A form of detectability, known as the input-output-to-state stability property, for singularly perturbed systems is examined in this work. This work extends the result of a paper by Christofides & Teel wherein they presented a notion of total stability for input-to-state stability with respect to singular perturbations. Analyzing singularly perturbed systems with outputs we show that if the boundary layer system is uniformly globally asymptotically stable and the reduced system is input-output-to-state stable with respect to disturbances, then these properties continue to hold, up to an arbitrarily small offset, for initial conditions in an arbitrarily large compact set and sufficiently small singular perturbation parameter over the time interval for which disturbances, their derivatives, and outputs remain in an arbitrarily large compact set. An application of the result is presented where we analyze the stability of a circuit with a nonlinear element through the measurement of only one of the variables of interest.

A form of detectability, known as the input-output-to-state stability property, for singularly perturbed systems is examined in this work. This work extends the result of a paper by Christofides & Teel wherein they presented a notion of total stability for input-to-state stability with respect to singular perturbations. Analyzing singularly perturbed systems with outputs we show that if the boundary layer system is uniformly globally asymptotically stable and the reduced system is input-output-to-state stable with respect to disturbances, then these properties continue to hold, up to an arbitrarily small offset, for initial conditions in an arbitrarily large compact set and sufficiently small singular perturbation parameter over the time interval for which disturbances, their derivatives, and outputs remain in an arbitrarily large compact set. An application of the result is presented where we analyze the stability of a circuit with a nonlinear element through the measurement of only one of the variables of interest.

In this dissertation, the robust stability of uncertain continuous and discrete singularly perturbed systems with multiple time delays is investigated. Firstly, the asymptotic stability for a class of linear continuous singularly perturbed systems with multiple time delays is investigated. A simple estimate of an upper bound of singular perturbation parameter is proposed such that the original system is asymptotically stable for any . Moreover, a delay-dependent criterion, but -independent, is proposed to guarantee the asymptotic stability of the original system. Secondly, we consider the robust stability problem of uncertain continuous singularly perturbed systems with multiple time delays. Two delay-dependent criteria are proposed to guarantee the robust stability of a class of uncertain continuous multiple time-delay singularly perturbed systems subject to unstructured perturbations. Thirdly, the robust D-stability of nominally stable discrete uncertain systems with multiple time delays is considered. Finally, the robust stability of nominally stable uncertain discrete singularly perturbed systems with multiple time delays subject to unstructured and structured perturbations is investigated. Some criteria, delay-dependent or delay-independent, will be proposed to guarantee the robust stability of the uncertain discrete multiple time-delay singularly perturbed systems. The improvements of our results over those in recent literature are also illustrated if the comparisons are possible. Some numerical examples will also be provided to illustrate our main results.

La commande à transmissions événementielles est une approche dans laquelle les instants de transmission sont définis selon un critère dépendant de l'état du système et non plus d'une horloge à l'instar des implantations périodiques. Dans cette thèse, nous nous concentrons sur la synthèse de telles lois de commande par retour de sortie. Les contributions sont les suivantes : (i) nous proposons une méthode de synthèse dite par émulation pour des systèmes non linéaires; (ii) nous présentons une méthode de synthèse jointe de la loi de commande et de la condition de déclenchement pour les systèmes linéaires; (iii) nous nous intéressons au cas de systèmes non linéaires singulièrement perturbés et nous construisons le contrôleur à partir d’approximation de la dynamique lente uniquement. / Event-triggered control is a sampling paradigm in which the sequence of transmission instants is determined based on the violation of a state-dependent criterion and not a time-driven clock. In this thesis, we deal with event-triggered output-based controllers to stabilize classes of nonlinear systems. The contributions of the presented material are threefold: (i) we stabilize a class of nonlinear systems by using an emulation-based approach; (ii) we develop a co-design procedure to simultaneously design the output feedback law and the event-triggering condition for linear systems; (iii) we propose stabilizing event-triggered controllers for nonlinear systems whose dynamics have two-time scales (in particular, we only rely on the knowledge of an approximate model of the slow dynamics)

Les travaux de cette thèse portent sur l’analyse de stabilité et la synthèse de commande pour les systèmes singulièrement perturbés. Dans une première partie, nous présentons et analysons une classe générale de systèmes linéaires hybrides singulièrement perturbés dans lesquels la nature lente et rapide des variables d’état dépend du mode de fonctionnement. L’analyse de stabilité est fondée sur des résultats classiques de la théorie de Lyapunov pour les systèmes singulièrement perturbés. Une deuxième partie de ce travail présente la conception d’une loi de commande décentralisée qui garantit la synchronisation des systèmes multi-agents singulièrement perturbés avec un coût global garanti. Afin de contourner l’utilisation d’informations centralisées liées à la structure du réseau d’interconnexion, le problème est résolu en reformulant le problème de synchronisation comme un problème de stabilisation d’un système linéaire incertain singulièrement perturbé / This PhD thesis is dedicated to the study of stability and control design for singularly perturbed systems. In the first part, we introduce and analyze a general class of singularly perturbed linear hybrid systems, in which the slow or fast nature of the variables is mode-dependent. Our stability analysis is based on classical results of Lyapunov’s theory for singularly perturbed systems. A second part of this work presents the design of a decentralized control strategy that allows singularly perturbed multi-agent systems to achieve synchronization with global performance guarantees. To avoid the use of centralized information related to the interconnection network structure, the problem is solved by rewriting the synchronization problem in terms of stabilization of a singularly perturbed uncertain linear system