The Geometry of the Reachability Set for Linear Discrete-time Systems with Positive Controls

In this paper we study the geometrical properties of the set of reachable states of a single input discrete-time linear time invariant (LTI) system with positive controls. This set is a cone and it can be expressed as the direct sum of a linear subspace and a proper cone. In order to give a complete geometrical characterization of the reachable set, we provide a formula to evaluate the dimension of the largest reachable subspace and necessary and sufficient conditions for polyhedrality of the proper cone in terms of eigenvalues location.