Sparse identification of nonlinear functions and parametric Set Membership optimality analysis

Sparse identification can be relevant in the automatic control field to solve several problems for nonlinear systems such as identification, control, filtering, fault detection. However, identifying a maximally sparse approximation of a nonlinear function is in general an NP-hard problem. The common approach is to use relaxed or greedy algorithms that, under certain conditions, can provide sparsest solutions. In this technical note, a combined l1-relaxed-greedy algorithm is proposed and conditions are given, under which the approximation derived by the algorithm is a sparsest one. Differently from other conditions available in the literature, the ones provided here can be actually verified for any choice of the basis functions defining the sparse approximation. A Set Membership analysis is also carried out, assuming that the function to approximate is a linear combination of unknown basis functions belonging to a known set of functions. It is shown that the algorithm is able to exactly select the basis functions which define the unknown function and to provide an optimal estimate of their coefficients.