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International audience; The authors present a novel nonlinear Thau-Luenberger observer for estimating Structure from Motion using a calibrated camera. Accurate reconstruction of the 3D structure of a scene relies on precise estimation of the camera’s translational and angular velocities, which can be challenging for cameras on mobile platforms. The proposed observer aims to estimate these velocities robustly in the presence of measurement noise, with stability characterized through Input to State Stability analysis and Lyapunov theory. The stability conditions are determined using optimization techniques based on Linear Matrix Inequalities. The performance of the proposed approach is validated through simulation and experimental data, demonstrating its effectiveness in recovering the depth of tracked features and its robustness against disturbances.

This paper presents novel stabilizability conditions for switched linear systems with arbitrary and uncontrollable underlying switching signals. We distinguish and study two particular settings: i) the \emph{robust} case, in which the active mode is completely unknown and unobservable, and ii) the \emph{mode-dependent} case, in which the controller depends on the current active switching mode. The technical developments are based on graph-theory tools, relying in particular on the path-complete Lyapunov functions framework. The main idea is to use directed and labeled graphs to encode Lyapunov inequalities to design robust and mode-dependent piecewise linear state-feedback controllers. This results in novel and flexible conditions, with the particular feature of being in the form of linear matrix inequalities (LMIs). Our technique thus provides a first controller-design strategy allowing piecewise linear feedback maps and piecewise quadratic (control) Lyapunov functions by means of semidefinite programming. Numerical examples illustrate the application of the proposed techniques, the relations between the graph order, the robustness, and the performance of the closed loop.

International audience; This paper addresses the problem of eventtriggered control (ETC) of discrete-time linear time-invariant (LTI) systems stabilized by neural network controllers. The event-triggering mechanisms (ETMs) is proposed to update only a portion of the layers required to maintain stability and satisfactory performance of the feedback system, thus reducing the computational cost associated with the evaluation of the neural network. Sufficient convex conditions in the form of linear matrix inequalities (LMIs) are provided to compute the triggering parameters and characterize an estimate of the domain of attraction for the feedback system. The formulation is based on the use of Lyapunov theory and a set of generalized sector constraints that deal with the nonlinear activation functions, represented in this case by the saturation function. Optimization procedures are also formulated to effectively reduce the amount of computation in the neural network. An example borrowed from the literature is used to illustrate the effectiveness of the proposal.

International audience; This paper investigates the problems of stability analysis and control design for continuoustime Lur'e systems with slope bounded nonlinearities. Starting from a stability analysis condition from the literature, based on the real positivity and a bound to the L 1 norm of a certain transfer function, new sufficient conditions are proposed in an augmented parameter space for the simultaneous existence of a stabilizing dynamic output-feedback controller and a Zames-Falb multiplier certifying the closed-loop stability. The matrices of the controller realization as well as of the Zames-Falb multiplier appear affinely in the conditions, being dealt with as optimization variables. Furthermore, no line search is required to enforce the L 1 norm constraint. An iterative algorithm is constructed to solve the problem through semidefinite programming, providing dynamic controllers of any given order. The controller can take into account both the output of the linear part of the system and of the nonlinearity. Numerical examples illustrate the results.

Infinite dimensional systems allow to model a large panel of physical phenomena for which the state variables evolve both temporally and spatially. This manuscript deals with the evaluation of the stability of their equilibrium point. Two case studies are treated in particular: the stability analysis of ODE-transport, and ODE-reaction-diffusion interconnected systems. Theoretical tools exist for the stability analysis of these infinite-dimensional linear systems and are based on an operator algebra rather than a matrix algebra. However, these existence results raise a problem of numerical constructibility. During implementation, an approximation is performed and the results are conservative. The design of numerical tools leading to stability guarantees for which the degree of conservatism is evaluated and controlled is then a major issue. How can we develop reliable numerical criteria to rule on the stability or instability of infinite-dimensional linear systems? In order to answer this question, one proposes here a new generic method, which is decomposed in two steps. First, from the perspective of Legendre polynomials approximation, augmented models are built and split the original system into two blocks: on the one hand, a finite-dimensional approximated system is isolated, on the other hand, the infinite-dimensional truncation error is preserved and modeled. Then, frequency and time tools of finite dimension are deployed in order to propose stability criteria that have high or low numerical load depending on the approximated order. In frequencies, with the aid of the small gain theorem, sufficient stability conditions are obtained. In temporal, with the aid of the Lyapunov theorem, an under estimate of the stability regions is proposed as a linear matrix inequality and an over estimate as a positivity test. Our two case studies have been treated with this general methodology. The main result concerns the case of ODE-transport interconnected systems, for which the approximation and stability analysis using Legendre polynomials leads to exponentially fast converging estimates of stability regions. The method developed in this manuscript can be adapted to other types of approximations and exported to other infinite-dimensional linear systems. Thus, this work opens the way to obtain necessary and sufficient finite-dimensional conditions of stability for infinite-dimensional systems., Les systèmes de dimension infinie permettent de modéliser un large spectre de phénomènes physiques pour lesquels les variables d'états évoluent temporellement et spatialement. Ce manuscrit s'intéresse à l'évaluation de la stabilité de leur point d'équilibre. Deux études de cas seront en particulier traitées : l'analyse de stabilité des systèmes interconnectés à une équation de transport, et à une équation de réaction-diffusion. Des outils théoriques existent pour l'analyse de stabilité de ces systèmes linéaires de dimension infinie et s'appuient sur une algèbre d'opérateurs plutôt que matricielle. Cependant, ces résultats d'existence soulèvent un problème de constructibilité numérique. Lors de l'implémentation, une approximation est réalisée et les résultats sont conservatifs. La conception d'outils numériques menant à des garanties de stabilité pour lesquelles le degré de conservatisme est évalué et maîtrisé est alors un enjeu majeur. Comment développer des critères numériques fiables permettant de statuer sur la stabilité ou l'instabilité des systèmes linéaires de dimension infinie ? Afin de répondre à cette question, nous proposons ici une nouvelle méthode générique qui se décompose en deux temps. D'abord, sous l'angle de l'approximation sur les polynômes de Legendre, des modèles augmentés sont construits et découpent le système original en deux blocs : d'une part, un système de dimension finie approximant est isolé, d'autre part, l'erreur de troncature de dimension infinie est conservée et modélisée. Ensuite, des outils fréquentiels et temporels de dimension finie sont déployés afin de proposer des critères de stabilité plus ou moins coûteux numériquement en fonction de l'ordre d'approximation choisi. En fréquentiel, à l'aide du théorème du petit gain, des conditions suffisantes de stabilité sont obtenues. En temporel, à l'aide du théorème de Lyapunov, une sous-estimation des régions de stabilité est proposée sous forme d'inégalité matricielle linéaire et une sur-estimation sous forme de test de positivité. Nos deux études de cas ont ainsi été traitées à l'aide de cette méthodologie générale. Le principal résultat obtenu concerne le cas des systèmes EDO-transport interconnectés, pour lequel l'approximation et l'analyse de stabilité à l'aide des polynômes de Legendre mène à des estimations des régions de stabilité qui convergent exponentiellement vite. La méthode développée dans ce manuscrit peut être adaptée à d'autres types d'approximations et exportée à d'autres systèmes linéaires de dimension infinie. Ce travail ouvre ainsi la voie à l'obtention de conditions nécessaires et suffisantes de stabilité de dimension finie pour les systèmes de dimension infinie.

International audience; This paper focuses on mixed [Formula: see text] fuzzy maximum power point tracking (MPPT) of photovoltaic (PV) system under asymmetric saturation and variations in climatic conditions. To maximize the power from the PV panel array, the DC–DC boost converter is controlled by its duty ratio which is practically saturated between 0 and 1. MPPT based on conventional control presents the problems of oscillations around maximum power point (MPP) and divergence under rapid climatic changes. In order to attenuate the effect of atmospheric condition variation and take into account asymmetric saturation of the duty ratio, we propose a novel robust saturated controller based on both [Formula: see text] performances and Takagi-Sugeno (T-S) representation of PV-boost nonlinear system. Within this approach, the nonlinear PV-boost system and its reference are first described by T-S fuzzy models. Second, the saturation effect is represented by a polytopic model. Then, a fuzzy integral state feedback controller is designed to achieve stable MPPT control. Based on Lyapunov function, the mixed [Formula: see text] stabilization conditions are derived in terms of linear matrix inequalities (LMIs). The optimization of the attraction domain of closed-loop system is solved as a convex optimization problem in LMI terms. Finally, the efficiency of the proposed controller under irradiance and temperature variations is demonstrated through the simulation results. The comparison with some existing controllers shows an improvement of MPPT control performance in terms of power extraction.

International audience; Peak Estimation aims to find the maximum value of a state function achieved by a dynamical system. This problem has been previously cast as a convex infinite-dimensional linear program on occupation measures, which can be approximately solved by a converging hierarchy of moment relaxations. In this letter, we present an algorithm to approximate optimal trajectories if the solutions to these relaxations satisfy rank constraints. We also extend peak estimation to maximin and safety analysis problems, providing a certificate that trajectories are bounded away from an unsafe set.

International audience; This paper deals with the design of output feedback event-triggered controllers equipped with generalized holding devices. Both emulation and co-design settings are explored. Specifically, a robust observer-based event-triggered controller with a dwell time logic is proposed to achieve closed-loop stability. The closed-loop system is modeled as a hybrid system and analyzed via Lyapunov theory for hybrid systems. Sufficient conditions in the form of matrix inequalities are given to ensure global exponential stability and input-to-state stability with respect to measurement disturbances for the closed-loop system. The proposed conditions enable the design of the controller gains, event-triggering mechanism, and of general holding devices, thereby including classical zero-order-holder devices. Convex optimization schemes address the implicit objective consisting of reducing the number of updates of the control input. The effectiveness of the conditions are illustrated through an illustrative example borrowed from the literature.

International audience; This article presents a novel methodology for exact convex rewriting of nonlinear systems that generalizes the well-known sector nonlinearity approach; it allows improving the capabilities of those methodologies based on polytopes and the direct Lyapunov method, which are still an active research area of fuzzy control systems. Depending on the computational burden, a tighter fitting of interdependent nonlinearities allows avoiding useless or damaging vertices, thus remarkably improving the feasibility chances of analysis and design, while preserving conditions in the form of linear matrix inequalities. A variety of examples give account of the advantages of the proposal.

IF=4.883; International audience; This brief is about robust estimation for a sampled output system with continuous dynamics using unknown input observers (UIOs). A novel worst case estimation error bound is developed using a particular form of Gronwall inequality. The motivation is to assess the accuracy of the estimator in real-world conditions, considering slow sensor sampling times and possible corruption of the measurements due to faults. The theoretical results are applied for robust velocity estimation of a braking train when wheel jamming faults affect odometric sensors. Simulations and experimental data analysis are used to evaluate the proposed approach.

International audience; Most research activities that utilize linear matrix inequality (LMI) techniques are based on the assumption that the separation principle of control and observer synthesis holds. This principle states that the combination of separately designed linear state feedback controllers and linear state observers, which are independently proven to be stable, results in overall stable system dynamics. However, even for linear systems, this property does not necessarily hold if polytopic parameter uncertainty and stochastic noise influence the system’s state and output equations. In this case, the control and observer design needs to be performed simultaneously to guarantee stabilization. However, the loss of the validity of the separation principle leads to nonlinear matrix inequalities instead of LMIs. For those nonlinear inequalities, the current paper proposes an iterative LMI solution procedure. If this algorithm produces a feasible solution, the resulting controller and observer gains ensure robust stability of the closed-loop control system for all possible parameter values. In addition, the proposed optimization criterion leads to a minimization of the sensitivity to stochastic noise so that the actual state trajectories converge as closely as possible to the desired operating point. The efficiency of the proposed solution approach is demonstrated by stabilizing the Zeeman catastrophe machine along the unstable branch of its bifurcation diagram. Additionally, an observer-based tracking control task is embedded into an iterative learning-type control framework.

International audience; This paper concerns the event-triggered dynamic output-feedback control of discrete-time linear parameter-varying (LPV) systems subject to saturating actuators. Two independent event-triggering schemes are introduced to determine whether the current signals should be transmitted (a) from the sensor to the controller and (b) from the controller to the actuator. As a result, the communication resources can be significantly saved. Both the emulation-based problem and the co-design problem are addressed. Sufficient conditions based on linear matrix inequalities (LMIs) are derived to ensure the regional asymptotic stability of the origin for the closed-loop system. A convex optimization procedure is proposed to determine the controller matrices and the event-triggering parameters aiming at reducing the number of updates on the independent channels sensor-to-controller and controllerto-actuator. At last, numerical examples are employed to testify to the validity of the proposed methods.

According to the World Health Organization, more than 70 million disabled people in the world need a wheelchair to aid their daily mobility needs. Providing access to low cost wheelchairs that covers the users basic needs is therefore a societal challenge that not only contribute to their well-being but also their dignity. Manual wheelchairs are widespread due to their low cost and high maneuverability. In order to mitigate the physical effects of the repeated efforts on the push rims, the company Autonomad Mobility developed an electrification kit. Among the available features, the so-called gyroscopic mode allows operating the wheelchair in an upright position with the caster wheels lifted off the ground. In this thesis, different control laws are proposed to swing-up the wheelchair from the grounded position to the unstable equilibrium position (gyroscopic mode). The specificity of the control problem is that the user is part of the system to be controlled. A mathematical model of the wheel chair and the user is built from the mechanical equations motion and its parameters are identified through adequate experiments. The wheelchair swingup is first casted as a reference tracking problem where the control associated with the trajectory is known. The system is modeled as a continuous Takagi-Sugeno descriptor model and the closed loop Lyapunov stability is formulated as a set on Linear Matrix Inequalities to be solved. The reference control and state trajectories are computed as a solution to an optimal control problem. The tuning of the criterion allows to tune the compromise between the control amplitude and the deviation from the wheelchair target position. Simulation and experimental results are provided and analyzed. Experimental results are globally similar to the simulations, thus demonstrating the validity of the model and the effectiveness of the control approach. Even though the trajectory is actually followed by the wheelchair, the quantification of some sensor readings combined with the numerical implementation of the continuous control law on a discrete system leads to some degradation of the control performances. To make improvements on the above control strategy, a second control approach, using a discrete time controller, is suggested. The wheelchair dynamics is formulated as an uncertain discrete time Takagi-Sugeno discrete descriptor model that accounts for different users' characteristics. The tracking problem is formulated and again, the control approach consists in tracking a given state trajectory but now it is generated by an a priori chosen model stabilized by an appropriate state feedback. The robust stabilization conditions are provided as a set of Linear Matrix Inequalities. Finally, simulation and experimental results are provided. Videos of the experiments are provided to illustrate the robustness of the control law to different users and the closed loop behavior when the user voluntary rocks its upper body.; D'après l'Organisation Mondiale de la santé, plus de 70 millions de personnes à mobilité réduite ont besoin d'un fauteuil roulant pour assurer leurs déplacements quotidiens. Garantir l'accès à des fauteuil abordables et capable de couvrir les déplacements de base est un challenge sociétal qui contribue non seulement au bien-être des personnes mais également à leur dignité. Les fauteuils manuels sont les plus rependus en raison de leur cout modeste et leur grande manœuvrabilité. Pour limiter les troubles musculaires liés aux appuis répétés sur les roues, la société Autonomad Mobility propose un kit d'électrification. Parmi les différentes fonctionnalités proposées, le mode gyroscopique permet à l'utilisateur de se déplacer en équilibre sur deux roues, les petites roues avant étant levées. Dans cette thèse, plusieurs lois de commande sont proposées pour assurer le passage du mode stable (avec les 4 roues posées sur le sol) à la position en équilibre instable du mode gyroscopique. Une des spécificités des lois de commandes proposées, est que l'utilisateur fait partie du système piloté. Un modèle mathématique du fauteuil et de l'utilisateur est construit en utilisant les lois de la mécanique, et ses paramètres sont identifiés grâce à des expérimentations spécifiques. Le passage à la position d'équilibre instable est formulé comme un problème de suivi de trajectoire où la commande utilisée pour générer la trajectoire est connue. Le système est écrit sous la fomre d'un descripteur flou de type Takagi Sugeno et la stabilité de la boucle fermée, au sens de Lyapunov, est formulée comme un ensemble d'inégalités linéaires matricielles. La trajectoire de référence est calculée comme la solution d'un problème de commande optimale. Les pondérations entre les différents termes du critère permettent de choisir le compromis entre l'amplitude de commande et le déplacement longitudinal du fauteuil. Les résultats de simulation et des expérimentations sont présentés et discutés. Globalement, les résultats expérimentaux sont similaires à ceux obtenus en simulation, démontrant ainsi la validité du modèle et la pertinence de l'approche utilisée. Même si la trajectoire de référence est bien suivie par le fauteuil, la quantification des mesures des capteurs et la mise en œuvre sur un calculateur numérique de la loi de commande formulée en temps continue, amène une dégradation des performances de la loi de commande. Afin d'améliorer les résultats obtenus, une deuxième approche, formulée en temps discret, est envisagée. La dynamique du fauteuil roulant est tout d'abord écrite sous la forme d'un descripteur flou de type Takagi-Sugeno. Des incertitudes sur les paramètres permettent de prendre en compte la morphologie de différents utilisateurs. Un problème de suivi de trajectoire est formulée de sorte à suivre une trajectoire d'état donné, mais cette fois ci, la trajectoire est généré en utilisant un modèle fixé a priori et stabilisé via un retour d'état approprié. L'écriture de la stabilisation robuste de la boucle fermée se résume à un ensemble d'inégalité matricielles linéaires. Finalement, des résultats de simulation et expérimentaux sont discutées. Des vidéos des expérimentations sont mises à dispositions. Différents utilisateurs ont testé la loi de commande proposée qui a été en mesure de réguler correctement le fauteuil en équilibre instable en dépit des brusques mouvement volontaires des utilisateurs.

This paper deals with the analysis reaction diffusion equation, which can be found in many applications such as in pharmaceutic or chemistry fields. The particularity of the present paper is to study the interconnection of this class of infinite-dimensional systems to a finite-dimensional systems. In this situation, stability is not straightforward to assess any more and one needs to look for dedicated tools to provide accurate numerical tests. Here, the objective is to provide a Lyapunov analysis leading to efficient and scalable stability criteria. This is made possible thanks to the Legendre orthogonal basis which allows building accurate Lyapunov functionals. Indeed this functional is expressed thanks to the state of the finite-dimensional system, the first Fourier-Legendre coefficients and the remainder of the Fourier-Legendre expansion of the infinite-dimensional state. Using this representation, efficient formulation of the Bessel and Wirtinger inequalities are provided leading to sufficient stability conditions expressed in terms of linear matrix inequalities. Numerical examples finally illustrate the accuracy and the potential of the stability result.

International audience; This paper deals with a control strategy for a DC-DC quadratic boost converter. In particular, a hybrid control scheme is proposed to encompass a control law and an observer for the estimation of the system states, based only on the measurements of the input and output voltages. Differently from classical control methods, where the controller is designed from a small-signal model, here the real model of the system is examined without considering the average values of the discrete variables. Using hybrid dynamical system theory, asymptotic stability of a neighborhood of the equilibrium point is established, ensuring practical stability of the origin, which contains estimation and tracking errors. Experimental results show the effectiveness of the proposed approach.

International audience; Aging attacks are a class of attacks targeting closed-loop control systems connected to a network. Such attacks consist in slightly modifying the control signals to increase the wear and tear of the physical system while maintaining the delivered service. In this paper, we exploit the technique of robust simulation of linear systems for forecasting and preventing these attacks by means of convex optimization. Besides, this technique allows the computation of the time at which the normal aging is not guaranteed anymore for a given control input set. The paper end highlights an example application upon an electrical machine.

International audience; The problem of stabilization of a system of coupled PDEs of the forth-order by means of boundary control is investigated. The considered setuparises from the classical Euler-Bernoulli beam model, and constitutes ageneralization of flexible mechanical systems. A linear feedback controlleris proposed, and using an abstract formulation based on operator semi-group theory, we are able to prove the well-posedness and the stability ofthe closed-loop system. The performances of the proposed controller areillustrated by means of numerical simulations.

International audience; This paper proposes a numerical approach to the stability analysis for a class of piecewise-affine systems with (possibly time-varying) parameter-dependent cells and dynamics. This class of model aims at allowing a better modelling of time-varying or parameter-varying nonlinearities of physical phenomena such as dry friction. We form the stability certification problem as the one of finding a Lyapunov function that is parameterised as a polynomial function of the variable parameter. The application of the well-known Lyapunov stability theorem together with the use of the generalised S-procedure reduces the problem to checking whether a certain set of matrices has the sum-of-squares property. The latter can be solved using well-documented numerical solvers, and we provide two examples of successful applications at the end of the paper.

International audience; In this study, the authors develop a new high-gain observer design method for non-linear systems. This new design provides a lower gain compared to both the high-gain and the enhanced high-gain observer. The idea is to combine the improved high-gain methodology with the linear matrix inequality based observer design technique to build a more general observer that allows one to exploit the benefits of both approaches. A numerical example is given to show the effectiveness of the proposed observer with different values of the Lipschitz constant and of the compromise index.

International audience; Observer design for a nonlinear system in which the process dynamics equation are composed of nonlinear vector functions of scalar combinations of the states is considered. Assuming that the nonlinear functions have bounded derivatives, an observer design algorithm that requires solving just a single linear matrix inequality for exponentially convergent state estimation is developed. The developed algorithm works effectively when the involved nonlinear functions are monotonic. However, it fails when all or even some of the system functions are non-monotonic. Analytical results are presented to show that no solutions exist when all process dynamics functions are non-monotonic, no matter how small the Lipschitz constant or the Jacobian bounds of the nonlinearities. To overcome this limitation, a switched gain observer that switches between multiple constant observer gains is developed that can provide global exponentially stability for systems with non-monotonic nonlinear functions. The application of the developed hybrid observer is demonstrated to a motion estimation application involving vehicle position tracking on local roads and highways.

International audience; This paper concerns the design of new positive filters for positive linear systems. In fact, we propose a new positive full order filter for positive linear systems subject to unknown inputs and bounded disturbances. The designed filter is always nonnegative at any time and converges asymptotically to the real state vector. This paper is among first attempts to design positive unknown input filters (PUIF) for positive linear systems. The proposed approach is based on the unbiasedness of the estimation error and by imposing the positivity of the design parameters; then a new method to avoid the derivative of the disturbance vector in the filtering error dynamics is proposed. Based on the extended strictly positive real (ESPR) design, this problem is solved by applying the Linear Matrix Inequalities (LMI)-based ESPR Lemma. Note that all structural constraints on filter matrices are addressed in terms of LMI formulation. An algorithm that summarizes the different steps of the proposed positive filter design is given. A numerical example is finally given to illustrate the effectiveness of the proposed method.

International audience; We propose a control oriented Wiener model for wet-clutches in filled conditions and we discuss the associated identification technique. We design a novel hybrid controller, which ensures zero steady-state error and a fast non-overshooting response. We show that the controller parameters can be conveniently obtained by solving a set of linear matrix inequalities. Finally, we test the proposed control strategy on the hydromechanical variable transmission developed by Dana-Rexroth Transmission Systems. The experiments show good performance and robustness with respect to modeling errors and noise.

International audience; This paper is concerned with finite-and fixed-time robust stabilization of uncertain multi-input nonlinear systems via the implicit Lyapunov function method. Instead of splitting the system into a linear nominal model and an additive perturbation which gathers nonlinearities, parametric uncertainties , and exogenous disturbances, the methodology hereby proposed preserves some nonlinear terms in the nominal system via an exact polytopic representation which leads to design conditions in the form of linear matrix inequalities. As a result, feasible solutions are found where former approaches fail; these solutions have more accurate settling-time estimates with reduced control effort. The corresponding control law includes well-known high-order sliding modes as a particular case. Numerical simulations are provided to illustrate the advantages of the proposal.

International audience; In this work we study the design of $H_2$, $H_\infty$ and $H_2$/$H_\infty$ mixed static output feedback controllers for Markov jump linear systems in a context of partial information. We assume that the controller has only access to an observable variable that plays the role of a detector of the jump process. We present suboptimal conditions based on the so-called two step procedure in order to design static output feedback controllers that depend only on the detector such that the closed-loop system is stochastically stable and its $H_2$ or/and $H_\infty$ norms are bounded. We also investigate the robust control case, in which the system matrices and transition probabilities are uncertain. An iterative algorithm that guarantees a non-increasing sequence of optimized costs is presented. Our results are illustrated by a numerical example in the context of systems subject to failures in which we show that the inherent conservatism of the mixed $H_2$/$H_\infty$ control and the two step procedure can be improved by the proposed method.

The aim of this dissertation is to present new results on analysis and control design of time-delay systems. On the first part, we extend the use of a finite order LTI system, called 'comparison system', to design a controller which depends not only on the output at the present time and maximum delay, but also on an arbitrary number of values between those. This approach allows us to increase the maximum stable delay without requiring any additional information. The methods presented here consider time-delay systems control design with classical numeric routines based on Hoo theory. The second part of this work deals with a new approach to develop an envelope that engulfs all poles of a time-delay system. Through LMIs, we are able to determine envelopes for retarded and neutral time-delay systems. The envelopes proposed are not only tighter than the ones in the literature but, with our procedure, they can also be applied to verify the stability of the system and design state-feedback controllers which cope with design requirements regarding alpha-stability and are robust in face of parametric uncertainties. Fractional systems are also discussed for both chapters mentioned above. The third and last part studies stochastic time-delay systems.First we discuss continuous-time systems that are subjected to Markov jumps. We define stability and obtain LMIs for the state-feedback control in such a way that the relation with the transition rates between the modes is affine, allowing, therefore, to treat the case in which the rates are uncertain. We then discuss positive systems with delays, both for the continuous case as for the discrete case. Equivalent systems are obtained and delay dependent stability is addressed. A fair amount of examples are presented throughout the dissertation.; Le but de cette thèse est de présenter de nouveaux résultats sur l'analyse et la synthèse de systèmes à retard. Dans la première partie, nous étendons l'utilisation du système invariant d'ordre fini, appelé "système de comparaison", à la conception d'un contrôleur qui dépend non seulement de la sortie à l'heure actuelle et du délai maximum, mais également d'un nombre arbitraire de valeurs entre celles-ci. Cette approche nous permet d'augmenter le délai maximal stable sans exiger d'informations supplémentaires. Les méthodes présentées ici concernent la conception de systèmes de contrôle avec des retards en utilisant des routines numériques classiques basées sur la théorie Hoo. La deuxième partie de ce travail traite d'une nouvelle approche pour développer une enveloppe englobant tous les pôles d'un système à retard. Grâce aux LMIs, nous sommes en mesure de déterminer les enveloppes pour les systèmes à retard du type retardé et du type neutre. Les enveloppes proposées sont non seulement plus étroites que celles de la littérature, mais, avec notre procédure, elles peuvent également être appliquées pour vérifier la stabilité du système et pour projeter contrôleurs de retour d'état qui répondent aux exigences de conception relatives à alpha-stabilité et sont robustes face aux incertitudes paramétriques. Les systèmes fractionnaires sont également discutés dans les deux chapitres mentionnés ci-dessus. La troisième et dernière partie étudie les systèmes stochastiques avec des retards. Nous discutons d'abord des systèmes à temps continu soumis à des sauts de Markov. Nous définissons la stabilité et obtenons des LMIs pour le contrôle par retour d'état de telle sorte que la relation entre les taux de transition entre les modes soit affine, ce qui permet donc de traiter le cas dans lequel les taux sont incertains. Nous discutons ensuite des systèmes positifs avec retards, tant pour le cas continu que pour le cas discret. Des systèmes équivalents sont obtenus et la stabilité dépendante du retard est abordée. De nombreux exemples sont illustrés tout au long de la thèse.

International audience; This study concerns the observer design for Lipschitz non-linear descriptor systems. It introduces a new observer structure called a generalised dynamic observer, which is more general than the proportional and proportional–integral observers. The originality of the proposed observer is that it provides additional degrees of freedom in the observer design, which can be used to increase steady-state accuracy and to improve robustness in estimation against modelling errors. Conditions for the existence and stability of this observer are given in terms of linear matrix inequalities. Reduced-order and full-order observers can be designed directly by using the proposed unified approach. The effectiveness of the developed method is illustrated by two numerical examples where impulse and impulse-free cases are considered, a comparison between the generalised dynamic observer, the proportional observer, and the proportional–integral observer is given to show the observers performances.

International audience; This paper will focus on static output feedback (SOF) control for continuous time Takagi-Sugeno (T-S) fuzzy systems. Based on the use of the concept of decay rate in the quadratic Lyapunov function, sufficient conditions for designing fuzzy SOF control are given in terms of linear matrix inequalities (LMIs). Moreover, the proposed method overcomes the drawback of the iterative algorithms based on LMI approach, and does not need any equality constraints; in addition, it can provide less conservative results. In the end, numerical and practical examples are given to demonstrate the effectiveness of the proposed method.

International audience; In this paper, we design periodically intermittent feedback controllers for Lur'e systems to achieve absolute stabilization. More precisely, our designed periodically intermittent feedback controller is able to cope with any unknown Lur'e-type nonlinearity within a given sector. First, by means of global exponential Lyapunov stability, a set of sufficient stability conditions on the controlled Lur'e system is derived. Subsequently, we give the controller design algorithm in terms of the necessary and sufficient conditions to the stability criteria. By using the LMI Control Toolbox in Matlab, it is easy to compute the control parameters, including the feedback gain matrix, the control period and the control width, involved in the stability criteria. A numerical example considering a Chua's oscillator under periodically intermittent feedback control is presented to illustrate the validity of our obtained theoretical results. Finally, further discussions close the paper along with some possible interesting topics for future research.

International audience; State-dependent jump linear systems (SJLSs) are a set of linear systems whose switching is determined by a finite state random process with state-dependent transition rates. In this study, a SJLS is considered with state multiplicative noise and stochastic perturbations. In particular, the jump process is regarded to have dissimilar transition rates among sets of state evolution space. The aim of this study is to consider stochastic $H_\infty$ control of such systems using state-feedback control input. Sufficient conditions for stochastic $H_\infty$ are obtained by solving linear matrix inequalities, which are validated by a simulation example.

Stability and stabilization for linear state feedback control systems in the presence of sensor quantization are studied. As the closed-loop system is described by a discontinuous right-hand side differential equation, Krasovskii solutions (to the closed-loop system) are considered. Sufficient conditions in the form of matrix inequalities are proposed to characterize uniform global asymptotic stability of a compact set containing the origin. Such conditions are shown to be always feasible whenever the quantization-free closed-loop system is asymptotically stable. Building on the obtained conditions, computationally affordable algorithms for the solution to the considered problems are illustrated. The effectiveness of the proposed methodology is shown in three examples., V1 fixes some minor typos in the published version. Corrections are colored in blue

Time-delay is a natural phenomenon that is present in most physical systems and engineering applications, thus, delay systems have been an active area of research in control engineering for more than 60 years. Observer design is one of the most important subject that has been dealt with, this is due to the importance of observers in control engineering systems not only when sensing is not sufficient but also when a sensing reliability is needed. In this work, the main goal was to design observers for different classes of nonlinear delayed systems with an arbitrary large delay, using different approaches. In the first part, the problem of observer design is addressed for a class of triangular nonlinear systems with not necessarily small delay and sampled output measurements. Another major difficulty with this class of systems is the fact that the state matrix is dependent on the un-delayed output signal which is not accessible to measurement. A new chain observer, composed of sub-observers in series, is designed to compensate for output sampling and arbitrary large delays.In the second part of this work, another kind of triangular nonlinear delayed systems was considered, where this time the delay was considered as a first order hyperbolic partial differential equation. The inverse backstepping transformation was invoked and a chain observer was developed to ensure its effectiveness in case of large delays. Finally, a new observer was designed for a class of nonlinear parabolic partial differential equations under point measurements, in the case of large delays. The observer was composed of several chained sub-observers. Each sub-observer compensates a fraction of the global delay. The stability analyses of the error systems were based on different Lyapunov-Krasovskii functionals. Also different mathematical tools have been used in order to prove the results. Simulation results were presented to confirm the accuracy of the theoretical results; Le retard est un phénomène naturel présent dans la majorité des systèmes physiques et dans les applications d’ingénierie, ainsi, les systèmes à retard ont été un domaine de recherche très actif en automatique durant les 60 dernières années. La conception d’observateur est un des sujets les plus importants qui a été étudié, ceci est dû à l’importance des observateurs en automatique et dans les systèmes de commande en absence de capteur pour mesurer une variable. Dans ce travail, l’objectif principal est de concevoir des observateurs pour différentes classes de systèmes à retard avec un retard arbitrairement large, et ce en utilisant différentes approches. Dans la première partie de cette thèse, la conception d’un observateur a été réalisée pour une classe de systèmes non linéaires triangulaires avec une sortie échantillonnée et un retard arbitraire. Une l’autre difficulté majeure avec cette classe de systèmes est le fait que la matrice d’état dépend du signal de sortie non-retardé qui est immesurable. Un nouvel observateur en chaine, composé de sous-observateurs en série est conçu pour compenser les retards arbitrairement larges. Dans la seconde partie de ce travail, un nouvel observateur a été conçu pour un autre type de systèmes non linéaires triangulaires, où le retard a été considéré, cette fois-ci, comme une équation aux dérivées partielles de type hyperbolique du premier ordre. La transformation inverse en backstepping et le concept de l’observateur en chaine ont été utilisés lors de la conception de cet observateur afin d’assurer son efficacité en cas de grands retards. Dans la dernière partie de cette thèse, la conception d’un nouvel observateur a été réalisée pour un type de système modélisé par des équations paraboliques non linéaires où les mesures sont issues d’un nombre fini de points du domaine spatial. Cet observateur est constitué d’une série de sous-observateurs en chaine. Chaque sous-observateur compense une fraction du retard global. L'analyse de la stabilité des systèmes d’erreur a été fondée sur différentes fonctionnelles Lyapunov-Krasovskii. Par ailleurs, différents instruments mathématiques ont été employés au cours des différentes preuves présentées. Les résultats de simulation ont été présentés dans le but de confirmer l'exactitude des résultats théoriques

Time-delay is a natural phenomenon that is present in most physical systems and engineering applications, thus, delay systems have been an active area of research in control engineering for more than 60 years. Observer design is one of the most important subject that has been dealt with, this is due to the importance of observers in control engineering systems not only when sensing is not sufficient but also when a sensing reliability is needed. In this work, the main goal was to design observers for different classes of nonlinear delayed systems with an arbitrary large delay, using different approaches. In the first part, the problem of observer design is addressed for a class of triangular nonlinear systems with not necessarily small delay and sampled output measurements. Another major difficulty with this class of systems is the fact that the state matrix is dependent on the un-delayed output signal which is not accessible to measurement. A new chain observer, composed of sub-observers in series, is designed to compensate for output sampling and arbitrary large delays.In the second part of this work, another kind of triangular nonlinear delayed systems was considered, where this time the delay was considered as a first order hyperbolic partial differential equation. The inverse backstepping transformation was invoked and a chain observer was developed to ensure its effectiveness in case of large delays. Finally, a new observer was designed for a class of nonlinear parabolic partial differential equations under point measurements, in the case of large delays. The observer was composed of several chained sub-observers. Each sub-observer compensates a fraction of the global delay. The stability analyses of the error systems were based on different Lyapunov-Krasovskii functionals. Also different mathematical tools have been used in order to prove the results. Simulation results were presented to confirm the accuracy of the theoretical results, Le retard est un phénomène naturel présent dans la majorité des systèmes physiques et dans les applications d’ingénierie, ainsi, les systèmes à retard ont été un domaine de recherche très actif en automatique durant les 60 dernières années. La conception d’observateur est un des sujets les plus importants qui a été étudié, ceci est dû à l’importance des observateurs en automatique et dans les systèmes de commande en absence de capteur pour mesurer une variable. Dans ce travail, l’objectif principal est de concevoir des observateurs pour différentes classes de systèmes à retard avec un retard arbitrairement large, et ce en utilisant différentes approches. Dans la première partie de cette thèse, la conception d’un observateur a été réalisée pour une classe de systèmes non linéaires triangulaires avec une sortie échantillonnée et un retard arbitraire. Une l’autre difficulté majeure avec cette classe de systèmes est le fait que la matrice d’état dépend du signal de sortie non-retardé qui est immesurable. Un nouvel observateur en chaine, composé de sous-observateurs en série est conçu pour compenser les retards arbitrairement larges. Dans la seconde partie de ce travail, un nouvel observateur a été conçu pour un autre type de systèmes non linéaires triangulaires, où le retard a été considéré, cette fois-ci, comme une équation aux dérivées partielles de type hyperbolique du premier ordre. La transformation inverse en backstepping et le concept de l’observateur en chaine ont été utilisés lors de la conception de cet observateur afin d’assurer son efficacité en cas de grands retards. Dans la dernière partie de cette thèse, la conception d’un nouvel observateur a été réalisée pour un type de système modélisé par des équations paraboliques non linéaires où les mesures sont issues d’un nombre fini de points du domaine spatial. Cet observateur est constitué d’une série de sous-observateurs en chaine. Chaque sous-observateur compense une fraction du retard global. L'analyse de la stabilité des systèmes d’erreur a été fondée sur différentes fonctionnelles Lyapunov-Krasovskii. Par ailleurs, différents instruments mathématiques ont été employés au cours des différentes preuves présentées. Les résultats de simulation ont été présentés dans le but de confirmer l'exactitude des résultats théoriques

This document describes our freely distributed Maple library {\sc spectra}, for Semidefinite Programming solved Exactly with Computational Tools of Real Algebra. It solves linear matrix inequalities with symbolic computation in exact arithmetic and it is targeted to small-size, possibly degenerate problems for which symbolic infeasibility or feasibility certificates are required., Significantly extended version

Designing a static state-feedback controller subject to structural constraint achieving asymptotic stability is a relevant problem with many applications, including network decentralized control, coordinated control, and sparse feedback design. Leveraging on the Projection Lemma, this work presents a new solution to a class of state-feedback control problems, in which the controller is constrained to belong to a given linear space. We show through extensive discussion and numerical examples that our approach leads to several advantages with respect to existing methods: first, it is computationally efficient; second, it is less conservative than previous methods, since it relaxes the requirement of restricting the Lyapunov matrix to a block-diagonal form., Comment: V1 fixes a few minor typos in the published version. V2 includes a clarification and fixes some minor typos in the proof of Theorem 2

The aim of this dissertation is to present new results on analysis and control design of time-delay systems. On the first part, we extend the use of a finite order LTI system, called 'comparison system', to design a controller which depends not only on the output at the present time and maximum delay, but also on an arbitrary number of values between those. This approach allows us to increase the maximum stable delay without requiring any additional information. The methods presented here consider time-delay systems control design with classical numeric routines based on Hoo theory. The second part of this work deals with a new approach to develop an envelope that engulfs all poles of a time-delay system. Through LMIs, we are able to determine envelopes for retarded and neutral time-delay systems. The envelopes proposed are not only tighter than the ones in the literature but, with our procedure, they can also be applied to verify the stability of the system and design state-feedback controllers which cope with design requirements regarding alpha-stability and are robust in face of parametric uncertainties. Fractional systems are also discussed for both chapters mentioned above. The third and last part studies stochastic time-delay systems.First we discuss continuous-time systems that are subjected to Markov jumps. We define stability and obtain LMIs for the state-feedback control in such a way that the relation with the transition rates between the modes is affine, allowing, therefore, to treat the case in which the rates are uncertain. We then discuss positive systems with delays, both for the continuous case as for the discrete case. Equivalent systems are obtained and delay dependent stability is addressed. A fair amount of examples are presented throughout the dissertation., Le but de cette thèse est de présenter de nouveaux résultats sur l'analyse et la synthèse de systèmes à retard. Dans la première partie, nous étendons l'utilisation du système invariant d'ordre fini, appelé "système de comparaison", à la conception d'un contrôleur qui dépend non seulement de la sortie à l'heure actuelle et du délai maximum, mais également d'un nombre arbitraire de valeurs entre celles-ci. Cette approche nous permet d'augmenter le délai maximal stable sans exiger d'informations supplémentaires. Les méthodes présentées ici concernent la conception de systèmes de contrôle avec des retards en utilisant des routines numériques classiques basées sur la théorie Hoo. La deuxième partie de ce travail traite d'une nouvelle approche pour développer une enveloppe englobant tous les pôles d'un système à retard. Grâce aux LMIs, nous sommes en mesure de déterminer les enveloppes pour les systèmes à retard du type retardé et du type neutre. Les enveloppes proposées sont non seulement plus étroites que celles de la littérature, mais, avec notre procédure, elles peuvent également être appliquées pour vérifier la stabilité du système et pour projeter contrôleurs de retour d'état qui répondent aux exigences de conception relatives à alpha-stabilité et sont robustes face aux incertitudes paramétriques. Les systèmes fractionnaires sont également discutés dans les deux chapitres mentionnés ci-dessus. La troisième et dernière partie étudie les systèmes stochastiques avec des retards. Nous discutons d'abord des systèmes à temps continu soumis à des sauts de Markov. Nous définissons la stabilité et obtenons des LMIs pour le contrôle par retour d'état de telle sorte que la relation entre les taux de transition entre les modes soit affine, ce qui permet donc de traiter le cas dans lequel les taux sont incertains. Nous discutons ensuite des systèmes positifs avec retards, tant pour le cas continu que pour le cas discret. Des systèmes équivalents sont obtenus et la stabilité dépendante du retard est abordée. De nombreux exemples sont illustrés tout au long de la thèse.

International audience; An observer for a nonlinear system may be required to satisfy multiple performance criteria such as minimum convergence rate and disturbance rejection, in addition to asymptotic stability. In such cases, the observer can no longer be designed using a linear matrix inequality. A bilinear matrix inequality (BMI) is needed instead and involves a non-separable product of the observer gain matrices and the Lyapunov positive definite matrix. This paper develops a technique to solve a BMI for such a multi-objective observer design problem. The BMI design condition is transformed into an eigenvalue problem and a convex-concave based sequential linear matrix inequality (LMI) optimization method is used to find a feasible solution to the BMI. The developed observer design method is applied to a robust automotive slip angle estimation problem, where the gain from the disturbance to the observer error is restricted be lower than 0.2 and the estimated states converges to the neighborhood of the real states within 0.3 s in the presence of uncertain vehicle dynamics.

International audience; This paper is concerned with stabilization of non-affine-in-control nonlinear systems, a problem which usually leads to algebraic loops usually broken via dynamical controllers. Conditions in the form of linear matrix inequalities for stabilization of such systems are found by introducing a novel dynamical control scheme based on an exact convex representation of the system nonlinearities. Examples, both physical and numerical, illustrate the proposed methodology as it solves a larger family of stabilization problems.

International audience; This paper deals with the stability analysis of time delay systems based on continuous-time approach. The originality of the present paper relies on the construction of several models for a same time-delay systems using the interconnection of an ordinary differential equation and a transport partial differential equation. The stability analysis is then performed using a Lyapunov functional. These models are constructed in order to first reduce potentially the complexity of the resulting stability conditions. Second several models are build in order to be interpreted as a discretization scheme as the one usually used in the Lyapunov functional. The proposed result can be seen as a generalized (N − M) discretization which consists in both a time-discretization of the delay interval into M sub-intervals, and the projection of the state function within each sub-interval on the Legendre polynomials of degree less than N. The efficiency of this novel approach is illustrated on an academic example.

International audience; Invariant set theory has been recognized as an important tool for control design of constrained systems subject to disturbances. Indeed, invariant sets as the minimal robustly positively invariant set help verify whether constraints are satisfied in closed-loop. This paper studies a class of discrete-time linear systems under additive disturbances. We propose a method to compute the linear state feedback gain that minimizes the impact of disturbances, related to the mRPI, on the constrained closed-loop dynamics. The method relies on the solution of an optimization problem that is formulated using set-theoretic methods and discrete-time sliding mode parametrization.

International audience; This paper is concerned with the building of application-oriented model-based Takagi-Sugeno models that arise from sector nonlinearity methodology. Despite its exactness, this modeling technique produces a polytopic embedding with a number of vertices of 2 raised to the number of nonlinearities which, if high, represents a rapid growth in the number of linear matrix inequalities (needed to solve even the simplest control and observation schemes). A solution for model reduction is proposed based on the similarity of Taylor approximations amongst nonlinearities, with a measurable bound on the approximation, which - as proven with a suitable application example - brings the methodology closer to real-time demands.

IF=7.671; International audience; Many of the recent advances on control and estimation of systems described by Takagi-Sugeno (TS) fuzzy models are based on matrix inversion, which could be a trouble in the case of real-time implementation. This paper is devoted to the development of alternative solutions to this matrix inversion problem in the discrete-time case. Two different methods are proposed: The first one relies on replacing the matrix inversion by multiple sums and the second methodology is based on an estimation of the matrix inversion by an observer structure. For the first methodology, a new class of controllers and observers are introduced which are called, respectively, the counterpart of an advanced TS-based (CATS) controller and the replica of an advanced TS-based (RATS) observer. Instead of relaxations for the linear matrix inequalities conditions, an original use of the membership functions is presented. In the second methodology, it is proposed the estimation-based control law for approximating TS-based (ECLATS) controller that uses a fuzzy state observer. The Lyapunov theory is used to ensure stability conditions for either the closed-loop system as well as the estimation error. Numerical examples and comparisons highlight the efficiency of the procedures that can be used to replace any inverted matrix in any advanced fuzzy controller or observer. Finally, advantages and drawbacks of the proposed method are discussed.

Drawing inspiration from the theory of linear "decomposable systems", we provide a method, based on linear matrix inequalities (LMIs), which makes it possible to prove the convergence (or consensus) of a set of interacting agents with polynomial dynamic. We also show that the use of a generalised version of the famous Kalman-Yakubovic-Popov lemma allows the development of an LMI test whose size does not depend on the number of agents. The method is validated experimentally on two academic examples., Comment: 6 pages, 5 figures

International audience; This paper addresses the stability problem of linear systems with a time-varying delay. Hierarchical stability conditions based on linear matrix inequalities are obtained from an extensive use of the Bessel inequality applied to Legendre polynomials of arbitrary orders. While this inequality has been only used for constant discrete and distributed delays, this paper generalizes the same methodology to time-varying delays. We take advantages of the dependence of the stability criteria on both the delay and its derivative to propose a new definition of allowable delay sets. It is shown that a light and smart modification in the definition of this set leads to relevant conclusions on the numerical results.

International audience; This study concerns the stability analysis of linear large-scale fractional-order systems and its functional observers design. First, based on the diffusive representation of the fractional-order derivative and the indirect Lyapunov approach, the stability of large-scale fractional-order interconnected systems is studied. Then, a sufficient stability condition of such systems is given in linear matrix inequality (LMI) formulation. In the second part of this work, the existence conditions of functional decentralised observers for this class of systems are given, and the asymptotic stability of estimation errors is investigated. The observers gains matrices are derived by solving the obtained LMI. Numerical examples are given to illustrate the validity of the proposed approach.

National audience; The vehicle is a very complex system, requiring strict control and efficient monitoring of its sensors. Recently, many solutions have been proposed by the introduction and development of new driver-assisted systems, such as ACC, ESP, and ABS ... These systems must ensure the stability of the vehicle in critical driving situations using the data provided by the sensors. Indeed, the automatic systems of the vehicle necessarily have a fault diagnosis system. In this study, we have developed a method for detection and isolation of sensor faults (FDI) and fault tolerance control (FTC), applied to the vehicle dynamics which are represented by the Takagi-Sugeno (T-S) fuzzy model. Using the LMI formulation, the fuzzy TS model of the vehicle's nonlinear dynamics is used to design an observer-based control. Then, an approach based on an observer's bank is then studied for the detection and isolation of sensor faults. The simulation results obtained show that sensor faults are detected immediately after their appearance and illustrate the proposed FTC strategy.; Le véhicule est un système très complexe, nécessitant un contrôle strict et une surveillance performante de ses capteurs. Récemment, de nombreuses solutions ont été proposées par l'introduction et le développement de nouveaux systèmes assistés par le conducteur, comme le ACC, ESP et ABS ... Ces systèmes doivent assurer la stabilité du véhicule dans des situations de conduite critiques en utilisant les données fournies par les capteurs. En effet, les systèmes automatiques du véhicule disposent nécessairement d'un système de diagnostic de pannes. Dans cette étude, nous avons élaboré une méthode de détection et d'isolation des défauts de capteurs (FDI) et de commande tolérance aux défauts (FTC), appliquée au véhicule dont sa dynamique est représentée par le modèle flou de Takagi-Sugeno (T-S). En utilisant la formulation LMI, le modèle flou de TS de la dynamique non linéaire du véhicule est utilisé pour concevoir une commande basée sur un observateur. Ensuite, une approche basée sur un banc d'observateurs est alors étudiée pour la détection et l'isolation des défauts de capteurs. Les résultats de simulation obtenus montrent que les défauts de capteurs sont détectés immédiatement après leur apparition et illustrent la stratégie FTC proposée.

International audience; In this paper we propose a time-varying observer for a linear continuous-time plant with asynchronous discrete-time measurements. The proposed observer is contextualized in the hybrid systems framework providing an elegant setting for the proposed solution. In particular some theoretical tools are provided, in terms of LMIs, certifying asymptotic stability of a certain compact set where the estimation error is zero. Moreover the case of asynchronous measurements is considered, i.e. when the measurements are not provided in well defined time instants, but they occur at an arbitrary time in a certain time interval. A design procedure based on the numerical solution of an infinite-dimensional LMI is also proposed, leading to a time-varying observer gain. Finally a numerical example shows the effectiveness of the proposed approach.

International audience; When using fuzzy Lyapunov functions for continuous-time Takagi-Sugeno fuzzy systems, it is common to have to deal with the membership functions’ time derivative. Several upper-bound inequalities have been proposed in the literature in order to deal with these derivatives in a Linear Matrix Inequality setting. In this article, we extend and compare some of these inequalities in the context of generating the largest estimate of the domain of attraction while synthesizing non-PDC control laws.

International audience; This paper concerns the simulation of a class of nonlinear discrete-time systems under a set of initial conditions described by an ellipsoid. We derive a procedure allowing the propagation of such ellipsoids through time, which makes it possible to set a guaranteed hard bound on the evolution of the state of the system for all the possible initial conditions. At the end of the paper, we show an application of the method through three academics examples, two of which are taken from the theory of fractals.

IF=7.11; International audience; This paper provides a computational method to study the asymptotic stability of piecewise multi-affine (PMA) systems. Such systems stem from a class of fuzzy systems with singleton consequents and can be used to approximate any smooth nonlinear system with arbitrary accuracy. Based on the choice of piecewise Lyapunov functions, stability conditions are expressed as a feasibility test of a convex optimization with linear matrix inequality constraints. The basic idea behind these conditions is to exploit the parametric expressions of PMA systems by means of Finsler's lemma. Numerical examples are given to point out the effectiveness of the proposed method.

International audience; Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the solution set of linear matrix inequalities or the Schur/Hurwitz stability domains. These sets often have very complicated shapes (non-convex, and even non-connected), which renders very difficult their manipulation. It is therefore of considerable importance to find simple-enough approximations of these sets, able to capture their main characteristics while maintaining a low level of complexity. For these reasons, in the past years several convex approximations, based for instance on hyperrect-angles, polytopes, or ellipsoids have been proposed. In this work, we move a step further, and propose possibly non-convex approximations , based on a small volume polynomial superlevel set of a single positive polynomial of given degree. We show how these sets can be easily approximated by minimizing the L1 norm of the polynomial over the semialgebraic set, subject to positivity constraints. Intuitively, this corresponds to the trace minimization heuristic commonly encounter in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of linear matrix inequality problems to generate these approximations, and we provide theoretically rigorous convergence results, in the sense that the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set. Two main applications of the proposed approach are considered. The first one aims at reconstruction/approximation of sets from a finite number of samples. In the second one, we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set. The efficiency of the proposed approach is demonstrated by different numerical examples.