On uniform global asymptotic stability of nonlinear discrete-time systems with applications

T. C. Lee, Zhong-Ping Jiang

Research output: Contribution to journalArticle

Abstract

This paper presents new characterizations of uniform global asymptotic stability for nonlinear and time-varying discrete-time systems. Under mild assumptions, it is shown that weak zero-state detectability is equivalent to uniform global asymptotic stability for globally uniformly stable systems. By employing the notion of reduced limiting systems, another characterization of uniform global asymptotic stability is proposed on the basis of the detectability for the reduced limiting systems associated with the original system. As a by-product, we derive a generalized, discrete-time version of the well-known Krasovskii-LaSalle theorem for general time-varying, not necessarily periodic, systems. Furthermore, we apply the obtained stability results to analyze uniform asymptotic stability of cascaded time-varying systems, and show that some technical assumptions in recent papers can be relaxed. Through a practical application, it is shown that our results play a similar role to the classic LaSalle invariance principle in guaranteeing attractivity, noting that reduced limiting systems are used instead of the original system. To validate the conceptual characterizations, we study the problem of sampled-data stabilization for the benchmark example of nonholonomic mobile robots via the exact discrete-time model rather than approximate models. This case study also reveals that in general, sampled-data systems may become non-periodic even though their original continuous-time system is periodic. A novel sampled-data stabilizer design is proposed using the new stability results and is supported via simulation results.

Original languageEnglish (US)
Pages (from-to)1644-1660
Number of pages17
JournalIEEE Transactions on Automatic Control
Volume51
Issue number10
DOIs
StatePublished - Oct 2006

Fingerprint

Asymptotic stability
Time varying systems
Continuous time systems
Invariance
Mobile robots
Byproducts
Stabilization

Keywords

  • Cascaded systems
  • Discrete-time systems
  • Mobile robots
  • Reduced limiting systems
  • Sampled-data controllers
  • Uniform global asymptotic stability
  • Weak zero-state detectability

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Electrical and Electronic Engineering

Cite this

On uniform global asymptotic stability of nonlinear discrete-time systems with applications. / Lee, T. C.; Jiang, Zhong-Ping.

In: IEEE Transactions on Automatic Control, Vol. 51, No. 10, 10.2006, p. 1644-1660.

Research output: Contribution to journalArticle

@article{56268573a3054218a67fc347fcd40eba,
title = "On uniform global asymptotic stability of nonlinear discrete-time systems with applications",
abstract = "This paper presents new characterizations of uniform global asymptotic stability for nonlinear and time-varying discrete-time systems. Under mild assumptions, it is shown that weak zero-state detectability is equivalent to uniform global asymptotic stability for globally uniformly stable systems. By employing the notion of reduced limiting systems, another characterization of uniform global asymptotic stability is proposed on the basis of the detectability for the reduced limiting systems associated with the original system. As a by-product, we derive a generalized, discrete-time version of the well-known Krasovskii-LaSalle theorem for general time-varying, not necessarily periodic, systems. Furthermore, we apply the obtained stability results to analyze uniform asymptotic stability of cascaded time-varying systems, and show that some technical assumptions in recent papers can be relaxed. Through a practical application, it is shown that our results play a similar role to the classic LaSalle invariance principle in guaranteeing attractivity, noting that reduced limiting systems are used instead of the original system. To validate the conceptual characterizations, we study the problem of sampled-data stabilization for the benchmark example of nonholonomic mobile robots via the exact discrete-time model rather than approximate models. This case study also reveals that in general, sampled-data systems may become non-periodic even though their original continuous-time system is periodic. A novel sampled-data stabilizer design is proposed using the new stability results and is supported via simulation results.",
keywords = "Cascaded systems, Discrete-time systems, Mobile robots, Reduced limiting systems, Sampled-data controllers, Uniform global asymptotic stability, Weak zero-state detectability",
author = "Lee, {T. C.} and Zhong-Ping Jiang",
year = "2006",
month = "10",
doi = "10.1109/TAC.2006.882770",
language = "English (US)",
volume = "51",
pages = "1644--1660",
journal = "IEEE Transactions on Automatic Control",
issn = "0018-9286",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
number = "10",

}

TY - JOUR

T1 - On uniform global asymptotic stability of nonlinear discrete-time systems with applications

AU - Lee, T. C.

AU - Jiang, Zhong-Ping

PY - 2006/10

Y1 - 2006/10

N2 - This paper presents new characterizations of uniform global asymptotic stability for nonlinear and time-varying discrete-time systems. Under mild assumptions, it is shown that weak zero-state detectability is equivalent to uniform global asymptotic stability for globally uniformly stable systems. By employing the notion of reduced limiting systems, another characterization of uniform global asymptotic stability is proposed on the basis of the detectability for the reduced limiting systems associated with the original system. As a by-product, we derive a generalized, discrete-time version of the well-known Krasovskii-LaSalle theorem for general time-varying, not necessarily periodic, systems. Furthermore, we apply the obtained stability results to analyze uniform asymptotic stability of cascaded time-varying systems, and show that some technical assumptions in recent papers can be relaxed. Through a practical application, it is shown that our results play a similar role to the classic LaSalle invariance principle in guaranteeing attractivity, noting that reduced limiting systems are used instead of the original system. To validate the conceptual characterizations, we study the problem of sampled-data stabilization for the benchmark example of nonholonomic mobile robots via the exact discrete-time model rather than approximate models. This case study also reveals that in general, sampled-data systems may become non-periodic even though their original continuous-time system is periodic. A novel sampled-data stabilizer design is proposed using the new stability results and is supported via simulation results.

AB - This paper presents new characterizations of uniform global asymptotic stability for nonlinear and time-varying discrete-time systems. Under mild assumptions, it is shown that weak zero-state detectability is equivalent to uniform global asymptotic stability for globally uniformly stable systems. By employing the notion of reduced limiting systems, another characterization of uniform global asymptotic stability is proposed on the basis of the detectability for the reduced limiting systems associated with the original system. As a by-product, we derive a generalized, discrete-time version of the well-known Krasovskii-LaSalle theorem for general time-varying, not necessarily periodic, systems. Furthermore, we apply the obtained stability results to analyze uniform asymptotic stability of cascaded time-varying systems, and show that some technical assumptions in recent papers can be relaxed. Through a practical application, it is shown that our results play a similar role to the classic LaSalle invariance principle in guaranteeing attractivity, noting that reduced limiting systems are used instead of the original system. To validate the conceptual characterizations, we study the problem of sampled-data stabilization for the benchmark example of nonholonomic mobile robots via the exact discrete-time model rather than approximate models. This case study also reveals that in general, sampled-data systems may become non-periodic even though their original continuous-time system is periodic. A novel sampled-data stabilizer design is proposed using the new stability results and is supported via simulation results.

KW - Cascaded systems

KW - Discrete-time systems

KW - Mobile robots

KW - Reduced limiting systems

KW - Sampled-data controllers

KW - Uniform global asymptotic stability

KW - Weak zero-state detectability

UR - http://www.scopus.com/inward/record.url?scp=33750143676&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750143676&partnerID=8YFLogxK

U2 - 10.1109/TAC.2006.882770

DO - 10.1109/TAC.2006.882770

M3 - Article

VL - 51

SP - 1644

EP - 1660

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 10

ER -