Sparse identification of nonlinear functions and parametric Set Membership optimality analysis

Identifying a sparse approximation of a function from a set of data can be useful to solve relevant problems in the automatic control field. However, finding a sparsest approximation 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 paper, a combined ℓ₁ -relaxed-greedy algorithm is proposed and a condition is given, under which the approximation derived by the algorithm is a sparsest one. Differently from other conditions available in the literature, the one provided here can be easily verified for any choice of the basis functions. 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. It must be remarked that exact basis function selection is performed for a finite number of data, whereas in standard system identification, a similar result can only be obtained for an infinite number of data. A simulation example, related to the identification of vehicle lateral dynamics, is finally presented.