A State Space Approach to Canonical Factorization with Applications

by Harm Bart, Israel Gohberg, Marinus A. Kaashoek, André C.M. Ran

Overview

The present book deals with canonical factorization problems for different classes of matrix and operator functions. Such problems appear in various areas of ma- ematics and its applications. The functions we consider havein common that they appear in the state space form or can be represented in such a form. The main results are all expressed in terms of the matrices or operators appearing in the state space representation. This includes necessary and sufficient conditions for canonical factorizations to exist and explicit formulas for the corresponding f- tors. Also, in the applications the entries in the state space representation play a crucial role. Thetheorydevelopedinthebookisbasedonageometricapproachwhichhas its origins in different fields. One of the initial steps can be found in mathematical systems theory and electrical network theory, where a cascade decomposition of an input-output system or a network is related to a factorization of the associated transfer function. Canonical factorization has a long and interesting history which starts in the theory of convolution equations. Solving Wiener-Hopf integral equations is closely related to canonical factorization. The problem of canonical factorization also appears in other branches of applied analysis and in mathematical systems theory, in H -control theory in particular.

Product Details

ISBN-13:	9783764387525
Publisher:	Birkhäuser Basel
Publication date:	06/04/2010
Series:	Operator Theory: Advances and Applications , #200
Edition description:	2010
Pages:	432
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

Convolution equations, canonical factorization and the state space method.- The role of canonical factorization in solving convolution equations.- The state space method and factorization.- Convolution equations with rational matrix symbols.- Explicit solutions using realizations.- Factorization of non-proper rational matrix functions.- Equations with non-rational symbols.- Factorization of matrix functions analytic in a strip.- Convolution equations and the transport equation.- Wiener-Hopf factorization and factorization indices.- Factorization of selfadjoint rational matrix functions.- Preliminaries concerning minimal factorization.- Factorization of positive definite rational matrix functions.- Pseudo-spectral factorizations of selfadjoint rational matrix functions.- Review of the theory of matrices in indefinite inner product spaces.- Riccati equations and factorization.- Canonical factorization and Riccati equations.- The symmetric algebraic Riccati equation.- J-spectral factorization.- Factorizations and symmetries.- Factorization of positive real rational matrix functions.- Contractive rational matrix functions.- J-unitary rational matrix functions.- Applications of J-spectral factorizations.- Application to the rational Nehari problem.- Review of some control theory for linear systems.- H-infinity control applications.

Editorial Reviews

From the reviews:

“This monograph develops a theory of canonical factorizations for various classes of matrix and operator functions, where the functions are represented in the form of a transfer function of an input-state-output linear system. … The book is self-contained and accessible for specialists in different areas of mathematics, science, and engineering.” (Dmitry Kaliuzhnyi-Verbovetskyi, Zentralblatt MATH, Vol. 1203, 2011)

From the Publisher