Nearest linear systems with highly deficient reachable subspaces

Abstract

We consider the 2-norm distance tau(r)(A, B) from a linear time-invariant dynamical system (A, B) of order n to the nearest system (A + Delta A(*), B + Delta 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 Popov-Belevitch-Hautus characterization, we derive the main result of this paper, which is a singular value optimization characterization for tau(r)(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.