On reachability of positive linear discrete-time systems with scalar controls

Positive systems are defined as systems in which the state trajectory is always positive (or at least non-negative), whenever the initial state is positive (nonnegative). Positive linear systems are defined on cones and not on linear spaces and that is why the reachability and controllability tests for linear systems prove to be false. In this paper necessary and sufficient condition for reachability of discrete-time positive linear systems with scalar input is proved. Criteria for recognising the reachability property of such systems are presented and complete characterisations of the generic structure of reachable non-negative pair (A, b) in both algebraic and graph-theoretic forms are developed. The paper gives a new general treatment of reachability properties of scalar-input positive linear systems