A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks

TL;DR

This paper introduces a passivity-based stability criterion for interconnected nonlinear systems, applicable to biochemical networks, extending previous criteria to more general structures and including robustness considerations.

Contribution

It develops a new stability test based on dissipativity matrices that generalizes existing criteria to broader network structures and models.

Findings

The stability criterion applies to biochemical reaction networks.

It extends the secant criterion to general interconnection graphs.

The method demonstrates robustness in systems with diffusion terms.

Abstract

This paper presents a stability test for a class of interconnected nonlinear systems motivated by biochemical reaction networks. One of the main results determines global asymptotic stability of the network from the diagonal stability of a "dissipativity matrix" which incorporates information about the passivity properties of the subsystems, the interconnection structure of the network, and the signs of the interconnection terms. This stability test encompasses the "secant criterion" for cyclic networks presented in our previous paper, and extends it to a general interconnection structure represented by a graph. A second main result allows one to accommodate state products. This extension makes the new stability criterion applicable to a broader class of models, even in the case of cyclic systems. The new stability test is illustrated on a mitogen activated protein kinase (MAPK) cascade model, and on a branched interconnection structure motivated by metabolic networks. Finally, another result addresses the robustness of stability in the presence of diffusion terms in a compartmental system made out of identical systems.