Distance to the loss of structural properties for linear systems under parametric uncertainties

In this paper, some techniques are proposed to determine the smallest perturbation that causes the loss of a structural property for a linear system with coefficients depending polynomially on a vector of parameters. In the case that the interest lies on a single structural property, the analysis is carried out using directly the state-space description of the system, whereas, in the case that the interest is to characterize either the pair (reachability and observability) or the pair (stabilizability and detectability), the analysis is carried out using the transfer matrix of the plant. Examples of application of the proposed methods are reported all throughout the paper to illustrate and validate the theoretical results.