Nonlinear Systems

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems

Levitation and control of particles with internal degrees of freedom

Levitodynamics is a fast growing field that studies the levitation and manipulation of micro- and nanoobjects, fuelled by both fundamental physics questions and technological applications. Due to the isolated nature of trapped particles, levitated systems are highly decoupled from the environment, and offer experimental possibilities that are absent in clamped nanomechanical oscillators. In particular, a central question in quantum physics is how the transition between the classical and quantum world materializes, and levitated objects represent a promising avenue to study this intermediate regime. In the last years, most levitation experiments have been restricted to optically trapped silica nanoparticles in vacuum, controlling the particle's position with intensity modulated laser beams. However, the use of optical traps severely constrains the experiments that can be performed, because few particle materials can withstand the optical absorption and resulting heating in vacuum. This completely prevents the use of objects with internal degrees of freedom, which---coupled to mechanical variables---offer a clear path towards the study of quantum phenomena at the macroscale. In this thesis, we address these issues by considering other types of trap and feedback schemes, achieving excellent control on the dynamics of optically active nanoparticles. With stochastic calculus, simulations and experiments, we study the dynamics of trapped particles in different regimes, considering also a hybrid quadrupole-optical trapping scheme. Then, using a Paul trap of our own design, we demonstrate the trapping, interrogation and feedback cooling of a nanodiamond hosting a single NV center in vacuum, a clear candidate to perform quantum physics experiments at the single spin level. Finally, we discuss and implement an optimal controller to cool the center of mass motion of an optically levitated nanoparticle. The feedback is realized by exerting a Coulomb force on a charged particle with a pair of electrodes, and thus requires no optics.La levitodinàmica és un camp de la física en ràpida expansió que estudia la levitació i manipulació de micro- i nano-objectes, empesa per la possibilitat de solucionar trencaclosques de física fonamental i de desenvolupar noves aplicacions tecnològiques. Gràcies al gran aïllament de les partícules en levitació, l’evolució dels sistemes levitodinàmics està molt desacoplada del seu entorn. Per consegüent, permeten fer experiments que no serien possibles en nanooscil·ladors mecànics sobre substrat. En particular, una qüestió central en física consisteix en entendre com es produeix la transició entre els mons clàssic i quàntic; els objectes en levitació permeten estudiar aquest règim intermedi de manera innovadora. En els últims anys, la majoria d’experiments de levitodinàmica s’han limitat a atrapar òpticament partícules de sílice en el buit, tot controlant la posició de la partícula amb feixos làser modulats. Tot i així, l’ús de trampes òptiques suposa un obstacle a l’hora d’exportar aquests experiments a règims més diversos perquè, a baixes pressions, pocs materials són capaços de suportar les altes temperatures resultants de l’absorció de llum làser. Això impedeix l’ús d’objectes amb graus de llibertat interns, que –acoplats a variables mecàniques– suposen un full de ruta clar per estudiar fenòmens quàntics a escala macroscòpica En aquesta tesi, adrecem aquestes qüestions tot considerant altres tipus de trampa i tècniques de feedback, i assolim un control excel·lent de la dinàmica de nanopartícules òpticament actives en levitació. Mitjançant càlcul estocàstic, simulacions i experiments, estudiem la dinàmica de les partícules en règims diversos, àdhuc considerant un esquema híbrid de trampa de Paul-òptica. A continuació, utilitzant una trampa de Paul, demostrem experimentalment l’atrapament, interrogació i feedback-cooling en el buit d’un nanodiamant que conté un únic NV− center, un clar candidat per a la realització d’experiments de física quàntica amb un únic spin. Finalment, estudiem i implementem un controlador òptim per a refredar el centre de massa d’una partícula òpticament levitada. El feedback es realitza exercint una força de Coulomb sobre una partícula carregada positivament mitjançant un parell d’elèctrodes, i per tant no requereix elements òptic

Systems reliability issues for future aircraft

The reliability of adaptive controls for future aircraft are discussed. The research, formulation, and experimentation for improved aircraft performance are considered

Combining reinforcement learning and optimal control for the control of nonlinear dynamical systems

This thesis presents a novel hierarchical learning framework, Reinforcement Learning Optimal Control, for controlling nonlinear dynamical systems with continuous states and actions. The adapted approach mimics the neural computations that allow our brain to bridge across the divide between symbolic action-selection and low-level actuation control by operating at two levels of abstraction. First, current findings demonstrate that at the level of limb coordination human behaviour is explained by linear optimal feedback control theory, where cost functions match energy and timing constraints of tasks. Second, humans learn cognitive tasks involving learning symbolic level action selection, in terms of both model-free and model-based reinforcement learning algorithms. We postulate that the ease with which humans learn complex nonlinear tasks arises from combining these two levels of abstraction. The Reinforcement Learning Optimal Control framework learns the local task dynamics from naive experience using an expectation maximization algorithm for estimation of linear dynamical systems and forms locally optimal Linear Quadratic Regulators, producing continuous low-level control. A high-level reinforcement learning agent uses these available controllers as actions and learns how to combine them in state space, while maximizing a long term reward. The optimal control costs form training signals for high-level symbolic learner. The algorithm demonstrates that a small number of locally optimal linear controllers can be combined in a smart way to solve global nonlinear control problems and forms a proof-of-principle to how the brain may bridge the divide between low-level continuous control and high-level symbolic action selection. It competes in terms of computational cost and solution quality with state-of-the-art control, which is illustrated with solutions to benchmark problems.Open Acces

Advances in Reinforcement Learning

Reinforcement Learning (RL) is a very dynamic area in terms of theory and application. This book brings together many different aspects of the current research on several fields associated to RL which has been growing rapidly, producing a wide variety of learning algorithms for different applications. Based on 24 Chapters, it covers a very broad variety of topics in RL and their application in autonomous systems. A set of chapters in this book provide a general overview of RL while other chapters focus mostly on the applications of RL paradigms: Game Theory, Multi-Agent Theory, Robotic, Networking Technologies, Vehicular Navigation, Medicine and Industrial Logistic

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”