A generalization of Kasovskii-LaSalle theorem for nonlinear time-varying systems: Converse results and applications

Ti Chung Lee, Zhong-Ping Jiang

Research output: Contribution to journalArticle

Abstract

This paper presents a practically applicable characterization of uniform (global) asymptotic stability (UAS and UGAS) for general nonlinear time-varying systems, under certain output-dependent conditions in the spirit of the Krasovskii-LaSalle theorem. The celebrated Krasovskii-LaSalle theorem is extended from two directions. One is using the weak zero-state detectability property associated with reduced limiting systems of the system in question to generalize the condition that the maximal invariance set contained in the zero locus of the time-derivative of the Lyapunov function is the zero set. Another one is using an almost bounded output-energy condition to relax the assumption that the time derivative of the Lyapunov function is negative semi-definite. Then, the UAS and UGAS properties of the origin can be guaranteed by employing these two improved conditions related to certain output function for uniformly Lyapunov stable systems. The proposed conditions turn out to be also necessary under some mild assumptions and thus, give a new characterization of UGAS (and UAS). Through an equivalence relation, the proposed detectability condition can also be verified in terms of usual PE condition. To validate the proposed results, the obtained stability criteria are applied to a class of time-varying passive systems and to revisit a tracking control problem of nonholonomic chained systems. For the latter, under certain persistency of excitation conditions, the K-exponential stability is achieved based on our approach.

Original languageEnglish (US)
Pages (from-to)1147-1163
Number of pages17
JournalIEEE Transactions on Automatic Control
Volume50
Issue number8
DOIs
StatePublished - Aug 2005

Fingerprint

Time varying systems
Lyapunov functions
Asymptotic stability
Derivatives
Stability criteria
Invariance

Keywords

  • Detectability
  • Nonholonomic systems
  • Nonlinear time-varying systems
  • Reduced limiting systems
  • Uniform asymptotic stability

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Electrical and Electronic Engineering

Cite this

@article{a127bd4f6c7345c28265ed58c167cf15,
title = "A generalization of Kasovskii-LaSalle theorem for nonlinear time-varying systems: Converse results and applications",
abstract = "This paper presents a practically applicable characterization of uniform (global) asymptotic stability (UAS and UGAS) for general nonlinear time-varying systems, under certain output-dependent conditions in the spirit of the Krasovskii-LaSalle theorem. The celebrated Krasovskii-LaSalle theorem is extended from two directions. One is using the weak zero-state detectability property associated with reduced limiting systems of the system in question to generalize the condition that the maximal invariance set contained in the zero locus of the time-derivative of the Lyapunov function is the zero set. Another one is using an almost bounded output-energy condition to relax the assumption that the time derivative of the Lyapunov function is negative semi-definite. Then, the UAS and UGAS properties of the origin can be guaranteed by employing these two improved conditions related to certain output function for uniformly Lyapunov stable systems. The proposed conditions turn out to be also necessary under some mild assumptions and thus, give a new characterization of UGAS (and UAS). Through an equivalence relation, the proposed detectability condition can also be verified in terms of usual PE condition. To validate the proposed results, the obtained stability criteria are applied to a class of time-varying passive systems and to revisit a tracking control problem of nonholonomic chained systems. For the latter, under certain persistency of excitation conditions, the K-exponential stability is achieved based on our approach.",
keywords = "Detectability, Nonholonomic systems, Nonlinear time-varying systems, Reduced limiting systems, Uniform asymptotic stability",
author = "Lee, {Ti Chung} and Zhong-Ping Jiang",
year = "2005",
month = "8",
doi = "10.1109/TAC.2005.852567",
language = "English (US)",
volume = "50",
pages = "1147--1163",
journal = "IEEE Transactions on Automatic Control",
issn = "0018-9286",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
number = "8",

}

TY - JOUR

T1 - A generalization of Kasovskii-LaSalle theorem for nonlinear time-varying systems

T2 - Converse results and applications

AU - Lee, Ti Chung

AU - Jiang, Zhong-Ping

PY - 2005/8

Y1 - 2005/8

N2 - This paper presents a practically applicable characterization of uniform (global) asymptotic stability (UAS and UGAS) for general nonlinear time-varying systems, under certain output-dependent conditions in the spirit of the Krasovskii-LaSalle theorem. The celebrated Krasovskii-LaSalle theorem is extended from two directions. One is using the weak zero-state detectability property associated with reduced limiting systems of the system in question to generalize the condition that the maximal invariance set contained in the zero locus of the time-derivative of the Lyapunov function is the zero set. Another one is using an almost bounded output-energy condition to relax the assumption that the time derivative of the Lyapunov function is negative semi-definite. Then, the UAS and UGAS properties of the origin can be guaranteed by employing these two improved conditions related to certain output function for uniformly Lyapunov stable systems. The proposed conditions turn out to be also necessary under some mild assumptions and thus, give a new characterization of UGAS (and UAS). Through an equivalence relation, the proposed detectability condition can also be verified in terms of usual PE condition. To validate the proposed results, the obtained stability criteria are applied to a class of time-varying passive systems and to revisit a tracking control problem of nonholonomic chained systems. For the latter, under certain persistency of excitation conditions, the K-exponential stability is achieved based on our approach.

AB - This paper presents a practically applicable characterization of uniform (global) asymptotic stability (UAS and UGAS) for general nonlinear time-varying systems, under certain output-dependent conditions in the spirit of the Krasovskii-LaSalle theorem. The celebrated Krasovskii-LaSalle theorem is extended from two directions. One is using the weak zero-state detectability property associated with reduced limiting systems of the system in question to generalize the condition that the maximal invariance set contained in the zero locus of the time-derivative of the Lyapunov function is the zero set. Another one is using an almost bounded output-energy condition to relax the assumption that the time derivative of the Lyapunov function is negative semi-definite. Then, the UAS and UGAS properties of the origin can be guaranteed by employing these two improved conditions related to certain output function for uniformly Lyapunov stable systems. The proposed conditions turn out to be also necessary under some mild assumptions and thus, give a new characterization of UGAS (and UAS). Through an equivalence relation, the proposed detectability condition can also be verified in terms of usual PE condition. To validate the proposed results, the obtained stability criteria are applied to a class of time-varying passive systems and to revisit a tracking control problem of nonholonomic chained systems. For the latter, under certain persistency of excitation conditions, the K-exponential stability is achieved based on our approach.

KW - Detectability

KW - Nonholonomic systems

KW - Nonlinear time-varying systems

KW - Reduced limiting systems

KW - Uniform asymptotic stability

UR - http://www.scopus.com/inward/record.url?scp=26244435293&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=26244435293&partnerID=8YFLogxK

U2 - 10.1109/TAC.2005.852567

DO - 10.1109/TAC.2005.852567

M3 - Article

VL - 50

SP - 1147

EP - 1163

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 8

ER -