Distributed Kalman Filters over Wireless Sensor Networks: Data Fusion, Consensus, and Time-Varying Topologies

Kalman filtering is a widely used recursive algorithm for optimal state estimation of linear stochastic dynamic systems. The recent advances of wireless sensor networks (WSNs) provide the technology to monitor and control physical processes with a high degree of temporal and spatial granularity. Several important problems concerning Kalman filtering over WSNs are addressed in this dissertation. First we study data fusion Kalman filtering for discrete-time linear time-invariant (LTI) systems over WSNs, assuming the existence of a data fusion center that receives observations from distributed sensor nodes and estimates the state of the target system in the presence of data packet drops. We focus on the single sensor node case and show that the critical data arrival rate of the Bernoulli channel can be computed by solving a simple linear matrix inequality problem. Then a more general scenario is considered where multiple sensor nodes are employed. We derive the stationary Kalman filter that minimizes the average error variance under a TCP-like protocol. The stability margin is adopted to tackle the stability issue. Second we study distributed Kalman filtering for LTI systems over WSNs, where each sensor node is required to locally estimate the state in a collaborative manner with its neighbors in the presence of data packet drops. The stationary distributed Kalman filter (DKF) that minimizes the local average error variance is derived. Building on the stationary DKF, we propose Kalman consensus filter for the consensus of different local estimates. The upper bound for the consensus coefficient is computed to ensure the mean square stability of the error dynamics. Finally we focus on time-varying topology. The solution to state consensus control for discrete-time homogeneous multi-agent systems over deterministic time-varying feedback topology is provided, generalizing the existing results. Then we study distributed state estimation over WSNs with time-varying communication topology. Under the uniform observability, each sensor node can closely track the dynamic state by using only its own observation, plus information exchanged with its neighbors, and carrying out local computation

Dynamics Days Latin America and the Caribbean 2018

This book contains various works presented at the Dynamics Days Latin America and the Caribbean (DDays LAC) 2018. Since its beginnings, a key goal of the DDays LAC has been to promote cross-fertilization of ideas from different areas within nonlinear dynamics. On this occasion, the contributions range from experimental to theoretical research, including (but not limited to) chaos, control theory, synchronization, statistical physics, stochastic processes, complex systems and networks, nonlinear time-series analysis, computational methods, fluid dynamics, nonlinear waves, pattern formation, population dynamics, ecological modeling, neural dynamics, and systems biology. The interested reader will find this book to be a useful reference in identifying ground-breaking problems in Physics, Mathematics, Engineering, and Interdisciplinary Sciences, with innovative models and methods that provide insightful solutions. This book is a must-read for anyone looking for new developments of Applied Mathematics and Physics in connection with complex systems, synchronization, neural dynamics, fluid dynamics, ecological networks, and epidemics

Subspace identification via convex optimization

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 88-92).In this thesis we consider convex optimization-based approaches to the classical problem of identifying a subspace from noisy measurements of a random process taking values in the subspace. We focus on the case where the measurement noise is component-wise independent, known as the factor analysis model in statistics. We develop a new analysis of an existing convex optimization-based heuristic for this problem. Our analysis indicates that in high-dimensional settings, where both the ambient dimension and the dimension of the subspace to be identified are large, the convex heuristic, minimum trace factor analysis, is often very successful. We provide simple deterministic conditions on the underlying 'true' subspace under which the convex heuristic provably identifies the correct subspace. We also consider the performance of minimum trace factor analysis on 'typical' subspace identification problems, that is problems where the underlying subspace is chosen randomly from subspaces of a particular dimension. In this setting we establish conditions on the ambient dimension and the dimension of the underlying subspace under which the convex heuristic identifies the subspace correctly with high probability. We then consider a refinement of the subspace identification problem where we aim to identify a class of structured subspaces arising from Gaussian latent tree models. More precisely, given the covariance at the finest scale of a Gaussian latent tree model, and the tree that indexes the model, we aim to learn the parameters of the model, including the state dimensions of each of the latent variables. We do so by extending the convex heuristic, and our analysis, from the factor analysis setting to the setting of Gaussian latent tree models. We again provide deterministic conditions on the underlying latent tree model that ensure our convex optimization-based heuristic successfully identifies the parameters and state dimensions of the model.by James Saunderson.S.M

Consensus problems and the effects of graph topology in collaborative control

In this dissertation, several aspects of design for networked systems are addressed. The main focus is on combining approaches from system theory and graph theory to characterize graph topologies that result in efficient decision making and control. In this framework, modelling and design of sparse graphs that are robust to failures and provide high connectivity are considered. A decentralized approach to path generation in a collaborative system is modelled using potential functions. Taking inspiration from natural swarms, various behaviors of the system such as target following, moving in cohesion and obstacle avoidance are addressed by appropriate encoding of the corresponding costs in the potential function and using gradient descent for minimizing the energy function. Different emergent behaviors emerge as a result of varying the weights attributed with different components of the potential function. Consensus problems are addressed as a unifying theme in many collaborative control problems and their robustness and convergence properties are studied. Implications of the continuous convergence property of consensus problems on their reachability and robustness are studied. The effects of link and agent faults on consensus problems are also investigated. In particular the concept of invariant nodes has been introduced to model the effect of nodes with different behaviors from regular nodes. A fundamental association is established between the structural properties of a graph and the performance of consensus algorithms running on them. This leads to development of a rigorous evaluation of the topology effects and determination of efficient graph topologies. It is well known that graphs with large diameter are not efficient as far as the speed of convergence of distributed algorithms is concerned. A challenging problem is to determine a minimum number of long range links (shortcuts), which guarantees a level of enhanced performance. This problem is investigated here in a stochastic framework. Specifically, the small world model of Watts and Strogatz is studied and it is shown that adding a few long range edges to certain graph topologies can significantly increase both the rate of convergence for consensus algorithms and the number of spanning trees in the graph. The simulations are supported by analytical stochastic methods inspired from perturbations of Markov chains. This approach is further extended to a probabilistic framework for understanding and quantifying the small world effect on consensus convergence rates: Time varying topologies, in which each agent nominally communicates according to a predefined topology, and switching with non-neighboring agents occur with small probability is studied. A probabilistic framework is provided along with fundamental bounds on the convergence speed of consensus problems with probabilistic switching. The results are also extended to the design of robust topologies for distributed algorithms. The design of a semi-distributed two-level hierarchical network is also studied, leading to improvement in the performance of distributed algorithms. The scheme is based on the concept of social degree and local leader selection and the use of consensus-type algorithms for locally determining topology information. Future suggestions include adjusting our algorithm towards a fully distributed implementation. Another important aspect of performance in collaborative systems is for the agents to send and receive information in a manner that minimizes process costs, such as estimation error and the cost of control. An instance of this problem is addressed by considering a collaborative sensor scheduling problem. It is shown that in finding the optimal joint estimates, the general tree-search solution can be efficiently solved by devising a method that utilizes the limited processing capabilities of agents to significantly decrease the number of search hypotheses

The Foundations of Infinite-Dimensional Spectral Computations

Spectral computations in infinite dimensions are ubiquitous in the sciences. However, their many applications and theoretical studies depend on computations which are infamously difficult. This thesis, therefore, addresses the broad question, “What is computationally possible within the field of spectral theory of separable Hilbert spaces?” The boundaries of what computers can achieve in computational spectral theory and mathematical physics are unknown, leaving many open questions that have been unsolved for decades. This thesis provides solutions to several such long-standing problems. To determine these boundaries, we use the Solvability Complexity Index (SCI) hierarchy, an idea which has its roots in Smale's comprehensive programme on the foundations of computational mathematics. The Smale programme led to a real-number counterpart of the Turing machine, yet left a substantial gap between theory and practice. The SCI hierarchy encompasses both these models and provides universal bounds on what is computationally possible. What makes spectral problems particularly delicate is that many of the problems can only be computed by using several limits, a phenomenon also shared in the foundations of polynomial root-finding as shown by McMullen. We develop and extend the SCI hierarchy to prove optimality of algorithms and construct a myriad of different methods for infinite-dimensional spectral problems, solving many computational spectral problems for the first time. For arguably almost any operator of applicable interest, we solve the long-standing computational spectral problem and construct algorithms that compute spectra with error control. This is done for partial differential operators with coefficients of locally bounded total variation and also for discrete infinite matrix operators. We also show how to compute spectral measures of normal operators (when the spectrum is a subset of a regular enough Jordan curve), including spectral measures of classes of self-adjoint operators with error control and the construction of high-order rational kernel methods. We classify the problems of computing measures, measure decompositions, types of spectra (pure point, absolutely continuous, singular continuous), functional calculus, and Radon--Nikodym derivatives in the SCI hierarchy. We construct algorithms for and classify; fractal dimensions of spectra, Lebesgue measures of spectra, spectral gaps, discrete spectra, eigenvalue multiplicities, capacity, different spectral radii and the problem of detecting algorithmic failure of previous methods (finite section method). The infinite-dimensional QR algorithm is also analysed, recovering extremal parts of spectra, corresponding eigenvectors, and invariant subspaces, with convergence rates and error control. Finally, we analyse pseudospectra of pseudoergodic operators (a generalisation of random operators) on vector-valued $l^p$ spaces. All of the algorithms developed in this thesis are sharp in the sense of the SCI hierarchy. In other words, we prove that they are optimal, realising the boundaries of what digital computers can achieve. They are also implementable and practical, and the majority are parallelisable. Extensive numerical examples are given throughout, demonstrating efficiency and tackling difficult problems taken from mathematics and also physical applications. In summary, this thesis allows scientists to rigorously and efficiently compute many spectral properties for the first time. The framework provided by this thesis also encompasses a vast number of areas in computational mathematics, including the classical problem of polynomial root-finding, as well as optimisation, neural networks, PDEs and computer-assisted proofs. This framework will be explored in the future work of the author within these settings

Causal decomposition of complex systems and prediction of chaos using machine learning

We live in a complex system. Therefore, it is essential to possess techniques to analyze and comprehend its intricate dynamics in order to improve decision making. The objective of this dissertation is to contribute to the research that enhances our ability to make these complex systems less intransparent to us. Firstly, we illustrate the impact on practical applications when nonlinearity - an often disregarded factor in causal inference - is taken into account. Therefore, we investigate the causal relationships within these systems, particularly shedding light on the distinction between linear and nonlinear drivers of causality. After developing the necessary methods, we apply them to a real-world use case and demonstrate that making slight adjustments to certain financial market frameworks can result in considerable advantages because of the resolution of the correlation-causation fallacy. Subsequently, once the linear and nonlinear causal connections are understood, we can derive governing equations from the underlying causality structure to enhance the interpretability of models and predictions. By fine-tuning the parameters of these equations through the phenomenon of synchronization of chaos, we can ensure that they optimally represent the data. Nevertheless, not all complex systems can be accurately described by governing equations. Therefore, the implementation of machine learning techniques like reservoir computing in predicting chaotic systems offers significant data-driven advantages. While their architecture is relatively simple, ensuring full interpretability and hardware realizations still relies on increased efficiency and reduced data requirements. This dissertation presents some of the necessary modifications to the traditional reservoir computing architecture to bring physical reservoir computing closer to realization.Wir leben in einem komplexen System. Daher ist es unerlässlich, über Techniken zur Analyse und zum Verständnis seiner verschleierten Dynamik zu verfügen, um die Entscheidungsfindung zu verbessern. Ziel dieser Dissertation ist es, einen Beitrag zur Forschung zu leisten, die unsere Möglichkeiten erweitert, diese komplexen Systeme für uns weniger intransparent zu machen. Zunächst wird aufgezeigt, welche Auswirkungen es auf praktische Anwendungen hat, wenn Nichtlinearität - ein oft vernachlässigter Faktor bei kausaler Inferenz - berücksichtigt wird. Daher untersuchen wir die kausalen Beziehungen innerhalb dieser Systeme und beleuchten insbesondere die Unterscheidung zwischen linearen und nichtlinearen Kausalitätsfaktoren. Nachdem wir die erforderlichen Methoden entwickelt haben, wenden wir sie auf einen realen Anwendungsfall an und zeigen, dass leichte Anpassungen bestimmter Finanzmarktmodelle durch die Auflösung des Korrelations-Kausalitäts-Fehlschlusses zu erheblichen Vorteilen führen können. Sobald die linearen und nichtlinearen Kausalzusammenhänge bekannt sind, können wir aus der zugrunde liegenden Kausalitätsstruktur die Differentialgleichungen ableiten, um die Interpretierbarkeit von Modellierungen und Vorhersagen zu verbessern. Durch die Feinjustierung der Parameter dieser Gleichungen durch das Phänomen der Synchronisierung von Chaos können wir sicherstellen, dass sie die Daten optimal darstellen. Allerdings lassen sich nicht alle komplexen Systeme durch Differentialgleichungen adäquat beschreiben. Daher bietet die Anwendung von Techniken des maschinellen Lernens wie Reservoir Computing bei der Vorhersage chaotischer Systeme erhebliche datenbasierte Vorteile. Obwohl ihre Architektur relativ einfach ist, ist die Gewährleistung einer vollständigen Interpretierbarkeit und Hardware-Realisierung immer noch von einer erhöhten Effizienz und reduzierten Datenanforderungen abhängig. In dieser Dissertation werden einige der notwendigen Änderungen an der traditionellen Architektur vorgestellt, um physikalisches Reservoir Computing näher an die Realisierung zu bringen

Stability radii of discrete-time stochastic systems with respect to blockdiagonal perturbations

We consider stochastic discrete-time systems with multiplicative noise which are controlled by dynamic output feedback and subjected to blockdiagonal stochastic parameter perturbations. Stability radii for these systems are characterized via scaling techniques and it is shown that for real data, the real and the complex stability radii coincide. In a second part of the paper we investigate the problem of maximizing the stability radii by dynamic output feedback. Necessary and sufficient conditions are derived for the existence of a stabilizing compensator which ensures that the stability radius is above a prespecified level. These conditions consist of parametrized matrix inequalities and a coupling condition. (C) 2000 Elsevier Science Ltd. All rights reserved