PTSC: A New Notion for Structural Controllability under Structured Perturbations

This paper proposes a novel notion for structural controllability under structured numerical perturbations, namely the perturbation-tolerant structural controllability (PTSC), on a single-input 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 structure prescribed by the perturbed entries. This new notion can characterize the generic property in controllability preservation under structured numerical perturbations. We give a necessary and sufficient condition for a single-input system to be PTSC, whose verification has polynomial time complexity. Our results can serve as some feasibility conditions for the conventional structured controllability radius problems from a generic view.