Positive discrete-time linear Lyapunov systems

The notion of positive discrete-time Lyapunov system is introduced. Solution of the Lyapunov state equation is derived and necessary and sufficient conditions for the positivity of Lyapunov system are established. Different necessary and sufficient conditions for the asymptotic stability of the positive Lyapunov systems are given. Using the Kronecker product of matrices necessary and sufficient conditions for reachability and controllability of the positive Lyapunov systems are established. The considerations are illustrated by numerical example.