Tensor-Based Dynamical Systems
Tensor-Based Dynamical Systems

Author: Can Chen

Publisher: Springer

Published: 2024-04-11

Total Pages: 0

ISBN-13: 9783031545047

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This book provides a comprehensive review on tensor algebra, including tensor products, tensor unfolding, tensor eigenvalues, and tensor decompositions. Tensors are multidimensional arrays generalized from vectors and matrices, which can capture higher-order interactions within multiway data. In addition, tensors have wide applications in many domains such as signal processing, machine learning, and data analysis, and the author explores the role of tensors/tensor algebra in tensor-based dynamical systems where system evolutions are captured through various tensor products. The author provides an overview of existing literature on the topic and aims to inspire readers to learn, develop, and apply the framework of tensor-based dynamical systems.

Manifolds, Tensor Analysis, and Applications
Manifolds, Tensor Analysis, and Applications

Author: Ralph Abraham

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 666

ISBN-13: 1461210291

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The purpose of this book is to provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists. The main goal is to provide a working knowledge of manifolds, dynamical systems, tensors, and differential forms. Some applications to Hamiltonian mechanics, fluid me chanics, electromagnetism, plasma dynamics and control thcory arc given in Chapter 8, using both invariant and index notation. The current edition of the book does not deal with Riemannian geometry in much detail, and it does not treat Lie groups, principal bundles, or Morse theory. Some of this is planned for a subsequent edition. Meanwhile, the authors will make available to interested readers supplementary chapters on Lie Groups and Differential Topology and invite comments on the book's contents and development. Throughout the text supplementary topics are given, marked with the symbols ~ and {l:;J. This device enables the reader to skip various topics without disturbing the main flow of the text. Some of these provide additional background material intended for completeness, to minimize the necessity of consulting too many outside references. We treat finite and infinite-dimensional manifolds simultaneously. This is partly for efficiency of exposition. Without advanced applications, using manifolds of mappings, the study of infinite-dimensional manifolds can be hard to motivate.

Tensor Calculus and Analytical Dynamics
Tensor Calculus and Analytical Dynamics

Author: John G. Papastavridis

Publisher: Routledge

Published: 2018-12-12

Total Pages: 435

ISBN-13: 1351411624

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Tensor Calculus and Analytical Dynamics provides a concise, comprehensive, and readable introduction to classical tensor calculus - in both holonomic and nonholonomic coordinates - as well as to its principal applications to the Lagrangean dynamics of discrete systems under positional or velocity constraints. The thrust of the book focuses on formal structure and basic geometrical/physical ideas underlying most general equations of motion of mechanical systems under linear velocity constraints. Written for the theoretically minded engineer, Tensor Calculus and Analytical Dynamics contains uniquely accessbile treatments of such intricate topics as: tensor calculus in nonholonomic variables Pfaffian nonholonomic constraints related integrability theory of Frobenius The book enables readers to move quickly and confidently in any particular geometry-based area of theoretical or applied mechanics in either classical or modern form.

Automated Synthesis of Low-rank Stochastic Dynamical Systems Using the Tensor-train Decomposition
Automated Synthesis of Low-rank Stochastic Dynamical Systems Using the Tensor-train Decomposition

Author: John Irvin P. Alora

Publisher:

Published: 2016

Total Pages: 83

ISBN-13:

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Cyber-physical systems are increasingly becoming integrated in various fields such as medicine, finance, robotics, and energy. In these systems and their applications, safety and correctness of operation is of primary concern, sparking a large amount of interest in the development of ways to verify system behavior. The tight coupling of physical constraints and computation that typically characterize cyber-physical systems make them extremely complex, resulting in unexpected failure modes. Furthermore, disturbances in the environment and uncertainties in the physical model require these systems to be robust. These are difficult constraints, requiring cyberphysical systems to be able to reason about their behavior and respond to events in real-time. Thus, the goal of automated synthesis is to construct a controller that provably implements a range of behaviors given by a specification of how the system should operate. Unfortunately, many approaches to automated synthesis are ad hoc and are limited to simple systems that admit specific structure (e.g. linear, affine systems). Not only that, but they are also designed without taking into account uncertainty. In order to tackle more general problems, several computational frameworks that allow for more general dynamics and uncertainty to be investigated. Furthermore, all of the existing computational algorithms suffer from the curse of dimensionality, the run time scales exponentially with increasing dimensionality of the state space. As a result, existing algorithms apply to systems with only a few degrees of freedom. In this thesis, we consider a stochastic optimal control problem with a special class of linear temporal logic specifications and propose a novel algorithm based on the tensor-train decomposition. We prove that the run time of the proposed algorithm scales linearly with the dimensionality of the state space and polynomially with the rank of the optimal cost-to-go function.

Tensor Product Model Transformation in Polytopic Model-Based Control
Tensor Product Model Transformation in Polytopic Model-Based Control

Author: Péter Baranyi

Publisher: CRC Press

Published: 2018-09-03

Total Pages: 262

ISBN-13: 1439818177

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Tensor Product Model Transformation in Polytopic Model-Based Control offers a new perspective of control system design. Instead of relying solely on the formulation of more effective LMIs, which is the widely adopted approach in existing LMI-related studies, this cutting-edge book calls for a systematic modification and reshaping of the polytopic convex hull to achieve enhanced performance. Varying the convexity of the resulting TP canonical form is a key new feature of the approach. The book concentrates on reducing analytical derivations in the design process, echoing the recent paradigm shift on the acceptance of numerical solution as a valid form of output to control system problems. The salient features of the book include: Presents a new HOSVD-based canonical representation for (qLPV) models that enables trade-offs between approximation accuracy and computation complexity Supports a conceptually new control design methodology by proposing TP model transformation that offers a straightforward way of manipulating different types of convexity to appear in polytopic representation Introduces a numerical transformation that has the advantage of readily accommodating models described by non-conventional modeling and identification approaches, such as neural networks and fuzzy rules Presents a number of practical examples to demonstrate the application of the approach to generate control system design for complex (qLPV) systems and multiple control objectives. The authors’ approach is based on an extended version of singular value decomposition applicable to hyperdimensional tensors. Under the approach, trade-offs between approximation accuracy and computation complexity can be performed through the singular values to be retained in the process. The use of LMIs enables the incorporation of multiple performance objectives into the control design problem and assurance of a solution via convex optimization if feasible. Tensor Product Model Transformation in Polytopic Model-Based Control includes examples and incorporates MATLAB® Toolbox TPtool. It provides a reference guide for graduate students, researchers, engineers, and practitioners who are dealing with nonlinear systems control applications.

Dynamic Network Representation Based on Latent Factorization of Tensors
Dynamic Network Representation Based on Latent Factorization of Tensors

Author: Hao Wu

Publisher: Springer Nature

Published: 2023-03-07

Total Pages: 89

ISBN-13: 9811989346

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A dynamic network is frequently encountered in various real industrial applications, such as the Internet of Things. It is composed of numerous nodes and large-scale dynamic real-time interactions among them, where each node indicates a specified entity, each directed link indicates a real-time interaction, and the strength of an interaction can be quantified as the weight of a link. As the involved nodes increase drastically, it becomes impossible to observe their full interactions at each time slot, making a resultant dynamic network High Dimensional and Incomplete (HDI). An HDI dynamic network with directed and weighted links, despite its HDI nature, contains rich knowledge regarding involved nodes’ various behavior patterns. Therefore, it is essential to study how to build efficient and effective representation learning models for acquiring useful knowledge. In this book, we first model a dynamic network into an HDI tensor and present the basic latent factorization of tensors (LFT) model. Then, we propose four representative LFT-based network representation methods. The first method integrates the short-time bias, long-time bias and preprocessing bias to precisely represent the volatility of network data. The second method utilizes a proportion-al-integral-derivative controller to construct an adjusted instance error to achieve a higher convergence rate. The third method considers the non-negativity of fluctuating network data by constraining latent features to be non-negative and incorporating the extended linear bias. The fourth method adopts an alternating direction method of multipliers framework to build a learning model for implementing representation to dynamic networks with high preciseness and efficiency.