Rank Robustness of Complex Matrices with Respect to Real Perturbations

Abstract

This paper examines the problem of computing a minimum norm real matrix perturbation that 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. The paper presents a simple condition on the complex matrix that eliminates this discontinuity. Specifically, the paper shows that the size of the smallest real rank-reducing perturbation is a continuous function of the complex matrix as long as the imaginary part of the complex matrix has full rank. The paper examines other aspects of the continuity of the problem. It also presents an algorithm that converges to a point satisfying a necessary condition for obtaining the smallest real rank-reducing matrix perturbation. A Lyapunov function approach is used to establish convergence of the algorithm. Some numerical examples are included illustrating the accuracy of the approach.