  For discrete-time systems, flatness is usually defined by replacing the
time-derivatives of the well-known continuous-time definition by
forward-shifts. With this definition, the class of flat systems corresponds
exactly to the class of systems which can be linearized by a discrete-time
endogenous dynamic feedback as it is proposed in the literature. Recently,
verifiable necessary and sufficient differential-geometric conditions for this
property have been derived. In the present contribution, we make an attempt to
take into account also backward-shifts. This extended approach is motivated by
the one-to-one correspondence of solutions of flat systems to solutions of a
trivial system as it is known from the continuous-time case. If we transfer
this idea to the discrete-time case, this leads to an approach which also
allows backward-shifts. To distinguish the classical definition with
forward-shifts and the approach of the present paper, we refer to the former as
forward-flatness. We show that flat systems (in the extended sense with
backward-shifts) still share many beneficial properties of forward-flat
systems. In particular, they still are reachable/controllable, allow a
straightforward planning of trajectories and can be linearized by a certain
subclass of dynamic feedbacks.
