Open AccessOn well‐definability of the L ∞ / L 2 Hankel operator and detection of all the critical instants in sampled‐data systems
Author(s) -
Hagiwara Tomomichi,
Inai Akira,
Kim Jung Hoon
Publication year - 2021
Publication title -
iet control theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.059
H-Index - 108
eISSN - 1751-8652
pISSN - 1751-8644
DOI - 10.1049/cth2.12069
Subject(s) - infimum and supremum , mathematics , computation , norm (philosophy) , hankel matrix , operator (biology) , uniform norm , hankel transform , discrete mathematics , pure mathematics , mathematical analysis , algorithm , law , biochemistry , chemistry , repressor , transcription factor , gene , bessel function , political science
Because sampled‐data systems have h ‐periodic nature with the sampling period h , an arbitrary Θ ∈ [ 0 , h ) is taken and the quasiL ∞ / L 2Hankel operator at Θ is defined as the mapping fromL 2 ( − ∞ , Θ )toL ∞ [ Θ , ∞ ) . Its norm called the quasiL ∞ / L 2Hankel norm at Θ is used to define theL ∞ / L 2Hankel norm as the supremum of their values over Θ ∈ [ 0 , h ) . If the supremum is actually attained as the maximum, then a maximum‐attaining Θ is called a critical instant and theL ∞ / L 2Hankel operator is said to be well‐definable. An earlier study establishes a computation method of theL ∞ / L 2Hankel norm, which is called a sophisticated method if our interest lies only in its computation. However, the feature of the method that it is free from considering the quasiL ∞ / L 2Hankel norm for any Θ ∈ [ 0 , h ) prevents the earlier study to give any arguments as to whether the obtainedL ∞ / L 2Hankel norm is actually attained as the maximum, as well as detecting all the critical instants when theL ∞ / L 2Hankel operator is well‐definable. This paper establishes further arguments to tackle these relevant questions and provides numerical examples to validate the arguments in different aspects of authors' theoretical interests.