This paper proposes a novel notion named perturbation-tolerant structural
controllability (PTSC) to study controllability preservation for a structured
linear system under structured 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 parameter
space, the corresponding 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 structured controllability radius
problems from a generic view.Comment: Fix some typos. arXiv admin note: substantial text overlap with
arXiv:2103.1190