A 'Mixed' small gain and passivity theorem for an interconnection of linear time-invariant systems

Wynita M. Griggs, Brian D.O. Anderson, Alexander Lanzon

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

8 Citations (Scopus)


We show that the negative feedback interconnection of two causal, stable, linear time-invariant systems with a 'mixed' small gain and passivity frequency domain property is guaranteed to be finite-gain stable. This 'mixed' small gain and passivity property refers to the characteristic that the frequency range -∞ < ω < ∞ can be divided into intervals for which the two systems in the interconnection are both: a) 'input and output strictly passive' (and one or both of the systems may or may not have 'gain less than one'); or b) 'input and output strictly passive and with gain less than one'; or c) 'with gain less than one' (and one or both of the systems may or may not be 'input and output strictly passive'). The 'mixed' small gain and passivity property is described mathematically using the notion of dissipativity of systems, and finite-gain stability of the interconnection is proven via a stability result for dissipative interconnected systems.

Original languageEnglish
Title of host publication2007 European Control Conference, ECC 2007
PublisherIEEE, Institute of Electrical and Electronics Engineers
Number of pages7
ISBN (Electronic)9783952417386
Publication statusPublished - 2007
Externally publishedYes
EventEuropean Control Conference 2007 - Kos, Greece
Duration: 2 Jul 20075 Jul 2007
Conference number: 9th
https://ieeexplore.ieee.org/xpl/conhome/7065133/proceeding (Proceedings)

Publication series

Name2007 European Control Conference, ECC 2007


ConferenceEuropean Control Conference 2007
Abbreviated titleECC 2007
Internet address


  • Dissipative systems
  • Finite gain
  • Linear systems
  • Negative feedback stability
  • Passivity

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