The structured distance to uncontrollability under multi-perturbations: An approach using multi-valued linear operators

In this paper we develop a unifying approach for computing the distance to uncontrollability of linear control systems. By using multi-valued linear operators in representing and estimating the system’s equations and matrices we are able to derive computable formulas of the distance from a controllable linear system to the nearest uncontrollable system under the assumption that the system’s matrices are subjected to structured multi-perturbations and measured by arbitrary operator norms. In the case of spectral norms, the obtained results unify and extend some previous works as well as a recent interesting result in [M. Karrow, D. Kressner, On the structured distance to uncontrollability, Systems Control Lett. 58 (2009) 128–132]. Some illustrating examples are given.