Robust stability of sampled-data systems under possibly unstable additive/multiplicative perturbations

Applies the FR-operator technique to the robust stability problem of sampled-data systems against additive/multiplicative perturbations, where a reasonable class of perturbations consists of unstable as well as stable ones. Assuming that the number of unstable modes of the plant does not change, we show that a small-gain condition in terms of the FR-operator representation (which is actually equivalent to a small-gain condition in terms of the L₂-induced norm) is still necessary and sufficient for the sampled-data system to be robustly stable against h-periodic perturbations, in spite of their possible instability. The result is derived by a Nyquist-type of arguments. Next, a necessary and sufficient condition for robust stability against linear time-invariant (LTI) perturbations is also given. Furthermore, we show that if the plant is either single-input or single-output, the condition can be reduced to a readily testable form. Finally, we clarify when the small-gain condition becomes a particularly poor measure for robust stability