This thesis compiles the publications I coauthored during my doctoral studies at University of Leipzig on the subject of optimizing thermodynamic systems, focusing on three optimization perspectives: maximum efficiency, maximum power, and maximum efficiency at given power. We considered two currently intensely studied models in finite-time thermodynamics, i.e., low-dissipation models and Brownian systems. The low-dissipation model is used to derive general bounds on the performance of real-world machines, while Brownian systems allow us to better understand the practical limits and features of small systems. First, we derived maximum efficiency at given power for various low-dissipation setups, with a particular focus on the behavior close to maximum power, which helps us to determine whether it is more beneficial to operate the system at maximum power, near maximum power or in a different regime. Then, we move to the design of maximum-efficiency and maximum-power protocols for Brownian systems under different boundary conditions. Particularly, when the constraints on control parameters are experimentally motivated, we presented a geometric method yielding maximum-efficiency and maximum-power protocols valid for systems with periodically scaled energy spectrum and otherwise arbitrary dynamics. Each chapter contains a short informal introduction to the matter as well as an outlook, pointing out the direction for our research in the future.

Ce mémoire de thèse traite de la stabilité en temps fini et de la stabilisation en temps fini des systèmes dynamiques. En effet, il est souvent important de garantir que pendant le régime transitoire, les trajectoires d'état ne dépassent pas certaines limites prédéfinies afin d'éviter les saturations et l'excitation des non-linéarités du système. Un système dynamique est dit stable en temps fini FTS si, pour tout état initial appartenant à un ensemble borné prédéterminé, la trajectoire d'état reste comprise dans un autre ensemble borné prédéterminé pendant un temps fini et fixé. Lorsque le système est perturbé, on parle de bornitude en temps fini FTB. Premièrement, des nouvelles conditions suffisantes assurant la synthèse d'un correcteur FTB par retour de sortie dynamique des systèmes linéaires continus invariants perturbés ont été développées via une approche descripteur originale. Le résultat a été établi par une transformation de congruence particulière. Les conditions obtenues sont sous forme de LMIs. Deuxièmement, l'utilisation de la notion d'annulateur combinée avec le lemme de Finsler, permet d’obtenir des nouvelles conditions sous formes LMIs garantissant la stabilité et la stabilisation en temps fini des systèmes non linéaires quadratiques. Enfin, pour obtenir des conditions encore moins pessimistes dans un contexte de stabilité en temps fini, de nouveaux développements ont été proposés en utilisant des fonctions de Lyapunov polynomiales / This dissertation deals with the finite time stability and the finite time stabilization of dynamic systems. Indeed, it is often important to ensure that during the transient regime, the state trajectories do not exceed certain predefined limits in order to avoid saturations and excitations of the nonlinearities of the system. Hence the interest is to study the stability of the dynamic system in finite time. A dynamic system is said to be stable in finite time (FTS) if, for any initial state belonging to a predetermined bounded set, the state trajectory remains within another predetermined bounded set for a finite and fixed time. When the system is disturbed, it is called finite time boundedness (FTB). In this manuscript, the goal is to improve the results of finite time stability used in the literature. First, new sufficient conditions expressed in terms of LMIs for the synthesis of an FTB controller by dynamic output feedback have been developed via an original descriptor approach. An original method has been proposed which consists in using a particular congruence transformation. Second, new LMI conditions for the study of finite time stability and finite time stabilization have been proposed for disturbed and undisturbed nonlinear quadratic systems. Third, to obtain even less conservative conditions, new developments have been proposed using polynomial Lyapunov functions

Cette thèse étudie le problème d'identification de la topologie d'un réseau de systèmes complexes dynamiques, dont les sous-systèmes sont décrits par des équations différentielles ordinaires (EDO) et/ou par des équations différentielles à retard (EDR). La première partie de ce travail porte sur l’identification des paramètres du réseau de systèmes linéaires. Ainsi, différentes classes de systèmes linéaires ont été traitées, à savoir les systèmes sans retard, les systèmes à retard commensurable et les systèmes à entrées inconnues. Un observateur impulsif est proposé afin d'identifier à la fois les états et les paramètres inconnus de la classe de système dynamique considérée en temps fini. Afin de garantir l'existence de l'observateur impulsif proposé, des conditions suffisantes sont déduites. Des exemples illustratifs sont donnés afin de montrer l'efficacité de l'observateur en temps fini proposé.La deuxième partie de ce travail traite le problème de l'identification de la topologie d'un réseau de systèmes dynamiques non linéaires. Dans nos considérations, les coefficients interconnexions de la topologie du réseau sont considérés comme des paramètres constants. Par conséquent, l'identification de la topologie est équivalente à l'identification des paramètres inconnus. Tout d’abord, nous avons déduit des conditions suffisantes sur l’identifiabilité des paramètres, puis nous avons proposé un différenciateur uniforme avec convergence en temps fini pour estimer les paramètres inconnus / This thesis investigates the topology identification problem for network of dynamical complex systems, whose subsystems are described by ordinary differential equations (ODE) and/or delay differential equations (DDE). The first part of this work focuses on the parameters identification of the network of linear systems. Thus, different classes of linear systems have been treated namely systems without delay, systems with commensurable delay and systems with unknown inputs. An impulsive observer is proposed in order to identify both the states and the unknown parameters of the considered class of dynamic system in finite time. In order to guarantee the existence of the proposed impulsive observer, sufficient conditions are deduced. An illustrative example is given in order to show the efficiency of the proposed finite-time observer.The second part of this work treats the topology identification of the network of nonlinear dynamic systems. In our considerations, the topology connections are represented as constant parameters, therefore the topology identification is equivalent to identify the unknown parameters. A sufficient condition on parameter identifiability is firstly deduced, and then a uniform differentiator with finite-time convergence is proposed to estimate the unknown parameters

Ce travail traite de l'évitement d'obstacles pour les robots mobiles à roues. D’abord, deux solutions sont proposées dans le cas d’un seul robot autonome. La première est une amélioration de la technique des champs de potentiel afin de contraster l’apparition de minima locaux. Le résultat se base sur l’application de la définition de l’ «Input-to-State Stability» pour des ensembles décomposables. Chaque fois que le robot mobile approche un minimum local l’introduction d’un contrôle dédié lui permet de l’éviter et de terminer la tâche. La deuxième solution se base sur l’utilisation de la technique du «Supervisory Control» qui permet de diviser la tâche principale en deux sous tâches : un algorithme de supervision gère deux signaux de commande, le premier en charge de faire atteindre la destination, le deuxième d’éviter les obstacles. Les deux signaux de commande permettent de compléter la mission en temps fini en assurant la robustesse par rapport aux perturbations représentant certaines dynamiques négligées. Les deux solutions ont été mises en service sur un robot mobile «Turtlebot 2». Pour contrôler une formation de type leader-follower qui puisse éviter collisions et obstacles, une modification de l’algorithme de supervision précédent a été proposée ; elle divise la tâche principale en trois sous-problèmes gérés par trois lois de commande. Le rôle du leader est adapté pour être la référence du groupe avec un rôle actif : ralentir la formation en cas de manœuvre d'évitement pour certains robots. La méthode proposée permet au groupe de se déplacer et à chaque agent d’éviter les obstacles, ou les collisions, de manière décentralisée / This dissertation work addresses the obstacle avoidance for wheeled mobile robots. The supervisory control framework coupled with the output regulation technique allowed to solve the obstacle avoidance problem and to formally prove the existence of an effective solution: two outputs for two objectives, reaching the goal and avoiding the obstacles. To have fast, reliable and robust results the designed control laws are finite-time, a particular class very appropriate to the purpose. The novelty of the approach lies in the easiness of the geometric approach to avoid the obstacle and on the formal proof provided under some assumptions. The solution have been thus extended to control a leader follower formation which, sustained from the previous result, uses two outputs but three controls to nail the problem. The Leader role is redesigned to be the reference of the group and not just the most advanced agent, moreover it has a active role slowing down the formation in case of collision avoidance manoeuvre for some robots. The proposed method, formally proven, makes the group move together and allow each agent to avoid obstacles or collision in a decentralized way. In addition, a further contribution of this dissertation, it is represented by a modification of the well known potential field method to avoid one of the common drawback of the method: the appearance of local minima. Control theory tools helps again to propose a solution that can be formally proven: the application of the definition of Input-to-State Stability (ISS) for decomposable sets allows to treat separate obstacles adding a perturbation which is able to move the trajectory away from a critic point

A ratchet is an asymmetric, non-equilibrated system that can produce a directed current of particles without the need for macroscopic potential gradients. In rocked quantum electron ratchets, tunnelling and wave-reflection can induce reversals in the direction of the net current as a function of system parameters. An asymmetric quantum point contact in a GaAs/GaAlAs heterostructure has been studied experimentally as a realisation of a quantum electron ratchet. A Landauer model predicts reversals in the direction of the net current as a function of temperature, amplitude of the rocking voltage, and Fermi energy. Artifacts such as circuit-induced asymmetry, also known as self-gating, were carefully removed from the experimental data, which showed net current and net differential conductance reversals, as predicted by the model. The model also predicts the existence of a heat current where the net electron current changes sign, as equal numbers of high and low energy electrons are pumped in opposite directions. An idealised quantum electron ratchet is studied analytically as an energy selective electron heat engine and refrigerator. The hypothetical device considered consists of two electron reservoirs with different temperatures and Fermi energies. The reservoirs are linked via a resonant state in a quantum dot, which functions as an idealised energy filter for electrons. The efficiency of the device approaches the Carnot value when the energy transmitted by the filter is tuned to that where the Fermi distributions in the reservoirs are equal. The maximum power regime, where the filter transmits all electrons that contribute positively to the power, is also examined. Analytic expressions are obtained for the power and efficiency of the idealised device as both a heat engine and as a refrigerator in this regime of operation. The expressions depend on the ratio of the voltage to the difference in temperature of the reservoirs, and on the ratio of the reservoir temperatures. The energy selective electron heat engine is shown to be non-endoreversible, and to operate in an analogous manner to the three-level amplifier, a laser based quantum heat engine. Implications for improving the efficiency of thermionic refrigerators and power generators are discussed.

Made available in DSpace on 2016-12-12T20:15:52Z (GMT). No. of bitstreams: 1 Rafael Marques.pdf: 14630401 bytes, checksum: 352c48d8c01e0db8ba6b41f5e8317ec8 (MD5) Previous issue date: 2015-03-11 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The main subject developed in this dissertation is the characterization of the dynamics of high-dimensional conservative systems using different statistical approaches. Looking at the conservative system phase-space, we can find chaotic and regular regions that are characterized by a random distribution of points and periodic structures formed by closed orbits, respectively. The nonlinearity parameter has a fundamental hole to the occurrence of chaotic trajectories that can get stuck for a finite time on the vicinity of regular regions. This phenomenon is known as stickiness effect and can be identified using different tools as the spectrum of finite time Lyapunov exponents or the recurrence time statistics (RTS), e.g. Throughout this dissertation, we propose to characterize this effect using such approaches and also apply a new methodology which uses the time series of the spectrum of finite time Lyapunov exponents to separate the dynamics in different regimes of motion. For this purpose, we study two conservative systems that are derived from standard map, a symplectic map extensively used to investigate the transition from regular to chaotic dynamic. The first system consists in a chain of coupled standard maps that originates a 2N-dimensional system, where N is the number of coupled maps. Using this system, from the definition of regimes of motion, we obtained the cumulative distribution of the consecutive time that the trajectory spends in a particular regime, which reproduces with a good precision the results obtained when using the RTS. The second system studied was the Modified Standard Map, which is obtained adding an action variable to the standard map. The coupling with an extra dimension allows the penetration of the regular structures by the trajectories, what was forbidden for the two-dimensional case. The application of the method of separation of regimes in this system enables a more detailed analysis of the stickiness effect, showing that only the trajectories located near the regular structures have Local Lyapunov exponents about zero. Thus, the development of this research contributes to a better understanding of the stickiness effect in high-dimensional conservative systems. / O tema principal desenvolvido nesta dissertação de Mestrado está relacionado com o estudo da dinâmica de sistemas conservativos, utilizando diferentes abordagens estatísticas. Ao analisarmos o espaço de fases de um sistema dinâmico pertencente a esta classe, podemos encontrar regiões caóticas e regulares que são caracterizadas pela distribuição aleatória de pontos e por estruturas periódicas formadas por órbitas fechadas, respectivamente. O parâmetro de não-linearidade tem um papel fundamental na existência de trajetórias caóticas que podem ser aprisionadas por um tempo finito nas proximidades das regiões regulares. Este fenômeno é conhecido como efeito stickiness, e pode ser identificado através da utilização de diferentes abordagens como, por exemplo, o espectro de Lyapunov calculado a tempo finito ou a estatística dos tempos de recorrência de Poincaré (ETR). No decorrer desta dissertação, propomos caracterizar o efeito stickiness utilizando tais abordagens, além de aplicar uma nova metodologia que consiste em analisar séries temporais do espectro de expoentes de Lyapunov afim de definir diferentes regimes de movimento. Para isso, estudamos dois sistemas conservativos multidimensionais derivados do mapa padrão, um mapa simplético muito utilizado para a investigação da transição da dinâmica regular para caótica. O primeiro deles consiste em uma rede de mapas padrão acoplados que dá origem a um sistema de 2N-dimensões, sendo N o número de mapas acoplados. Utilizando este sistema, a partir da definição de regimes de movimento, foi possível determinar a distribuição cumulativa do tempo consecutivo que a trajetória permanece em um determinado regime, sendo que os resultados obtidos por meio da análise desta quantidade podem reproduzir de forma satisfatória aqueles obtidos quando utilizamos a ETR. O segundo sistema estudado foi o Mapa Padrão Modificado (MPM), resultante do acoplamento entre uma variável ação extra e o mapa padrão tradicional. O acoplamento com uma dimensão extra permite que trajetórias penetrem nas regiões de regularidade, o que antes era proibido para o caso bidimensional. A aplicação da técnica de separação de regimes neste sistema permite uma análise mais detalhada do efeito stickiness, mostrando que apenas trajetórias que se encontram em torno das estruturas de regularidade possuem expoentes de Lyapunov Locais com valores próximos a zero. Desta forma, o desenvolvimento desta pesquisa contribui para o melhor entendimento do efeito stickiness em sistemas conservativos de alta dimensionalidade.

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In this dissertation, we consider two different strategies for improving the projection-based reduced order model (ROM) accuracy: (I) adding closure terms to the standard ROM; (II) using Lagrangian data to improve the ROM basis. Following strategy (I), we propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS-ROM closure terms, i.e., the terms representing the interactions among the three types of scales. In the third step, we use available data to model the VMS-ROM closure terms. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMS-ROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. We test the new data-driven VMS-ROM in the numerical simulation of four test cases: (i) the 1D Burgers equation with viscosity coefficient $nu = 10^{-3}$; (ii) a 2D flow past a circular cylinder at Reynolds numbers $Re=100$, $Re=500$, and $Re=1000$; (iii) the quasi-geostrophic equations at Reynolds number $Re=450$ and Rossby number $Ro=0.0036$; and (iv) a 2D flow over a backward facing step at Reynolds number $Re=1000$. The numerical results show that the data-driven VMS-ROM is significantly more accurate than standard ROMs. Furthermore, we propose a new hybrid ROM framework for the numerical simulation of fluid flows. This hybrid framework incorporates two closure modeling strategies: (i) A structural closure modeling component that involves the recently proposed data-driven variational multiscale ROM approach, and (ii) A functional closure modeling component that introduces an artificial viscosity term. We also utilize physical constraints for the structural ROM operators in order to add robustness to the hybrid ROM. We perform a numerical investigation of the hybrid ROM for the three-dimensional turbulent channel flow at a Reynolds number $Re = 13,750$. In addition, we focus on the mathematical foundations of ROM closures. First, we extend the verifiability concept from large eddy simulation to the ROM setting. Specifically, we call a ROM closure model verifiable if a small ROM closure model error (i.e., a small difference between the true ROM closure and the modeled ROM closure) implies a small ROM error. Second, we prove that a data-driven ROM closure (i.e., the data-driven variational multiscale ROM) is verifiable. For strategy (II), we propose new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct new Lagrangian ROMs. We show that the new Lagrangian ROMs are orders of magnitude more accurate than the standard Eulerian ROMs, i.e., ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs' accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis. / Doctor of Philosophy / Reduced order models (ROMs) are popular in physical and engineering applications: for example, ROMs are widely used in aircraft designing as it can greatly reduce computational cost for the aircraft's aeroelastic predictions while retaining good accuracy. However, for high Reynolds number turbulent flows, such as blood flows in arteries, oil transport in pipelines, and ocean currents, the standard ROMs may yield inaccurate results. In this dissertation, to improve ROM's accuracy for turbulent flows, we investigate three different types of ROMs. In this dissertation, both numerical and theoretical results show that the proposed new ROMs yield more accurate results than the standard ROM and thus can be more useful.

Exploring the concepts of long range aerial transport of microorganisms is the main motivation of this study. For this purpose we use theories and concepts of dynamical systems in the context of geophysical fluid systems. We apply powerful notions such as finite-time Lyapunov exponent (FTLE) and the associated Lagrangian coherent structures (LCS) and we attempt to provide mathematical explanations and frameworks for some applied questions which are based on realistic concerns of atmospheric transport phenomena. Accordingly, we quantify the accuracy of prediction of FTLE-LCS features and we determine the sensitivity of such predictions to forecasting parameters. In addition, we consider the spatiotemporal resolution of the operational data sets and we propose the concept of probabilistic source and destination regions which leads to the definition of stochastic FTLE fields. Moreover, we put forward the idea of using ensemble forecasting to quantify the uncertainty of the forecast results. Finally, we investigate the statistical properties of localized measurements of atmospheric microbial structure and their connections to the concept of local FTLE time-series. Results of this study would pave the way for more efficient models and management strategies for the spread of infectious diseases affecting plants, domestic animals, and humans. / Ph. D.