Computation of the Real Controllability Radius and Minimum-Norm Perturbations of Higher-Order, Descriptor, and Time-Delay LTI Systems

A linear time-invariant (LTI) system´s controllability radius measures the norm of the smallest parametric perturbation such that the perturbed system is uncontrollable, and is of practical importance. In this note, we study the real controllability radii of i) higher-order systems; ii) descriptor systems; and iii) time-delay systems, where the perturbations are restricted to the set of real values, and the spectral norm is considered. Formulas for these radii are presented using a framework involving generalized real perturbation values, which has certain computational advantages over other formulations found in the literature. In particular, the formulas are readily more computable, especially for higher-dimensional systems, and a minimum-norm perturbation can also easily be obtained. Numerical examples are presented.