Measuring Rank Robustness of General Complex System Matrices with Respect to Real Matrix Perturbations

This paper examines the problem of computing the real matrix perturbation having smallest norm which causes a general complex (system) matrix to drop rank. Given the state model describing a linear time-invariant system, the norm of this matrix perturbation helps to determine the robustness of several system properties with respect to real parameter variations. The norm of this perturbation, or the real-restricted singular value of the complex matrix, is known to be a discontinuous function of the complex matrix. This paper presents some theoretical results characterizing the nature of this discontinuity and presents an experimental algorithm which may be used to compute the smallest real rank-reducing matrix perturbation and its norm. Some numerical examples are included.