Diagonal stability of a class of cyclic systems and its connection with the secant criterion

Murat Arcak, Eduardo D. Sontag

Research output: Contribution to journalArticlepeer-review

147 Scopus citations


We consider a class of systems with a cyclic interconnection structure that arises, among other examples, in dynamic models for certain biochemical reactions. We first show that a "secant" criterion for local stability, derived earlier in the literature, is in fact a necessary and sufficient condition for diagonal stability of the corresponding class of matrices. We then revisit a recent generalization of this criterion to output strictly passive systems, and recover the same stability condition using our diagonal stability result as a tool for constructing a Lyapunov function. Using this procedure for Lyapunov construction we exhibit classes of cyclic systems with sector nonlinearities and characterize their global stability properties.

Original languageEnglish (US)
Pages (from-to)1531-1537
Number of pages7
Issue number9
StatePublished - Sep 2006

All Science Journal Classification (ASJC) codes

  • Electrical and Electronic Engineering
  • Control and Systems Engineering


  • Biochemical reactions
  • Diagonal stability
  • Passivity
  • Storage functions
  • Vector Lyapunov functions


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