Nouveaux schémas de commande et d'observation basés sur les modèles de Takagi-Sugeno

This thesis addresses the estimation and controller design for continuous-time nonlinear systems. The methodologies developed are based on the Takagi-Sugeno (TS) representation of the nonlinear model via the sector nonlinearity approach. All strategies intend to get more relaxed conditions.The results presented for controller design are split in two parts. The first part is about standard TS models under control schemes based on: 1) a quadratic Lyapunov function (QLF); 2) a fuzzy Lyapunov function (FLF); 3) a line-integral Lyapunov functions (LILF); 4) a novel non-quadratic Lyapunov functional (NQLF). The second part concerns to TS descriptor models. Two strategies are proposed: 1) within the quadratic framework, conditions based on a general control law and some matrix transformations; 2) an extension to the nonquadratic approach based on a line-integral Lyapunov function (LILF) using non-PDC control law schemes and the Finsler’s Lemma; this strategy offers parameter-dependent linear matrix inequality (LMI) conditions instead of bilinear matrix inequality (BMI) constraints for second-order systems. On the other hand, the problem of the state estimation for nonlinear systems via TS models is also addressed considering: a) the particular case where premise vectors are based on measured variables and b) the general case where premise vectors can be based on unmeasured variables. Several examples have been included to illustrate the applicability of the obtained results.Cette thèse aborde l'estimation et la conception de commande de systèmes non linéaires à temps continu. Les méthodologies développées sont basées sur la représentation Takagi-Sugeno (TS) du modèle non linéaire par l'approche du secteur non-linéarité. Toutes les stratégies ont l'intention d'obtenir des conditions plus détendu. Les résultats présentés pour la conception de commande sont divisés en deux parties. La première partie est environ sur les modèles TS standard au titre des schémas de commande basés sur: 1) une fonction de Lyapunov quadratique (QLF); 2) une fonction de Lyapunov floue (FLF); 3) une fonction de Lyapunov intégrale de ligne (LILF); 4) un nouveau fonctionnelle de Lyapunov non-quadratique (NQLF). La deuxième partie concerne des modèles TS descripteurs. Deux stratégies sont proposées: 1) dans le cadre quadratique, des conditions basées sur une loi de commande général et quelques transformations de matrices; 2) une extension de l'approche non quadratique basée sur LILF utilisant un schéma de commande non-PDC et le lemme du Finsler; cette stratégie offre conditions sur la forme d’inégalité matricielles linéaires (LMI) dépendant des paramètres au lieu des contraintes sur la forme d’inégalité matricielles bilinéaires (BMI) pour les systèmes de second ordre. D'autre part, le problème de l'estimation de l'état pour les systèmes non linéaires via modèles TS est également abordé considérant: a) le cas particulier où les vecteurs prémisses sont basées sur les variables mesurées et b) le cas général où les vecteurs prémisse peuvent être basés sur des variables non mesurées. Plusieurs exemples ont été inclus pour illustrer l'applicabilité des résultats obtenus

New control and observation schemes based on Takagi-Sugeno models

Cette thèse aborde l'estimation et la conception de commande de systèmes non linéaires à temps continu. Les méthodologies développées sont basées sur la représentation Takagi-Sugeno (TS) du modèle non linéaire par l'approche du secteur non-linéarité. Toutes les stratégies ont l'intention d'obtenir des conditions plus détendu. Les résultats présentés pour la conception de commande sont divisés en deux parties. La première partie est environ sur les modèles TS standard au titre des schémas de commande basés sur: 1) une fonction de Lyapunov quadratique (QLF); 2) une fonction de Lyapunov floue (FLF); 3) une fonction de Lyapunov intégrale de ligne (LILF); 4) un nouveau fonctionnelle de Lyapunov non-quadratique (NQLF). La deuxième partie concerne des modèles TS descripteurs. Deux stratégies sont proposées: 1) dans le cadre quadratique, des conditions basées sur une loi de commande général et quelques transformations de matrices; 2) une extension de l'approche non quadratique basée sur LILF utilisant un schéma de commande non-PDC et le lemme du Finsler; cette stratégie offre conditions sur la forme d’inégalité matricielles linéaires (LMI) dépendant des paramètres au lieu des contraintes sur la forme d’inégalité matricielles bilinéaires (BMI) pour les systèmes de second ordre. D'autre part, le problème de l'estimation de l'état pour les systèmes non linéaires via modèles TS est également abordé considérant: a) le cas particulier où les vecteurs prémisses sont basées sur les variables mesurées et b) le cas général où les vecteurs prémisse peuvent être basés sur des variables non mesurées. Plusieurs exemples ont été inclus pour illustrer l'applicabilité des résultats obtenus.This thesis addresses the estimation and controller design for continuous-time nonlinear systems. The methodologies developed are based on the Takagi-Sugeno (TS) representation of the nonlinear model via the sector nonlinearity approach. All strategies intend to get more relaxed conditions.The results presented for controller design are split in two parts. The first part is about standard TS models under control schemes based on: 1) a quadratic Lyapunov function (QLF); 2) a fuzzy Lyapunov function (FLF); 3) a line-integral Lyapunov functions (LILF); 4) a novel non-quadratic Lyapunov functional (NQLF). The second part concerns to TS descriptor models. Two strategies are proposed: 1) within the quadratic framework, conditions based on a general control law and some matrix transformations; 2) an extension to the nonquadratic approach based on a line-integral Lyapunov function (LILF) using non-PDC control law schemes and the Finsler’s Lemma; this strategy offers parameter-dependent linear matrix inequality (LMI) conditions instead of bilinear matrix inequality (BMI) constraints for second-order systems. On the other hand, the problem of the state estimation for nonlinear systems via TS models is also addressed considering: a) the particular case where premise vectors are based on measured variables and b) the general case where premise vectors can be based on unmeasured variables. Several examples have been included to illustrate the applicability of the obtained results

Non-quadratic stabilization of second order continuous Takagi-Sugeno descriptor systems via line-integral Lyapunov function

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Decoupled Nested LMI conditions for Takagi-Sugeno Observer Design

International audienceThis work extends recent investigations on control design of continuous-time nonlinear models to non-quadratic observer design. The models under consideration are exactly rewritten in the Takagi-Sugeno form. By means of the Finsler's Lemma or a Tustin-like transformation, the progressively complex observer gains and the Lyapunov function are decoupled, thus providing the flexibility of using a quadratic Lyapunov functions while preserving the non-quadratic nature of the observer. Conditions obtained are expressed as linear matrix inequalities which are efficiently solved by convex optimization techniques. Examples are provided to show the effectiveness of the proposed approach

Improvements on Non-quadratic Stabilization of Takagi-Sugeno Models via Line-Integral Lyapunov Functions

International audienceThis paper introduces two different generalizations on non-quadratic stabilization of Takagi-Sugeno models via line-integral Lyapunov functions. Inspired in recent developments which have proved to relax the conservativeness in the quadratic framework via suitable matrix transformations using a more general class of Lyapunov functions, the results here provided outperform the existing ones for line-integral Lyapunov functions with a linear matrix inequalities (LMI) form instead of bilinear matrix inequalities (BMI) in second order systems. Examples are provided to illustrate the advantages of the proposed technique over the existing approach

Improvements on non-PDC controller design for Takagi-Sugeno models

International audienceThis paper presents a novel approach for controller design of Takagi-Sugeno models based on decoupling a quadratic Lyapunov function from a progressively more complex non-parallel distributed compensation control law. Inspired in recent results from the linear parameter varying literature, which provide a way to find parameter-dependent linear matrix inequalities, the proposed generalization includes former PDC quadratic cases as well as some recent approaches. Examples are provided to illustrate the aforementioned contribution

Eliminating the parameter-dependence of recent LMI results on controller design of descriptor systems

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<inline-formula> <tex-math notation="LaTeX">H∞H\infty</tex-math> </inline-formula> LMI-Based Observer Design for Nonlinear Systems via Takagi–Sugeno Models With Unmeasured Premise Variables

International audienceThis paper is concerned with state estimation for continuous-time nonlinear systems. By means of a polytopic exact representation of the system using interpolation functions whose premise variables can be unmeasured, the differential mean value theorem and a quadratic Lyapunov function are used to provide conditions in the form of linear matrix inequalities, guaranteeing the state estimation error to be asymptotically driven to zero. A robust extension of the proposed observer design performing H∞ disturbance rejection is also presented. Examples illustrating the effectiveness of the developed novelties against former results are include

Asymptotically necessary and sufficient conditions for Takagi–Sugeno models using generalized non-quadratic parameter-dependent controller design

International audienceResults in this paper generalize recent approaches on non-quadratic controller design for Takagi–Sugeno models based on a matrix transformation. Conditions thus obtained depend on a single parameter which allows an efficient search of feasible solutions via linear matrix inequalities; moreover, they include and outperform former advances, such as Polya's approaches on the same framework. Some examples are included to illustrate the proposed generalization

A non-quadratic Lyapunov functional for H∞ control of nonlinear systems via Takagi–Sugeno models

International audienceThis paper presents a novel non-quadratic Lyapunov functional with time delay for controller design and H∞ disturbance rejection for nonlinear systems. Based on an exact Takagi–Sugeno representation of the nonlinear model, a proper integral structure for Lyapunov-based design allows circumventing the well-known problem of handling the time derivatives of the membership functions without imposing additional requirements. The methodology proposed that both increase the complexity of the classical quadratic LMI design and the control law structure, but in a reasonable manner. The main advantages are that the conditions in the proposed approach are expressed as linear matrix inequalities and proven to outperform previous results on the same subject. Illustrative examples are included to point out the aforementioned contribution