The Necessity of the Small-Gain Theorem for Time-Varying and Nonlinear Systems

Jeff S. Shamma

doi:10.1109/9.90227

The Necessity of the Small-Gain Theorem for Time-Varying and Nonlinear Systems

Jeff S. Shamma^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

32 Scopus citations

Abstract

It is shown that an appropriately modified version of the small-gain theorem which employs the “asymptotic gain” of a fading-memory system leads to a necessary and sufficient condition for the stable invertibility of certain feedback operators. This modified small-gain theorem is then used to obtain conditions for robust finite-energy input-output stability of 1) nonlinear plants subject to nonlinear time-varying perturbations and 2) linear time-varying plants subject to linear time-varying perturbations. These conditions are shown to be necessary as well as sufficient.

Original language	English (US)
Pages (from-to)	1138-1147
Number of pages	10
Journal	IEEE Transactions on Automatic Control
Volume	36
Issue number	10
DOIs	https://doi.org/10.1109/9.90227
State	Published - Oct 1991
Externally published	Yes

ASJC Scopus subject areas

Control and Systems Engineering
Computer Science Applications
Electrical and Electronic Engineering

Access to Document

10.1109/9.90227

Cite this

@article{5dca2b3bddef44a78e67a7a3427e0002,

title = "The Necessity of the Small-Gain Theorem for Time-Varying and Nonlinear Systems",

abstract = "It is shown that an appropriately modified version of the small-gain theorem which employs the “asymptotic gain” of a fading-memory system leads to a necessary and sufficient condition for the stable invertibility of certain feedback operators. This modified small-gain theorem is then used to obtain conditions for robust finite-energy input-output stability of 1) nonlinear plants subject to nonlinear time-varying perturbations and 2) linear time-varying plants subject to linear time-varying perturbations. These conditions are shown to be necessary as well as sufficient.",

author = "Shamma, {Jeff S.}",

year = "1991",

month = oct,

doi = "10.1109/9.90227",

language = "English (US)",

volume = "36",

pages = "1138--1147",

journal = "IEEE Transactions on Automatic Control",

issn = "0018-9286",

publisher = "Institute of Electrical and Electronics Engineers Inc.",

number = "10",

}

TY - JOUR

T1 - The Necessity of the Small-Gain Theorem for Time-Varying and Nonlinear Systems

AU - Shamma, Jeff S.

PY - 1991/10

Y1 - 1991/10

N2 - It is shown that an appropriately modified version of the small-gain theorem which employs the “asymptotic gain” of a fading-memory system leads to a necessary and sufficient condition for the stable invertibility of certain feedback operators. This modified small-gain theorem is then used to obtain conditions for robust finite-energy input-output stability of 1) nonlinear plants subject to nonlinear time-varying perturbations and 2) linear time-varying plants subject to linear time-varying perturbations. These conditions are shown to be necessary as well as sufficient.

AB - It is shown that an appropriately modified version of the small-gain theorem which employs the “asymptotic gain” of a fading-memory system leads to a necessary and sufficient condition for the stable invertibility of certain feedback operators. This modified small-gain theorem is then used to obtain conditions for robust finite-energy input-output stability of 1) nonlinear plants subject to nonlinear time-varying perturbations and 2) linear time-varying plants subject to linear time-varying perturbations. These conditions are shown to be necessary as well as sufficient.

UR - http://www.scopus.com/inward/record.url?scp=0026239616&partnerID=8YFLogxK

U2 - 10.1109/9.90227

DO - 10.1109/9.90227

M3 - Article

AN - SCOPUS:0026239616

SN - 0018-9286

VL - 36

SP - 1138

EP - 1147

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

IS - 10

ER -

The Necessity of the Small-Gain Theorem for Time-Varying and Nonlinear Systems

Abstract

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this