M Abdellah

Salama

Ismaël Louis

i Tout d'abord, je tiens à remercier mon directeur de thèse, qui m'a accepté au sein de l'équipe de recherche Commande Véhicule (CoVE) pour mon doctorat, M. Ahmed El Hajjaji. Je lui dois ma reconnaissance pour son aide concernant les démarches administratives que j'ai eu à effectuer et pour son soutien lors des périodes difficiles durant ma thèse. Je tiens également à remercier M. Olivier Pages, mon co-encadrant, qui a toujours été très minutieux et m'a corrigé dans les moindres détails mes résultats et mes publications. J'espère pouvoir continuer à collaborer avec vous dans le futur.

Je remercie M. Thierry Guerra, et M. Michel Zasadzinski, les rapporteurs de ce mémoire qui ont accepté de consacrer une partie de leur précieux temps pour me faire part de leurs remarques et questions. Je remercie également M. Driss Mehdi, et

Introduction Générale et Résume

Avant-propos

La thèse a été menée au sein de l'équipe de recherche COVE (Commande et Vehicles) du laboratoire MIS (Modélisation, Information et Systèmes) réalisée dans le cadre d'un doctorant. Le sujet de la thèse porte sur la « Stabilité et stabilisation des systèmes de retard à paramètres linéaires et variables dans le temps avec saturation des actionneurs ». Nous nous concentrons sur la résolution des problèmes de stabilité et de stabilisation des systèmes LPV et quasi-LPV incluant certaines contraintes de performances (retard variable dans le temps, saturation des actionneurs, variations de paramètres, perturbations externes, etc.).

Contexte et motivations

Les systèmes physiques contiennent des non-linéarités et des dynamiques variant dans le temps. Il est possible d'approximer le comportement non linéaire d'un état du système à partir de la linéarisation. Les méthodes de linéarisation envisagées pour les systèmes non linéaires pourraient être divisées en trois catégories [START_REF] Vidyasagar | Nonlinear Systems Analysis[END_REF]): 1-Linéarisation autour d'un équilibre; 2-Linéarisation globale; et 3-Linéarisation autour d'une trajectoire d'état. Cette représentation des systèmes LPV sera étudiée ici.

L'intérêt des techniques LPV consiste en une approximation des systèmes non linéaires par des systèmes dépendant de paramètres avec l'hypothèse où l'ensemble des paramètres est supposé compact. Le comportement du système non linéaire est linéarisé localement autour de la trajectoire des paramètres variant dans le temps. Sur la base de ces hypothèses, l'analyse des critères de robustesse, de stabilité et de performance des systèmes LPV sont déduites. L'analyse de la robustesse des incertitudes dynamiques (non-linéarités, dynamique négligée, etc.) et des paramètres incertains (connaissance incertaine sur les grandeurs physiques simulées) a reçu une attention considérable. La théorie de la commande linéaire à paramètres variables (LPV) joue un rôle clé dans la gestion des incertitudes ou des inexactitudes. Les transformations fractionnaires linéaires (LFT) et les inégalités matricielles linéaires (LMI) ont été développées pour traiter l'analyse robuste de la stabilité et les performances des systèmes LPV et quasi-LPV. D'une manière générale, la fonction de Lyapunov quadratique (QLF) joue un rôle central dans l'analyse de la stabilité et la stabilisation des systèmes LPV via les LMI. Cependant, une fonction de Lyapunov quadratique peut être trop conservatrice pour une analyse de stabilité robuste car elle impose l'existence d'une seule matrice définie positive de Lyapunov vérifiant un ensemble de LMI. Il en résulte une dégradation des performances (conservatisme) pour des exigences multi-objectifs, par exemple, contraintes sur les entrées et les sorties, saturation des actionneurs, retard variant dans le temps, etc. Suite à cet argument, une condition de stabilité robuste dérivée du lemme réel borné à l'échelle est vi généralement moins conservatrice que le QLF. Une question intéressante ressort de cette discussion : comment pourrions-nous exploiter plus d'informations sur le système et améliorer la flexibilité des conditions de conception ? Ainsi, la première motivation consiste à assouplir les conditions de stabilité basées sur les QLF.

En raison de la nature des systèmes paramétriques, un problème dans la technique d'analyse LPV repose sur les formulations dérivées associées à l'ensemble compact de paramètres réellement représentés comme des conditions de dimension infinie. Ainsi, des méthodes de relaxation des LMI paramétrés ont été proposées pour formuler efficacement des problèmes d'analyse base sur l'optimisation convexe impliquant des contraintes LMI de finie dimension. En conséquence, une synthèse des méthodes de relaxation pour les LMI paramétrés a été proposée dans [START_REF] Apkarian | Parameterized LMIs in Control Theory[END_REF][START_REF] Gahinet | Affine parameter-dependent Lyapunov functions and real parametric uncertainty[END_REF][START_REF] Tuan | Relaxations of parameterized LMIs with control applications[END_REF].

De manière générale, la stabilité robuste utilisant une fonction de Lyapunov dépendante des paramètres (PDLF, qui semble plus adaptée à la synthèse de contrôle LPV) a été bien étudiée. Cependant, le problème de conception de la commande n'est pas entièrement résolu en présence de contraintes de saturation des actionneurs. Ce problème crucial sera étudié en détail et constitue l'une des principales contributions.

Dans l'analyse de la conception des systèmes de commande, un phénomène observé dans de nombreux systèmes d'ingénierie est la saturation des actionneurs. À première vue, l'effet de cette non-linéarité peut paraître simple, mais une analyse inappropriée ou ignorant ses effets peuvent entraîner une dégradation des performances ou une instabilité du système. La saturation des actionneurs est inévitable dans l'ingénierie des systèmes dynamiques pratiques concernant les limites physiques (vitesse, tension, cycle, etc.) et les contraintes de sécurité (pression, température, puissance, consommation d'énergie, etc.). C'est pourquoi nous devons trouver une méthode de stabilisation consistant à replacer les points de fonctionnement du système d'asservissement dans la région sans élément sature. Au cours des dernières décennies, une attention considérable a été accordée aux systèmes LTI soumis à la saturation des actionneurs, voir par exemple [START_REF] Hu | Control Systems with Actuator Saturation[END_REF][START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF][START_REF] Zaccarian | Modern Anti-windup Synthesis: Control Augmentation for A tuator Saturation[END_REF] et les références qui s'y trouvent. D'une manière générale, il existe deux approches principales pour effectuer l'analyse de stabilisation dans la littérature. La première considère les bornes de saturation dans la stratégie de conception. Dans la seconde part, une synthèse stabilisante asymptotique proposée pour un système en boucle fermée ne tient pas compte des bornes de commande. Ensuite, une stratégie de conception appropriée analysera pour compenser la saturation, telle que Direct Linear Anti-windup (DLAW) ou Model Recovery Anti-windup (MRAW). Le domaine anti-windup a fait l'objet de discussions approfondies au cours des dernières décennies. Nous pouvons nous rendre compte que l'analyse de stabilisation en boucle fermée pour DLAW et MRAW est plus compliquée en tenant compte de l'effet du comportement non linéaire et de la dynamique incertaine. Néanmoins, la construction DLAW dépasse le cadre de la thèse, elle ne sera donc pas incluse.

Enfin, une autre contribution porte sur les retards, des phénomènes de retard temporel sont observés dans divers systèmes d'ingénierie tels que les procédés chimiques, les transmissions mécaniques, les transmissions hydrauliques, les processus métallurgiques et les vii systèmes de contrôle en réseau. La stabilité et la stabilisation des systèmes retards (TDS) ont reçu une attention considérable dans la pratique et la théorie du contrôle. Le retard peut être classé en différentes approches en fonction de la caractéristique ou du comportement de réponse du retard au système. La littérature sur la stabilité et la stabilisation des systèmes retardés porte sur les systèmes LTI, les systèmes LPV, le domaine temporel l'analyse de stabilité basée sur Lyapunov et l'approche basée sur les valeurs propres.

De plus, les conditions dépendantes du retard dérivées de l'analyse de stabilité et de stabilisation via la technique LMI basée sur la fonction de Lyapunov-Krasovskii sont rencontrées dans de nombreux articles de la littérature. La majorité des résultats de la littérature concerne les systèmes LPV/quasi-LPV avec retards sans prise en compte des effets de saturation des actionneurs. Peu de techniques de synthèse sont disponibles pour la stabilisation robuste des systèmes LPV retards avec des actionneurs contraints ce sera notre contribution. De plus, les conditions de stabilisation de ces classes de systèmes sont généralement des inégalités matricielles non linéaires (NMI ou NLMI), qui sont généralement des problèmes polynomiaux non déterministes (NP-hard).

Une structure appropriée de LKF peut faire référence aux termes intégraux supplémentaires, aux vecteurs d'état croissants et aux approches de partitionnement de retard/fragmenté qui se sont révélées extrêmement efficaces pour réduire la résolution des conditions de stabilité. Il convient de noter que plus les matrices de variables d'écart utilisées sont nombreuses, plus l'analyse des conditions de stabilité dépendantes au retard est compliquée. En conséquence, ces approches sont le compromis entre la relaxation de la condition de stabilité et celle du calcul.

Le troisième objectif de cette thèse est de fournir la stratégie de conception de contrôle la moins restrictive pour les systèmes LPV et quasi-LPV avec des contraintes de retard variant dans le temps et de saturation. Outre l'utilisation de LKF appropriés, de variables d'écart et de bornes de saturation, pour obtenir des conditions plus flexibles, la méthode proposée est un équilibre entre conservatisme et réduction de la complexité de calcul.

Plan de thèse

Cette thèse est organisée selon les chapitres suivants :

Le Chapitre 1 donne une introduction générale et un résumé de la thèse.

Le Chapitre 2 donne un aperçu des représentations de la famille des systèmes LPV. Ensuite, les propriétés dépendantes des paramètres implicites dans les LMIs dérivées sont linéarisées par les méthodes de relaxation. La convergence asymptotique peut être obtenue lors de la résolution d'un ensemble de conditions d'inégalité matricielle. Enfin, une synthèse détaillée de la stabilité des approches non quadratiques de Lyapunov donne une approche pour stabiliser les systèmes LPV.

Le Chapitre 3 est consacré à l'analyse de la stabilité et de la stabilisation robuste de systèmes dépendant de paramètres sans contraintes de saturation. La première contribution sur l'algorithme d'itération optimale concave utilisant des blocs de paramètres diagonaux est présentée et comparée à la littérature existante. viii Le Chapitre 4 traite l'analyse et la synthèse du contrôleur de programmation de gain saturé avec des inégalités de stabilisation plus strictes basées sur la fonction paramétrique de Lyapunov telle que PDLF et la fonction floue de Lyapunov (FLF). Les résultats ont été obtenus par la méthode de relaxation appliquée aux conditions PDLMI obtenues. Le chapitre se termine par la troisième contribution de l'analyse de stabilisation pour les systèmes LPV avec contraintes de saturation.

Le Chapitre 5 traite de l'analyse de stabilité et de la conception de contrôle pour les systèmes LPV avec retard utilisant une fonction convexe appropriée basée sur la fonction de Lyapunov-Krasovskii. Une nouvelle condition de stabilité dépendante du retard est donnée à l'aide de la fonctionnelle de Lyapunov-Krasovskii dépendante des paramètres (PDLKF) combinée à la bounding technique. Cette approche fournit une inégalité plus étroite pour délimiter l'intégrale quadratique d'un vecteur étendu. Plusieurs types de stabilité dans les cadres de valeur de retard de mémoire exacte et de retard approximatif sont étudiés. La condition de stabilité du retard de mémoire incertaine est considérée comme convenant à l'exigence de mise en oeuvre. Enfin, l'efficacité des conditions PDLMI proposées est démontrée par des résultats d'analyse de stabilité par rapport aux méthodes existantes pour les systèmes linéaires invariants dans le temps et LPV.

Le Chapitre 6 contribue à la stabilisation des systèmes à retard variable dans le temps LPV avec saturation de l'actionneur. En incluant le délai de mémoire approché, un contrôleur par retour d'état et par retour de sortie dynamique sont présenté. Ensuite, des conditions PLMI nécessaires et suffisantes ont été proposées pour garantir une stabilisation résiliente à mémoire respectant les contraintes de saturation. Par rapport aux résultats existants récents, cette méthode fournit une performance améliorée avec une borne supérieure du retard. La discussion finale démontre les caractéristiques efficaces de cette stabilisation.

Table of Content

Remerciements i Introduction Générale et Résume v

Table of Content ix

List of Figures xii

List of Tables xiv

Notation xv

Acronyms xvi

Chapter 1. General Introduction and Summary

List of Figures

General Introduction and Summary

Foreword

The thesis was conducted at the "COVE" research team of MIS laboratory throughout a Ph.D. research. The subject of the dissertation is "Stability and Stabilization of linear parameter-varying and time-varying delay Systems with Actuators Saturation." We focus on solving the stability and stabilization problems for the LPV and quasi-LPV systems including some performance constraints (time-varying delay, actuator saturation, parameter variations, external disturbances, etc.).

Context and motivations

Since the last decade of the 20th century, the guarantee of the stability and the robustness analysis of the system robustly against the influence of dynamical uncertainties (nonlinearities, neglected dynamics, etc.) and uncertain parameters (uncertain knowledge about simulated physical values, the time variations of these values during operation) have received considerable attention of engineering control community. The robustness analysis and linear parameter-varying (LPV) control theory plays a key role in handling uncertainties or inaccuracies. The Linear Fractional Transformations (LFTs) and Linear Matrix Inequalities (LMIs) have been developed to deal with the robust stability and performance analysis for both LPV and quasi-LPV systems.

Generally speaking, the quadratic Lyapunov function (QLF) plays a central role in analyzing the stability and stabilization of LPV systems via LMIs. However, a quadratic Lyapunov function may be too conservative for robust stability synthesis because it imposes the existence of a single Lyapunov positive definite matrix verifying a set of LMIs. But, due to some particular properties of the implementation, a considerable approach has been developed or analyzed the ℋ∞ gain-scheduling controller with this modest case. And, because it cannot characterize slow variation parameters, it results in a degradation performance (conservatism) for multi-objective requirements, for example, hard time constraints, actuators saturation, time-varying delay, etc. Followed this argument, a robust stability condition derived from the scaled-bounded real lemma is typically less conservative than the QLF. An interesting question arises from this discussion: how could we exploit more system information and improve the flexibility of design conditions? So, the first motivation engages in relaxing the QLF-based stability conditions by taking advantage of the parametric properties of the scaling structure.

Due to the nature of parametric systems, one critical issue in the LPV analysis technique is the derived formulations associated with the compact set of parameters actually represented as infinite dimension LMI conditions. Thereby, popular relaxation methods of Parameterized LMIs have been proposed to efficiently formulate analysis problems as convex optimization problems involving a finite LMI constraints. Generally speaking, the robust stability using a parameter-dependent Lyapunov function (PDLF, which seems more suitable for LPV control synthesis) is well-studied and well-advanced. However, the control design issue is not completely resolved in the presence of actuator saturation constraints. This crucial problem will be studied in detail and constitutes one of the main motivations of this thesis.

It is well known that actuator saturation can lead to performance degradation or even instability in several engineering systems. As a result, actuator saturation is considered as an exciting challenge in control system design. The problem of the saturated feedback controller design for these classes of dynamics systems constitutes an interesting problem for both theoretical and practical reasons. Stability analysis and control synthesis of saturated LPV systems are generally divided into two different strategies: (1) Anti-windup scheme, (2) Saturation nonlinearities.

It can be seen that the conventional input constraint imposed by a small-gain theorem is related to the input-output approach in a strict manner. So, the less conservative method, such as generalized sector condition (GSC), gives an extra degree of freedom in the stabilization synthesis used in this literature. Besides, the GSC is appropriate for the parameter-dependent LMI conditions and well suits extension for delay-dependent stabilization.

On the other side, the delay-dependent conditions derived from the stability and stabilization analysis via the Lyapunov-Krasovskii functional based LMI technique is encountered in many papers of literature. The majority of the results in the literature concerns LPV/quasiLPV systems with time delay without taking into account the actuator saturation effects. Few synthesis techniques are available for the robust stabilization of timed LPV systems with constrained actuators. Moreover, the stabilization conditions for these classes of systems are usually nonlinear matrix inequalities (NMI or NLMI), which are usually nondeterministic (NP-hard) polynomial problems. Therefore, the stability analysis and stabilization of saturated LPV/quasi-LPV systems with time delay become a more attractive challenge.

The third objective of this thesis is to propose less restrictive control design strategy for LPV and quasi-LPV systems with time-varying delay and saturation constraints. Besides, the use of appropriate LKFs, slack-variables, saturation bounds, to obtain more flexible design conditions, the proposed method is a balance between conservatism and computational complexity reduction.

Chapter 3 is devoted to the analysis of Robust Stability and Stabilization of parameter-dependent systems without saturation constraints. An improved solution of the Robust T-S Fuzzy controller stabilizing analysis with ℒ2 norm-bounded input constraints is presented. The first contribution about the concave optimal iteration algorithm using diagonal parameter blocks is presented and compared with the existing literature.

Chapter 4 discusses the analysis and synthesis of the saturated gain-scheduling controller with tighter stabilizing inequalities based on the parametric Lyapunov function such as PDLF and fuzzy Lyapunov function (FLF). The less conservative results were attained by the proposed relaxation method applied to the designed PDLMI conditions. The chapter is concluded with the third contribution of stabilization analysis for LPV systems with saturation constraints.

Chapter 5 deals with stability analysis and control design for LPV time-delay systems using an appropriated Lyapunov-Krasovskii functional based convex function. A new delay-dependent stability condition is addressed using parameterdependent Lyapunov-Krasovskii functional (PDLKF) combined with the advanced bounding technique. This approach provides a tighter inequality for bounding the quadratic integral of an extended vector. Several types of stability in both exact-memory delay value and approximate delay frameworks are studied. The uncertain-memory delay stability condition is considered suiting the implementation requirement. Finally, the effectiveness of the proposed PDLMI conditions are demonstrated through stability analysis results compared with the existing methods for both linear time-invariant (LTI) and LPV time-delay systems.

Chapter 6 contributes to the stabilization of LPV time-varying delay systems with actuator saturation. Including the approximated-memory delay, a more general controller is introduced for both state feedback and dynamic output feedback controllers. Then, necessary and sufficient PLMI conditions have been proposed 1.4. Publications to guarantee a memory-resilient stabilization respecting the saturation constraints (sector nonlinearities). Besides, the optimization of the estimation of the domain of attraction (DOA) is analyzed. The proposed method is validated by considering several numerical examples. Compared to the recent existing results, this method provides an enhanced performance conforming to a higher upper bound of the delay value. The final discussion demonstrates the efficient characteristics of saturation stabilization.

Publications

The thesis is based on the following publications and other studies in the process of being submitted.

Chapter 2. Overview Linear Parameter-Varying Systems

Overview Linear Parameter-Varying Systems

Physical systems are practically involved with nonlinearities and time-varying dynamics. It is possible to approximate the nonlinear behavior of a system state in the range of nominal operating conditions, usually referred to as linearization [START_REF] Isidori | Nonlinear Control Systems[END_REF][START_REF] Khalil | Nonlinear Systems[END_REF]. The linearization methods considered for nonlinear systems could be supposedly characterized into three categories [START_REF] Vidyasagar | Nonlinear Systems Analysis[END_REF]): 1-Linearization around an equilibrium; 2-Global linearization; and 3-Linearization around a state trajectory. § Linear Time-Invariant (LTI) system as a representative for first method related with the simplest analysis and synthesis techniques, which is expressed by linearizing the dynamic systems around the neighborhood of equilibrium points. During the operation, the presence of nonlinearities with a wide range of variation (include dynamical uncertainty, saturation, and inaccurate knowledge of dynamics....) causes the inaccurate linearization occurs over equilibrium conditions. § The multi-model representation of Linear Time Varying (LTV) systems also called Linear Differential Inclusions (LDI) is used to represent trajectories of a nonlinear system by a set of trajectories of LDI in the entire operating range. Nevertheless, this linearization method could be conservative because the approximated trajectories that are sometimes not actual trajectories of the given system. § Parameter Dependent systems as a typical for third method where the nonlinear system can be approximated by a family of linearization or the parameterized linearization. Since the proposed method is valid around a state trajectory rather than a single equilibrium point, then it can characterize a nonlinear system in a wider range of operating conditions than LTI. This representation of the LPV systems that will intensively study here.

This chapter provides a non-exhaustive overview of the linear parameter-varying (LPV) systems used to approximate nonlinear systems according to the trajectories of parameters. Depending on the characteristics of the parameter, it can classify as linear time-varying (LTV), LPV systems [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF][START_REF] Lim | Parameter-Varying Systems[END_REF]Mohammadpour & Scherer, 2012;[START_REF] Shamma | Analysis and design of gain scheduled control systems[END_REF][START_REF] Wu | Control of linear parameter varying systems[END_REF], or quasi-linear parameter varying (quasi-LPV or qLPV) systems, for example, the representation of T-S fuzzy systems [START_REF] Lam | Polynomial Fuzzy Control Systems Stability Analysis and Control Synthesis Using Membership Function-dependent[END_REF][START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF][START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. In addition, LPV systems can also classify according to the parameter properties, e.g., the intrinsic (endogenous) or extrinsic (exogenous), the physical properties (e.g., parametric uncertainty, and dynamical parameters), and mathematical significance (continuous/discrete, smooth or non-smooth, continuous derivative, etc.).

The benefit of LPV techniques consists in an approximation of the characterizations of the nonlinear systems to the parameter-dependent systems. Where the compact sets of parameters and their derivatives are the prerequisite of the system design hypothesis. The behavior of the nonlinear system is linearized locally around the trajectory of the timevarying parameters. Based on these assumptions, the analysis of robustness, stability, and performance criteria of the LPV systems thus simplifies as on LTV or LTI systems (Apkarian, [START_REF] Apkarian | Parameter-Dependent Lyapunov Functions for Robust Control of Systems with Real Parametric Uncertainty[END_REF]Apkarian & Gahinet, 1995;[START_REF] Gahinet | Affine parameter-dependent Lyapunov functions and real parametric uncertainty[END_REF].

In the first part, in section 2.1, we discuss commonly used framework to represent LPV systems along with some applications. Corresponding to each representation is a characteristic approach for the stability analysis provided in section 2.2. The stability synthesis for the LPV system via the Lyapunov function results in the parametric conditions. In which the convex optimization linear matrix inequality (LMI) tools cannot solve these conditions directly. As a result, a synthesis of relaxing methods for the parameterized LMIs has been methodically discusses in [START_REF] Apkarian | Parameterized LMIs in Control Theory[END_REF][START_REF] Gahinet | Affine parameter-dependent Lyapunov functions and real parametric uncertainty[END_REF][START_REF] Tuan | Relaxations of parameterized LMIs with control applications[END_REF] including: § The gridding technique with uniform density [START_REF] Wu | Control of linear parameter varying systems[END_REF][START_REF] Wu | Induced L2-norm control for LPV systems with bounded parameter variation rates[END_REF], and the meshing parametric affine [START_REF] Apkarian | Parametrized LMIs in control theory[END_REF][START_REF] Lim | Parameter-Varying Systems[END_REF]. § The convex combination of multi-LTI systems is known as the polytopic paradigm (Apkarian, Gahinet, et al., 1995;[START_REF] Apkarian | Parameterized LMIs in Control Theory[END_REF][START_REF] Gahinet | Affine parameter-dependent Lyapunov functions and real parametric uncertainty[END_REF], and the T-S fuzzy model [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF][START_REF] Tuan | New Fuzzy Control Model and Dynamic Output Feedback Parallel Distributed Compensation[END_REF]Tuan, Apkarian, et al., 2001). § The multiplier-based linear fraction transformation (LFT) use D-scaling to capture the behavior of the parameter with the additional inequality on the multiplier quadratic in the scheduling block (Apkarian & Gahinet, 1995;[START_REF] Packard | Gain scheduling via linear fractional transformations[END_REF]. The matrix algebraic transformations of LFT LPV based on S-procedure can be found at [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF][START_REF] Pólik | A Survey of the S-Lemma[END_REF]. § The sum of squares (SoS) relaxation-based LPV stability synthesis on the SoS decomposition for multivariate polynomials can be efficiently computed using semidefinite programming [START_REF] Parrilo | Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[END_REF][START_REF] Parrilo | Semidefinite programming relaxations for semialgebraic problems[END_REF].

The parameterized linear matrix inequality (PLMI) can be converted into finite-dimensional inequalities. Where the feasible solution is obtained by solving the LMI conditions. The methodologies and numerical examples of LPV stability synthesis are introduced at the end of section 2.2. It should be notice that the design requirements conformed to the LPV framework such as, robust stability and performance e.g., ℋ∞ criterion, or the fullblock S-procedure (FBSP) will be presented in Appendix B.

In section 2.3, we recall some fundamental definitions like the region of linearity, region of attraction and regions of asymptotic stability and the developments concerned within the dissertation, with respect to main problems of the stability analysis and the stabilization of linear parameter-varying and time-delay systems with saturating inputs. The characterization of sets of admissible initial states and admissible disturbances plays a central role in stability analysis as well as in the synthesis of stabilizing control laws when saturation occurs.

Introduction of LPV/quasi-LPV systems

Let us introduce a generalized expression of the LPV system that is studied throughout this dissertation under the forms of a non-autonomous non-stationary system of ordinary differential equations: where vectors ,,,, nprmd xtytztutwt  RRRRR are respectively the state of the system, the measured output, the regulated output, the control input and the external disturbance. The behavior of LPV system (2.1) depends on the behavior of the parameters. From the point of view of the physical meaning of parameters (e.g., measurability, endogenous or exogenous parameters, etc.), or mathematical properties (i.e., continuous/discontinuous parameters, differentiable/non-differentiable parameter, etc.), that provides a classification for LPV modeling or the appropriate method for system stability analysis and control system design strategy.

Endogenous Parameters and Exogenous Parameters

Let consider a time-continuous state space system: (2.3) where the exogenous parameters 12 sin2, cos/31, 1 tttt    are in- dependent of the system state. It can be noticed that these parameters have continuous and bounded derivatives.

On the other hand, the time-varying parameters can also characterize the states of the nonlinear system, for instance (2.5) In this case, when the parameters are functions of states, they are usually classified as endogenous parameters. The representation systems are referred to as a quasi-LPV system (Briat, 2015a;[START_REF] Hoffmann | A survey of linear parameter-varying control applications validated by experiments or high-fidelity simulations[END_REF][START_REF] Lovera | LPV Modelling and Identification: An Overview[END_REF][START_REF] Rotondo | Quasi-LPV modeling, identification and control of a twin rotor MIMO system[END_REF][START_REF] Shamma | An Overview of LPV Systems[END_REF]. It is interesting to note that systems (2.3) and (2.5) have a similar LPV representation.

Continuous (discontinuous) parameter values with continuous (discontinuous) derivative

In addition, parameter behaviors can also be classified by mathematical properties such as discrete or continuous value, smooth or non-smooth functions, and differentiability or non-differentiability. Let's consider an example:

: 0,1 t     RB  (2.6)
where B is the image set of function t maps from t    R to :0,1.  B In this case, the parameter trajectory is a continuous switching between the piecewise constants. The systems involved in the function described the discrete value parameter could be considered as Hybrid Systems (deterministic and stochastic switched cases). The interested readers can refer to the literature [START_REF] Alwan | Theory of Hybrid Systems: Deterministic and Stochastic[END_REF][START_REF] Boukas | Stochastic Switching Systems[END_REF][START_REF] Briat | Stability analysis and state-feedback control of LPV systems with piecewise constant parameters subject to spontaneous poissonian jumps[END_REF][START_REF] Briat | Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers[END_REF][START_REF] Chatterjee | Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functions[END_REF][START_REF] Colaneri | Dwell time analysis of deterministic and stochastic switched systems[END_REF][START_REF] Teel | Stability analysis for stochastic hybrid systems: A survey[END_REF] Generally, the bounding derivative of the parameters often interferes with stability analysis. The dynamics of state are theoretically unbounded, but with the definition, the parameters do. Nonetheless, if considered that functions mapping from state to parameter domain are bounded on each state, it is a too strong assumption. Let borrow a simple example from [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF] to make this clear: in the synthesis of stabilization conditions of single input single output SISO LPV system is ensued in condition , . xtab   && Then, by implementing the closed-loop system with the found controller, the system's trajectory exhibits a stable characteristic, but the behavior of the derivative of the state goes outside the bounded region, e.g., 1, 1. ab So, it is not reasonable to confirm that the closed-loop system is stable in the domain , , ab  and then the analysis should start over with an expansion of the bounds on the derivative of the state, e.g., 2, 2. ab  But, the increase of the limits also leads to the conservative results. The compact parameter sets can be symbolized as follows ,conv. p tt    UU & Hereafter, we inherit the mathematical definition of convex optimization from [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF][START_REF] Boyd | Convex Optimization[END_REF]) and use the computational toolboxes such as LMI toolbox MATLAB [START_REF] Gahinet | LMI Control Toolbox For Use with MATLAB[END_REF], Yalmip toolbox [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF], semidefinite programming problems -SDP Sedumi toolbox [START_REF] Sturm | Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones[END_REF], cone programming Mosek toolbox (E D [START_REF] Andersen | On implementing a primal-dual interiorpoint method for conic quadratic optimization[END_REF][START_REF] Andersen | The Mosek Interior Point Optimizer for Linear Programming: An Implementation of the Homogeneous Algorithm[END_REF] to solve the convex optimization problems.

In the following sections, the presentations of the LPV/qLPV paradigms are provided along with the summarizing stability synthesis of LPV systems based on Lyapunov technique expressing in the parameterized LMIs.

Affine LPV Systems

An affine LPV (ALPV) system is linear parameter-varying systems whose matrices are affine functions of the scheduling parameters. Considering a dynamic system depend affinely on the parameter vector For the sake of simplicity, parametric dependent matrix expression At   are reduced to . A  This is one of the most common LPV system formulations encountered in control synthesis, where the affine-dependent system leads to a low degree of conservatism of stability conditions.

Polynomial Systems

The parameter polynomial formulations are widely concerned to the modeling and control system design of the LPV systems. A polynomial system relating to the parameter-dependent state-space representation could express by the following form: It should note that a polynomial system is directly approximated by Taylor's expansion of the nonlinear expressions. A general formulation could find in the literature [START_REF] Parrilo | Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[END_REF][START_REF] Parrilo | Semidefinite programming relaxations for semialgebraic problems[END_REF][START_REF] Sato | Robust stability/performance analysis for linear timeinvariant polynomially parameter-dependent systems using polynomially parameter-dependent Lyapunov functions[END_REF].

Polytopic Systems

The polytopic LPV formulation is a linear combination of a convex set of dependent parameters, widely used in the framework for control synthesis of LPV systems. Introduced the early 90s in the literatures [START_REF] Apkarian | Parameterized LMIs in Control Theory[END_REF][START_REF] Feron | Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions[END_REF]Gahinet et al., 1994[START_REF] Gahinet | Explicit controller formulas for LMI-based H ∞ synthesis[END_REF] which address robust stability and robust performance of the uncertainty systems. Generally, a polytopic system is defined by the following equations: Example 2.1.1. [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF]) Considering an affine system with 2 parameters: [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF], the author has used this example as a simple way to demonstrate the drawback of the polytopic approach. As seen in Figure 2-1.b, the polytope lost its parametric dependence since the two vertices of the system are not related to the parameter domain 2 , tt   resulting in the conservative stability conditions. Consequently, a uncertain polytope method (Appendix A. 1.4) that reshapes the quasi-convex vertex containing the parameter curve presented in [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF] and [START_REF] Gonçalves | New Approach to Robust$ cal D$-Stability Analysis of Linear Time-Invariant Systems With Polytope-Bounded Uncertainty[END_REF] to reduce this conservatism.

Takagi-Sugeno Fuzzy Systems

Since introduced by [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF], the T-S fuzzy model has shown the effective linearization of the nonlinear systems by using logical rules (fuzzy sets). Following this research direction, [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF] presented the control and observation analysis for nonlinear systems that could reformulate to the T-S fuzzy framework. Like the LPV system, the T-S fuzzy systems grow the influence in multidisciplinary applications of the robust stability control analysis. Now, let's consider a nonlinear system:

pn i C   R
The aggregation of the subsystems is based on membership function j t

 in ij M 1 , 0, , jjjjjjiji j ttttt      with 1 0, 1 and 1. l N ii i tt      (2.20)
A similarity can be found in this expression of the membership function and the polytopic coordinate system (2.13). Then, the defuzzification process of this model can derive by: for more details refer to [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF].

    , 1 , 1 , 1,2,,. l l N iiwi i l N iiwi i xttAxtBwt iN zttHxtJwt                     & K (2.21)
The fuzzification is convenient for modeling dynamic systems, complex non-smooth nonlinear systems, chaotic systems, etc. The choice of distribution law and member functions is dependent on the purpose of system design, more specific of the fuzzification and defuzzification referred to [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]). In the next section, we briefly describe the mathematical formula of membership function i t   of a bounded nonlinear system (local sector) and unbounded nonlinear system (global sector).

Sector Nonlinearity

As discussed in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF], T-S fuzzy modeling has two typical approaches.

In the one hand, the fuzzy identification modeling using input-output relation is difficult to analyze the control design for physical models. On the other hand, the nonlinear dynamic models obtained from the Lagrange equation or the Newton-Euler theorem are fuzzificated by "sector nonlinearity." The sector nonlinearity method introduced by [START_REF] Kawamoto | An approach to stability analysis of second order fuzzy systems[END_REF] and generalized by [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF] Usually, these constraints are often involved in the stability synthesis with the parameter-dependent Lyapunov functions (i.e., fuzzy Lyapunov function -FLF).

A computational load requirement for the sector nonlinearity approach is to simplify the original nonlinear system with fewer model rules (vertices) as possible so that the reduction of the effort for analysis and design of control systems does not degrade the performance of the modeling process. In the work of [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF] a local approximation in fuzzy partition spaces was used to simplify the system with a significant decrease of the model rules. In the other direction, the combination of fuzzification with the monomial formulation will balance less conservative condition and computational synthesis complexity.

Polynomial Fuzzy Model

The polynomial fuzzy proposed for the SoS-based state-space approach in [START_REF] Tanaka | A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems[END_REF] is more effective for representing nonlinear control systems and providing less conservative stability analysis and synthesis than the traditional T-S fuzzy model. The main difference between T-S fuzzy model (2.21)

Example

In this section, a nonlinear system is used to illustrate the representation to all the discussed types of LPV/quasi-LPV systems.

Example 2.1.3. Let's consider a nonlinear system [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]):

T-S Fuzzy Model

Let's designate the local sector nonlinearities: This quasi-LPV representation is expressed by a convex combination of the four vertices of the nonlinear system (2.25) in the specified domain. The linear combinational representation is convenient for stability and stabilization analysis. As stated in the discussion of Polytopic systems, the convexity conditions are directly delivered. The conservatism of the convex envelope is illustrated in Figure 2-1.b, and the computational burden increases exponentially with the number of vertices of the system (i.e., the number of membership functions).

In the other aspect, the polynomial formulation more accurately describes the nonlinear behaviors. But it depends on the selection of the monomial vector Zx  to have a reasonable number of polynomial order (a higher degree leads to complexity in the stability analysis and increased computational load). Accordingly, a fuzzy polynomial system balances conservatism with numerical complexity is considered as follows.

Polynomial Fuzzy Model

Using the same monomial vector .

ii i xtxAxZx      & (2.35)
The vertices of system (2.35) is halved compared to system (2.32), which also has a simpler polynomial matrix structure than system (2.29).

An appropriate stability analysis method will be delivered for each presented model in section 2.2, including the advantages and disadvantages suitable for the implementation. Then, in section 2.2.5 with these representation, the appropriate stability conditions will be developed on each LPV model corresponding to an LMI relaxation method.

Applications

Linear Parameter-Varying (LPV) systems have been extensively studied over the last three decades to approximate the nonlinear systems and provide a systematic design framework of gain-scheduled controllers robust again the uncertain information. Their applications have been found in various fields such as automotive systems, aircraft systems, robotic manipulators, mechatronic systems see, for example, (Briat, 2015a;[START_REF] Giri | Robust Control and Linear Parameter Varying Approaches[END_REF][START_REF] Hoffmann | A survey of linear parameter-varying control applications validated by experiments or high-fidelity simulations[END_REF], etc. From a theoretical perspective, in addition to choosing an appropriate LPV model, the online measurement feature also plays a decisive role in accurately restructuring the system trajectories and linearizing the behavior of the nonlinear dynamics. However, in some LPV control applications, not all scheduling parameters are available for measuring. The high-precision engineering systems are even more demanding on requirements and system performances, e.g., aerospace application, Unmanned aircraft systems [START_REF] Marcos | Development of Linear-Parameter-Varying Models for Aircraft[END_REF][START_REF] Marcos | LPV modeling, analysis and design in space systems: Rationale, objectives and limitations[END_REF]. It can be found that increasing the number of scheduling parameters enhances the simulation accuracy and the system validation process. However, it also burdens the computational load, increases memory requirements, and growths the synthesis complexity, see, e.g., (Hoffmann et al., 2014;Hoffmann & Werner, 2014, 2015). So, depending on the design requirements, we have appropriate analysis and synthesis tools. Now, let's discuss some applications of the LPV modeling.

Automotive Chassis Systems

In the last decades, LPV control synthesis has been addressed the robust stability and performance of the lateral dynamic stabilizing system integrated on-road vehicles, see, for example, in [START_REF] Dahmani | Observer-Based Robust Control of Vehicle Dynamics for Rollover Mitigation in Critical Situations[END_REF][START_REF] Doumiati | Integrated vehicle dynamics control via coordination of active front steering and rear braking[END_REF][START_REF] Ono | Theoretical approach for improving the vehicle robust stability and maneuverability by active front wheel steering control[END_REF][START_REF] Ono | Robust stabilization of the vehicle dynamics by gain-scheduled H/sub /spl infin// control[END_REF][START_REF] Zhang | Robust gain-scheduling energy-to-peak control of vehicle lateral dynamics stabilisation[END_REF]. The simplified lateral dynamic stabilization system (illustrated in Figure 2-3.b) can be described by the following equations:

22 22 yyffyrrx zyfffyrrr mvtFtFtmtvt ItFtlFtl            & & && (2.36)
where the full description of the physical parameters is detailed in Appendix E.2. This dynamic system depends on the lateral friction forces , yfyr FF   (also called cornering forces), described by the nonlinear equations of the tire slip angle at the contact point of front tires f t  (rear tires ) r t  and road surface, see, e.g., [START_REF] Bakker | A New Tire Model with an Application in Vehicle Dynamics Studies[END_REF][START_REF] Dugoff | An analysis of tire traction properties and their influence on vehicle dynamic performance[END_REF][START_REF] Kiencke | Automotive Control Systems[END_REF]. If longitudinal speed x v is constant, then the vehicle system is regarded with LTI system [START_REF] Farrelly | Estimation of vehicle lateral velocity[END_REF][START_REF] Fukada | Slip-angle estimation for vehicle stability control[END_REF][START_REF] Guldner | Analysis of automatic steering control for highway vehicles with look-down lateral reference systems[END_REF]. (the variation cornering stiffness shows in Figure 2-3.a), then it is characterized by LTV systems discussed in [START_REF] Ono | Robust stabilization of the vehicle dynamics by gain-scheduled H/sub /spl infin// control[END_REF][START_REF] Zhang | Robust gain-scheduling energy-to-peak control of vehicle lateral dynamics stabilisation[END_REF] that results in a quasi-LPV formulization.   are intrinsic parameters of these qLPV systems, see, e.g., [START_REF] Bui Tuan | Robust Observer-Based Control for TS Fuzzy Models Application to Vehicle Lateral Dynamics[END_REF][START_REF] Dahmani | Observer-Based Robust Control of Vehicle Dynamics for Rollover Mitigation in Critical Situations[END_REF][START_REF] Dahmani | Detection of impending vehicle rollover with road bank angle consideration using a robust fuzzy observer[END_REF]Dahmani, Pages, & El Hajjaji, 2015;[START_REF] El Hajjaji | Observer-based robust fuzzy control for vehicle lateral dynamics[END_REF][START_REF] Latrech | Vehicle dynamics decentralized networked control[END_REF].

LTI and LTV systems

The coordinate parameters i   are usually accompanied by the assumption that the scheduling parameters can be measured. However, the stabilization chassis control system (2.40) is involved with an unmeasurable dynamic -tire sideslip angle . t  The prob- lem of an inexact parameter î   has been addressed for the LPV system, see, e.g., [START_REF] Daafouz | On inexact LPV control design of continuous-time polytopic systems[END_REF][START_REF] Heemels | Observer-based control of discrete-time LPV systems with uncertain parameters[END_REF][START_REF] Rotondo | Robust state-feedback control of uncertain LPV systems: An LMI-based approach[END_REF][START_REF] Sato | Gain-scheduled output-feedback controllers using inexact scheduling parameters for continuous-time LPV systems[END_REF] and the T-S fuzzy system [START_REF] Li | Fault Detection for T-S Fuzzy Systems With Unknown Membership Functions[END_REF][START_REF] Yang | Finite-Time Convergence Adaptive Fuzzy Control for Dual-Arm Robot with Unknown Kinematics and Dynamics[END_REF]Zhang & Wang, 2017).

Using the estimates of membership function improve the performance of the control system but increases the complexity of the stability condition.

Now, let's consider a time-varying longitudinal velocity .

x tvt   A compact set of the parameter transformed to the polytopic formulation discussed in [START_REF] Bosche | An output feedback controller design for lateral vehicle dynamic[END_REF][START_REF] Nguyen | Simultaneous Estimation of Vehicle Lateral Dynamics and Driver Torque using LPV Unknown Input Observer[END_REF][START_REF] Zhang | Robust gain-scheduling energy-to-peak control of vehicle lateral dynamics stabilisation[END_REF]Zhang et al., , 2015)).

                                             & &&& 4 1 , i  (2.41)
In this case, the representation is considered as an extrinsic LPV system because x vt is a state-independent parameter. If a global chassis system is used [START_REF] Kiencke | Automotive Control Systems[END_REF][START_REF] Poussot-Vassal | Commande Robuste LPV Multivariable de Châssis Automobile[END_REF], this parameter is now an implicit function of the states 1 , xt 

2

. xt So, the definition is quite abstract and depends on the specific design purpose. In addition, x vt is measurable, so it suits the gain-scheduling controller technique.

Generally, depending on the system requirements (such as robust stability, robust performance, fast response, good tracking, etc.) and the accessibility of state variables and parameters, we have an appropriate approach for the analysis and design of control systems. Another application of LPV control relates to the improvement of performance and comfort of the automotive chassis system. The recent advances technique can find for instance in [START_REF] Do | Approche LPV pour la commande robuste de la dynamique des véhicules : amélioration conjointe du confort et de la sécurité[END_REF][START_REF] Do | An LPV control approach for semi-active suspension control with actuator constraints[END_REF][START_REF] Doumiati | Gain-scheduled LPV/H∞ controller based on direct yaw moment and active steering for vehicle handling improvements[END_REF][START_REF] Giri | Robust Control and Linear Parameter Varying Approaches[END_REF][START_REF] Nguyen | LPV approaches for modelling and control of vehicle dynamics : application toa small car pilot plant with ER dampers[END_REF][START_REF] Poussot-Vassal | Commande Robuste LPV Multivariable de Châssis Automobile[END_REF][START_REF] Savaresi | Semi-Active Suspension Control Design for Vehicles[END_REF][START_REF] Tuan | Nonlinear H/sub ∞/ control for an integrated suspension system via parameterized linear matrix inequality characterizations[END_REF][START_REF] Vu | Active anti-roll bar control using electronic servo valve hydraulic damper on single unit heavy vehicle[END_REF], and the reference therein.

Aircrafts Systems

The physical properties of the flight dynamics (high velocity, large number of degrees of freedom, aerodynamic influence, etc.) characterize the aviation control systems. The requirements, therefore, are more demanding in terms of robustness, system performance, and stability compared to the ground vehicles. The LPV theory is suitable for enhancing performance, robustness, and ensuring accuracy and safety during operation against the influence of aerodynamics. See, for example, analysis robustness margins [START_REF] Schug | Robustness Margins for Linear Parameter Varying Systems[END_REF], robust ℋ∞ control [START_REF] Papageorgiou | Taking robust LPV control into flight on the VAAC Harrier[END_REF], high performance on F-16 Aircraft System [START_REF] Shin | Blending approach of linear parameter varying control synthesis for F-16 aircraft[END_REF] on F-14 and F-18 Aircraft System (Balas et al., n.d., 1997), LPV modeling and controller design for Boeing 747-100/200 [START_REF] Ganguli | Reconfigurable LPV control design for Boeing 747-100/200 longitudinal axis[END_REF][START_REF] Marcos | Linear parameter varying modeling of the boeing 747-100/200 longitudinal motion[END_REF], 2004), and developments of LPV controllers for an unmanned air vehicle (UAV) [START_REF] Chen | Robust LPV Control of UAV with Parameter Dependent Performance[END_REF][START_REF] Natesan | Design of static H∞ linear parameter varying controllers for unmanned aircraft[END_REF][START_REF] Rotondo | LPV model reference control for fixed-wing UAVs[END_REF]. The application of gain-scheduling applies to missile autopilot LPV systems [START_REF] Pellanda | Missile autopilot design via a multichannel LFT/LPV control method[END_REF][START_REF] Shamma | Linear parameter varying approach to gain scheduled missile autopilot design[END_REF][START_REF] Wu | LPV control design for pitch-axis missile autopilots[END_REF].

Let's consider a qLPV modeling of Boeing 747-100/200 longitudinal motion [START_REF] Marcos | A Linear Parameter Varying Model of the Boeing 747-100/200 Longitudinal Motion[END_REF][START_REF] Marcos | Development of Linear-Parameter-Varying Models for Aircraft[END_REF] as follows: [START_REF] Marcos | A Linear Parameter Varying Model of the Boeing 747-100/200 Longitudinal Motion[END_REF].

      , , ,,,, ,,,,.

Mechatronics and Robotics Systems

Another application of the LPV control system concerns the stabilization of the nonlinear robotic arm system depending on the parameter dynamics and nonlinearities. Showing in Figure 2-4.a is the force diagram of another example of an inverted pendulum analyzed in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF][START_REF] Wang | A Course in Fuzzy Systems and Control[END_REF]. The nonlinear state-space equations governing the two-link dynamic system are given by: can consider as an uncertainty, then followed the definition of [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF], we have the following membership functions: (2.45) As presented before, the local sector linearization method could transform the nonlinear system (2.44) into the qLPV (T-S fuzzy) system as follows:

  1 , 1,2,,. l N ijiil i xtxAxtButiN      &K (2.46)
However, this T-S fuzzy system representation requires 4 2 l N  linear combination of membership functions (2.45). This number increase exponentially in control design analysis (fuzzy PDC controller). The computational burden is a hindrance of the implementation. In addition, the following LPV system (2.47) is based on assumptions about the bounded region of the parameter , t but qLPV system (2.46) is involved with the con- straints bounding on the state 1 .

xt This association cause trouble in the design prerequi- sites such as initial condition 0 xand the upper bound of the derivative /.

ij dxdt   Illustrated in Figure 2-4.b is the diagram of the physical parameters and force diagram of the arm-driven inverted pendulum (ADIP) system [START_REF] Kajiwara | LPV techniques for control of an inverted pendulum[END_REF]Canudas-de-Wit et al., 1996). ut is control signal of the electric motor, 1212 ,,, mmllare respectively the mass and the half of the length of the arm and the pendulum. The control objective is to maintain the pendulum in a reference vertical position (inverted pendulum motion) using the generated torque from the arm. This moment is regulated by a motor power amplifier voltage, with , aa KTare constant mechanic-electric parameters.

To facilitate for developing a gain-scheduling controller, t and zt are assumed to be measurable in this two-link robot manipulator example. This intrinsic parameter is state independent. The further discussions about robot dynamics models can find at (Canudasde-Wit et al., 1996), for the LPV application on robot-arm control, see, e.g., [START_REF] Halalchi | Flexible-Link Robot Control Using a Linear Parameter Varying Systems Methodology[END_REF][START_REF] Robert | An H∞ LPV design for sampling varying controllers: Experimentation with a T-inverted pendulum[END_REF][START_REF] Sename | A LPV approach to control and realtime scheduling codesign: Application to a robot-arm control[END_REF], for the fuzzy application and control [START_REF] Roose | Fuzzy-logic control of an inverted pendulum on a cart[END_REF][START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF][START_REF] Wang | A Course in Fuzzy Systems and Control[END_REF][START_REF] Yi | Stabilization fuzzy control of inverted pendulum systems[END_REF].

Other applications

From the above discussion, the LPV modeling and control design provide wide-range applications such as automobile engine systems, photovoltaic systems, electronic circuit systems, Etc. Another application that we would like to mention here is the LPV modeling and control for the tokamak fusion reactor [START_REF] Ariola | Magnetic Control of Tokamak Plasmas[END_REF][START_REF] Wesson | Tokamaks[END_REF]. A control-oriented distributed model discussed in [START_REF] Witrant | A control-oriented model of the current profile in tokamak plasma[END_REF], then an approximation of an LPV system deployed in [START_REF] Bribiesca Argomedo | Polytopic control of the magnetic flux profile in a tokamak plasma[END_REF] from heterogeneous transport partial differential equation (PDE) model of dynamics of the poloidal flux. Thereby, developing a polytopic feedback control law for the non-inductive lower hybrid current drive (LHCD) is proposed [START_REF] Bribiesca Argomedo | Polytopic control of the magnetic flux profile in a tokamak plasma[END_REF], that regulates the current and heat source on the plasma. This method shows an efficient reduction in computational cost, and easier to integrate the saturated constraint than to seek for the weight matrices for the linear-quadratic regulator (LQR) method by solving the algebraic Riccati equation (ARE) [START_REF] Bribiesca Argomedo | Safety Factor Profile Control in a Tokamak[END_REF], 2010). Readers interested in this topic can refer to the monographs [START_REF] Ariola | Magnetic Control of Tokamak Plasmas[END_REF][START_REF] Bribiesca Argomedo | Safety Factor Profile Control in a Tokamak[END_REF]. More discussion and analysis on nonlinear dynamics motion, and methods of identification and linearizing LPV models refers to [START_REF] Tóth | Modeling and Identification of Linear Parameter-Varying Systems[END_REF].

Stability of LPV/quasi-LPV Systems

Before going into the stability analysis of the parameter-dependent system, let's revise the fundamental control system theory by analyzing the stability of the equilibrium point at the origin of a pendulum system (2.48) with ,. ab   R The system stability in the sense of Lyapunov function is to find a con- tinuously differentiable energy functionVxt  that decreases over time, and the solutions of the system starting in the vicinity of the equilibrium point will be nearby or converge to the equilibrium point as the time approaches infinity. For more details on stability of dynamical systems, the reader could refer to [START_REF] Khalil | Nonlinear Systems[END_REF][START_REF] Scherer | Linear matrix inequalities in control[END_REF][START_REF] Vidyasagar | Nonlinear Systems Analysis[END_REF].

Theorem 2.2.1: Lyapunov Theorem [START_REF] Khalil | Nonlinear Systems[END_REF] [START_REF] Khalil | Nonlinear Systems[END_REF] along the 1 x  axis, we can only confirm that the system is stable at origin using the Lyapunov function (2.50). However, the phase portrait of the dynamic 1 [START_REF] Khalil | Nonlinear Systems[END_REF], chapter 4 system exhibits asymptotically convergence to this equilibrium. So let include a quadratic term to the previous Lyapunov function: [START_REF] Khalil | Nonlinear Systems[END_REF], the failure of Lyapunov function does not imply that the equilibrium of the system is stable nor asymptotically stable. It can only emphasize that the stability of the system cannot be guaranteed with these inappropriate Lyapunov candidates. Let's move forward to discuss the necessary and sufficient Lyapunov conditions addressed for the stability of a linear system. 4 Abba  in the open right-half plane for all ,0. ab Thus, this equilibrium is unstable. It can be checked that there will be no matrix 0 P f satisfies 22 0, ,0.

T APPAab  p

Quadratic Stability and Non-Quadratic Stability Analysis

The stability analysis for the LTI system using the Lyapunov function can be easily performed. But the linearizing characteristic of the nonlinear system in the form of an LTI system does not accurately describe the behaviors of the original system.

In this case, by approximating the nonlinear system as a parametric affine is much more convenient for the analysis and synthesis of system design. In the early 90s, the term parametric uncertainties or parameter-dependent is related to two commonly analytical methods, robust stability for LTV system [START_REF] Dullerud | A Course in Robust Control Theory[END_REF][START_REF] Khargonekar | Robust stabilization of uncertain linear systems: quadratic stabilizability and H/sup infinity / control theory[END_REF][START_REF] Zhou | Robust and Optimal Control[END_REF][START_REF] Zhou | [END_REF][START_REF] Zhou | Robust stabilization of linear systems with normbounded time-varying uncertainty[END_REF] and quadratic stability (Apkarian & Gahinet, 1995;[START_REF] Becker | Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback[END_REF]. To ensure stability and performance against uncertainty parameters, Apkarian has presented two different approaches in the middle 90s. On one side, according to the linear fraction dependence and linear fractional transformation (LFT) technique (Apkarian & Gahinet, 1995), the parametric-dependency extraction from the nominal plant using an uncertainty structure [START_REF] Doyle | Structured uncertainty in control system design[END_REF][START_REF] Doyle | Review of LFTs, LMIs, and μ[END_REF]. Followed the scaling commuting structure, LFT approach shows the efficiency in dealing with uncertainties and provides more relaxation than old-fashioned conditions. Besides, the bounded real lemma [START_REF] Feron | Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions[END_REF]Gahinet & Apkarian, 1994;[START_REF] Scherer | The Riccati inequality and state-space H∞-optimal control[END_REF]) also plays a central role in robust ℋ∞ performance synthesis during this time.

However, these stability conditions capture only fast variation parameters that become very conservative with slowly varying parameters. So, the affine quadratic Lyapunov formulation [START_REF] Apkarian | Parameter-Dependent Lyapunov Functions for Robust Control of Systems with Real Parametric Uncertainty[END_REF]Gahinet et al., 1994) is proposed to address the stability of the LPV system. Where the scaled-bounded real lemma can enhance the robustness and performance requirements. The parameter-dependent Lyapunov functions (PDLF) consider for stability analysis and gain-scheduling controller synthesis of LPV systems that better describe the behavior of parameters, see for example, in the literature [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF][START_REF] Apkarian | Parameterized LMIs in Control Theory[END_REF][START_REF] Gahinet | Affine parameter-dependent Lyapunov functions and real parametric uncertainty[END_REF][START_REF] Gahinet | LMI Control Toolbox For Use with MATLAB[END_REF][START_REF] Lim | Analysis of LPV systems using a piecewise affine parameter-dependent Lyapunov function[END_REF][START_REF] Tuan | Relaxations of parameterized LMIs with control applications[END_REF][START_REF] Wu | Induced L2-norm control for LPV systems with bounded parameter variation rates[END_REF], for the qLPV T-S fuzzy systems [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF][START_REF] Tanaka | A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems[END_REF][START_REF] Tuan | New Fuzzy Control Model and Dynamic Output Feedback Parallel Distributed Compensation[END_REF]Tuan, Apkarian, et al., 2001), for the LPV time-delay systems [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF][START_REF] Briat | Stability analysis and control of a class of LPV systems with piecewise constant parameters[END_REF] and the references therein.

The leverage of the development of convex optimization programing and the relaxation PLMI methods, it makes the use of PDLF widespread in the stability and stabilization analysis for the parameter-dependent systems. The following sections are devoted to the stability analysis for the LPV/quasi-LPV systems using Lyapunov function via parameterized LMI conditions.

Stability of Polytopic Systems

This issue was excavated decades ago because of its effectiveness in analyzing for robust stability and robust performance [START_REF] Feron | Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions[END_REF]Gahinet et al., 1994[START_REF] Gahinet | Affine parameter-dependent Lyapunov functions and real parametric uncertainty[END_REF]. The quadratic [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF][START_REF] Scherer | Linear matrix inequalities in control[END_REF] and non-quadratic stability [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF][START_REF] Apkarian | Parameterized LMIs in Control Theory[END_REF]Tuan, Apkarian, et al., 2001;[START_REF] Tuan | Relaxations of parameterized LMIs with control applications[END_REF] are discussed for the polytopic systems. Generally, the open loop polytopic LPV system is obtained under the form:

W

However, considering only a common matrix to guarantee the quadratic stability for the multi-convex system is conservative. In some cases, it doesn't exist a candidate matrix n P   S that satisfies stability conditions (2.55). Hence, it makes more sense to consider the parameter-dependent Lyapunov function PDLF or the piece-wise Lyapunov function approach introduces in [START_REF] Johansson | Piecewise Linear Control Systems[END_REF] to relax the conservativeness. Theorem 2.2.4: (Briat, 2015a) Polytopic system (2.54) is robust stability, if there exist symmetric positive matrices

1 ,, l N n iii i PPP       S such that 11 0, l N T i ji ij j APPAtP          & p (2.57) hold for 1,2,,, l iN  with 1 l N iii AtA     is a convex combination.
When the relationship between the parameters and its derivations is undetermined, these conditions will be more difficult to deal with. By assuming the derivatives of parameters

  1 / jijji t  
upper bound the rate variation of the parameter. An exciting transformation method introduced in [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF] uses the differential-algebraic equation to transfer the coordinate of the vertices enclosed in the unknown derivate parameters.

Stability of Polynomial Parameter-Dependent Systems

One of the most effective ways to approximate the nonlinear systems is to represent as polynomials of the Taylor expansion. The polynomial form of state and time-varying parameters is a regular representation in a company with trigonometric forms, for example, diodes system [START_REF] Khalil | Nonlinear Systems[END_REF], jet engines [START_REF] Azuma | A new LMI approach to analysis of linear systems depending on scheduling parameter in polynomial forms[END_REF][START_REF] Fakhri | Application of Polytopic Separation Techniques to Nonlinear Observer Design[END_REF][START_REF] Watanabe | Hinf Control of Gasturbine Engines for Helicopters Helicopters Control of Gasturbine Engines for Gasturbine Engines for Helicopters[END_REF], and the academic applications [START_REF] Sala | The polytopic/fuzzy polynomial approach for non-linear control: advantages and drawbacks[END_REF][START_REF] Sala | Stability analysis of LPV systems: Scenario approach[END_REF][START_REF] Sala | Polynomial fuzzy models for nonlinear control: A taylor series approach[END_REF]Sato & Peaucelle, 2007a;[START_REF] Scherer | LMI relaxations in robust control[END_REF][START_REF] Tanaka | A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems[END_REF]Wu & Prajna, 2004), etc. Let's introduce a parameter-dependent system expressed as a polynomial formulation:

0 1 0 1 , 0, p N N j i j iji xtAtxtAAtxt xx         & (2.58)
and a parameter-dependent Lyapunov function: (2.60)

Proof.

The derivative of PDLF (2.59) along the trajectories of LPV system (2.58) is given by:

      1 00 111111 0 1 0 0 111 0000 0 ppp pp NNNNNN jjjj ijiijiijii ijijij NNNN i T T jj ijiiji jij T i ij j T ji T dV AtPtPtAt dt AAAP Pt AAtPPtPtt PPtAAt APPAPP                         1 0 1 0 p NN j i i ij j t PA      1 ,1,111 0. pp NNNN jljj klklikijii ikjli T ijij j PPt A tPtt A   

Stability of T-S Fuzzy Systems

The stability analysis for this class of nonlinear systems based on the Lyapunov theory expanded via LMI conditions. The overview of the PDC controller strategy analyzes systematically for this class of systems are introduced in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. The design analysis of the fuzzy control system at this stage is essentially a matter of finding a quadratic Lyapunov function for all stability conditions, see, e.g., [START_REF] Tanaka | Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs[END_REF][START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF] and the references therein. The relaxing stabilization conditions methods [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF][START_REF] Tuan | Nonlinear H/sub ∞/ control for an integrated suspension system via parameterized linear matrix inequality characterizations[END_REF] or the non-quadratic Lyapunov function NQLF approach [START_REF] Rheex | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF][START_REF] Sala | The polytopic/fuzzy polynomial approach for non-linear control: advantages and drawbacks[END_REF][START_REF] Sala | Polynomial fuzzy models for nonlinear control: A taylor series approach[END_REF][START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF] are considered to reduce the conservatism of the design conditions. However, the proposed conditions do not exactly describe the behavior of membership function rate but instead are subdivision into local stability by the linear combinatorial method applied to non-quadratic Lyapunov functions.

Let's consider an open-loop T-S fuzzy system:

1 0 ,dim,1,2,,, 0. l N iiil i xttAxtAniN xx        &K (2.62)
Both quadratic and non-quadratic stability is delivered and well-developed and studied in the framework of the T-S fuzzy system. It can state that the quadratic stability condition of T-S fuzzy system is essentially the same formulation as condition (2.55). It frequently encounters in the analyzing stability of the nonlinear systems approaching via linear combination. These conditions are solved at each vertex of a convex polyhedron encapsulates the behavior of the parameters. As mentioned, there does not always exist a global solution n P   S that can satisfy all LMI conditions (2.55). So, the relaxation of LMI condition has been considered to reduce conservativeness, see, for example [START_REF] Tanaka | Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs[END_REF]Tuan, Apkarian, et al., 2001). In this period, the piecewise Lyapunov functions effectively reducing the conservativeness of stability and stabilization problems have received attention, see e.g., [START_REF] Johansson | Piecewise Linear Control Systems[END_REF][START_REF] Xie | Piecewise Lyapunov functions for robust stability of linear time-varying systems[END_REF]. Based on this approach, the Fuzzy Lyapunov Function (FLF) is delivered for the non-quadratic stability analysis, provides the less conservative condition.

Most of the challenges are related to the derivative development of the membership functions (constrains this value also leads to limitation of the system dynamics). Perhaps, for this reason, much effort of the early work of the stabilizing implementations for T-S fuzzy systems focused on quadratic stability. The fuzzy Lyapunov function is implemented later in [START_REF] Mozelli | Reducing conservativeness in recent stability conditions of TS fuzzy systems[END_REF][START_REF] Sala | Polynomial fuzzy models for nonlinear control: A taylor series approach[END_REF][START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF][START_REF] Tanaka | A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems[END_REF], Etc.

Non-quadratic Stability (Bounded Parameters)

The PDLF is expanded as a fuzzy Lyapunov function under form: [START_REF] Guerra | A way to escape from the quadratic framework[END_REF][START_REF] Mozelli | Reducing conservativeness in recent stability conditions of TS fuzzy systems[END_REF][START_REF] Sala | The polytopic/fuzzy polynomial approach for non-linear control: advantages and drawbacks[END_REF][START_REF] Sala | Stability analysis of LPV systems: Scenario approach[END_REF][START_REF] Sala | Polynomial fuzzy models for nonlinear control: A taylor series approach[END_REF].

Non-quadratic Stability (Sum of Squares)

A generalization of fuzzy modeling and control presented in [START_REF] Lam | Polynomial Fuzzy Control Systems Stability Analysis and Control Synthesis Using Membership Function-dependent[END_REF][START_REF] Tanaka | A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems[END_REF] shows the refinement of nonlinearities as the fuzzy polynomial model. In which the stability and stabilizability conditions can convert to the SoS problem based on polynomial Lyapunov function [START_REF] Guerra | A way to escape from the quadratic framework[END_REF][START_REF] Jaadari | Continuous quasi-LPV Systems: how to leave the quadratic Framework?[END_REF][START_REF] Sala | Polynomial fuzzy models for nonlinear control: A taylor series approach[END_REF] develop the local sector for the polynomial-fuzzy system. The polynomial condition can be checked as an SoS or consider as the parametric polynomial matric inequality.

The defuzzification process discussed in section 2.1.4.2 can applied as follows: for the polynomial system. The preliminary concepts and the prerequisites will be covered in section 2.2.4.2. According to the SoS argument, the stability analysis for quasi-LPV system (2.65) is usually characterized by the following results. Theorem 2.2.7: [START_REF] Tanaka | A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems[END_REF] for 0, x  then the equilibrium is asymp- totical stable. If Px  is a constant matrix, the stability condition holds globally. Hence, the polynomial SoS relaxation method provides a more general condition. But the coefficients associated with irreducible polynomials in the solutions ensues in the polynomial dependencies with utopian exponents.

Relaxation of the Parameterized Linear Matrix Inequality

The parameter-dependent characterization of the stability conditions is an infinite set of LMIs across the parameter space domain. For delivering the convexity argument, the commonly used relaxations of PLMIs condition include: § Gridding technique [START_REF] Wu | Control of linear parameter varying systems[END_REF] is fragmented N-finite parameter operation range, and affine meshing parameter space [START_REF] Apkarian | Parameterized LMIs in Control Theory[END_REF][START_REF] Lim | Parameter-Varying Systems[END_REF]. § Convex combination or multi-convexities [START_REF] Gahinet | Affine parameter-dependent Lyapunov functions and real parametric uncertainty[END_REF]Tuan & Apkarian, 2002) -Fuzzy Lyapunov Function [START_REF] Guerra | A way to escape from the quadratic framework[END_REF][START_REF] Sala | The polytopic/fuzzy polynomial approach for non-linear control: advantages and drawbacks[END_REF][START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF]. § Generalization of Filner's Lemma: (1)-Sum of squares (SoS) decomposition for polynomial systems [START_REF] Papachristodoulou | On the construction of Lyapunov functions using the sum of squares decomposition[END_REF][START_REF] Parrilo | Semidefinite programming relaxations for semialgebraic problems[END_REF] -for fuzzy systems [START_REF] Lam | Polynomial Fuzzy Control Systems Stability Analysis and Control Synthesis Using Membership Function-dependent[END_REF][START_REF] Sala | Polynomial fuzzy models for nonlinear control: A taylor series approach[END_REF][START_REF] Tanaka | A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems[END_REF]. ( 2)-Slack variable or S-variable [START_REF] Sato | Robust stability/performance analysis for linear timeinvariant polynomially parameter-dependent systems using polynomially parameter-dependent Lyapunov functions[END_REF][START_REF] Sato | Robust stability/performance analysis for uncertain linear systems via multiple slack variable approach: Polynomial LTIPD systems[END_REF].

The gridding technique is simple and can be deployed directly on the parameter dependence condition. Following the argument of (Apkarian & Tuan, 2000a[START_REF] Lim | Parameter-Varying Systems[END_REF][START_REF] Wu | Control of linear parameter varying systems[END_REF]: with finite intervals, it is impossible to verify whether it captures all of the critical points or describes the nonlinear behavior of the parameters t within their defined boundaries , .   In the work of [START_REF] Apkarian | Parametrized LMIs in control theory[END_REF][START_REF] Lim | Parameter-Varying Systems[END_REF], the author has introduced an approach solving the parameterized LMIs (PLMIs), which only need to grid a surface of lower dimension whenever the function is quasi-convex or convex along some direction. But it costs in high computation load.

During this time, the relaxation based on the sum of squares decomposition has also gained considerable attention that separates of the polynomial parameters and can be cast as a semidefinite programming problem, see, e.g., the convex optimization SoS toolbox [START_REF] Papachristodoulou | SOSTOOLS: Sum of squares optimization toolbox for MATLAB[END_REF][START_REF] Prajna | Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach[END_REF]. This approach promises less conservative results for the relaxation of the PLMI conditions. It is also widely employed for the analysis of stabilizing analysis of the qLPV & T-S fuzzy systems in the literature such as [START_REF] Gahlawat | Control and verification of the safetyfactor profile in Tokamaks using sum-of-squares polynomials[END_REF][START_REF] Sala | The polytopic/fuzzy polynomial approach for non-linear control: advantages and drawbacks[END_REF][START_REF] Sala | Polynomial fuzzy models for nonlinear control: A taylor series approach[END_REF][START_REF] Tanaka | A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems[END_REF]. But the fractional form involving the polynomial gains is an obstacle of the implementation. For example: a feedback gain Then, how to perform the inverse of Xt   ?

An alternative method is to convert the polynomial parameter-dependent condition into a Slack-Variable formulation [START_REF] Ebihara | S-Variable Approach to LMI-Based Robust Control[END_REF][START_REF] Hosoe | S-variable approach to robust stabilization state feedback synthesis for systems characterized by random polytopes[END_REF]Sato & Peaucelle, 2007a[START_REF] Sato | Robust stability/performance analysis for uncertain linear systems via multiple slack variable approach: Polynomial LTIPD systems[END_REF]. This method, also based on Finsler's generalization, shows a computational advantage over SoS decomposition. Furthermore, the S-variable allows to manipulate , Xtt    X X is t independent variable matrix, and t    is a column of parameters. A numerical computation comparison of the two methods is given in (Sato & Peaucelle, 2007a[START_REF] Sato | Robust stability/performance analysis for uncertain linear systems via multiple slack variable approach: Polynomial LTIPD systems[END_REF], and a numerical comparison of the optimization of polynomial methods provides in Section 3.2.2. More details on the S-variable application to the robustness and stability analysis of the system based on LMI conditional developments refers to [START_REF] Ebihara | Robust H2 Performance Analysis Of Uncertain Lti Systems Via Polynomially Parameter-Dependent Lyapunov Functions[END_REF][START_REF] Ebihara | S-Variable Approach to LMI-Based Robust Control[END_REF][START_REF] Hosoe | S-variable approach to robust stabilization state feedback synthesis for systems characterized by random polytopes[END_REF]Dimitri Peaucelle & Ebihara, 2014).

The proposed methods have distinctive advantages and disadvantages, which may be the trade-off between conditional conservatism and computational complexity. For example, the gridding method simplifies the parameter-dependent conditions into a set of LMI conditions, but the weakness of this linearization is whether it covers all the critical points or how to accurately describe specific characteristics of parameters on operating conditions. On the other hand, the SoS method is characterized by the polynomial Lyapunov functional formulation. Giving more tight relaxation on variation of the S-procedure constraints are exchanged with complexity in parametric decomposition. Finally, the multiconvexities Lyapunov function is a linear combination between vertices, covering the trajectories of the parameter within the convex domain for analyzing controller design.

The synthesis of relaxing PLMI methods discusses in the literature [START_REF] Apkarian | Parameterized LMIs in Control Theory[END_REF]El Ghaoui & Niculescu, 2000;[START_REF] Tuan | Relaxations of parameterized LMIs with control applications[END_REF]. The promising fruition of the LPV control system in this time inherited the growth of linear programming or convex optimization tools (Erling D [START_REF] Andersen | The Mosek Interior Point Optimizer for Linear Programming: An Implementation of the Homogeneous Algorithm[END_REF][START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF][START_REF] Boyd | Convex Optimization[END_REF][START_REF] Gahinet | LMI Control Toolbox For Use with MATLAB[END_REF][START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF][START_REF] Papachristodoulou | SOSTOOLS: Sum of squares optimization toolbox for MATLAB[END_REF]. Which converts the linear matrix inequality (LMI) constraint derived from the stability analysis into barrier function conditions. Readers interested in the interior-point methods or the other contemporary methods, for example, conjugate gradient, golden section, the wider scope of polynomial-time complexity, can refer to the monographs [START_REF] Bertsimas | Nonlinear programming[END_REF][START_REF] Bertsimas | Introduction to Linear Optimization[END_REF][START_REF] Konno | Optimization on Low Rank Nonconvex Structures[END_REF]. Bertsimas describes in more detail algorithm problems (i.e., gradient descent, update step size, etc.). For optimization problems and global optimization can be found in [START_REF] Tuy | Convex Analysis and Global Optimization[END_REF]. The work of literature (Hiriart-Urruty & Lemaréchal, 1993a[START_REF] Hiriart-Urruty | Convex Analysis and Minimization Algorithms II[END_REF] provides multiple sections appropriately devoted to readers.

Relaxation of Parametrized LMIs by Discretization

It is possible to refer the finite discretizing methods over time-varying intervals [START_REF] Apkarian | Parametrized LMIs in control theory[END_REF][START_REF] Lim | Parameter-Varying Systems[END_REF] and time-independent intervals [START_REF] Wu | Control of linear parameter varying systems[END_REF]. The LMI relaxation proposed by [START_REF] Wu | Control of linear parameter varying systems[END_REF], well-known as a "gridding" method, illustrates in the following example. its determination remains a difficult question. It is difficult to assert an ideal density of a parameter domain because their unfeasible regions are ambiguous information. The unfeasible set is estimated only when the infeasibility of the problem finds. That can visualize by adjusting the sound of an instrument that has to hit a chord to adjust the correct range. We cannot fine-tune before playing an instrument, and we cannot copy the tuning of one musical accessory to another. Like the lament of Briat: "This paradox shows that probably no method to find a perfect gridding would develop someday."

On the other hand, the piecewise affine parameter-dependent (PAPD) approaches introduced as multi-switch partitioned parameter space, see, e.g., [START_REF] Apkarian | Parametrized LMIs in control theory[END_REF][START_REF] Lim | Parameter-Varying Systems[END_REF], could provide less conservative stability conditions. The parametric switch subsystem is recalled in Appendix 1.4.2. But the number of LMI condition that must check is overwhelming. For example, given a LPV system depend on p parameters, each parameter is partitioned into i N subspace, so the number of conditions that need to be checked is about

1 2. p p ii N 

Relaxation of Parametrized LMIs by Sum-of-Squares Decomposition

Generally, the stability conditions based on the parameter-dependent Lyapunov function has difficulty expressing the derivative expansion of the parameter. But the polynomials allow for easier development and expand the partial derivation of state with the endogenous parameter polynomial formulation. Let's recall some mathematical premises mainly introduced by [START_REF] Parrilo | Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[END_REF][START_REF] Parrilo | Semidefinite programming relaxations for semialgebraic problems[END_REF][START_REF] Prajna | Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach[END_REF]. Lemma 2.2.1. [START_REF] Parrilo | Semidefinite programming relaxations for semialgebraic problems[END_REF] The canonical decomposition of a polynomial is expanded to SoS that can be converted to a semidefinite programming problem. The analysis of polynomials are introduced by [START_REF] Papachristodoulou | SOSTOOLS: Sum of squares optimization toolbox for MATLAB[END_REF][START_REF] Papachristodoulou | On the construction of Lyapunov functions using the sum of squares decomposition[END_REF][START_REF] Prajna | Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach[END_REF] could handle the stabilization PLMI condition as SoS expressions.

Lemma 2.2.2. [START_REF] Prajna | Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach[END_REF]) Let Fx  be an NN  polynomial matrix degree 2d in .

n

x  R Moreover, let Zx  be a column vector whose entries are all monomials in x with degree no greater than d, then the following statements

(1) 0 Fx  ± for all . n x  R
(2) T vFxv  is a sum of squares, where . The polynomial LMI stability are the infinite-dimensional parameter-dependent conditions. Following the SOS-based polynomial method [START_REF] Parrilo | Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[END_REF][START_REF] Parrilo | Semidefinite programming relaxations for semialgebraic problems[END_REF], the monomial in x (variables) are apart from its coefficients (decision variables) in the polynomial conditions. Then, some slack variables are injected during the S-procedure, which converts the infinite parametric conditions into a finite LMIs (that able to be solved by the interiorpoint method with solvers such as Mosek, Sedumi, SDPT3, SDPA, etc.). The variation of the S-procedure constraints provides significantly relaxed condition than the existing approaches (discretization in section 2.2.4.1, or convex combination in section 2.2.1 and 2.2.3.1). In addition, there are other methods to decompose the polynomial matrix inequalities, see, for example the S-variable approach [START_REF] Ebihara | Robust H2 Performance Analysis Of Uncertain Lti Systems Via Polynomially Parameter-Dependent Lyapunov Functions[END_REF][START_REF] Sato | Robust stability/performance analysis for linear timeinvariant polynomially parameter-dependent systems using polynomially parameter-dependent Lyapunov functions[END_REF], 2007a).

Example

According to the representation for each model (i.e., LPV, T-S fuzzy, Polynomial, Polynomial Fuzzy) discussed in Section 2.1.5, the characteristic stability analysis will be deployed by a suitable approach (e.g., gridding, convex combination, SoS, etc.).

Relaxation of Parameter-Dependent Lyapunov Function

It should be noted that the stability analysis using the Lyapunov quadratic form i.e., condition (2.55) of Theorem 2.2.2 fails to verify the stability of T-S fuzzy system (2.32).

Example 2.2.2. Let's consider the representations of nonlinear system (2.25), including affine system (2.28), polynomial system (2.29), T-S fuzzy system, (2.32) and fuzzy polynomial (2.35).

LPV model & Gridding

The stability for LPV system (2.28) is analyzed using the compact set of the parameters defined in (2.71) As discussed in section 2.2, these conditions can either handle by the sum of squares decomposition method, or directly treated by the SoS toolbox. Both methods are based on a generalization of Finsler's lemma. In this example, the polynomial matrix conditions are converted to the SoS expressions and solved by SoS toolbox. The higher number of the monomial degree increases the computational complexity, so if the polynomials with the highest degree are fuzzification, then the fuzzy polynomial model is theoretically more advantageous than the original polynomial model.

Polynomial model & SoS

Followed the line of Theorem 2.2.7, polynomial system (2.29) 2.301101.009100.4532.194100.7493.881, 1.304102.849100.2841.412107.157101.566 2.036101.822100.4361.576100.3640.827, 9.019102.715100.2071.329106.498101.482, 2.994102.267103.475103 1.846100.506, 4.215102.074100.1921.700100.3650.682 1.960108.877101.0767.6639.17620.89, 6.401102.215100.4483.1970.0731.877, 2.845101.209101.0441.806106.5426 1.904101.719100.3083.4891.8502.095, 3.469102.125100.5641.215102.2002.305, 3.781101.529101.1712.798102.1864.34

xxxxxx Pxxxxxx Pxxxxxx       7.
The computational time is taken to solve the conditions by Gridding and convex combinations methods is 14.4237 and 0.2297 seconds, respectively, while the time spent on sum-squared polynomials is 19.1659 and 24.6257 seconds. The characteristic convergence of the region of stability is shown in Figure 2-5. However, we cannot draw any further conclusions. Each relaxation method has its own advantages and disadvantages, which are compatible with each type of LPV representation.

Theoretically, the SoS decomposition should give the least conservative results. But up to now, the limitations of numerical computation (e.g., SoS toolbox), have not allowed to take full advantage of the lossless transformation. Specifically, the time takes to process sorting, separating the variables, and solving the conditions will increase exponentially.

Besides, this numerical computation tool is sensitive to complex conditions (e.g., a large scale PLMI condition). On the other hand, the convex polyhedron method (such as the stability conditions of T-S fuzzy or polytopic systems) provides solutions with a reasonable algorithmic time is 0.2297s compared to 24.6257s of the polynomial-fuzzy SoS method. Finally, the gridding method is straightforward to handle the parameter-dependent conditions by discretizing the parameter domain. However, the computational time to treat the LMI conditions also increases exponentially with the number of parameters.

There's always a price to pay!

The Conservativeness of Fuzzy and Fuzzy Polynomial Lyapunov Functions

In the last section, the relaxation of PLMI stability conditions analyzed for LPV systems has delivered satisfactory results (except for the quadratic Lyapunov, which returns an infeasible result). Right after, an example of [START_REF] Sala | Polynomial fuzzy models for nonlinear control: A taylor series approach[END_REF] is used to show conservatism of the parameter-dependent LMI conditions where traditional relaxation methods do not work (as illustrated in Table 2.1). The approximations of the nonlinear model to the LPV representations can be fulfilled similarly to section 2.1.5. For details of the transformation can be found in the literature cited below. If this designed accuracy radius is achieved, then info.numerr is set to 0 [START_REF] Sturm | Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones[END_REF]. A way out of these misunderstandings is to "transform" all semi-definite inequalities into definite inequalities [START_REF] Labit | SEDUMI INTERFACE 1.02: a tool for solving LMI problems with SEDUMI[END_REF]Dimitri Peaucelle et al., 2002). In this work, we introduce a positive definite scalar in all inequalities s.t., 000 0, ,

nn n PPPI    ±R± (2.87)
with a small arbitrary constant .   ò This constraint transformation is expected do not add some extra conservatism and tune the solver convergence without strongly modifying the feasibility radius. It should be noted that a strict inequality 0 0 P f is not recommended on the current numerical computation tools such as Yalmip. So, numerical adjustment (2.87) provides an appropriate modification to the linear matrix inequalities (e.g., the stability condition of the T-S fuzzy multi-convex system), but it doesn't fit the SoS constraints. There are two reasons, first it can be asserted that there is no solution, and the second is that the polynomial matrix inequality can be positive but not a sum of squares. Accordingly, we applied a refinements of the sum of squares polynomials proposed by [START_REF] Sala | Polynomial fuzzy models for nonlinear control: A taylor series approach[END_REF] 4.688101.249102.398100.31822.463107.40110, 8.674100.31818.474104.0101.141106.98710, 0.31813.134101 Though the old-fashioned sum of squares decomposition method failed to approve the stability of the fuzzy polynomial system, the obtained results demonstrate the effectiveness of the Positivstellensatz multipliers polynomial approaches. However, this method is also sensitive to the numerical problem, which regularly entails unsatisfactory results related to, e.g., the complex stabilization conditions [START_REF] Furqon | An SOS-Based Control Lyapunov Function Design for Polynomial Fuzzy Control of Nonlinear Systems[END_REF]Sala & Ariño, 2008). On the other hand, the arbitrary uniform discretization over the parameter domain (i.e., gridding) delivers promising results (as observed in Table 2.1), with the advantage of being straightforward to perform on the polynomial LMI conditions.

Through these two examples, it can be emphasized that all the stabilization conditions developed for saturated LPV systems in this thesis are formulated by the parameter dependency matrix inequalities, e.g., condition (2.73), which are solved by the gridding over the defined domain of parameters. The decision matrices such as Lyapunov candidate will be chosen polynomial form, for example:

22 011221212111222 . P PPtPtPttPtPt         
This polynomial expression is simpler to unify than other nonlinear forms such as trigonometric forms.

The Saturation Nonlinearity -Stabilization Analysis

In control system design analysis, a phenomenon observed in many engineering systems, chemical processes, biology, and even economics, is saturation in actuators. At first glance, the effect of this nonlinearity is quite simple, but analysis inappropriately or ignoring its effects can lead to performance degradation or system instability. Actuator saturation is unavoidable in engineering practical dynamics systems concerning physical limits (velocity, voltage, cycle, etc.) and safety constraints (pressure, temperature, power, energy consumption, etc.). Besides, the saturation effect characterized as nonlinearity cannot linearize. That must find a stabilization method of replacing the operating points of the feedback control system into the region without element saturates. In the last decades, considerable attention has been devoted to LTI systems subject to actuator saturation, see for instance [START_REF] Hu | Control Systems with Actuator Saturation[END_REF][START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF][START_REF] Zaccarian | Modern Anti-windup Synthesis: Control Augmentation for A tuator Saturation[END_REF] and the references therein.

There are two main approaches to carry out the stabilization analysis in the literature. The first one considers the saturation bounds like a prerequisite in the design strategy (João Manoel Gomes da Silva & Tarbouriech, 1999;[START_REF] Henrion | Control of linear systems subject to time-domain constraints with polynomial pole placement and LMIs[END_REF][START_REF] Henrion | LMI relaxations for robust stability of linear systems with saturating controls[END_REF][START_REF] Hu | An analysis and design method for linear systems subject to actuator saturation and disturbance[END_REF][START_REF] Hu | Composite quadratic Lyapunov functions for constrained control systems[END_REF][START_REF] Tarbouriech | Stability analysis and stabilization of systems presenting nested saturations[END_REF]. On the other side, an asymptotical stabilizing synthesis proposed for a closed-loop system disregards the control bounds. Then, a suitable design strategy will analyze to compensate for the saturation, such as Direct Linear Anti-windup (DLAW) or Model Recovery Antiwindup (MRAW). The anti-windup domain has been discussed thoroughly in recent decades, see for example [START_REF] Galeani | Reduced order linear anti-windup augmentation for stable linear systems[END_REF][START_REF] Galeani | A Tutorial on Modern Anti-windup Design[END_REF][START_REF] Gomes Da Silva | Antiwindup design with guaranteed regions of stability: An LMI-based approach[END_REF][START_REF] Grimm | Antiwindup for Stable Linear Systems with Input Saturation: An LMI-Based Synthesis[END_REF][START_REF] Hu | Anti-windup synthesis for linear control systems with input saturation: Achieving regional, nonlinear performance[END_REF][START_REF] Wu | Anti-windup controller design using linear parameter-varying control methods[END_REF][START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF], 2005[START_REF] Do | Approche LPV pour la commande robuste de la dynamique des véhicules : amélioration conjointe du confort et de la sécurité[END_REF] and the references therein. It can realize that the closed-loop stabilization analysis for DLAW and MRAW is more complicated in considering the effect of the nonlinear behavior and uncertain dynamics. Nonetheless, the DLAW construction is beyond the scope of the dissertation, so it will not be included. Instead, controversial analyzes of the Anti-windup compensator issue using the differential-algebraic equations to constrain the DOF controller are presented in (Bui Tuan et al., 2021).

It can be emphasized that many cited research papers and books are devoted to LTI sys-tems. Obviously, the saturated synthesis for the LPV/quasi-LPV systems isn't fit in stabilization analysis for saturated LTI systems. However, just several works in the literature subject to the saturation problems analyzed for nonlinear systems or parametric dependence systems, for example, the LPV systems [START_REF] Cao | Set invariance analysis and gain-scheduling control for LPV systems subject to actuator saturation[END_REF][START_REF] Forni | Model based, gain-scheduled anti-windup control for LPV systems[END_REF], 2010;Kapila & Grigoriadis, 2002;[START_REF] Nguyen | Gain-scheduled static output feedback control for saturated LPV systems with bounded parameter variations[END_REF][START_REF] Roos | On-Ground Aircraft Control Design Using an LPV Anti-windup Approach[END_REF][START_REF] Theis | Observer-based LPV control with anti-windup compensation: A flight control example[END_REF][START_REF] Wu | Anti-windup controller design using linear parameter-varying control methods[END_REF] or T-S fuzzy systems [START_REF] Benzaouia | Advanced Takagi-Sugeno Fuzzy Systems: Delay and Saturation[END_REF][START_REF] Dey | Stability and Stabilization of Linear and Fuzzy Time-Delay Systems[END_REF]. So, let's discuss one stability analysis tool used for the saturated LPV systems and LPV time-delay systems.

Sector Nonlinearity Model

A representation of LPV systems with actuator saturation gives under the forms: [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], with the assumptions that openloop poles are located in the closed left-half plane, and the set of admissible-initial conditions is explicitly defined. But it can notice that the stabilizing condition analysis for system (2.1) is typically localized corresponding to the assumptions about the compact set of the parameters (2.8)-(2.9). Similar to the LPV system, the T-S fuzzy system is characterized by the operating ranges of the fuzzificated functions. Perceptibly, the global stabilization condition does exist in control synthesis for this class of systems. But when saturation limits are involved, the local stabilization condition is more reasonable.

Saturation nonlinearity.

Following this approach, it is generally classified into three representations used for closed-loop system with saturated actuators:

(1)-Polytopic models,

(2)-Sector nonlinearity models, and

(3)-Regions of saturation models.

Giving a control input vector 12 ,,,

            (2.90)
The notations :max,:min iiii uutuut  are used for the purpose of simplifying the presentation. As analyzed in the literature [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], the polytopic bounds technique and regions of saturation might result in higher computational complexity, respectively, with 2, m and 3 m conditions compared with sector nonlinearity has only m conditions. Furthermore, it worthy to note that the region of stability of the first two approaches seem to be equally scaled (under the same primary assumptions), if not to say the local sector bounding provides better performance. Since mentioned works are commonly imposed on a symmetric saturation (there is rare work realizing for asymmetric saturation) so the lower bounded is typically set by , with 0.

iii uuu  which can be recognized as the prototype GSC condition in [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. However, the existence of vector vt in the GSC condition and polyhedral set It is possible to find the feedback control laws (e.g. ) utKxt   so that the unsatu- rated closed-loop system is stable (i.e. ABK   is Hurwitz ).

p t  U Due to actuator saturation, there exist initial conditions that could lead to divergence of the closed-loop system, an incorrect convergence of the equilibrium point away from the origin, or a destabilization [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. However, determining the initial conditions such that the trajectories of the closed-loop system are asymptotical stable is not simple as it seems. Therefore, the estimation of R A related with the admissible set of initial conditions is generally encountered in saturation control synthesis, see for examples [START_REF] Cao | An antiwindup approach to enlarging domain of attraction for linear systems subject to actuator saturation[END_REF][START_REF] Gomes Da Silva | Local stabilization of discretetime linear systems with saturating controls: an LMI-based approach[END_REF], 2005;[START_REF] Hu | An analysis and design method for linear systems subject to actuator saturation and disturbance[END_REF][START_REF] Hu | Anti-windup synthesis for linear control systems with input saturation: Achieving regional, nonlinear performance[END_REF]. In this thesis, a stability analysis tool for the dynamic system primarily deploys by considering the various forms of Lyapunov function. The associated level sets are given by the characterized domains corresponding to:

  1 1 ,, , , l N T xii i PxtVxtxtPxt           ED (2.103)   1 0, 1,1,,,1. l l N N lili i iNx     U R K (2.104) Polynomial Fuzzy Lyapunov function (PFLF) candidate:   1 1 ,, , . l N T xiil i PxtVtxtPxxt         EDU (2.105)
Then, the problem is formulated by finding the matrices P are defined in (2.100)-(2.105) so that the given level sets , P   E are regions of asymptotical stability for the closed-loop system. Since the parameter dependency form (2.102) able to present both (2.103), (2.105), and if t is constant then it yields to quadratic form, so this formulation is more general. Henceforth, this parameter-dependent elliptical domain is employed mostly, the remaining forms are only considered in specific cases.

Among the Lyapunov functions are derived for stability analysis of LPV system (2.1)

(without external disturbance), the quadratic formulation generally leads to strict conditions (illustrated by the smallest ellipsoid, as seen in Figure 2-7.a). The next figure shows the piecewise Lyapunov function commonly delivered for switching LTI systems. In visualization, we can see that the estimates of the region of asymptotic stability of the parametric Lyapunov functions (FLF and PFLF) are significantly larger. Nevertheless, the optimization problems such as the size or performance criteria results in the parameterized conditions. The relaxation of these PLMIs depends on the structure of transformation used for the stabilization condition.

Optimization problems.

The elliptic domain optimization discussed in [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] [START_REF] Gomes Da Silva | Local stabilization of discretetime linear systems with saturating controls: an LMI-based approach[END_REF]Tarbouriech, , 1999;;[START_REF] Hu | An analysis and design method for linear systems subject to actuator saturation and disturbance[END_REF][START_REF] Hu | Anti-windup synthesis for linear control systems with input saturation: Achieving regional, nonlinear performance[END_REF] or a convex combination piecewise Lyapunov function [START_REF] Hu | Composite quadratic Lyapunov functions for constrained control systems[END_REF]. Without loss generality, we might consider the unit level set   1, to simplify the optimization problems involved in bilinear couple between  and P. In Chapter 4, the maximization of the minor axis carries out for the parametric ellipsoidal set (2.102). The optimization problem is even more interesting in implemented on an LPV time-delay system with stability analysis based on the Lyapunov-Krasovskii function discussed in Chapter 6.

On the other side, in consideration of the effect of the external disturbances and the initial condition, the optimization of performance requirement consisted of finding the L2-gain scheduling controller for saturated system (2.1) so that the criterion is satisfied:

22 22 11 0 ztwt    LL (2.106)
where 0  is maximum of the non-null admissible initial conditions 0 . , P   E And, an energy bounded exogenous signal such as:

2 1 ,|. d wwwt     L W R
Actually, the optimization problem is the trade-off between the estimation of the region of asymptotical stability (RAS), the level of attenuated disturbances, and the region of linear behavior (characterized by the unsaturated regulation region).

Saturated Feedback Control Synthesis

The feedback controller structures illustrated in Figure 2-8 are proposed to stabilize the saturated LPV systems. The PLMI stabilization conditions are derived from stability analysis using parameter-dependent Lyapunov (2.102). Concerning constraints defined by polyhedrons   ,, u  S the analysis of the saturation bounds address as follows:

21 2 ,, 1,2,,, T s s t Vtttt uxx m Px ts       K (2.107) if 11 0 0: , 0,0 , w x tVtt x V    (2.108) if 2 1 1 2 1 0 0: , 0, . 0 w xxwt tVttV           L (2.109)
The necessary condition (2.107) is set directly for each auxiliary controller . s t   And, the sufficient conditions (2.108)-( 2.109) related to the stabilization condition (with and without the influence of disturbance), it is guaranteed that the closed-loop system trajectories is confined within the level set of the ellipsoidal domain , t P   E from the initial conditions are belonged to this domain. The satisfaction of the necessary and sufficient conditions means that the ellipsoid is included in the polyhedron set   ,. u  S From Corollary 2.3.1, the following GSC condition holds

1 0. T uTuttu    (2.110)
So, depending on the feedback controller structure , ut we will choose the appropriate auxiliary controller structure , t so that the combination of these two vectors in the latter GSC condition is convenient for the designed purpose. In the following sections, the necessary conditions are specifically designed for each controller structure (state feedback, observer-based feedback, static output feedback, and dynamic output feedback) correspond to each optimization method.

Parameterized State Feedback Controller

The simplest method of robust stabilizing and performance analysis for the LPV system is evident the development of the state feedback law. Where the stabilization problem can express directly as the PLMI condition. The studies have been coherently discussed for the continuous and discrete LTI system in the work of (J.M. Gomes da Silva [START_REF] Gomes Da Silva | Local stabilization of discrete-time linear systems with saturating controls: an LMI-based approach[END_REF][START_REF] Gomes Da Silva | Local stabilization of linear systems under amplitude and rate saturating actuators[END_REF][START_REF] Gomes Da Silva | Contractive polyhedra for linear continuous-time systems with saturating controls[END_REF][START_REF] Hu | An analysis and design method for linear systems subject to actuator saturation and disturbance[END_REF][START_REF] Hu | Stability and performance for saturated systems via quadratic and nonquadratic Lyapunov functions[END_REF][START_REF] Hu | Anti-windup synthesis for linear control systems with input saturation: Achieving regional, nonlinear performance[END_REF][START_REF] Tarbouriech | Stability analysis and stabilization of systems presenting nested saturations[END_REF][START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF].

Considering compact sets of parameters (2.8)-( 2.9) and a gain-scheduled state feedback controller ,sat. utKxtuKxtu   Then, the closed-loop system is ob- tained by substitute this gain-scheduled state feedback controller to saturated LPV system (2.89) as follows: In Section 4.1.2, the optimization problems involve in the sets of the admissible initial condition 0, the estimation of the ellipse domain , the upper bound of the disturbance  -1 and the rejection disturbance level  is investigated. But, the minimization of disturbance attenuation entailing the decline of the linear operating area (region of unsaturated control signals), and predominantly effects on the generalized sector condition. So, an enhancement of the control system's performance could be done by utilizing the D-stable LMI method to relocate the pole of the closed-loop system.

Parameterized Static Output Feedback Controller

Since a relevant design case impractical control engineering as some states is unmeasurable, the full-state feedback controller is not appropriate for the implementation. But, solving the stabilization condition of static output feedback (SOF) is much more difficult, usually leading to a nonconvex, bilinear matrix inequality (BMI) [START_REF] Sadabadi | From static output feedback to structured robust static output feedback: A survey[END_REF][START_REF] Syrmos | Static output feedback -A survey[END_REF].

On the one side, the iterative LMI algorithm [START_REF] Cao | Static output feedback stabilization: An ILMI approach[END_REF][START_REF] He | An Improved ILMI Method for Static Output Feedback Control With Application to Multivariable PID Control[END_REF], the algebraic equation [START_REF] Gossmann | Parameter dependent static output feedback control-An LPV approach[END_REF][START_REF] Syrmos | Static output feedback -A survey[END_REF], the iterative global optimization CCL algorithm [START_REF] El Ghaoui | A cone complementarity linearization algorithm for static output-feedback and related problems[END_REF], the two-steps algorithm with output structural constraints (D. [START_REF] Peaucelle | An efficient numerical solution for H2 static output feedback synthesis[END_REF], the congruence transformation [START_REF] Dong | Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties[END_REF][START_REF] Prempain | Static output feedback stabilisation with H∞ performance for a class of plants[END_REF] and the S-variable method [START_REF] Ebihara | S-Variable Approach to LMI-Based Robust Control[END_REF][START_REF] Pipeleers | Extended LMI characterizations for stability and performance of linear systems[END_REF] have been proposed to cope with SOF control design problem. Besides, the other unsaturated SOF controller synthesis could refer to [START_REF] Chang | New Results on Output Feedback <formula formulatype="inline"> <tex Notation="TeX">$H_{\infty} $</tex></formula> Control for Linear Discrete-Time Systems[END_REF][START_REF] Gossmann | Parameter dependent static output feedback control-An LPV approach[END_REF][START_REF] Kau | Robust H∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties[END_REF][START_REF] Nguyen | Static output feedback design for a class of constrained Takagi-Sugeno fuzzy systems[END_REF][START_REF] Nguyen | Gain-scheduled static output feedback control for saturated LPV systems with bounded parameter variations[END_REF][START_REF] Qiu | Static-output-feedback H∞ control of continuoustime T-S fuzzy affine systems via piecewise lyapunov functions[END_REF], where a gain-scheduled static output feedback (SOF) controller law is generally designed with the form . utKyt   However, there is still room for researching and developing the satu- rated SOF structure implemented in the LPV systems.

On the other side, a controller gain could be considered by

1 , KYW    
and a congruent transformation is deployed like [START_REF] Dong | Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties[END_REF]Nguyen et al., 2018). Let's consider SOF controller 1 utYWyt   for saturated LPV system (2.89), then a closed-loop system is represented as a parametric dependence formulation:

    1 , , ,
.

w w xtABKCxtBuBwt ztHJKCxtJuJwt ytCxt utKytYWCxt          & (2.113)
The stabilization PLMI condition is derived from a congruence transformation treated the bilinear structure matrix 1 YWC    without using the strong mathematical constraints. Due to this specific construction, some scalar variable injects into the design condition. Generally, the gridding logarithm searches linearly on a scale in the interval, e.g., 10,10, nn   withn   N is a positive integer. In Section 4.2, we use this controller structure to analyze the local stabilization involved in the expansions of the polyhedron set provided in conditions (2.107)-(2.109).

Parameterized Observer-Based Output Feedback Controller

In the branch using measurement outputs to design control systems, the dynamic output feedback (DOF3 ) and observer-based feedback controller (OBF) have gained considerable attention in recent decades. Each approach has its benefits and drawbacks. The observer-based controller structure is an exceptional form of DOF, introduced by [START_REF] Cristi | Dynamic Output Feedback by Robust Observer and Variable Structure Control[END_REF], provided a simple construction and easier to implement. Alternatively, the DOF full-block design analysis was presented by (Chilali et al., 1996;[START_REF] Gahinet | Affine parameter-dependent Lyapunov functions and real parametric uncertainty[END_REF]Gahinet & Apkarian, 1994;[START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF] for the LTI systems proposed a congruence transformation with new substitution of variables. Both approaches have been analyzed and implemented in a wide range of system engineering.

Besides, it can mention that there are two common approaches for observer-based controller synthesis: the 2-step separated strategy and the full-block observer-based output feedback control framework. Whereas one method separates the design of the observer from the controller, it is common to apply to accurately known systems (the states are not influenced by the uncertainties).

On the other hand, the simultaneous design method of the extended system includes system dynamics, and the estimated error could handle the parameter uncertainty. In this case, the observation error involves in the input control and state of the plant with the feedback controller . ûtKxt  The second method focused in this thesis shows many difficulties and more challenges for control design strategy. For readers interested, more details about observer-based controller develop for LTI systems [START_REF] Lien | Robust observer-based control of systems with state perturbations via LMI approach[END_REF], for nonlinear Lipschitz systems [START_REF] Ahmad | Observer-based robust control of one-sided Lipschitz nonlinear systems[END_REF][START_REF] Ibrir | Observer-based control of discrete-time Lipschitzian non-linear systems: Application to one-link flexible joint robot[END_REF][START_REF] Zemouche | Robust observerbased stabilization of Lipschitz nonlinear uncertain systems via LMIs -discussions and new design procedure[END_REF][START_REF] Zemouche | A new LMI based H∞ observer design method for Lipschitz nonlinear systems[END_REF]Zemouche & Boutayeb, 2013), for nonlinear systems represented by T-S fuzzy model [START_REF] Benzaouia | Advanced Takagi-Sugeno Fuzzy Systems: Delay and Saturation[END_REF][START_REF] Bui Tuan | Robust TS-Fuzzy observer-based control for Quadruple-Tank system[END_REF][START_REF] Bui Tuan | Robust Observer-Based Control for TS Fuzzy Models Application to Vehicle Lateral Dynamics[END_REF][START_REF] Dahmani | Observer-Based Robust Control of Vehicle Dynamics for Rollover Mitigation in Critical Situations[END_REF]Dahmani, Pages, El Hajjaji, et al., 2015;[START_REF] Gassara | Design of polynomial fuzzy observer-controller for nonlinear systems with state delay: sum of squares approach[END_REF], for LPV system (Briat, 2015a;[START_REF] Heemels | Observer-based control of discrete-time LPV systems with uncertain parameters[END_REF], for the LTI saturated system [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], and the references therein. This section targets to deliver necessary conditions of feedback control based on the structure observer as follows: The stabilization of closed-loop system (2.115) is analysis like unsaturated observerbased feedback control system, with an additional reform in the constrained control conditions that deploy the same in section 2.3.2.1. Now, considering an ellipsoid as region of asymptotic stability for system (2.115),

 , ˆˆ, xtAxtButBuLCxtCxt utKxt     & ( 2
    121 1 2 ,,, ,, , , T xe
  L 1 1 1 0         (2.119)
However, the traditional approach of analytical stabilization conditions for this controller has a major drawback (the conservatism will expose in Section 4.2 and 4.5.1).

Parameterized Dynamic Output Feedback Controller

The full-block output-feedback control law framework has also earned a lot of interest in a wide range of control syntheses. The early methodologies of the dynamic output feedback controller synthesis could mention (Chilali et al., 1996;[START_REF] Gahinet | Explicit controller formulas for LMI-based H ∞ synthesis[END_REF]Gahinet & Apkarian, 1994;[START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF] employed on the LTI systems; [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF]Apkarian & Gahinet, 1995;[START_REF] Apkarian | Parameterized LMIs in Control Theory[END_REF][START_REF] Lim | Parameter-Varying Systems[END_REF]Tuan & Apkarian, 2002) deployed for uncertain parameters and LPV systems, the LPV time-delay systems discussed in [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF](Briat, , 2015a)), and a dynamic parallel distributed compensation (DPDC) analyzed for the T-S fuzzy systems in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF] via LMI conditions for both cubic, quadratic and linear parameterizations. The synthesis methods is presented by Gahinet, but the ℋ∞ performance synthesis is widely known by [START_REF] Scherer | A full block S-procedure with applications[END_REF] especially for the LTI systems. In essence, the stabilization problem for saturated DOF controllers is alternatively approached for the anti-windup strategy (DLAW or MRAW). In monograph [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] From initial condition 0 , E the analysis of saturation constraints on auxiliary vector t  are expanded similarly to the previous sections.

2 21 2 21 , T ss s T N uuttt t N                        Y G é 52 CHAPTER 2. OVERVIEW LINEAR PARAMETER-VARYING SYSTEMS if 0 11 0 0 0 , 0 0: 0 TT TT t w NN t NN t                YY éé if 2 2 00 0 00. 0: TT TT w t NN twt N t N            YY L éé 1 1 1 0         (2.127)
The nonlinear terms in the stabilization conditions related to system construction (2.121),

or the heterogeneous form with the occurrence of é in the latter conditions, will be thor- oughly handled thanks to a congruence transform presented in Section 4.4.

Conclusions

In this chapter, the fundamental concepts of LPV/quasi-LPV systems have been recapitulated. Thereby, three usual approximation forms: polytopic system, polynomial system, and polynomial-fuzzy system have been delivered, corresponding to the stability analysis and synthesis for each representation.

The stability conditions derived from the analysis of the parameter dependent Lyapunov function are presented as parametric linear matrix inequalities. Then, the relaxation methods of parameterized LMI: the gridding, the sum of squares, and the convex combination that convert the infinite-dimension conditions into the finite-dimension constraints as linear matrix inequalities. It can be solved by numerical mathematic tools (SDP, CP, etc.). Then, the design specifications and requirements for the saturated control system are discussed and analyzed based on the Lyapunov technique. The necessary and sufficient conditions deliver for the stabilization of saturation LPV systems corresponding to constrained feedback control systems.

Chapter 3. Quadratic Stabilization Analysis for LPV/quasi-LPV Systems with Actuators Saturation

Quadratic Stabilization Analysis for LPV/quasi-LPV Systems

In this chapter, the stability of LPV systems is solved by using an observer-based feedback control law corresponds to the ℋ∞ performance criterion. The first section devotes to the stabilizing analysis of LPV/quasi-LPV systems including: § The observer-based feedback control stabilization delivers for the LPV systems considering the influence of disturbance and uncertain parameter. The non-convex problem related to coupling variable matrices is handled by the generalized of Young's inequality. This controller design improves the results of the quadratic conditions [START_REF] Dahmani | Observer-Based Robust Control of Vehicle Dynamics for Rollover Mitigation in Critical Situations[END_REF](Dahmani et al., , 2015) ) and presents an adjustment to optimize the concave condition using the scaling-commuting sets, that is addressed in sections 3.1.3 and 3.2.3.2. The results show a significant enhancement in the stabilization condition application of Young's inequality (relates to the scalars considering as the weighted geometric mean). § The stabilization condition delivered on the parametric linear matrix inequalities (PLMIs) formulation are infinite dimension, which could not solve directly by semidefinite programming (SDP) or cone programming (CP). So, the relaxation of PLMI methods is represented in section 3.2 to reformulate the design conditions to finite convex optimization problems.

A raised question about the content focuses on the stabilization analysis of the observerbased controller for the LPV system. First, it can realize that feedback control synthesis for this class of systems is well-investigated and developed. However, the conventional approach of the observer-based feedback design is conservative. In addition, the analysis and synthesis of controllers (i.e., state feedback, new observer-based feedback, output static and dynamic feedback) for the LPV saturation system will deliver in the next chapter using parameter-dependent Lyapunov functions. Therefore, this chapter is devoted to addressing the concave problem relating to quadratic stabilizing conditions and presenting the relaxation methods of the parametrized LMI conditions. A global optimization method cone complementarity linearization (CCL) has effectively reduced the gaps in Young's inequality and enhanced the system's performance. A quadratic Lyapunov candidate is applied to demonstrate the conservative relaxation using the CCL method for the performance and robustness requirements. Accordingly, the scaling parameters method combined with the CCL algorithm provides smaller optimal disturbance rejection values confirming the system performance improvement. In addition, the parameter-dependent conditional relaxation methods such as Gridding, parametric matrix polynomials (S-variable, Sum of squares), convex combination (polytopic/T-S fuzzy) are presented along with the further discussions. Finally, the illustrated examples, a design PDC controller validates on a vehicle lateral stabilization and a quadruple-tank process systems. Bases on this ellipsoidal set, a inputs with bounded L2-norm impose by small-gain theorem [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. The norm-bounded conditions set directly to the input control ,, diagXII  for condition (3.10) yields to:

Stabilization Analysis

1 11 2 2 1 0. 0 0 T T s sss s P K Y P u K Y                  ± (3.12)
However, the remnant of s K    in these conditions leads to a heterogeneous form. By deploying a matricial generalization of Young's inequality to eliminate these non-convex problems, we have:

1 11 11 00000000 0000. 00 00

T T TT T sss s IIIPIP YY KK                              ± (3.13)
Substituting (3.13) into (3.12), continually applying Schur's Lemma that entails in condition LMI (3.8).

It concludes the proof.

W

The LMIs (3.8) are conservative conditions, and there is a small probability of finding a feasible solution that satisfies the stabilization problems with a preselected  value. However, better results could be obtained if this scalar were a variable. Besides, the satisfaction of (3.8) only guarantees a necessary condition. Sufficient condition (3.11) is satisfied if the derivative of Lyapunov function (3.6) along the trajectory of the closed-loop system is negative. Further analysis will deliver in section 3.1.2.

Remark 3.1.1. The norm-bounded conditions are given in Lemma 3.1.1, and the observer-based control stabilization had been addressed in (Bui [START_REF] Bui Tuan | Robust Observer-Based Control for TS Fuzzy Models Application to Vehicle Lateral Dynamics[END_REF]. But, the bounds on the control input has not been properly treated.

Observer-based control stabilization

In consideration of the effect of external disturbances the robustness and performance requirements of system (3.1) are consisted on finding the L2-gain scheduling controller K   and observer L   given in (3.4) that guarantee a stabilization condition concerns to a minimize disturbance rejection level 0   across the frequency domain as follows:

22 1 ztwt    LL (3.14)
Use quadratic Lyapunov function (3.6) for stabilizing analysis associated with this ℋ∞ performance that leads to the following result.

ip   U ,1,2,4, i  R
such that the PLMI:

0,         p (3.15)
is satisfied. Where the scheduling controller and observer gains are For the sake of simplicity, the time-varying " t " and parameter-dependent expressions "   " are omitted from next inequality. Takes the derivative of the Lyapunov function (3.6) iii X   

The cone complementarity linearization CCL method [START_REF] El Ghaoui | A cone complementarity linearization algorithm for static output-feedback and related problems[END_REF] has been employed to handle with concave problems involved with the strategic SOF or observerbased controller designs. Nonetheless, this iterative optimization algorithm could deliver positive results if the constraints are not too complex, other than there are no guarantees for the larger scale of stability conditions.

Concave Nonlinearity -Cone Complementarity Linearization

The nonconvex or quasi-convex problem relate to nonlinearity matrix structure is frequently encountered in the stabilization condition via LMI synthesis. These problems are not easy to handle directly or cannot solve successfully by standard techniques (convex programming). A relaxation method reformates as a global optimization problem -cone complementary linearization Algorithm [START_REF] El Ghaoui | A cone complementarity linearization algorithm for static output-feedback and related problems[END_REF] provides a useful tool that consists in linearizing a nonconvex problem to a canonical form: minimizing a linear function over a difference of two convex sets. This method is commonly found in the robust control system design involving SOF synthesis or parameter uncertainty such as [START_REF] Cao | Static output feedback stabilization: An ILMI approach[END_REF][START_REF] Moon | Delay-dependent robust stabilization of uncertain state-delayed systems[END_REF][START_REF] Sun | Delay-dependent stability and stabilization of neutral time-delay systems[END_REF]. Most of them could solve practically only problem instances of very limited size, as would be expected from the NPhardness of these problems. In another aspect, for a more exhaustive and comprehensive analysis of this issue and other approaches of global optimization should refer to the following monographs [START_REF] Konno | Optimization on Low Rank Nonconvex Structures[END_REF][START_REF] Tuy | Convex Analysis and Global Optimization[END_REF].

On the other hand, Young's inequality plays an essential role in the observer-based controller analysis method for nonlinear systems (Zemouche et al., 2017;Zemouche & Boutayeb, 2013), most of which are pre-selected or gridding within reasonable intervals with these variables. But it is tricky to pick the appropriate ranges of the scalars to deliver good results, which may lead to conservative design conditions.

Iterative algorithm

We now discuss a concave problem encountered in robust stabilizing conditions of the observer-based control analysis (3.15), related to bi-linearity forms

1 , . XX   CHAPTER 3. QUADRATIC STABILIZATION ANALYSIS FOR LPV/QUASI-LPV SYSTEMS WITH ACTUATORS SATURATION 1 0 0, 0 I X X                  ± L OMM (3.22)
As discussed in Appendix A.4, the role of slack-variable  is to tighten the gaps of these inequalities and is essential to the relaxation of Young's inequality. Then, an iterative algorithm is provided to seek for the globally optimal value of the stabilization condition for the substitutions shown in (Bui [START_REF] Bui Tuan | Robust TS-Fuzzy observer-based control for Quadruple-Tank system[END_REF][START_REF] Bui Tuan | Robust Observer-Based Control for TS Fuzzy Models Application to Vehicle Lateral Dynamics[END_REF].

Gives a new scalar Now we analyze this problem by the approach of (Bui [START_REF] Bui Tuan | Robust TS-Fuzzy observer-based control for Quadruple-Tank system[END_REF], then nonconvex problems (3.22) Step 2: Assigning above solutions be initial set 0000000 ,,,,,, XUSUS   then set 0. k 

Step 3: Find new solution at th k by solving LMI problem given as:

Minimize   1,2,3, Trace 0.25 kkk JJJ  (3.29) subject to (3.22)-(3.25), with   1, 2, 3, , ,
.

kn kk kkkkk kkkkkkkkkkkkk kkkkkkkkkkkk JI JUUUUSSSS JXUXUXUXU XSXSXSXS        
Setting th k optimal solution of (3.29)

Step 4: Fix a positive scalar ,   ¡ and a sufficient small tolerance ò , if   By seeking for the new variables ,,,,,, XUUSS  at each loop such that the conditions (3.22)-(3.25) hold. A better value   is achieved by increasing (or decreasing) their value each time the conditions (3.30) are satisfied. But it does not guarantee the accurate convergence of the solution. So, we only use this method just to find the epsilon coefficients, and then deploy a conjoint algorithm to efficiently achieve a better local optimization.

Algorithm 3.2. Enhanced CCL & Local Optimization

Step 1: Choose on purpose initial values ,  Ks.t. the conditions (3.22)-(3.25) are feasible.

Step 2: Solve steps 2 through 4 of Algorithm 3.1 to find a solution k  that satisfies conditions (3.30), then set 2 .

k  

Step 3: Optimizing  under design stabilization LMI conditions.

The proposed iterative algorithm able to converge directly a better local region for design condition, e.g., better performance attenuation … It should remind that the above CCL algorithms have been improved to be better adapt to the concave structure and to converge faster to the optimal solution. As illustrated in the example section, the varying scalars reduces effectively the conservation of stabilization conditions that some existing approaches are confused to find a feasible solution.

Reduction to Finite-Dimensional Problems

The purpose of this section is to present the scaling-parameter method that can be combined with the optimization algorithms in the previous section to increase the performance of the control system. As discussed in chapter two, the PLMI relaxation methods could be straightforwardly applied for stabilization condition (3.15) in Theorem 3.1.1. Hereafter, the design conditions for the saturated LPV system in the next chapters will only be presented in the parametric formulation.

It should remind that the proposed stabilizing conditions expresses as parametric dependence matric inequalities, which are infinite-dimensional problems characterized by infinite space in the range of parameters. The relaxation PLMI methods generally reformulate design condition to finitely LMIs. One of the widely known relaxation methods is the finite-dimensional meshing -gridding techniques, or the affine parameter-dependence distributed over each parameter subspace. The limitation of the methods is how approximate the behavior of the parameters in their operating range, including the critical points with the smallest number of discrete points.

Besides, the parameterized LMI conditions could represent as multiconvexities by polytopic or T-S fuzzy representations or tensor product transformation, that shows efficiency in reducing the numerical computation, the number of iterations, etc. An alternative approach uses the polynomial expression converted to a SoS problem is treated effectively by the SoS toolbox. The semi-definite programming (SDP) problems derived from the polynomial matrix and SoS constraints could be solved by the interior-point techniques. However, the computational time and numerical resources reserved for this method is enormous. This is a trade-off between conservative and reasonable computational efforts.

In the first part, the piecewise-affine parameter, the sum of squares, and the convex combination (fuzzification) are manipulated to relax the parametric LMIs.

Finite Discretization of Parametrized LMIs via

The first discussion deserves for the discretization of parametrized LMIs into finite multiple subspaces without knowledge of the density of the parameters. Define a parameter set as follows: It should mention that the parametric-dependent conditions are converted effortlessly into the multiple LMIs, but there is a probability of missing essential information (critical points). Precisely, the gridding points must be enormous to cover exactly the behavior of the parameters. That entails in the exponentially increasing number of the solved conditions. Alternatively, the piecewise affine parameter-dependent is introduced in the works [START_REF] Apkarian | Robust control via concave minimization local and global algorithms[END_REF][START_REF] Lim | Parameter-Varying Systems[END_REF][START_REF] Tuan | Relaxations of parameterized LMIs with control applications[END_REF] assigning continuous subspace domains.

Let us now introduce a simplification of the method piecewise switching-dependent functional. Giving the set of parameters , p t U then it is distributed into m subspace pa- rameter domains as follows:

 

Polynomial Parameter-Dependent LMIs via Sum-of-Squares

As introduced in Chapter 2, a polynomial of parameter dependence matrices inequality could address by the SoS approach or Slack-Variable (SV) approach. This method decomposes the sum-of-squares polynomials, then converts them to a convex coordinate transformation (related to the coefficients of the polynomials over the defined domain of the parameter). This parametrized relaxation converts the parameter-dependence matrix inequalities to the Sum-of-Squares expression, which means positive definite (guaranteed with a minor deviation). As mentioned by [START_REF] Prajna | Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach[END_REF], the scalar variables defined by a set of polynomial inequalities (vars and decvars) concerned the variation of the S-procedure constraints where the pre-selected scalars. In such an approach, the parameter values are not constant, giving more relaxation in stability conditions.

The sum of squares decomposition has limitations in the experiment application, even for a simple structure of the polynomial gain-scheduling e.g., 2 012 , XXXX   the feasible solution of the stabilization conditions results in higher orders of the polynomials. For example, a structured controller classically decomposes by 1 , KYX    that ensues in a complicated fraction form in monomials. This method is evident an illustration of trade-off between computational burden and conservativeness.

SV-LMI-based control design

The slack-variable method has shown convenience for control synthesis and applicability in analyzing the parametric polynomial dependency matrix structure. This relaxation method based on the generalization of Finsler's lemma characterized via quadratic relations generalized S-Lemma (so-called S-procedure). In this aspect, a comparison between the SoS formulation and the S-variable approach has been delivered in (Sato & Peaucelle, 2007a[START_REF] Sato | Robust stability/performance analysis for uncertain linear systems via multiple slack variable approach: Polynomial LTIPD systems[END_REF] The S-variable approach provides better results, and closes to the theoretical technique (error 12 ,).

ee The interesting point is that the S-variable is solved at the boundaries (two points) also gives a positive result where error 1 e approximates to error 2 e obtained by 19 points gridding in the range 6,12.  When solving simple parameter dependency conditions, the S-variable gives better convergence results than SoS decomposition. However, for large-scale conditions, the SoS toolbox handles the PLMI conditions more delicately with fewer additional slack variables.

Besides, the mathematical programming software such as MATLAB ® effortlessly returns the invertible matrix as fractions of the parameters with large polynomial exponents. But, the possibility of the implementation in practice is questionable. Another approach, more conservative, the LMI conditions are directly delivered by the linear combination of the vertices of the parametric convex domain. The two well-known methods are the T-S fuzzy and Polytopic models. There was no difference in the representations of the two systems but a distinction in the stabilizing control synthesis for the PDC scheme and polytopic gain-scheduling, respectively.

Parametrical Dependent LMIs via Convex Combination

Given compact parameter set .

p t U There exists a linear parallel to transform the basis conversion bilinear mapping and linear conservation from the parametric dependent function

1 0 :, ,| , 1,, p N ipp ttiN    P FCU R K to   1 0 :, ,| , 1,,. p N ipp ttiN     K Q FCU R (3.51) where 0, 1,1,,. p N qip iN   K U R
The conversion of the parametric coordinate system to the convex coordinate system

1 , 1, 1,, l N iijjl j ttttjN      K (3.52)
has been discussed in Chapter 2. However, the reverse transformation is rarely mentioned in many documents. Such an algebraic transformation method is presented in Appendix A.1.4.1, which allows for the generalization of the coordinates expression of the multiconvex system. In the next section, a Lyapunov quadratic is considered for the stability analysis of a T-S fuzzy system.

T-S Fuzzy Controller Stabilization Construction

The fuzzification and defuzzification are applied to convert the parametric dependent conditions to the linear combination forms. Let's introduce a convex combination of affine system (3.5) and a feedback parallel distributed compensation controller by using rules (3.52) as follows:

, , where ,,,,,.

1 l N iijijijij ij ij iiiijiiiij ABKABKBKBK xtxt tt ALCBKALCBK etet                  & & ,, 1
    1, 22, ,, 2 1,, , ,,1,,1, 22, , 1, 0 ,
ii ijiii Biii ii X diagIIIIX NYI                        Proof.
The demonstration is directly inferred from the parameter-dependent condition in Theorem 3.1.1 with the use of linear convex combinations (3.52). We now look for a stabilization feedback control

l l N N ijij ijiil ij ij i ttttijN            K  p (3.56)
Then, following the property of convex combinations, if each condition at each vertex is satisfied, condition (3.56) holds. Next, by applying a relaxation of the stabilization condition approach [START_REF] Tanaka | Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs[END_REF]) that yields to (3.55).

W

The stabilizability of PDC controller and relaxing LMI conditions can find in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]Tuan et al., 2001) for quadratic Lyapunov function and for fuzzy Lyapunov function [START_REF] Tanaka | A multiple Lyapunov function approach to stabilization of fuzzy control systems[END_REF]. It should note that the fuzzy Lyapunov functions (FLF) reduces conservatism in design conditions. And, the derivative of the membership function relates to

1 l k N k k X X t          & &
could expand as method of [START_REF] Sala | Polynomial fuzzy models for nonlinear control: A taylor series approach[END_REF]. This issue is encountered in the next chapter. Now, let impose the constraints for the PDC control law of system (3.53) (3.55) to demonstrate the effectiveness of the relaxation method -CCL algorithm on a T-S fuzzy system and compares to the generalized sector condition. In the result section, we show a comparison of the pre-selection epsilon method [START_REF] Benzaouia | Advanced Takagi-Sugeno Fuzzy Systems: Delay and Saturation[END_REF][START_REF] Dahmani | Observer-Based Robust Control of Vehicle Dynamics for Rollover Mitigation in Critical Situations[END_REF]Dahmani et al., , 2015;;[START_REF] El Hajjaji | Observer-based robust fuzzy control for vehicle lateral dynamics[END_REF][START_REF] Kheloufi | On LMI conditions to design observer-based controllers for linear systems with parameter uncertainties[END_REF]Zemouche et al., 2017) with the CCL iteration algorithms (combining with scaling parameters).

Problem 3.2.1. For the control synthesis based on convex combination framework:

What is an essential distinction between the stabilized condition structure analyzed for a T-S fuzzy system and a Polytopic system? (In another view, what is the difference between a structure PDC controller and a polytopic gain-scheduling controller in control system design?)

Let's consider a state feedback controller construction

1 1 , 1, 0, 1, 1,,. l l N N iiiil i i uttKxtttiN      K (3.59)
At first glance, it does not show difference in the control analysis of the two approaches. Supposes the parameters t  belongs to , ,   then the rules set on local system:

T-S Fuzzy: In the conventional approaches using CCL, authors often simplify scalars .

1 , l N iijj j xtAxtBtKxt     & , , clijiij AABK  (3.
i   R In this section, an enhancement of the CCL condition associated with a set of uncertain parameters develops via the linear combinations.

In robust control theory, the set of scaling associated with the uncertain structure is considered to be compatible with the LPV control synthesis. In this approach, the scaling small-gain theorem or structured-robust stability [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF]Apkarian & Gahinet, 1995) is probably the most commonly known, which delivers a less conservative stabilizing condition associated with the scaling matrices depends on uncertain parameters. We employ the scaled parameters to conditions (3.18)

Example

In this section, the numerical results are implemented to demonstrate the performance and robustness improvements using the CCL method, a quadratic Lyapunov candidate is applied on the most conservative condition -observer-based control stabilization.

We employ the design controller based on the structure observer to stabilize the quadruple-tank process system, and the vehicle lateral dynamic system with the influence of external disturbance and uncertain parameters. The PLMI stabilization conditions in Theorem 3.1.1 (without saturation constraint) is relaxed to the stabilizing LMIs, respectively, conditions of Theorem 3.2.1. Firstly, we discuss on the results of (Bui [START_REF] Bui Tuan | Robust TS-Fuzzy observer-based control for Quadruple-Tank system[END_REF], demonstrating the effectiveness of the proposed method when using Young's inequality in combination with the global optimal algorithm CCL.

In the following analysis, the results of (Bui Tuan et al., 2021;[START_REF] Bui Tuan | Robust Observer-Based Control for TS Fuzzy Models Application to Vehicle Lateral Dynamics[END_REF] will be reproduced to compare with the other results. In this section, At the same time, we deploy the stabilization condition of Theorem 3.2.1 associated with norm bounded inputs of Lemma 3.2.2 for system (3.70) compared with the stabilization condition of feedback control. By pre-selecting the initial conditions and using Algorithm 3.1 (CCL1) & Algorithm 3.2 (CCL2) to minimize the performance criterion ,  the comparison are given.

Quadruple-tank process system

Let's consider a quadruple-tank process system [START_REF] Johansson | Relay Feedback and Multivariable Control[END_REF] The fuzzy controller gains given in Appendix E.1 obtained by using toolbox Yalmip [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF] with solver Sedumi [START_REF] Sturm | Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones[END_REF]

                                 L L L L (3.67)
where MP is minimum phase setting [START_REF] Johansson | Relay Feedback and Multivariable Control[END_REF] The minimization of disturbance effects on the system is enhanced by using the CCL algorithm. As we can see in Figure 3-3 and Figure 3-4 (left), the error estimate of liquid level in tanks 1 st , 2 nd is smaller than 0.6 mm, corresponding to level in tank 3 rd being less than 3 mm and level tank 4 is less than 5 mm. And the stabilized control signal of the observer-based controller design for the quadruple-tank process system gives in Figure 3-4 (right). With the limited voltage of each pump is 12 (V), the maximum flow of each pump is 40 (ml. s -1 ) corresponding to 2.4 (l. mn -1 ). The illustrative simulation results show the high performances of the proposed design technique.

The norm-bounded constraint [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] increases the conservatism of the stabilization condition, rendering unsatisfactory results obtained from the preselected scalars methods. The scaling scalars method does indeed relax the proposed conditions. However, the above results do not expose the effectiveness of scaling-parameter transformation combined with the CCL algorithm. So, the optimal disturbance rejection for each design case is discussed just below, where the L2 norm-bounded condition is compared with the other works.

Vehicle chassis stabilization system

Let's recall an example discussed in Section 2.1.6. [START_REF] Dahmani | Observer-Based Robust Control of Vehicle Dynamics for Rollover Mitigation in Critical Situations[END_REF](Dahmani et al., , 2015;;[START_REF] El Hajjaji | Observer-based robust fuzzy control for vehicle lateral dynamics[END_REF][START_REF] Kheloufi | On LMI conditions to design observer-based controllers for linear systems with parameter uncertainties[END_REF]Kheloufi et al., , 2013a)).

Cone Complementarity Linearization and Scaling-Parameters

As can be seen in Table 3.1, if the scalars i  are pre-selected, these stabilization inequalities become more conservative, causing a decline in system performance (with a larger value of  or even infeasible conditions). When the variables i   are considered, as a way of envisioning that makes appropriately rotates the self-selected barriers to reduce the inequality gap (as illustrated in On one hand, performance degradation can observer in Table 3.1 as a result of applying a quadratic Lyapunov function to enforce the stabilizing condition of the parameter-dependent system. For example, let's consider a diagonal slack variable matrix tighten the gaps of Young's inequality. Besides, it shows an adaptation to integrate with structure parameterization, from simple form i  to parameter-depend- ent form i   and affine scaling form . i   The slack-variable reduces conservatism and reach to better optimal performance level. § Algorithm CCL2 searches for the better local minimum value and faster than the iterative global optimization -algorithm CCL1. However, CCL2 returns positive results based on the solutions of parameters inherited from CCL1. § Reduction -Redn (simplified) conditions shows better results thanks to a more compact and simpler conditional structure. However, if the slack-variable of the Young's inequalities reach their optimal values, both approaches (the original and the reduced form) approximately converge to the same optimal solutions (for example, in the case of affine scaling). § Finally, the less conservative results observed in the 2 nd catalog versus the 3 rd and 4 th catalog in Table 3.1.

In this example, the selection of T-S fuzzy model (3.71) with a distinction between the local linear systems leads to a disparate outcome in the 2 nd catalog versus the 3 rd and 4 th catalogs in Table 3.1. The quadratic Lyapunov function used for stabilization analysis for the LPV system led to conservatism. It makes randomly choosing a set of scalars i  as [START_REF] Dahmani | Observer-Based Robust Control of Vehicle Dynamics for Rollover Mitigation in Critical Situations[END_REF](Dahmani et al., , 2015;;[START_REF] El Hajjaji | Observer-based robust fuzzy control for vehicle lateral dynamics[END_REF][START_REF] Kheloufi | On LMI conditions to design observer-based controllers for linear systems with parameter uncertainties[END_REF]Zemouche et al., 2017) could be unsolvable.

For example, 3 rd catalog of Table 3 It may not be an exaggeration to say that have no manipulation that can select this set of parameters to achieve optimal values (evenly gridding method, fine selection, etc.). Most of the work involves parametric uncertainty, in which these scalars often are randomly pre-selected. When dealing with numerical example (3.71), it is miserably hard to pick up a set of ij  for the feasible solution. Thereby, it authenticates the effectiveness of the proposed CCL algorithm.

Conclusions

In this chapter, we analyzed the conservation of the stabilization conditions delivered for the observer-based feedback system. The design structure of this controller is typically based on the application of Young's inequality combined with a quadratic Lyapunov matrix. Both mentioned properties will end in strict stabilization conditions. Based on the CCL algorithm, an improvement using scaling-dependent sets has enhanced the performance of the closed-loop systems. The numerical simulation results and reducing disturbance optimization values have demonstrated the effectiveness of the proposed method. However, applying a quadratic Lyapunov function to analyze the stabilization of the parameter-dependent system is very conservative. So, in the next chapter, we present a synthesizing method of the parameter-dependent conditions derived from the stabilizing analysis for LPV/quasi-LPV system subject to actuator saturation.

Furthermore, the L2-norm bounded input does not accurately guarantee the saturation limit of the actuator. Therefore, in the next chapter, the saturated LPV controller stabilization is ensured by generalized sector condition.

Chapter 4. Stabilization Synthesis for LPV/quasi-LPV Systems with Actuators Saturation

Stabilization Synthesis for LPV/quasi-LPV Systems with Actuators Saturation

In this chapter, the sector bounding condition is predominantly used to enforce the bounds on the saturated control, and the parameter-dependent Lyapunov function is considered in stabilization analysis to enhance system performance. The following sections devote to the synthesis of controllers for saturated LPV systems including: § The state feedback (section 4.1), the static output feedback (section 4.2), the observerbased feedback (section 4.3), and the dynamic output feedback (section 4.4) controllers are considered for LPV systems in the influence of disturbance and the uncertain parameter. Where the generalized sector condition (GSC) condition is employed to ensure the saturation limits. The saturated gain-scheduling controllers are obtained by the feasible solution of the parametric LMI stabilization conditions. § The developments of observer-based control and dynamic output controller in sections 4. 3 & 4.4 involve with the non-convex forms in the stabilization matrix inequality and the saturation condition. The bilinear terms will be handled by a less conservative congruence transformation.

It should note that of the feedback controller design architectures, the observer-based controller usually results in the most conservative stabilization condition. As a result, the scaling sets are considered for this control design strategy in chapter 3. However, the problem is still not fully covered, so a new design strategy is presented. Finally, the numerical simulation results in sections 4.1.2 and 4.5 will demonstrate the effectiveness of the proposed methods.

State Feedback Stabilization

Sector Nonlinearity Models

As discussed in section 2.3. S such that the following matrix inequalities hold: (2) for 0, wt  the ellipsoid E is a region of asymptotical stability (RAS) for satu- rated LPV system (2.111).

  2 0, 0 T w T d wr symABYX YZTBT BI HXJYJTJ X I                              & p (4.1) (a). 0 1 0 0 0 xX           ± , (b). 2 0, 1,2,..,, s s X u sm Z               ± (4.2)

Proof.

On one side, as analyzed in section 2.3.2, the PLMIs (4.2) are equivalent to:

(4.2).a  1 000 001, 0, T xPxV     (4.3) (4.2).b     

  

   hold. However, this sub-optimization is conservative. Especially, the latter condition is expressed by a bilinear form relating to parameter-dependent conditions (4.1), (4.2). Furthermore, the minimization of  results in the decline of the linear operating area (regions of unsaturated control signal), which predominantly affects the constraints like GSC (that is discussed in section 4. 1.2.3).

Example

In the first part, the stabilization conditions design for the saturated system considering the effect disturbance and the time-varying parameters reveals the relaxation of GSC constraints (on the single input systems). Three relaxation methods of PLMI conditions will apply to Theorem 4. 1.1 (i.e.,gridding,SoS,and polytope). Then, the relaxation of multimodel (T-S fuzzy) is used to compare with the works in the literatures (A. T. [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF][START_REF] Vafamand | A robust L1 controller design for continuous-time TS systems with persistent bounded disturbance and actuator saturation[END_REF].

Concerning the stabilization of the saturated MIMO systems, some discussion is delivered about the high-gain problem of the bounding sector constraints. A typical example used with the difference between the vertices of the local linear systems and the disproportion of the actuator's limits aggravates this numerical problem. As a result, a modification is proposed via LMI formulation to overcome the limitation.

Ellipsoidal regions of stability

            & & (4.11)
The saturation bound is The LMI conditions are solved by the convex optimization Mosek (Erling D [START_REF] Andersen | The Mosek Interior Point Optimizer for Linear Programming: An Implementation of the Homogeneous Algorithm[END_REF] combined with the numerical calculation toolbox Yalmip [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF]. The second column is obtained from using the SoS toolbox [START_REF] Papachristodoulou | SOSTOOLS: Sum of squares optimization toolbox for MATLAB[END_REF] to solve the parameter matrices polynomial according to the SoS decomposition approach with solver Sedumi [START_REF] Sturm | Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones[END_REF]. And provided in catalog number 3, by deploying a conditional relaxation of Theorem 4. The optimization results are obtained by solving Theorem 4.1.1 with Problem 4.1.1 by these relaxed PLMI methods corresponding to the weighting scalar 12 1.

  It can be seen that the multi-convexities (T-S fuzzy) returns the best results with the optimal solve time. Meanwhile, the complexity in the conditional structure of the SoS decomposition method leads to a higher computation time. The finite-discreteness over the parameter domain provides a satisfactory result with reasonable time (this approach is the simplest relaxation of PLMI).

A largest invariant ellipsoid contained in a polytope [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]

X   E and 1 4 . X   E
Nonetheless, the condition (4.15) is very conservative constraint, and it is not always guaranteed that all of the trajectories of saturated closed-loop system is bounded in the definite domain. In this case, the hard-bounds imposed on the states are better fulfilled by the barrier function, see, e.g., [START_REF] Ngo | Integrator backstepping using barrier functions for systems with multiple state constraints[END_REF][START_REF] Nguyen | Exponential Control Barrier Functions for enforcing high relative-degree safety-critical constraints[END_REF][START_REF] Tee | Barrier Lyapunov Functions for the control of output-constrained nonlinear systems[END_REF].

The comparison of conservativeness of conditions

Example 4.1.2: Consider a T-S fuzzy system modified by (A. T. [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF] expressed by the following local linear matrices:



   guarantees that the ellipsoid E is a region of asymptotical stability for saturated qLPV system (4.17).

Colored dash-dotted trapezoid (A. T. [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF] Violet dotted trapezoid [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF] Green solid trapezoid [START_REF] Vafamand | A robust L1 controller design for continuous-time TS systems with persistent bounded disturbance and actuator saturation[END_REF] Small blue circle (Corollary 4.1.1) XX  ± The feasi- ble region exhibits the proposed condition to be much less conservative than other approaches. In the work of [START_REF] Vafamand | A robust L1 controller design for continuous-time TS systems with persistent bounded disturbance and actuator saturation[END_REF], although the author intends to test on a smaller conditional region 2, 11,6, 16, ab  it can be objectively recognized that there is a big difference compared with the feasibility region of the saturation constraint of Corollary 4.1.1. Besides, the optimization results of the stabilization conditions of Theorem 4.1, solved for the system (4.14), are also superior to the results of (A. T. [START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF]. However, the comparison will be more reasonable in analyzing with the same structure as the DOF control system provided in Section 4.4.

Generalized Sector Condition

As analyzed on the feasibility of the saturated LPV systems, the GSC shows a better performance. Now, let's study the numerical problem related to GSC for a two-rule T-S fuzzy system (example 3.4.2). The peculiarity of this example is that the saturation limits on the control inputs vary greatly. For simplified analysis, the parameter-dependent stabilization conditions (4. At first glance, we can realize a numerical problem using the quadratic Lyapunov form to stabilize the uncertain parameter system corresponding to the norm bounded and generalized sector constraints. However, why are only high gain problems observed in Table 4.2 with GSC conditions?

To understand this better, let's first remind some preliminary definitions of the generalized sector condition. Given a vector auxiliary control 12 ,,, It's worthy to note that if conditions (4.21) hold, then condition (4.23) is kept, but the reverse is not correct. The problem does not reveal when the similarity limits enforced in each control input are considered. In monographs [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] Let's recall the stabilization problems discussed on LTI systems with saturated actuators. The closed-loop poles are pushed away from the imaginary axis to the left at the optimized cancellation values. The features such as fast convergence and good response are ineffective when the control signal reaches its limit. The dead-zone nonlinear behavior can cause the system to become unstable and convergent about equilibria other than the origin (Gomes da Silva, 1997; Gomes da [START_REF] Gomes Da Silva | Local stabilization of linear systems under amplitude and rate saturating actuators[END_REF][START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. As a result, a deteriorated control performance (the chattering phenomenon) is observed. And presented in Figure 4-2.b, the poles get closer to the imaginary axis avoids chattering in the input signal control but the response characteristics are slower. This problem is alleviated by the method of displacing poles into the D-stable region [START_REF] Chilali | Robust pole placement in LMI regions[END_REF]Chilali & Gahinet, 1996), a tradeoff between the system performance and the linear characteristic region. This approach could be implemented directly on design LMI conditions by relocating the eigenvalues of the matrix into region D. The better control performance and the improved results are obtained.

Remark 4.1.1. This example was intentionally chosen to discover the limitation of the weakness of GSC condition in applied to the MIMO systems. It could remind that the structure of GSC conditional stabilization using quadratic Lyapunov was similar between Polytopic and PDC-fuzzy analyses.

Remark 4.1.2. The state-feedback controller construction yields a simplified control construction and a better characteristic of closed-loop system response. But it may be unimplemented in practice due to metrological and economic reasons. In this case, the more suitable approach would be the output feedback.

Static Output Feedback Stabilization

The static output design strategies often suffer from a bilinear structure due to the form of the measurement matrix. In this section, we present a decomposition structure of feedback controller gain that allows using simple congruence transformations. Besides, a more general problem could be included by considering a parameter-dependent formulation in the measurement matrix. By choosing the set of admissible initial conditions with 0 0,   and polyhedral set (2.92) related to level sets of ellipsoidal domains (2.102), the stability analysis of LPV system (2.113) to the following theorem. S such that the following matrix inequalities satisfy:

Proof.

In a repetitive manner, conditions (4.36).a brings to the initial condition:

1 10200 1 0 0000. 0 TT xPxePeV          (4.37)
And the saturation constraints on the additional vector are derived from: A relaxing method for the stabilization condition using Young's inequality presents in Chapter 3. But the conservation of this inequality is also related to the organization of the right side. As observed in condition , 0,0 X   f the right-hand side of this inequality is always positive, whereas the left side (the primary solution domain) can be positive or negative. That leads to the conservatism in the design conditions.

    2 2 2

Generalization of Finsler's Lemma

The conservatism of OBF controllers using Young's inequality will reveal in the comparison section. To overcome the problem in Theorem 4.3.1, we introduce a new observerbased control structure using the generalization of Finsler's lemma. Then, the stabilization condition is stated as follows. In a repetitive manner, conditions (4.43) proceed from the initial condition and saturation constraints on the additional controls such that:

1 10200 000 , 0 TT xPxePe    (4.44)     2 2 2 1 2 1 1 0 0 . T s ss ttt etet xxPx et uGG P                      (4.45)
Where the GSC condition can recall as follows:

1 12 0. T xt TGG Ku et uK                            (4.

Dynamic Output Feedback Stabilization

As described in section 2.3.2.4, we now address the necessary and sufficient conditions for the stabilization of closed-loop system (2.121) using dynamic control system (2.120) combined with auxiliary controller (2.126) The crux of the parameter-dependent stabilization synthesis is the derivatives of the parameter lied on the design conditions. The approach of [START_REF] Gahinet | Explicit controller formulas for LMI-based H ∞ synthesis[END_REF][START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF]

Examples

In this section, the presented results relate to the following methodological arguments: § The conservatism and implementation of the designed saturation controllers: the optimal level of reduced disturbance and the size criterion are addressed consistent to each proposed method (the state feedback and the output feedback strategy). In addition, the responses of the closed-loop systems governed by the design controllers expose an overall features and characteristics of the saturated control synthesis. § The performance degradation and instability of the nominal (unconstrained) control system when the actuator saturates, compared with designed saturated systems. § The analysis of the stabilization conditions using the parameter-dependent Lyapunov function shows the performance superiority over the quadratic Lyapunov condition.

In the first part, the stabilization analysis of a lateral axis dynamics for the L-1011 aircraft are deployed by the state feedback -SF (Theorem 4.1.1), the static output feedback -SOF (Theorem 4.2.1), the observer-based feedback -OBF (Theorem 4.3.1&Theorem 4.3.2), and the dynamic output feedback -DOF (Theorem 4.4.1). Based on these optimization results, we choose the appropriate feedback control structures to continue for the succeeding comparison. In which the gain-scheduling controllers stabilize the closed-loop LPV system demonstrate the effectiveness of the proposed method (ensures the stability of the corresponding designed saturation limits, and improves the system performance when the actuator reaches the saturation threshold).

Finally, the parameter-dependent stabilization conditions deployed for a quadratic Lyapunov stabilization condition and a non-quadratic Lyapunov function (NQLF). The latter discussion provides the less conservative stabilizing conditions of the parameter-dependent Lyapunov function toward the quadratic formulation.

Saturated Feedback Controller Comparison

Example 4.5.1: Consider the lateral axis dynamics of the L-1011 aircraft. The state-space representation for the L-1011 aircraft associated with the yaw rate, side slip angle, bank angle, and roll rate dynamics borrowed from [START_REF] Andry | Modalized Observers[END_REF][START_REF] Galimidi | The Constrained Lyapunov Problem and Its Application to Robust Output Feedback Stabilization[END_REF] and modified in (A. T.   Through the prompt so- lutions, the comparison of the design stabilization conditions for the feedback controllers is delivered. It is interesting to point out that the stabilization conditions of the saturated gain-scheduling SOF (Theorem 4.2.1) and DOF (Theorem 4.4.1) roughly attain the same disturbance rejection optimization values with the SF (Theorem 4.1.1). However, as can be expected about conservativeness of the old-fashioned observer-based controller stabilization conditions. Briefly, it can explain that this approach has two critical drawbacks:

1.

Young's inequality is typically deployed for the strategy design of observer-based controllers. But, using this bounding technique loses the equivalence of the solution domain of the conditions before and after applying the inequality (explained in more detail in the Appendix A.4).

2.

If it compares with the other structure of output feedback strategies, the use of a block-matrix diagonal Lyapunov candidature associated with the observer-based controller design is conservative. Besides, a marked improvement on the results of the new OBF design method can be appreciated (compare 3 rd and 4 th catalogs). Solving problem (4.13) does not directly yield the optimums of the disturbance rejection levels or the size criterion (Volume Maximization, Minor Axis Maximization, or Trace Minimization, Etc.). But, based on the simultaneous minimization of opt  and , opt  we acquire comparable optimal values of the state feedback, static output feedback, and dynamic output feedback approaches. On the contrary, the traditional approach for the observer-based control is not nearly as feasible. Therefore, this method (Theorem 4.3.1) is excluded in the following comparisons.

The simulations start from initial condition 0/8 /4 /3 /20 T x    that belongs to the estimate of RAS. Using Yalmip LMI toolbox [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF]) integrated with interior-point optimizer Mosek ® (E D [START_REF] Andersen | On implementing a primal-dual interiorpoint method for conic quadratic optimization[END_REF][START_REF] Andersen | The Mosek Interior Point Optimizer for Linear Programming: An Implementation of the Homogeneous Algorithm[END_REF] to solve optimization problem (4.13). Then, the parametric dependent forms of decision matrices are given corresponding to the design strategies:

Saturated Dynamic Output Feedback Controller

The optimal values achieve in Table 4.3 by solving the stabilization conditions of Theorem 4.4.1 with problem (4.13). Nonetheless, the high-gains cause the computation burden relating to the numerical simulation (it takes almost 10 minutes to complete a 10-second simulation). So, we slightly increase   0.3753  0.4528. Where, the dynamic gains and compensator gain are expressed, respectively, in Eqs. ( 4 Applying the gains scheduling to the LPV system (4.74), we obtain the simulation results. The signals of the dynamic output and state feedback control system exceed the saturation limits during the effect of exogenous signal (given in Figure 4-3) while the SOF controller and the new OBF exhibit a decent response corresponding to the design bound. The project lemma method demonstrates outstanding performance in designing static output controllers, which directly deliver a convex condition without applying mathematical constraints. When observing the time-evolution of the closed-loop states, the SOF system regulated a reasonable control signal conforming to the saturation limit. On the contrary, the convergence rate of the OBF control signal is too low (it can observe in the third and the fourth states). Furthermore, there exists a quasi-convex in condition (4.47) so this controller is also not deployed in the next comparison.

The robust performance and the stabilizability of the designed controllers evaluate under the influence of external disturbance from a non-zero initial condition. The time-evolution of the dynamical states shows in the 1 st frame to the 4 th frame and the time-varying There is no difference between states 12 , xtxt governing by state feedback, static output feedback, and dynamic output feedback controllers. But it should note that the 3 rd and 4 th states are measurable, which leads to a distinction in the response of the respective systems. As seen in Figure 4-4, the disturbance (sine function with amplitude 1) affects the system during the 4 th to 6 th second minimized corresponding to the L2-norm values given in Table 4.3. The control systems show the effectiveness of ensuring saturation limits and reinforcing performance. Now, let's discuss the size criteria associated with the ellipsoidal domain. The volume maximization delivers a linear and convex form relating to the decision variables. But it results in a disproportionate scale of the ellipsoidal region (that could entail an inaccurate estimate of the domain of attraction). The trace minimization and the maximization along certain direction method provide a multi-directional minimization of the ellipsoid characterized for the stability region. Nonetheless, the minor axis maximization exhibits the simplicity and the integrability with the stabilizing conditions. It should point out that the DOF stabilization conditions have a more complex structure. So, let's study the following minor axis maximization problem employed, respectively, for the saturated SF, SOF, and DOF control systems:

Problem 4.5.  ensues in a quasi-convex condition, where N   is supposed to be an implicit slack-variable that reappears in the optimal conditions. One solution for this is to preset a full-rank matrix , N and then solve the design LMI conditions. Note that in this simulation, we consider the case where either M or N is a full-rank constant matrix, and the remaining matrix is parameter-dependent. Similar to the first case, we could optimize eta and beta values at the same time. But, for this time, we fixed 10   and varied  from 0.01 to 0.1 to optimize .

 Note that the smaller the value of ,  the larger the linear behavior region.

It can observe in Table 4.3 and Table 4.4 that the static and dynamic output feedback controller solved by the conditions of Theorem 4.2.1 and Theorem 4.4.1 show good disturbance attenuation levels and reasonable estimations of RAS compared to state feedback controller. There is an insignificant difference of the optimal values. But it should be noted that the output feedback design method provides a practical implementation.

Evaluation of the performance and stability of the saturated system

In Example 4.5.1, only two states are measurable in four states of the system dynamic. But the vector spaces larger than two ensues in difficulty to exhibit the ellipsoidal domain.

Using two out of four states in combination with the corresponding basis matrix (A. T. Nguyen et al., 2018) is incorrect to characterize the estimate of the convergence region. So, let's study the following example to discuss more the domain of attraction of the closed-loop systems governed by the proposed saturated feedback controllers and the performance deteriorate in the unsaturation design control systems.

Example 4.5.2: Consider the dynamic of the pitch-axis motion of an autopilot for a missile model. The aircraft associated with the angle of attack, the pitch rate discussed many times in the literature [START_REF] Biannic | Missile autopilot design via a modified LPV synthesis technique[END_REF][START_REF] Daafouz | On inexact LPV control design of continuous-time polytopic systems[END_REF][START_REF] Pellanda | Missile autopilot design via a multichannel LFT/LPV control method[END_REF][START_REF] Wu | LPV control design for pitch-axis missile autopilots[END_REF] p N  points uniformly spaced over parameter range, we attain the optimal results as given in Table 4.3. As noticed in the above table, the SOF controller is exhausted in this example, both in the two optimization categories. By varying coefficient SOF  from 3 10  to 3 10 , the optimal values are obtained at In the first comparison of the state feedback systems, the performance degradation can be 4.5.2. Evaluation of the performance and stability of the saturated system 107 discerned on the nominal system (does not include saturation constraints). An enlargement of the frame from 4 th to 7 th second (Figure 4-5.a&b) clarifies the instability of states corresponding to the chattering effect of the controller shown in Figure 4-5. In the opposite direction, the feedback controller designed by Theorem 4.1.1 exhibits a good performance and enforces stability when the actuator is saturated. In the second comparison, an instability behavior with increasing amplitude of oscillation is observed on the response of the nominal SOF control system, corresponding to this control signal being saturated all the simulated time. Once again, the SOF controller designed by Theorem 4.2.1 has demonstrated the enhanced performance compared to the nominal system without integrating saturation conditions. It should be noted that the phase plane diagrams of the nominal SOF closed-loop system are ellipses extending to infinity (the instability trajectories are given in Figure 4567).

Through these simulation results, the control signals are regulated from stabilized control systems obtained by design conditions with no saturation conditions (e.g., sector bounding GSC). It could be noticed the performance degradation and system instability when the actuator is saturated. On the opposite, from similar initial conditions, the designed controllers ensure stability and enhance system performance corresponding to the optimal values obtained in Table 4.

Quadratic and Non-Quadratic Stabilization

As mentioned in Chapter 3, there is a performance deteriorate of the stabilization conditions using quadratic Lyapunov function (QLF) candidature against the parameter-dependent form. In this section, Example 4.5.1 is adopted to deliver the optimization results

Conclusions

We have developed several feedback controls law to stabilize the saturated LPV/qLPV systems. The control system design strategy related to the desired performance ensures that the operation agrees to the actuator capacity. Accordingly, the necessary and sufficient stabilization conditions via the PLMI formulation are addressed for the feedback controllers conforming to the design requirements (i.e., the admissible set of the initial conditions, the estimated region of the asymptotic convergence domain, the robust stability against uncertain dynamics, and time-varying parameters, and the system performance with the influence of input disturbance, Etc.). Besides, the nonlinearities and concave problems involved the generalized sector condition converted to the tractable conditional forms. The extension of the gains-scheduling technique has been addressed for the saturated LPV systems. Then, specific criteria are compared based on optimization results.

Performance degradation and instability trajectories on a control system without a saturation design occurred when the actuator reached saturation bounds. The presented results persuade the essential of saturation design for control system strategy. In addition, the relaxation of the design conditions is claimed by the comparison between the quadratic Lyapunov and PDLF stabilization conditions.

It is worth noting that the design stabilization is presented in the generalized form as an expression of the parameterized linear matrix inequality (PLMI). So, it can adapt and develop for each specific strategy or a suitable relaxation method of the PLMI. Furthermore, it remains an open problem in the LPV control framework.

Chapter 5. Stability Analysis of the LPV/Quasi-LPV Time-Delay Systems

Stability Analysis of the LPV/Quasi-LPV Time-Delay Systems

Time-delay phenomena observe in various engineering systems such as chemical processes, mechanical transmissions, hydraulic transmissions, metallurgical processes, and networked control systems. They are often a source of instability and poor control performance. The stability and stabilization of the time-delay systems (TDS) have received considerable attention in the practice and control theory. The time delay can sort into different approaches depending on the characteristic or the response behavior of the delay to the system. The framework of time-delay systems represents by functional differential equations classified into four types: discrete delay, distributed delay, neutral delay, and scale delay [START_REF] Briat | Stability analysis and control of a class of LPV systems with piecewise constant parameters[END_REF]. The literature on stability and stabilization of time-delay systems is exhaustive and could find in the monographs for LTI systems [START_REF] Dey | Stability and Stabilization of Linear and Fuzzy Time-Delay Systems[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF] for LPV systems [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF][START_REF] Briat | Stability analysis and control of a class of LPV systems with piecewise constant parameters[END_REF], time domain-based Lyapunov stability analysis [START_REF] Fridman | Tutorial on Lyapunov-based methods for time-delay systems[END_REF][START_REF] Sipahi | Stability and Stabilization of Systems with Time Delay[END_REF], and the eigenvalue based approach [START_REF] Michiels | Continuous pole placement for delay equations[END_REF][START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF].

Frequency-domain approaches dedicated to linear time-invariant (LTI) systems addressed a few cases of model transformations or varying delays. The stability of a system is verified from the distribution of the roots of its characteristic equation or the solutions of a complex Lyapunov matrix function equation. Interested readers can consult in more depth on this issue in the literature [START_REF] Gu | Stability of Time-Delay Systems[END_REF][START_REF] Michiels | Stability and stabilization of time-delay systems[END_REF]Michiels et al., 2005;[START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF][START_REF] Michiels | An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type[END_REF][START_REF] Schoen | Stability and stabilization of time-delay systems[END_REF]. But the approach meets with difficulties in analyzing the robust performance of dynamic system with the uncertainty, disturbances, and nonlinearities. In this case, the time-domain analysis technique is more suitable for dealing with the control challenge of this class of LPV time-delay system.

During the last decade, significant effort has addressed the problem of stability analysis and controller design for time-delay systems. Based on the time-domain approach, the Lyapunov stability is deployed primarily by two famous stability theorems: namely (1) Lyapunov-Krasovskii (LK) theorem and (2) Lyapunov-Razumikhin (LR) theorem. A generalized analysis for both approaches outlines in the works of [START_REF] Briat | Stability analysis and control of a class of LPV systems with piecewise constant parameters[END_REF][START_REF] Fridman | Tutorial on Lyapunov-based methods for time-delay systems[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF] and references therein. Generally, there are the major approaches to carrying out the stability analysis of TDS, depending upon the size and bound of the delay as follows: § Delay-Independent Stability condition § Delay-Dependent Stability condition § Time-Varying-Delay and Delay-Range Stability condition

Recently, the primary research trend performed by the Lyapunov-Krasovskii functional analysis usually focuses on seeking less conservative (LC) stability conditions. There are two fundamental conservatism reduction approaches: (1) the reformulation of Lyapunov-Krasovskii functional and ( 2) the bounding techniques of its derivatives.

On the one hand, the tighter bounding integral inequalities can mention as the Wirtingerbased I (WBI) [START_REF] Seuret | On the use of the Wirtinger inequalities for timedelay systems[END_REF], the Wirtinger-based II (WBII) (M. Park et al., 2015;[START_REF] Seuret | Wirtinger-based integral inequality: Application to time-delay systems[END_REF], the Free-matrix-based II (FMBII) [START_REF] Zeng | Free-Matrix-Based Integral Inequality for Stability Analysis of Systems With Time-Varying Delay[END_REF][START_REF] Zeng | Hierarchical stability conditions of systems with time-varying delay[END_REF] On the other hand, a suitable structure of LKF can refer to the additional integral terms, the increasing state vectors, and delay-partitioning/fragmented approaches showed superbly efficient on reducing the conservativeness of the stability conditions. It should note that the more slack-variable matrices used, the more complication is in analyzing the delay-dependent stability conditions. Accordingly, these approaches are the trade-off of the relaxation of the stability condition with the computational complexity.

In section 5.1, we discuss the Lyapunov-Krasovskii stability analysis for the time-delay LPV/qLPV system. An essential key point to relax the parameter-dependent stability condition founds on appropriate LKFs combined with reasonable bounding inequalities. In section 5.1.2, the convex function features, such as the auxiliary-function-based method (P. G. Park et al., 2015;[START_REF] Van Hien | Refined Jensen-based inequality approach to stability analysis of time-delay systems[END_REF][START_REF] Zhao | A new double integral inequality and application to stability test for time-delay systems[END_REF], fragmented/discretized Lyapunov functional (Y. [START_REF] Chen | Robust Stabilization for Uncertain Saturated Time-Delay Systems: A Distributed-Delay-Dependent Polytopic Approach[END_REF][START_REF] Fridman | Descriptor discretized Lyapunov Functional method[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF][START_REF] Han | A Delay Decomposition Approach to Stability of Linear Neutral Systems[END_REF] are employed to tackle with the conservatism of Jensen's inequality.

Besides, a well-known problem in control design is capturing or measuring the exactdelay value. The input-output approach proposed in [START_REF] Gu | Stability of Time-Delay Systems[END_REF] provides a methodology where the delay treats as uncertain dynamics of the LTI system. Then, an improvement for the LPV time-delay system is presented in [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF][START_REF] Briat | Delay-scheduled state-feedback design for time-delay systems with time-varying delays-A LPV approach[END_REF][START_REF] Briat | Memory-resilient gain-scheduled statefeedback control of uncertain LTI/LPV systems with time-varying delays[END_REF]. This approach purposely converts delay into the uncertain parameter, so it can also deploy by the different LMI-based stability designs (e.g., LFT framework). Based on the delay approximation, also known as a -memory-resilient, a stability condition derives from a Lyapunov-Krasovskii functional presented in section 5.2. Moreover, the auxiliaryfunction-based method provides a less conservative stability condition than the traditional Jensen-based inequality (Briat, 2015a;[START_REF] Briat | Memory-resilient gain-scheduled statefeedback control of uncertain LTI/LPV systems with time-varying delays[END_REF].

Introduction to LPV Time-Delay Systems

The first part of the chapter will reserve the definitions used in the rest of the thesis, such as delay space, convex function, etc. As discussed in the previous chapters, we are interested in the class of the parameter-dependent systems represented by LPV or quasi-LPV models (declaring properties such as time-continuous and having definite derivatives in the specific domain). Second, the time-varying delay only considers the case of small and slow-varying values.

Representation of LPV System with Single Delay

Let's introduce a generic LPV time-delay system of the form: Vt has been deployed in some delay-independent stability conditions. Since it doesn't contain information on the implemented delay, the approach is excessively conservative, especially when the delay is small, see, e.g., [START_REF] Fridman | Tutorial on Lyapunov-based methods for time-delay systems[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF]. Subsequently, the delay-dependent condition involved in the stability analysis contains the addition of a quadratic doubleintegral term 3 , Vt which capture the upper bound of delay. The derivative of 3 Vt  is expressed by

2 3 . t TT th VthxtRxthxRxd     & &&&& (5.4)
entails an obstacle with the integral term on the right side. At first glance, it seems like a complicated integration and should find ways to cancel out rather than tackling it directly.

During the first decade of the 21st century, considerable efforts and attention devoted to the study of delay-dependent stability can mention such as: the model transformations method (Descriptor, Parameterization, Cross-Term Bounding, and Free-Weighting Matrices, etc.), the input-output method (Delay Operators, Small gain Theorem, etc.), and the discretized Lyapunov-Krasovskii functional method. The outline of the methods and their pros and cons are discussed in more detail in Appendix C. Besides, one of the limitations of the delay space 0 H and LKF (5.3) is that it does not include the lower bound of the delay. Which in practice this condition sometimes does not exactly characterize nonzero hysteresis systems. The delay-range stability research has attracted a lot of attention in recent decades for LTI systems such as [START_REF] He | Delay-range-dependent stability for systems with time-varying delay[END_REF][START_REF] Liu | Stability analysis of systems with time-varying delays via the second-order Bessel-Legendre inequality[END_REF][START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF][START_REF] Tian | A new multiple integral inequality and its application to stability analysis of time-delay systems[END_REF][START_REF] Briat | Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers[END_REF]; X. M. Zhang et al., 2017), for T-S fuzzy systems such as [START_REF] Datta | Improved stabilization criteria for Takagi-Sugeno fuzzy systems with variable delays[END_REF][START_REF] Dey | Improved delay-dependent stabilization of time-delay systems with actuator saturation[END_REF][START_REF] Li | Stability and Stabilization with Additive Freedom for Delayed Takagi-Sugeno Fuzzy Systems by Intermediary-Polynomial-Based Functions[END_REF][START_REF] Lian | Stability and Stabilization of T-S Fuzzy Systems with Time-Varying Delays via Delay-Product-Type Functional Method[END_REF][START_REF] Lian | Stability and stabilization for delayed fuzzy systems via reciprocally convex matrix inequality[END_REF][START_REF] Peng | Delay-range-dependent robust stabilization for uncertain T-S fuzzy control systems with interval time-varying delays[END_REF][START_REF] Tian | Stability Analysis and Generalized Memory Controller Design for Delayed T-S Fuzzy Systems via Flexible Polynomial-Based Functions[END_REF]Wang & Lam, 2018a[START_REF] Wang | A new approach to stability and stabilization analysis for continuous-time takagi-sugeno fuzzy systems with time delay[END_REF][START_REF] Sala | Stability analysis of LPV systems: Scenario approach[END_REF], etc. However, in this dissertation, we do not cover this issue but focus more on relaxing the stability conditions and handling the stabilization for the saturated control system.

Recently, the convex function inequalities have significantly contributed to the more accurate estimation of the upper bound of the derivative of the Lyapunov function. Accompanying these developments is the suitable modification of the LKF candidate from an old-style form (5.3) to an extended vector with double and triple integrals. Generally, these studies are based on the application of analytic functions (see e.g., Appendix C.1).

The approaches should meet the requirements: reduce the conservativeness and optimize the numbers of decision-variable matrices. The following section is devoted to the analyses of the recent studies concerned with the less conservatism of Jensen's inequality.

Delay-Dependent LKF Stability -Convex function

We would deliver the alternative improvement for the tighter bounding inequalities involved with the LKF stability analysis via LMIs. All developments and postulates are based on characteristic analysis of convex functions such as Jensen, Wirtinger, Bessel-Legendre inequalities, etc., presented in Appendix C.1.

Jensen's Inequality and Extended Approach

There are considerable famous inequalities derived from original Jensen's inequality are characterized by convex function or variations of convexity. The definition and some properties of convex functions of higher order, the definitions of convex domain properties refer to Appendix A.1 or [START_REF] Boyd | Convex Optimization[END_REF]Mitrinović et al., 1993a). Among studies, the integral version of Jensen's inequality is frequently employed in control delay theory in the last decades. Jensen's inequality improvement also can be found in the mathematical literature [START_REF] Fink | Jensen inequalities for functions with higher monotonicities[END_REF]Mitrinović et al., 1993b). . Accordingly, the most effective of these bounding conditions is accompanied by a reasonable choice of expansion vectors with single integral, double integral and triple integral, respectively. Let's consider the following LKF candidates associated to LPV time-delay system (5.1) as follows: § The single integral case for application of Wirtinger-based inequality (WBII): Vt  Correspondingly, inequalities (5.9), (5.10) are used to estimate the lower bound of the derivative 4 .

The generalizations of

Vt  The effectiveness of the convex function is demonstrated in the works of [START_REF] Datta | Improved stabilization criteria for Takagi-Sugeno fuzzy systems with variable delays[END_REF]P. G. Park et al., 2015;[START_REF] Tian | A new multiple integral inequality and its application to stability analysis of time-delay systems[END_REF][START_REF] Briat | Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers[END_REF][START_REF] Zhao | A new double integral inequality and application to stability test for time-delay systems[END_REF] with the significant improvement in the stability conditions. In section 5.2, the maximum allowable hysteresis are compared with recent work on stability analysis of LTI and LPV time-varying delay systems.

Discretized Convex Function

Along with the auxiliary function method, the n-convex discretization method also shows the effectiveness in relaxing the conservatism. The analyses of n-convex can find in the monographs [START_REF] Boyd | Convex Optimization[END_REF][START_REF] Fink | Jensen inequalities for functions with higher monotonicities[END_REF]Mitrinović et al., 1993a). It is interesting to emphasize that the gap of Jensen's inequality significantly decreases corresponding to the number of segments. The reduction of the inequality gap is obtained by discretizing, respectively, 1, 2, and 3 segments (in Appendix C.1.

2) It shows that the higher fragmentation, the less conservative Jensen-based inequality. Using the simple Lyapunov-Krasovskii functional (5.3) cannot attain to the analytical delay limit (including using free-matrix conjoint with the decision matrices). So, let's consider a discretized Lyapunov-Krasovskii functional candidate associated to LPV timedelay system (5.1) under the form: (Jensen and Wirtinger). The minimizing gap in inequalities, making the condition least conservative. Some result has shown almost closer with analytical prediction [START_REF] Briat | Delay-Scheduled State-Feedback Design for Time-Delay Systems with Time-Varying Delays[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF][START_REF] Han | A Delay Decomposition Approach to Stability of Linear Neutral Systems[END_REF]. The interesting point about the discretizing delay intervals is that it suits to all advanced bounding techniques (section 5.1.2).

Delay-Dependent Stability -Input-Output Approach

As discussed, the delay decomposition approach provides less conservative results for stability analysis and controller design. But the method is effective for systems that access the exact knowledge of the delay, which is ideal for numerical computation in practical design. Actually, the identifications or estimations of the continuous-time delay phenomenon in practice are tough challenges, see, e.g., [START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF][START_REF] Belkoura | Parameters estimation of systems with delayed and structured entries[END_REF][START_REF] Chen | Robust identification of continuous-time models with arbitrary time-delay from irregularly sampled data[END_REF][START_REF] Ren | Online identification of continuous-time systems with unknown time delay[END_REF][START_REF] Zheng | Delay identification for nonlinear timedelay systems with unknown inputs[END_REF]. It should to mention that almost all the works from the literature analyzes the stabilization problem with memoryless (conservative) or exact-memory (non-implementable) controllers. In this case, the uncertainty (approximation) delay method discussed in Section 8.6 [START_REF] Gu | Stability of Time-Delay Systems[END_REF] shows to be more suitable for implementing the strategy of control system design. Specifically, the time-varying delay that is not accurately known at the time of analysis and design is considered as the dynamical uncertainties of nominal system. The inputoutput approach is very convenient in analyzing the stability based on the reformulation of the original system to feedback interconnection with additional inputs and outputs of auxiliary systems. Based on this approach, the stability is formulated in the input-output framework where the characterized LMI conditions obtain by the Scaled Small-Gain theorem [START_REF] Briat | Delay-scheduled state-feedback design for time-delay systems with time-varying delays-A LPV approach[END_REF][START_REF] Hmamed | Stability analysis of linear systems with time varying delay: An input output approach[END_REF] or supply function [START_REF] Briat | Memory-resilient gain-scheduled statefeedback control of uncertain LTI/LPV systems with time-varying delays[END_REF].

The objective of this section is to deliver a delay-dependent stability condition with an uncertain knowledge of the implemented delay.

zt zt  L L .
Temporarily ignore the effect of control input, TDS system (5.1) is transformed to the following differential equation using the internal topology with input-output structure: The equivalent between the Scaled Small-Gain and Lyapunov-based technique is discussed in [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF][START_REF] Boyd | Structured and simultaneous lyapunov functions for system stability problems[END_REF][START_REF] Doyle | Review of LFTs, LMIs, and μ[END_REF][START_REF] Zhang | Stability of time-delay systems: Equivalence between Lyapunov and scaled small-gain conditions[END_REF][START_REF] Zhou | [END_REF] for LTI/LTV systems. Both constant and time-varying approximate delay approaches will be more detailed in Appendix C.3. The quadratic supply rate (5.18) will be integrated in the delay-dependent stability analysis so-called m  memory resilient (in Section 5.2.4), and developed for the saturated control design of the LPV time-delay system (in Chapter 6).

Stability Analysis of Lyapunov-Krasovskii functional

In this section, the stability of the time-varying delay LPV/quasi-LPV system is verified by using the parameter-dependent Lyapunov-Krasovskii function (PDLKF) candidate given in (5.3), (5.11), and (5.13). Besides, the advanced bounding techniques provide a better relaxation for the stability condition but return in the conditional complexity. That requires a delicate manipulation to decouple the bilinear components encountered in the control design strategy.

Single Delay-Dependent LKF Stability and associated relaxation

Jensen's Inequality

From the above discussion, PDLKF in Eqs. (5.3) is used to deliver a stability condition of system (5.1) combining with ℋ∞ performance that leads to the following results.

Lemma 5. From this point, a linearizing transformation with the slack-variables concerning a generalization of Finsler's lemma decouples the decision matrix variables and maintains the parametric characterization structure (provides flexibility for the LMI condition without the additional assumptions). Now, by using the projection lemma that results in the associated relaxation of PLMI condition. produce extra-degrees of freedom for the designed condition (less conservative). But, the inclusion of condition (5.29) entails the unnecessary constraints for decision matrices. In the view of the LMI-Based relaxation methods, e.g., the slack-variable method [START_REF] Ebihara | LMI approach to linear positive system analysis and synthesis[END_REF][START_REF] Zope | Delay-Dependent Output Feedback Control of Time-Delay LPV Systems[END_REF]

Auxiliary Function-Based Integral Inequality

In this section, the stability condition delivers for the LPV time-delay system using an AFBII. As shown in the result section 5.2.5, this approach is the superior improvement of system performance compared to WBII and Jensen Inequality. Using the LKF equation (5.13) for stability analysis for the dynamical system (5.1), we have the following result. The LPV time-delay system (5.1) is asymptotical stable if the derivative of LKF (5.13) along the trajectories of system satisfies:

Decomposition Lyapunov-Krasovskii Functional Stability

In the last sections, the tighter bounding techniques has delivered a less conservative condition with the augmented LKF. So, how could one improve the system performance by using simple Lyapunov-Krasovskii functional? The necessary and sufficient conditions are derived from the discretized delay method in the works [START_REF] Gu | A further refinement of discretized Lyapunov functional method for the stability of time-delay systems[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF] for the LTV systems, then refined to the LKF decomposition [START_REF] Han | A Delay Decomposition Approach to Stability of Linear Neutral Systems[END_REF].

First, let recall a discretized Lyapunov-Krasovskii function associated to LPV time-delay system (5. The sketch of demonstration is based on the lines of Lemma 5.2.4 combined with the use of Wirtinger-based inequality (5.23) in Lemma 5.2.2. The stability delay-dependent is verified if the conditions:

                             & M L L L MMMO LL 0,                    p (5.
                            MO L LL LL LL MMMMMMMO LLL 0,              p (5.
1 1 2 tiht N TT i i tiht d VttPtxQxd dt          & &L 1 1 0. tiht N TT i i tih d tPthxsRxsdsd dt            & && (5.45)
holds along the trajectories of LPV system (5.1). The key role of development lies in the application of the WBII inequality to the second term of the following expansion .

tihtih t NN TTT iii ii tihtih d hxsRxsdsdhxtRxthxRxd dt                                  
  1 1 22 1 1 1 0 ,0, 00 r TT N I R Vtztwttt R                                              
N ii NNNN R N h h iiN R NNNN h h TTT wwwd R R R R BPBPBPI                                              & OO & L OOMO LL LLL ,                  

Uncertain Delay-Dependent Lyapunov-Krasovskii Functional Stability

As discussed about the obstacle of knowing the exact-delay value in implementation of control system design, let's consider an uncertain time-delay system as follows:

, ,

hdw hdw xtAxtAxthtAxtdtBwt ztHxtHxthtHxtdtJwt     &
(5.50)

with delay features as specified in the previous section. In the absence of exact knowledge of the delay , ht a robust stability addresses for the control and observation design strat- egy involved with two-delays [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF]. Concerning some design requirements such that the admissible maximum of the delay value, the permissible estimate (margin robust uncertain delay

), m tdtht   and the optimization of ℋ∞ performance criterion level might consider for the optimization problems.

A parameter-dependent Lyapunov-Krasovskii function is associated with system (5.50) as follows: m   By reformatting the integral limits, it is possible to capture the un- certainty variation of the approximate delay instead of just tracking the maximal relation. More specifics could find in section 5.7 [START_REF] Gu | Stability of Time-Delay Systems[END_REF] and section 4.7 [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF] Proof. This section is represented in D.3.

Memory-Resilient Delay-Dependent Lyapunov-Krasovskii Functional Stability

Now, let's address uncertain delay-dependent as the input disturbance by using the relation of bounded delay operator (5.16). Then, the L2 scaled bounded real lemma is applied to ensure robust stability for the uncertain structure (satisfies a well-connected property (5.18)). The interesting point is the approximation of the exact-delay value varying within an uncertain ball, defined by an algebraic inequality . m tdtht  

Jensen's Inequality

This methodology so-called -memory resilient where the stability PDLMI conditions are derived from the development of Lyapunov-Krasovskii function (5.14).

Lemma 5.2.7. (Briat, 2015a)

S sym 

Proof. A sketch proof is given in the D.4, a the full version can find in the literature (Briat, 2015a).

Wirtinger-Based Inequality

Let's employ a bounding technique WBII (5.6) for the resilient delay-dependent stability analysis for LPV system (5.50) involved the L2-norm bounded delay operator (5.16) that results in the following theorem.

Proof.

Take the derivative the Lyapunov-Krasovskii functional (5.11) along trajectories of LPV time-delay system (5.50) and combined with L2-norm performance on the controlled output, that implies an accustomed delay-dependent stability condition:  Applying the SSG lemma, with substituting

1 1 1 0 0, 0 TTTT r I Vtztztwtwttt R                    & (5.
(5.59) into condition (5.58) that yields the following parameter dependent LMI condition: Similarly, rearranging condition (5.64), we obtain stability condition (5.57).

1 1 0000 1 0 0, 0 TTTTT r I tTTtztLLLztwtLwt R                 

W

Remark 5.2.5. It should be noted that the satisfaction of parametric condition (5.61) implies the fulfillment of statements (5.64) and (5.65), but the reverse is not correct. By choosing L   sufficiently small when 0, m   the approximate delay stability conditions

(5.61) suggest to PDLMI (5.25). So this development provides a more general conditional form for delay-dependent stability analysis. Besides, the augmented Lyapunov-Krasovskii functional (as discussed in Sections 5.2.1 and 5.2.2) could consider delivering a further improvement of the inequality gap.

Remark 5.2.6. The feasible solutions of inequalities (5.57) depend on the parameters and its variation rates ,, tt   & nonetheless the slack-variable X   in condition (5.61) only 5.2.5. Example 135 depends on . t That causes a degradation in the equivalent characterization of the con- gruence transformation. However, the obtained condition is convenient for the stabilization synthesis. It should be emphasized that the associated relaxation of delay-dependent stability conditions (5.25) and (5.61) will be thoroughly utilized in the controller design strategy for the saturated LPV time-delay system.

Example

In the first section, the two well-known examples in the domain of delay-dependent stability analysis for LTI time-delay system is used to deliverer a concise comparison of the proposed conditions with other works. Then, the proposed PLMI stability conditions is relaxed to the multiconvexities forms (linear combination, T-S fuzzy, as discussed in previous chapters) to compare with the literature of delay-dependent stability analysis for fuzzy systems. First, let's consider the simplifications of the time-delay system (5.1) with:

Example 5.2.1: [START_REF] Gu | Stability of Time-Delay Systems[END_REF] [START_REF] Dey | Improved delay-dependent stabilization of time-delay systems with actuator saturation[END_REF] 2.2594 1.8502 2.3370 5n 2 +2n Theorem 1 [START_REF] Seuret | Delay-Dependent Reciprocally Convex Combination Lemma for the Stability Analysis of Systems with a Fast-Varying Delay[END_REF] 2.2130 18.5n 2 +5.5n Theorem 1 [START_REF] Zhao | A new double integral inequality and application to stability test for time-delay systems[END_REF] 3.1544 7.0463 9.5n 2 +3.5n Theorem 1 [START_REF] Kwon | Improved results on BIBLIOGRAPHY 225 stability of linear systems with time-varying delays via Wirtinger-based integral inequality[END_REF] 2.4203 1.6962 9n 2 +3n Theorem 1 (T. H. Lee & Park, 2017) 3.1555 2.4963 114n 2 +18n Proposition 1 (X. M. Zhang et al., 2017) As can be observed from the above table, the delay-dependent reciprocally convex combination method [START_REF] Seuret | Delay-Dependent Reciprocally Convex Combination Lemma for the Stability Analysis of Systems with a Fast-Varying Delay[END_REF][START_REF] Zeng | Hierarchical stability conditions of systems with time-varying delay[END_REF]C. K. Zhang et al., 2017;[START_REF] Zhang | An improved reciprocally convex inequality and an augmented Lyapunov-Krasovskii functional for stability of linear systems with time-varying delay[END_REF] considers amazed decision variables. However, the effectiveness is not as impressive as using the auxiliary convex function (Lemma 5.2.3). Actually, most of the maximal values of the upper bound delay (the bold values) solved by this lemma show superiority in all categories. So, the extending application of the bounding technique in conditions (5.5)-(5.10) Lemma 5.2.3 has expressively enhanced the MAUB. Now, by using the multi-convexities conditional relaxation form of PLMI for Lemma 5. Where the time-varying delay 0 ,, p htxt   HU and the membership function:

21 121 1, 1. xt xtextxt    
Applying LMI relaxation methods similar to those in previous chapters to verify the stability for system (5.68) that yields the MAUB values in the catalogs 1 and 2. 2.4293 2.0616 3.7638 3.0913 10n 2 +4n Theorem 1 [START_REF] Yang | New delay-dependent stability analysis and synthesis of T-S fuzzy systems with time-varying delay[END_REF] 0.4995 0.4988 58n 2 +4n Theorem 1 [START_REF] Zeng | Improved delay-dependent stability criteria for T-S fuzzy systems with time-varying delay[END_REF] 0.7584 0.7524 16.5n 2 +6.5n Theorem 1 [START_REF] Lian | Further robust stability analysis for uncertain Takagi-Sugeno fuzzy systems with time-varying delay via relaxed integral inequality[END_REF] 1.3123 1.2063 51.5n 2 +9.5n Theorem 1 [START_REF] Li | Stability and Stabilization with Additive Freedom for Delayed Takagi-Sugeno Fuzzy Systems by Intermediary-Polynomial-Based Functions[END_REF] 1.4819 Recently, the stability and stabilization analysis for the delayed LPV/Quasi-LPV system has adopted the novel LKF constructions for the LTI time-delay system and the advanced bounding techniques s.t. Wirtinger-based II [START_REF] Zeng | Improved delay-dependent stability criteria for T-S fuzzy systems with time-varying delay[END_REF][START_REF] Zhang | New stability and stabilization conditions for T-S fuzzy systems with time delay[END_REF], free-matrix-based II [START_REF] Lian | Stability analysis for T-S fuzzy systems with time-varying delay via free-matrix-based integral inequality[END_REF], auxiliary-function-based II [START_REF] Datta | Improved stabilization criteria for Takagi-Sugeno fuzzy systems with variable delays[END_REF][START_REF] Li | Stability and Stabilization with Additive Freedom for Delayed Takagi-Sugeno Fuzzy Systems by Intermediary-Polynomial-Based Functions[END_REF][START_REF] Tian | Stability Analysis and Generalized Memory Controller Design for Delayed T-S Fuzzy Systems via Flexible Polynomial-Based Functions[END_REF], reciprocal convex combination [START_REF] Lian | Stability analysis for T-S fuzzy systems with time-varying delay via free-matrix-based integral inequality[END_REF][START_REF] Lian | Stability and Stabilization of T-S Fuzzy Systems with Time-Varying Delays via Delay-Product-Type Functional Method[END_REF], then combines with the relaxation methods of the PLMI condition (Wang & Lam, 5.2.5. Example 137 2018a[START_REF] Wang | A new approach to stability and stabilization analysis for continuous-time takagi-sugeno fuzzy systems with time delay[END_REF][START_REF] Sala | Stability analysis of LPV systems: Scenario approach[END_REF] to deliver a less conservative condition. In recent years much effort has been devoted to the delay-partitioning LKFs and augmented LKFs, and fruitful results have been achieved, see for example [START_REF] Li | Stability and Stabilization with Additive Freedom for Delayed Takagi-Sugeno Fuzzy Systems by Intermediary-Polynomial-Based Functions[END_REF][START_REF] Lian | Stability and Stabilization of T-S Fuzzy Systems with Time-Varying Delays via Delay-Product-Type Functional Method[END_REF][START_REF] Tian | Stability Analysis and Generalized Memory Controller Design for Delayed T-S Fuzzy Systems via Flexible Polynomial-Based Functions[END_REF] and the reference therein. In order to further reduce the conservatism of the stability results, the new auxiliary polynomial-based functions (APFs) (S. Y. Lee et al., 2017;[START_REF] Lee | A novel Lyapunov functional for stability of timevarying delay systems via matrix-refined-function[END_REF], the Intermediary-Polynomial-Based Functions (IPFs) [START_REF] Li | Stability and Stabilization with Additive Freedom for Delayed Takagi-Sugeno Fuzzy Systems by Intermediary-Polynomial-Based Functions[END_REF] and Flexible Polynomial-Based Functions (FPFs) (Y. [START_REF] Tian | Stability Analysis and Generalized Memory Controller Design for Delayed T-S Fuzzy Systems via Flexible Polynomial-Based Functions[END_REF] are studied by introducing a set of orthogonal polynomials. These advanced techniques (i.e., generalized parameter-dependent reciprocally convex inequality) are proposed to better estimate the triple integral inequalities. However, it is possible to realize a significant improvement in system performance when comparing Lemma 5.2.3 with the mentioned works (as seen in the catalogs 1 and 2 of Table 5.2).

It can objectively acknowledge that the construction of polynomial-based functions aims to address the delay-range stability conditions. While the designed stability conditions in Lemma 5.2.2 and Lemma 5.2.3 are quite simple and have fewer computational complexities. With this advantage, these conditions would effortlessly adapt to the extension of the stabilized design strategy with the saturation constraints, approximation delay, etc. It's worth noting that both Lemma 5.2.1-Lemma 5.2.3 have validated delay-dependent stability for system (5.68) with a MAUB greater than 100s, corresponds to slowly varying delay cases 0, 0.1. hthht  & Shown in Figure 5-2 is the evolution of the system dynamic with a slow-varying time function Currently, the numerical simulation tool Simulink ® does not allow integrals in the variable interval, so we reformat the LKF as follows: TT fxxQxgxxRx   &&& Then, the approximation of LKF is given in the third frame of Figure 5-2. Besides, another popular example has been used in the last decade in demonstrating the effectiveness of the proposed stability conditions for the T-S fuzzy system is given below. The results of the MAUB delay values are provided in the catalogs 3 and 4 of Table 5.2 respectively with Lemma 5.2.1-Lemma 5.2.3. However, the majority of studies use this example deal with the problem of delay-range stability. So, we could not deliver a further comparison result. It's worthy to note that Wirtinger-based inequality (Lemma 5.2.2) could be considered as an exceptional case of the auxiliary-function-based method (Lemma 5.2.3). The vector expansion of this lemma employs only one single integral, showing an adequate trade-off between the number of variables to be solved (computational complexity) and the maximum value of the upper bound delay (conservatism). It provides a less conservative stability condition than traditional Jensen conditions, with an integrable conditional structure (without too much decoupling of the decision matrices to the dynamic system).

Furthermore, as mentioned about the multiple production complexity of the stability conditions using LKF formulation (5.13), in order to be compatible with control strategy designs for LPV time-delay systems subject to the saturated actuator, this appropriate LKF takes precedence over conservatism. Specifically, the analysis to the approximate delay and the memoryless gain-scheduled feedback controller with the saturation constraints are based on condition (5.25) in Theorem 5.2.1. It exhibits a well-proportioned stabilization condition between conservatism and numerical burden. These issues will be continued to discuss in detail in the next chapter.

Conclusions

In this chapter, the preliminary premises of the LPV delay system have been delivered. The delay-dependent stability is addressed on time-domain based on the Lyapunov Krasovskii functional technique. Where the convex properties of the analytical function are generalized the application to improve the stability conditions. Corresponding to each bounding technique (Jensen-based integral inequality, Wirtinger-based integral inequality, auxiliary-function-based integral inequality, etc.) is an appropriate selection of the augmented LKF to achieve the highest efficiency.

These delay-dependent stability conditions are analyzed for the parametric dependent system adopting the new structures LKF with double integral and triple integral. The parametric LMI stability conditions can be relaxed into LMI conditions and effectively solved by common convex optimization algorithms (barrier function, interior-point method, etc.). Along with the original stabilizing LFK conditions via PLMI, are associated-relaxation presented to decouple the multiple-production of the decision matrices and the dynamical matrices of state system. Furthermore, the transformation could adapt to all generalization of the proposed auxiliary-function condition structures were analyzed for the delay-dependent stability. A simple linearization method directly delivers a tractable condition that is suitable with the stabilization synthesis. The comparison results illustrate the effectiveness of the designed stability condition.

Chapter 6. Stabilization Synthesis for the LPV/quasi-LPV

Time-delay Systems with Actuators Saturation

Stabilization Synthesis for the LPV/quasi-LPV

Time-delay Systems with Actuators Saturation

This chapter inherits the implementations of the LPV saturation system in the previous chapter to develop the controller for the LPV delay system with constrained actuators. The analysis of the LPV delay control system typically entangles in multiple productions (for example, the stability conditions in sections 5.2.1, 5.2.4 are related to the matrix product expressions PA and RA with P, R being the decision variable matrices). The approaches such as descriptor or free weighting matrix will be more problematic when considering saturation constraints in the stabilization condition. Thus, this methodology settles these problems agreeing with the following design strategy:

1 -Develop the stability conditions for open-loop system (unsaturated)

2 -Employ the saturation conditions imposing on controller (generalized sector bounding), then multiple-productions will be decoupled using Finsler's lemma.

3 -Substitute the variable into the closed-loop expression (using the congruence transformations and setting the variable to obtain the parameter-dependent LMI condition).

4 -Relax the associated PLMI conditions, corresponding to the design requirements, then the PLMI conditions are converted to finite dimension LMIs by gridding, convex combination, S-Variable methods, etc.

The "keywords" of modern control technology related to the control theory of LPV systems, the time-delay LPV systems, and the saturated system analysis, respectively, have been fully annotated in chapters 2, 4 and 5 with respective references. It should be noted that the characteristic of exact-memory controllers is non-implementable in practice due to the difficulty in estimating delays. Therefore, the uncertain delay gain-scheduled controller is more suitable for the delay-dependent stabilization condition. The features of the control system are: § The delay considers to vary in a range or approximate, and thereby more applicable in practice. § In the framework of modified sector condition, a suitable auxiliary controller strategy not only gives a more accurate estimation of the lower bound of LKF but also relaxes the saturation constraints. § The use of Wirtinger inequality reduces the gap in Jensen's inequality, it has shown a reducing conservatism of the stability and stabilization delay-dependent conditions analysis. The improvements compare with the existing results by using fewer number of decision matrix variables.

In the first section, the rudimentary definitions such as the admissible set of the initial condition, saturation constraints, and structure of delay-dependent controllers will be present. The corresponding stabilization conditions then deliver for each design strategy.

Problem Formulation and Preliminaries

Sector Nonlinearity Model Approach

A time-delay LPV system with actuator saturation presents under the forms:

      sat,

Saturation nonlinearity.

Control input vector 12 ,,,

Tm m utututut    K R constrains by the limits of satura- tion , ,1,2,,.
iii utuuim   K Let's recall a dead-zone nonlinearity associated with a symmetric saturation function sat uutu  : signif 1,2. 0if

iiiii i i i uuuuu uim u u           K (6.2)
Define an auxiliary control law t belong to the polyhedral set

    ,,:, 1,2. m ii uuuim   K S R (6.3)

Region of Attraction.

The saturation limits of actuators make the control design of the time delay LPV system more challenging. The system (6.1) attains global stability if the trajectories asymptotically converge to the origin from all initial conditions ,,0, h   without effect of disturbance 0. wt  Nonetheless, this condition is hard to satisfy in practice. Instead of having to assurance all the initial conditions, an estimation of the region of attraction determines the initial conditions to which the system will converge asymptotically. The key issue relating to the estimate of the Region of Attraction (or Domain of Attraction -DoA) belongs to Banach space of continuous vector function of initial:

  2 2 1 0 12 2 2 ,0,0 |sup,sup. ,0, n hh h       X C R & (6.4)
Ellipsoidal Set of Stability.

Lyapunov-Krasovskii functional candidate is used as a primary stability analysis tool for the dynamic systems. The estimations of DoA are associated to the following LKF:



In the next sections, the delay-dependent stabilization conditions derive for the state feedback, and the dynamic output feedback controller related to the determination of the delay in the system.

Parameterized State Feedback Controller

Considering a controller law is based on the compact sets of parameters :

t  , d utKxtKxtdt   (6.6)
with scheduling gain , d KK   are sought to stabilize the time delay system (6.1), and dt is the delay approximation of the system delay ht as presented in section 5.2.4.2.

Let's recall the permissible delay estimate: Most delay-dependent control strategies are typically concerned with the first case to simplify the design and be suitable for implementation (where delay values are unavailable for measurement). Since the gain d K   does not include in the feedback controller struc- ture, this approach is conservative. The second case allows relaxation of the stabilization conditions but is inapplicable in practice. In the last case, the delay approximation (within the robust margin) indicates a practical implementation and is less conservative than a memoryless controller.

In the framework of saturation control using the generalized sector condition, the restriction on the feedback control law can be wiped off with the auxiliary controller. Specifically, let's consider the following controllers t S associated with ut (which clcl AABKHHJK  The design requirements related to the ℋ∞-performance criterion is to search for bounded feedback controllers and auxiliary controllers, which guarantees the region stability of system (6.11). From initial condition (6.4), then condition (6.3) states briefly as follows: for the design saturation limits i u look for the auxiliary control law t that satisfy ,1,2,,. ii uim  K Generally, to ensure the stability of the closed-loop system under the influence of disturbance, the following necessary and sufficient conditions need to be fulfilled:

must
2 2 , 1 , i i V t xtxt u        & (6.12) if 1 , 0 , 0: wtVV xtxt      & (6.13) if 2 2 1 , 0 0 : . w t t twt V x Vx     & L (6.14)
From the view of condition (6.12), the appropriate selection of an auxiliary controller ensues a better estimate of the lower bound of the Lyapunov function. That makes the stabilization conditions of the saturated control system less conservative. But conditions (6.13)-(6.14) are complicated to enforce directly for the time-delay LPV system unlike the method proposed in Chapter 4. Without loss of generality, we can assume that the energy bound of the disturbance is known

2 1 2 , wt    L
and the set of admissible initial condition is defined by the upper bounds in (6.4).

Optimization problems.

Combined with the estimation of DoA (6.4), the performance criterion, memory resilient and the upper-lower bounds of delay value, we formulate the following optimization problems: § Given ,,, m h  then maximize the size of DoA. § Given ,, m h  and a set of admissible initial condition then minimize  (optimization disturbance rejection level). § Given ,  a set of admissible initial condition then optimize h (the maximal upper bound on the delay value) or maximize m  (the allowable delay approximation).

It should be noted that the above cases consider a supposition on energy bounded exogenous signals (a L2-bound on the admissible disturbances defined by ).  In addition, the optimization problems such as minimizing energy-to-energy index ,  maximizing the up- per bound of the delay value , h and maximizing the delay approximation value m  are all convex problems. These values can be derived from a sub-optimization method or an iterative algorithm. However, the admissible set of the initial conditions usually relates to the concave problem. Therefore, we focus more on seek for the largest estimate of DoA 6. 1.3. Parameterized Dynamic Output Feedback Controller 145 that satisfies the designed delay-dependent stabilization condition.

Parameterized Dynamic Output Feedback Controller

Let's consider a dynamic controller feedback system with approximate memory:

    0 ,, 0 ,0, 0 ,

DD

It is confusing to deploy the delay-dependent stability condition directly for the saturated LPV system considering a dynamic output feedback controller. Unlike the state feedback controller analysis, the stabilization analysis for closed-loop system (6.16) involves nonlinear structures. The use of congruence transformation only exacerbates the problem because of the coupling of the decision matrix LKF. Moreover, it can be seen that the quasiconvex related to the saturation conditions did not completely resolve in previous work (even the controller analysis for LTI systems). Inspired by the research of [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF][START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF], we propose a new approach to solving the problem sequentially in the following steps: § First, deliver a delay-dependent stability condition for unsaturated system (6.16) with inputs u  andwt (employ Theorem 5.2.2 to address the stability condition with approximated delay value). § Then, include the saturation conditions and integrate the GSC condition (developed similarly to Section 4.4). § Lastly, use a congruence transformation to return the tractable condition.

It can see that stability analysis for resilient memory DOF controllers is more challenging than for exact-memory DOF controllers. The variable substitution is more problematic when there is no match in the matrices h A and d A concerning A (which shows the impos- sibility of setting the variable as the method in section 4.4). The problem synthesis will be discussed and presented in Section 6.3.

State Feedback Controllers

This section concerns the synthesis of saturated state-feedback control laws with a memoryless and approximate delay value. The stabilization of closed-loop is addressed based on the delay-dependent stability conditions given in sections 5.2.1.2 and 5.2.4.2, respectively, for the single delay and approximate-delay case. These conditions are approached based on the application of the Wirtinger inequality, which are less conservative than the Jensen inequality (the comparison has shown in section 5.2.5).

Memoryless Delay-Dependent Stabilization

In the first case, we do not include the exact memory gain in the feedback controller structure, but the auxiliary controller t could employ with formulation (6.9) to relax the saturation condition. Now, the closed-loop system is obtained from the general form (6.11) with 0

                                                % %% %%% 0,                p (6.

Proof.

A sketch of the proof is presented sequentially as follows. First, by using Lyapunov-Krasovskii functional candidate (6.5), the parametric LMI condition (6.19) suggests the lower bound of the saturation constraints on auxiliary controller (6.9) are developed similar to Theorem 4.1.1:

(6.19)     1 2 1 23 2 1 , 22 T II T ii i h PhR GGGG PhRPQR            ° Lemma 5.1.1 1 2 1 23 2 1 . 22 TT ii T i h PhR ttVt PhRPQR           (6.20) with 1,2.
im  K Then, in the view of the initial bounding set conditions (6.4), we have: The development of the delay-dependent stability is based on Lemma 5.2.2 by using Lyapunov-Krasovskii functional candidate (6.5) combined with ℋ∞ performance criterion, and GSC condition (6.22) that results in: UU Finally, if the stabilization condition (6.18) fulfills then the derivative of Lyapunov function (6.5) holds along the trajectories of closed-loop system (6.17), that ensues § When 0, wt  then   Then, the DoA size optimization problem is formulated to seek the minimum value of the upper bound on condition (6.21). In contrast, the upper limit is indisposed to set by 1   because it also relates to the energy bounded disturbance and the attenuation level 1 .

  22 1 1 t T I tht w u tu VtzttT KGxGxd                   & 1 1 0 0, 0 TTT r I tttt R       

 

Since conditions (6.18)-(6.19) are linear matrix inequalities, seeking the upper bound of time-varying delay is not much of a challenge. However, condition (6.21) yields the nonconvex formulations relating to decision matrix matrices ,, QXQXRXRX  %% Kmake it not always possible to attain a good estimate of the initial condition domains. In Section 6.2.3, a linearization is proposed to handle the concave problem.

Approximated Delay-Dependent Stabilization

As discussed in section 5.2.4.2, the uncertain delaydt provides a more general form of stability condition, in which a delay-dependent stability condition with memoryless or exact memory can derive from an approximate delay condition. By substituting the closed-loop system (6.11) into the relaxed LMI condition (5.61) and repeat the same analysis as in Theorem 6.2.1 that leads to following result.

Proof.

This part will be omitted because it is similar to the development of memoryless control laws in the last theorem. By substituting closed-loop system (6.17) into delay-dependent stability condition (2.194) gives in (6.9). W Remark 6.2.1. The delay approximation transformation is derived from the delay-dependent stability condition in Theorem 5.2.2. The difference between conditions (6.27) and (6.18) lies in the fourth and eleventh row and column. As a result, the approximated memory state-feedback controller provides the implementation feasibility compared to the exact-memory controller and gives less conservative stabilizing conditions than the memoryless controller.

Optimization problem -Maximization Domain of Attraction

The optimization problems usually involve to the size criteria of the admissible initial condition and the maximum allowable level of the disturbances. Specifically, let's consider the following criteria relate to inequalities (6.13), (6.14), that includes: § Preselect , then minimizing  As discussed, the optimization problems such as minimizing energy-to-energy index ,  maximizing the upper bound of the delay value , h and maximizing the delay approxima- tion value m  are all affine conditions. These values can be found by optimization methods or by iterative algorithms. However, the expressions of the admissible set of the initial conditions are involved in the bilinear forms (e.g., condition (6.21)). In this section, we seek for the largest estimate of DoA that satisfies the delay-dependent stabilization conditions in Theorem 6. . PI  °So, optimization problem (6.30) also means looking for the minimization of the greatest eigenvalue of variable matrices ,, PQR of Lyapunov-Krasovskii functional (6.5). By such expansions, the concave problem is transformed into a linear optimization problem with the objective of finding variables ,1,2,3,4,

i i    R
such that the cost function (6.30) reaches the smallest value corresponding to the design stabilization conditions and the linearity conditions (6.31).

Example

This section is devoted to the analysis of the results and discussion of the proposed method. First, an example of the LPV delay system discussed in [START_REF] Briat | Memory-resilient gain-scheduled statefeedback control of uncertain LTI/LPV systems with time-varying delays[END_REF][START_REF] Zhang | Delay-dependent stability analysis and H∞ control for state-delayed LPV system[END_REF] is applied to demonstrate the performance of a designed controller with the approximate delay value (in the case of with and without external disturbance). Second, the efficiency of the saturation constraints is validated on an unstable open-loop LPV system with the bounded signal control (due to physical limitation or safety mechanism, etc.). Then, the performance degradation and instability will observe in the systems without the saturated control design.

In the next part -section 6.2.4.2, we borrow a well-known example (Fridman et al., 2003;[START_REF] Gomes Da Silva | Stabilisation of neutral systems with saturating control inputs[END_REF] to show the adaptation of the proposed method. In which the local stabilization enforces the saturated LTI time-delay system. After that, a comparison of the maximizing estimation of the domain of attraction is provided. The optimal disturbance reduction levels solving by the stabilization condition of Theorem 6.2.1 and Theorem 6.2.2 deliver for unsaturated system (6.32) that will compare with other unsaturated control systems. So, the constrained saturation on controller temporarily is ignored. Specifically, condition (6.19), the fifth columns and row of condition (6.18) of Theorem 6.2.1, condition (6.28), the sixth columns and row of condition (6.27) in Theorem 6.2.2 will be not included in the first controller comparison. Parameter t is gridded over 41 p N  points uniformly spaced 1,1  . The optimal re- sults are solved by the modified conditions of Theorem 6.2.1 and Theorem 6.2.2 consistent with given delay values in Table 6.1 and the assumption of zero initial conditions. The proposed stabilizations show the flexibility when it can comfortably apply to the unsaturated controller system analysis. Moreover, performance improvement has been attained with the memoryless and memory-resilient controllers compared to [START_REF] Briat | Linear Parameter-Varying and Time-Delay Systems[END_REF][START_REF] Briat | Memory-resilient gain-scheduled statefeedback control of uncertain LTI/LPV systems with time-varying delays[END_REF], demonstrated the relaxation of the proposed method. It is interesting to point out that the stabilization conditions with an approximated delay value could deliver the same disturbance rejection optimization levels with the exact-memory and memoryless, respectively, when 0 m   and .

m h   However, the effects of saturation are less attractive analyzing the stabilizable system (LPV time delay system (6.32) has h AA   Hurwitz). So, let's consider a modification of the previous example. Example 6.2.2: Let's introduce a quasi-LPV time-delay system: The responses of the bounded controllers correspond to saturation limit 5, u  with 1,0.9. h  

In this example, the simulation starts from safe initial conditions that belongs to the estimate of DoA. Then, the disturbance amplitude gradually increases but does not exceed the maximum allowable bounded energy The purpose of this process is to estimate the evolutions of the state-dependent parameter as well as the saturation response of the unbounded feedback controllers. It clears to realize that t  & is bounded between 5,5.  But most of the response characteristics are in the range of 3,3  (even when system is influenced by a large magnitude external disturbance). So we proceed to solve the proposed theorems with parameter values selected in the assumed range.

The actuator saturation usually takes place in many practical applications where its existence may lead to the degradation performance or even cause the instability of the closedloop system. In the monographs [START_REF] Hu | Control Systems with Actuator Saturation[END_REF][START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], the authors thoroughly explained all the behavior of the closed-loop states frozen when the actuator is saturated. In which the state dynamics can destabilize or converge to a parasitic equilibrium instead of toward to the origin. The gain-scheduling controllers solved by the proposed methods could be compared with the system without the saturation conditions. The constrained conditions (6.19), and (6.28) ensure the local stabilization contrasting to the system without a saturated design.

Example 155

However, these results may be less objective than in Table 6.1, when we reconstruct stabilization conditions of Theorem 8.1.5 and Theorem 4.2 of the literature [START_REF] Briat | Linear Parameter-Varying and Time-Delay Systems[END_REF][START_REF] Briat | Memory-resilient gain-scheduled statefeedback control of uncertain LTI/LPV systems with time-varying delays[END_REF] to implement controls applicable for the saturated LPV time-delay system (6.33). Nevertheless, the scheduling gains of the proposed theorems have shown the stability regulation efficiency for the LPV time-delay system conforming to actuator limits. The parametric dependent gains are given by:

Memoryless Saturated Controller

Solving the stabilization condition of Theorem 6.



In the bounded controller framework, the signal is sensitive to saturation limit if the designed gain is too high. Explicitly, the higher rejection disturbance level ensues in smaller linear behavior region. As a way of repeating, it is a trade-off between system performance, estimation of domain asymptotical stability, and the admissible set of the initial conditions.

Maximization of the set of admissible initial conditions

Now, we present the results of the optimization method discussed in Section 6.2.3 for estimating the allowable initial conditions. There are inconsiderable studies on this aspect for the LPV time-varying delay system with the saturated actuator, so we employ the The more slack-matrix variables usually lead to high computation complexity. The proposed method provides good results with less conservative stabilization conditions for a reasonable number of variables to be determined. To solve the stabilization conditions for system (6.37), we only use 27 variables compared to 36, 37 variables of the reciprocally convex combination (RCC) and free matrix based (FMB) method [START_REF] Dey | Improved delay-dependent stabilization of time-delay systems with actuator saturation[END_REF], and 82, 35 variable of the free-weighting matrix (FWM) method (Chen et al., 2015[START_REF] Chen | Robust Stabilization for Uncertain Saturated Time-Delay Systems: A Distributed-Delay-Dependent Polytopic Approach[END_REF], respectively. The methods outlined in Table 6.2 focus on the LTI system. The trade-off of conditional relaxation with the computational complexity of the condition makes these approaches hard to implement on a parameter-dependent stabilization. The proposed condition balances the number of variable matrices, the conservatism, and the scalability of the control synthesis. To the best of our knowledge, the stabilization problem for LPV/quasi-LPV time-delay system subject to actuator saturation has not been well addressed, although the problematics have significant practical applications. In the next section, the proposed structure conditions are used to deliver the stabilization of the dynamic output controller. Then, the optimization problems for LPV time-varying delay systems relate to the size criteria of the admissible initial condition and the maximum disturbances attenuation are provided in section 6.

Proof.

A sketch of the proof is provided as follows. Firstly, using the transformation (6.41), we have LMI (6.51).a equivalent to the following condition: We consider the Lyapunov-Krasovskii functional (6.5) for extended system (6.38) consistent with the dynamic state . t  It can realize that (6.53) are the saturation constraints involved in feedback controller (6.39) and auxiliary controller (6.40). By inversing variable assignments (6.49), we get the dynamic output controller (6.52). The rest of the demonstration will follow the lines as in Theorem 6.2.1.

W

For using the congruence transformation, the slack-matrix W is chosen as a symmetric matrix. Besides, as discussed in the previous sections 12 ,,, W  must be parameter-in- dependent variable matrices to avoid nonlinear components appearing in the condition. Analysis of stability conditions for system (6.38) takes full advantage of the commensurate structure between matrices A and , will suppress the bilinear terms and deliver a convex condition. Nonetheless, it loses the generality and interestingness of the approximated delay method. In the next section, Young's inequality is recalled to deal with these nonlinear problems.

Approximate Delay-Dependent Stabilization

Let's introduce an input control Applying Young's inequality for above PLMI condition, then using Schur to rearrange the matrix inequality that entails in the stability condition (6.66). The rest follows the same lines as the one of Theorem 6.3.1.

,

It can be noticed the difference between the analysis of a dynamic output controller with an approximated delay and a state feedback controller. Where the exact-delay value is still stuck in the controller structure (6.57). We could assume a simplification of output measurement, but the hysteresis also affects the measured signals in practice. Thus, system (6.15) presents a more general form. But, to deal with the concave problem, we have to use Young's inequality for condition (6.68), which has limitation.

Optimization problem -Maximization Domain of Attraction

As shown in the results section 6.2.4, the minimum values of attenuation criterion can be obtained by directly optimizing  in PLMI stabilization conditions. The maximum allow- able value of the delay margin h can be found by the iterative sweeping technique. It should note that both approaches carry out with preselected values of 0 ,,,  or , h etc.

As we discussed in section 6.2.3, the estimates of the domain of attraction are concerned with the coupling of the variable matrices in the conditions. In which minimizing ,,, PQR %%% concomitant with maximizing X % makes the global optimization sometimes not converge to the correct solution. Nonetheless, so far, the recent methods have solved the optimization problem of the stabilization condition according to this approach. By substituting 1  in place of X and slightly modifies Problem 6. The methodology avoids concave-problem and singular matrix forms. But, it has a drawback in approaching the dynamic output controller stabilization. That will be revealed in section 6.3.4.2.

Example

The stabilization implementations analyzed in this chapter use the generalized sector bounding condition to enforce the control saturation. Nevertheless, the saturation constraints set for the state feedback controller in Theorem 6.2.2, distinguishes from the dynamic output controller in Theorem 6. In addition, most of the work on developing stabilization conditions with delay and actuator saturation is generally applied to LTI systems and employs a quadratic Lyapunov function. In section 6.3.4.1, the parameter-dependent stabilization conditions are addressed for an LPV time-delay system. Then, the comparison of gain-scheduled controllers, and the maximizing the estimation of DoA concern to the parameter-dependent Lyapunov function will be provided in section 6. [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF] with solver Mosek [START_REF] Andersen | On implementing a primal-dual interiorpoint method for conic quadratic optimization[END_REF].

Following the definition of domain of attraction 0 , X provides the maximal values of On the contrary, if the initial conditions do not belong to 0 , X then the trajectories of the closed-loop system won't leave the RAS corresponding to the optimal value 1 . opt   It is noteworthy that compared with the results in section 6. 2.4.1 (optimizing ),  the scheduling gains (6.75) are smaller. The system trades between performance and stability in the size criterion optimization (e.g., minimizing eta). As explained in Chapter 4, since the linear response region expands, the feedback control signal rarely exceeds the saturation threshold. These properties will illustrate in the next figures.

Saturated State Feedback Controller with Approximate Memory

Solving the stabilization condition of Theorem 6. The dynamic controller gains schedule online consistent with the expressions given in (6.49). From the allowable set of the initial conditions, we can observe from Figure 6-3 that the control feedback signals exceed the saturation threshold from the 0 th to the 1 st second. The state feedback controller continues to surpass the control limit from the 3.5 rd to the 5.5 th second. That can be referred to as the states of the feedback control system in Figure 6-2. If there is no saturation bound, then the system will converge asymptotically 6. The evolutions of the system state regulated by the state feedback controller and the dy- The behaviors of the dynamic system governed by state feedback controller are underdamped (in comparison to the dynamic output feedback controller). In this example, an exchange between system performance and the set of the admissible initial conditions could deduce from expression: 111 0 . opt    is pre-set for both theorems, and a disturbance input sin wtt effects from the 6 th to the 10 th second. Explicitly, both LPV time-delay systems ensure stable asymptotes consistent with the design values. The time-varying delay expressed like the one in Eqs. (6.36) and the approximate delay are presented in the bottom frame of Figure 6-2. The feedback controller gains (6.75) with an uncertain delay value obtained from the solution of Theorem 6.2.2. Besides, the scheduling gains of the dynamic control system (6.52) are indirectly found from the variable matrices (6.77), which attain from the feasible solution of Theorem 6.3.1. The results demonstrate the effectiveness of the proposed method with the guarantee of local stability for the delayed LPV system respect to the saturated constraints. It should note that in practice, it is difficult to reach all states (cause the limitations of measurement, the effect of delay on measurement output, etc.). Therefore, the approach of Theorem 6.3.1 shows more practical significance.

Optimization Problems

We deliver a comparison of the minor axis maximization of the ellipsoid , the minimization of disturbance attenuation index , the maximization of the upper bounds of the    X

The comparison between the two theorems are detailed in Table 6.3.

As can be easily visualized, Theorem 6.2.2 dominates Theorem 6.3.1 in all optimization categories. From the estimation of RAS, the rejection disturbance criterion or the maximum allowable value of the delay are all better. It is reasonable because the stabilizability synthesis of the state feedback controller always attains the best relaxation compared to the output feedback controller. It should note that the difficulty of optimization problems increases from categories 1 to 4. In which the optimizations of  and  could achieve directly from the stabilization conditions. And the maximum delay is solved by an incremental loop algorithm. But, maximizing the set of the admissible initial conditions must meet the satisfaction of Problem 6.2.1 and Problem 6.3.1, respectively, combines with the stabilization condition of Theorem 6.2.2 and Theorem 6.3.1. Let's recall the upper bound expression of the initial condition (6.21) combined with the analysis of the stabilization with and without the influence of external disturbance (6.13)-(6.14), we have: In the simulation with the feedback controller (relating to Theorem 6.2.2, illustrating in Figure 6-4.a&b), we chose 010,1030,30, x and 0,0,0. h   In con- trast to Chapter 4, the characterization of the asymptotic stable domain of an LPV timedelay system meets difficulties with the extended vector and the initial condition. It can notice that the level set of ellipsoids (6.79) could not completely cover the behavior of Vt  (cause that does not characterize the integrals). So, it could not accurately describe the domain of attraction. Specifically, some initial conditions outside the ellipsoidal set, such as 06,22, x7,20, 8,18  Kwill converge to origin. The estimates of the domain of attraction presented in Figure 6-4.c&d are the solution of solving Theorem 6.3.1&Problem 6.3.1. It is apparent to recognize that the region of attraction of the closed-loop system regulated by the state feedback controller is larger than that of the dynamic output feedback controller.

The optimal results of Theorem 6.3.1 given in Table 6.3 are approached similarly to Theorem 6.2.2. However, inequality (6.70) contains a quasi-convex form, if 2  is present in Using the second approach, we obtain a disc of DOA with a diameter of 0.2985 and an ellipsoidal domain shown in Figure 6-4.c&d.

Conclusions

We have addressed the feedback control laws to stabilize the LPV time-delay systems with the saturated actuators. The control analysis and synthesis with the state feedback and the dynamic output feedback structure is provided. Where the nonlinearities and concave problems in the stabilization conditions handle effectively by using a simple transformation. The multi-criteria optimizations implement both memoryless, approximation delay controllers subject to the control saturation. The results show an enhancement of system performance and robust stability against the effect of the external disturbance and time-varying parameters.

Chapter 7.

Conclusions and Perspectives

Summary

Through the contents discussed in this thesis, there are main contributions which are briefly summarized by the following conclusions:

 PLMI conditions have been considered to solve the analysis and design problems for LPV and quasiLPV systems. The derived conditions have a general formulation which is convenient for various design purpose. Some numerical results in Chapter 2 have given to illustrate the advantage this methodology.

 By considering the scaling structure, the non-convex optimization related to the robust controller design is linearized into the multiple-convex optimization problem through an iterative algorithm CCL. The relaxation results given in Chapter 3 have shown the effectiveness of the proposed method.

 In Chapter 4, the control synthesis conforming to saturation constraints is investigated to address the stabilization of the feedback controller design for saturated LPV systems. The derived PLMI formulations allow relaxation for an individual implementation. The simulation results emphasize the reducing conservativeness of the presented condition compared to the existing works and provide an LPV analysis tool for gain-scheduling feedback controller subject to saturation constraints.



The following chapter has focused on the stability of time-delay systems based on Lyapunov-Krasovskii functional. This chapter is a slightly different construction in which an appropriate stability analysis method is addressed for the LPV delay systems before proceeding with the stabilization analysis. The comparisons presented in chapter 5 demonstrate the less conservative results of refined LKF conditional forms using expanded vectors. The best results are obtained by the investigated method compared with the recent works in the literature.

 Based on this design strategy, the saturation conditions combined with the delaydependent stabilization scheme allow the balance between the conservatism and computational efforts. The estimation of attraction domains shown in the last chapter has demonstrated this point of view. Furthermore, the simulation results also reveal the system's enhancement when actuators are saturated. In addition, the resilientmemory controller has shown good performance with respect to saturation limits and the robustness with uncertain knowledge of time-varying delay values. Finally, a linearization method fruitfully is converted the nonlinear matrix inequality constraints involving the optimization of the DOA of dynamic output feedback controllers to the tractable LMI conditions. The estimation of DOA also exposes some characterized ellipsoidal domains associated with LKF.

Remaining problems and future work

The work of this dissertation is covered by five chapters related to stability analysis and stabilization for LPV systems and time-varying delay subject to actuator saturation. The main objectives of the thesis have been achieved through theoretical results. In which the proposed methods have quite productively treated the problems corresponding to specific contexts. However, it is possible to point out some open problems involving in the particular case as follows:

 The CCL algorithm in Chapter 3 is convenient for linearizing design problems using QLF. However, if the more general forms i.e., PDLF are considered, it must be quite confusing to deploy this algorithm. On the other hand, the application of CCL algorithm mainly involves refining the slack scalars in Young's inequality. Unfortunately, the use of this inequality to decompose uncertain structures or bilinear matrices is a very conservative manner. Similar to the concave optimization problem discussed in Chapter 6, mathematical tools are needed to properly treat these nonlinear matrix structures.

 Regarding the developed saturation conditions, limits only consider the symmetry case. In fact, there are many asymmetric bounded saturation systems for which the design constraints (GSC) needs to be developed to be more suitable for the general case. Moreover, the guarantee of regional stability conditions corresponding to the state constraints has not really been completely solved. One of the promising approaches that can be integrated with the saturation condition via LMI technique is the shaping Lyapunov function (control barrier function) method.

 Almost all the examples consider time-delay systems are stabilizable/detectable for 0 , 0, hth  H which makes the designed conditions not accurately describe (conservative) the stability characteristic of the system with 0. h  In these cases, to take into account the information of the lower bound of delay, a delay-range-dependent condition makes more sense.

1 : sjj j sxxx         (A.3)

A.1.2 Convex sets

The convex set contains all points such that the segment connecting any two points of the set is stay inside the convex set. And the convex hull is the smallest outer boundary that contains of this set.  then as defined in [START_REF] Boyd | Convex Optimization[END_REF]) the convex set is the set of all convex combinations of points in :

  112233123 , 1 , ppi xxxxx     conv R (A.4)

A.1.3 Convex functions

Now, let's recall some definitions provided in the literature [START_REF] Boyd | Convex Optimization[END_REF]Mitrinović et al., 1993a) as follows:

Definition A. TTT abaRabRb   (A.9)

Further discussion with the application of this inequality will be presented in the following sections.

A.1.4 Polytope Partition

"Polytopic" extensively used in robust analysis and control strategy for the LPV system in recent decades. The application of the quadratic Lyapunov function is straightforward for stabilization conditions but results in conservatism. Since then, many methods proposed to improve the performance of LPV systems. The representation of the system as a polytopic model using all the vertices of the convex hull covering the parameter domain directly yields a multi-LTIs formulation. However, in some case, it might cause a conservativeness and numerical burden. The partitioning illustrated in The representation of the LPV system as the uncertain polytope is proposed by [START_REF] Gonçalves | New Approach to Robust$ cal D$-Stability Analysis of Linear Time-Invariant Systems With Polytope-Bounded Uncertainty[END_REF] known as Polytope-Bounded Uncertainty method. It promises to reduce computational load but increase complexity with parametric uncertainties. An application can be mentioned, see for example, a relaxing result of ℋ2/ℋ∞ performance condition presented in the articles (H. [START_REF] Zhang | Robust gain-scheduling energy-to-peak control of vehicle lateral dynamics stabilisation[END_REF]Zhang et al., , 2015)).

A.1.4.1 Switching Multiple-Affine

Now, let's discuss a generalization of converting the coordinate parameter dependence to the fundamental coordinate system of the convex domain. Given compact parameter set .

p t U There exists a linear mapping to transform the basis linear/bilinear conserva- tion from the parametric dependent function

ij   if i j  A and , 1 ij   if i j  B so that 11 , 1 1, 1,,, 2. l p ii N N iijjkljpl i kl tttttjNN       AB K (A.12)
Coordinate Eqs. (A.12) presents a generalization of the combinatorial convex from the defined parameter hyper-rectangle set. Furthermore, the combinatorial formulation introduced in Eq. (A.10) is convenient for the expansion of derivatives that makes the relaxation method [START_REF] Guerra | A way to escape from the quadratic framework[END_REF][START_REF] Sala | Polynomial fuzzy models for nonlinear control: A taylor series approach[END_REF] to deploy more efficiently. On the other hand, the piecewise affine parameter-dependent (PAPD) approaches introduced as multi-switch partitioned parameter space, see, e.g., [START_REF] Apkarian | Parametrized LMIs in control theory[END_REF][START_REF] Lim | Parameter-Varying Systems[END_REF], could provide less conservative stability conditions. The parametric switch subsystem is illustrated in the following figure. [START_REF] Lim | Parameter-Varying Systems[END_REF].

For simplicity, we assume the affine system depends on the parameter vector 12 T ttt    U Then the piecewise discretization of the parameters is given by: approach leads to less conservative stability conditions. But the number of LMI condition that must check is overwhelming. For example, given a LPV system depend on p parameters, each parameter is partitioned into i N subspace, so the number of conditions that need to be checked is about

Linear Matrix Inequality

In recent decades there has been a wide variety of problems in system control theory related to convex optimization expressed in the linear matrix inequalities LMIs. For a more detailed history of the linear matrix inequality, readers can refer to [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Since these resulting optimization problems can be solved numerically very efficiently using interior-point methods (also referred to as barrier methods), and are very convenient compared to seeking an analytic or frequency-domain solution. So, many constraints including convex quadratic inequalities, matrix norm inequalities, and Lyapunov function, can be expressed as convex optimization problems The reader intends to study in-depth about interior points, refers to the following monograph [START_REF] Nesterov | Interior-Point Polynomial Algorithms in Convex Programming[END_REF].

A.2.1 Linear Programming

The problem statement and optimality condition of the linear programming relates to minimization of linear function subject to linear constraints can represented in the inequality form where positive scalar  is called the barrier parameter. For a sequence of monotonically decreasing and sufficiently small values of , there exists an associated sequence x   called barrier trajectory (or central path) that converges to the feasible solution x  from the strict interior of the feasible region [START_REF] Wright | Interior methods for constrained optimization[END_REF]. (A.35) For specified value p y,k will find the corresponding value p x,k and p z,k .

Minimize

A.2.2 The Semidefinite Programming Problem

Semidefinite programming may be viewed as a generalization of linear programming, but they are not much harder to solve. Semidefinite programming unifies some standard problems and could be found in many applications of control system theory and combinatorial optimization, for details, see the survey [START_REF] Vandenberghe | Semidefinite Programming[END_REF] Presenting the SDP in this form is quite similar to the standard-form LP problem. In the work [START_REF] Nesterov | Interior-Point Polynomial Algorithms in Convex Programming[END_REF], the author has shown that the function logdet X  is a self-concordant barrier function for the semidefinite programming problem, which can be solved in polynomial time via a sequence of barrier parameter :

& Peaucelle, 2016), etc.). Moreover, this bounded technique is also encountered in deployments of stability of time-delay system. In this aspect, this generalization known as cross term bounding technique [START_REF] He | Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays[END_REF][START_REF] Moon | Delay-dependent robust stabilization of uncertain state-delayed systems[END_REF][START_REF] Park | A delay-dependent stability criterion for systems with uncertain time-invariant delays[END_REF]M. Wu et al., 2004) did not include the scaling scalar but rather free weighting matrices (FWM). As has been analyzed by (Briat, 2015a;[START_REF] Han | Absolute stability of time-delay systems with sector-bounded nonlinearity[END_REF]Han, , 2005a)), these methods do not seem to yield satisfactory results.

A.5 Finsler's Lemma

Yakubovich's S-lemma is a consequence quadratic result known as non-strict Finsler's lemma. In its original form it is widely used in optimization and control theory. A comprehensive state-of-the-art review of the S-Lemma and its applications was given by Polik and Terlaky in [START_REF] Pólik | A Survey of the S-Lemma[END_REF]. Following the well-known concepts in optimization, relaxation methods, and functional analysis in the work of the literature [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF][START_REF] Cimprič | Finsler's Lemma for matrix polynomials[END_REF][START_REF] Pólik | A Survey of the S-Lemma[END_REF][START_REF] Skelton | A Unified Algebraic Approach To Control Design[END_REF][START_REF] Tuy | Generalized S-Lemma and strong duality in nonconvex quadratic programming[END_REF], the lemma was briefly represented in a general form as follows.

Lemma A.5. For further discussions and demonstrations can be found in the mentioned documents.

Generalization of Finsler's Lemma

The following statement, known as Projection Lemma (or also as Elimination Lemma), is particularly related to robust control and linear matrix inequalities Lemma A.5.2. Projection Lemma (Gahinet & Apkarian, 1994)

B.1 Bounded Real Lemma -ℋ∞ performance

In the presence of external disturbance, giving a so-called energy-to-energy performance criterion  involved in the stability analysis for LPV time-delay system (B.2) which meets the following requirements: § for 0, wt  the LPV system (B.2) is asymptotically stable. § for 0, wt  guarantees L2-norm on output of (B.2) bounded with 0:

  22 22 1 ztwt    LL (B.3)
This margin robustness and performance are typically required for the design specification of the systems affected by external disturbance and parametric uncertainty. The bounded real lemma [START_REF] Scherer | The Riccati inequality and state-space H∞-optimal control[END_REF]) is a well-known criterion allowing for the computation of the ℋ∞-norm, coincides with the L2-norm, that is, the highest input-output gain for finite energy. Generally speaking, the sensitiveness of the disturbance input wt on the regulated output zt is evaluated by the performance norm Energy-to-energy index    also refers to a level of rejection disturbance. This criterion is usually encountered in the design objective, e.g., the robust stability and performance.

B.2 Block-Structured Uncertainty -Scaled Bounded Real Lemma

The representation of LPV system in a linear fractional transformation (LFT) form, the original system is separate in two interconnected subsystems Figure B-1. Where the sta-bility analysis of nominal system affected by the dynamical uncertainties, and the parametric dynamic is reformulated in the input-output framework. The characterized LMI stability conditions are derived from Scaled Small-Gain theorem [START_REF] Doyle | Structured uncertainty in control system design[END_REF][START_REF] Doyle | Review of LFTs, LMIs, and μ[END_REF][START_REF] Dullerud | A Course in Robust Control Theory[END_REF][START_REF] Feron | Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions[END_REF][START_REF] Packard | The complex structured singular value[END_REF]).

Let's introduce a LPV plant with dynamical uncertain governed by state-space equations of the generalized form: Where the block-diagonal uncertainty set i  is subset of the unit ball in L2 space. The commuting of the uncertainty is devoted to reduce the conservatism of the small gain condition [START_REF] Doyle | Structured uncertainty in control system design[END_REF][START_REF] Dullerud | A Course in Robust Control Theory[END_REF][START_REF] Zhou | Robust and Optimal Control[END_REF] Following the arguments of (Apkarian & Gahinet, 1995;[START_REF] Packard | The complex structured singular value[END_REF] This formulation offers an extra degree of freedom provides more relaxation than the small-gain theorem. The necessary and sufficient condition are delivered for LTI systems could find in [START_REF] Packard | The complex structured singular value[END_REF] and (Apkarian & Gahinet, 1995;Briat, 2015a) for LPV systems. Following the linear fractional transformation, the uncertain structures appear to detach from the plant. The stability problem of an LPV system based on this approach does not require much essential information from uncertain parameters, except for its range of variations. It should note that the LMI (B.12) is a quadratic stability, in which P is independent-parameter decision matrix. So, this condition in some circumstances is a strict condition.

B.3 Full-Block S-Procedure

Finsler's lemma has been widely recognized in control system theory by the well-known -projection lemma introduced in the early 1990s [START_REF] Gahinet | Explicit controller formulas for LMI-based H ∞ synthesis[END_REF]Gahinet & Apkarian, 1994;[START_REF] Iwasaki | All controllers for the general H∞ control problem: LMI existence conditions and state space formulas[END_REF][START_REF] Skelton | A Unified Algebraic Approach To Control Design[END_REF]. Another generalization can be mentioned as the S-procedure (or S-lemma) provides an efficient polynomial-time powerful approach for system analysis and synthesis via convex optimization problems. The analytic solutions can be losslessly reformulated as the feasibility of SDPs and deliver an alternative approach to the stability control synthesis based-LMI condition. A comprehensive state-of-the-art review is given in the monographs [START_REF] Boyd | Convex Optimization[END_REF][START_REF] Pólik | A Survey of the S-Lemma[END_REF]. Now, we would like to discuss the S-Variable LMI-based method [START_REF] Ebihara | S-Variable Approach to LMI-Based Robust Control[END_REF] using a generalization of Finsler's lemma to carry the unconventional conditions of the ℋ∞-performance. From the expression of system (B. ) reverts to ℋ∞-performance (B.5), so this condition provides the more general stability formulation. The main advantage is the decoupling between the decision variable and the dynamic system (P and ,, w AB  respectively). This transformation is the key for conservatism reduction of the new robust stability analysis. In the work of literature [START_REF] Ebihara | S-Variable Approach to LMI-Based Robust Control[END_REF] shows a relaxation of the robust stabilization condition compared with the traditional LMI based Lyapunov method.

It is worth recognizing that it is also used for the stability analysis of time-delay systems known as the free-weighting matrix. In section 5.2, this decoupling technique is applied essentially to handle with the couples between the Lyapunov-Krasovskii matrices with the dynamics system in the stability analysis of the LPV time-delay system.

The full-block S-procedure provides a general result and comprehends the (scaled) smallgain, ℋ∞-norm results of the previous sections. The term full-block comes from the fact that the scaling involved are general matrices, as opposed to the block-diagonal scaling in (B.9), (B.12). Besides, a variety of methods has been developed within the area of robust control (includes all the previous results), which can be reformulated to fall within the framework of integral quadratic constraints IQCs [START_REF] Megretski | System analysis via integral quadratic constraints[END_REF][START_REF] Pfifer | Integral quadratic constraints for delayed nonlinear and parameter-varying systems[END_REF]. This important mathematical object that can implicitly characterize the operators in an input/output framework, e.g., small-gains theorem, bounded real lemma, the full-block S-procedure, etc.

Definition B.3.1. [START_REF] Scherer | LPV control and full block multipliers[END_REF] The latter condition can be recognized as a more general form when both conditions (B.12) and (B.16) are included. Therefore, it is expected to cover a wider class of systems with a reduced conservatism.

B.4 Pole-Placement LMI regions

Following the performance constraints of ℋ∞ synthesis [START_REF] Chilali | Robust pole placement in LMI regions[END_REF]Chilali & Gahinet, 1996) has proposed an effective pole placement method based on the LMI formulation of Lyapunov stability condition. The objective is looking for the pole clustering in suitable stability sub-regions consistent to good behavior such as rational controller dynamics, well damping, fast decay, etc. This LMI-based representation of D-stability regions is characterized by relocating its poles in sub-region D of the complex plane (halfplanes, disks, sectors, vertical/horizontal stripes, and any intersection (as illustrated in (Chilali & Gahinet, 1996 Since condition (B.24) is parameter-dependent, according to the analysis in reference [START_REF] Chilali | Robust pole placement in LMI regions[END_REF] it is recommended to check overall characteristic function for each region i D in order to be more efficient and less conservative to test the robust D-stability.

B.5 ℋ2 Performance

The ℋ2-norm is more effective in dealing with stochastic characteristics such as measurement noise and random disturbance (tremors, wind loads, wind gusts, surface profiles, turbulent, etc.). Rather than bounding the output energy, it may be desirable to keep the are satisfied. These conditions restrict the limits of the impulse response of the system to be smaller than some admissible values conforming to the design specification. Indeed, that implies in a less conservative condition.

B.6 Generalized ℋ2 Performance

The multi-objective typically relates to the conflicting requirements e.g., satisfying timedomain hard constraints, capturing the peak amplitude of the output signal over all unitenergy inputs, etc. So, the energy-to-peak strategy shows a reasonable choice to relax the conservation of the stability condition involved in the robust control design. Similarly, supposing 0, D    then the so-called generalized ℋ2-norm is defined for LPV system (B.2) as L2-L∞ induced norm: This modified condition keeps the peak amplitude of the output zt bounded by an allowable value corresponding to the design specifications, e.g., to guarantee the safety constraints, to avoid the actuator saturations (saving energy), etc. The generalized ℋ2-norm uses to ensure robust stability that appears less conservative than the ℋ∞-norm (since it bounds the peak amplitude of the output over the input disturbances -white noise or impulse). However, the specification of time-domain hard constraints is sometimes in the sense of probability.

In order to better estimate the lower bound on the quadratic integral term derived from the stability conditions, the application of the following inequalities plays a core role in the developments:

0 , XY Z      f (C.2)
then the inequality hold by choosing matrix 0, M  the above condition returns to its basic form.

C.1.1 Jensen's Inequality and Extensions Approach

There are considerable famous inequalities are derived from original Jensen's inequality to some applied convex function or variations of characterize convexity. Among studies, the integral version of Jensen's inequality is frequently employed in control delay theory in the last decades.

Lemma C.1.3. (Mitrinović et al., 1993a) that entails in the original form of Jensen inequality that might be found in the time delay literature [START_REF] Briat | Stability analysis and control of a class of LPV systems with piecewise constant parameters[END_REF][START_REF] Fridman | Tutorial on Lyapunov-based methods for time-delay systems[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF] and references mentioned therein. Jensen's inequality improvement also can be found in the mathematical literature [START_REF] Fink | Jensen inequalities for functions with higher monotonicities[END_REF]Mitrinović et al., 1993c). Thereby, it provides a judicious generalization of this inequality including the finite n-segments on the specified interval ,.

ab  The convexity application is proposed to reduce the conservation of Jensen integral inequality (C.4).

In addition, the higher monotonicity derive from the use of Chebyshev's inequality. These consist essentially of functions for which several derivatives are also convex. In the literature [START_REF] Fink | Jensen inequalities for functions with higher monotonicities[END_REF], they have invested the "best possible" of Jensen's inequality for all  convex and  is end-positive. The article also covers the case of function  is nconvex, which is widely applied in two development directions. Those are the discrete interval integrals [START_REF] Gu | Stability of Time-Delay Systems[END_REF][START_REF] Han | A Delay Decomposition Approach to Stability of Linear Neutral Systems[END_REF] and high-order convex functions (Park et al., 2015;[START_REF] Tian | A new multiple integral inequality and its application to stability analysis of time-delay systems[END_REF][START_REF] Zhao | A new double integral inequality and application to stability test for time-delay systems[END_REF].

Lemma C.1.4. [START_REF] Fink | Jensen inequalities for functions with higher monotonicities[END_REF]) For ,, abI and an integrable function Proof. The demonstration is referred to [START_REF] Fink | Jensen inequalities for functions with higher monotonicities[END_REF][START_REF] Tian | A new multiple integral inequality and its application to stability analysis of time-delay systems[END_REF].

Lemma C.1.5. (Park et al., 2015) Gives an integrable function :, , Proof. The proof is given in appendix 2.2 or referred to (Park et al., 2015). Inequality (C.6) is essentially a generalization of Jensen's inequality from which the following consequences can be deduced.

Corollary C.1.1. Jensen's Inequality [START_REF] Gu | Stability of Time-Delay Systems[END_REF] For all continuous integrable function :, , Another proof of Corollary C.1.4 can be found at [START_REF] Van Hien | Refined Jensen-based inequality approach to stability analysis of time-delay systems[END_REF]. Recently, a general case of the double integral inequalities deployed for the 2 nd order formulation (similar to Corollary C.1.4) and the 3 rd order formulation provided in the literature [START_REF] Zhao | A new double integral inequality and application to stability test for time-delay systems[END_REF] are attracting works.

Lemma C.1.6. [START_REF] Zhao | A new double integral inequality and application to stability test for time-delay systems[END_REF] This method significantly enhances the stability analysis of time-delay systems with the least conservative results by minimizing gap in matrix inequalities constraint.

C.2 Model Transformation

There are considerable researches using model Transformation methods for analyzing delay-dependent stability conditions using the Lyapunov-Krasovskii function as outlined in [START_REF] Briat | Stability analysis and control of a class of LPV systems with piecewise constant parameters[END_REF][START_REF] Fridman | Tutorial on Lyapunov-based methods for time-delay systems[END_REF]Fridman & Shaked, 2003). Actually, it can be recapitulated in two approaches: explicit model transformations and implicit model transformations [START_REF] Gu | Stability of Time-Delay Systems[END_REF]. The model transformation methods and the improvement inequalities were proposed to reduce the conservatism of delay-dependent conditions of the LTI delay system. According to the analysis in [START_REF] Briat | Stability analysis and control of a class of LPV systems with piecewise constant parameters[END_REF][START_REF] Fridman | Tutorial on Lyapunov-based methods for time-delay systems[END_REF]Fridman & Shaked, 2003;[START_REF] Gu | Stability of Time-Delay Systems[END_REF], nonetheless these methods have yet to be thoroughly improved. Let's consider the features of model transformations.

C.2.1 Model transformation I -Newton-Leibniz Model Transformation

The delayed terms xth  in the LTI delay systems are substituted to yield the fixed model transformations:

. The following Lyapunov-Krasovskii functional is used to determine a delay-dependent stability condition for system (C.12) and (C. ity problem analysis is replaced by original system, but that entails in the persisted delay terms . xth  The inconsistent elimination of the integral terms and delay terms in this explicit model transformation leads to conservativeness. On the other hand, the use of parametrized model transformation method produces a new matrix parameter in crossterm, where the compensation of the additional dynamics is reformatted into variable matrix optimization problem. However, the basic bounding inequality applied for crossterms resulting in conservatism, whereas the application of Park's inequality to PMT stability condition as analysis (Fridman & Shaked, 2003) shows no more improvement in conditional relaxation.

C.2.2 Model transformation II -Neutral type Transformation

As an alternative introduction to system (C.12), we have the following LTI delay system: . The aim of above model transformations is to convert the integral term into the functional differential equation so as to produce both cross terms and quadratic integral terms in the derivative of a Lyapunov-Krasovskii functional along the trajectories of the systems. But because of the following disadvantages that leads to their supersession by other methods.

1 -The original system is recovered in (C.17) then the effect of the transformed model would be lost.

2 -Both transformation methods introduce additional dynamics into the system, then the transformed system is not equivalent to the original one.

3 -By applied the bounding inequality for cross-term (C.17) entails in conservatism since the right-hand side of (C.18), (C.19) are always positive.

4 -Eliminating the integral expression in the derivative of LKF has also lost important information in negative definite stability conditions.

As point out by [START_REF] Briat | Stability analysis and control of a class of LPV systems with piecewise constant parameters[END_REF]Fridman & Shaked, 2003;[START_REF] Gu | Further remarks on additional dynamics in various model transformations of linear delay systems[END_REF] the use of the Newton-Leibniz model transformation might lead to more conservative stability conditions and lose its generality for the reasons outlined above. In order to improve these results certainly count on either by using less restrictive model-transformations (or even no model-transformation at all) or by employing more accurate bounding techniques (as given in Lemma C. 1.1 and Lemma C.1.2). The application these inequalities reduces the conservative LMI stability conditions, and can be incorporated with others model transformations. However, when the computational complexity of the conditions increase (parameter dependent Lyapunov function PDLF, optimization of the attraction domain -saturation control, robust stability...) then the generality of the inequality lost.

C.2.3 Model transformation III -Descriptor (Fridman) Transformation

The following descriptor model transformation is proposed by [START_REF] Fridman | New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems[END_REF] ). Even though the descriptor method relies on a non-conservative model transformation and make an interesting result, however, it still has some limitations.

1 -It is still based on cross-terms bounding inequality ensuing to conservatism.

2 -Inconsistent substitution of delay term in system and in stability condition and the adding zero-equivalent term with weighting matrices lead to the requirement of determining optimization of the fixed weighting matrices.

The use of Moon's inequality (Lemma C.1.2) in combination with this transformation method yield to less conservativeness delay-dependent conditions that seems useful in stability, stabilization analysis and control synthesis. Besides, the computational complexity of the above result can be improved by relaxation method [START_REF] Briat | Stability analysis and control of a class of LPV systems with piecewise constant parameters[END_REF]) that provide a better construction of the variable matrices.

C.2.4 Model transformation IV -Free Weighting Matrix Approach

In the above sections, we have analyzed the transformation methods and the inequalities that enhance the LK dependent stability conditions. As has been mentioned, the distribution of heterogeneous matrix weight functions corresponding to variables , xt  , xth  and t th xd    & (there are a relationship between them), that can interfere with solution of stability conditions. Consequently, the Free Weighting Matrix (FWM) method is introduced in (He et al., 2004a[START_REF] He | Delay-range-dependent stability for systems with time-varying delay[END_REF] generally called by lifting-variables that gives more degree of freedom in design condition stability. In which the fixed weighting matrices are replaced by appropriate dimensional slack matrices by the following null algebraic matrix equations: A methodological improvement is found in the works [START_REF] Briat | Stability analysis and control of a class of LPV systems with piecewise constant parameters[END_REF][START_REF] Wu | Stability analysis and robust control of time-delay systems[END_REF], however there are the matrix variables to be determined along with decision matrices of C.3. Input-Output Approach 201 stability conditions including P-system dynamic, Q and R-delay dependent. And as pointed out [START_REF] Briat | Stability analysis and control of a class of LPV systems with piecewise constant parameters[END_REF], after using the reduction in computational complexity method for FWM inequality that leads to in similar results to the inequality introduced hereafter.

C.3 Input-Output Approach

As discussed, the delay decomposition approach is less conservative results for stability analysis and controller design. But the method is effective for systems that access the exact knowledge of the delay, which is ideal for numerical computation in practical design. The identifications or estimations of the continuous-time delay phenomenon in practice are tough challenges, see, e.g., [START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF][START_REF] Belkoura | Parameters estimation of systems with delayed and structured entries[END_REF]Chen et al., 2015;[START_REF] Ren | Online identification of continuous-time systems with unknown time delay[END_REF][START_REF] Zheng | Delay identification for nonlinear timedelay systems with unknown inputs[END_REF]. In this case, the uncertainty (approximation) delay method discussed in Section 8.6 [START_REF] Gu | Stability of Time-Delay Systems[END_REF] shows to be more suitable for implementing the strategy of control system design. Specifically, the timevarying delay that is not accurately known at the time of analysis and design is considered as the dynamical uncertainties of a nominal system. Based on this approach, the stability is formulated in the input-output framework where the characterized LMI conditions obtain by the Scaled Small-Gain theorem [START_REF] Briat | Delay-scheduled state-feedback design for time-delay systems with time-varying delays-A LPV approach[END_REF][START_REF] Hmamed | Stability analysis of linear systems with time varying delay: An input output approach[END_REF] or supply function [START_REF] Briat | Memory-resilient gain-scheduled statefeedback control of uncertain LTI/LPV systems with time-varying delays[END_REF]. The equivalent between the Scaled Small-Gain and Lyapunov-based technique is discussed in [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF][START_REF] Boyd | Structured and simultaneous lyapunov functions for system stability problems[END_REF][START_REF] Doyle | Review of LFTs, LMIs, and μ[END_REF][START_REF] Zhang | Stability of time-delay systems: Equivalence between Lyapunov and scaled small-gain conditions[END_REF][START_REF] Zhou | [END_REF] for LTI/LPV systems.

C.3.1 Approximate Delay-Range Approach

The input-output approach is very convenient in analyzing the stability based on the representation of the original system to feedback interconnection with additional inputs and outputs of auxiliary systems. The analysis of uncertain LPV systems deployed by the small-gain theorem has attracted much interest in the literature in the last century. Besides, the robust stability and performance problems could reformulate in LMIs formulation can efficiently solve by convex programming.

The approximate delay value around the nominal values derived from the uncertain delayrange-dependent approach [START_REF] Gu | Stability of Time-Delay Systems[END_REF], or the time-varying approximation delay [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF]. Temporarily ignore the effect of disturbance, the TDS system is transformed to the following differential equation: The system (C.25) is represented by a generalization [START_REF] Gu | Stability of Time-Delay Systems[END_REF], where the operators are defined as From condition (C.36) an alternative proof may be assigned.

C.3.2 Uncertain Delay-Dependent Approach

The previous stability of the comparison system is analyzed at a constant approximated delay in the varying dependency range, that purposely approached by the small gain theorem. Followed this approach, a time-varying approximation delaydt was discussed in [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF][START_REF] Briat | Memory-resilient gain-scheduled statefeedback control of uncertain LTI/LPV systems with time-varying delays[END_REF] tegration limits which are generalized in the approximate delay-dependent stability and stabilization analysis. Note that xt has derivatives across the defined domain, the condition  needs to be validated at the critical extreme time , dtht that is explained detail in Appendix D.1. Besides, the delay approximation analyzed in [START_REF] Briat | Delay-scheduled state-feedback design for time-delay systems with time-varying delays-A LPV approach[END_REF][START_REF] Briat | Memory-resilient gain-scheduled statefeedback control of uncertain LTI/LPV systems with time-varying delays[END_REF] is more general than that introduced of [START_REF] Gu | Stability of Time-Delay Systems[END_REF] and enhances stability condition. But there is a restriction when dt is involved in the system with two dependent delays, that need more design memory for the observer and controller structure.

In the framework of non-small delay, the upper bound on gains wtzt  given by [START_REF] Shustin | On delay-derivative-dependent stability of systems with fast-varying delays[END_REF] with assumption of zero-initial condition as follows

C.3.3 Delay-Scheduled LFT Approach

In the framework of input-output stabilizing, the delay operators could let the time-varying delay play a role as the parametric uncertain, where the characterized stability synthesis of the approximate dependent delay can be addressed by the Lyapunov technique or by the small-gain theorem. In addition, similar to the uncertain structure framework, time-varying delay values can be cast as a gain-scheduled parameter. The gain scheduling in the analytical robust stability control framework for LTI/LPV systems are well-developed. Unlikely, the delay scheduling analysis for delay-dependent LPV systems has not been adequately studied and analyzed, see, for example [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF][START_REF] Briat | Stability analysis and control of a class of LPV systems with piecewise constant parameters[END_REF][START_REF] Briat | Delay-Scheduled State-Feedback Design for Time-Delay Systems with Time-Varying Delays[END_REF][START_REF] Briat | A LFT/ℋ<inf>∞</inf> state feedback design for linear parameter varying time delay systems[END_REF]). An LPV plant with linear fractional depends on parameter and delay can represent as an upper LFT interconnection structure with delay operators  and parameter sets  (illustrated in Figure B-1). Let's consider the following operators. , with 1,2. i ztzti  LL .

The operator mentioned in Equation (C.47) has a singularity at zero, so the delay dependent is limited in the interval   minmax , hh does not contain zero. Besides, operator (C.48) is quite similar to operator (C.42) with a defined interval conforming to the delay space 0 H and transformed L2-norm encouraging for tightening conditions. By using structural LFT, the stabilizing condition is resolved via small gain theorem in [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF]. Henceforward, for the sake of simplicity, the time-dependent expressions " t ", and scheduling parameter "   " are omitted in equations, denote :,   By using Schur complement [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]

D.2 The proof of Lemma 5.2.4.

The stability delay-dependent of LPV system (5.1) is ensured if the condition holds along the trajectories of system. The stability LMIs are derived from the derivative expansion of Lyapunov-Krasovskii functional (5.51) along trajectories of LPV time-delay system (5.50). Firstly, by expanding the 0

Vt  & and combining with L2-norm performance on the controlled output (5.50), that entails a similar condition with Lemma 5.2.1 as follows: Derivative the simple Lyapunov-Krasovskii functional along trajectories of LPV timedelay system (5.50) and integrating with L2-norm performance on the controlled output (5.50), that implies accustomed stability delay-dependent condition: where the membership functions are given by the expression: The generalized method is based on the Levenberg-Marquardt algorithm combined with the least square method. The details of the fuzzy parameters are given Table E.1. This assumption reduced the number of the membership functions from 8 to 2. Areas of outlet in tanks 1,3 (m 2 ) 7.1 × 10 -6 a s2 , a s4

Areas of the outlet in tanks 2,4 (m 2 ) 5.7 × 10 -6 k vi

Coefficient of pump-i, i =1,2 (ml V -1 s -1 )

3.33, 3.35 h i

The liquid level in tank-i (m) γ i

The value scaling of flow at valve-i v i

The voltage control signal at pump-i (V)

Remark 1. Time-varying parameters , i t  i = 1, 2 is the flow rate value of valve-i correspond tanks, by assumed to be measurable and transformed to the convex combination parameters as follows:

k k t    
The gains of observer-based controller derived from the design conditions in Theorem 3.2.1 solving by toolbox Matlab Yalmip [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF] with solver Mosek [START_REF] Andersen | On implementing a primal-dual interiorpoint method for conic quadratic optimization[END_REF] are given by:

The fuzzy controller gains:

E.2 Chapter 3 -Lateral Vehicle Dynamic

The accurate estimation of the tire friction's side force plays an important role in the design vehicle stability system. Among the static tire models, it can be mentioned the HSRI model [START_REF] Dugoff | An analysis of tire traction properties and their influence on vehicle dynamic performance[END_REF], the semi-empirical Pacejka model [START_REF] Pacejka | Tire Characteristics and Vehicle Handling and Stability[END_REF], and the Kiencke model [START_REF] Kiencke | Automotive Control Systems[END_REF], etc. Since, the Pacejka model describes the lateral forces that could have an effective/linear cornering stiffness and nonlinear characteristics. In this literature, the nonlinear forces are modeled by TS-fuzzy method as in (Bui [START_REF] Bui Tuan | Robust Observer-Based Control for TS Fuzzy Models Application to Vehicle Lateral Dynamics[END_REF][START_REF] Dahmani | Observer-Based Robust Control of Vehicle Dynamics for Rollover Mitigation in Critical Situations[END_REF][START_REF] Dahmani | Detection of impending vehicle rollover with road bank angle consideration using a robust fuzzy observer[END_REF]Dahmani, Pages, et al., 2015;[START_REF] El Hajjaji | Observer-based robust fuzzy control for vehicle lateral dynamics[END_REF] The membership functions satisfy the following properties The lower bound of the longitudinal velocity (m s -1 ) 10

x v
The upper bound of the longitudinal velocity (m s -

Figure 3 - 2 .Figure 3 - 3 .

 3233 Figure 3-2. The diagram of quadruple-tank process model. ..........................................Figure 3-3. The evolution of dynamic states and estimations under disturbance and uncertain dynamics. ..................................................................................... Figure 3-4. The observer performance -error estimations (left); and the DC-Motor pumps signal control (right). ....................................................................... Figure 4-1. The estimate of feasibility regions with 25, 20,0, 25. ab  ..........

Figure 4 - 2 .Figure 5 - 2 .

 4252 Figure 4-2. The improvement of the controller performance with LMI D-Region, via Polytopic quadratic stability condition. ...................................................... Figure 4-3. a. The simulated constrained responses of the state-feedback SF controller (colored solid line), the static-output feedback SOF controller (colored dash-dotted line), the observer-based feedback OBF controller (colored dashed line), and the dynamic output feedback DOF controller (colored dotted line). b. The time-evolution of external disturbance. ...................... Figure 4-4. The stabilized states of the closed-loop systems are regulated, respectively, by the gain-scheduled SF controller (colored solid line), SOF controller (colored dash-dotted line), OBF controller (colored dashed line), DOF controller (colored dotted line) conforming to the simulated parameter (green solid line in the last frame). ............................................................ Figure 4-5. The comparison of the closed-loop systems regulated, respectively, by the saturated gain-scheduled SF controller (colored solid line), and nominal state feedback controller (colored dotted line). ......................................... Figure 4-6. The comparison of the closed-loop systems regulated, respectively, by the saturated gain-scheduled SOF controller (colored solid line), and nominal static output feedback controller (colored dash-dotted line). ................... Figure 4-7. Region of stability. Trajectories of the closed-loop systems respond from the same initial conditions during 10 seconds of simulation........................... Figure 5-1. Graph of a convex function. .......................................................................

Figure 2 - 1 .

 21 Figure 2-1. Polytopic parametrization.

Figure 2 - 2 .

 22 Figure 2-2. Illustrates fuzzy model construction by sector nonlinearity.

Figure 2 - 3 .v

 23 Figure 2-3. Vehicle lateral dynamics.

Figure 2 - 4 .

 24 Figure 2-4. LPV modelling and control of robotic systems.

  in accordance with the relationships (1)  (2)  (3).

Figure 2 - 5 .

 25 Figure 2-5. Time transient of Lyapunov function.

Figure 2 - 7 .

 27 Figure 2-7. The level set of ellipsoidal domains. Quadratic Lyapunov function (QLF) candidate:   1 , .

Figure 2 - 8 .

 28 Figure 2-8. The closed-loop diagram of saturated feedback controllers.

Theorem 3 . 1 . 1 :

 311 In presence of parametric uncertainties and disturbances, for the positive scalars 3 ,,  then closed-loop system (3.5) is robustly asymptotical stable corre- sponding to energy-to-energy index(3.14), if there exist matrices

Figure 3 - 2 .

 32 Figure 3-2. The diagram of quadruple-tank process model.

Figure 3 - 3 .

 33 Figure 3-3. The evolution of dynamic states and estimations under disturbance and uncertain dynamics. The time-evolutions of the liquid level in tanks including the measurable states 12 ,, hh  the unmeasurable states 34 ,, hh  respectively, and its estimations shown in Figure 3-3 that have demonstrated the good performance of the designed observer-based control strategy.The minimization of disturbance effects on the system is enhanced by using the CCL algorithm. As we can see in Figure3-3 and Figure3-4 (left), the error estimate of liquid level in tanks 1 st , 2 nd is smaller than 0.6 mm, corresponding to level in tank 3 rd being less than 3 mm and level tank 4 is less than 5 mm. And the stabilized control signal of the observer-based controller design for the quadruple-tank process system gives in. With the limited voltage of each pump is 12 (V), the maximum flow of each pump is 40 (ml. s -1 ) corresponding to 2.4 (l. mn -1 ). The illustrative simulation results show the high performances of the proposed design technique.

Figure 3 - 4 .

 34 Figure 3-4. The observer performance -error estimations (left); and the DC-Motor pumps signal control (right).

  Figure A-5). The feasible solutions are obtained at the permissible errors of iterative Algorithm 3.1 CCL1. Then, taking advantage of the new sought scalars i  to locally optimize (using iterative Algorithm 3.2 CCL2) that affords a better minimum value. For found values , i   the best relaxations are attained for stabili- zation conditions corresponding to each design case. Besides, the conservation drops significantly by considering the slack variables corresponding from the 2 nd catalog to the 4 th catalog. It can realize that the stabilization conditions are infeasible with fixing constant values , i   however the CCL algorithm provides the feasible solutions. Moreover, CCL1 (Algorithm 3.1 -global optimization) always returns to a higher rejection level than CCL2 (Algorithm 3.2 -local optimization). But algorithm CCL2 achieves good results inheriting from the set of optimal values , i   de- rived from algorithm CCL1. Without these data, algorithm CCL2 cannot converge to the local optimal region.

  im  K and using a PDLF associated with ellipsoidal domains(2.102) to analyze the stability of closed-loop system(2.111). Then, the stabilization condition combined with the ℋ∞-norm condition and the GSC condition (Corollary 2.3.1) are expressed as follows. Theorem 4.1.1: In the presence of disturbance, for the positive scalars 0 ,,,, s u  if ex- ists continuously differentiable matrices function :,

,

  From above analysis, the trade-offs between the performance requirement ,   the esti- mation of RAS (ellipsoidal domain) -,  and the set of the admissible initial conditions - 0 is a mandatory condition. The selection of appropriate constants combined with opti- mization of the remaining variables depends on the design purposes. For instance, preselecting 0 ,  then minimizing  such that conditions (4.1), (4

Example 4. 1 . 1 :

 11 Consider a nonlinear open-loop unstable system introduced by (A. T.[START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF] is presented by the following equations:

Figure 4 - 1 .

 41 Figure 4-1. The estimate of feasibility regions with 25, 20,0, 25. ab  By setting 1,   the feasibility of T-S fuzzy system (4.17) is checked using the proposed relaxation (polytope) Corollary 4.1.1 for all 1,196 points of a 4626 rectangle uniformly grid over the space 25, 20,0, 25 ab  with assumption 12 0. XX  ± The feasi-

Figure 4 - 2 .

 42 Figure 4-2. The improvement of the controller performance with LMI D-Region, via Polytopic quadratic stability condition. It could realize that the magnitude of the first and second row of the gains solved by the

Theorem 4. 2 . 1 :

 21 For the positive scalars 0 ,,,,, s u  if there exist continuously matrices function :,:,:,:, nmpmnpp pppp XYZW    UUUU SRRR and a diagonal matrix function :, m p T   U

Figure 4 - 3 .

 43 Figure 4-3. a. The simulated constrained responses of the state-feedback SF controller (colored solid line), the static-output feedback SOF controller (colored dash-dotted line), the observer-based feedback OBF controller (colored dashed line), and the dynamic output feedback DOF controller (colored dotted line). b. The time-evolution of external disturbance.

Figure 4 - 4 .

 44 Figure 4-4. The stabilized states of the closed-loop systems are regulated, respectively, by the gain-scheduled SF controller (colored solid line), SOF controller (colored dash-dotted line), OBF controller (colored dashed line), DOF controller (colored dotted line) conforming to the simulated parameter (green solid line in the last frame).

  STABILIZATION SYNTHESIS FOR LPV/QUASI-LPV SYSTEMS WITH ACTUATORS SATURATION

  a-b. The time-evolution of state dynamics. c. The simulated constrained responses of the SF controller Theorem 4.1.1, and the nominal controller.

Figure 4 - 5 .

 45 Figure 4-5. The comparison of the closed-loop systems regulated, respectively, by the saturated gain-scheduled SF controller (colored solid line), and nominal state feedback controller (colored dotted line). The time-varying parameters 1,1, 1, tt   & and the saturation limit sets on actuator 5. utu  Similar to the discussed optimization methods, by solving Theorem 4.1.1, Theorem 4.2.1, and Theorem 4.4.1 gridded over 121

  compare the design controller with GSC condition with the corresponding nominal controller. The control systems are set up for simulation of LPV system (4.82) with bounded actuators. Given in Figure4-5 and Figure4-6 are the comparisons of state-feedback and static output feedback controllers. The simulations start at the same initial condition 0100 0 T x regardless of the effect of disturbances.

  a-b. The time-evolution of state dynamics. c. The simulated constrained responses of the SOF controller Theorem 4.2.1, and the nominal controller.

Figure 4 - 6 .

 46 Figure 4-6. The comparison of the closed-loop systems regulated, respectively, by the saturated gain-scheduled SOF controller (colored solid line), and nominal static output feedback controller (colored dash-dotted line).

  STABILIZATION SYNTHESIS FOR LPV/QUASI-LPV SYSTEMS WITH ACTUATORS SATURATION

  3. a. Destabilize trajectories governed by a nominal SOF controller. b. Stabilize trajectories regulated by design saturated SOF controller.

Figure 4 - 7 .

 47 Figure 4-7. Region of stability. Trajectories of the closed-loop systems respond from the same initial conditions during 10 seconds of simulation.

  conditions for LPV saturation systems analyzed with a quadratic Lyapunov function compared to a parameter-dependent Lyapunov functional.

  Jensen-based inequality such as Wirtinger-based inequality (Appendix C.1.1 Corollary C1.2) and auxiliary functions show significantly the relaxation of the stability conditions. By substituting the function xs  & forws in these corollaries, that yields the following results. conditions (5.6), (5.7), and (5.9) is inferred directly from Appendix C.1.1 Corollary C1.1-4, and extended conditions (5.8), (5.10) can consult at (Y.[START_REF] Tian | A new multiple integral inequality and its application to stability analysis of time-delay systems[END_REF][START_REF] Zhao | A new double integral inequality and application to stability test for time-delay systems[END_REF]. It should note that these methods provide better estimates of the lower bound of the expression

  is applied to derive a better estimate of the lower bound of expression derivative of the function 3 .

  a. The chord of the parabola expresses convex inequality. b. The integral region of the parabola is in the interval. c. The integral domain fragmentation.

Figure 5 - 1 .

 51 Figure 5-1. Graph of a convex function.

     and N is the number of divisions in the interval, i.e.,   ,0. h  This method significantly enhances the stability analysis of time-delay systems with the continuous and piecewise Lyapunov matrices. Then, the integral term t T th xsRxsds   && involving in the derivative of function 3 Vt bounded by the Jensen-based inequalities 5.1.3. Delay-Dependent Stability -Input-Output Approach 117

  complement yields to (5.43).

W

  the scaling sets and uncertain norm-bounded operator 0 .

Figure 5 - 2 .

 52 Figure 5-2. The evolutions of quasi-LPV time-delay system (5.68) with the slowvarying delay 20.4917, 0.1. htht  &

  understanding of dt versus nominal delay value , ht we have the follow- ing controller design strategies: as a memoryless controller. § If , dtht  the controller refers to as an exact-memory controller. § If ||, m dtht    the controller labels to as a -memory-resilient controller.

1 :

 1 -loop system into the relaxed LMI condition (5.25) in section 5.2.1.2, with the dead-zone nonlinearity related in the GSC condition treated as in Chapter 4. Then, the memoryless state feedback controller attains by solving the following delaydependent stability conditions. Theorem 6.2.For time-varying delay 0 , ht H parameter ,, p tt    & UU posi- tive scalars ,,, ij u and presence of the L2-bound disturbance, if there exist continu- ously differentiable matrices function 1 :,,,:,

W

  If the disturbance is not taken into account, we can choose 1.

  the certain allowable level of the initial conditions (it should note that in the cases 11 , opt    then we adjust the bounds on the external disturbances). § Preselect , then minimizing 

6. 2

 2 .4.1. Memory-resilient saturated controller synthesis 6.2.4. Example 153

Figure

  Figure 6-1. The responses of the bounded controllers correspond to saturation limit 5, u  with 1,0.9. h  

hA

  so the variable substitutions are not much of a problem. However, the uncertain delay value typically leads to incompatibility between the

3 . 1 .

 31 What lies in the feedback signal of the deadzone function u  considered the input of the controller system (6.15). And dynamic gain c E   plays the role of the saturated compensation and enhances the performance of 6

  perform in Figure 6-2.

Figure 6 - 3 .

 63 Figure 6-3. The responses of the stabilizing gain-scheduling controllers solved by Theorem 6.2.2, and Theorem 6.3.1 corresponding to saturation limit 5, u  with 1,0.5, h   and 2.5.  

  a. DoA -Theorem 6.2.2. b. Ellipsoidal sets -Theorem 6.2.2. c. DoA -Theorem 6.3.1. d. Ellipsoidal sets -Theorem 6.3.1.

Figure 6 - 4 .

 64 Figure 6-4. Example 6.3.1 -designed controllers. The estimates of region of stability using sector nonlinearities approaches with different criteria. Asymptotically stabilized (green solid-lines) and destabilized trajectories (brown dotted-lines) of the closed-loop systems from the initial conditions (o).

  As a result, minimizing the estimation of the set of the admissible initial conditions leads N to be a singular matrix. To avoid this, based on the relationship YX  ,T n NMI  we propose two solutions: § Choose a small value of  such that the stability conditions of Theorem 6.3.1 to have a solution, then use this matrix N to solve the optimal Problem 6.3.1. § Use the optimal value of opt  to calculate the upper bound of 1 0 ,   then indirectly ob- tains the boundary of the DoA estimation domain

  4

Figure A- 2 .Example A. 1 . 1 .

 211 Figure A-2. Illustrates some simple convex and nonconvex sets 2 . R Example A.1.1. In the plane px  (Figure A-2.a) the smallest convex hull containing three points 123 , ,, xxxpx   is the triangle domain 123 , xxxpx  then as defined in[START_REF] Boyd | Convex Optimization[END_REF]) the convex set is the set of all convex combinations of points in :

  Figure A-3 significantly decreases the vertices of the polytope and results in the less conservative condition.

Figure A- 3 .

 3 Figure A-3. Two steps of three possible subdivisions of a triangle (Gonçalves et al., 2006). (a) Division by two, or bisection. (b) Division by three. (c) Division by four, or edgewise subdivision.

  the transient binary switching (A.12) to converts the LPV system to the following switched LTI systems:

  Figure A-4. Visualization PALPV model discretization[START_REF] Lim | Parameter-Varying Systems[END_REF].

  and real symmetric matrices .

  system model. b. Block-Structured LPV.

Figure B- 1 .

 1 Figure B-1. The interconnection structure.

Figure B- 2 .

 2 Figure B-2.Pole-placement of LMI regions(Chilali & Gahinet, 1996).

  and only if there exists an energy-to-peak performance index 0

Lemma C. 1 . 1 .

 11 [START_REF] Park | A delay-dependent stability criterion for systems with uncertain time-invariant delays[END_REF] Given vector functions ,:. n ab  a¡ Then above condition returns to its basic form.Lemma C.1.2.[START_REF] Moon | Delay-dependent robust stabilization of uncertain state-delayed systems[END_REF] Assume that : if the following condition is satisfied:

  1.3. Auxiliary-Function-Based Inequality(Park et al., 2015) For a positive matrix n R   R and any continuous integrable function :

Figure C- 1 .

 1 Figure C-1. Graph of a convex function.

1 .

 1 to be directly suppressed by model transformation. By imposed the bounding cross-term inequality we have: Model transformation given in (C.13) has the same purpose as the fixed model transformation (C.12). Where the term T hxRx   && produced in derivative stabil-

  constraint of Newton-Leibniz formula and associate with information on maximizing the upper bound on delay value.

  auaa thththhhhh  The distributed delay is now considered as an input disturbance. The stability analysis for system (C.25) would infer to system , h xtAxtAxth  & (C.26)but without the initial condition there does not guarantee an equivalence.By using the internal topology with input-output structure:

1 .

 1 These operators enjoy the following immediate property §  is L2 input-output stable and satisfies the scaled small gain condition.Proof.Let's consider a set of block-diagonal appropriate dimension matrices (C.29) to the left side of (C.32) that results in:   & and assuming the zero-initial condition. It should be reminded the derivate uncertain delay . max,min0,, aaa hhh   then by changing order of integration in double integrals as illustrated in Figure C-2.a, with assumption of the zero initial condition we obtain:

Figure C- 2 .

 2 Figure C-2. Changing order of integration of double integrals.

  an asymmetric bound of in-

  ,[START_REF] Briat | Memory-resilient gain-scheduled statefeedback control of uncertain LTI/LPV systems with time-varying delays[END_REF] deliverers an operator to represent the interconnection input-output and have implemented both on LK and LFT stabilization approaches for delay-dependent LPV systems. operator 0  satisfies SSG condition is addressed the same as proposition. The interconnected system is well-defined and input-output mapping stable if the following Integral Quadratic Constraints (IQC) satisfy: limits of integration into the domain of double integrals are described in Figure C-2.b. With assumption of the zero initial condition, we get:

  These operators enjoy the property: i  are L2 input-output stable and satisfies the bounded small gain constraint 22 00

  trajectories of system. From the above condition, xt  &are substituted by the original dynamic system, the derivative of the time-varying delay is bounded, the derivative of the matrix dependent Pt   analyzed as in Chapter 2, while the integral of the positive real function is organized by Jensen's inequality as follows. verify the well-posed of the inequality (D.2) at extreme point 0, ht  it is necessary to prove the defined domain included point 0 (D.2) is defined for all xt and ht satisfying the initial assumption.

  external perturbations, combined the performance constraint (B.3) with the PDLKF stability condition (D.1) that results in the following stability condition:

  using Schur complement and rearrange rows and columns that results (5.41).Q.E.D.

Case 2 :

 2 (Uncertain delay) the study involved when the two delays were different j dt  . j ht  Consequently, by expanded of conditions (D.10) and (D.11), we obtain:



  use operator (C.42) and interconnection (C.43) to express the relation between h x and d x as input perturbed delay 0 w is bounded by the IQC (C.44). Accordingly, we have a transformation of coordinate: the scaling commuting sets  D and uncertain bounded operator 0 .  Then applied the scaled bounded real lemma for uncertainty, with substituting (D.15) into condition (D.14) that lead to the following parameter dependent LMI condition: Schur-complement that results in the PDLMI condition: ftht of the liquid level in tank-k, k = 1...4 is described in two regions as following rule-based system:

  generalization of the Cauchy distribution (also known as the Bell MF), which is specified by three parameters {a j , b j , c j }:

  , with the front and rear lateral forces are represented by stiffness in front and rear of SUV E-Class car model are fuzzificated as the local linear gain between tire sideslip angle and tire force. For the simplicity, expression ˆ, f r   may be shorten to only .  

Figure E- 1 .

 1 Figure E-1. The forces acting on two-tracks 3D vehicle model.

  Figure 2-1. Polytopic parametrization. ..........................................................................Figure 2-2. Illustrates fuzzy model construction by sector nonlinearity. ....................... Figure 2-3. Vehicle lateral dynamics. ............................................................................. Figure 2-4. LPV modelling and control of robotic systems. ........................................... Figure 2-5. Time transient of Lyapunov function. ..........................................................

. Figure 2-7. The level set of ellipsoidal domains. ............................................................

Figure 2-8. The closed-loop diagram of saturated feedback controllers. ......................

Figure 3-1. The extreme points of the polynomial in the interval 6,12.  ...................

  -LPV time-delay system (5.68) with the slow-varying delay 20.4917, 0.1. htht  & .............................................................. 137 Figure 6-1. The responses of the bounded controllers correspond to saturation limit 5, u  with 1,0.9. h   ......................................................................... 154 Figure 6-2. The evolutions of the LPV time-delay system regulate by saturated state feedback and dynamic output feedback controllers. ................................. 167 Figure 6-3. The responses of the stabilizing gain-scheduling controllers solved by ................................................................ 168 Figure 6-4. Example 6.3.1 -designed controllers. The estimates of region of stability using sector nonlinearities approaches with different criteria. Asymptotically stabilized (green solid-lines) and destabilized trajectories (brown dotted-lines) of the closed-loop systems from the initial conditions (o). .............................................................................................................. 170

	Theorem 6.2.2, and Theorem 6.3.1 corresponding to saturation limit with 1,0.5, h   and 2.5.  	u 	5,

xiv

List of Tables

Table 2 .

 2 1. The Conservativeness of Parametrized LMI Conditions. ..............................Table 3.1. Observer-Based Feedback Robust Stabilization and Performance with Saturation actuators. ................................................................................... Table 4.1. Multi-objective optimization -Problem 4.1.1 ............................................... Table 4.2. Robust Pole Placement in LMI Regions. ....................................................... Table 4.3. Multi-objective optimization and . ................................................. Table 4.4. Minor Axis Maximization 10.    ........................................................... Table 4.5. Actuator Saturation -Optimization Problems. ........................................... Table 4.6. Multi-objective optimization -Example 4.5.1. ............................................ Table 5.1. The maximum admissible upper bound MAUB for delay 0 . ht H ..........

optopt  

Table 5 . 2 .

 52 The maximum admissible upper bound for delay 0 . ht H ......................

Table 6 . 1 .

 61 The optimization of ℋ∞ performance criterion . ................................. Table 6.2. Domain of attraction, with 1. h  ...............................................................

opt  

Table 6 . 3 .

 63 Multi-criteria optimization. .........................................................................

	xv

1 A  Inverse of matrix A T A  Transpose of the inverse of A A  Moore-Penrose pseudoinverse of A q zt  L Lq-norm of the function : n zt   RR defined as

General Introduction and Summary

  ).

	Let's consider a continuous parameter with discontinuous trajectory:
	sin2, 1cos2,21 k 2 ttkk tk ttkk      	. N	(2.7)
	There is no existence of the continuous derivative function discontinuous at times . tk 	.  & This function is ftt 

 Instead, it takes left and right values by -1 and 1, respectively.

  Nonetheless, to unify the symbolizations on the control system in this dissertation,ht is signified for the time-varying delay, and zt  is used for the regulated output. And parameters i t   is a monochromatic transfor-

	14	CHAPTER 2. OVERVIEW LINEAR PARAMETER-VARYING SYSTEMS
	denoted by etc. mation from ,, iil hzttrN    , jij tM  

It should note that the commonly used notations in the fuzzy logic control community are

  In this 2.1.5. Example case, nonlinear systems are approximated around a local bound on each state. However, there are no prerequisites requirements for the dynamics of system states, e.g., the upper bound of the derivative ?. xt  &

					xt 
	As shown in Figure 2-2 is the Cartesian coordinate system Oxy  representing nonlinear function fxt  is locally or globally bounded by two linear function 22 yxtaxt   and 11 . yxtaxt   Where Figure 2-2.a, the construction of T-S fuzzy model for the nonlin-ear system fxt  is illustrated at point 11 , xfx  as follows:
	111211121 11, fxyxyxaxax   	with			1 211 1 211 afxfx aafx      	.	(2.22)

can interpret as follows: in the specified time domain of state , x xt  D R give a finite nonlinear function , fxt   R such that the upper and lower bound functions of fxt  are finite over certain time-domain of . Similarly, Figure 2-2.b shows a fuzzyfication of the system in local region ,, xtdd  where the fuzzification of the function fxt  is so-called local sector linearization.

  and a polynomial fuzzy model is indicated in the following formulation of a local linear model.

	if	11 i tM  	and … and pip tM  				(2.23)
	then	, , iiwi iwi xtAxtZxtBxtwt ztHxtZxtJxtwt     &	,	1,2,, iN  K	l	(2.24)
	Where trices in AxtCxtBxt  , ,,, nNpN iiwi   RRand . xt  The term 121212 , wi Jxt   are the polynomial ma-22 . T Z x xtt xxtxxtx t t        

N

Zxt   R signifies a column vector of monomial in , n xt  R e.g., a second order monomial

  It should be noted that system (2.28) depends affinely on the time-varying parameters and is presented as a first-order polynomial matrix. The expression t can be implicitly con-

			CHAPTER 2. OVERVIEW LINEAR PARAMETER-VARYING SYSTEMS
	Then, system (2.25) is represented by LPV formulation as follows:
		01122 xtAxtxtAAxtAxtxt     ,     &	(2.28)
	with	1 AxtAA 01 2 1 1001 ,,, 1 0100 xt xt                     	and 2 A	00 . 10     
	Polynomial formulation
	Now, by defining a monomial vector in x : (2.25) is expressed by the following polynomial equation: 1212 Zxtt xxxtx t    T 	,	then system
		. xtAxZx   &		(2.29)
	with	Ax  	2 2 22 11 10 31 xt xtxt        	.
	3 112 33 1212 xtxtxt xtxtxtxt 3          1 2 xt xt       & & with state vector 12 , , T xt xtxt      and 12 , . xtxt belong to domain	(2.25) 1,1. x  D 
	Affine system
	Let's define a parameters vector 2 112 2 , x xtt xt xx xtxtxt 12 ,, 2        U R with 2 1 2 3. tt x t       	(2.26)
	are constrained by:
		12 11,04. xtxt      	(2.27)

sidered as an uncertain parameter (either endogenous parameters or exogenous parameters). Whereas system (2.29) is an explicit state-dependent polynomial.

  Considering linear cornering forces, i.e., , .

	yfffyrrr FCtFCt     could approximate by the following expression: , fr tt , y fff dynamics xx vt t tlt vtvt        & and . y rr xx vt t l vtvt       &	Where the (2.37)

  ).

	2222 2222 2222 frfrffrrffrr xx ffrrffrrffrrffrr zxzx CCCCClClClCl mvmv yy x ClClClClClClClCl IvIv v vtvt tt                  & &&&	2 2 CCl ff fff z CC m I      . f t     (2.39)

a. Tire sideslip angle -local sectors b. Single-track parameter description

  The approximation of this LPV model is presented as follows:

	with	11 xx ztzt ztzt Aut 0 0 0 a a K T rtrt tt          &   ,        &&& &   &&&  111122 3 4 :2sin,:2cos,:, xyx rtlttrtltztrtlt       2 1 0100 00 0000 310 10100001,  0000 01 4 000 00 a T At l                  	(2.47)

  In this case, it is not possible to determine the stability feature at the origin by this linearization system. Similarly, linearizing the system (2.48) at the equilibrium 2

	on the imaginary axis for eq x  b  0, T 0.  we obtain matrix 2 2 0 01 , T eq x fxt A ab xt          
	has one of the eigenvalues	2	11 22	2
	Theorem 2.2.2: Let us consider the time-continuous LTI system
	0 xtAxt 0 xx     &				(2.53)
		n PQ  S such that the Lyapunov equation
			T APPAQ 	0,
	is satisfied. 5 -There exist matrices , PQ  n S such that the Lyapunov inequality holds.
			T APPA 	p	0,
	By linearizing the system (2.48) around the origin 1 0 0, T eq x  the Jacobian matrix: 1 1 0 01 , eq x fxt A ab xt          is a Hurwitz matrix
	with the eigenvalues that this equilibrium is asymptotically stable for ,0. 2 11 1 22 4 Abba  have negative real part. It can conclude ab But, the eigenvalues will slide

is obtained as the linearization around an equilibrium point. The following statements are equivalent 1 -The system (2.53) is globally asymptotically stable. 2 -The system (2.53) is globally exponentially stable.

3 -The matrix A is Hurwitz (i.e., the eigenvalues of matrix A have negative real part).

4 -There exist matrices ,

Example 2.2.1. Let

	consider polynomial system (2.28): 1111 01122 2222 1 1 xtxtxtxt AAxtAxt xtxtxtxt                     100100 ,, and . 010010 AAA        Since the LPV system is affine in 12 ,, xx  then choosing a polynomial matrix Lya-. punov as follows: 01122 0. PxPPxPx    f (2.69) Following condition (2.61), the LPV system (2.68) is robustly stable if and only if the matrix inequality   01122011221122 0, symPPxPxAAxAxPP           p (2.70) hold, for 2 012 ,, PPP S and 12 , xx   within the specified range: 22 12 ,1,10,4, ||, 1,2. ii xxxi    U RR && (2.71) Indeed, the LMI conditions are derived from (2.70) by equally discretized Ni points in the intervals of parameters (2.27). If the conditions hold throughout the parameter domain, then the stability of LPV system (2.68) is ensured robustly in the presence of the time-varying parameters.  with 012 This approach bases on the discretizing parameter space. So, what is a "good" density to be able to cover most of critical points? The critical points are a set of , i x     U for which the LMI is unfeasible in Dx. For example, system (2.68) is unstable in the interval of parameter 12 1. xx    In addition, what are the appropriate parameter density, and

  Let fx  be a polynomial in

	n x  R of degree 2d. And let Zx  be a column vector whose entries are all monomials in x with degree no greater than d. Then fx  is a sum of squares if and only if there exists a positive semidefinite matrix Q such that
	. fxZxQZx   T	(2.72)

  .78410

	Pxxxxxx Pxxxxxx Pxxxx      22 22  62652 33112122

  .958,

	Pxxxxxx Pxxxxxx Pxxxxxx       3242 22112122 P 52442 23112122
	42342 33112122

Table 2 .1. The

 2 Conservativeness of Parametrized LMI Conditions.

		Quadratic LF				Parameter-Dependent LF
	LPV System		T-S fuzzy	Affine	T-S fuzzy	Polynomial	Poly-fuzzy
	Stability	Theorem 2.2.3 Theorem 2.2.5 Theorem 2.2.6 Theorem 2.2.5 Theorem 2.2.7
	Relaxation			Gridding	Multi-convex	SoS	SoS
	Example 2.1.3		(2.32)	(2.28)	(2.32)	(2.29)	(2.35)
			infeasible	feasible	feasible	feasible	feasible
	Example 2.2.3		(2.83)	(2.81)	(2.83)	(2.82)	(2.84)
			infeasible	feasible	infeasible	infeasible	infeasible
	Imprv		[ref1]				[ref2]	[ref3]
			infeasible				feasible	feasible
	0		1	2		3	4	5
	Imprv is the improvement of the respective stability conditions combining with the following methods:
	[ref1] -Relaxed Stability LMI conditions (Tuan, Apkarian, et al., 2001),
	[ref2] -Locality and Shape-Dependent Conditions (Sala, 2009),
	[ref3] -Positivstellensatz Relaxation (Furqon et al., 2017; Sala & Ariño, 2009).
	Example 2.2.3. Let's consider a nonlinear system (Sala & Ariño, 2009):
	112 2212 30.5 23sin xtxtxt xtxtxtxt     & &	(2.80)
	with state vector	12 , xtxt     T xt 	,	and 12 , xtxt belong to domain	1,1. x  D 
	First, an affine system gives as follows:
	30.5 023 xtxt t        .  &   with parameter 1 sin0.8415, 0.8415, x tt t     1, 1.   &	(2.81)
	Then, using a third-order polynomial of Taylor series of the sinusoid around 1 0, x  we get a polynomial system:
	3 11 30.5 0230.5 xtxt xtxt      .    &	(2.82)
	By applying the fuzzy modeling (Sala & Ariño, 2009), we obtain a representation of T-S
	fuzzy system
	12 30.530.5 0501 xtttxt           .        &	(2.83)

  to verify the stability of the polynomial systems (2.82),(2.83).

	Corollary 2.2.1. Giving a set | n xi xtfFfxin 1 12 ,,,, n Ffxftft R K 1 , 0, 1,,,  K D R and arbitrary polynomials 2 ,1,, j gxjn K are composed of products of F. Then, a suf-then define a region ficient condition for polynomials x  are positive , x x  D if there exist multiplier SoS polynomial 2 0,1,,, j qxjn  K such that the following expressions:
	 vv x  2 1 n j T j j xqxg   	,	(2.88)
	are sum of squares for arbitrary appropriate dimension vector . v
	Using a third order Taylor expansion method similar to (Sala & Ariño, 2009), combining
	the relaxed SoS condition (2.88) with the PLMI stability condition analyzed for fuzzy
	polynomial system (2.84), we obtain a second degree decision matrix:
	12 2 2 P P  11 P        4244255 0, for , x x  D f  where 11112122 P x  
	8267266 12112122
	276 221121

  .18110

	Pxxxxxx Pxxxxxx Pxxxx       9276 22 5.805101.024101.39710, xx  
	with local constraints multipliers of second degree 1122 1,1, 22 gxxgxx 	and the polynomial Positivstellensatz
	1 2 qx qx    	9 9  10 10 	11 22 11 22 5.35660 0.42542.5644 4.69830 . 0.052766968 T T xx xx xx xx                    	0, 0. ± ±
	And, even with the quadratic case that returns a feasible solution
	6 10  10  10    0.2101 0.21011.963 7 3 9.014     9 11 2 , I ò f for 22 9 11 22 6.25790 1.14142.6545 5.68230 . 0.082380597 T T xx xx xx xx      2 P 1 qx qx                ± ò 0.  0, 10,  7       ±	and arbitrary polynomials

 Figure 2-6. Dead

  -zone nonlinearity.

	that satisfy, for
	sat0. 0,sat0, , iiii ii iiiiiiiiiii 0, uu uu uuuuuvvu     	.
	Then, for T  for all 0 i u  there exist functions : ip T U 1 sat0, iiii uTu  	 R such that , iii uuim   1,2,,.  K	(2.96)
	Similarly, for
	sat0, ,sat0. iiiiiiiii 0, iiiiii 0, ii uuuvvu uu uuuu     	.
	Hence, for T uTu  0 i u  there exist functions : ip T U 1 sat0, iiii   for all	 R such that , iii  uuim  1,2,,.  K	(2.97)
	It follows immediately from condition (2.96), (2.97), there exists a diagonal matrix func-tion 12 :, :diag,,, mm pm TTTTT     K U such that (2.93) holds. R W
	By substituting sat, uutututvt  for the condition (2.93), we get , 1 0, T uTuvt 
	Accordingly, let's introduce with the dead-zone nonlinearity associated with a symmetric saturation function sat uutu  : signif 1,2. 0if iiiii i i i uuuuu uim u u    K (2.91)
	Given a vector tutvt vtvtvtvt  12 ,,, Tm m  K R and defined an auxiliary control vector , belong to the polyhedral set     ,,: , 1,2. m iiiii uvuuvuuvuim   S R K (2.92)
	Then, a generalized sector condition (GSC) is introduced as follows.
	Lemma 2.3.1. Given nonlinearity uutu ,, utt  S and a diagonal matrix function :, mm p T    U R sat  satisfies the following inequality	then
	1 T uTut  sat0.  			(2.93)
	Proof.
	Consider the dead-zone nonlinear function (2.91) conceding the following properties:
	0if 0if iii iii uuu uuu  	0 with 0	1,2. im  K	(2.94)
	This is illustrated in Figure 2-6 (left), where nonlinear function line, and linear function i u is the green dashed line. For all i   define i u  is the solid blue 0, , iii vu   we look for a vector   12 ,,,, T m uvu   K s.t.  S , 1, with 1,2,,, iiiiiiii uuuvuim   K (2.95)



2.3.1.2. Region of Attraction.

	 uvu  , uration constraints. Instead, the GSC condition is reformatted by finding a vector  S would lead to the unnecessarily difficult in analyzing sat-, t a feedback control law , ut and a nonlinear function . u 
	Corollary 2.3.1 Given nonlinearity uutu ,, utt  S and a diagonal matrix function :, mm p T    U R sat  satisfies the following inequality	then
	1 T uTuttu   	0.		(2.98)
	From now on, this formulation will be essentially implemented for the saturated LPV and
	LPV time-delay system in the following chapters. Where the definition of the polyhedron   ,u  S is recalled as an alternative of   ,. uvu  S
	The defined domain of states of the system (2.1) is denoted by . x D Without effect of dis-turbance 0, wt  the region of attraction of the LPV system is defined as a set of xt  x D such that from specified initial condition, the trajectory ,0 xtx  converges asymptot-ically to the origin.
	 Ax xtxtxt ,00 as n   RD R	.	(2.99)
	2.3.1.3. Ellipsoidal Set of Stability.

  has presented and discussed the two separate step design for the LTI systems. Now, let's introduce a dynamic controller feedback law derived for LPV system (2.89) by the following equation:

	cccc ccc xtAxtByt utCxtDyt     &			(2.120)
	with output measurement conditional stabilizing structure with proposed DOF (2.120). By replacing the controller , w ytCxtDwt  the goal is to seek a saturation
	in the system (2.89), the extended closed-loop system is obtained:
	12 12 ttuwt zttuwt    , .  & ABB CDD , T TT c txtxt    cccww (2.121)  12 12 ,,, 0 ,,. cccw cccww ABDCBCBDDB B   with BCABD HJDCJCJJDDJ                A  B B   C DD
	TT NM PP NM            YX éé			(2.122)
	whereé we don't care,
	 PVttPtt txtxt   1 , ,,, T TTT c        EU	p		.
				(2.123)
	A level set of admissible initial conditions,
	 000000 ,000   1 00, TTTT cp PxxP    EU (2.124)  .
	Then, expanding the structure of the DOF controller as follows:
	 cwcc , with u ttDDwtDCC        KK	,		(2.125)
	and defining an auxiliary dynamic controller
	 ttGCF , with cc    GG   . 			(2.126)

, .

From this point, the stabilization analysis for saturated DOF is quite challenging, in which the system (2.121) contains nonlinear couples concerned with the variable matrices. Now, let's assign the following parameter-dependent Lyapunov variables: 1 , , 2n P   S is Lyapunov matrix belong to ellipsoid as region of asymptotic stability of system (2.121),

  Imposing the constraints on observerbased feedback controller law(3.4) leads to the following result.

	3.1.1. Norm-Bounded Input	55
	such that K Lemma 3.1.1. Giving a set of initial condition, positive scalars , s 2 2 max:, 1,2,,. sss um u tuts    u  and a feedback con-trol law (3.4), then the constraints on input signal control are enforced   0, t  if the following matrix inequalities hold
	1 22 1 1 0 0 0 xP PeP            1 11 2 1 11 0 ± 1 0 s T ss PY uY P                   with 1 s Y    symbolizes s-th row of 1 , , 0, PI 2 1 P    1     ±± Y   gains controller 0, 12 , . 1 TT xtetVtxtPxtetPet     ED	(3.7) (3.8) and ellipsoid 1,2,,, sm  K 1 1 , P Y   with K  
	Proof.
	Condition (3.7) directly implies the initial condition:
	12 0000 TT xPxePe  	1.	(3.9)
	The limit constraints	2 ss 2 ut u  	are guaranteed if the following conditions hold:
	2 1 s u	 KKxeet  2 12 TT s xt xtPttP et        ,        	(3.10)
	1212 0000 TTTT xtPttPtxPxePe xee    	1.	(3.11)
	Let's set transformation by YKX   By using Schur's complement and then doing a congruence .

  The stability development for the feedback control system is guaranteed robustly against the influence of external disturbance and the model uncertainty conforming to the selected design values (e.g., ,,1,,4). The transformation(3.19) provides a tractable condition without imposing strong constraints like the rank of the matrix. But the use of the Young inequality (bounding lemma) usually leads to conservation. That can point out two weaknesses of this inequality. First, the upper bound is always positive, and this disadvantage is at the primary of the transform method (which cannot be improved). Second, as we can see that this approach is dependent on a wise choice of scalars .

	3.1.3. Concave Nonlinearity -Cone Complementarity Linearization	59
	SYSTEMS WITH ACTUATORS SATURATION
	along the trajectories of LPV system (3.5), combines with ℋ∞ performance (3.14) (Young's inequality) for this non-convex problem that results in: that exposes: 11 22 22 11 12 00 . 00 00 00 00 00 00 T T CC XMXM YMYM PMPM                            Parametric LMI (3.17) contains bilinear forms K perform a congruence transformation. Continue applies the tedious matrix transformation  presented in places that is difficult to    (3.18)  It completes the proof.
	11 22 00 22 00 TT xABKABKBKBKxxBB XX ww ww eALCBKALCBKeeBB PP             1 0. 0000 T T ww T ee xx HJKJKJHJKJKJ eeww HH ww             (3.16) 11 11 2121 111 33 11 00 . 00 00 00 00 00 T T K XBBYXBBY II PBYPBY XX JYJY                        °      (3.19) Reorganizing stabilization condition (3.16) yields to: 11 22 312 1 0 0, 0 000 e T K wd wr er BPI HXJYJI HI                  p (3.17) Combining conditions (3.18), (3.19) with PLMI (3.17), consecutively deploys congru-ence transformation by ,,,,, diagXIIII  and Schur's complement that ensues in: 1112 22 0,      p (3.20)    where 11 :   gives in (3.15), and   with 1 122 , , XPYPL        1 11 11 2 1 31 2222 1 2 0 , w , 0 ,, 0 0 0 0 T T Kw XBBK symXAXBK I PBK symBX JK PAYC XM P sym                                                    11 2221 12 1 00 00 , 00000 00000 000000 TT ABC T W MNXNYXNBBY PMYMIPBY N JY                      111 22112233 ,,,,,. diagIIIIXX        2 0 00 . 0 00 0 000 0 000 TT TT ABC C T W NNKN YM M FtFt N              The uncertainties can be eliminated similarly as follows:                                  The uncertainties can be suppressed using Young's inequality (Appendix A.4): 12 0000 00 0000 0000 TT TTTT ABABCC TT WW NNKNNKNN NN                   ° 1 2122 1 4 1 0 0 0 000 000 T T B BYMMM PBYPMPMP symsymFt NY                                               M MMM °2 0 0 T M          M   4 11 00 00 00 T TT BB NYNY               MM  (3.21)

Substitutes inequality

(3.21)

into condition

(3.20)

, then repeatedly applies Shur's complement that results in

(3.15)

.

W i i   K Remark 3.

1.2. i   If it was set as variables which turn conditions (3.18), (3.19) and (3.21) are nonconvex with bilinearity 1 ,,.

  are transformed into the optimization problem as follows:

	subject (3.22) and the inequalities,
	1  0, 0, n XI UI U U    n         ±±±	0.	(3.28)
	Then, the Cone Complementary Algorithm allows converting concave problems (3.24),
	(3.25) (or reduced conditions (3.27), (3.28)) into the following iterative optimization.
	Minimize	 IUUSSXUXS Trace0.25 n   	 	(3.24)
	subject (3.22) and the inequalities,
	1 0,        ±±±±± 0, 0, 0, nn nn XIXI UISI USSU      	0.	(3.25)
	where matrix d is as a pseudo of	1 ,
	0, 0.50.5 ± :, :. 0 0 U   I X   L       OMM    with :, UX   and   	(3.26) Pursuant to this formulation that makes condi-
	Minimize Trace	0.5 IUUXU n  	(3.27)

 d etc.

Furthermore, to reduce the complexity of problem

(3.24)

and the number of LMI constraints

(3.25)

, then conditions

(3.22)

can also be reformatted to an alternative approach: tions

(3.24)

and (3.25) decrease computational efforts by almost half. We obtain a reducing optimization version of original problem (3.24): Algorithm 3.1. Adapted CCL & Multi-optimization Objective Step 1: Choose any initial values ,  K s.t. the conditions (3.22)-(3.25) are feasible.

  ultaneously verify the feasibility at N-discretized points of the parameter range.

	3.2.2. Polynomial Parameter-Dependent LMIs via Sum-of-Squares	63
	belonged to a subset	/2, /2   (or t	1,1).  These conditions need to sim-t
	1 :, 0 pp tt ,:. p N   FCU R	(3.31)
	This approach is applicable for almost all types of parameterized LMIs conditions.
	Example 3.2.1 Let us consider the PDLF candidates such as:
	012 sincos, XXXX  with or in the form 012 XXXtXt  t   /2, /2, 2 ,  with t  1, 1.	(3.32) (3.33)
	We now seek for symmetric matrices 012 ,, XXXsuch that	0  f for X 	p U has t 

  ). We seek the maximum value of this polynomial by using the S-variable method then compare it with the SoS method and theoretical one. The problem rewrites into the following statement:We now compare the results in turn using the following methods: § The S-variable LMI polytope 1

	1000 § The S-variable LMI Gridding-N X: -2.385   2 SVopt minimize  s.t., , 0,, 0,, : i   with 1,,1, iN  K and interval /. N    QQQ °°° X: 8.348 Y: 1600	0,	(3.47)
	-1000 0 § The sum of squares decomposition SoSopt Y: 241.5 X: 1.579 Y: 44.23   minimize  s.t., ,0, a   M ±	:	with	,.  	(3.48)
	§ The theoretical approach:
	-5 Based on the SoS decomposition, this methodology cast the polynomial expression as a 0 5 -2000 , sup,. T a    M (3.49)  10 semidefinite programming problem. Let assign Z   are vector of monomials in 12 ,,, p Np   U K (vars) and coefficients , ij a   are decision variables (decvars). Then, the parameter dependent matrix involving in constructing vectors of monomials With 6,12, solving SDPs (3.46), we obtain slack-variable matrix   13.907340.07162.4979 , 22.76632.29470.3019      P (3.50)   introduced as follows: 0 ,. p N jj j aaaZ    MMM  (3.37) where , n j a  M S and . { 22 , 11024361 ,,650 1 T a a         and the optimal values, respectively,              M 1 1600.358242052000, (3.42) 6 11 1 1 2.77232652479142410, 1600.358239279673,  14444244443 M 7 22 2 1 2.19239154830575010, 1600.358242271239,
	Applying Lemma 3.2.1 to PLMI (3.42), we have a following condition: 1 3 2.670134199433960 1600.358215350658,
	T   a  M The condition (3.38) is satisfied f , TT  U if exits the symmetric matrix , p  0, a   MPP ° where Pis slack-variable matrix and 0.    One reasonable choice is given by (3.38) ,, c aa  M (3.43) R de-composed in (3.37), and a slack-variable matrix P with appropriate dimensions such that the following parameterized condition 0. TT a   MPP f (3.39) 10 . 01           (3.44)
	hold, where   is an orthogonal matrix gathering monomials occurring in the PLMIs. With an appropriate choice of   could yield to . p N j jj    Although the polynomial (3.43) has a global extreme, we cannot solve the above optimi-zation problem on the whole domain of . For simplicity, we choose a local domain ,    so that could cover all 3 extreme points ,/0. a  In combining with  M PLMI (3.43) and optimization problem (3.41), we get:
	Example 3.2.2. Consider the following polynomial 432 43210 43210 ,, with 1, 10, 18, 86, aaaaaa aaaaa    102,  M Since, 4 minimize  with ,,   such that: 102436 1010 . R ,65 0101 1 T        T          QPP °(3.45) (3.40) 0,  a  the coefficient of highest order monomial is negative and the derivative ,/0 a  has three distinct solutions, then polynomial (3.40) exists a global ex- M treme. minimize  such that, ,,, 0, . aa   MM R (3.41) SVopt :   minimize  s.t., , 0,, 0.   QQ °° (3.46)

c a  R Lemma 3.2.1. (Apkarian & Tuan, 2000) Let us consider a parameterized linear matrix (3.51) is represented as a spectral formulation of polynomial matrix inequality constraint 0, Figure 3-1. The extreme points of the polynomial in the interval 6,12.  Uses the decomposition method for polynomial (3.40), then the optimization problem (3.41) is rewrite in the spectral form:

  The control analysis of the Polytopic system seems to be the switching of the individual stabilized LTI systems corresponding to the operating regions of the parameter coordinates. The T-S fuzzy system is an alternate combination of LTI systems that in each operating area is a stabilized PDC control rule (even gain control j K are sometimes not stabilization law of local linear systems ,,

	Polytopic: ,, T cli cli APP A 	p	0,	1,, iN  K	p	.	(3.63)
	ii AB with  ± but 11 ,,,, ). ij  Specifically, in some case ,, 0, T clijclij APPA 22 0, TT clijcljiclijclji APAPPAPA  p that gives more re-laxation of the stabilizing conditions.
	3.2.3.2. Reducing Conservatism -Parameter-Dependent Scaling
	It can realize that the scalars i   related to the Young inequality parameterized in the design conditions of theorems Theorem 3.1.1, e.g., inequalities (3.18), (3.19), and (3.21).
								60)
	Polytopic:	, xtAxtBKxt  iii &		cliiii AABK 	(3.61)
	Based on Lyapunov quadratic stability analysis, we have the following statements: If there exists a common positive definite matrix 0 P f such that
	T-S Fuzzy:  11   ,,,, 22 TT clijcljiclijclji APAPPAPAj i  0, with , i  p	,1,,. p jN  K	(3.62)

, .

  with diagram shown in Figure 3-2.

	Example 3.3.1: The continuous-time nonlinear dynamical system is represented by the
	following equations:
	,1,31,1 1131 ,1,1,1 ,2,42,2 2242 ,2,2,2 ,32,2 332 ,3,3 ,41 44 ,4 22, 22, 1 2, 1 2 ssv sss ssv sss sv ss sv s aatk htghtghtVt AAA aatk htghtghtVt AAA atk htghtVt AA atk htght A    ,1 1 ,4 , s Vt A            	(3.64)
	where i ht are the liquid level in tank ,1,2,3,4; ii  tank ; i  , si a are the outlet cross-sectional area of tank ;0, 1,1,2 , si A are the cross sectional area of j itj   are time- varying valve flow ratio; sumed proportional to level measurement cm  of tank 1 and tank 2. For further descrip-tion of the parameters of tank process model and fuzzyfication analysis, refer to Appendix
	E.1. Based on this T-S fuzzy rules and the set of membership function, the quadruple-
	tank process can be represented as follows:
	 ,,  . iiiiw xtAAtxtButBwt 8 1 i ytCCtxt        &	(3.65)
	The flow disturbance from pump to tanks is presented by the following equation:
	w	  sin30cos20sin26sin31 ttt t t      T  	.	(3.66)
	The nominal constant matrices	4442 ,, ii AB   RR	and

j Vt are the voltage control signal of Pump j  with the corresponding coefficient , . vj k The measurement voltage V  on output signal , p yt ¡ are as- 24 C   R are given: SYSTEMS WITH ACTUATORS SATURATION ,1,,

  to solve optimization Algorithm 3.1 and the stabilization condition in Theorem 3.2.1 (with

	3.2.3. Parametrical Dependent LMIs via Convex Combination	73
	The uncertain parameters are the unmodeled and neglected dynamics, and error lineari-zation, modified by varying 1, 100, 1..8. i i   We consider the flow distribution rate in the tanks placed by the position of the regulating valve 1 t  and 2 , t  is adjusted ac-cording to the following rules:
	if if if if	020 20120 120220 220300 ts sim sim sim sim ts ts ts	then then then then	        1, 2 0, 1 1, 2 0, 1 ttMP 12 12 12 12 ttNMP ttMP ttNMP
				i B 

, ,, iwi JJ , , wi B  and 0).

Table 3 . 1 .

 31 Observer-Based Feedback Robust Stabilization and Performance with Saturation actuators.

	1, with the dynamic states govern the





is 2842 Nm.

    5.6668 reduced to    5.0952, respectively, by replacing the block variable matrices (3.73) with(3.76) ones. Moreover, the reduced formulation derived in (3.26)-(3.28) corresponding to Theorem 3.2.1 associated with Lemma 3.2.2 solved by algorithm CCL1 shows a slight enhancement from    5.2818 to    5.1002. Finally, through the above analysis, some conclusions can state: § The algorithm CCL provides more flexible stabilizing conditions by letting variables

	,1,2,3, i i  
	111 123123 ,,,,,, diagIIXIIX   				(3.72)
	1  to 11, 2 i j tj ,with j    catalog). In the second case -Polytope, diagonal slack-variables matrix (3.72) represents 1,2, provides significant relaxation (from 2 nd to 3 rd
	as convex combination formulation:
	111 1,2,31,2,3 ,,,,,, jjjj diagIIXIIXj   	1,2.			(3.73)
	Precisely, the attenuation optimum of stabilization condition of Theorem 3.2.1 associated
	with Lemma 3.2.2 and algorithm CCL1 is considerably declined from 53.7486 to only
	5.6668 by using the proposed method.
	On the other hand, consider the set of positive definite similarity scaling associated with
	structure uncertainty (Apkarian & Adams, 1998; Apkarian & Gahinet, 1995):
	   K , ,,1, : , ,,0 ijij ijijijk ttt diag     f		,	(3.74)
	where associates with the scaling set of parameters enjoys ,, , , 1,2,1,2, and 1,2,,rank. ij ijk   K Then, given a symmetric matrix Yt  ijij YtYt  we have:
	 0,    0.50.50.50.5 ,,,, 1 , 1 , , . T TT ijijijij TTTTTT ij TT ij XYZXYZ XYtZZYtXXXZYtYtZ XXZZ      mm  	(3.75)

in LMI conditions

(3.55)

in Table 3.1. It can interpret that each epsilon 123 ,,  must be satisfied all LMIs ij  corresponding to all vertex of condition (3.55), which makes the stabilization conditions claustrophobic. In this case, the parameterization is a more reasonable choice for the parameter-dependent conditions. As discussed in section 3.2.3.2, the improvement includes by changing e.g., And the last case -scaling, we have the following change in the variable matrix (3.72) by applying about development: 111 123123 ,,,,,, 1,2. jjjj diagXIXj    (3.76) By deploying the affine scaling formulation to the stabilization conditions (3.56)-(3.58) leads to a better performance from 

Table 4 . 1 .

 41 Multi-objective optimization -Problem 4.1.1

	1.1 by convex combination of four vertices

The process of fuzzification of nonlinear system (4.11) and membership functions can be found in more detail at (A. T.

[START_REF] Nguyen | Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi-Sugeno systems[END_REF]

.

  is recalled to impose the constraints on the state. By defining a polyhedron :| 1

	encloses the ellipsoidal set 0. T x x Xh h        ±	E	nT xx xthxt  P R 1  1 :| nT xtxtXxt   that yields: R (4.15)
	In this example, a vector ing conditions (4.13), (4.15) with the stabilization conditions solved with the relaxation   1 01.5 x h   is chosen to enforce norm 2 2 2 1.5. Combin-xt  methods at 0.25   and 2.75, we obtain the regions of asymptotic stability of the   LPV systems.
	For the gridding and SoS methods, the discretization of ellipsoidal domain (at level set 1  E ) by uniformly grids 20 values of t over interval 1.5, 1.5  corresponding to 20 ellipses in the bounded region 2 1.5. On the other hand, 20 ellipsoids of FLF xt  4 1 1 :| 1. nT ii i xttxtXxt    E R (4.16)   is obtained by discretized the membership function 1 t  uniformly over interval 0, 1.  It can be noticed that these ellipsoids are bounded by 1 1

  In the second example, the objective is to compare the conservatives of the novel procedure for considering saturation constraint enforced by the GSC condition with those of the different approaches. The controller design conditions are applied for saturated T-S fuzzy systems without disturbance input. To facilitate comparison with work in the literature. A compact (no input disturbance) stabilization version of conditions (4.1)-(4.2) is presented as a relaxation of parameterized LMI as follows.

	1212, 210510.1 ,,,,, 2012120.1 wi ab AABBB       ,, 10, 0, 0, with 1,2. iiiiwiwi CHDJDJi 	(4.17)
	Corollary 4.1.1: For the positive scalars ,, s u  if there exist matrices , mn   R and diagonal matrix , m T  S such that the following LMIs satisfy: kjkjk ,, n XYZ   S
	  p  ijkjikjkiikjkijkji 2 2 , 1 1 6 0 1 22 j jk j k k X X Zu Zu            ± with jk  0,    12 2 iijk ijk T jkjkji k symABYXX YZT X BT j    with ,         1 , ,, ,1 ,,2 k i x jk X    where with   E a bounded rate of membership function , ijjk  then state feedback controller gains are given by 2 1 jkkk j k tX KY     (4.18) (4.19) 1, t & 1 

  1)-(4.2) are employed by a quadratic Lyapunov function compare to a state-feedback controller constrained by norm bounded (Lemma 3.1.1).

	12 12 ,1,1 10.392518.91200.051519.9951 , 0.741615.23110.00340.0752 117.333600.58400 ,, 94.36270.00040.46970.0004 0.54590117.33360. , 00.372094.3627 ww AA BB BB                  545900.5840 , , 00.37200.4697     2, , 0, 0.

iwi HIJJ  (4.20)

Table 4 . 2 .

 42 and references therein usually consider only single control input signal, or two control inputs with similarity bounds (e.g., 12 ||5,||15). utut  Robust Pole Placement in LMI Regions.

	ℋ∞		Quadratic Stability
			Polytopic						PDC-Fuzzy
			NB	GSC				GSC
	opt 		2.8887	6.104		10 	3		6.444		10 	3
				High gain			High gain
	opt 	r	20,5   1.3054	& D-Stable 30,15 r   0.0977	r	30,15   0.0978
	0		1							2
	NB -Norm bounded; GSC -generalized sector condition;
	In this case, however, the bounds 1 u		,  and 2 2820 0.0524 d ra u 	Nm  differ greatly.
	2075 58.8411549.8932.7810960. , KK       	,
	and the GSC condition:
	99 12 4.19423.2400.31950.2468 10, 10. 15.07216.88114.11317.626 KK      

Hence, the problem becomes more severe in optimizing the reduction disturbance level with the stabilization conditions approached by the convex combination. For example, the fuzzy-feedback gains obtained corresponding to 1 st catalog of Table

4

.2 are given, respectively, with the Norm-Bounded (NB) constraint: 12 0.00210.10150.01660.a. GSC constraint without D-Stable c. GSC constraint with D-Stable

Observer-based Feedback Stabilization 4.3.1. Generalization of Young's Inequality As

  Now, given any matrices form P  : KerP, Q  : KerQ, base of the null spaces of P and Q. For example, let's consider: By using projection lemma (a generalization of Finsler's lemma), the nonlinearities in the stabilizing condition are converted to the affine parameter condition. analyzed in section 2.3.2.3, the stabilization of the closed-loop system (2.115) using a parametric dependent ellipsoidal domain(2.116) corresponding to feedback controller(2.114) and auxiliary controller(2.118) is stated as follows. wt  the ellipsoid E is a region of asymptotical stability for saturated LPV system (2.115) and error estimate et converges asymptotically in the domain E.

	4.1.2. Example	89
	1 2 0000 1 2 1 2 000 0 00 w d wr WCW BYAXBYCXBTB XC X YYCZT I JYHJYCJTJI                              & Followed the projection lemma (see in Appendix A.5), the solvability of inequality (4.29) 0, s ym for   is equivalent to the feasibility of the underlying LMIs: p kerker 0, T  (4.30)  PP p kerker 0, T   QQ p (4.31)   (4.24) (a). 0 1 0 0, 0 xX           ± (b). 2 0, s s X Zu           with . YKW  Then, condition (4.30) is rearranged as follows:  ± (4.25)   1 , xX    E then PDC feedback controller with gains 1 KYW    guaran-tees that (1) for 0, wt  the ellipsoid E is a region of asymptotical stability for saturated LPV system (2.113). (2) for bounded disturbance , wt  from 0 0 x   EE the trajectories of saturated closed-loop system (2.113) stay in the enclosed domain ellipsoid E corresponding to performance index . 1 2 1 2 1 2 0 00 0, 000 w w B X AXBKCXXBT KCXZT sym I HJKCXJTJI                            & p (4.32) Applying a congruent transformation by 1 ,,, diagPTII    Theorem 4.3.1: In presence of disturbances, for the positive scalars 0 ,,,,. s u  If there exist continuously differentiable matrices function 2112 ,:, ,,: n pp XPYZZ   UU S 2 ,:, mnnp p Y   U RR and a diagonal matrix function :, m p T   U S such that the follow-ing PLMI conditions:   for the latter matrix in-equality that results in 11 2 0, 0 s w T w T PABKP TKGBPT BPI HJKJ C J C C I                              & p (4.33) TTT tt xw u      and its transposition that entails to the condition: 0,       p (4.35)   (a). 0 1 0 22 00 0 ,0 0 0 xX PeP                   ± (b). 1 22 2 0, 0 s s T s XZ PG u              (4.36) ±   hold, 2 , ,,,, , , 1.., x sm eXP     E then feedback controller and observer with gains 11 122 , KYXLPY    ensures that
	1 000 001,0, T xPxV     (4.25).b      22 11 . T s sss s P u uGG       (4.27) (4.26) 2 , tV t 1 , and . GZPPX    And, the GSC condition can be presented by: 1 0. T ut TKCxG u x t           (4.28) Then, PDLMI (4.24) is reorganized by: with 0, TTT   PQQP p (4.29) where , W         0, 0000, T T TTT IKBKKJI       PQ 1 2 1 2 1 2 0000 00 000 0. 0000 0 w d wr CW AXBYCXBTB YCZ XC T sym I HJYCJTI X J                                              & algebraic manipulation effectively permutes nonlinearities terms K X C       to the SOF controller analysis with substitution KCY X CK    CC XW      and slack-variable . W  In addition, if     related The p 1 22 2 TTT w TTT tPttttP xABKCxxPxxBw xB t tPutttt zzww            & (1) for 0,  1 2 0. T TKCxG t x utu          (4.34) Combining with condition (4.28), we have the derivative of PDLF (2.102) along the tra-jectories of system (2.113) satisfies 22 1 0. Vtztt w       & (2) for bounded disturbance , wt  from 0 0,0 xe    EE the trajectories of satu-rated closed-loop system (2.115) stay in the domain ellipsoid E corresponding to performance index .  That ensues in the statements: § When 0, wt  then 2 1 0 Vtzt     & ensures that , P    E are the region of asymptotical stability of system (2.113) from 00 . x  EE  § When \0, wt      W then (4.34) claims from 00 , x  EE  the trajectories of LPV system (2.113) do not leave the set , ,, P   E with 2 2 1 0 VtwtV          L L 11 0 .    where, 1 ,,,,, dnn diagIIIXX        1 222 112 0, 2 S S T AXBYX PAYCP YZTBT G             &       &         11 2 1 00 00. 000 T w T wen T BHXYBY PB T J HI Y J                  
	000 000 ker, and ker. 0000 0000 0000 0000 000 0000 TT BKI I KI I I I JKI I                                 PQ



Proof.

Following the lines of Theorem 4.

1.1, inequalities (4.25)

directly imply the constraints of initial conditions and saturation condition on the additional control vector: (4.25).a  Firstly, by applied Schur's complement for above inequality, then pre-and post-multiplying with , Remark 4.2.1.

1 W  does not involve in the controller structure, the stabilization conditions return to the general case of SOF controller synthesis for LPV systems. So, this approach provides a more general condition. Based on this development, a parameter-dependent SOF and a fuzzy PDC controller are addressed in the work of literature

(Bui Tuan et al., 2021a)

.

4.3.

Theorem 4.3.2:

  In presence of disturbances, for the positive scalars wt  the ellipsoid E is a region of asymptotical stability for saturated LPV system(2.115), and error estimate et converges asymptotically in the domain E.

	0 ,,,,. s u  If there exist continuously differentiable matrices function 12112 ,:,,,: n pp PPYGG   UU S 2 ,:, mnnp p Y   U RR a matrix function 3 ,:, mm p XY    U R and a diagonal matrix func-tion :, m p T   U S such that the following PLMI conditions:
	1122 133 1 312 2 12 0 00 00 00000 m w S T T T TT wd w en r X PBY Y IYGGT BPBPI JHJI HI                                              (a). 1 00 00 1 0 1 22 0 0 0, 0 PxP PeP                     ± (b). 11 22 2 0, 0 s s T T s PG PG u                ±	0, (4.42) p (4.43)
	hold, gains	 ,, 12 xePP  ,,   E 11 1 ,..,, sm  then feedback controller and observer with 122 , KXYLPY     ensures that
	(1) for	0,
	(2) for bounded disturbance rated closed-loop system (2.115) are contractive in the domain ellipsoid E corre-, wt  from 0 0,0 xe    EE the trajectories of satu-sponding to performance index .  where,   221133222 , . S S PAPPAYCP         &&
	Proof.

  . A decomposition uses for the gain-scheduling DOF design strategy with the transformation matrix variables:

	12 , 00 TT II MN         XY   1212 , T P I     , I      X Y	.	(4.52) (4.53)

  Based on the new transformation, we get the following stabilization conditions for the saturated system (2.121) with feedback control law(2.125).

	4.3.2. Generalization of Finsler's Lemma	97
	1 11 2 T symPP   & A 1 2 ˆ, 1 2 ˆÂBABC sym YAC         & & XCXD ABY 2122 , and , TT ww ww BBDB B BDB                 D Y Y BB 1 ˆ, HJHJC   C XCD 11 ,, ˆˆˆˆĈ C        GKG  (4.55) (4.56) (4.57) FGCFDG (4.58) where the change of controller variables:   ˆ, ˆ, ˆ, ˆ, ˆTT c T cc cc T cc c cc NMABDCNAM BCMNBC BDNB DCCM D GCF             && AYXYX YX BY CX D FX , T M   ˆ. c G   G (2) for bounded disturbance , wt  from 0 0,0 c xx    EE the trajectories of satu-Continuously, by using substitutions (4.54)-(4.58) that entails directly stabilization con-rated closed-loop system (2.121) converge in the enclosed domain ellipsoid E cor-responding to performance index .  where, ˆˆˆ, ˆˆˆˆ2 S TS TT AB ABCYAC TBCTBT                  & & ditions (4.60).a. Whereas, condition (4.60).b is set to guarantee the definitively symmet-rical positive of the matrix function , P  can indirectly determine by 11 . T P   Non-zero scalar  ensures that matrix   XY is nonsingular. XCX AD G Y F Y This end of the proof. B CD ˆ, ˆT ww T ww T w BDBHJ BDBHJC DTJ                   DXC T TB     Y relates to the expression 12 T T       B causes condition (4.60).a to no longer affine, because both   Y andT   are already presented in this condition. In this case, the dead-zone nonlinearity is now included in dynamic YD controller (2.120) by following expression: D and ˆ. T dww r IJDJ I          D , . ccccc ccc xtAxtBytEu utCxtDyt    & (4.69)   That entails a slight modification in extended system (2.121) such as: Proof. 1 . c B E        B  (4.70)   (4.59) Then, the following the development, the bilinear problem is reformatted as:
	Theorem 4.4.1: In presence of disturbances, for the positive scalars exist continuously differentiable matrices function ,:, n  If there 0 ,,,,. s u p   XY U S ˆ:,: nn p   AB U R ˆˆˆ, :,,:, npmnmp ppp   ,CFDG UUU RRR and diagonal matrix , m T   S such that the following conditions: 1 0 1 2020 0 0, 10010 00 0 T T P                 (4.64) 21 . T c BT T BTNET              Y B (4.71) ± 11 2 0, 1,..,. 00 0101 s T s s T P u sm                 G G (4.65) Now, let's denote a new variable ˆ, c BTNET     EY then the stabiliza-tion in Theorem 4.4.1 returns to convex form, where the dynamic gain is obtained by: ± It can realize that these are the initial condition and the saturation bounding on the con-11 ˆ.      EY (4.72)
	meets the difficulties with the expansion of derivative P   & involved in the deriv-ative of . é To avoid this problem, we can take advantage of the relationship (4.53) s.t., 111211 , TTT T T PP YMN NM            &&& &&& &&& XX YXY (4.54) with . TT YXNMYXNM   &&&& From the definition of state-space system (2.121), the production of variable matrices are deployed as follow: (a). 0,         p (b). 0, I I         X Y ± (4.60) 0 0 0 1 0 0 0 0, 00 c x xN I x     troller. Now, applying Corollary 2.3.1 to the feedback control laws (2.125), we have: 1 ˆ0. T w uTtutDwt    KG (4.66) Remark 4.4.2. It should notice that the observer-based output feedback development in the previous section is a particular case of dynamic output feedback by setting: D               X YY ± (4.61) 2 ˆ ˆ 0, T TT s s s I C u               ± XF YG (4.62) are satisfied,  , 1,..,, ,, c xP sm txt    E then the controller gains are given by 1 1 ˆ, ˆˆ, ˆ, ˆ, c T c c T c T D CCM BNB ANNMC YBABCM               && L BYD AYXBX CYDX (4.63) ensures that, (1) for 0, system (2.121). wt  the ellipsoid E is a region of asymptotical stability for saturated LPV 1 112122 2 1 1 2 1 112 2 0. 0 ˆ0 000 T w PPT TD sym I TI                  CDD  & KG p (4.68) D ABB CDX By pre-and post-multiply 1 ,,, T diagTII    for above condition: D Repeating tedious mathematical transformations, combining the ℋ∞ performance, GSC     :; :; :; :. ĉcccc xtxtAALCBKBLDCCKK   (4.73) condition (4.66) and derivative of Lyapunov function (2.123) along the trajectories of closed-loop system (2.121) for all ,, tP    E that yields to: 1 12 2 111 1 2 1 12 2 . 0 ˆ0 0 0 0 0 w PPPP TTTD sym I I                  CDD  & D ABB p (4.67) KG Besides, we only present the ordinary case where the order of the dynamical output-feed-back controller equals to the plant system. The square matrices , NM   provide the unique solutions for Eqs. (4.63) and (4.72).

Inherently, the above conditions are interpreted the same as previous theorems. Firstly, the saturation constraints involved in feedback controller (2.125) and auxiliary controller (2.126) will be analyzed. By using the congruence transformation (4.52), (

4

.53), and (4.58) we have LMIs (4.61)-(4.62) equivalent to the following conditions: W Remark 4.4.1. Bilinearity c ENTB  

Table 4 . 3 .

 43 Multi-objective optimization and .

					optopt 
		Parameter-dependent Stabilization & Actuator Saturation
		State Feedback			Output Feedback
		SF	SOF	OBF	# OBF	DOF
	opt  opt 	Theorem 4.1.1 Theorem 4.2.1 Theorem 4.3.1 Theorem 4.3.2 0.3656 0.3661 6 1.065210  0.7094  3 2.268210   3 2.882710   4 4.470910   7 3.989610  	Theorem 4.4.1 0.3753 3 2.450910  
	0	1	2		3	4	5

SF: State feedback; SOF: Static output feedback; OBF: Observer-based; DOF: Dynamic output feedback.

Table 4 .

 4

			4. Minor Axis Maximization	  	10.
					Parameter-dependent Stabilization & Actuator Saturation
					State Feedback		Output Feedback
					SF	SOF	DOF
	   	0.01 0.1	 	opt  opt 	Theorem 4.1.1 Problem 4.5.1 1.3106 0.7324	Theorem 4.2.1 Problem 4.5.1 1.3108 0.7344	Theorem 4.4.1 Problem 4.5.2 1.4373 0.8361
		0			1	2	3

SF: State feedback; SOF: Static output feedback; DOF: Dynamic output feedback.

  is represented by the state-space as follows:

	 ,,,1.520,  0.1190.01 130.80 w ABBC 0.890.891 142.6178.250 t t                   0, 01, 1, 0.

ww DDHJJ   (4.82)

Table 4 .

 4

					Multi-objective optimization
					State Feedback		Output Feedback
					SF	SOF	DOF
					Theorem 4.1.1	Theorem 4.2.1	Theorem 4.4.1
					Problem 4.1.1	Problem 4.1.1	Problem 4.1.1
				opt  opt 	2 1.044810   4 2.622610  	2 5.382110   4 4.228110  	2 1.045110   4 2.602010  
					Minor Axis Maximization	  	10.
					Problem 4.5.1	Problem 4.5.1	Problem 4.5.2
	   	0.01 0.1	 	opt  opt 	1.9632 0.8316	4.3569 * 1.6935 *	1.9664 0.8337
		0			1	2		3

5. Actuator Saturation -Optimization Problems. SF: State feedback; SOF: Static output feedback (*: ε=0.02); DOF: Dynamic output feedback.

Table 4 .

 4 6. Multi-objective optimization -Example 4.5.1.Implementing the minimization methods as in Table4.3 and Table 4.4 for the quadratic optimization conditions, we obtain the results in Table 4.6. It can observe a performance degradation using a quadratic Lyapunov functional for the stabilization conditions as the optimization values opt  increase approximately five times and ten times for opt

			Quadratic Stabilization & Actuator Saturation
					State Feedback	Output Feedback
					SF	SOF		DOF
			opt  opt 	Theorem 4.1.1 Problem 4.1.1 1.7416 2 1.807110   Minor Axis Maximization Theorem 4.2.1 Problem 4.1.1 3.8731 2 7.622510     	10.	Theorem 4.4.1 Problem 4.1.1 1.7416 2 1.806910  	Compare with	Table 4.3
	   	0.01 0.1	 	opt  opt 	Problem 4.5.1 10.4753 6.1052	Problem 4.5.1 16.3353 14.9865		Problem 4.5.2 15.4075 8.7692	Table 4.4
		0			1	2			3
	adjustments: other two theorems. XXXTT  ,0,.  &	following Then, analyze in a similar way for the

SF: State feedback; SOF: Static output feedback; DOF: Dynamic output feedback. The quadratic stabilization expansion based on Theorem 4.1.1, Theorem 4.2.1, and Theorem 4.4.1 using the Lyapunov function (2.120) can interpret briefly as follows: considering the parameter-dependent condition (4.1)-(4.2) in Theorem 4.1.1, and the  (the comparison between Table

4

.6 QLF vs. Table

4

.3 PDLF). Similarly, we also have conservatism for the minor axis optimization conditions (the comparison between Table

4

.6 QLF vs. Table

4

.4 PDLF) where the value opt  also growths approximately ten times.

  , the Auxiliary-function-based II (AFBII) (P. G.Park et al., 2015), the reciprocal convex combination[START_REF] Datta | Improved stabilization criteria for Takagi-Sugeno fuzzy systems with variable delays[END_REF][START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF] C. K. Zhang et al., 2017;[START_REF] Zhang | An improved reciprocally convex inequality and an augmented Lyapunov-Krasovskii functional for stability of linear systems with time-varying delay[END_REF], and the generalized vectors-based multiple integral inequalities (Y.[START_REF] Tian | A new multiple integral inequality and its application to stability analysis of time-delay systems[END_REF][START_REF] Briat | Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers[END_REF][START_REF] Van Hien | Refined Jensen-based inequality approach to stability analysis of time-delay systems[END_REF][START_REF] Zhao | A new double integral inequality and application to stability test for time-delay systems[END_REF] significantly improve the asymptotic stability of TDS systems. The fundamental methodology of bounding techniques is to estimate better the lower bounds of the quadratic integral terms in the derivative of the Lyapunov-Krasovskii functional.

  It should note that this stability analysis approach simplifies the stability PLMI condition, which concerns only the three decision matrices variables P(),Q and R from Lyapunov-Krasovskii functional (no slack matrices are included). Then, to avoid the use of the old-style techniques (e.g., cross-term inequality, model transformation), Jensen-based inequality (5.5) is employed to bound the integral . The coupling decision matrices and system matrices in inequality (5.20) are more complicated than one in inequality (5.19). Consider condition (5.22), the expansion of2 If the double and triple integrals include in LKF, then the design stabilization conditions are really in trouble.

	5.2.1. Single Delay-Dependent LKF Stability and associated relaxation 5.2.1. Single Delay-Dependent LKF Stability and associated relaxation	119 121
	11 T wd 0 1 T hwr 0 hw APRQR h BPI HHJI hRAhRAhRBR                  . p N i i i P symPAQR   with: 11        Proof. The proof is detailed in Appendix D.1. 12 23 , . T t TT T th PtP txtxsdsPt PP            For the sake of simplicity, denote :, h real function is rearranged by Wirtinger-Based inequality WBII (5.6): xxtht  The integral of the positive 0,     p (5.19) VttPttxtQxthtxthtQxtht   && & 2 0, t TTT th tPtthxtRxthxRxd   hold along the trajectories of system, with   & &&&& (5.22) tiple productions terms 12 , T PAPA   prevents performing a congruence transfor-mation with the inverse of matrices 12 ,. T PP 
	Remark 5.2.1. t th xsRxsds T   && 5.2.1.2. Wirtinger-Based Inequality As we know, Wirtinger-Based inequality (5.6) has a better estimate of the lower bound of the expression t T 1122 2 3, 426/ 46/. 12/ t t TT tht th WBII TT T tt hh tt thth hxRxdhxRxd     &&&&  h RR ht xx RRRh xx RRh xsdsxsds Rh             (5.23)        where, 12 2 , . t thth tht xxxxxsds ht     th xsRxsds The well-posed problem of the inequality (5.23) at extreme point 0
	Lemma 5.2.2. For positive scalars ,, h  delay space system (5.1) is asymptotically stable corresponding to ℋ∞ performance index, if there 0 , ht H and , p t U then LPV exist a continuously differentiable matrix function 1 :, n p P   U S positive symmetric ma-trices 3 ,, PQR n   S and a matrix 2 , nn P   R such that the following PLMIs fulfill 1 1 1 0 0, 0 TTTTT r I Vtztztwtwttttt R           & (5.24)
	2.1. Given positive scalars ,, h  delay and parameter belong to the sets   1 0 1 0 ,0, :||1, , ,, ||, , 1,,. p N iii piiiip hthhh t tiN        & & K HC C U RR R Then, the LPV time-delay system (5.1) is asymptotically stable corresponding to designed 2 11 21 12 3132 12 , 14 0, 0 0 00 h TT wwd hwr hw QR R BPBPI HHJI hRAhRAhRBR                        where & p (5.20) 12 23 0, T PP P PP         f (5.21) 11121 2112 6 3123 6 3223 ,4, 12, , 1. p N ii i TT h T h T h h symPAPPQR APPR PAPR PAPR        & with 1 2 12 12 66 12 2323 1 , 0 , 0 ||4 24 0 i t T TTTT th th hw hw P S i i TT h TT h hh h TT ww xxxsdswt HHJ hRAhRAhRB PAPQR APPRQR PAPRPAPRR BPB                               & 2 d PI            with notation 2233 1,1,1, PPPPQQ    and :. S sym  Then, using Schur's complement yields to (5.20).   ℋ∞ performance attenuation, if there exist a continuously differentiable matrix function :, n p P   U S matrices ,, n QR  such that the parameter-dependent matrix inequal-Proof.  S ity satisfies Consider LKF (5.11) for LPV time-delay system (5.1). Then, this dynamic system is de-lay-dependent stable if the conditions: T hxtRxt  && results in multiple product terms RA() and RA h () etc., which could avoid if the conditional vector is expanded with . xt  & But, the existence of the mul-	,



 &&than the traditional Jensen's inequality (5.5). The manipulation of this inequality accompanies a slight change in the LKF formulation from (5.3) to (5.11) combined with an augmented vector that yields the following lemma. t xxt  :. i ht  at time i tt  is validated as analyzed in D.1. Then, considering the influence of external disturbance, performance constraint included in stability condition (5.22) and combines with condition (5.23) that entails in the following PLMI condition: W Remark 5.2.2.

  XXXX  but it reduces the interestingness of the SV method. By the way, one slack matrix is concerned in PDLMI (5.25).

	5.2.1. Single Delay-Dependent LKF Stability and associated relaxation	123
	with system's matrices	the slack-variable matrices couple AA  , h  and .	123 ,, diagXXX 

w B 

Three slack-variables yields a more relaxed condition, and the second LMI condition kerker0 T   QQ p is always feasibility.

Nonetheless, too much coupling hinders the scalability of the controller design strategy.

A linearization could derive from choosing 2131 ,,

  N  the latter condition reverts to stability condition (5.19) in Lemma 5.2.1. So, if we discretize the auxiliary convex function in the conditions of Lemma 5.2.2 and Lemma 5.2.3, that would yield the least conservative results. Now, let consider a discretization of extended Lyapunov-Krasovskii functional (5.11):

	23 VttPttVtVt  T  , , T t TTn 2 12 23 T th PP txtxsdsP PP              S  By using this Lyapunov-Krasovskii functional, we now study the asymptotic stability for (5.42) where .
	LPV time-delay system (5.1) based on Lemma 5.2.2 as follows.
	Lemma 5.2.5. For time-delay integer number N, then LPV system (5.1) is asymptotically delay-dependent stable with 0 , ht H parameter , p t U a positive scalar ,  and an corresponding to design L2 norm performance, if there exist matrices 3 ,,,, n ii PQR   S 1,2,,, iN  K a continuously differentiable matrix function 1 :, n p P   U S and a matrix
	holds, where: 1111 1 ,/, p N jj j symPAPQR   &   1 2 22 1 11 11 , 0 00 1, 1,,1, i NN iiiiii R R QQRRiN            O      O     MOO     L K	41)
	and the decomposition of delay spaces into N-subsets
	1 :,0,: ||1. hththht 0    HC RR &
	Proof. The demonstration is delivered in D.2.

W

It is worth mentioning that when 1,

  Similar to the development in Appendix D.1, by combined the ℋ∞ performance criteria with the PDLKF stability condition (5.45), we have the following condition:

	128	CHAPTER 5. STABILITY ANALYSIS OF THE LPV/QUASI-LPV TIME-DELAY SYSTEMS
	12 23 23 23 23 00 00 00 T T T T PP PP AAB 0 00 0 0 00 00 hw tPttt TT II PP PP PP            MM        ,              LL & MM LL	(5.48)
		&&&&&&
	By denoting 0 ::,:,:. tiN xxxtxxtihtxxtht   The latter integrals of the positive real functions are reorganized by WBII:
	11 11 11 2 426/ 46/ 12/ tihtiht TT ii tihtiht T ii iii ii tihttih ii i tihttiht hxRxdhxRxd xx RRRh xx RRh xdxd Rh          &&&&                            t           (5.46) .
	It should be noted that ttiht N 1 1 i thtiht dd xdxd dtdt                  1 1111 N i ihtxtihtihtxtiht      &&	(5.47)
	Denoting :1, II    and giving an extended vector: 11 ttNht 1 T TTTTT tNN thttht xxxxxdxdw               LL	.
	In the view of uniform distribution, we have:

  . It should realize that the uncertain delay involves in the two following cases. At the times of ,,

	 1 hdu  11 22 , 00 33 T T T wd hdwr 0 00 hdw mumuhmudmuwu APR h APR BPI HHHJI hRAhRAhRAhRBR RARARARBR           &                          11 , 1 0 0 0 0 T hdu T wd hwr hdw mumuhdmuwu 0, p AAPRQQR BPI HHJI hRAhRAAhRBR RARAARBR          &                where 11 1 ,, p N iiu i symPAPQQR   &   2233 1,11, uduu QRRQR  	(5.54) 0, p (5.55)
	and the delay space	1	,0,: ||1.
	ij tt belongs to the specified domain such that condition reforms to similar condition of a single delay dependent. And in the second ,0, iii dthtt   then the stability case when , jj dtht that ensues on the following result.
	Lemma 5.2.6. (Briat, 2008) For positive scalars ,, h  parameter time-varying delay (5.50) is asymptotically stable with 0 , ht H uu QRQR uously differentiable matrix function :, n p P   U S such that the following conditions are , p t U then LPV , n   S and a contin-satisfied

d dt H correspond- ing to H∞ performance criterion, if there exist matrices ,,, dd ddd    & HC RR

  For positive scalar,,, m

	5.2.4. Memory-Resilient Delay-Dependent Lyapunov-Krasovskii Functional Stability
					131
	with	11  77 31 22 21 411 72 22 77 33 44 7 53 2 ,, , , , , 1, , p i N P S ii T mhm T hd T w hd mm mh PAQR APR AAPR BP hRAA QRL H                  & 	22   32 52 62 63 73  7 2 7 2 7 2 1, 1, , , , , m hd hd mh mh QR QR HH LAA LA hRA           
	and notation	:.
	h  parameter ht H the LPV time-delay (5.50) is asymptotically stable consistent the ℋ∞ perfor-, p t U and delay 0 , mance, if there exist continuously differentiable matrix function ,:, n p PL   U S and pos-itive matrices ,, n QR   S such that the following matrix inequality satisfies
		11 2122 313233 41 5253 6263 7273 , 0 0 HJI 00 0 d wr w w I LALBL hRAhRBR           &                 	p	0,	(5.56)

  The satisfaction of small-gain stability for operator (5.16) is associated with finding a resilient-stable trajectory for an approximated delaydt constrained within a ball of diameter m  centered along the trajectory of . ht  It can be shown that if 0Let's introduce the matrices form base of the null spaces of P, and Q respectively,

	5.2.4. Memory-Resilient Delay-Dependent Lyapunov-Krasovskii Functional Stability
	133
	with
	Remark 5.2.4. m   ( i dt  is approached close to ) i ht and L   sufficiently small, then inequality (5.57) brings about stability condition (5.20). This condition is therefore more general than the memory-delay dependent stability condition in Lemma 5.2.2. Now, similar to Theorem 5.2.1, an associated relaxation condition of Theorem 5.2.2 is provided in next result. Theorem 5.2.3: For positive scalar ,,,, m h  delay 0 , ht H and parameter , p t U then the LPV time-delay (5.1) is asymptotically stable conforming to the energy-to-energy index, if there exist continuously matrices function :, :, nnn pp XL    UU RS a con-tinuously differential matrices function 1 :, n p P   U S positive matrices 3 ,,, n PQR   S and a matrix 2 , nn P   R such that the following PLMIs satisfy 2 122 3233 424344 12 2525354 7374 2 0 , 0 0 00000 00 000000 0000000 000000 T h d wr P PR I HJI LL hRR PL                     &     1 00000, 000000000. hw hRP IAABI I                        P  Q	,
	   0 11 21 3132 6 234243 7 2 2 7 4 12 2 12 0, (5.60)  , 4 4 , 00 hdmhw m T h h TT wwd LztLALAALALBt with T QR QR TT PAPRR BPBPI                         7 2 7 2 0 . 0 hdmhw hdmhw HHHHJ T hRAhRAAhRAhRB       First, stability condition (5.61) is rearranged as follows:  Proof.     & 2 2122 313233 41424344 12 2525354 61 7374 , 0000 00 000000 0000000 S T h d wr X PR I HJI LL hR                  & 21 0, 0 0000 R XPLhRP                      p 00000 000000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 , 00000000 00000000 00 00000000 00000000 00000000 00000000 hw AABI I I I I I I I I I I I I I I                      PQ 000000 00000000 00000000 00000000 I I I I                  (5.61) 12 23 0. T PP P PP      f Followed the projection lemma, the feasibility of (5.61) entails the feasibility of the un-(5.62) derlying conditions:   with: 1 33 211 31 7 41 2 61 2221 322 7 422 2 6 523 1 , , , , ,4, 12, 12, , i T T hd T mh T w P S i i T T m h Q PAX AAX AX BX PQRP PR PR PR                       & 7 43 2 6 533 73 2 7 44 4 7 6 543 2 7 74 2 4, 14, 1, , 14, 1, . m h hd m m h mh R QR PR HH QRL PR H             kerker 0, T   PP (5.64) p kerke r0, T   QQ p (5.65)	.
	Finally, using Schur-complement to rearrange condition (5.60) that results PDLMI (5.57). W 0, TTT XX   PQQP p (5.63)

Table 5 .1. The maximum admissible upper bound MAUB for delay 0 . ht H Delay-dependent Stability LTI System Example 5.2.1 Example 5.2.2

 5

	2010 , 00.91 .0 1 h AA       	.							(5.66)
	The analytical maximal delay value for which system (5.66) is asymptotically stable
	6.17.
	0100 , 1211 h AA       where the time-varying delay 0 . . ht H								(5.67)
		 	0.5	 	0.9	 	0.3	 	0.5	NoV
	Lemma 5.2.1	1.5874 1.1798 2.1756			1.5n 2 +1.5n
	Lemma 5.2.4 (N = 2)	2.3200 1.2012 2.3025			2.5n 2 +2.5n
	Lemma 5.2.4 (N = 3)	5.0553 4.2626 5.9301			3.5n 2 +3.5n
	Lemma 5.2.2	2.1111 1.7576 2.1798			3n 2 +2n
	Lemma 5.2.3	5.2312 3.9416 7.5882 6.1862	9.5n 2 +3.5n
	Theorem 1

analytics h 

Example 5.2.2:

[START_REF] Kharitonov | On the stability of linear systems with uncertain delay[END_REF]

  2.1-Lemma 5.2.3 for analyzing the delay-dependent stability of the following delayed quasi-LPV system:

	Example 5.2.3: (Wu & Li, 2007) Let's consider a T-S fuzzy time-delay system with the
	local linear matrices are given by:
	11 22 3.201.00.9 , 0.02.10.02.0 1.00.00.90.0 , 1.03.01.01.6 .6 h h AA AA              	, .	(5.68)

Table 5 .

 5

	2. The maximum admissible upper bound for delay	0 . ht H
	Delay-dependent Stability quasi-LPV System
		Example 5.2.3	Example 5.2.4
		 	0.5	 	0.9	 	0.5	 	0.9	NoV
	Lemma 5.2.1 (Multi-convexities)	0.4917 0.4743 1.5782 0.9116	2n 2 +2n
	Lemma 5.2.2 (Multi-convexities)	0.9603 0.5113 1.6960 1.2316	3.5n 2 +2.5n
	Lemma 5.2.3 (Multi-convexities)

  R is state vector with initial condition and unknown time varying delay ht assumes belong to spaces 0 .H

		sat, sat, hw ,,0 hw hw xtAxtAxthtBuBwt ytCxtCxthtDuDwt ztHxtHxthtJuJwt xh      &   	(6.1)
	where	xt	n

p yt R and r zt R are outputs; d wt R is dis- turbance inputs; and nonlinear saturation function sat. u The parameters t  and its un- known rate of variation belong to parameter spaces (2.8)-(2.9).

  Similar to the analysis for the feedback controller with uncertain delay value cases, when dtht  refers to an exact-memory DOF controller. If dc

		 cccdcccc  . . ccdccc hw xtAxtAxtdtBytEu utCxtCxtdtDyt ytCxtCxthtDwt     , &  	(6.15)
	A   dt H we have a resilient memory DOF 0, dc C   that repre-d controller. Replacing the controller in the system (6.1), the extended closed-loop system sents a memoryless DOF controller, and if
	is given by:
	with	12 12 ttthtduwt hd hd zttthtduwt    , .  & AAABB CCCDD , TTT c txtxt  and	(6.16)

Theorem 6.2.2:

  For time-varying delay 0 ,, d htdt  HH positive scalars ,,,,

	i u  %%%% R continuously 23 ,,,,, nn XPPQR   differentiable matrices function 1 :,,,,:, nmn and L2-bound disturbances. If there exist matrices pp dh PYYZZ    % UU SR and a diagonal matrix function :, m p T   U S s.t. the following PDLMIs satisfy:
	2 12 02 41424344 2122 313233 2525354 6162 828384 00000 00 0000000 00000000 h dI T wd T r S w X PR YZT BI JTJI LL hRR                               %% %% %% 21 00 0000 T XPLhRP                         %%%%   1 1 23 2 2 22 0, 1,2, T h iIii PhR PhRPQRim ZZu                 %% %%%%% K f   where	0, (6.27) p (6.28)
	 211 31 7 41 2 61 22211 1  322 7 422 2 6 523 62 , , , 4, 12, 12, , / p T T hd T mh T N S jj j T TT m h AXBYP AXBY AX BT PPQRP PR PR PR YZ              %             %%   %%%%% %% %%   	82 84 543 2 2 7 7 83 2 7 44 4 6 533 2 43 7 33 14,	, 6

  2.1 & Theorem 6.2.2 conforming to the design delay space 0 . By utilized the minor axis maximization, the non-convex problems are converted to the following linear criteria.

	152	CHAPTER 6. STABILIZATION SYNTHESIS FOR THE LPV/QUASI-LPV TIME-DELAY
				SYSTEMS WITH ACTUATORS SATURATION
		1 .
	Problem 6.2.1. Given delay space the definition of the initial set 0 . XFind the variable decision matrices ,,, 0 , ht Hcompact sets of parameters t U and , p XPQR %%% such that the initial conditions meet condition (6.21), and the following statement fulfills:
		Minimize	23 1234 10.5 fhhh  	(6.30)
	subject to (6.18), (6.19), 4 II 123 , , PIQIRI  %%% ppp 2 0. T XXI       ±  for pre-selected scalar ,  weighting scalar . 	,	and	(6.31)
	It's worth noting that inequality matrix . nn X  R So, the constraint (6.31) implies XIXI   ± always hold for all scalar 0 T 121 TT XXXXII    R , 4 ,   °°shows less conservatism than a directly imposed condition 1 4 . XI   ° This method de-pends on the selection of ,   R and condition 2 TT XXXXI   ± for all ,, X  but the opposite holds only when 2 0. T XXI   ± Thus, it is possible to miss the solution belong to the negative side: 2 0. T XXI   p
	Besides, we have	11 224 , TT QXQXXXI     % °°	and similarly for	34 , RI  °14
					H
	Followed the definition of the domain of attraction 0 , X maximizing the size of DoA means minimizing of the greatest eigenvalue of nonlinear matrices 11 ,, TT XPXXQX  %%%%

T XRX  %

Table 6 . 1 .

 61 The optimization of ℋ∞ performance criterion .

	Example 6.2.1: Let's consider an LPV time-delay system:
	01sinsin0.1 23sin0.2sin0.3 sin 0.2 , 0.1sin 0.2 0100 . 000.1 tt xtxtxtht tt t utwt t ztxtut          &                   0.2,0.1.   By setting parameter sin tt that implies   & (6.32) ,1,1 tt and assuming time-varying delay belong to where 0 . ht  H
												opt 
						Delay-Dependent Stabilization
		Example 6.2.1	h 	 	0.5, 0.5	h 	 	10 0.9			h 	 	3 0.99	h 	 	3.3 0.99
	(Briat, 2010)	Theorem 4.1 Theorem 4.2	m   m   h 0	1.9089 12.8799  13.0604  04.1658	(Briat, 2015)	Corollary 8.1.3 Theorem 8.1.5	10.2210 10.2210   03.4691 m h   0 m	  
		Theorem 6.2.1		1.0821 03.1444			0 3.4924 4.5415
		Theorem 6.2.2	m   1.0821 03.1444 h 0 m   1.0761 02.6648			m   m   02.1131 2.2758 h 03.4924 4.5415 0
		0			1		2		3		4	5

: does not include.

  SYSTEMS WITH ACTUATORS SATURATIONproposed implementation methods for an LTI system.

	Example 6.2.3: Consider the following linear time-delay system (LTDS):
	11.50110 , , 0.32001 h AABu , and       	15.	(6.37)
	The stabilizations of LTDS (6.37) address by Theorem 6.2.1 and Theorem 6.2.2, respec-tively, for a delay bound of 1,0, h   and 0.1. The estimates of DoA are carried   out by the optimization Problem 6.2.1.
	The estimate domain of attraction solving by Theorem 6.2.1 (a stabilization of memor-yless controller) bounds by 225 12 11.31252.002210   (for the case of 12 )   and   86.6633 (for the case of 12 )   . The maximum radius of the stability ball stabilizing state-feedback controller obtains for all delays that are less than or equal to 1 h  corre-sponding to 0,   and 0.1.

Table 6 .

 6

	2. Domain of attraction, with	h 	1.
			Delay-Dependent Stabilization
	Example 6.2.3 Theorem 3 (Dey, 2014)	0.1 12     	0.0 12     106.2856	Number of Variables 2,1 nm  722 nnmm 	37	Parameters ,  11 10  
	Theorem 1 (Chen, 2017)		092.5966	2 Nnn 21 2   32 Nmm n 	35	9 0.4,0, 1 N   		3
	Theorem 6.2.1 Theorem 1 (Chen, 2015)	84.8793 086.6634 2331 nnmn  27  084.6074 5274 nnmn  82	10 1,10,1.15   9 0.4,10  
	(Gomes da Silva, 2011) 83.55 	2 22 3 11 nnmnm  2	22	
	Theorem 2 (Dey, 2014)		080.3239	722 nnm 	36	 	10	9
	Th1 (Fridman, 2003)		079.43	2 323 22 nnmn  2	33	
	0	1	2		3		4
	: does not include; Th1: Theorem 1.

  2.1, we have the following result.

	Problem 6.3.1. Given delay space the initial condition definitions 0 . ht Hcompact sets of parameters 0 , XFind the variable decision matrices 1 ,,, PQR , p t U and  %%% such that the following statement satisfies:
	Minimize	23 1234 10.5 fhhh  			(6.69)
	subject to (6.50), (6.51), 4222 123 , , PIQIRI  %%% ppp 212 2 0 2 0. nnn TT n II I          ±  for pre-selected scalar ,    R weighting scalar    R With 242 1 ,,, and . nnn QRP    %%% SSR	, .	and	(6.70)
	Condition (6.70) is interpreted similar to condition (6.31), with expansions:
	1212 T  ± infers 0,			(6.71)
	121 1121224 2, TTT I    °°			(6.72)

Table 6 . 3 .

 63 Multi-criteria optimization.

				Delay-Dependent Stabilization
	Example 6.3.1 Theorem 6.2.2	h 	 	1.0 0.5   opt  1 332.6817 	2.5	h 	 	1  0.4852 1.0 0.5   opt 	10	1 0.510 2.5      h 2.8908	h 	 	1      12 1.0 0.5   0.7740	40 2.5
	Theorem 6.3.1			41.0729				2.0741		1.0612			0.2985
	0			1				2		3			4
	Theorem 6.2.2: State Feedback Controller; Theorem 6.3.1: Dynamic Output Feedback Controller.



  The generalization of proposed methods have been developed for: systems with un-combination of p x slide on the plane px  (shows in Figure A-1.a) is given:

	112233 xx : p x x     44 ,, p x  x C xpx  and the set of parameters  fine combination of s With x in space 3 R (shows in Figure A-1.b) is obtained: 4 1 i ii     1, R	(A.2) then the af-
	certain parameters ˆ, t  systems with asymmetrical limits on derivative , t   & systems with delayed parameters . tht  Or consider an extension of the parameter-dependent decision matrices such that , QtRt   are other per-spectives that will be developed.

 Application for dynamic systems such as vehicle body stability system (ESC, ESP, ADS, etc.), remote robot control for slave -master system (e.g., for medical service), or other engineering systems. affine

  At the optimal solution, there is an estimate of the Lagrange multipliers for the equality constraints y  such that: Fxhhh  is the third differential of Fx  taken at x along the collection of direction 123 ,,. hhh  Using the concept of self-concordance, new barrier functions have been devised for certain convex programming problems, e.g., semidefinite programming.

	A.2. Linear Matrix Inequality		181
	of vector x. where 3	,,
	A.2.1.1 The Primal Barrier Method for Linear Programming
	Consider a standard-form linear program, then a barrier sub-problem associated with lin-ear problem (A.22) is A.2.1.2 The Primal-Dual Barrier Method for Linear Programming
	Minimize In the last decades of the last century, innumerable papers have been written about the 1 ln m T j j cxx   subject to A , xb  (A.25) interior revolution relates to the primal-dual family. Both methods are based on applying   with the assumptions (1) the set of x satisfies A , xb  and 0 x  Newton's method, but the Primal barrier algorithm is formulated in terms of only seeking primal variables x. Let's consider the optimal solution of the sub-problem (A.25) that satisfies (A.27) for some vector y. By defining a vector z such that f is nonempty, (2) given the set , yz  satisfies , T Ayzc  and 0 z  f is nonempty, and (3) rank . Am , . T cAyz Xz    1 (A.32)  Make the use of Lagrange multiplier estimate to seek dual variables y and z satisfying the following barrier trajectory
	11 TT gxcXAycXAy   , B  11 The barrier trajectory for standard-from LP (A.22) is defined by vectors . , for 0, , for 0, . kk T kkk kk Axbx Ayzcz Xz     1 f f	, kk xy 	(A.26) (A.33)
	1 kk kk , for . T Axbx XAyc     1 Starting from 0, x f using gradient and barrier Hessian give in (A.24), the Newton equa-0, f (A.27) tion of sub-problem (A.25) corresponding to Newton step p k is 1 2 1 . 0 T k kk kk k k p Xc XA y Axb A                     1 (A.28) , , , 00 0 . 0 xkk TT ykkk kkzkkkk ApbAx AIpcAyz ZXpXZ             (A.34)    11   using the similar transformation as (A.29)-(A.30) to eliminate p x,k and p z,k , we have  In which the Newton step p k satisfies 111 , . TT kkykkkkkkk AZXApAZXcAyXbAx 
	21 kkkkkk XpXAy T    1	1 , 	(A.29)
	for some Lagrange multiplier vector y k+1 . Using the relation plying the latter equation with 2 , AX we have k 2 1 , T kkkk AXAyAXXc    1	,0, kk AxbAp  by multi-(A.30)
	has a unique solution y k+1 since rank, Am  and step is defined in term of y k+1 as follows	2 T k AXA is positive definite. So, the new
	12 11 T kkkkk pxXAyc     .	(A.31)
	In the late 1980s, the analysis of polynomial-time complexity for an interior method
	should recognize the work of (Nesterov & Nemirovskii, 1994) with the contribution of
	the defining of a self-concordant barrier function. A convex function	:
	12 , , BB , gxcXHxX    1 1,,1, and T X  1 K means the diagonal matrix whose diagonal elements are those . (A.24) (2) for all where 323/2 ,: ,,2. nT xhFxhhhhFxh   R

The gradient of barrier function and barrier Hessian have simple forms:

nn Fx RR a is  self-concordant in  if (1) Fx  is three time continuously differentiable in , and

With the presence of nonlinear problem in the third equation. Following Newton's method, we obtain a linear system for Newton step p k in x, y, and z:   1

  and reference therein. The semidefinite programming problem could be expressed as follows:

	Minimize trace CX 		(A.36)
	subject to trace, ii AXbimX 1,,,  K with an appropriated real matrix , C a symmetric matrix 0, ±	, A  S and a vector n i	.

  that leads to the following results.

	Proposition B.2.1. Supposing M is bounded LTI operator mapping w  z and Ais Hur-witz matrix. Then the uncertainty structure , u M   is robustly well-connected, and the system (B.6) is asymptotically stable if one of the following conditions hold:
	§ there exists L § there exists matrix   D such that , n P   S   T T w T w H PAAPPB LML   2 1 L such that 0. w LHJ J L            p	1.	(B.9)

  The formulation (B.15) is dependent on the decision variable P and slack variables. It should remark that the equivalence of the two conditions (B.14) and (B.15) is still held in the case of taking into account the PDLF

	T ww T T PH IJHJ      P             Then, following the argument of S-variable LMI based on Finsler's lemma that yields:
	0,  UUU 123 TTTT    p TT with an injected matrix dimension. . are slack variable (SV) matrices of adequate (B.15) P  By applying Schur's com-plement for the latter inequality that entails in the LMI condition:
	111213 2223 33 TTTT 0 w TTT wwww T UAAUUBAUUAUPH IUBBUUBUJ UU I                	0. (B.16) p
	The stability condition (B.14) is satisfied if condition (B.16) is fulfilled. Furthermore, the
	matrices ,1,2,3
				2), take full advantage of null can-
	celation equation:
	 ABIxtwtxt   ker TTTT  w     & 144424443 144424443	0.	(B.13)
	Consider a quadratic Lyapunov function candidate
	kerker T Vxt  &	p	, 0		(B.14)

T VxtxtPxt for LPV system (B.2). Then, the derivation of this function along the trajectories of system dynamics combining with ℋ∞-norm can be represented by: i Ui  are slack variables in the above inequality, do not require definite positive symmetry like P. And, if 123 ,0,0, UPUU  then condition (B.16

  Let us now rewrite the full-block problem formulation with the S-variable approach by taking the derivative of the Lyapunov function candidate

	B.4. Pole-Placement LMI regions		191
		J	0    T T QS wsws ds zszs SR          	0.	(B.17)
	T VxtxtPxt  along the trajectories of system dynamics combining with IQC (B.17) as follows
	   kerker kerker  T TTT  Vxt   && J  ker, T TTT xtxtwt   &  0000 0, p 00000 00 00000 000 0000 000 0000 . 00 0000 00 00 T T T ww (B.18) T T ww QS II IPI HJHJ IPI SR P II P II QS II HJHJ SR                                 Similar to the previous argument, there exists an SV matrix such that where
	with	0,  AIBUUU  TT  p   123 , TTTT  w  	.	(B.19)
	Gives two signals mance specification from the channel , wz  if the following condition holds 2 , wz L satisfies the IQC perfor-

  ).(LMI regions -Chilali & Gahinet, 1996) A subset D of complex plane is called an LMI region if there exist a real symmetric matrix L, real matrix M such that Ais said D-stable when its spectrum :

	Definition B.4.1. : zfzzMzM   D 0, DL C where fz  D is characteristic function of D. Given below the examples of LMI regions (B.20) 1 -Half-plane Re:20. zfzzz    D 2 -Disk centered at ,0 q   with radius :0, rqz rfz qzr       D p   and some dynamic e.g., oscillations Re, ; zzr    bandwidth 12 Re; z    horizontal strip Im; z    and damping cone RetanIm. zz   
	Example B.4.1. Giving a dynamic system:
	. xtAxt   &	(B.21)
	Then, matrix Ai A    belongs to region D. Let's consider a Lyapunov characterization of LMI stability of system (B.21):
	T PAAP 	p	0,	(B.22)
	Then, a disk of radiusr and center if there exists a symmetric matrix P such that: ,0 q   is an LMI region with characteristic function f D 0, with 0. T rPqPA fPAP qAPr       D pf (B.23)   Pole clustering in LMI regions can formulate as a more general region, e.g., ,,. r   S
	Let's analysis the ℋ∞-performance LMI constraint (B.5) belong to this disk region. Spe-cifically, system (B.2) is quadratically D-stable in LMI region disk of center ,0 q   and radius , r if there exists a symmetric matrix P such that condition (B.24) holds, . 0 0 0 w T T dw r rPqPPAPB rPH IJ I                p (B.24)
	where:	L	{ 1 011 000 T M   , and rq M qr     	 {  2 01. M

  B.6. Generalized ℋ2 Performance 193 peak amplitude of the output below a certain level. The ℋ2-norm of a system measures the output energy in the impulse responses of the system. Suppose LPV system (B.2) with

	0   is asymptotically stable, if and only if exists a performance index D  symmetric positive matrix , nr PQ   SS such that the following LMIs:	0,   and a
	2 0,0,trace. TT APPAPBPC Q IQ           f p	(B.25)

  Let I be an interval in R . For ,,

	abI then an abif the following inequality IR is said to be J-convex on  integrable function :,  a , , bbbb aaaa fddfdd         (C.4)   is valid for all integrable function . fdom  
	By choosing	,, fxfxQfxfxx T   & with	, and	,

nn xQ   IS

1.2 Discretized Convex Function We

  For an auxiliary scalar function :,, use an example to show the effective reduction of the conservation of inequality by the discretizing method of the n-convex function.  ¡ is J-convex functions satisfying the definitions given in Appendix A.. The gap in Jensen's inequality is characterized by a positive difference between integrals 0 ,1,,.

	C.2. Model Transformation			197
	And,
	2 22 2 3322 2 202 2 3 . 22 3 22 565. 33 2 . 3164812 ab ab b b a a II ccc xx x baba cccbacba babaaabb                                   2
	Similarly, by dividing interval   , ab into 3 equal segments, we have integral
	 12   111      33 222 12 33 22 333 ababb aabab Icxdxcxdxcxdx 3 bababa            	2	,
	and 3 th -order integral inequality gap
	12 33 12 33 22 2 222 33222422 303 3 3 . 333 . 3 222 741407423 343 ababb b a aabab II cccc xxx x bababa cc baabaabbab ba                             
	n (C.11) and any continuous inte-xab   . wwddspdds  0, aaij bb dds   a positive matrix n R   R grable function satisfies :, , n wxab  1 1 , bb TT ii as i i wRwddswRw p  holds with 2 , iiii asas   bbbb       R the following equality  R 33 2 . 10812 cbacba     3
	a. The chord of the parabola	b. The integral region of the	c. The integral domain
	expresses convex inequality.	parabola is in the interval.	fragmentation.
	Example C.1.1: By considering the following integrals:
	  111 2   1 222 1 2 2 22 012 2 122 ,,and . bbabb aaaab IcxdxIcxdxIcxdxcxdx bababa              where 2 , fxcxc
	ii IIiN  K 33 2 . 341212 analyze the first three orders 1,2,3, as follows: i  01 2 2 3 3333 3 2 b b a a II cc x x ba cccbacba babaabba    1                 1	Let's

C.



  The bounding of this cross terms eliminates the quadratic integral terms in the derivative of the Lyapunov-Krasovskii functional (the one related to the derivative of the double integral 2Vt  &

	The Lyapunov-Krasovskii functional is employed for this class of system:
	  , VxQxdVt xt { 1 2 23 0 0 00 T t T tht E P xtxt I PP ytyt           .         14243 P After derivative of LKF equation (C.22) similar to the previous methods, we obtained the (C.22)
	following cross terms:
	0 2. T t T th h xt Pyd A yt              	(C.23)
	t hh th xtytytAAxtAyd     &	(C.21)

has attracted significant attentions in last decade:

,.

  along with a specified LK function to be able to capture both the maximal nominal delay value max, hht and the maximal resilient delay max. htdt   This delay-dependent approach is based on Lyapunov-Krasovskii

	technique with a resilient uncertain By consider an additional Lyapunov-Krasovskii functions as follow:   , t   is introduced by relation	.  dthtt 
		 unumu tt TT tdttht VtxQxdxRxdd       &&	.	(C.39)
	with	max,||.

m

  The derivative of this LK function results to the following integral inequalities with approximately limited integral range:

&

  As latter discussion, from the above inequality we have the two possibilities: at time , .4. The proof of Lemma 5.2.7.209(D.10) with (D.11) will gather into the following form.By using the Schur complement that leads to the condition (5.54).

	1 01111111 00 1 00 r TTTTT u I VVzzwwR R            &&    where time expression t is dropped 1 , T TTT th xxw     , hdw hdw mmhdmw HHHJ hAhAAhB AAAB               1 1 , ,1. (D.12) 0 p N T iu i i T dhu T d P PAAPQQR AAPRQQR EPI                    
	1 0000 1 0 0 TTTTT r I Vtztztwtwttttt R        ,  &  where 0 , , hdw T TTTT thd hdw HHHJ t xxxwt hAhAhAhB         and    0 || ,. 1 00 00 T i i i T h T d T wd P PAAPQR APRQR AP BPI                       &  &	(D.10)
	On the other hand, applying the analysis in (C.40) we have
	 11   2 TTTT  tht tutdudmttmu tht VtxQxdtxQxxRxhtxsRxsds       &&& &&&&   2 11. TTTT tutdtudmtutdhudh xQxxQxxRxxxRxx   && (D.11)
	the delay approximatesdt is close to so T dhdh xxRxx  approaching 0. And, the second case is . ht According to analysis in (D.3) when .	t  t	i 0

jj dtht  Case 1: (Time uniformity) , ii dtht then the derivative LKF conditions associate D

Table E . 1 .

 E1 The Parameter of Membership Functions. The coefficient of fuzzy set in region M1 2.389 × 10 3C fz2The coefficient of fuzzy set in region M2 0.8149Assumption 1: By sharing the same fuzzy structure and same nonlinear function characteristics, so the proposed rules could determine by only  

	Parameters	Description	Values
	a 1 , a 2 b 1 , b 2 c 1 , c 2 C fz1	The width of MFs (Standard deviation) The center of MFs (mean)	0.0021, 0.3078 0.7219, 5.3137 -0.1656, 0.3155
				1 , j h	j = 1, 2.

Table E . 2 .

 E2 The Nominal Parameter Value of Tank Process.

	Parameters	Description	Values
	A s1 , A s3 A s2 , A s4 a s1 , a s3	Areas of tanks 1,3 (m 2 ) Areas of tanks 2,4 (m 2 )	2.8 × 10 -3 3.2 × 10 -3

Table E . 3 .

 E3 Coefficient of Pacejka tire model: 265/75 R16.

	Tire characteristics	Parameter of the Membership Function
	Front tires	Rear tires	Front tires	Rear tires
	B1 = 9.2916 B2	= 9.3449	Cf1 = 1.0748e5 Cr1 = 8.2913e4
	C1 = 1.6307 C2 = 1.6307	Cf2 = 534.98	Cr2 = 407.68
	D1 = 4339.4 D2 = 3393.2	a1 = 0.0801	a2 = 112.64
	E 1 = 0.2290 E2	= 0.2292	b1 = 0.7309	b2 = 676.78
				c1 = 0.0202	c2 = 112.61

Table E .

 E 4. Simulation vehicle parameters -SUV E-Class car model.

  When simplifying the model of system based on two vertices of the front wheel slip angle that consider only

												1 )	40
		Iz	Yaw moment of inertia at center of gravity (CG) (kg m 2 ) 2988
		l r	Distance from CoG to rear axle (m)	1.77
		lf	Distance from CoG to front axle (m)	1.18
		Mz	Driving/Braking force control (Nm)
		δf	Front steering angle (rad)
		δsw	Steering wheel angle (deg)
		iS	Steering ration							20.0312
	The local time-varying matrices	i A	 R	22 , 	B	i	 R	22 , 	and	C	12   R are given as:
	i A	22 01 22 12 xx zxzx iix mvmv ii IvIv CCvt CC         	,	i B	, , 21 2 0 , zz fj fjf m II C Cl     	B 	, i	, fj , fjf C Cl 2 2 z m I     	and	d B	1 0 z 1 0 . m I     
	where 0 CCC  ifjrk if , jk  then 1,4 i  and if jk  then , 1 ifjfrkr CClCl  i  , and 2,3, CClCl  22 2 ifjfrkr with ,1,2, and , jki  and i follows the rule: 1,2,3,4.
	It should be noted that the combination of membership function (E.7) is:
		ˆ, ijfkr   	with ,1,2, and jki 	1,2,3,4.	(E.9)

ˆ:, iifir   then 1,2. i 

& & & & &

&, then parametric matrix inequality (2.57) are multi-affine in 0,1,1,2,,. il tiN   Actually, there is no method to adequately describe the

CHAPTER 2. OVERVIEW LINEAR PARAMETER-VARYING SYSTEMS

& &

& p(2.61)The stability condition is presented as a matrix polynomial inequality that can be casted in terms of convex problems LMIs by the gridding method (meshing affine) or Sum of Square decomposition technique.

& &(2.68)

The polynomial condition is executed on Matlab v2020b, using the SoS toolbox v4.00 with Interior-point solvers, Sedumi v1.03 and Mosek v9.3 respectively.

The Degree of Freedom is denoted as DoF throughout this dissertation.

& & & &

& & %%% & %%%%%

Remerciements

Acknowledgments

I would like to express my sincere gratitude to the anonymous reviewers for taking the time to review and provide comments. Your criticism and suggestions help me significantly progress the results and improve the writing style of the PhD dissertation.

Acknowledgments iv

The work resulting in this thesis has been funded by the Ministry of Education and Training of Vietnam for three years (2016-2019) under grant project 911, along with the tuition fee and insurance policy supported by the Ministry of Foreign Affairs of the French Republic for five years is gratefully acknowledged.

Viet Long. v

Contributions

Appendix A Linear algebra

Linear algebra

This section is not intended to provide complete definitions of matrix algebra, ring fields, matrix determinants, eigenvalues, and etc. Instead, it is mainly concerned with the essential algebraic mathematical techniques relating to system and control theory. Specifically, the postulates and properties of convex functions, integrable functions, and convex optimization involved in the analysis of system and control expressed by linear/bilinear matrix inequality conditions.

A.1 Affine space

The rudimental concepts (i.e. convex sets, convex functions, and linear combinations, etc.) involved in this work could be illustrated by n-dimensional vector space in the following figure. in n-dimensional coordinate space is expressed linearly independent based on the basis set and the corresponding coordinate parameters

It is conceivable that the coordinates of a point belonging to subspace n   R can be expressed by linear expressions (convex or quasi-convex). Now, let's review some basis of the linear formulations. From this point, it can be seen that the similarity is quite clear between last equations and the one in linear programming problem (A.33).

A.1.1 Affine sets

A.3 Schur Complement

The Schur complement is a fundamental and core mathematical tool used in matrix analysis in the field of theoretical control systems. In the context of LMIs formulation, the conditions for positive definiteness and semi-definiteness that can be expressed by: Lemma A.3.1. The following statements are equivalent:

(1) Let a real symmetric 0.

From this view, it could be realized that the nonlinear matrix inequalities in statements 2 and 3 also deliver convex problems in the form of affine LMI (statement 1).

A.4 Young's inequality

Let's recall Young's inequality (Mitrinović et al., 1993b) and its matricial generalization [START_REF] Ando | Matrix Young Inequalities[END_REF] for further discussion about its application for the LMI analysis.

Lemma A.4.1. (Mitrinović et al., 1993b) Let a continuous function :0, f  a¡ is increasing function defined for nonnegative real numbers , x with initial condition 00. f  Give , abare positive real numbers such thata is in the domain of f and b is in the image of f . Then

W

Actually, these conditions are the matricial generalization of Young's inequality [START_REF] Ando | Matrix Young Inequalities[END_REF]. These inequalities are well-known in the control system theory, while condition (A.42) is regularly used to eliminate uncertainty matrices, condition (A.43) is typically encountered in general in output feedback control design (SOF and OBF, see for example [START_REF] Benzaouia | Advanced Takagi-Sugeno Fuzzy Systems: Delay and Saturation[END_REF][START_REF] He | Output Feedback Stabilization for a Discrete-Time System With a Time-Varying Delay[END_REF][START_REF] Leibfritz | An LMI-Based Algorithm for Designing Suboptimal Static $\cal H_2/\cal H_\infty$ Output Feedback Controllers[END_REF][START_REF] Peng | Delay-range-dependent robust stabilization for uncertain T-S fuzzy control systems with interval time-varying delays[END_REF] This lemma is usually used in signal and system control analysis (common for both time and frequency domains) based on the LMI technique. In this thesis, we could only utilize a small application for dealing with the nonlinear structures in controller synthesis for LPV saturated systems.

Appendix B Robust Stability and Performances Analysis via LMIs

Robust Stability and Performances Analysis via LMIs

In this section, the design specifications and requirements analyze on the state-space using the Lyapunov stability via linear matrix inequality formulation. The performance and robustness analysis for the LTI system systematically introduced by [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] deploys on the concept of the input-output properties. The terms such as L2-norm, L2-gain and ℋ∞-norm have been widespread in the stability and performance evaluation consistent with the (Scaled) Small-Gain Theorem [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF][START_REF] Doyle | Structured uncertainty in control system design[END_REF][START_REF] Zhou | Robust and Optimal Control[END_REF], the (Scaled) Bounded Real Lemma (Apkarian & Gahinet, 1995;Gahinet & Apkarian, 1994;[START_REF] Scherer | The Riccati inequality and state-space H∞-optimal control[END_REF], the full-block S-procedure [START_REF] Scherer | LPV control and full block multipliers[END_REF][START_REF] Scherer | A full block S-procedure with applications[END_REF][START_REF] Scherer | Linear matrix inequalities in control[END_REF], or the Integral Quadratic Constraints (IQC) [START_REF] Jönsson | Stability analysis with Popov multipliers and integral quadratic constraints[END_REF][START_REF] Megretski | System analysis via integral quadratic constraints[END_REF]. This issue is also considered for the LPV delay systems in [START_REF] Briat | Commande et Observation Robuste des Systemes LPV Retardés[END_REF](Briat, , 2015a)), and the TS-fuzzy systems in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. We briefly recall the general definitions for control system design criteria discussed in the work of [START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF] and references therein.

The approximate deviations, and the uncertain knowledge about the parameters in the modeling are often referred to as the parametric uncertainty. Whereas the nonlinearities components are considered as dynamical uncertainty. The distinction demarcates between robust control theory and the LPV gain-scheduling technique (typically related to measurable parameters). Since the scheduling controller design are introduced by [START_REF] Shamma | Analysis and design of gain scheduled control systems[END_REF][START_REF] Shamma | Analysis of gain scheduled control for nonlinear plants[END_REF]) that shown an efficient way to analyze and synthesize for this class of system. In recent decades, applications of LPV gain-scheduling controllers pervades in a wide range of system engineering from aeronautical engineering (aircraft, missiles, helicopters, AUVs...) to road traffic engineering (cars, trucks, trains), robotics, energy engineering (renewable energy systems), etc. It's also easy to find the applications in the same areas are addressed by the robust stability theory. We would discuss here the alternative improvement for the tighter bounding inequalities relate to the LKF stability conditions. Some definitions of convex domain properties refer to Appendix A.1.3.

Appendix D Demonstration

Demonstration

D.1 The proof of Lemma 5.2.1.

The LPV system (5.1) is delay-dependent stable if the conditions:

,, 0 01

Abstract

The dissertation is devoted to developing a methodology of stability and stabilization for the linear parameter-dependent (PD) and time-delay systems (TDSs) subject to control saturation. In the industrial process, control signal magnitude is usually bounded by the safety constraints, the physical cycle limits, and so on. For this reason, a suitable synthesis and analysis tool is needed to accurately describe the characteristics of the saturated linear parameter-varying (LPV) systems.

In the part one, a parameter-dependent form of the generalized sector condition (GSC) is considered to solve the saturated stabilization problem. Several feedback control strategies are investigated to stabilize the saturated LPV/qLPV systems. Necessary and sufficient stabilization conditions via the parameterized linear matrix inequality (PLMI) formulation proposed for the feedback controllers conforming to the design requirements (i.e., the admissible set of the initial conditions, the estimated region of the asymptotic convergence domain, the robust stability and performance with the influence of perturbations, Etc.). The relaxation of the designed PLMIs is shown through the comparison results using a parameter-dependent Lyapunov function (PDLF).

In the second part, the delay-dependent stability developments based on Lyapunov-Krasovskii functional (LKF) are presented. The modern advanced bounding techniques are utilized with a balance between conservatism and computational complexity. Then, saturation stabilization analyzes for the gain-scheduling controllers are proposed.

Inspired by uncertain delay system methods, a novel stabilization condition is derived from the delay-dependent stabilizing analysis for the LPV time-delay system subject to saturation constraints. In this aspect, the stabilizing gainscheduling feedback controllers improve the performance and stability of the saturated system and provide a large attraction domain. It can be emphasized that the derived formulation is general and can be used for the design control of many dynamic systems. Finally, to maximize the attraction region while guaranteeing the asymptotic stability of the closed-loop system, an optimization problem is included to the proposed control design strategy.

Key-Words: LPV/quasi-LPV Systems, Actuators Saturation, Time-Delay Systems, Robust Control, Parametrized LMIs.

Résumé

La thèse est consacrée au développement d'une méthodologie de stabilité et de stabilisation pour les systèmes linéaires à paramètres variables (LPV) et à retard soumis à la saturation de la commande. Dans les procédés industriel, l'amplitude du signal de commande est généralement limitée par les contraintes de sécurité, etc. Une synthèse de commande est donc nécessaire pour les systèmes saturés à paramètres linéaires variables.

Dans la première partie, une nouvelle expression de la condition de secteur généralisée (GSC) est considérée pour résoudre le problème de stabilisation saturée. Plusieurs stratégies de contrôle sont étudiées pour stabiliser les systèmes LPV/quasi-LPV saturés. Des conditions de stabilisation nécessaires et suffisantes via la résolution des inégalités matricielles linéaires paramétrées sont proposées pour les contrôleurs par retour d'état respectant les exigences de conception (c'est-à-dire l'ensemble admissible des conditions initiales, la région estimée du domaine de convergence asymptotique, la stabilité et les performances robustes sous l'influence des perturbations, etc.). La relaxation des conditions LMI paramétrées est illustrée par des résultats proposant une fonction de Lyapunov dépendant des paramètres.

Dans la deuxième partie, les conditions de stabilité dépendant du retard basé sur la fonction de Lyapunov-Krasovskii (LKF) sont présentées. Les techniques modernes et avancées de délimitation sont utilisées avec un compromis entre conservatisme et complexité de calcul. Ensuite, des conditions de stabilisation avec saturation sont proposés pour les contrôleurs à gain préprogrammé. Inspirée des méthodes de systèmes à retard incertains, une nouvelle condition de stabilisation est dérivée pour le systèmes à retard LPV soumis à des contraintes de saturation. Les contrôleurs de rétroaction à gain préprogrammé améliorent les performances et la stabilité du système saturé et fournissent un grand domaine d'attraction. Nous pouvons souligner que la formulation dérivée des conditions est générale et peut être utilisée pour le contrôle de nombreux systèmes dynamiques. Enfin, pour maximiser la région d'attraction tout en garantissant la stabilité asymptotique du système en boucle fermée, un problème d'optimisation est inclus dans la stratégie de conception de commande proposée.

Mots-clés : Systèmes LPV/quasi-LPV, Saturation des Actionneurs, Systèmes à Retard, Commande Robuste, LMI Paramétrés.