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International audience; An optimal control problem for a system of linear Itˆo differential equations withtwo fast time scales and Markovian jumping is considered. The asymptoticstructure for the stabilizing solution satisfying a sign condition for the coupledstochastic algebraic Riccati equations is derived. Furthermore, a near optimalcontrol whose gain matrices do not depend upon small parameters is discussed.

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.

International audience; In this paper, it is proposed to apply the strategy of system identification to modeling relationships in Water Dynamical Networks (WDN). System identification is issued from control theory where a model is computed from acquired data of a given system. Nevertheless, unlike in machine-learning, the model produced is constrained to belong to a predefined model class which embedding the available a priori knowledge of studied system. When this a priori knowledge is directly derived from mechanistic laws, the strategy is referred to as gray-box approaches. This paper exposes the pitfalls to avoid in gray-box models, with a detailed example on a tank level estimation. Moreover, when applying data-driven strategies to WDN, black-box modeling strategies are often used (water demand estimation), even for well-known relationships (e.g tanks level). In this setup, how to infer a priori knowledge? This is usually left to some automated optimisation based on fitting score only. This paper, while recognizing the power of black-box approaches, emphasizes that this optimisation is linked to the quality of the data at hand while forgetting the physical intuition. It shows that taking this knowledge in account, even in black-box models, leads to better and more plausible prediction capabilities.

International audience; We consider a two players linear quadratic differential game associated to a positive theta-periodic dynamical system with jump Markov perturbations. We introduce the concept of stabilizing solution and strong stabilizing solution of the associated system of coupled matrix Riccati differential equations and we will point out the relation between these kinds of solutions. Moreover, we provide a set of conditions which guarantee the existence of strong stabilizing and theta-periodic solution (stabilizing and theta-periodic solution, respectively) of the considered set of coupled matrix Riccati differential equations.

International audience; A state-dependent jump linear system (SDJLS) is a particular type of stochastic switching system, where the transition rates or probabilities of the underlying random jump process depend on the state variable. In this paper, a continuous-time SDJLS subject to external disturbances with finite energy is considered. Regarding the underlying random jump process, the transition rates are assumed to have different values across different sets where the state of the system evolves. By evaluating the infinitesimal generator for the stochastic Lyapunov function of SDJLS, a state-feedback H infinity controller is synthesized that guarantees disturbance rejection while ensuring stochastic stability via solving a set of sufficient linear matrix inequalities. The overall approach is illustrated with a numerical example.

International audience; This paper addresses the problem of optimal H2H2 filtering for a class of continuous-time periodic stochastic systems with periodic sampled measurements. The class of admissible filters consist of deterministic continuous-time periodic systems with finite jumps. The optimal solution of the considered optimization problem is obtained by integrating a suitable generalized continuous-time Riccati equation with finite jumps. To illustrate the proposed filtering strategy, we consider a problem of field monitoring where sensors are distributed on a rectangular region in order to estimate the state of a diffusion process. We consider that the sensors and the filter communicate over a communication channel and we assume that the communication channel induces some communication constraints. More specifically, we consider a medium access constraint under which the shared network can only accommodate a limited number of simultaneous communications between components.

International audience; This paper addresses the problem of solving a class of periodic discrete-time Riccati equation with an indefinite sign of it's quadratic term. Such an equation is closely related to the so called full information $H_\infty$ control of discrete-time periodic systems. A globally convergent iterative algorithm with local quadratic convergence rate is proposed for this purpose. An application to the problem of $H_\infty$ filtering of discrete-time periodic systems is also developed and illustrated via a numerical example.

International audience; This paper design a method for fault detection and isolation based on observers for systems modelled as Descriptor-Linear Parameter Varying (D-LPV) with Unmeasurable Scheduling Functions (USF). The first contribution of the paper is deal with the USF problem by transforming the into an uncertain D-LPV system with an estimated scheduling parameter. As a second contribution, a robust LPV observer is designed with H_{infty} performance to supply a robust state estimation against uncertainties provided by USF. Sufficient conditions to guarantee the robustness and convergence are obtained via linear matrix inequalities (LMIs). Finally, an observer bank based on robust H_{infty} observers is used to generate residuals and perform sensor fault detection and isolation. A numerical example demonstrates the effectiveness of our methods.

International audience; This note addresses the problem of optimal $\mathcal{H}_{2}$ filtering for a class of continuous-time time-varying stochastic systems. The time-variation character is considered to be periodic. The state space representation of the optimal filter is designed based on the unique periodic solution of a suitable Lyapunov differential equation and the periodic and stabilizing solution of a suitable generalized periodic Riccati differential equation. Some numerical experiments are also added to show the effectiveness of the proposed method.

Despite the evident interaction between FDI and reconfiguration algorithms, it is true that the research on FDI and reconfiguration methods has often evolved separately, certainly because of the difficulty of each of these problems. The main contribution of this work is to use a mathematical model that includes in the same analysis framework the FDI and reconfiguration algorithms. Such a model belongs to the class of Markovian jump linear systems. In this class of systems, two random processes are defined: the first represents system components failures and the second represents the FDI process. The first problematic considered in this thesis is related to the synthesis of output feedback controllers that stochastically stabilize this class of systems subject to Brownian motion. The developed results are based essentially on Lyapunov theory and Supermartingale notion. The different synthesis conditions are formulated as nonlinear matrix inequalities problematic. Noncovex optimization algorithms were then proposed to solve these conditions. The second problematic addressed in this work concerns the multi-objective control of this class of Markovian jump systems. The specifications and objectives under consideration include stochastic stability, H_{2} and H_{infinity} performances. Output feedback controllers synthesis conditions were also proposed in term of LMI, BMI and NLMI. Finally, we have addressed the discrete-time counterpart and proposed H_{2}/H_{infinity} synthesis conditions. The developed results were applied to the problematic of control of networked systems subject to delays, packet loss and failures., Dans cette thèse, nous nous sommes intéressés aux contraintes résultants de l'intégration d'un module de diagnostic de pannes et d'un module de reconfiguration de lois de commandes. Contraintes pouvant conduire à une perte de performances, voir une instabilité, du système. La formalisation mathématique de cette problématique nous a amené à nous intéresser à une classe de systèmes hybrides stochastiques à sauts markoviens. La première partie du travail de thèse a été consacrée à la synthèse de lois de commande, par retour de sortie, stabilisant stochastiquement cette classe de systèmes à des bruits multiplicatifs. Les approches développées sont basées sur la théorie de Lyapunov et de Supermartingale. Les différentes conditions de synthèse sont données en termes d'inégalités matricielles non linéaires. Des algorithmes d'optimisation non convexe nt alors été proposés pour la résolution de ces différentes conditions. En deuxième partie de thèse, nous nous sommes intéressés au problème de commande multi-performances de cette classe de systèmes. Plus particulièrement, nous avons considéré des critères H_{infinity} et des critères H_{2}. Là aussi, nous avons proposé des conditions sous forme LMI, BMI et NLMI pour la résolution de ce problème. En dernière partie de thèse, nous nous sommes intéressés au cas des systèmes à temps discret. Nous avons là aussi considéré des problèmes de stabilisation stochastique et de commande multi-objectifs, pour lesquels des conditions sous forme LMI et NLMI ont été établies. Nous avons ensuite appliqué ces résultats à la problématique de commande de systèmes en réseaux sujets à des retards, des pertes de paquets et d'éventuels pannes.