Recursive rank one perturbations for pole placement and cone reachability

Abstract

The role of rank one perturbations in transforming the eigenstructure of a matrix has long been considered in the context of applications, especially in linear control systems. Two cases are examined in this paper: First, we propose a practical method to place the system eigenvalues (poles) in desired locations via feedback control that is computed in terms of recursive rank one perturbations. Second, a choice of feedback control is proposed in order to achieve that a trajectory eventually enters the nonnegative orthant and remains therein for all time thereafter. The latter situation is achieved by imposing the strong Perron-Frobenius property and involves altering the eigenvalues, as well as left eigenvectors via rank one perturbations.