Numerical Methods For Large Scale Linear Time Varying Control Systems And Related Differential Matrix Equations
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| Author | : Norman Lang |
| Publisher | : Logos Verlag Berlin GmbH |
| Total Pages | : 232 |
| Release | : 2018 |
| Genre | : Mathematics |
| ISBN | : 3832547002 |
This thesis is concerned with the linear-quadratic optimal control and model order reduction (MOR) of large-scale linear time-varying (LTV) control systems. In the first two parts, particular attention is paid to a tracking-type finite-time optimal control problem with application to an inverse heat conduction problem and the balanced truncation (BT) MOR method for LTV systems. In both fields of application the efficient solution of differential matrix equations (DMEs) is of major importance. The third and largest part deals with the application of implicit time integration methods to these matrix-valued ordinary differential equations. In this context, in particular, the rather new class of peer methods is introduced. Further, for the efficient solution of large-scale DMEs, in practice low-rank solution strategies are inevitable. Here, low-rank time integrators, based on a symmetric indefinte factored representation of the right hand sides and the solution approximations of the DMEs, are presented. In contrast to the classical low-rank Cholesky-type factorization, this avoids complex arithmetic and tricky implementations and algorithms. Both low-rank approaches are compared for numerous implicit time integration methods.
| Author | : Biswa Datta |
| Publisher | : Academic Press |
| Total Pages | : 737 |
| Release | : 2004 |
| Genre | : Mathematics |
| ISBN | : 0122035909 |
Numerical Methods for Linear Control Systems Design and Analysis is an interdisciplinary textbook aimed at systematic descriptions and implementations of numerically-viable algorithms based on well-established, efficient and stable modern numerical linear techniques for mathematical problems arising in the design and analysis of linear control systems both for the first- and second-order models. Unique coverage of modern mathematical concepts such as parallel computations, second-order systems, and large-scale solutions Background material in linear algebra, numerical linear algebra, and control theory included in text Step-by-step explanations of the algorithms and examples
| Author | : Eugenius Kaszkurewicz |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 279 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461213460 |
This monograph presents a collection of results, observations, and examples related to dynamical systems described by linear and nonlinear ordinary differential and difference equations. In particular, dynamical systems that are susceptible to analysis by the Liapunov approach are considered. The naive observation that certain "diagonal-type" Liapunov functions are ubiquitous in the literature attracted the attention of the authors and led to some natural questions. Why does this happen so often? What are the spe cial virtues of these functions in this context? Do they occur so frequently merely because they belong to the simplest class of Liapunov functions and are thus more convenient, or are there any more specific reasons? This monograph constitutes the authors' synthesis of the work on this subject that has been jointly developed by them, among others, producing and compiling results, properties, and examples for many years, aiming to answer these questions and also to formalize some of the folklore or "cul ture" that has grown around diagonal stability and diagonal-type Liapunov functions. A natural answer to these questions would be that the use of diagonal type Liapunov functions is frequent because of their simplicity within the class of all possible Liapunov functions. This monograph shows that, although this obvious interpretation is often adequate, there are many in stances in which the Liapunov approach is best taken advantage of using diagonal-type Liapunov functions. In fact, they yield necessary and suffi cient stability conditions for some classes of nonlinear dynamical systems.
| Author | : Amit Bhaya |
| Publisher | : SIAM |
| Total Pages | : 297 |
| Release | : 2006-01-01 |
| Genre | : Mathematics |
| ISBN | : 9780898718669 |
Control Perspectives on Numerical Algorithms and Matrix Problems organizes the analysis and design of iterative numerical methods from a control perspective. The authors discuss a variety of applications, including iterative methods for linear and nonlinear systems of equations, neural networks for linear and quadratic programming problems, support vector machines, integration and shooting methods for ordinary differential equations, matrix preconditioning, matrix stability, and polynomial zero finding. This book opens up a new field of interdisciplinary research that should lead to insights in the areas of both control and numerical analysis and shows that a wide range of applications can be approached from, and benefit from, a control perspective.
| Author | : Patrick DeWilde |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 457 |
| Release | : 2013-03-09 |
| Genre | : Science |
| ISBN | : 1475728174 |
Complex function theory and linear algebra provide much of the basic mathematics needed by engineers engaged in numerical computations, signal processing or control. The transfer function of a linear time invariant system is a function of the complex vari able s or z and it is analytic in a large part of the complex plane. Many important prop erties of the system for which it is a transfer function are related to its analytic prop erties. On the other hand, engineers often encounter small and large matrices which describe (linear) maps between physically important quantities. In both cases similar mathematical and computational problems occur: operators, be they transfer functions or matrices, have to be simplified, approximated, decomposed and realized. Each field has developed theory and techniques to solve the main common problems encountered. Yet, there is a large, mysterious gap between complex function theory and numerical linear algebra. For example, complex function theory has solved the problem to find analytic functions of minimal complexity and minimal supremum norm that approxi e. g. , as optimal mate given values at strategic points in the complex plane. They serve approximants for a desired behavior of a system to be designed. No similar approxi mation theory for matrices existed until recently, except for the case where the matrix is (very) close to singular.
| Author | : Claude Brezinski |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 296 |
| Release | : 2013-12-01 |
| Genre | : Mathematics |
| ISBN | : 1461302617 |
Many devices (we say dynamical systems or simply systems) behave like black boxes: they receive an input, this input is transformed following some laws (usually a differential equation) and an output is observed. The problem is to regulate the input in order to control the output, that is for obtaining a desired output. Such a mechanism, where the input is modified according to the output measured, is called feedback. The study and design of such automatic processes is called control theory. As we will see, the term system embraces any device and control theory has a wide variety of applications in the real world. Control theory is an interdisci plinary domain at the junction of differential and difference equations, system theory and statistics. Moreover, the solution of a control problem involves many topics of numerical analysis and leads to many interesting computational problems: linear algebra (QR, SVD, projections, Schur complement, structured matrices, localization of eigenvalues, computation of the rank, Jordan normal form, Sylvester and other equations, systems of linear equations, regulariza tion, etc), root localization for polynomials, inversion of the Laplace transform, computation of the matrix exponential, approximation theory (orthogonal poly nomials, Pad6 approximation, continued fractions and linear fractional transfor mations), optimization, least squares, dynamic programming, etc. So, control theory is also a. good excuse for presenting various (sometimes unrelated) issues of numerical analysis and the procedures for their solution. This book is not a book on control.
| Author | : Achim Ilchmann |
| Publisher | : Springer |
| Total Pages | : 343 |
| Release | : 2014-12-04 |
| Genre | : Mathematics |
| ISBN | : 3319110500 |
The present volume comprises survey articles on various fields of Differential-Algebraic Equations (DAEs), which have widespread applications in controlled dynamical systems, especially in mechanical and electrical engineering and a strong relation to (ordinary) differential equations. The individual chapters provide reviews, presentations of the current state of research and new concepts in - Observers for DAEs - DAEs in chemical processes - Optimal control of DAEs - DAEs from a functional-analytic viewpoint - Algebraic methods for DAEs The results are presented in an accessible style, making this book suitable not only for active researchers but also for graduate students (with a good knowledge of the basic principles of DAEs) for self-study.
| Author | : Athanasios C. Antoulas |
| Publisher | : SIAM |
| Total Pages | : 489 |
| Release | : 2009-06-25 |
| Genre | : Mathematics |
| ISBN | : 0898716586 |
Mathematical models are used to simulate, and sometimes control, the behavior of physical and artificial processes such as the weather and very large-scale integration (VLSI) circuits. The increasing need for accuracy has led to the development of highly complex models. However, in the presence of limited computational accuracy and storage capabilities model reduction (system approximation) is often necessary. Approximation of Large-Scale Dynamical Systems provides a comprehensive picture of model reduction, combining system theory with numerical linear algebra and computational considerations. It addresses the issue of model reduction and the resulting trade-offs between accuracy and complexity. Special attention is given to numerical aspects, simulation questions, and practical applications.
| Author | : National Aeronautics and Space Adm Nasa |
| Publisher | : |
| Total Pages | : 26 |
| Release | : 2018-09-27 |
| Genre | : |
| ISBN | : 9781724087232 |
A variable order method of integrating initial value ordinary differential equations that is based on the state transition matrix has been developed. The method has been evaluated for linear time variant and nonlinear systems of equations. While it is more complex than most other methods, it produces exact solutions at arbitrary time step size when the time variation of the system can be modeled exactly by a polynomial. Solutions to several nonlinear problems exhibiting chaotic behavior have been computed. Accuracy of the method has been demonstrated by comparison with an exact solution and with solutions obtained by established methods. Bartels, Robert E. Langley Research Center NASA/TM-2002-211776, NAS 1.15:211776, L-18185...
| Author | : William M. McEneaney |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 268 |
| Release | : 2006 |
| Genre | : Language Arts & Disciplines |
| ISBN | : 9780817635343 |
The central focus of this book is the control of continuous-time/continuous-space nonlinear systems. Using new techniques that employ the max-plus algebra, the author addresses several classes of nonlinear control problems, including nonlinear optimal control problems and nonlinear robust/H-infinity control and estimation problems. Several numerical techniques are employed, including a max-plus eigenvector approach and an approach that avoids the curse-of-dimensionality. The max-plus-based methods examined in this work belong to an entirely new class of numerical methods for the solution of nonlinear control problems and their associated Hamilton–Jacobi–Bellman (HJB) PDEs; these methods are not equivalent to either of the more commonly used finite element or characteristic approaches. Max-Plus Methods for Nonlinear Control and Estimation will be of interest to applied mathematicians, engineers, and graduate students interested in the control of nonlinear systems through the implementation of recently developed numerical methods.
