Computing the radius of controllability for state space systems

In this paper, we discuss the problem of computing the nearest uncontrollable system to a given control system represented by a matrix pair (A,B)(A,B). In order to do so, we construct a sequence of structured matrices from given system matrices AA and BB. Controllability of the pair (A,B)(A,B) is equivalent to a condition on the null-space dimension of an appropriate matrix in this sequence. We show that the dimension of the reachability space is also related to the above condition. Further, it is shown that the nearest Structured Low Rank Approximation (SLRA) of this structured matrix corresponds to a nearest uncontrollable system to the pair (A,B)(A,B).