Library-based adaptive observation through a sparsity-promoting adaptive observer

This paper proposes an adaptive observer for a class of nonlinear system with linear parametrization. The main novelty of the technique is that the regressor vector is considered to be unknown. Instead, a library of candidate nonlinear functions is implemented, which transforms the original parameter vector into a new one that is characterized by being sparse. In such problem, it is shown that standard adaptive observers cannot recover the original vector due to a lack of persistence of excitation. Instead, a parameter-adaptation with an implicit l1 regularization is implemented. It is shown that this new observer can recover the parameter vector under standard assumptions of sparse signal recovery. The results are validated in a numerical simulation.