Theorem 3.2. This one is based on the application of the Peterson Lemma

	To illustrate this point, note that with the approximated T-S
	model of an inverted pendulum considered in Subsection 4.4.1, it took 12.97 seconds to solve
	the conditions of Theorem 4.2 (we use for this test a late 2016 HP laptop having a 2.6GHz Intel
	Core I7 processor and 16GB of memory). Otherwise, to solve the conditions of Theorem 4.1

with the full T-S matching descriptor model of the inverted pendulum (given in Section 4.4.2), the computational burden was significantly more extensive, i.e. an average of 45 minutes per tests, with the same computer.

Many other points can merit some attention to further reduce the conservatives of the proposed LMI-based conditions for the T-S model-based sampled-data controller design. First, to handle the

ᾱξ(t)(3.20)

0 0 ⋆ -η R22αᾱ 0 0 0 ⋆ ⋆ Λ33 α ᾱ 0 0 ⋆ ⋆ ⋆ -R11αᾱ 0 ⋆ ⋆ ⋆ ⋆ Λ55 α ᾱ

Acknowledgements

, I would like to thank the Region Grand Est and the University of Reims Champagne-Ardenne for funding this research project. I would like to acknowledge the support received from Ms. Lolita Lebrun, from the International Office of the University The time spent in France would not be so memorable without speaking the French language,

From Figure 3.1 it is possible to observe that the trajectories of the states variables for the stabilization problem with the designed SD controller happen to be smoother than the response obtained with the DT controller (applied to the original continuous-time system), especially for the speed variable (x 2 = θ(t)). Moreover, it is verified that the equilibrium point (98% of the final value, i.e., 0.0288) is reached about 34% faster with the SD controller, which takes 1.03s , than with the DT controller, which takes 1.55s.

Furthermore, we investigate the behavior of the Lyapunov functions obtained with the design of both controllers. In this context, it is important to recall that, from Theorem 3.1, it is ensured that the whole LKF (3.8) is continuous and positive at each sampling time t k and monotonously decreasing for all t. So, taking benefit of the fact that, at each sampling instant t = t k , we have To assert this fact, we also propose to evaluate the euclidean norm of the control input signals Hence, ∀x(t) ∈ D x , applying the sector nonlinearity approach on each f ℓ (x(t)), we have:

fℓfℓ ≥ 0, and where andso w 2 3 (x(t)) andw 2 4 (x(t))), as shown in Figure 4.8. This fact allows to fairly reduce the number of vertices from r = 2 4 to r = 2 3 , to obtain a tighter closely matching T-S descriptor (4.5) to represent the nonlinear dynamics (3.52) (than the approximated two rules T-S fuzzy models considered in Subsections 3.5.2 and 4.4.1), with the membership functions:

. and the vertices given by:

Based on the obtained descriptor model, our goal is now to design a sampled-data controller (4.7) for the stabilization of the nonlinear model of the inverted pendulum (3.52). To do so, we will apply Theorem 4.1 together with Theorem 3.2, but remind that they need, as parameter, an estimation of the bounds of the time-derivatives of the membership functions. As quoted in Remark 4.2, these bounds can be approximated as:

It is worth to highlight that, to estimate the bounds φ k , only an assumption on the input saturation ū of the inverted pendulum is needed. Hence, to illustrate this fact and to evaluate its effect on the conservatism of the conditions, two cases are considered in the sequel with ū = 150N and ū = 600N . From these results, Figure 4.9 shows the closed-loop domains of attraction obtained re-Again, with the same parameters but now with η = 1 ms, we obtained the following From Figure 4.10, we clearly observe enlargements of the guaranteed closed-loop region of attraction (with ū = 600N ) regarding to the previous case (where ū = 150N ). Moreover, it is straightforward that the region of attraction obtained with η = 1ms is significantly larger that the one obtained with the maximal admissible value η = 8ms found for this case. This motivates the use of the gain scheduled event-triggering mechanism, proposed in Subsection 4.3.3, in order to take benefit of large domain of attraction for initial conditions taken as far as possible from the origin, then gradually reducing the computational demand by increasing the sampling intervals while getting closer to the equilibrium.