On The Numerical Solution Of Continuous Coupled Algebraic Riccati Equations
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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.
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.
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.
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.
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.
Lyapunov Matrix Equation In System Stability And Control
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Author : Zoran Gajic
language : en
Publisher: Courier Corporation
Release Date : 2008-01-01
Lyapunov Matrix Equation In System Stability And Control written by Zoran Gajic and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-01-01 with Mathematics categories.
This comprehensive treatment provides solutions to many engineering and mathematical problems related to the Lyapunov matrix equation, with self-contained chapters for easy reference. The authors offer a wide variety of techniques for solving and analyzing the algebraic, differential, and difference Lyapunov matrix equations of continuous-time and discrete-time systems. 1995 edition.
A Collection Of Benchmark Examples For The Numerical Solution Of Algebraic Riccati Equations
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Author : Peter Benner
language : en
Publisher:
Release Date : 1995
A Collection Of Benchmark Examples For The Numerical Solution Of 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 1995 with 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.
A Collection Of Benchmark Examples For The Numerical Solution Of Algebraic Riccati Equations
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Author : Peter Benner
language : en
Publisher:
Release Date : 1995
A Collection Of Benchmark Examples For The Numerical Solution Of 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 1995 with categories.
On The Numerical Solution Of The Discrete Time Algebraic Riccati Equation
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Author : Thrasyvoulos Nicholaou Pappas
language : en
Publisher:
Release Date : 1979
On The Numerical Solution Of The Discrete Time Algebraic Riccati Equation written by Thrasyvoulos Nicholaou Pappas and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1979 with Discrete-time systems categories.