Nonlinear system identification in Sobolev spaces

We consider the problem of approximating an unknown function from experimental data, while approximating at the same time its 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 improved performance in several endeavours, such as in multi-step prediction, simulation, Nonlinear Model Predictive Control, and control design in general. In this paper, we propose a novel approach based on convex optimisation, 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 three numerical examples, concerned with univariate function identification, multi-step prediction of the Chua chaotic circuit, and control of the inverted pendulum.