A Lie-algebraic condition for stability of switched nonlinear systems

Michael Margaliot, Daniel M Liberzon

Research output: Contribution to journalConference article

Abstract

We present a stability criterion for switched nonlinear systems which involves Lie brackets of the individual vector fields but does not require that these vector fields commute. A special case of the main result says that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields whose third-order Lie brackets vanish is globally uniformly asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [1]. To prove the result, we consider an optimal control problem which consists in finding the "most unstable" trajectory for an associated control system, and show that there exists an optimal solution which is bang-bang with a bound on the total number of switches. By construction, our criterion also automatically applies to the corresponding relaxed differential inclusion.

Original languageEnglish (US)
Pages (from-to)4619-4624
Number of pages6
JournalProceedings of the IEEE Conference on Decision and Control
Volume5
StatePublished - Dec 1 2004
Event2004 43rd IEEE Conference on Decision and Control (CDC) - Nassau, Bahamas
Duration: Dec 14 2004Dec 17 2004

Fingerprint

Switched Systems
Lie Brackets
Nonlinear systems
Vector Field
Nonlinear Systems
Switched Linear Systems
Globally Asymptotically Stable
Stability criteria
Differential Inclusions
Commute
Asymptotically Stable
Stability Criteria
Linear systems
Optimal Control Problem
Vanish
Open Problems
Switch
Optimal Solution
Unstable
Control System

Keywords

  • Differential inclusion
  • Global asymptotic stability
  • Lie bracket
  • Maximum principle
  • Optimal control
  • Switched nonlinear system

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

Cite this

A Lie-algebraic condition for stability of switched nonlinear systems. / Margaliot, Michael; Liberzon, Daniel M.

In: Proceedings of the IEEE Conference on Decision and Control, Vol. 5, 01.12.2004, p. 4619-4624.

Research output: Contribution to journalConference article

@article{ba580d4790834dfab29aa1e6ffb924ce,
title = "A Lie-algebraic condition for stability of switched nonlinear systems",
abstract = "We present a stability criterion for switched nonlinear systems which involves Lie brackets of the individual vector fields but does not require that these vector fields commute. A special case of the main result says that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields whose third-order Lie brackets vanish is globally uniformly asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [1]. To prove the result, we consider an optimal control problem which consists in finding the {"}most unstable{"} trajectory for an associated control system, and show that there exists an optimal solution which is bang-bang with a bound on the total number of switches. By construction, our criterion also automatically applies to the corresponding relaxed differential inclusion.",
keywords = "Differential inclusion, Global asymptotic stability, Lie bracket, Maximum principle, Optimal control, Switched nonlinear system",
author = "Michael Margaliot and Liberzon, {Daniel M}",
year = "2004",
month = "12",
day = "1",
language = "English (US)",
volume = "5",
pages = "4619--4624",
journal = "Proceedings of the IEEE Conference on Decision and Control",
issn = "0191-2216",
publisher = "Institute of Electrical and Electronics Engineers Inc.",

}

TY - JOUR

T1 - A Lie-algebraic condition for stability of switched nonlinear systems

AU - Margaliot, Michael

AU - Liberzon, Daniel M

PY - 2004/12/1

Y1 - 2004/12/1

N2 - We present a stability criterion for switched nonlinear systems which involves Lie brackets of the individual vector fields but does not require that these vector fields commute. A special case of the main result says that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields whose third-order Lie brackets vanish is globally uniformly asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [1]. To prove the result, we consider an optimal control problem which consists in finding the "most unstable" trajectory for an associated control system, and show that there exists an optimal solution which is bang-bang with a bound on the total number of switches. By construction, our criterion also automatically applies to the corresponding relaxed differential inclusion.

AB - We present a stability criterion for switched nonlinear systems which involves Lie brackets of the individual vector fields but does not require that these vector fields commute. A special case of the main result says that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields whose third-order Lie brackets vanish is globally uniformly asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [1]. To prove the result, we consider an optimal control problem which consists in finding the "most unstable" trajectory for an associated control system, and show that there exists an optimal solution which is bang-bang with a bound on the total number of switches. By construction, our criterion also automatically applies to the corresponding relaxed differential inclusion.

KW - Differential inclusion

KW - Global asymptotic stability

KW - Lie bracket

KW - Maximum principle

KW - Optimal control

KW - Switched nonlinear system

UR - http://www.scopus.com/inward/record.url?scp=14544305191&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=14544305191&partnerID=8YFLogxK

M3 - Conference article

AN - SCOPUS:14544305191

VL - 5

SP - 4619

EP - 4624

JO - Proceedings of the IEEE Conference on Decision and Control

JF - Proceedings of the IEEE Conference on Decision and Control

SN - 0191-2216

ER -