Keywords: discrete time linear systems, difference algebraic conditions, rank condition, reachability, controllability
A generalisation of Fuhrmann's rank condition for discrete dynamic systems
For discrete-time linear systems, controllability and reachability are not equivalent. Instead of the well-known Kalman's rank condition, which characterises 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 generalised 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.