Measuring how far a controllable system is from an uncontrollable one

The main goal of this note is to explore some of the properties of a controllable system which, in a precise sense to be defined, is near an uncontrollable pair. To do this, we first discuss the notion of the distance between a controllable system and the nearest uncontrollable one in both real and complex cases [2]-[5]. When this distance is relatively small, we show to what extent this distance corresponds to certain properties of the uncontrollability matrix and the reachability Grammian.