Nearest Linear Systems with Highly Deficient Reachable Subspaces

We consider the 2-norm distance τr(A,B) from a linear time-invariant dynamical system (A,B) of order n to the nearest system (A + ∆A∗, B + ∆B∗) whose reachable subspace is of dimension r < n. We first present a characterization to test whether the reachable subspace of the system has dimension r, which resembles and can be considered as a generalization of the Popov-Belevitch-Hautus test for controllability. Then, by exploiting this generalized PopovBelevitch-Hautus characterization, we derive the main result of this paper, which is a singular value optimization characterization for τr(A,B). A numerical technique to solve the derived singular value optimization problems is described. The numerical results on a few examples illustrate the significance of the derived singular value characterization for computational purposes.