Nonlinear small-gain condition covering iISS systems: Necessity and sufficiency from a Lyapunov perspective

Hiroshi Ito, Zhong Ping Jiang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

This paper presents conditions for stability of interconnected nonlinear systems consisting of integral input-to-state stable subsystems. Both the necessity and the sufficiency are investigated using a Lyapunov formulation. The ultimate result is a necessary and sufficient condition for integral input-to-state stability (iISS) of the interconnection, which is believed to be the first of its kind. This paper derives two sufficient conditions. One is the iISS-ISS small-gain condition which has been proposed earlier by one of the authors. Its sufficiency is strengthened to cover situations more general than the earlier result. The other condition is a universal representation of the nonlinear small-gain condition which has been a popular tool for dealing with input-to-state stable (ISS) subsystems. This paper proves that, for establishing iISS property of the interconnection, the universal representation can deal with iISS subsystems as much as the iISS-ISS small-gain condition if supply rate functions are smooth everywhere and analytic at zero. Furthermore, two necessary conditions this paper derives demonstrate that the nonlinear small-gain condition in the universal form is also necessary for the iISS property.

Original languageEnglish (US)
Title of host publicationProceedings of the 45th IEEE Conference on Decision and Control 2006, CDC
Pages355-360
Number of pages6
StatePublished - Dec 1 2006
Event45th IEEE Conference on Decision and Control 2006, CDC - San Diego, CA, United States
Duration: Dec 13 2006Dec 15 2006

Publication series

NameProceedings of the IEEE Conference on Decision and Control
ISSN (Print)0191-2216

Other

Other45th IEEE Conference on Decision and Control 2006, CDC
CountryUnited States
CitySan Diego, CA
Period12/13/0612/15/06

    Fingerprint

ASJC Scopus subject areas

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

Cite this

Ito, H., & Jiang, Z. P. (2006). Nonlinear small-gain condition covering iISS systems: Necessity and sufficiency from a Lyapunov perspective. In Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC (pp. 355-360). [4177198] (Proceedings of the IEEE Conference on Decision and Control).