On necessary conditions for stability of interconnected iISS systems

Hiroshi Ito, Zhong Ping Jiang

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


This paper investigates necessary conditions for stability of nonlinear systems made of feedback interconnection of two iISS subsystems. The integral input-to-state stability(iISS) is a dissipative property which includes the input-to-state stability(ISS) as a special case. It is proved that at least one subsystem needs to be ISS for global asymptotic stability of the feedback loop when two subsystems are not completely specified and only their supply rates are available. In contrast, the situation where ODEs describing subsystems are available does not exclude pairs of subsystems which are only iISS. This paper further investigates necessary conditions for global asymptotic stability, and proves that the nonlinear small-gain condition is necessary in the case of unspecified iISS subsystems described with supply rates. When one of the two subsystems is given as a specific ODE, the nonlinear small-gain condition is no longer necessary, while the linear small-gain condition is necessary in the case of linear systems. For nonlinear systems, the necessity recovers only if the supply rate fits the known subsystem tightly in the shape as well as the magnitude.

Original languageEnglish (US)
Title of host publicationProceedings of the 2006 American Control Conference
Number of pages6
StatePublished - 2006
Event2006 American Control Conference - Minneapolis, MN, United States
Duration: Jun 14 2006Jun 16 2006

Publication series

NameProceedings of the American Control Conference
ISSN (Print)0743-1619


Other2006 American Control Conference
Country/TerritoryUnited States
CityMinneapolis, MN

ASJC Scopus subject areas

  • Electrical and Electronic Engineering


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