  In this paper, we consider the problem of designing an asymptotic observer
for a nonlin-ear dynamical system in discrete-time following Luenberger's
original idea. This approach is a two-step design procedure. In a first step,
the problem is to estimate a function of the state. The state estimation is
obtained by inverting this mapping. Similarly to the continuous-time context,
we show that the first step is always possible provided a linear and stable
discrete-time system fed by the output is introduced. Based on a weak
observ-ability assumption, it is shown that picking the dimension of the stable
auxiliary system sufficiently large, the estimated function of the state is
invertible. This approach is illustrated on linear systems with polynomial
output. The link with the Luenberger observer obtained in the continuous-time
case is also investigated.
