Stability of switched systems: A Lie-algebraic condition

Daniel Liberzon, João P. Hespanha, A. Stephen Morse

Research output: Contribution to journalArticlepeer-review


We present a sufficient condition for asymptotic stability of a switched linear system in terms of the Lie algebra generated by the individual matrices. Namely, if this Lie algebra is solvable, then the switched system is exponentially stable for arbitrary switching. In fact, we show that any family of linear systems satisfying this condition possesses a quadratic common Lyapunov function. We also discuss the implications of this result for switched nonlinear systems.

Original languageEnglish (US)
Pages (from-to)117-122
Number of pages6
JournalSystems and Control Letters
Issue number3
StatePublished - Jul 1999
Externally publishedYes


  • Quadratic common Lyapunov function
  • Switched system
  • Uniform exponential stability

ASJC Scopus subject areas

  • Control and Systems Engineering
  • General Computer Science
  • Mechanical Engineering
  • Electrical and Electronic Engineering


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