Numerical Solution Of Projected Algebraic Riccati Equations
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Numerical Solution Of Projected Algebraic Riccati Equations
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Author : Peter Benner
language : en
Publisher:
Release Date : 2013
Numerical Solution Of Projected Algebraic Riccati Equations written by Peter Benner and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with categories.
Abstract: We consider the numerical solution of projected algebraic Riccati equations using Newton\'s method. Such equations arise, for instance, in model reduction of descriptor systems based on positive real and bounded real balanced truncation. We also discuss the computation of low-rank Cholesky factors of the solutions of projected Riccati equations. Numerical examples are given that demonstrate the properties of the proposed algorithms.
Algebraic Riccati Equations
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Author : Peter Lancaster
language : en
Publisher: Clarendon Press
Release Date : 1995-09-07
Algebraic Riccati Equations written by Peter Lancaster and has been published by Clarendon Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995-09-07 with Mathematics categories.
This book provides a careful treatment of the theory of algebraic Riccati equations. It consists of four parts: the first part is a comprehensive account of necessary background material in matrix theory including careful accounts of recent developments involving indefinite scalar products and rational matrix functions. The second and third parts form the core of the book and concern the solutions of algebraic Riccati equations arising from continuous and discrete systems. The geometric theory and iterative analysis are both developed in detail. The last part of the book is an exciting collection of eight problem areas in which algebraic Riccati equations play a crucial role. These applications range from introductions to the classical linear quadratic regulator problems and the discrete Kalman filter to modern developments in HD*W*w control and total least squares methods.
Numerical Solution Of Algebraic Riccati Equations
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Author : Dario A. Bini
language : en
Publisher: SIAM
Release Date : 2012-03-31
Numerical Solution Of Algebraic Riccati Equations written by Dario A. Bini and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-03-31 with Mathematics categories.
This treatment of the basic theory of algebraic Riccati equations describes the classical as well as the more advanced algorithms for their solution in a manner that is accessible to both practitioners and scholars. It is the first book in which nonsymmetric algebraic Riccati equations are treated in a clear and systematic way. Some proofs of theoretical results have been simplified and a unified notation has been adopted. Readers will find a unified discussion of doubling algorithms, which are effective in solving algebraic Riccati equations as well as a detailed description of all classical and advanced algorithms for solving algebraic Riccati equations and their MATLAB codes. This will help the reader gain an understanding of the computational issues and provide ready-to-use implementation of the different solution techniques.
Riccati Equations
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Author : Aleksandr Ivanovič Egorov
language : en
Publisher: Pensoft Publishers
Release Date : 2007
Riccati Equations written by Aleksandr Ivanovič Egorov and has been published by Pensoft Publishers this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Mathematics categories.
Presents the necessary auxiliary facts from algebra, functional analysis and Lie group analysis. This book illustrates theory with solutions of numerous examples. It also presents the matrix Riccati equations. It deals with theoretical questions concerning matrix and operator equations based on various applied problems from mathematical physics.
Numerical Solution Of The Coupled Algebraic Riccati Equations
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Author : Prasanthan Rajasingam
language : en
Publisher:
Release Date : 2013
Numerical Solution Of The Coupled Algebraic Riccati Equations written by Prasanthan Rajasingam and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with categories.
In this paper we develop new and improved results in the numerical solution of the coupled algebraic Riccati equations. First we provide improved matrix upper bounds on the positive semidefinite solution of the unified coupled algebraic Riccati equations. Our approach is largely inspired by recent results established by Liu and Zhang. Our main results tighten the estimates of the relevant dominant eigenvalues. Also by relaxing the key restriction our upper bound applies to a larger number of situations. We also present an iterative algorithm to refine the new upper bounds and the lower bounds and numerically compute the solutions of the unified coupled algebraic Riccati equations. This construction follows the approach of Gao, Xue and Sun but we use different bounds. This leads to different analysis on convergence. Besides, we provide new matrix upper bounds for the positive semidefinite solution of the continuous coupled algebraic Riccati equations. By using an alternative primary assumption we present a new upper bound. We follow the idea of Davies, Shi and Wiltshire for the non-coupled equation and extend their results to the coupled case. We also present an iterative algorithm to improve our upper bounds. Finally we improve the classical Newton's method by the line search technique to compute the solutions of the continuous coupled algebraic Riccati equations. The Newton's method for couple Riccati equations is attributed to Salama and Gourishanar, but we construct the algorithm in a different way using the Frechet derivative and we include line search too. Our algorithm leads to a faster convergence compared with the classical scheme. Numerical evidence is also provided to illustrate the performance of our algorithm.
Numerical Algebra Matrix Theory Differential Algebraic Equations And Control Theory
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Author : Peter Benner
language : en
Publisher: Springer
Release Date : 2015-05-09
Numerical Algebra Matrix Theory Differential Algebraic Equations And Control Theory written by Peter Benner and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-05-09 with Mathematics categories.
This edited volume highlights the scientific contributions of Volker Mehrmann, a leading expert in the area of numerical (linear) algebra, matrix theory, differential-algebraic equations and control theory. These mathematical research areas are strongly related and often occur in the same real-world applications. The main areas where such applications emerge are computational engineering and sciences, but increasingly also social sciences and economics. This book also reflects some of Volker Mehrmann's major career stages. Starting out working in the areas of numerical linear algebra (his first full professorship at TU Chemnitz was in "Numerical Algebra," hence the title of the book) and matrix theory, Volker Mehrmann has made significant contributions to these areas ever since. The highlights of these are discussed in Parts I and II of the present book. Often the development of new algorithms in numerical linear algebra is motivated by problems in system and control theory. These and his later major work on differential-algebraic equations, to which he together with Peter Kunkel made many groundbreaking contributions, are the topic of the chapters in Part III. Besides providing a scientific discussion of Volker Mehrmann's work and its impact on the development of several areas of applied mathematics, the individual chapters stand on their own as reference works for selected topics in the fields of numerical (linear) algebra, matrix theory, differential-algebraic equations and control theory.
On The Numerical Solution Of Differential And Algebraic Riccati Equations And Related Matters
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Author : Luca Dieci
language : en
Publisher:
Release Date : 1990
On The Numerical Solution Of Differential And Algebraic Riccati Equations And Related Matters written by Luca Dieci and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990 with Riccati equation categories.
A Collection Of Benchmark Examples For The Numerical Solution Of Algebraic Riccati Equations Ii Discrete Time Case
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Author : Peter Benner
language : en
Publisher:
Release Date : 1998
A Collection Of Benchmark Examples For The Numerical Solution Of Algebraic Riccati Equations Ii Discrete Time Case written by Peter Benner and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with categories.
On The Numerical Solution Of Continuous Coupled Algebraic Riccati Equations
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Author : Prasanthan Rajasingam
language : en
Publisher:
Release Date : 2016
On The Numerical Solution Of Continuous Coupled Algebraic Riccati Equations written by Prasanthan Rajasingam and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016 with Evolution equations, Nonlinear categories.
In this dissertation we first derive a new unified upper solution bound for the continuous coupled algebraic Riccati equation, which arises from the optimal control of a Markovian jump linear system. In particular, we address the issue of rank deficiency with the control matrices. In the case of rank deficiency the existing matrix upper bounds are inapplicable. Moreover, our new result is not restricted to rank deficiency cases only. It now contains the existing results as special cases. Next, an iterative refinement is presented to improve our new unified matrix upper solution bounds. In particular, this iterative refinement determines a monotonically decreasing sequence of upper bounds for the solution of the continuous coupled algebraic Riccati equation. We formulate a new iterative algorithm by modifying this iterative refinement. We also prove that this new algorithm generates a monotonically decreasing sequence of matrix upper solution bounds that converges to the maximal solution of the continuous coupled algebraic Riccati equation. Furthermore, we prove the convergence of an accelerated Riccati iteration which computes a positive semidefinite solution of the continuous coupled algebraic Riccati equation. In particular, we establish sufficient conditions for the convergence of this algorithm. We also prove that for particular initial values this algorithm determines a monotonically increasing sequence of positive semidefinite matrices that converge to the minimal solution of the continuous coupled algebraic Riccati equation. Additionally, we show that for specific initial values this algorithm generates a monotonically decreasing sequence that converges to the maximal solution of the continuous coupled algebraic Riccati equation. In addition, we prove that this accelerated Riccati iteration converges faster than the Riccati iteration. Finally, we formulate a weighted modified accelerated Riccati iteration which is a more generalized Riccati type iteration. All of the existing Riccati iterations are now the special cases of this algorithm. Furthermore, we establish sufficient conditions for the convergence of this algorithm and we prove the monotonic convergence of the sequence generated by this algorithm. We also discuss how the weight and other quantities affect the rate of convergence of this algorithm. Illustrative numerical examples are also presented.
On The Solution Of Large Scale Algebraic Riccati Equations By Using Low Dimensional Invariant Subspaces
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Author : Peter Benner
language : en
Publisher:
Release Date : 2014
On The Solution Of Large Scale Algebraic Riccati Equations By Using Low Dimensional Invariant Subspaces written by Peter Benner and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with categories.
Abstract: This article discusses an approach to solving large-scale algebraic Riccati equations (AREs) by computing a low-dimensional stable invariant subspace of the associated Hamiltonian matrix. We give conditions on AREs to admit solutions of low numerical rank and show that these can be approximated via Hamiltonian eigenspaces. We discuss strategies on choosing the proper eigenspace that yields a good approximation, and different formulas for building the approximation itself. Similarities of our approach with several other methods for solving AREs are shown: closely related are the projection-type methods that use various Krylov subspaces and the qADI algorithm. The aim of this paper is merely to analyze the possibilities of computing approximate Riccati solutions from low-dimensional subspaces related to the corresponding Hamiltonian matrix and to explain commonalities among existing methods rather than providing a new algorithm.