Controllability and Reachability of Dynamic Discrete-Time Linear Systems

For discrete-time linear systems, controllability and reachability are not equivalent. Instead of the well-known Kalman´s rank condition, which characterizes reachability, controllability to origin of the time invariant, discrete-time linear system is equivalent to the Fuhrmann´s rank condition. In the first part of this paper we prove that controllability to origin of time varying discrete-time linear systems, under a difference-algebraic condition, is equivalent to a generalized Fuhrmann´s rank condition. In the second part we prove that reachability and observability for time varying discrete-time linear systems are equivalent to a structured Kalman´s rank condition, under the difference algebraic independence of the structure matrices.