Perturbation-Tolerant Structural Controllability for Linear Systems.

This paper proposes a novel notion named perturbation-tolerant structural controllability (PTSC) to study controllability preservation for a structured linear system under structured numerical perturbations. To be precise, we consider a structured system whose entries can be classified into three categories: fixed zero entries, unknown generic entries whose values are fixed but unknown, and perturbed entries that can take arbitrary complex values. Such a system is PTSC if, for almost all values of the unknown generic entries in the corresponding space, the controllable system realizations can preserve controllability under arbitrary complex-valued perturbations with their zero/nonzero structure prescribed by the perturbed entries. It is proven genericity exists in this notion, that is, depending on the structure of the structured system, for almost all of its controllable realizations, either there exists an addable structured perturbation prescribed by the perturbed entries so that the resulting system is uncontrollable, or there is not such a perturbation. We give a decomposition-based necessary and sufficient condition for a single-input linear system, ensuring PTSC, whose verification has polynomial time complexity; we then present some intuitive graph-theoretic conditions for PTSC. For the multi-input case, we provide some necessary conditions for PTSC. As an application, our results can serve as some feasibility conditions for the conventional {\emph{structured controllability radius}} problems from a generic view.