Restricted real perturbation values with applications to the structured real controllability radius of LTI systems

In this paper, the concept of restricted real perturbation values of a complex matrix triplet is introduced, and a formula for computing lower bounds of these values is presented. Restricted real perturbation values are a generalization of the real perturbation values introduced in, which is a key concept in evaluating various robustness radii found in the control literature, such as the real controllability/observability radius, the real decentralized fixed-mode radius, the real minimum-phase radius, etc. The generalization to restricted real perturbation values is needed for an extension of these radii to account for more general system perturbation structures. As an example, we used the results of this paper to compute the true value of the structured real controllability radius of the multi-link inverted pendulum system. Also, we numerically investigate cases of when the provided lower bounds are achievable.