Parameter-dependent Lyapunov functions for linear systems with constant uncertainties

Peter J Seiler Jr, Ufuk Topcu, Andy Packard, Gary Balas

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

Robust stability of linear time-invariant systems with respect to structured uncertainties is considered. The small gain condition is sufficient to prove robust stability and scalings are typically used to reduce the conservatism of this condition. It is known that if the small gain condition is satisfied with constant scalings then there is a single quadratic Lyapunov function which proves robust stability with respect to all allowable time-varying perturbations. In this technical note we show that if the small gain condition is satisfied with frequency-varying scalings then an explicit parameter dependent Lyapunov function can be constructed to prove robust stability with respect to constant uncertainties. This Lyapunov function has a rational quadratic dependence on the uncertainties.

Original languageEnglish (US)
Pages (from-to)2410-2416
Number of pages7
JournalIEEE Transactions on Automatic Control
Volume54
Issue number10
DOIs
StatePublished - Sep 24 2009

Fingerprint

Lyapunov functions
Linear systems
Robust stability
Uncertainty

Keywords

  • Parameter-dependent Lyapunov function (PDLF)

Cite this

Parameter-dependent Lyapunov functions for linear systems with constant uncertainties. / Seiler Jr, Peter J; Topcu, Ufuk; Packard, Andy; Balas, Gary.

In: IEEE Transactions on Automatic Control, Vol. 54, No. 10, 24.09.2009, p. 2410-2416.

Research output: Contribution to journalArticle

Seiler Jr, Peter J ; Topcu, Ufuk ; Packard, Andy ; Balas, Gary. / Parameter-dependent Lyapunov functions for linear systems with constant uncertainties. In: IEEE Transactions on Automatic Control. 2009 ; Vol. 54, No. 10. pp. 2410-2416.
@article{4e7952e07f464ae5b01efec4b505904f,
title = "Parameter-dependent Lyapunov functions for linear systems with constant uncertainties",
abstract = "Robust stability of linear time-invariant systems with respect to structured uncertainties is considered. The small gain condition is sufficient to prove robust stability and scalings are typically used to reduce the conservatism of this condition. It is known that if the small gain condition is satisfied with constant scalings then there is a single quadratic Lyapunov function which proves robust stability with respect to all allowable time-varying perturbations. In this technical note we show that if the small gain condition is satisfied with frequency-varying scalings then an explicit parameter dependent Lyapunov function can be constructed to prove robust stability with respect to constant uncertainties. This Lyapunov function has a rational quadratic dependence on the uncertainties.",
keywords = "Parameter-dependent Lyapunov function (PDLF)",
author = "{Seiler Jr}, {Peter J} and Ufuk Topcu and Andy Packard and Gary Balas",
year = "2009",
month = "9",
day = "24",
doi = "10.1109/TAC.2009.2029294",
language = "English (US)",
volume = "54",
pages = "2410--2416",
journal = "IEEE Transactions on Automatic Control",
issn = "0018-9286",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
number = "10",

}

TY - JOUR

T1 - Parameter-dependent Lyapunov functions for linear systems with constant uncertainties

AU - Seiler Jr, Peter J

AU - Topcu, Ufuk

AU - Packard, Andy

AU - Balas, Gary

PY - 2009/9/24

Y1 - 2009/9/24

N2 - Robust stability of linear time-invariant systems with respect to structured uncertainties is considered. The small gain condition is sufficient to prove robust stability and scalings are typically used to reduce the conservatism of this condition. It is known that if the small gain condition is satisfied with constant scalings then there is a single quadratic Lyapunov function which proves robust stability with respect to all allowable time-varying perturbations. In this technical note we show that if the small gain condition is satisfied with frequency-varying scalings then an explicit parameter dependent Lyapunov function can be constructed to prove robust stability with respect to constant uncertainties. This Lyapunov function has a rational quadratic dependence on the uncertainties.

AB - Robust stability of linear time-invariant systems with respect to structured uncertainties is considered. The small gain condition is sufficient to prove robust stability and scalings are typically used to reduce the conservatism of this condition. It is known that if the small gain condition is satisfied with constant scalings then there is a single quadratic Lyapunov function which proves robust stability with respect to all allowable time-varying perturbations. In this technical note we show that if the small gain condition is satisfied with frequency-varying scalings then an explicit parameter dependent Lyapunov function can be constructed to prove robust stability with respect to constant uncertainties. This Lyapunov function has a rational quadratic dependence on the uncertainties.

KW - Parameter-dependent Lyapunov function (PDLF)

UR - http://www.scopus.com/inward/record.url?scp=70350347457&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350347457&partnerID=8YFLogxK

U2 - 10.1109/TAC.2009.2029294

DO - 10.1109/TAC.2009.2029294

M3 - Article

VL - 54

SP - 2410

EP - 2416

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 10

ER -