Exponential Stabilisation of Continuous-time Periodic Stochastic Systems by Feedback Control Based on Periodic Discrete-time Observations

Since Mao in 2013 discretised the system observations for stabilisation problem of hybrid SDEs (stochastic differential equations with Markovian switching) by feedback control, the study of this topic using a constant observation frequency has been further developed. However, time-varying observation frequencies have not been considered. Particularly, an observational more efficient way is to consider the time-varying property of the system and observe a periodic SDE system at the periodic time-varying frequencies. This study investigates how to stabilise a periodic hybrid SDE by a periodic feedback control, based on periodic discrete-time observations. This study provides sufficient conditions under which the controlled system can achieve pth moment exponential stability for p > 1 and almost sure exponential stability. Lyapunov's method and inequalities are main tools for derivation and analysis. The existence of observation interval sequences is verified and one way of its calculation is provided. Finally, an example is given for illustration. Their new techniques not only reduce observational cost by reducing observation frequency dramatically but also offer flexibility on system observation settings. This study allows readers to set observation frequencies according to their needs to some extent

On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters

Copyright [2002] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this paper, we investigate the stochastic stabilization problem for a class of bilinear continuous time-delay uncertain systems with Markovian jumping parameters. Specifically, the stochastic bilinear jump system under study involves unknown state time-delay, parameter uncertainties, and unknown nonlinear deterministic disturbances. The jumping parameters considered here form a continuous-time discrete-state homogeneous Markov process. The whole system may be regarded as a stochastic bilinear hybrid system that includes both time-evolving and event-driven mechanisms. Our attention is focused on the design of a robust state-feedback controller such that, for all admissible uncertainties as well as nonlinear disturbances, the closed-loop system is stochastically exponentially stable in the mean square, independent of the time delay. Sufficient conditions are established to guarantee the existence of desired robust controllers, which are given in terms of the solutions to a set of either linear matrix inequalities (LMIs), or coupled quadratic matrix inequalities. The developed theory is illustrated by numerical simulatio

Discretisation of continuous-time stochastic optimal control problems with delay

In the present work, we study discretisation schemes for continuous-time stochastic optimal control problems with time delay. The dynamics of the control problems to be approximated are described by controlled stochastic delay (or functional) differential equations. The value functions associated with such control problems are defined on an infinite-dimensional function space. The discretisation schemes studied are obtained by replacing the original control problem by a sequence of approximating discrete-time Markovian control problems with finite or finite-dimensional state space. Such a scheme is convergent if the value functions associated with the approximating control problems converge to the value function of the original problem. Following a general method for the discretisation of continuous-time control problems, sufficient conditions for the convergence of discretisation schemes for a class of stochastic optimal control problems with delay are derived. The general method itself is cast in a formal framework. A semi-discretisation scheme for a second class of stochastic optimal control problems with delay is proposed. Under standard assumptions, convergence of the scheme as well as uniform upper bounds on the discretisation error are obtained. The question of how to numerically solve the resulting discrete-time finite-dimensional control problems is also addressed

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

Stochastic Parameterization: A Rigorous Approach to Stochastic Three-Dimensional Primitive Equations

The atmosphere is a strongly nonlinear and infinite-dimensional dynamical system acting on a multitude of different time and space scales. A possible problem of numerical weather prediction and climate modeling using deterministic parameterization of subscale and unresolved processes is the incomplete consideration of scale interactions. A stochastic treatment of these parameterizations bears the potential to improve the simulations and to provide a better understanding of the scale interactions of the simulated atmospheric variables. The scientific community that is dealing with stochastic meteorological models can be divided into two groups: the first one uses pragmatic approaches to improve existing complex models. The second group pursues a mathematical rigorous way to develop stochastic models, which is currently limited to conceptual models. The overall objective of this work is to narrow the gap between pragmatic approaches and the mathematical rigorous methods. Using conceptual climate models, we point out that a stochastic formulation must not be chosen arbitrarily but has to be derived based on the physics of the system at hand. Equally important is a rigorous numerical implementation of the resulting stochastic model. The dynamics of sub grid and unresolved processes are often described by time continuous stochastic processes, which cannot be treated with deterministic numerical schemes. We show that a stochastic formulation of the three-dimensional primitive equations fits in the mathematical framework of abstract stochastic fluid models. This allows us to utilize recent results regarding existence and uniqueness of solutions of such systems. Based on these theoretical results we propose a Galerkin scheme for the discretization of spatial and stochastic dimensions. Using the framework of mild solutions of stochastic partial differential equations we are able to prove quantitative error bounds and strong mean square convergence. Under additional assumptions we show the convergence of a numerical scheme which combines the Galerkin approximation with a temporal discretization.Stochastische Parametrisierung: Ein Rigoroser Ansatz für die Stochastischen Drei-Dimensionalen Primitiven Gleichungen Die Atmosphäre ist ein von starken Nichtlinearitäten geprägtes, unendlich-linebreak dimensionales dynamisches System, dessen Variablen auf einer Vielzahl verschiedener Raum- und Zeitskalen interagieren. Ein potentielles Problem von Modellen zur numerischen Wettervorhersage und Klimamodellierung, die auf deterministischen Parametrisierungen subskaliger Prozesse beruhen, ist die unzureichende Behandlung der Interaktion zwischen diesen Prozessen und den Modellvariablen. Eine stochastische Beschreibung dieser Parametrisierungen hat das Potential die Qualität der Simulationen zu verbessern und das Verständnis der Skalen-Interaktion atmosphärischer Variablen zu vertiefen. Die wissenschaftlich Gemeinschaft, die sich mit stochastischen meteorologischen Modellen beschäftigt, kann grob in zwei Gruppen unterteilt werden: die erste Gruppe ist bemüht durch pragmatische Ansätze bestehende, komplexe Modelle zu erweitern. Die zweite Gruppe verfolgt einen mathematisch rigorosen Weg, um stochastische Modelle zu entwickeln. Dies ist jedoch aufgrund der mathematischen Komplexität bisher auf konzeptionelle Modelle beschränkt. Das generelle Ziel der vorliegenden Arbeit ist es, die Kluft zwischen den pragmatischen und mathematisch rigorosen Ansätzen zu verringern. Die Diskussion zweier konzeptioneller Klimamodelle verdeutlicht, dass eine stochastische Formulierung nicht willkürlich gewählt werden darf, sondern aus der Physik des betrachteten Systems abgeleitet werden muss. Ebenso unabdingbar ist eine rigorose numerische Implementierung des resultierenden stochastischen Modells. Diesem Aspekt wird besondere Bedeutung zu Teil, da dynamische subskalige Prozesse oftmals durch zeitabhängige stochastische Prozesse beschrieben werden, die sich nicht mit deterministischen numerischen Methoden behandeln lassen. Wir zeigen auf, dass eine stochastische Formulierung der dreidimensionalen primitiven Gleichungen im mathematischen Rahmen abstrakter stochastischer Fluidmodelle behandelt werden kann. Dies ermöglicht die Anwendung kürzlich gewonnener Erkenntnisse bezüglich Existenz und Eindeutigkeit von Lösungen. Wir stellen einen auf dieser theoretischen Grundlage basierenden Galerkin Ansatz zur Diskretisierung der räumlichen und stochastischen Dimensionen vor. Mit Hilfe sogenannter milder Lösungen der stochastischen partiellen Differentialgleichungen leiten wir quantitative Schranken der Diskretisierungsfehler her und zeigen die starke Konvergenz des mittleren quadratischen Fehlers. Unter zusätzlichen Annahmen leiten wir die Konvergenz eines numerischen Verfahrens her, das den Galerkin Ansatz um eine zeitliche Diskretisierung erweitert

Quantum Linear Systems Theory

This paper surveys some recent results on the theory of quantum linear systems and presents them within a unified framework. Quantum linear systems are a class of systems whose dynamics, which are described by the laws of quantum mechanics, take the specific form of a set of linear quantum stochastic differential equations (QSDEs). Such systems commonly arise in the area of quantum optics and related disciplines. Systems whose dynamics can be described or approximated by linear QSDEs include interconnections of optical cavities, beam-splitters, phase-shifters, optical parametric amplifiers, optical squeezers, and cavity quantum electrodynamic systems. With advances in quantum technology, the feedback control of such quantum systems is generating new challenges in the field of control theory. Potential applications of such quantum feedback control systems include quantum computing, quantum error correction, quantum communications, gravity wave detection, metrology, atom lasers, and superconducting quantum circuits. A recently emerging approach to the feedback control of quantum linear systems involves the use of a controller which itself is a quantum linear system. This approach to quantum feedback control, referred to as coherent quantum feedback control, has the advantage that it does not destroy quantum information, is fast, and has the potential for efficient implementation. However, the design of coherent quantum feedback controllers remains a major challenge. This paper discusses recent results concerning the synthesis of H-infinity optimal controllers for linear quantum systems in the coherent control case. An important issue which arises both in the modelling of linear quantum systems and in the synthesis of linear coherent quantum controllers is the issue of physical realizability. This issue relates to the property of whether a given set of QSDEs corresponds to a physical quantum system satisfying the laws of quantum mechanics. The paper will cover recent results relating the question of physical realizability to notions occurring in linear systems theory such as lossless bounded real systems and dual J-J unitary systems.Research supported by the Australian Research Council (ARC)