  We consider the problem of deriving from experimental data an approximation
of an unknown function, whose derivatives also approximate the unknown function
derivatives. Solving this problem is useful, for instance, in the context of
nonlinear system identification for obtaining models that are more accurate and
reliable than the traditional ones, based on plain function approximation.
Indeed, models identified by accounting for the derivatives can provide a
better performance in several tasks, such as multi-step prediction, simulation,
Nonlinear Model Predictive Control, and control design in general. In this
paper, we propose a novel approach based on convex optimization, allowing us to
solve the aforementioned identification problem. We develop an optimality
analysis, showing that models derived using this approach enjoy suitable
optimality properties in Sobolev spaces. The optimality analysis also leads to
the derivation of tight uncertainty bounds on the unknown function and its
derivatives. We demonstrate the effectiveness of the approach with two
numerical examples. The first one is concerned with approximation of an
univariate function, the second one with multi-step prediction of the Chua
chaotic circuit.
