This technical note presents a feedback design approach to minimise state ultimate-bounds (UBs) in linear time-invariant discrete-time systems subject to non-vanishing bounded disturbances. A target closed-loop eigenstructure is first constructed to achieve the lowest possible UBs for one or more state components of the system. Structural conditions on the system matrices are then derived to guarantee that the target eigenstructure can be assigned by state feedback. When the required structural conditions are satisfied, an eigenvector assignment procedure can be applied to simultaneously minimise multiple UBs in systems with multiple inputs. The results are illustrated by a numerical example drawn from a distributed voltage control problem in inverter-based microgrids.