A State Space Approach to Canonical Factorization with Applications

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Springer Science & Business Media, Feb 9, 2011 - Mathematics - 432 pages
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The present book deals with canonical factorization problems for di?erent 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 su?cient 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 di?erent ?elds. 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.
 

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Contents

Chapter 0 Introduction
1
Part I Convolution equations canonical factorization and the state space method
7
Chapter 1 The role of canonical factorization in solving convolution equations
8
Chapter 2 The state space method and factorization
19
Part II Convolution equations with rational matrix symbols
34
Chapter 3 Explicit solutions using realizations
37
Chapter 4 Factorization of nonproper rational matrix functions
57
Part III Equations with nonrational symbols
75
Part V Riccati equations and factorization
217
Chapter 12 Canonical factorization and Riccati equations
218
Chapter 13The symmetric algebraic Riccatiequation
233
Chapter 14 Jspectral factorization
248
Part VI Factorizations and symmetries
289
Chapter 15 Factorization of positive realrational matrix functions
291
Chapter 16 Contractive rational matrix functions
301
Chapter 17 Junitary rational matrix functions
313

Chapter 5 Factorization of matrix functions analytic in a strip
76
Chapter 6 Convolution equations and the transport equation
115
Chapter 7 WienerHopf factorization and factorization indices
143
Part IV Factorization of selfadjoint rational matrix functions
168
Chapter 8 Preliminaries concerning minimal factorization
171
Chapter 9 Factorization of positive definite rational matrix functions
180
Chapter 10 Pseudospectral factorizations of selfadjoint rational matrix functions
197
Chapter 11 Review of the theory of matrices in indefinite inner product spaces
211
Part VII Applications of Jspectral factorizations
346
Chapter 18 Application to the rational Nehari problem
349
Chapter 19 Review of some control theory for linear systems
371
Chapter 20 Hinfinity control applications
379
Bibliography
405
List of symbols
415
Index
419
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