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.

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.

Conceived by Count Jacopo Francesco Riccati more than a quarter of a millennium ago, the Riccati equation has been widely studied in the subsequent centuries. Since its introduction in control theory in the sixties, the matrix Riccati equation has known an impressive range of applications, such as optimal control, H? optimization and robust stabilization, stochastic realization, synthesis of linear passive networks, to name but a few. This book consists of 11 chapters surveying the main concepts and results related to the matrix Riccati equation, both in continuous and discrete time. Theory, applications and numerical algorithms are extensively presented in an expository way. As a foreword, the history and prehistory of the Riccati equation is concisely presented.

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.

Many physical problems are most naturally described by systems of differential and algebraic equations. This book describes some of the places where differential-algebraic equations (DAE's) occur. The basic mathematical theory for these equations is developed and numerical methods are presented and analyzed. Examples drawn from a variety of applications are used to motivate and illustrate the concepts and techniques. This classic edition, originally published in 1989, is the only general DAE book available. It not only develops guidelines for choosing different numerical methods, it is the first book to discuss DAE codes, including the popular DASSL code. An extensive discussion of backward differentiation formulas details why they have emerged as the most popular and best understood class of linear multistep methods for general DAE's. New to this edition is a chapter that brings the discussion of DAE software up to date. The objective of this monograph is to advance and consolidate the existing research results for the numerical solution of DAE's. The authors present results on the analysis of numerical methods, and also show how these results are relevant for the solution of problems from applications. They develop guidelines for problem formulation and effective use of the available mathematical software and provide extensive references for further study.

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.