  The identification of a nonlinear dynamic model is an open topic in control
theory, especially from sparse input-output measurements. A fundamental
challenge of this problem is that very few to zero prior knowledge is available
on both the state and the nonlinear system model. To cope with this challenge,
we investigate the effectiveness of deep learning in the modeling of dynamic
systems with nonlinear behavior by advocating an approach which relies on three
main ingredients: (i) we show that under some structural conditions on the
to-be-identified model, the state can be expressed in function of a sequence of
the past inputs and outputs; (ii) this relation which we call the state map can
be modelled by resorting to the well-documented approximation power of deep
neural networks; (iii) taking then advantage of existing learning schemes, a
state-space model can be finally identified. After the formulation and analysis
of the approach, we show its ability to identify three different nonlinear
systems. The performances are evaluated in terms of open-loop prediction on
test data generated in simulation as well as a real world data-set of unmanned
aerial vehicle flight measurements.
