Model reduction is an important engineering problem in which one aims to replace an elaborate model by a simpler model without undue loss of accuracy. The accuracy can be mathematically measured in several possible norms and the Hankel norm is one such. The Hankel norm gives a meaningful notion of distance between two linear systems: roughly speaking, it is the induced norm of the operator that maps past inputs to future outputs. It turns out that the engineering problem of model reduction in the Hankel norm is closely related to the mathematical problem of finding solutions to the sub-optimal Nehari-Takagi problem, which is called "the sub-optimal Hankel norm approximation problem" in this book. Although the existence of a solution to the sub-optimal Hankel norm approximation problem has been known since the 1970's, this book presents explicit solutions and, in particular, new formulae for several large classes of infinite-dimensional systems for the first time.
Author(s): A. Sasane
Series: Lecture Notes in Control and Information Sciences O277
Publisher: Springer
Year: 2002
Language: English
Pages: 144
Lecture Notes in Control and Information Sciences......Page 1
Hankel Norm Approximation for Infinite-Dimensional Systems......Page 3
Preface......Page 5
Contents......Page 7
Standard notation......Page 0
Introduction......Page 9
The sub-optimal Nehari problem......Page 12
The sub-optimal Hankel norm approximation problem......Page 13
The Hankel norm of systems......Page 14
Hankel norm approximation and model reduction......Page 16
Organization of the book......Page 18
Classes of well-posed linear systems......Page 21
Well-posed linear systems......Page 22
Duality theory......Page 24
The Pritchard-Salamon class......Page 28
The analytic class......Page 33
Transfer function algebras......Page 60
Compactness and nuclearity of Hankel operators......Page 71
Compactness of Hankel operators......Page 74
Nuclearity of Hankel operators......Page 82
Loo-error of sub-optimal Hankel norm approximants......Page 88
The sub-optimal Hankel norm approximation problem......Page 92
Main assumptions and a few useful consequences......Page 96
All solutions to the sub-optimal Hankel norm approximation problem......Page 102
A comparison with existing results......Page 106
J-spectral factorization for the Pritchard-Salamon class......Page 107
J-spectral factorization for the analytic class......Page 110
The non-exponentially stable case......Page 115
A few useful results......Page 116
The class of systems and the main result......Page 117
State-space solutions......Page 120
Reciprocal systems......Page 124
The sub-optimal Hankel norm approximation problem......Page 126
The sub-optimal Nehari problem......Page 128
Coda......Page 131
Bibliography......Page 134
Index......Page 141
Standard notation......Page 143
