  Willems et al. showed that all input-output trajectories of a discrete-time
linear time-invariant system can be obtained using linear combinations of time
shifts of a single, persistently exciting, input-output trajectory of that
system. In this paper, we extend this result to the class of discrete-time
single-input single-output flat nonlinear systems. We propose a data-based
parametrization of all trajectories using only input-output data. Further, we
use this parametrization to solve the data-based simulation and output-matching
control problems for the unknown system without explicitly identifying a model.
Finally, we illustrate the main results with numerical examples.
