Internal stability of linear consensus processes

Ji Liu, A. Stephen Morse, Angelia Nedic, Tamer Basar

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


In a network of n agents, consensus means that all n agents reach an agreement on a specific value of some quantity via local interactions. A linear consensus process can typically be modeled by a discrete-time linear recursion equation or a continuous-time linear differential equation, whose equilibria include nonzero states of the form a1 where a is a constant and 1 is a column vector in n whose entries all equal 1. Using a suitably defined semi-norm, this paper extends the standard notions of uniform asymptotic stability and exponential stability from linear systems to linear recursions and differential equations of this type. It is shown that these notions are equivalent just as they are for conventional linear systems. The main contributions of this paper are first to provide a simple, direct proof of the necessary graph-theoretic condition given in [1] for a discrete-time linear consensus process to be exponentially stable, and second to derive a necessary graph-theoretic condition for a piecewise time-invariant continuous-time linear consensus process to be exponentially stable.

Original languageEnglish (US)
Title of host publication53rd IEEE Conference on Decision and Control,CDC 2014
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages6
ISBN (Electronic)9781479977468
StatePublished - 2014
Event2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014 - Los Angeles, United States
Duration: Dec 15 2014Dec 17 2014

Publication series

NameProceedings of the IEEE Conference on Decision and Control
ISSN (Print)0743-1546
ISSN (Electronic)2576-2370


Other2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014
Country/TerritoryUnited States
CityLos Angeles

ASJC Scopus subject areas

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


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