The Lq/Lp Hankel Norms of Positive Systems

Yoshio Ebihara, Bohao Zhu, James Lam

Research output: Contribution to journalArticle

Abstract

This letter is concerned with the Lq/Lp Hankel norms of linear time-invariant positive systems, where the Hankel norms are defined as the induced norms from vector-valued Lp -past inputs to vector valued Lq -future outputs. The Lq/Lp Hankel norms are studied in detail for general (nonpositive) LTI systems, and closed-form formulas have been derived for their characterization. However, some of them unavoidably include the absolute integral of impulse responses and/or implicit functions in their characterization and hence there remain difficulties in practical computation. In this letter we show that such difficulties are circumvented for positive systems, where positivity drastically facilitates the characterization of the past input that attains the Hankel norm. We also provide linear-programming- and semidefinite-programming-based characterizations of the Lq/Lp Hankel norms of positive systems, which are useful in the case where we analyze the Lq/Lp Hankel norms of positive systems affected by parametric uncertainties.

Original languageEnglish
Article number8895893
Pages (from-to)462-467
Number of pages6
JournalIEEE Control Systems Letters
Volume4
Issue number2
DOIs
Publication statusPublished - Apr 2020
Externally publishedYes

Fingerprint

Positive Systems
Hankel
Norm
Impulse response
Linear programming
Implicit Function
Parametric Uncertainty
Semidefinite Programming
Impulse Response
Positivity
Linear Time
Closed-form
Invariant
Output

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Control and Optimization

Cite this

The Lq/Lp Hankel Norms of Positive Systems. / Ebihara, Yoshio; Zhu, Bohao; Lam, James.

In: IEEE Control Systems Letters, Vol. 4, No. 2, 8895893, 04.2020, p. 462-467.

Research output: Contribution to journalArticle

Ebihara, Yoshio ; Zhu, Bohao ; Lam, James. / The Lq/Lp Hankel Norms of Positive Systems. In: IEEE Control Systems Letters. 2020 ; Vol. 4, No. 2. pp. 462-467.
@article{d654bbf4f6b141688808c58457ebacc2,
title = "The Lq/Lp Hankel Norms of Positive Systems",
abstract = "This letter is concerned with the Lq/Lp Hankel norms of linear time-invariant positive systems, where the Hankel norms are defined as the induced norms from vector-valued Lp -past inputs to vector valued Lq -future outputs. The Lq/Lp Hankel norms are studied in detail for general (nonpositive) LTI systems, and closed-form formulas have been derived for their characterization. However, some of them unavoidably include the absolute integral of impulse responses and/or implicit functions in their characterization and hence there remain difficulties in practical computation. In this letter we show that such difficulties are circumvented for positive systems, where positivity drastically facilitates the characterization of the past input that attains the Hankel norm. We also provide linear-programming- and semidefinite-programming-based characterizations of the Lq/Lp Hankel norms of positive systems, which are useful in the case where we analyze the Lq/Lp Hankel norms of positive systems affected by parametric uncertainties.",
author = "Yoshio Ebihara and Bohao Zhu and James Lam",
year = "2020",
month = "4",
doi = "10.1109/LCSYS.2019.2952622",
language = "English",
volume = "4",
pages = "462--467",
journal = "IEEE Control Systems Letters",
issn = "2475-1456",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
number = "2",

}

TY - JOUR

T1 - The Lq/Lp Hankel Norms of Positive Systems

AU - Ebihara, Yoshio

AU - Zhu, Bohao

AU - Lam, James

PY - 2020/4

Y1 - 2020/4

N2 - This letter is concerned with the Lq/Lp Hankel norms of linear time-invariant positive systems, where the Hankel norms are defined as the induced norms from vector-valued Lp -past inputs to vector valued Lq -future outputs. The Lq/Lp Hankel norms are studied in detail for general (nonpositive) LTI systems, and closed-form formulas have been derived for their characterization. However, some of them unavoidably include the absolute integral of impulse responses and/or implicit functions in their characterization and hence there remain difficulties in practical computation. In this letter we show that such difficulties are circumvented for positive systems, where positivity drastically facilitates the characterization of the past input that attains the Hankel norm. We also provide linear-programming- and semidefinite-programming-based characterizations of the Lq/Lp Hankel norms of positive systems, which are useful in the case where we analyze the Lq/Lp Hankel norms of positive systems affected by parametric uncertainties.

AB - This letter is concerned with the Lq/Lp Hankel norms of linear time-invariant positive systems, where the Hankel norms are defined as the induced norms from vector-valued Lp -past inputs to vector valued Lq -future outputs. The Lq/Lp Hankel norms are studied in detail for general (nonpositive) LTI systems, and closed-form formulas have been derived for their characterization. However, some of them unavoidably include the absolute integral of impulse responses and/or implicit functions in their characterization and hence there remain difficulties in practical computation. In this letter we show that such difficulties are circumvented for positive systems, where positivity drastically facilitates the characterization of the past input that attains the Hankel norm. We also provide linear-programming- and semidefinite-programming-based characterizations of the Lq/Lp Hankel norms of positive systems, which are useful in the case where we analyze the Lq/Lp Hankel norms of positive systems affected by parametric uncertainties.

UR - http://www.scopus.com/inward/record.url?scp=85075631396&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075631396&partnerID=8YFLogxK

U2 - 10.1109/LCSYS.2019.2952622

DO - 10.1109/LCSYS.2019.2952622

M3 - Article

AN - SCOPUS:85075631396

VL - 4

SP - 462

EP - 467

JO - IEEE Control Systems Letters

JF - IEEE Control Systems Letters

SN - 2475-1456

IS - 2

M1 - 8895893

ER -