Robust stabilization relative to the unweighted H∞norm is generically unattainable in the presence of singular plant perturbations

Recent work by Vidyasagar gives a sufficient condition under which a singularly perturbed system can be robustly stabilized relative to the unweighted
H^{\infty}
norm. In this note we show that the same condition is also necessary. In addition, we prove that this condition is generically untrue with respect to the class of all linear singularly perturbed systems, as well as the class of all singular perturbations of any fixed, stabilizable, and detectable system.