Geometric synthesis of a hybrid limit cycle for the stabilizing control of a class of nonlinear switched dynamical systems

International audienceThis paper proposes a new constructive method for synthesizing a hybrid limit cycle for the stabilizing control of a class of switched dynamical systems in IR 2 , switching between two discrete modes and without state discontinuity. For each mode, the system is continuous, linear or nonlinear. This method is based on a geometric approach. The first part of this paper demonstrates a necessary and sufficient condition of the existence and stability of a hybrid limit cycle consisting of a sequence of two operating modes in IR 2 which respects the technological constraints (minimum duration between two successive switchings, boundedness of the real valued state variables). It outlines the established method for reaching this hybrid limit cycle from an initial state, and then stablizing it, taking into account the constraints on the continuous variables. This is then illustrated on a Buck electrical energy converter and a nonlinear switched system in IR 2. The second part of the paper proposes and demonstrates an extension to IR n for a class of systems, which is then illustrated on a nonlinear switched system in IR 3

Event-triggered control for piecewise affine discrete-time systems

In the present work, we study the problems of stability analysis of piecewise-affine (PWA) discrete-time systems, and trigger-function design for discrete-time event-triggered control systems. We propose a representation for piecewise-affine systems in terms of ramp functions, and we rely on Lyapunov theory for the stability analysis. The proposed implicit piecewise-affine representation prevents the shortcomings of the existing stability analysis approaches of PWA systems. Namely, the need to enumerate regions and allowed transitions of the explicit representations. In this context, we can emphasize two benefits of the proposed approach: first, it makes possible the analysis of uncertainty in the partition and, thus, the transitions. Secondly, it enables the analysis of event-triggered control systems for the class of PWA systems since, for ETC, the transitions cannot be determined as a function of the state variables. The proposed representation, on the other hand, implicitly encodes the partition and the transitions. The stability analysis is performed with Lyapunov theory techniques. We then present conditions for exponential stability. Thanks to the implicit representation, the use of piecewise quadratic Lyapunov functions candidates becomes simple. These conditions can be solved numerically using a linear matrix inequality formulation. The numerical analysis exploits quadratic expressions that describe ramp functions to verify the positiveness of extended quadratic forms. For ETC, a piecewise quadratic trigger function defines the event generator. We find suitable parameters for the trigger function with an optimization procedure. As a result, this function uses the information on the partition to reduce the number of events, achieving better results than the standard quadratic trigger functions found in the literature. We provide numerical examples to illustrate the application of the proposed representation and methods.Ce manuscrit présente des résultats sur l’analyse de stabilité des systèmes affines par morceaux en temps discret et sur le projet de fonctions de déclenchement pour des stratégies de commande par événements. Nous proposons une représentation pour des systèmes affines par morceaux et l’on utilise la théorie de stabilité de Lyapunov pour effectuer l’analyse de stabilité globale de l’origine. La nouvelle représentation implicite que nous proposons rend plus simple l’analyse de stabilité car elle évite l’énumération des régions et des transitions entre régions tel que c’est fait dans le cas des représentations explicites. Dans ce contexte nous pouvons souligner deux avantages principaux, à savoir I) la possibilité de traiter des incertitudes dans la partition qui définit le système et, par conséquent des incertitudes dans les transitions, II) l’analyse des stratégies de commande par événements pour des systèmes affines par morceaux. En effet, dans ces stratégies les transitions ne peuvent pas être définies comme des fonctions des variables d’état. La théorie de stabilité de Lyapunov est utilisée pour établir des conditions pour la stabilité exponentielle de l’origine. Grâce à la représentation implicite des partitions nous utilisons des fonctions de Lyapunov quadratique par morceaux. Ces conditions sont données par des inégalités dont la solution numérique est possible avec une formulation par des inégalités matricielles linéaires. Ces formulations numériques se basent sur des expressions quadratiques décrivant des fonctions rampe. Pour des stratégies par événement, une fonctions quadratique par morceaux est utilisée pour le générateur d’événements. Nous calculons les paramètres de ces fonctions de déclenchement a partir de solutions de problèmes d’optimisation. Cette fonction de déclenchement quadratique par morceaux permet de réduire le nombre de d’événementsen comparaison avec les fonctions quadratiques utilisées dans la littérature. Nous utilisons des exemples numériques pour illustrer les méthodes proposées.No presente trabalho, são estudados os problemas de análise de estabilidade de sistemas afins por partes e o projeto da função de disparo para sistemas de controle baseado em eventos em tempo discreto. É proposta uma representação para sistemas afins por partes em termos de funções rampa, e é utilizada a teoria de Lyapunov para a análise de estabilidade. A representação afim por partes implícita proposta evita algumas das deficiências das abordagens de análise de estabilidade de sistemas afins por partes existentes. Em particular, a necessidade de anumerar regiões e transições admissíveis das representações explícitas. Neste contexto, dois benefícios da abordagem proposta podem ser enfatizados: primeiro, ela torna possível a análise de incertezas na partição, e, assim, nas transições. Segundo, ela permite a análise de sistemas de controle baseado em eventos para a classe de sistemas afins por partes, já que, para o controle baseado em eventos, as transições não podem ser determinadas como uma função das variáveis de estado. A representação proposta, por outro lado, codifica implicitamente a partição e as transições. A análise de estabilidade é realizada com técnicas da teoria de Lyapunov. Condi- ções para a estabilidade exponencial são então apresentadas. Graças à representação implícita, o uso de funções candidatas de Lyapunov se torna simples. Essas condições podem ser resolvidas numéricamente usando uma formulação de desigualdades matriciais lineares. A análise numérica explora expressões quadráticas que descrevem funções de rampa para verificar a postivividade de formas quadráticas extendidas. Para o controle baseado em eventos, uma função de disparo quadrática por partes define o gerador de eventos. Parâmetros adequados para a função de disparo sãoencontrados com um procedimento de otimização. Como resultado, esta função usa informação da partição para reduzir o número de eventos, obtendo resultados melhores do que as funções de disparo quadráticas encontradas na literatura. Exemplos numéricos são fornecidos para ilustrar a aplicação da representação e mé- todos propostos

Commande dynamique de robots déformables basée sur un modèle numérique

This work focuses on modeling and control of soft robots. It covers the entire development of the controller, from the modeling step to the practical experimental validation.From a theoretical point a view, large-scale dynamical systems along with model reduction algorithms are studied. In addition to the theoretical studies, different experimental setups are used to illustrate the results. A cable-driven soft robot and a pressurized soft arm are used to test the control algorithms. Through these different setups, we show that the method can handle different types of actuation, different geometries and mechanical properties. This emphasizes one of the interests of the method, its genericity.Cette thèse s'intéresse à la modélisation et à la commande de robots déformables (robots dont le mouvement se fait par déformation). Nous nous intéressons à la conception de lois de contrôle en boucle fermée répondant aux besoins spécifiques du contrôle dynamique de ces robots, sans restrictions fortes sur leur géométrie. La résolution de ce défi soulève des questions théoriques qui nous amènent au deuxième objectif de cette thèse: développer de nouvelles stratégies pour étudier les systèmes de grandes dimensions

Nonlinear control of wheeled mobile robots

The purpose of this project is to implement an autonomous navigation system using nonlinear control techniques to control a wheeled mobile robot (WMR) to follow a preplanned trajectory and track a path. Two other aspects of navigation are studied: path planning and obstacle avoidance. Those three aspects are integrated into a navigation strategy that manages navigation and prevents deadlocks. Two nonlinear control techniques for path tracking and trajectory following have been developed and implemented. In the first approach, a fuzzy logic controller is used to drive the robot through a set of waypoints leading to the destination. In another approach, a controller derived from a Lyapunov function is used to track a reference trajectory that is time dependent. For path planning, a novel optimization technique based on dynamic programming has been developed. The curvature velocity method has been used for obstacle avoidance. The testing was conducted on a P3-AT all-terrain mobile robot equipped with encoders, a gyroscope, and sonar sensors for localization and environment perception. The test results validate the effectiveness of the different approaches that have been developed

A port-Hamiltonian formulation of flexible structures. Modelling and structure-preserving finite element discretization

Despite the large literature on port-Hamiltonian (pH) formalism, elasticity problems in higher geometrical dimensions have almost never been considered. This work establishes the connection between port-Hamiltonian distributed systems and elasticity problems. The originality resides in three major contributions. First, the novel pH formulation of plate models and coupled thermoelastic phenomena is presented. The use of tensor calculus is mandatory for continuum mechanical models and the inclusion of tensor variables is necessary to obtain an equivalent and intrinsic, i.e. coordinate free, pH description. Second, a finite element based discretization technique, capable of preserving the structure of the infinite-dimensional problem at a discrete level, is developed and validated. The discretization of elasticity problems requires the use of non-standard finite elements. Nevertheless, the numerical implementation is performed thanks to well-established open-source libraries, providing external users with an easy to use tool for simulating flexible systems in pH form. Third, flexible multibody systems are recast in pH form by making use of a floating frame description valid under small deformations assumptions. This reformulation include all kinds of linear elastic models and exploits the intrinsic modularity of pH systems

Moment-sum-of-squares hierarchies for set approximation and optimal control

This thesis uses the idea of lifting (or embedding) a nonlinear controlled dynamical system into an infinite-dimensional space of measures where this system is equivalently described by a linear equation. This equation and problems involving it are subsequently approximated using well-known moment-sum-of-squares hierarchies. First, we address the problems of region of attraction, reachable set and maximum controlled invariant set computation, where we provide a characterization of these sets as an infinite-dimensional linear program in the cone of nonnegative measures and we describe a hierarchy of finite-dimensional semidefinite-programming (SDP) hierarchies providing a converging sequence of outer approximations to these sets. Next, we treat the problem of optimal feedback controller design under state and input constraints. We provide a hierarchy of SDPs yielding an asymptotically optimal sequence of rational feedback controllers. In addition, we describe hierarchies of SDPs yielding approximations to the value function attained by any given rational controller, from below and from above, as well as a hierarchy of SDPs providing approximations from below to the optimal value function, hence obtaining performance certificates for the designed controllers as well as for any given rational controller. Finally, we describe a method to verify properties of a closed loop interconnection of a nonlinear dynamical system and an optimization-based controller (e.g., a model predictive controller) for deterministic and stochastic nonlinear dynamical systems. Properties such as global stability, the $\ell_2$ gain or performance with respect to a given infinite-horizon cost function can be certified. The methods presented are easy to implement using freely available software packages and are documented by a number of numerical examples

Imitation Learning of Motion Coordination in Robots:a Dynamical System Approach

The ease with which humans coordinate all their limbs is fascinating. Such a simplicity is the result of a complex process of motor coordination, i.e. the ability to resolve the biomechanical redundancy in an efficient and repeatable manner. Coordination enables a wide variety of everyday human activities from filling in a glass with water to pair figure skating. Therefore, it is highly desirable to endow robots with similar skills. Despite the apparent diversity of coordinated motions, all of them share a crucial similarity: these motions are dictated by underlying constraints. The constraints shape the formation of the coordination patterns between the different degrees of freedom. Coordination constraints may take a spatio-temporal form; for instance, during bimanual object reaching or while catching a ball on the fly. They also may relate to the dynamics of the task; for instance, when one applies a specific force profile to carry a load. In this thesis, we develop a framework for teaching coordination skills to robots. Coordination may take different forms, here, we focus on teaching a robot intra-limb and bimanual coordination, as well as coordination with a human during physical collaborative tasks. We use tools from well-established domains of Bayesian semiparametric learning (Gaussian Mixture Models and Regression, Hidden Markov Models), nonlinear dynamics, and adaptive control. We take a biologically inspired approach to robot control. Specifically, we adopt an imitation learning perspective to skill transfer, that offers a seamless and intuitive way of capturing the constraints contained in natural human movements. As the robot is taught from motion data provided by a human teacher, we exploit evidence from human motor control of the temporal evolution of human motions that may be described by dynamical systems. Throughout this thesis, we demonstrate that the dynamical system view on movement formation facilitates coordination control in robots. We explain how our framework for teaching coordination to a robot is built up, starting from intra-limb coordination and control, moving to bimanual coordination, and finally to physical interaction with a human. The dissertation opens with the discussion of learning discrete task-level coordination patterns, such as spatio-temporal constraints emerging between the two arms in bimanual manipulation tasks. The encoding of bimanual constraints occurs at the task level and proceeds through a discretization of the task as sequences of bimanual constraints. Once the constraints are learned, the robot utilizes them to couple the two dynamical systems that generate kinematic trajectories for the hands. Explicit coupling of the dynamical systems ensures accurate reproduction of the learned constraints, and proves to be crucial for successful accomplishment of the task. In the second part of this thesis, we consider learning one-arm control policies. We present an approach to extracting non-linear autonomous dynamical systems from kinematic data of arbitrary point-to-point motions. The proposed method aims to tackle the fundamental questions of learning robot coordination: (i) how to infer a motion representation that captures a multivariate coordination pattern between degrees of freedom and that generalizes this pattern to unseen contexts; (ii) whether the policy learned directly from demonstrations can provide robustness against spatial and temporal perturbations. Finally, we demonstrate that the developed dynamical system approach to coordination may go beyond kinematic motion learning. We consider physical interactions between a robot and a human in situations where they jointly perform manipulation tasks; in particular, the problem of collaborative carrying and positioning of a load. We extend the approach proposed in the second part of this thesis to incorporate haptic information into the learning process. As a result, the robot adapts its kinematic motion plan according to human intentions expressed through the haptic signals. Even after the robot has learned the task model, the human still remains a complex contact environment. To ensure robustness of the robot behavior in the face of the variability inherent to human movements, we wrap the learned task model in an adaptive impedance controller with automatic gain tuning. The techniques, developed in this thesis, have been applied to enable learning of unimanual and bimanual manipulation tasks on the robotics platforms HOAP-3, KATANA, and i-Cub, as well as to endow a pair of simulated robots with the ability to perform a manipulation task in the physical collaboration

Commande hiérarchique de réseaux multi-terminaux à courant continu et haute tension

This thesis focuses on the hierarchical control for a multi-terminal high voltage direct current (MT-HVDC) grid suitable for the integration of large scale renewable energy sources. The proposed control scheme is composed of 4 layers, from the low local control at the power converters in the time scale of units of ms; through distributed droop control (primary control) applied in several terminals in the scale of unit of seconds; and then to communication based Model Predictive Control (MPC) that assures the load flow and the steady state voltage/power plan for the whole system, manage large scale storage and include weather forecast (secondary control); finally reaching the higher level controller that is mostly based on optimization techniques, where economic aspects are considered in the same time as longer timespan weather forecast (tertiary control).Concerning the converters' level, special emphasis is placed on DC/DC bidirectional converters. In this thesis, three different topologies are studied in depth: two phases dual active bridge (DAB), the three phases DAB, and the use of the Modular Multilevel Converter (MMC) technology as DC/DC converter. For each topology a specific non-linear control is presented and discussed. In addition, the DC/DC converter can provide other important services as its use as a direct current circuit breaker (DC-CB). Several operation strategies are studied for these topologies used as DC-CB.With respect to primary control, which is the responsible to maintain the DC voltage control of the grid, we have studied several control philosophies: master/slave, voltage margin control and droop control. Finally we have chosen to use droop control, among other reasons, because the communication between nodes is not required. Relative to the secondary control, its main goal is to schedule power transfer between the network nodes providing voltage and power references to local and primary controllers, providing steady state response to disturbances and managing power reserves. In this part we have proposed a new approach to solve the power flow problem (non-linear equations) based on the contraction mapping theorem, which gives the possibility to use more than one bus for the power balance (slack bus) instead of the classic approach based on the Newton-Raphson method. Secondary control plays a very important role in practical applications, in particular when including time varying power sources, as renewable ones. In such cases, it is interesting to consider storage devices in order to improve the stability and the efficiency of the whole system. Due to the sample time of secondary control is on the order of minutes, it is also possible to consider different kinds of forecast (weather, load,..) and to achieve additional control objectives, based on managing storage reserves. All these characteristics encourage the use of a model predictive control (MPC) approach to design this task. In this context, several possibilities of optimization objective were considered, like to minimize transmission losses or to avoid power network congestions.The main task of tertiary control is to manage the load flow of the whole HVDC grid in order to achieve economical optimization. This control level provides power references to the secondary controller. In this thesis we were able to maximize the economic profit of the system by acting on the spot market, and by optimizing the use of storage devices. In this level it is again used the MPC approach.With the aim of implementing the hierarchical control philosophy explained in this thesis, we have built an experimental test bench. This platform has 4 terminals interconnected via a DC grid, and connected to the main AC grid through VSC power converters. This DC grid can work at a maximum of 400 V, and with a maximum allowed current of 15 A.Cette thèse traite de la commande hiérarchique de réseaux à courant continu multi-terminaux à haute tension (MT-HVDC) intégrant des sources d'énergie renouvelables à grande échelle. Le schéma de contrôle proposé est composé de quatre ‘couches’ : le contrôle local où se trouvent les convertisseurs de puissance, avec une échelle de temps de l’ordre de la milliseconde ; le contrôle primaire qui est décentralisé et appliqué à plusieurs terminaux avec une échelle du temps de l’ordre de la seconde ; un niveau de commande où la communication est prise en compte et où l’approche de Modèle du Commande Prédictive (MPC) assure la planification de la tension et de la puissance à leur état d'équilibre, pour l'ensemble du système; enfin, le contrôleur de niveau supérieur, qui est principalement basé sur les techniques d'optimisation, où les aspects économiques sont pris en compte (il s’agit du réglage dit tertiaire).Au niveau des convertisseurs, un accent particulier est mis sur les convertisseurs bidirectionnels DC/DC. Dans cette thèse, trois topologies différentes sont étudiées en profondeur: deux phases Dual Active Bridge (DAB), trois phases DAB, et l’utilisation de la technologie Modular Multilevel converter (MMC) comme convertisseur DC/DC. Pour chaque topologie, une commande non-linéaire spécifique est discutée. D’autre part une nouvelle fonction pour le convertisseur DC/DC est étudiée. Il s’agit de son utilisation comme disjoncteur à courant continu (DC-CB). En ce qui concerne le contrôle primaire, qui permet de maintenir le niveau de tension continue dans le réseau, nous avons étudié trois philosophies de contrôle: celle de maître/esclave, celui du contrôle « voltage margin control » et celle de la commande du statisme (droop control). Enfin, nous avons choisi d'utiliser le droop control, entre autres, parce que la communication entre les nœuds n’est pas nécessaire. Concernant la commande secondaire, son principal objectif est de planifier le transfert de puissance entre les nœuds du réseau, qui fournissent la tension et la puissance de référence aux contrôleurs locaux et primaires, même lorsque des perturbations apparaissent. Dans cette partie, nous avons proposé une nouvelle approche pour résoudre les problèmes de flux de puissance (équations non-linéaires) basée sur le théorème du point fixe de l’application contractive. Ceci permet d'utiliser plus d'un slack bus, contrairement à l’approche classique basée sur la méthode de Newton-Raphson. Par ailleurs, le réglage secondaire joue un rôle très important dans les applications pratiques, en particulier lorsque les sources d'énergie renouvelables (variables dans le temps). Dans de tels cas, il est intéressant de considérer des dispositifs de stockage afin d'améliorer la stabilité de tout le système. Il est également possible d'envisager différents types de prévisions (météo, charge, ..) basées sur la gestion des réserves de stockage. Toutes ces caractéristiques ont suggéré l'utilisation d'une approche MPC. Dans ce contexte, plusieurs critères d'optimisation ont été considérés, en particulier la minimisation des pertes de transmission ou des congestions dans le réseau.La tâche principale de réglage tertiaire est de d'atteindre l'optimisation économique de l'ensemble du réseau. Dans cette thèse, nous avons pu maximiser le profit économique du système en agissant sur le marché réel, et en optimisant l'utilisation des périphériques de stockage. Dans le but de mettre en œuvre la philosophie de contrôle hiérarchique présentée dans cette thèse, nous avons construit un banc d'essai expérimental. Cette plate-forme dispose de quatre terminaux reliés entre eux par l'intermédiaire d'un réseau à courant continu, et connectés au réseau principal de courant alternatif. Ce réseau DC peut fonctionner à un maximum de 400 V, et avec une courant maximal de 15 A

Préparation et stabilisation de systèmes quantiques

This thesis tackles the problem of preparing and stabilizing highly non classical states of quantum systems. We consider specific models based on current experiments in cavity quantum electrodynamics, Josephson circuits and ultra-fast coherent quantum control. The problem is posed in the framework of control theory where we search for a control law which prepares or stabilizes a desired target state.Of particular interest to us are target states with no classical analog: superposition and entangled states. More generally, we propose a scheme for the stabilization of a manifold of quantum states, thus introducing some new ideas for autonomous quantum error correction in a cavity. Close collaborations with experimentalists helped us in the design of control protocols which are readily employable in the laboratory. Experimental demonstrations are currently being implemented and preliminary measurements are in good agreement with the theory introduced in this thesis.Cette thèse s'intéresse au problème de préparation et de stabilisation de systèmes quantiques. Nous considérons des modèles correspondant à des expériences actuelles en électrodynamique quantique en cavité, circuits Josephson, et de contrôle quantique cohérent par laser femtoseconde. Nous posons les problèmes dans le contexte de la théorie du contrôle et nous proposons des lois de commande qui préparent ou stabilisent des états cibles. En particulier, nous nous intéressons à des états cibles qui n'ont pas d'analogue classique: des états superpositions et intriqués. De plus, nous proposons une commande pour la stabilisation d'un sous-espace de l'espace des états, contribuant ainsi au domaine de la correction d'erreur quantique. Ces résultats ont été obtenu en étroite collaboration avec des expérimentateurs. Des mesures expérimentales préliminaires sont en bon accord avec certaines prédictions théoriques de cette thèse

Modeling and control of thermoacoustic instabilities

This work deals with modeling and control of thermoacoustic combustion instabilities in lean premixed combustion systems. Because of the complex interactions present in thermoacoustic systems, a network modeling approach is used. The model of each network element or subsystem is obtained analytically, numerically, or by making use of experimental techniques. The dynamics of a network system are determined experimentally by making use of a transfer matrix measurement technique. The transfer functions of a premixed flame have been determined experimentally on an atmospheric combustion test facility with a full-scale gas turbine burner, for a wide variety of operating conditions. An analytical model of the dynamic behavior of the reaction zone was made. In this model, the heat release fluctuations are assumed to be caused by fluctuations of the mass fraction of fuel and by fluctuations in the burning velocity. The model proved to be in good agreement with experimental results. Wave propagation in complex three-dimensional geometries is modeled by making use of a modal expansion technique. The modes used for the modal expansion can be obtained analytically for relatively simple geometries, or numerically (finite element method) for geometries of any complexity. By representing the modal expansion in state-space, a very numerically efficient and robust model is obtained. The thermoacoustic network model combines the state-space representations of the sub-systems in one system. The system can be analyzed in the time domain or in the frequency domain. The stability analysis is straightforward and does not require a numerical search. Non linear elements can easily be incorporated in the time domain simulation. This novel method has been validated by comparison with analytic solutions of simple thermoacoustic systems found in literature, by comparison with Finite Element codes, and by comparison with experimental results. An excellent agreement was found for all comparisons. When including non-linear elements in an annular system, a rotating acoustic field is predicted, which corresponds to experimental observations. This result has been verified analytically. Based on network models, a model based controller has been obtained using H∞ optimization. This controller has been tested in simulation and experiment on a single burner rig and proved to suppress acoustic levels by more than 25dB. An adaptive controller, based on a genetic algorithm, has been developed that does not require any knowledge about the system. This controller has been tested and proved to have similar performance as the model-based controllers. An active control system for multi-burner configurations has been developed and proved to perform well in simulations