  This paper is concerned with a robust instability analysis for the
single-input-single-output unstable linear time-invariant (LTI) system under
dynamic perturbations. The nominal system itself is possibly perturbed by the
static gain of the uncertainty, which would be the case when a nonlinear
uncertain system is linearized around an equilibrium point. We define the
robust instability radius as the smallest $H_\infty$ norm of the stable linear
perturbation that stabilizes the nominal system. There are two main theoretical
results: one is on a partial characterization of unperturbed nominal systems
for which the robust instability radius can be calculated exactly, and the
other is a numerically tractable procedure for calculating the exact robust
instability radius for nominal systems parametrized by a perturbation
parameter. The results are applied to the repressilator in synthetic biology,
where hyperbolic instability of a unique equilibrium guarantees the persistence
of oscillation phenomena in the global sense, and the effectiveness of our
linear robust instability analysis is confirmed by numerical simulations.
