1. M. Arcak and E.D. Sontag. Passivity-based Stability of Interconnection Structures. In V. Blondel, S. Boyd, and H. Kimura, editors, Recent Advances in Learning and Control, volume Volume 371, pages 195-204. Springer-Verlag, NY, 2008. [PDF] [doi:10.1007/978-1-84800-155-8_14] Keyword(s): passive systems, secant condition, reaction networks. Abstract:

   In this expository paper, we provide a streamlined version of the key lemma on stability of interconnections due to Vidyasagar and Moylan and Hill, and then show how it its hypotheses may be verified for network structures of great interest in biology.




2. M. Arcak and E.D. Sontag. Diagonal stability of a class of cyclic systems and its connection with the secant criterion. Automatica, 42:1531-1537, 2006. [PDF] Keyword(s): passive systems, systems biology, reaction networks, cyclic feedback systems, secant condition, nonlinear stability, dynamical systems. Abstract:

   This paper considers a class of systems with a cyclic structure that arises, among other examples, in dynamic models for certain biochemical reactions. We first show that a 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.




3. E.D. Sontag. Passivity gains and the ``secant condition'' for stability. Systems Control Lett., 55(3):177-183, 2006. [PDF] Keyword(s): cyclic feedback systems, systems biology, reaction networks, nonlinear stability, dynamical systems, passive systems, secant condition, reaction networks. Abstract:

   A generalization of the classical secant condition for the stability of cascades of scalar linear systems is provided for passive systems. The key is the introduction of a quantity that combines gain and phase information for each system in the cascade. For linear one-dimensional systems, the known result is recovered exactly.

1. L. Scardovi, M. Arcak, and E.D. Sontag. Synchronization of interconnected systems with an input-output approach. Part I: Main results. In Proc. IEEE Conf. Decision and Control, Shanhai, Dec. 2009, pages 609-614, 2009. Note: First part of conference version of journal paper.Keyword(s): passive systems, secant condition, reaction networks, systems biology. Abstract:

   See abstract and link to pdf in entry for Journal paper.




2. L. Scardovi, M. Arcak, and E.D. Sontag. Synchronization of interconnected systems with an input-output approach. Part II: State-Space result and application to biochemical networks. In Proc. IEEE Conf. Decision and Control, Shanhai, Dec. 2009, pages 615-620, 2009. Note: Second part of conference version of journal paper.Keyword(s): passive systems, secant condition, reaction networks, systems biology. Abstract:

   See abstract and link to pdf in entry for Journal paper.

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