NSM constrained approximation of Lipschitz functions from data

The problem of approximating a function from a set of its values in the presence of constraints is considered. Solving this problem is useful in system identification, state estimation, and control. The typical approach to this problem consists in (1) deriving an approximation without considering the constraints, and (2) using the constraints to clip the approximation. However, using this approach may give poor results and, in any case, no optimality properties can be guaranteed. In this paper, a Nonlinear Set Membership method is developed, allowing optimal constrained approximation. The effectiveness of the proposed method is shown through its application to the problem of model predictive control approximation.