A Formal Power Series Approach to Multiplicative Dynamic and Static Output Feedback

The goal of the paper is two-fold. The first of which is to derive an explicit formula to compute the generating series of a closed-loop system when a plant, given in a Chen-Fliess series description is in multiplicative output feedback connection with another system given in Chen-Fliess series description. In addition, the multiplicative dynamic output feedback connection has a natural interpretation as a transformation group acting on the plant. The second of the two-part goal of this paper is same as the first part albeit when the Chen-Fliess series in the feedback is replaced by a memoryless map, so called multiplicative static feedback connection

Entropy of Generating Series for Nonlinear Input-Output Systems and their Interconnections

This paper has two main objectives. The first is to introduce a notion of entropy that is well suited for the analysis of nonlinear input-output systems that have a Chen-Fliess series representation. The latter is defined in terms of its generating series over a noncommutative alphabet. The idea is to assign an entropy to a generating series as an element of a graded vector space. The second objective is to describe the entropy of generating series originating from interconnected systems of Chen-Fliess series that arise in the context of control theory. It is shown that one set of interconnections can never increase entropy as defined here, while a second set has the potential to do so. The paper concludes with a brief introduction to an entropy ultrametric space and some open questions

Formal Power Series Approach to Nonlinear Systems with Static Output Feedback

The goal of this paper is to compute the generating series of a closed-loop system when the plant is described in terms of a Chen-Fliess series and static output feedback is applied. The first step is to reconsider the so called Wiener-Fliess connection consisting of a Chen-Fliess series followed by a memoryless function. Of particular importance will be the contractive nature of this map, which is needed to show that the closed-loop system has a Chen-Fliess series representation. To explicitly compute the generating series, two Hopf algebras are needed, the existing output feedback Hopf algebra used to describe dynamic output feedback, and the Hopf algebra of the shuffle group. These two combinatorial structures are combined to compute what will be called the Wiener-Fliess feedback product. It will be shown that this product has a natural interpretation as a transformation group acting on the plant and preserves the relative degree of the plant

Wiener-Fliess Composition of Formal Power Series: Additive Static Feedback and Shuffle Rational Series

The problem statement for this dissertation is two-fold. The first problem considered is when does a Chen-Fliess series in an additive static feedback connection with a formal static map yield a closed-loop system with a Chen-Fliess series expansion? This work proves that such a closed-loop system always has a Chen-Fliess series representation. Furthermore, an algorithm based on the Hopf algebras for the shuffle group and the dynamic output feedback group is designed to compute the generating series of the closed-loop system. It is proved that the additive static feedback connection preserves local convergence and relative degree, but a counterexample shows that the additive static feedback does not preserve global convergence in general. This dissertation then pivots to the second problem considered, the shuffle rationality problem. The notion of shuffle rationality and shuffle recognizability are first defined, akin to the traditional notion of rational series in bilinear systems theory. It is proved that shuffle rationality and shuffle recognizability coincide, similar to Schutzenberger’s theorem. An equivalent characterization of shuffle rational series is provided in terms of a canonical state space realization. Specifically, it is shown that a shuffle rational series corresponds to a realization of a nilpotent bilinear system cascaded with a static rational map

Physics-based Machine Learning Approaches to Complex Systems and Climate Analysis

Komplexe Systeme wie das Klima der Erde bestehen aus vielen Komponenten, die durch eine komplizierte Kopplungsstruktur miteinander verbunden sind. Für die Analyse solcher Systeme erscheint es daher naheliegend, Methoden aus der Netzwerktheorie, der Theorie dynamischer Systeme und dem maschinellen Lernen zusammenzubringen. Durch die Kombination verschiedener Konzepte aus diesen Bereichen werden in dieser Arbeit drei neuartige Ansätze zur Untersuchung komplexer Systeme betrachtet. Im ersten Teil wird eine Methode zur Konstruktion komplexer Netzwerke vorgestellt, die in der Lage ist, Windpfade des südamerikanischen Monsunsystems zu identifizieren. Diese Analyse weist u.a. auf den Einfluss der Rossby-Wellenzüge auf das Monsunsystem hin. Dies wird weiter untersucht, indem gezeigt wird, dass der Niederschlag mit den Rossby-Wellen phasenkohärent ist. So zeigt der erste Teil dieser Arbeit, wie komplexe Netzwerke verwendet werden können, um räumlich-zeitliche Variabilitätsmuster zu identifizieren, die dann mit Methoden der nichtlinearen Dynamik weiter analysiert werden können. Die meisten komplexen Systeme weisen eine große Anzahl von möglichen asymptotischen Zuständen auf. Um solche Zustände zu beschreiben, wird im zweiten Teil die Monte Carlo Basin Bifurcation Analyse (MCBB), eine neuartige numerische Methode, vorgestellt. Angesiedelt zwischen der klassischen Analyse mit Ordnungsparametern und einer gründlicheren, detaillierteren Bifurkationsanalyse, kombiniert MCBB Zufallsstichproben mit Clustering, um die verschiedenen Zustände und ihre Einzugsgebiete zu identifizieren. Bei von Vorhersagen von komplexen Systemen ist es nicht immer einfach, wie Vorwissen in datengetriebenen Methoden integriert werden kann. Eine Möglichkeit hierzu ist die Verwendung von Neuronalen Partiellen Differentialgleichungen. Hier wird im letzten Teil der Arbeit gezeigt, wie hochdimensionale räumlich-zeitlich chaotische Systeme mit einem solchen Ansatz modelliert und vorhergesagt werden können.Complex systems such as the Earth's climate are comprised of many constituents that are interlinked through an intricate coupling structure. For the analysis of such systems it therefore seems natural to bring together methods from network theory, dynamical systems theory and machine learning. By combining different concepts from these fields three novel approaches for the study of complex systems are considered throughout this thesis. In the first part, a novel complex network construction method is introduced that is able to identify the most important wind paths of the South American Monsoon system. Aside from the importance of cross-equatorial flows, this analysis points to the impact Rossby Wave trains have both on the precipitation and low-level circulation. This connection is then further explored by showing that the precipitation is phase coherent to the Rossby Wave. As such, the first part of this thesis demonstrates how complex networks can be used to identify spatiotemporal variability patterns within large amounts of data, that are then further analysed with methods from nonlinear dynamics. Most complex systems exhibit a large number of possible asymptotic states. To investigate and track such states, Monte Carlo Basin Bifurcation analysis (MCBB), a novel numerical method is introduced in the second part. Situated between the classical analysis with macroscopic order parameters and a more thorough, detailed bifurcation analysis, MCBB combines random sampling with clustering methods to identify and characterise the different asymptotic states and their basins of attraction. Forecasts of complex system are the next logical step. When doing so, it is not always straightforward how prior knowledge in data-driven methods. One possibility to do is by using Neural Partial Differential Equations. Here, it is demonstrated how high-dimensional spatiotemporally chaotic systems can be modelled and predicted with such an approach in the last part of the thesis

The SLH framework for modeling quantum input-output networks

Many emerging quantum technologies demand precise engineering and control over networks consisting of quantum mechanical degrees of freedom connected by propagating electromagnetic fields, or quantum input-output networks. Here we review recent progress in theory and experiment related to such quantum input-output networks, with a focus on the SLH framework, a powerful modeling framework for networked quantum systems that is naturally endowed with properties such as modularity and hierarchy. We begin by explaining the physical approximations required to represent any individual node of a network, eg. atoms in cavity or a mechanical oscillator, and its coupling to quantum fields by an operator triple $(S,L,H)$. Then we explain how these nodes can be composed into a network with arbitrary connectivity, including coherent feedback channels, using algebraic rules, and how to derive the dynamics of network components and output fields. The second part of the review discusses several extensions to the basic SLH framework that expand its modeling capabilities, and the prospects for modeling integrated implementations of quantum input-output networks. In addition to summarizing major results and recent literature, we discuss the potential applications and limitations of the SLH framework and quantum input-output networks, with the intention of providing context to a reader unfamiliar with the field.Comment: 60 pages, 14 figures. We are still interested in receiving correction

Computational Analysis of Antipode Algorithms for the Output Feedback Hopf Algebra

The feedback interconnection of two systems written in terms of Chen-Fliess series can be described explicitly in terms of the antipode of the output feedback Hopf algebra. At present, there are three known computational approaches to calculating this antipode: the left coproduct method, the right coproduct method, and the derivation method. Each of these algorithms is defined recursively, and thus becomes computationally expensive quite quickly. This motivates the need for a more complete understanding of the algorithmic complexity of these methods, as well as the development of new approaches for determining the Hopf algebra antipode. The main goals of this thesis are to create an implementation in code of the derivation method and compare the computational performance against existing code for the two coproduct methods in Mathematica. Both temporal and spatial complexity are examined empirically, and the main conclusion is that the derivation method yields the best performance

Aeronautical engineering: A special bibliography, supplement 29, March 1973

This special bibliography lists 410 reports, articles, and other documents introduced into the NASA scientific and technical information system in February 1972