On the reachable set for third-order linear discrete-time systems with positive control: The case of complex eigenvalues

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 a complex conjugate pair of 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.