Practical Mittag-Leffler stability of quasi-one-sided Lipschitz fractional order systems

This paper focuses on the global practical Mittag-Leffler feedback stabilization problem for a class of uncertain fractional-order systems. This class of systems is a larger class of nonlinearities than the Lipschitz ones. Based on the quasi-one-sided Lipschitz condition, firstly, we provide sufficient conditions for the practical observer design. Then, we exhibit that practical Mittag-Leffler stability of the closed loop system with a linear, state feedback is attained. Finally, a separation principle is established and we prove that the closed loop system is practical Mittag-Leffler stable

Filtrage et commande basée sur un observateur pour les systèmes stochastiques

This thesis deals with the filtering and control of nonlinear systems described by Itô stochastic differential equations whose diffusion is controlled by a noise which is multiplied with the state vector. The noise is a Wiener process, also known as Brownian motion. When the noise is multiplied with the state in a differential equation, it can stabilize or destabilize the system, which is not the case when the noise occurs additively with respect to the state. In addition, there are several types of stability for the systems described by stochastic differential equations, some being more conservative than others. In this manuscript, the goal is to relax the conditions of stability used in the literature using the almost sure exponential stability, also called exponential stability with probability equal to one. Three main fields are treated in this manuscript :(i) observers synthesis, (ii) stability and stabilization of stochastic systems, (iii) bounded real lemma for stochastic algebro-differential systems.A new theorem on the almost sure exponential stability of the equilibrium point of a class of triangular nonlinear stochastic systems is proposed : the stability of the whole system is ensured by the stability of each decoupled subsystem. The proof of this result is based on the boundedness of the Lyapunov exponents. It was shown that the problem of filtering of stochastic systems with multiplicative noises by imposing the almost sure exponential stability of the observation error can not be solved by using the Lyapunov type approaches available in the literature. This difficulty was overcome by using the triangular structure, associated with this filtering problem, which allows to split the original observer design problem into two decoupled subproblems : (i) demonstrate the stability of the stochastic differential equation describing the dynamics of the state to be estimated, (ii) stabilize the stochastic differential equation describing the dynamics of the observation error. This approach is based on the new theorem on the almost sure exponential stability of a class of Lipschitz triangular nonlinear stochastic systems mentioned above. This has been extended to nonlinear stochastic systems with one-sided Lipschitz nonlinearities. To ensure the stability of the observation error, a polytopic approach was proposed with a “descriptor” formalism (or algebro-differential). The results presented above have been extended to the synthesis of robust observers in the presence of parametric uncertainties. Conditions for asymptotic rejection of perturbations occurring in a stochastic differential equation with multiplicative noises have been proposed. The considered stability is the almost sure exponential one. A bound of the Lyapunov exponent ensures the almost sure convergence rate to zero for the state of the system. A bang-bang control law is synthesized for a class of stochastic nonlinear systems in two cases : (i) state feedback and (ii) measured output feedback with an observer. The used stability is the almost sure exponential one. A version of the bounded real lemma is developed for stochastic algebro-differential systems (also called singular systems or descriptor systems) with multiplicative noises. This work required the development of Itô formula in the case of nonlinear stochastic algebro-differential equations. This approach has been used for the synthesis of an H∞ measured output feedback control law with the exponential mean square stability. An observer for nonlinear stochastic algebro-differential systems was proposed using the almost sure exponential stability.Ce mémoire de thèse traite du filtrage et de la commande des systèmes non linéaires décrits par des équations différentielles stochastiques au sens d’Itô dont la diffusion est commandée par un bruit qui intervient de manière multiplicative avec l’état. Ce bruit est un processus de Wiener, aussi appelé mouvement brownien. Lorsque le bruit agit de manière multiplicative avec l’état dans une équation différentielle, il peut stabiliser ou déstabiliser le système, ce qui n’est pas le cas lorsque le bruit intervient de manière additive. Il y a plusieurs types de stabilité pour les systèmes décrits par des équations différentielles stochastiques, certaines étant plus pessimistes que d’autres. Dans ce manuscrit, nous avons cherché à relaxer les conditions de stabilité utilisées dans la littérature en employant la stabilité exponentielle presque sûre, aussi appelée stabilité exponentielle avec une probabilité de un. Trois domaines principaux sont traités dans ce manuscrit :(i) synthèse d’observateurs, (ii) commande des systèmes stochastiques,(iii) lemme borné réel pour les systèmes stochastiques algébro-différentiels.Un nouveau théorème sur la stabilité exponentielle presque sûre du point d’équilibre d’une classe de systèmes stochastiques non linéaires triangulaires est proposé : la stabilité de l’ensemble du système est assurée par la stabilité de chaque sous-système considéré isolément. La preuve de ce résultat est basée sur la majoration des exposants de Lyapunov. On a montré que le problème du filtrage des systèmes stochastiques avec des bruits multiplicatifs en imposant la stabilité exponentielle presque sûre de l’erreur d’observation ne peut pas être résolu en appliquant les approches de type Lyapunov disponibles dans la littérature. Cette difficulté a été surmontée en proposant d’exploiter la structure triangulaire associée à ce problème de filtrage, ce qui nous a permis de décomposer la synthèse de l’observateur en deux sous-problèmes découplés : (i) démontrer la stabilité de l’équation différentielle stochastique décrivant la dynamique de l’état à estimer, (ii) stabiliser l’équation différentielle stochastique décrivant la dynamique de l’erreur d’observation. Cette approche est basée sur le nouveau théorème sur la stabilité exponentielle presque sûre d’une classe de systèmes stochastiques non linéaires triangulaires et lipschitziens évoquée ci- dessus. Ce résultat a été étendu aux systèmes stochastiques non linéaires ayant des non linéarités de type one-sided Lipschitz. Pour garantir la stabilité de l’erreur d’observation, une approche de type polytopique a été proposée avec un formalisme “descripteur” (ou algébro-différentiel). Les résultats présentés ci-dessus ont été étendus à la synthèse d’observateurs robustes en présence d’incertitudes paramétriques. Des conditions pour le rejet asymptotique des perturbations intervenant dans une équation différen- tielle stochastique avec des bruits multiplicatifs ont été proposées. La stabilité considérée est la stabilité exponentielle presque sûre. Une borne de l’exposant de Lyapunov permet de garantir le taux de conver- gence vers zéro de l’état du système. Un correcteur de type bang-bang est synthétisé pour une classe de systèmes non linéaires stochastiques dans deux cas : (i) par retour d’état et (ii) par retour de sorties mesurées avec un observateur. Le type de stabilité utilisé est la stabilité exponentielle presque sûre. Une version du lemme borné réel est élaborée pour les systèmes stochastiques algébro-différentiels (ou singuliers, ou descripteurs) avec des bruits multiplicatifs. Ce travail a nécessité le développement de la formule d’Itô dans le cas des équations stochastiques algébro-différentielles non linéaires. Cette approche a été utilisée pour la synthèse d’un correcteur H∞ par retour de sorties en utilisant la stabilité exponentielle en moyenne quadratique. Un observateur pour les systèmes stochastiques algébro-différentiels non linéaires a été proposé avec la stabilité exponentielle presque sûre

Filtering of SPDEs: The Ensemble Kalman Filter and related methods

This paper is concerned with the derivation and mathematical analysis of continuous time Ensemble Kalman Filters (EnKBFs) and related data assimilation methods for Stochastic Partial Differential Equations (SPDEs) with finite dimensional observations. The signal SPDE is allowed to be nonlinear and is posed in the standard abstract variational setting. Its coefficients are assumed to satisfy global one-sided Lipschitz conditions. We first review classical filtering algorithms in this setting, namely the Kushner--Stratonovich and the Kalman--Bucy filter, proving a law of total variance. Then we consider mean-field filtering equations, deriving both a Feedback Particle Filter and a mean-field EnKBF for nonlinear signal SPDEs. The second part of the paper is devoted to the elementary mathematical analysis of the EnKBF in this infinite dimensional setting, showing the well posedness of both the mean-field EnKBF and its interacting particle approximation. Finally we prove the convergence of the particle approximation. Under the additional assumption that the observation function is bounded, we even recover explicit and (nearly) optimal rates

Robust Fault Detection for a Class of Uncertain Nonlinear Systems Based on Multiobjective Optimization

A robust fault detection scheme for a class of nonlinear systems with uncertainty is proposed. The proposed approach utilizes robust control theory and parameter optimization algorithm to design the gain matrix of fault tracking approximator (FTA) for fault detection. The gain matrix of FTA is designed to minimize the effects of system uncertainty on residual signals while maximizing the effects of system faults on residual signals. The design of the gain matrix of FTA takes into account the robustness of residual signals to system uncertainty and sensitivity of residual signals to system faults simultaneously, which leads to a multiobjective optimization problem. Then, the detectability of system faults is rigorously analyzed by investigating the threshold of residual signals. Finally, simulation results are provided to show the validity and applicability of the proposed approach

Infinite series representation of fractional calculus: theory and applications

This paper focuses on the equivalent expression of fractional integrals/derivatives with an infinite series. A universal framework for fractional Taylor series is developed by expanding an analytic function at the initial instant or the current time. The framework takes into account of the Riemann-Liouville definition, the Caputo definition, the constant order and the variable order. On this basis, some properties of fractional calculus are confirmed conveniently. An intuitive numerical approximation scheme via truncation is proposed subsequently. Finally, several illustrative examples are presented to validate the effectiveness and practicability of the obtained results

Euler's method applied to the control of switched systems

Hybrid systems are a powerful formalism for modeling and reasoning about cyber-physical systems. They mix the continuous and discrete natures of the evolution of computerized systems. Switched systems are a special kind of hybrid systems, with restricted discrete behaviours: those systems only have finitely many different modes of (continuous) evolution, with isolated switches between modes. Such systems provide a good balance between expressiveness and controllability, and are thus in widespread use in large branches of industry such as power electronics and automotive control. The control law for a switched system defines the way of selecting the modes during the run of the system. Controllability is the problem of (automatically) synthezing a control law in order to satisfy a desired property, such as safety (maintaining the variables within a given zone) or stabilisation (confinement of the variables in a close neighborhood around an objective point). In order to compute the control of a switched system, we need to compute the solutions of the differential equations governing the modes. Euler's method is the most basic technique for approximating such solutions. We present here an estimation of the Euler's method local error, using the notion of " one-sided Lispchitz constant " for modes. This yields a general control synthesis approach which can encompass several features such as bounded disturbance and compositionality