Robust Instability Radius for Multi-agent Dynamical Systems with Cyclic Structure

This paper is concerned with robust instability analysis for linear multi-agent dynamical systems with cyclic structure. This relates to interesting and important periodic oscillation phenomena in biology and neuronal science, since the nonlinear phenomena often occur when the linearized model around an equilibrium point is unstable. We first make a problem setting on the analysis and define the notion of robust instability radius (RIR) as a quantitative measure for maximum allowable stable dynamic perturbation in terms of the H-infinity norm. After showing lower bounds of the RIR, we derive the exact RIR, which is analytic and scalable, for first order time-lag agents. Finally, we make a remark on the potential applicability to some classes of higher order systems.