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

This paper applies the frequency response (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. Under certain mild assumptions, 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 internally stable in the sense of exponential stability, in spite of possible instability of perturbations. The result is derived by a Nyquist-stability-criterion type of arguments. For the case where a stronger assumption is made that the perturbations are linear time-invariant, a necessary and sufficient condition for robust stability is 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