Dynamics of discrete-time linear systems with decaying disturbances

Dynamical systems in the real world are always subject to various disturbances. This paper studies the dynamics of delayed discrete-time linear systems with decaying disturbances. It is first shown that if an unforced linear system is exponentially stable, then the disturbed system has a dynamical property like the exponential stability if the disturbance decays at an exponential rate, and has a dynamical property like the asymptotic stability if the disturbance asymptotically approaches to zero.