Essential Reachability of Positive Periodic Discrete-Time Systems

In this work the essential reachability property of positive N -periodic discrete-time linear systems is studied. A characterization, by means of the colored union of directed graphs of the periodic matrices, and a canonical form for the esential reachability property is shown. Key–Words:Positive invariant systems, positive periodic systems, essential reachability, directed graphs. 1 Background and Preliminaries We consider a positive N -periodic discrete-time linear control system, (F (·), G(·))N ≥ 0, given by x(k + 1) = F (k)x(k) +G(k)u(k), k ∈ Z, (1) where F (k) = F (k + N) ∈ Rn×n + , G(k) = G(k + N) ∈ Rn×m + , x(k) ∈ R+ and u(k) ∈ R+ . The system (1) is said to be reachable at time s (from 0) if for any nonnegative state xf ∈ R+ there exists a nonnegative input sequence transferring the state of the system from the origin at time s, x(s) = 0, to xf in a finite time. It is reachable if it is reachable at time s, for all s ∈ Z. The system (1) is said to be essentially reachable at time s if for every positive final state xf 0, there exists a nonnegative input sequence transferring the state of the system from the origin at time s, x(s) = 0 to xf in a finite time. It is essentially reachable if it is essential reachable at time s, for all s ∈ Z. Note that the system (1) is essentially reachable if all nonnegative states which cannot be reached in a finite time are limit of nonnegative states reachable in a finite time. It is known that the positive N -periodic system (1) has associated a positive invariant cyclically augmented ∗ Supported by spanish grant DGES PB97-0334. system (see [1]), (Fe, Ge) ≥ 0, which is given by z(k + 1) = Fez(k) +Geue(k), (2)