Robust stability analysis for LTI systems with generalized frequency variables and its application to gene regulatory networks

A class of networked dynamical systems with multiple homogeneous agents can be represented by a linear system with a generalized frequency variable. This paper is concerned with robust stability analysis of such class of systems under perturbations in the dynamics of agents. The perturbed agents no longer share the same dynamics and thus the analysis encompasses heterogeneous multi-agent systems consisting of slightly different agents. We first present nominal stability conditions for the system of interconnected MIMO agents. We then focus on robust stability, where we consider three types of perturbations, leading to homogeneous agents, heterogeneous agents with and without uncertain interconnections. For each case, we show that a necessary and sufficient condition for robust stability is given by an H _∞ -norm bound on a set of transfer functions parametrized by the eigenvalues of the interconnection matrix, provided the nominal interconnections have a certain structure. The robust stability condition is interpreted as the requirement that the eigenvalues of the interconnection matrix be located in a particular region on the complex plane. The usefulness of the results is demonstrated through an application to robust stability analysis for gene regulatory networks.