  The robust instability of an unstable plant subject to stable perturbations
is of significant importance and arises in the study of sustained oscillatory
phenomena in nonlinear systems. This paper analyzes the robust instability of
linear discrete-time systems against stable perturbations via the notion of
robust instability radius (RIR) as a measure of instability. We determine the
exact RIR for certain unstable systems using small-gain type conditions by
formulating the problem in terms of a phase change rate maximization subject to
appropriate constraints at unique peak-gain frequencies, for which stable
first-order all-pass functions are shown to be optimal. Two real-world
applications -- minimum-effort sampled-data control of magnetic levitation
systems and neural spike generations in the FitzHugh--Nagumo model subject to
perturbations -- are provided to illustrate the utility of our results.
