On the reachable set for third-order linear discrete-time systems with positive control

In this paper we study the geometrical properties of the set of reachable states of a single-input third-order discrete-time linear system with positive controls. This set is a cone and we give a complete geometrical characterization of this set when the system has all real eigenvalues. More in detail, we give necessary and sufficient conditions for properness and polyhedrality of the cone and provide the number of its edges in terms of eigenvalue locations. Moreover, we provide necessary and sufficient conditions for finite time reachability of every reachable state and characterize the minimum number of steps needed to reach every state in terms of eigenvalue locations.