The Structured Distance to the Nearest System Without Property $\mathcal {P}$

For a system matrix M, this paper explores the smallest (Frobenius) norm additive structured perturbation δM for which a system property P (e.g., controllability, observability, stability, etc.) fails to hold, i.e., δM is the structured perturbation with smallest Frobenius norm such that there exists a property matrix R E P for which M-δM-R drops rank. The Frobenius norm is used because of its direct dependence on the magnitude of each entry in the perturbation matrix. Necessary conditions on a locally minimum norm structured rank-reducing perturbation δM and associated property matrix R are set forth and proven. An iterative algorithm is also set forth that computes a locally minimum norm structured perturbation and associated property matrix satisfying the necessary conditions. Algorithm convergence is proven using a discrete Lyapunov function.