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

Authors:
Murat Arcak;Eduardo D. Sontag
Affiliations:
Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA;Department of Mathematics, Rutgers University, New Brunswick, NJ, USA
Venue:
Automatica (Journal of IFAC)
Year:
2006

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Abstract

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.