Set membership identification of nonlinear systems

We investigate the problem of finding upper and lower bounds of a real valued function of several variables, on the base of a set of noise corrupted values of the function evaluated at a given set of variables and on some assumptions on function regularity and on noise bounds. Several set membership linear and nonlinear identification problems can be recast into the above problem. Two solutions are proposed. The first one is quite straightforward and leads to the definition of bounds that are the tightest ones but, in high dimensional spaces, computationally expensive. The second solution, relying on approximation properties of neural networks, leads to the evaluation of somewhat more conservative bounds, whose computational complexity is significantly lower than for the optimal bounds. A numerical example, related to the identification and prediction of a Lorenz chaotic system is presented to show the effectiveness of the proposed approach