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The presented paper tackles the problem of modeling an unknown function, andits first $r-1$ derivatives, out of scattered and poor-quality data. Theconsidered setting embraces a large number of use cases addressed in theliterature and fits especially well in the context of control barrierfunctions, where high-order derivatives of the safe set are required topreserve the safety of the controlled system. The approach builds on a cascadeof high-gain observers and a set of Gaussian process regressors trained on theobservers' data. The proposed structure allows for high robustness againstmeasurement noise and flexibility with respect to the employed sampling law.Unlike previous approaches in the field, where a large number of samples arerequired to fit correctly the unknown function derivatives, here we suppose tohave access only to a small window of samples, sliding in time. The paperpresents performance bounds on the attained regression error and numericalsimulations showing how the proposed method outperforms previous approaches.

Data-Driven Analytic Differentiation via High Gain Observers and Gaussian Process Priors